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arXiv:2403.19491v1 [gr-qc] 28 Mar 2024
arXiv:2403.19491v1 [gr-qc] 2024 年 3 月 28 日

Effects of lunisolar perturbations on TianQin constellation: An analytical model
阴阳扰动对天琴星座的影响:一个解析模型

Bobing Ye yebb5@mail.sysu.edu.cn    Xuefeng Zhang zhangxf38@sysu.edu.cn MOE Key Laboratory of TianQin Mission, TianQin Research Center for Gravitational Physics
&\&& School of Physics and Astronomy, Frontiers Science Center for TianQin,
Gravitational Wave Research Center of CNSA, Sun Yat-sen University (Zhuhai Campus), Zhuhai 519082, People’s Republic of China
(May 2, 2024) (5月 2, 2024)
Abstract 抽象

TianQin is a proposed space-based gravitational-wave observatory mission that critically relies on the stability of an equilateral-triangle constellation. Comprising three satellites in high Earth orbits of a 105superscript10510^{5}10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT km radius, this constellation’s geometric configuration is significantly affected by gravitational perturbations, primarily originating from the Moon and the Sun. In this paper, we present an analytical model to quantify the effects of lunisolar perturbations on the TianQin constellation, derived using Lagrange’s planetary equations. The model provides expressions for three kinematic indicators of the constellation: arm-lengths, relative line-of-sight velocities, and breathing angles. Analysis of these indicators reveals that lunisolar perturbations can distort the constellation triangle, resulting in three distinct variations: linear drift, bias, and fluctuation. Furthermore, it is shown that these distortions can be optimized to display solely fluctuating behavior, under certain predefined conditions. These results can serve as the theoretical foundation for numerical simulations and offer insights for engineering a stable constellation in the future.
“天琴”是一项拟议的天基引力波天文台任务,它严重依赖于等边三角形星座的稳定性。该星座由三颗位于半径 1 105superscript10510^{5}10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT 公里的地球高轨道上的卫星组成,其几何结构受到引力扰动的显着影响,引力扰动主要源自月球和太阳。在本文中,我们提出了一个解析模型来量化阴阳扰动对天琴星座的影响,该模型使用拉格朗日行星方程推导。该模型提供了星座的三个运动学指标的表达式:臂长、相对视距速度和呼吸角。对这些指标的分析表明,阴阳扰动会扭曲星座三角形,导致三种不同的变化:线性漂移、偏差和波动。此外,结果表明,在某些预定义条件下,这些失真可以优化为仅显示波动行为。这些结果可以作为数值模拟的理论基础,并为未来设计稳定的星座提供见解。

preprint: APS/123-QED
预印本:APS/123-QED

I Introduction I 引言

The successful detection of gravitational waves (GWs) by the ground-based observatory LIGO [1] has opened up the era of GW astronomy. To detect GWs in the millihertz range (0.1 mHz–1 Hz), known for its rich sources and to circumvent the impact of seismic noise, space-based GW observatories are highly favored [2, 3]. For such observatories, proposed projects include LISA [4, 5], DECIGO [6], TianQin [7], Taiji [8], etc. Among these, TianQin is a geocentric space-based GW observatory mission that consists of three drag-free controlled satellites with an orbital radius of 105superscript10510^{5}10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT km [7]. The three satellites form a nearly equilateral-triangle constellation, standing almost vertical to the ecliptic, and they employ high-precision laser-ranging interferometry to measure distance changes between satellites for the detection of GWs. The mission will bring rich science prospects to GW astronomy [9, 10, 11].
地面天文台 LIGO [1] 成功探测到引力波 (GW),开启了 GW 天文学的时代。为了探测以丰富的来源而闻名的毫赫兹范围(0.1 mHz–1 Hz)的 GW 并规避地震噪声的影响,天基 GW 天文台受到高度青睐 [23]。对于此类天文台,拟议的项目包括 LISA [45]、DECIGO [6]、天琴 [7]、太极 [8] 等。其中,“天琴”是一项地心天基GW天文台任务,由三颗轨道半径为 105superscript10510^{5}10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT 公里的无阻力受控卫星组成[7]。这三颗卫星形成一个几乎等边三角形的星座,几乎垂直于黄道,它们采用高精度激光测距干涉测量法测量卫星之间的距离变化,以检测 GW。该任务将为 GW 天文学带来丰富的科学前景 [91011]。

TianQin, as well as other three-satellite GW missions, relies crucially on the stability of an equilateral-triangle constellation [3, 7]. Unequal variations in the three arm-lengths of the constellation prevent the cancellation of laser frequency noise, which has a profound impact on the design of frequency stabilization systems and requires time-delay interferometry (TDI) [12, 13, 14]. The relative line-of-sight velocities between satellites induce Doppler frequency shifts, affecting phase meter bandwidth and ultra-stable oscillator design [15]. Moreover, changes in the three breathing angles of the triangle directly influence the design of telescopes and beam pointing mechanisms [7]. It is crucial to minimize variations in the triangular constellation, as indicated by these three kinematic indicators.
“天琴”以及其他三颗卫星的 GW 任务在很大程度上依赖于等边三角形星座的稳定性 [37]。星座三个臂长的不相等变化阻止了激光频率噪声的抵消,这对稳频系统的设计有深远的影响,需要时延干涉测量法 (TDI) [121314]。卫星之间的相对视距速度会引起多普勒频移,影响相位计带宽和超稳定振荡器设计 [15]。此外,三角形三个呼吸角的变化直接影响望远镜和光束指向机构的设计 [7]。正如这三个运动学指标所表明的那样,尽量减少三角形星座的变化至关重要。

Analytical analysis of satellite motion and constellation variations holds significant importance [16, 17, 18, 19, 20, 17]. To identify orbits with minimal variations in the constellation, extensive efforts have been dedicated to numerical orbit optimization and analysis (for a review, see Ref. [19]). The use of analytical models, as opposed to numerical simulations, allows for deeper physical insights and often yields clearer solutions for issues related to satellite motion [16, 17]. Moreover, these analytical models provide the basis for further numerical simulations, enhancing orbit optimization efficiency [18, 19]. They also enable theoretical studies on inter-satellite optical links and light propagation [20, 17].
卫星运动和星座变化的分析具有重要意义[161718, 192017]。为了确定星座中变化最小的轨道,人们投入了大量精力进行数值轨道优化和分析(综述见参考文献 [19])。与数值模拟相反,使用分析模型可以获得更深入的物理见解,并且通常可以为与卫星运动相关的问题提供更清晰的解决方案[16,17]。 此外,这些解析模型为进一步的数值模拟提供了基础,提高了轨道优化效率 [1819]。它们还有助于对星间光链路和光传播进行理论研究 [2017]。

Concerning analytical efforts, Ref. [21] first presented the analytical coordinates of the TianQin satellites, based on unperturbed Keplerian orbits, which showed that the arm-lengths of the constellation remain constant when orbital eccentricities are ignored. Furthermore, the leading-order effect of the third-body perturbation was considered to derive expressions for both arm-lengths and breathing angles [22]. These expressions were constructed iteratively, assuming circular orbits, and they were used to study the impact of initial orbit errors. Moreover, the effect of the Earth’s non-spherical gravitational perturbation was analyzed in [23], with a particular focus on its influence on inter-satellite range acceleration noise.
在分析工作方面,参考文献[21]首先提出了天琴卫星的解析坐标,该坐标基于未受扰动的开普勒轨道,结果表明,当忽略轨道偏心率时,星座的臂长保持不变。此外,第三体扰动的超前序效应被认为可以推导出臂长和呼吸角的表达式 [22]。这些表达式是迭代构建的,假设圆轨道,它们被用来研究初始轨道误差的影响。此外,在 [23] 中分析了地球非球面引力扰动的影响,特别关注其对星间距离加速度噪声的影响。

The analytical investigation into the influence of gravitational perturbations on the TianQin constellation is incomplete. Existing models have neglected the satellite’s orbital eccentricity, a crucial factor for constellation stability [24, 25]. Moreover, relying solely on the leading-order lunar perturbation is insufficient to address the high-altitude TianQin orbits. These issues highlight the necessity for an analytical study to develop a more explicit and higher-precision model.
关于引力扰动对天琴星座影响的分析研究尚不完整。现有的模型忽略了卫星的轨道偏心率,这是星座稳定性的一个关键因素[24,25]。 此外,仅仅依靠超前阶月球扰动不足以解决高空天琴轨道的问题。这些问题凸显了分析研究开发更明确和更高精度模型的必要性。

In the exploration of three-satellite constellations in heliocentric GW missions, such as LISA [26, 20, 27, 28, 29, 30, 18] and Taiji [17, 31], expressions for these three indicators have been derived and analyzed using either Keplerian orbits or perturbation solutions of satellite orbits. Valuable references are also found in geocentric satellite formation missions, including NASA’s four-satellite Magnetospheric Multiscale (MMS) mission [16], and extensive studies on third-body perturbations in general satellites (see [32] and references therein). Perturbation solutions for third-body effects can be derived by solving Lagrange’s planetary equations [33], where the perturbative potential depends on the orbital elements of both the satellite and perturbing bodies. To directly obtain solutions with instantaneous elements, perturbation methods [33, 34, 35, 36], especially the mean element method [34, 35], are utilized. This method employs a slowly precessing elliptical orbit as a reference, effectively reducing errors in analytical solutions.
在日心GW任务中对三星星座的探索中,如LISA [26202728293018] 和太极[1731],已经推导出了这三个指标的表达式,并使用开普勒轨道或卫星轨道的扰动解进行了分析。在地心卫星编队任务中也发现了有价值的参考资料,包括NASA的四颗卫星磁层多尺度(MMS)任务[16],以及对通用卫星中第三体扰动的广泛研究(见[32]和其中的参考文献)。第三体效应的扰动解可以通过求解拉格朗日行星方程 [33] 来推导出,其中扰动势取决于卫星和扰动体的轨道元件。为了直接获得瞬时单元的解,使用了微扰法 [33343536],尤其是均元法 [3435]。该方法采用慢进椭圆轨道作为参考,有效减少了解析解中的误差。

In this work, we will construct an analytical model for the TianQin constellation. To address its near-circular, high Earth orbits, we utilize singularity-free Lagrange equations while accounting for lunar, solar perturbations, and Earth’s J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT perturbation. This model will then be used to analyze and optimize the three kinematic indicators. Additionally, to facilitate the perturbation-inclusive study, the unperturbed Keplerian orbits of TianQin satellites will also be presented.
在这项工作中,我们将为天琴座构建一个解析模型。为了解决其近圆形的高地球轨道问题,我们利用了无奇点的拉格朗日方程,同时考虑了月球、太阳的扰动和地球 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 的扰动。然后,该模型将用于分析和优化三个运动指标。此外,为了促进包含微扰的研究,还将介绍天琴卫星的未扰动开普勒轨道。

The paper is organized as follows. In Sec. II, we introduce the Keplerian orbits of the satellites and present the design of the nominal equilateral-triangle constellation. The gravitational perturbations on the constellation are studied in Sec. III. In Sec. IV, we make the concluding remarks.
本文的组织结构如下。在第二部分,我们介绍了卫星的开普勒轨道,并介绍了名义等边三角形星座的设计。星座上的引力扰动在第三节中进行了研究。 在第四节中,我们做了结束语。

II Fundamentals of stable TianQin constellation
、稳定天琴座的基本原理

In this section, we describe the motion of TianQin satellites in the geocentric ecliptic coordinate system and present the orbit design of satellites for a stable equilateral-triangle constellation.
本节描述了天琴卫星在地心黄道坐标系中的运动,并介绍了稳定等边三角形星座的卫星轨道设计。

II.1 Keplerian orbits of satellites
II.1 卫星的开普勒轨道

Within the central gravitational field of the Earth, satellite moves in a Keplerian orbit, as illustrated in Fig. 1.
在地球的中心引力场内,卫星在开普勒轨道上移动,如图 1 所示。1.

Refer to caption
Figure 1: Depiction of the TianQin constellation in the geocentric ecliptic coordinate system. The ecliptic plane is spanned by the x𝑥xitalic_x and y𝑦yitalic_y axes, with the x𝑥xitalic_x-axis directed toward the vernal equinox. The figure also illustrates the orbital coordinate system {X,Y,Z}𝑋𝑌𝑍\{X,Y,Z\}{ italic_X , italic_Y , italic_Z } for SC1, where the X𝑋Xitalic_X-axis points toward the perigee of the satellite’s orbit, and the Z𝑍Zitalic_Z-axis (not shown) is perpendicular to the orbital plane. The angles i𝑖iitalic_i, ΩΩ\Omegaroman_Ω, ω𝜔\omegaitalic_ω, and ν𝜈\nuitalic_ν denote the orbital inclination, longitude of ascending node, argument of perigee, and true anomaly, respectively. Specifically, i=94.7𝑖superscript94.7i=94.7^{\circ}italic_i = 94.7 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT and Ω=210.4Ωsuperscript210.4\Omega=210.4^{\circ}roman_Ω = 210.4 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT are set to orient the TianQin detector plane toward the reference source, the white-dwarf binary RX J0806.3+1527.
图 1: 天琴座在地心黄道坐标系中的描述。黄道平面由 x𝑥xitalic_xy𝑦yitalic_y 轴跨越, x𝑥xitalic_x 其中 -轴指向春分。该图还说明了 SC1 的轨道坐标系 {X,Y,Z}𝑋𝑌𝑍\{X,Y,Z\}{ italic_X , italic_Y , italic_Z } ,其中 X𝑋Xitalic_X -轴指向卫星轨道的近地点, Z𝑍Zitalic_Z 而 -轴(未显示)垂直于轨道平面。角度 i𝑖iitalic_iΩΩ\Omegaroman_Ωω𝜔\omegaitalic_ων𝜈\nuitalic_ν 分别表示轨道倾角、升交点的经度、近地点的参数和真实异常。具体来说, i=94.7𝑖superscript94.7i=94.7^{\circ}italic_i = 94.7 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT 并将 Ω=210.4Ωsuperscript210.4\Omega=210.4^{\circ}roman_Ω = 210.4 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT 天琴探测器平面定向到参考源,即白矮星双星 RX J0806.3+1527。

{X,Y,Z}𝑋𝑌𝑍\{X,Y,Z\}{ italic_X , italic_Y , italic_Z } is the orbital right-handed coordinate system, with the origin at the Earth’s center of mass. The satellite’s orbital plane is same as the X𝑋Xitalic_X-Y𝑌Yitalic_Y plane, where the X𝑋Xitalic_X-axis points toward the perigee. In this system, the satellite’s Cartesian coordinates (X,Y,Z,X˙,Y˙,Z˙)𝑋𝑌𝑍˙𝑋˙𝑌˙𝑍(X,Y,Z,\dot{X},\dot{Y},\dot{Z})( italic_X , italic_Y , italic_Z , over˙ start_ARG italic_X end_ARG , over˙ start_ARG italic_Y end_ARG , over˙ start_ARG italic_Z end_ARG ) can be denoted as [35, 33]:
{X,Y,Z}𝑋𝑌𝑍\{X,Y,Z\}{ italic_X , italic_Y , italic_Z } 是轨道右手坐标系,原点位于地球质心。卫星的轨道平面与 X𝑋Xitalic_X - Y𝑌Yitalic_Y 平面相同,其中 X𝑋Xitalic_X -轴指向近地点。在这个系统中,卫星的笛卡尔坐标 (X,Y,Z,X˙,Y˙,Z˙)𝑋𝑌𝑍˙𝑋˙𝑌˙𝑍(X,Y,Z,\dot{X},\dot{Y},\dot{Z})( italic_X , italic_Y , italic_Z , over˙ start_ARG italic_X end_ARG , over˙ start_ARG italic_Y end_ARG , over˙ start_ARG italic_Z end_ARG ) 可以表示为 [3533]

{X=rcosν=a(cosEe),Y=rsinν=a1e2sinE,Z=0,X˙=μarsinE,Y˙=μar1e2cosE,Z˙=0,cases𝑋𝑟𝜈𝑎𝐸𝑒𝑌𝑟𝜈𝑎1superscript𝑒2𝐸𝑍0˙𝑋𝜇𝑎𝑟𝐸˙𝑌𝜇𝑎𝑟1superscript𝑒2𝐸˙𝑍0\displaystyle\left\{\begin{array}[]{l}X=r\cos\nu=a(\cos E-e),\vskip 3.0pt plus% 1.0pt minus 1.0pt\\ Y=r\sin\nu=a\sqrt{1-e^{2}}\sin E,\vskip 3.0pt plus 1.0pt minus 1.0pt\\ Z=0,\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \dot{X}=-\frac{\sqrt{\mu a}}{r}\sin E,\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \dot{Y}=\frac{\sqrt{\mu a}}{r}\sqrt{1-e^{2}}\cos E,\vskip 3.0pt plus 1.0pt % minus 1.0pt\\ \dot{Z}=0,\end{array}\right.{ start_ARRAY start_ROW start_CELL italic_X = italic_r roman_cos italic_ν = italic_a ( roman_cos italic_E - italic_e ) , end_CELL end_ROW start_ROW start_CELL italic_Y = italic_r roman_sin italic_ν = italic_a square-root start_ARG 1 - italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG roman_sin italic_E , end_CELL end_ROW start_ROW start_CELL italic_Z = 0 , end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_X end_ARG = - divide start_ARG square-root start_ARG italic_μ italic_a end_ARG end_ARG start_ARG italic_r end_ARG roman_sin italic_E , end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_Y end_ARG = divide start_ARG square-root start_ARG italic_μ italic_a end_ARG end_ARG start_ARG italic_r end_ARG square-root start_ARG 1 - italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG roman_cos italic_E , end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_Z end_ARG = 0 , end_CELL end_ROW end_ARRAY (7)

with r𝑟ritalic_r representing the geocentric radius, ν𝜈\nuitalic_ν the true anomaly, a𝑎aitalic_a the semimajor axis, e𝑒eitalic_e the orbital eccentricity, μ=GMe𝜇𝐺subscript𝑀e\mu=GM_{\text{e}}italic_μ = italic_G italic_M start_POSTSUBSCRIPT e end_POSTSUBSCRIPT the Earth’s gravitational constant, and E𝐸Eitalic_E the eccentric anomaly. E𝐸Eitalic_E satisfies Kepler’s equation,
代表 r𝑟ritalic_r 地心半径、 ν𝜈\nuitalic_ν 真实异常、 a𝑎aitalic_a 长半轴、 e𝑒eitalic_e 轨道偏心率、 μ=GMe𝜇𝐺subscript𝑀e\mu=GM_{\text{e}}italic_μ = italic_G italic_M start_POSTSUBSCRIPT e end_POSTSUBSCRIPT 地球引力常数和 E𝐸Eitalic_E 偏心异常。 E𝐸Eitalic_E 满足 Kepler 方程,

EesinE=M,𝐸𝑒𝐸𝑀\displaystyle E-e\sin E=M,italic_E - italic_e roman_sin italic_E = italic_M , (8)

where M𝑀Mitalic_M denotes the mean anomaly. Specifically, M𝑀Mitalic_M is given by M=n(ttp)𝑀𝑛𝑡subscript𝑡pM=n\,(t-t_{\text{p}})italic_M = italic_n ( italic_t - italic_t start_POSTSUBSCRIPT p end_POSTSUBSCRIPT ) in the two-body problem, with the mean motion n𝑛nitalic_n and the passing time of the perigee tpsubscript𝑡pt_{\text{p}}italic_t start_POSTSUBSCRIPT p end_POSTSUBSCRIPT. Equation (8), which is a transcendental equation, can be solved iteratively, resulting in the following expression [21]:
其中 M𝑀Mitalic_M 表示均值距平。具体来说, M𝑀Mitalic_MM=n(ttp)𝑀𝑛𝑡subscript𝑡pM=n\,(t-t_{\text{p}})italic_M = italic_n ( italic_t - italic_t start_POSTSUBSCRIPT p end_POSTSUBSCRIPT ) 在双体问题中给出,具有近地点的平均运动 n𝑛nitalic_n 和通过时间 tpsubscript𝑡pt_{\text{p}}italic_t start_POSTSUBSCRIPT p end_POSTSUBSCRIPT 。方程 (8) 是一个超越方程,可以迭代求解,得到以下表达式 [21]

E=M+esinM+e2cosMsinM+𝒪(e3).𝐸𝑀𝑒𝑀superscript𝑒2𝑀𝑀𝒪superscript𝑒3\displaystyle E=M+e\sin M+e^{2}\cos M\sin M+\mathcal{O}(e^{3}).italic_E = italic_M + italic_e roman_sin italic_M + italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_cos italic_M roman_sin italic_M + caligraphic_O ( italic_e start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) . (9)

By substituting Eq. (9) into Eq. (7), one can obtain the explicit coordinates (X,Y,Z,X˙,Y˙,Z˙)𝑋𝑌𝑍˙𝑋˙𝑌˙𝑍(X,Y,Z,\dot{X},\dot{Y},\dot{Z})( italic_X , italic_Y , italic_Z , over˙ start_ARG italic_X end_ARG , over˙ start_ARG italic_Y end_ARG , over˙ start_ARG italic_Z end_ARG ).
通过将方程 (9) 代入方程 (7),可以得到显式坐标 (X,Y,Z,X˙,Y˙,Z˙)𝑋𝑌𝑍˙𝑋˙𝑌˙𝑍(X,Y,Z,\dot{X},\dot{Y},\dot{Z})( italic_X , italic_Y , italic_Z , over˙ start_ARG italic_X end_ARG , over˙ start_ARG italic_Y end_ARG , over˙ start_ARG italic_Z end_ARG )

The orbital planes may not be identical for the three TianQin satellites. Thus, the geocentric ecliptic coordinate system {x,y,z}𝑥𝑦𝑧\{x,y,z\}{ italic_x , italic_y , italic_z } is also employed, where the x𝑥xitalic_x-y𝑦yitalic_y plane is the ecliptic plane. The x𝑥xitalic_x-axis points toward the vernal equinox, and the z𝑧zitalic_z-axis is normal to the ecliptic plane. The coordinates (x,y,z)𝑥𝑦𝑧(x,y,z)( italic_x , italic_y , italic_z ) and (x˙,y˙,z˙)˙𝑥˙𝑦˙𝑧(\dot{x},\dot{y},\dot{z})( over˙ start_ARG italic_x end_ARG , over˙ start_ARG italic_y end_ARG , over˙ start_ARG italic_z end_ARG ) in this system can be obtained by (X,Y,Z)𝑋𝑌𝑍(X,Y,Z)( italic_X , italic_Y , italic_Z ) and (X˙,Y˙,Z˙)˙𝑋˙𝑌˙𝑍(\dot{X},\dot{Y},\dot{Z})( over˙ start_ARG italic_X end_ARG , over˙ start_ARG italic_Y end_ARG , over˙ start_ARG italic_Z end_ARG ) through the following transformation [35, 33]:
三颗天琴卫星的轨道平面可能并不相同。因此,还采用了地心黄道坐标系 {x,y,z}𝑥𝑦𝑧\{x,y,z\}{ italic_x , italic_y , italic_z } ,其中 x𝑥xitalic_x - y𝑦yitalic_y 平面是黄道平面。 x𝑥xitalic_x -轴指向春分, z𝑧zitalic_z -轴垂直于黄道平面。在这个系统中,坐标 (x,y,z)𝑥𝑦𝑧(x,y,z)( italic_x , italic_y , italic_z )(x˙,y˙,z˙)˙𝑥˙𝑦˙𝑧(\dot{x},\dot{y},\dot{z})( over˙ start_ARG italic_x end_ARG , over˙ start_ARG italic_y end_ARG , over˙ start_ARG italic_z end_ARG ) 可以通过以下变换 [3533] 获得 (X,Y,Z)𝑋𝑌𝑍(X,Y,Z)( italic_X , italic_Y , italic_Z ) (X˙,Y˙,Z˙)˙𝑋˙𝑌˙𝑍(\dot{X},\dot{Y},\dot{Z})( over˙ start_ARG italic_X end_ARG , over˙ start_ARG italic_Y end_ARG , over˙ start_ARG italic_Z end_ARG )

[xyz]=Rz(Ω)Rx(i)Rz(ω)[XYZ],matrix𝑥𝑦𝑧subscript𝑅𝑧Ωsubscript𝑅𝑥𝑖subscript𝑅𝑧𝜔matrix𝑋𝑌𝑍\displaystyle\begin{bmatrix}x\\ y\\ z\end{bmatrix}=R_{z}(-\Omega)R_{x}(-i)R_{z}(-\omega)\begin{bmatrix}X\\ Y\\ Z\end{bmatrix},[ start_ARG start_ROW start_CELL italic_x end_CELL end_ROW start_ROW start_CELL italic_y end_CELL end_ROW start_ROW start_CELL italic_z end_CELL end_ROW end_ARG ] = italic_R start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( - roman_Ω ) italic_R start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( - italic_i ) italic_R start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( - italic_ω ) [ start_ARG start_ROW start_CELL italic_X end_CELL end_ROW start_ROW start_CELL italic_Y end_CELL end_ROW start_ROW start_CELL italic_Z end_CELL end_ROW end_ARG ] , (10)
[x˙y˙z˙]=Rz(Ω)Rx(i)Rz(ω)[X˙Y˙Z˙],matrix˙𝑥˙𝑦˙𝑧subscript𝑅𝑧Ωsubscript𝑅𝑥𝑖subscript𝑅𝑧𝜔matrix˙𝑋˙𝑌˙𝑍\displaystyle\begin{bmatrix}\dot{x}\\ \dot{y}\\ \dot{z}\end{bmatrix}=R_{z}(-\Omega)R_{x}(-i)R_{z}(-\omega)\begin{bmatrix}\dot{% X}\\ \dot{Y}\\ \dot{Z}\end{bmatrix},[ start_ARG start_ROW start_CELL over˙ start_ARG italic_x end_ARG end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_y end_ARG end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_z end_ARG end_CELL end_ROW end_ARG ] = italic_R start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( - roman_Ω ) italic_R start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( - italic_i ) italic_R start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( - italic_ω ) [ start_ARG start_ROW start_CELL over˙ start_ARG italic_X end_ARG end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_Y end_ARG end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_Z end_ARG end_CELL end_ROW end_ARG ] , (11)

where ΩΩ\Omegaroman_Ω, i𝑖iitalic_i, and ω𝜔\omegaitalic_ω denote the satellite’s longitude of the ascending node, inclination, and argument of perigee, respectively. Additionally, Rz(γ)subscript𝑅𝑧𝛾R_{z}(\gamma)italic_R start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_γ ) and Rx(γ)subscript𝑅𝑥𝛾R_{x}(\gamma)italic_R start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_γ ) are the rotation matrices that rotate vectors by an angle γ𝛾\gammaitalic_γ about the zlimit-from𝑧z-italic_z - or xlimit-from𝑥x-italic_x -axis,
其中 ΩΩ\Omegaroman_Ω ,、 i𝑖iitalic_iω𝜔\omegaitalic_ω 分别表示卫星的升交点经度、倾角和近地点参数。此外, Rz(γ)subscript𝑅𝑧𝛾R_{z}(\gamma)italic_R start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_γ )Rx(γ)subscript𝑅𝑥𝛾R_{x}(\gamma)italic_R start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_γ ) 是将向量绕 zlimit-from𝑧z-italic_z - or xlimit-from𝑥x-italic_x - 轴旋转一定角度 γ𝛾\gammaitalic_γ 的旋转矩阵,

Rz(γ)=[cosγsinγ0sinγcosγ0001],subscript𝑅𝑧𝛾matrix𝛾𝛾0𝛾𝛾0001\displaystyle R_{z}(\gamma)=\begin{bmatrix}\cos\gamma&\sin\gamma&0\\ -\sin\gamma&\cos\gamma&0\\ 0&0&1\end{bmatrix},italic_R start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_γ ) = [ start_ARG start_ROW start_CELL roman_cos italic_γ end_CELL start_CELL roman_sin italic_γ end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL - roman_sin italic_γ end_CELL start_CELL roman_cos italic_γ end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL end_ROW end_ARG ] , (12)
Rx(γ)=[1000cosγsinγ0sinγcosγ].subscript𝑅𝑥𝛾matrix1000𝛾𝛾0𝛾𝛾\displaystyle R_{x}(\gamma)=\begin{bmatrix}1&0&0\\ 0&\cos\gamma&\sin\gamma\\ 0&-\sin\gamma&\cos\gamma\end{bmatrix}.italic_R start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_γ ) = [ start_ARG start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL roman_cos italic_γ end_CELL start_CELL roman_sin italic_γ end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL - roman_sin italic_γ end_CELL start_CELL roman_cos italic_γ end_CELL end_ROW end_ARG ] . (13)

Combining Eqs. (7) and (9)-(13), the position vector and velocity vector of SCk𝑘kitalic_k (k=1𝑘1k=1italic_k = 1, 2, 3), 𝐫k=(xk,yk,zk)subscript𝐫𝑘subscript𝑥𝑘subscript𝑦𝑘subscript𝑧𝑘\mathbf{r}_{k}=(x_{k},y_{k},z_{k})bold_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) and 𝐫˙k=(x˙k,y˙k,z˙k)subscript˙𝐫𝑘subscript˙𝑥𝑘subscript˙𝑦𝑘subscript˙𝑧𝑘\mathbf{\dot{r}}_{k}=(\dot{x}_{k},\dot{y}_{k},\dot{z}_{k})over˙ start_ARG bold_r end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = ( over˙ start_ARG italic_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over˙ start_ARG italic_y end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over˙ start_ARG italic_z end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ), are given by
组合方程。(7) 和 (9)-(13),则 SC k𝑘kitalic_kk=1𝑘1k=1italic_k = 1 , 2, 3) 𝐫k=(xk,yk,zk)subscript𝐫𝑘subscript𝑥𝑘subscript𝑦𝑘subscript𝑧𝑘\mathbf{r}_{k}=(x_{k},y_{k},z_{k})bold_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )𝐫˙k=(x˙k,y˙k,z˙k)subscript˙𝐫𝑘subscript˙𝑥𝑘subscript˙𝑦𝑘subscript˙𝑧𝑘\mathbf{\dot{r}}_{k}=(\dot{x}_{k},\dot{y}_{k},\dot{z}_{k})over˙ start_ARG bold_r end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = ( over˙ start_ARG italic_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over˙ start_ARG italic_y end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , over˙ start_ARG italic_z end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) 的位置矢量和速度矢量由下式给出

{xk=ak[cosΩkcosλkcosiksinΩksinλk+12ek(cosΩkfc1kcosiksinΩkfs1k)]+𝒪(ek2),yk=ak[sinΩkcosλk+cosikcosΩksinλk+12ek(sinΩkfc1k+cosikcosΩkfs1k)]+𝒪(ek2),zk=ak(siniksinλk+12eksinikfs1k)+𝒪(ek2),x˙k=μak[cosΩksinλkcosiksinΩkcosλk+ek(cosΩkfs2kcosiksinΩkfc2k)]+𝒪(ek2),y˙k=μak[sinΩksinλk+cosikcosΩkcosλk+ek(sinΩkfs2k+cosikcosΩkfc2k)]+𝒪(ek2),z˙k=μak(sinikcosλk+eksinikfc2k)+𝒪(ek2),\displaystyle\left\{\begin{array}[]{l}\displaystyle x_{k}=a_{k}[\cos\Omega_{k}% \cos\lambda_{k}-\cos i_{k}\sin\Omega_{k}\sin\lambda_{k}\vskip 3.0pt plus 1.0pt% minus 1.0pt\\ \displaystyle\phantom{x_{k}=}+\frac{1}{2}e_{k}(\cos\Omega_{k}\,f_{\text{c1}k}-% \cos i_{k}\sin\Omega_{k}\,f_{\text{s1}k})]+\mathcal{O}(e^{2}_{k}),\vskip 3.0pt% plus 1.0pt minus 1.0pt\\ \displaystyle y_{k}=a_{k}[\sin\Omega_{k}\cos\lambda_{k}+\cos i_{k}\cos\Omega_{% k}\sin\lambda_{k}\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \displaystyle\phantom{y_{k}=}+\frac{1}{2}e_{k}(\sin\Omega_{k}\,f_{\text{c1}k}+% \cos i_{k}\cos\Omega_{k}\,f_{\text{s1}k})]+\mathcal{O}(e^{2}_{k}),\vskip 3.0pt% plus 1.0pt minus 1.0pt\\ \displaystyle z_{k}=a_{k}(\sin i_{k}\sin\lambda_{k}+\frac{1}{2}e_{k}\sin i_{k}% \,f_{\text{s1}k})+\mathcal{O}(e^{2}_{k}),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \displaystyle\dot{x}_{k}=\frac{\sqrt{\mu}}{\sqrt{a_{k}}}[-\cos\Omega_{k}\sin% \lambda_{k}-\cos i_{k}\sin\Omega_{k}\cos\lambda_{k}\vskip 3.0pt plus 1.0pt % minus 1.0pt\\ \displaystyle\phantom{\dot{x}_{k}=}+e_{k}(-\cos\Omega_{k}\,f_{\text{s2}k}-\cos i% _{k}\sin\Omega_{k}\,f_{\text{c2}k})]+\mathcal{O}(e^{2}_{k}),\vskip 3.0pt plus % 1.0pt minus 1.0pt\\ \displaystyle\dot{y}_{k}=\frac{\sqrt{\mu}}{\sqrt{a_{k}}}[-\sin\Omega_{k}\sin% \lambda_{k}+\cos i_{k}\cos\Omega_{k}\cos\lambda_{k}\vskip 3.0pt plus 1.0pt % minus 1.0pt\\ \displaystyle\phantom{\dot{y}_{k}=}+e_{k}(-\sin\Omega_{k}\,f_{\text{s2}k}+\cos i% _{k}\cos\Omega_{k}\,f_{\text{c2}k})]+\mathcal{O}(e^{2}_{k}),\vskip 3.0pt plus % 1.0pt minus 1.0pt\\ \displaystyle\dot{z}_{k}=\frac{\sqrt{\mu}}{\sqrt{a_{k}}}(\sin i_{k}\cos\lambda% _{k}+e_{k}\sin i_{k}\,f_{\text{c2}k})+\mathcal{O}(e^{2}_{k}),\end{array}\right.{ start_ARRAY start_ROW start_CELL italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT [ roman_cos roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_cos italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - roman_cos italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_sin roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_sin italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( roman_cos roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT c1 italic_k end_POSTSUBSCRIPT - roman_cos italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_sin roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT s1 italic_k end_POSTSUBSCRIPT ) ] + caligraphic_O ( italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , end_CELL end_ROW start_ROW start_CELL italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT [ roman_sin roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_cos italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + roman_cos italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_cos roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_sin italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( roman_sin roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT c1 italic_k end_POSTSUBSCRIPT + roman_cos italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_cos roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT s1 italic_k end_POSTSUBSCRIPT ) ] + caligraphic_O ( italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , end_CELL end_ROW start_ROW start_CELL italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( roman_sin italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_sin italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_sin italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT s1 italic_k end_POSTSUBSCRIPT ) + caligraphic_O ( italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG square-root start_ARG italic_μ end_ARG end_ARG start_ARG square-root start_ARG italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG end_ARG [ - roman_cos roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_sin italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - roman_cos italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_sin roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_cos italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL + italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( - roman_cos roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT s2 italic_k end_POSTSUBSCRIPT - roman_cos italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_sin roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT c2 italic_k end_POSTSUBSCRIPT ) ] + caligraphic_O ( italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_y end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG square-root start_ARG italic_μ end_ARG end_ARG start_ARG square-root start_ARG italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG end_ARG [ - roman_sin roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_sin italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + roman_cos italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_cos roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_cos italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL + italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( - roman_sin roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT s2 italic_k end_POSTSUBSCRIPT + roman_cos italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_cos roman_Ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT c2 italic_k end_POSTSUBSCRIPT ) ] + caligraphic_O ( italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , end_CELL end_ROW start_ROW start_CELL over˙ start_ARG italic_z end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG square-root start_ARG italic_μ end_ARG end_ARG start_ARG square-root start_ARG italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG end_ARG ( roman_sin italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_cos italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_sin italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT c2 italic_k end_POSTSUBSCRIPT ) + caligraphic_O ( italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , end_CELL end_ROW end_ARRAY (24)

where 哪里

λk:=Mk+ωk,assignsubscript𝜆𝑘subscript𝑀𝑘subscript𝜔𝑘\displaystyle\lambda_{k}:=M_{k}+\omega_{k},italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT := italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (25)

fc1k:=cos(2λkωk)3cosωkassignsubscript𝑓c1𝑘2subscript𝜆𝑘subscript𝜔𝑘3subscript𝜔𝑘f_{\text{c1}k}:=\cos(2\lambda_{k}-\omega_{k})-3\cos\omega_{k}italic_f start_POSTSUBSCRIPT c1 italic_k end_POSTSUBSCRIPT := roman_cos ( 2 italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - 3 roman_cos italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, fs1k:=sin(2λkωk)3sinωkassignsubscript𝑓s1𝑘2subscript𝜆𝑘subscript𝜔𝑘3subscript𝜔𝑘f_{\text{s1}k}:=\sin(2\lambda_{k}-\omega_{k})-3\sin\omega_{k}italic_f start_POSTSUBSCRIPT s1 italic_k end_POSTSUBSCRIPT := roman_sin ( 2 italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - 3 roman_sin italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, fc2k:=cos(2λkωk)assignsubscript𝑓c2𝑘2subscript𝜆𝑘subscript𝜔𝑘f_{\text{c2}k}:=\cos(2\lambda_{k}-\omega_{k})italic_f start_POSTSUBSCRIPT c2 italic_k end_POSTSUBSCRIPT := roman_cos ( 2 italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ), and fs2k:=sin(2λkωk)assignsubscript𝑓s2𝑘2subscript𝜆𝑘subscript𝜔𝑘f_{\text{s2}k}:=\sin(2\lambda_{k}-\omega_{k})italic_f start_POSTSUBSCRIPT s2 italic_k end_POSTSUBSCRIPT := roman_sin ( 2 italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ). Define σ{a,e,i,Ω,ω,λ}𝜎𝑎𝑒𝑖Ω𝜔𝜆\sigma\in\{a,e,i,\Omega,\omega,\lambda\}italic_σ ∈ { italic_a , italic_e , italic_i , roman_Ω , italic_ω , italic_λ }, σk(t)subscript𝜎𝑘𝑡\sigma_{k}(t)italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) in Eq. (24) are straightforwardly determined in the two-body problem by
fc1k:=cos(2λkωk)3cosωkassignsubscript𝑓c1𝑘2subscript𝜆𝑘subscript𝜔𝑘3subscript𝜔𝑘f_{\text{c1}k}:=\cos(2\lambda_{k}-\omega_{k})-3\cos\omega_{k}italic_f start_POSTSUBSCRIPT c1 italic_k end_POSTSUBSCRIPT := roman_cos ( 2 italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - 3 roman_cos italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT fc2k:=cos(2λkωk)assignsubscript𝑓c2𝑘2subscript𝜆𝑘subscript𝜔𝑘f_{\text{c2}k}:=\cos(2\lambda_{k}-\omega_{k})italic_f start_POSTSUBSCRIPT c2 italic_k end_POSTSUBSCRIPT := roman_cos ( 2 italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )fs1k:=sin(2λkωk)3sinωkassignsubscript𝑓s1𝑘2subscript𝜆𝑘subscript𝜔𝑘3subscript𝜔𝑘f_{\text{s1}k}:=\sin(2\lambda_{k}-\omega_{k})-3\sin\omega_{k}italic_f start_POSTSUBSCRIPT s1 italic_k end_POSTSUBSCRIPT := roman_sin ( 2 italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - 3 roman_sin italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPTfs2k:=sin(2λkωk)assignsubscript𝑓s2𝑘2subscript𝜆𝑘subscript𝜔𝑘f_{\text{s2}k}:=\sin(2\lambda_{k}-\omega_{k})italic_f start_POSTSUBSCRIPT s2 italic_k end_POSTSUBSCRIPT := roman_sin ( 2 italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) .定义 σ{a,e,i,Ω,ω,λ}𝜎𝑎𝑒𝑖Ω𝜔𝜆\sigma\in\{a,e,i,\Omega,\omega,\lambda\}italic_σ ∈ { italic_a , italic_e , italic_i , roman_Ω , italic_ω , italic_λ }σk(t)subscript𝜎𝑘𝑡\sigma_{k}(t)italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) 在方程(24)中直接由下式确定

σk(t)=σ0k+διnk(tt0),for Keplerian orbit,subscript𝜎𝑘𝑡subscript𝜎0𝑘subscript𝛿𝜄subscript𝑛𝑘𝑡subscript𝑡0for Keplerian orbit\displaystyle\sigma_{k}(t)=\sigma_{0k}+\delta_{\iota}n_{k}(t-t_{0}),\qquad% \text{for Keplerian orbit},italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) = italic_σ start_POSTSUBSCRIPT 0 italic_k end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , for Keplerian orbit , (26)

where 哪里

σ0k:=σk(t0),δι={1,σ=λ,0,σλ.formulae-sequenceassignsubscript𝜎0𝑘subscript𝜎𝑘subscript𝑡0subscript𝛿𝜄cases1𝜎𝜆0𝜎𝜆\sigma_{0k}:=\sigma_{k}(t_{0}),\qquad\delta_{\iota}=\left\{\begin{array}[]{ll}% 1,&\sigma=\lambda,\vskip 3.0pt plus 1.0pt minus 1.0pt\\ 0,&\sigma\neq\lambda.\end{array}\right.italic_σ start_POSTSUBSCRIPT 0 italic_k end_POSTSUBSCRIPT := italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT = { start_ARRAY start_ROW start_CELL 1 , end_CELL start_CELL italic_σ = italic_λ , end_CELL end_ROW start_ROW start_CELL 0 , end_CELL start_CELL italic_σ ≠ italic_λ . end_CELL end_ROW end_ARRAY (27)

Note that Eq. (24) remains valid even when considering gravitational perturbations, with the only change being the replacement of σk(t)subscript𝜎𝑘𝑡\sigma_{k}(t)italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) in Eq. (26) with the corresponding perturbation solution.
请注意,即使考虑引力扰动,方程(24)仍然有效,唯一的变化是用相应的扰动解替换了方程(26σk(t)subscript𝜎𝑘𝑡\sigma_{k}(t)italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) 中的方程。

II.2 Orbit design of the TianQin constellation
二.2 天琴座的轨道设计

The TianQin constellation is composed of three satellites in geocentric orbits, forming a triangular configuration which continuously evolves in geometry over time. The closer the configuration change approaches an equilateral triangle, the more it aids in alleviating design constraints on measurement system instruments. Therefore, it is essential to find a constellation design with minimal variations.
天琴星座由三颗位于地心轨道上的卫星组成,形成一个三角形结构,随着时间的推移,几何结构不断演变。配置更改越接近等边三角形,就越有助于减轻测量系统仪器的设计限制。因此,找到变化最小的星座设计至关重要。

The constellation is considered more stable if it is closer to an equilateral triangle. There are three main kinematic indicators to characterize the stability, namely, the arm-length Lijsubscript𝐿𝑖𝑗L_{ij}italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT, relative line-of-sight velocity between satellites vijsubscript𝑣𝑖𝑗v_{ij}italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT, and breathing angle αksubscript𝛼𝑘\alpha_{k}italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT,
如果星座更靠近等边三角形,则认为它更稳定。有三个主要的运动学指标来表征稳定性,即臂长 Lijsubscript𝐿𝑖𝑗L_{ij}italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT 、卫星 vijsubscript𝑣𝑖𝑗v_{ij}italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT 之间的相对视线速度和呼吸角 αksubscript𝛼𝑘\alpha_{k}italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT

Lijsubscript𝐿𝑖𝑗\displaystyle L_{ij}italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT =|𝐫i𝐫j|,absentsubscript𝐫𝑖subscript𝐫𝑗\displaystyle=\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|,= | bold_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_r start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | , (28)
vijsubscript𝑣𝑖𝑗\displaystyle v_{ij}italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT =L˙ij,absentsubscript˙𝐿𝑖𝑗\displaystyle=\dot{L}_{ij},= over˙ start_ARG italic_L end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT , (29)
αksubscript𝛼𝑘\displaystyle\alpha_{k}italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT =arccosLki2+Lkj2Lij22LkiLkj,absentsuperscriptsubscript𝐿𝑘𝑖2superscriptsubscript𝐿𝑘𝑗2superscriptsubscript𝐿𝑖𝑗22subscript𝐿𝑘𝑖subscript𝐿𝑘𝑗\displaystyle=\arccos\frac{L_{ki}^{2}+L_{kj}^{2}-L_{ij}^{2}}{2L_{ki}L_{kj}},= roman_arccos divide start_ARG italic_L start_POSTSUBSCRIPT italic_k italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_L start_POSTSUBSCRIPT italic_k italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_L start_POSTSUBSCRIPT italic_k italic_i end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT italic_k italic_j end_POSTSUBSCRIPT end_ARG , (30)

where i𝑖iitalic_i, j𝑗jitalic_j, and k𝑘kitalic_k take values 1, 2, or 3 and ijk𝑖𝑗𝑘i\neq j\neq kitalic_i ≠ italic_j ≠ italic_k. Substituting Eq. (24) into Eqs. (28)-(30), one can obtain the explicit expressions, for these three kinematic indicators, with forms Lij(σi(t),σj(t))subscript𝐿𝑖𝑗subscript𝜎𝑖𝑡subscript𝜎𝑗𝑡L_{ij}(\sigma_{i}(t),\sigma_{j}(t))italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) , italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ), vij(σi(t),σj(t))subscript𝑣𝑖𝑗subscript𝜎𝑖𝑡subscript𝜎𝑗𝑡v_{ij}(\sigma_{i}(t),\sigma_{j}(t))italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) , italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ), and αk(σk(t),σi(t),σj(t))subscript𝛼𝑘subscript𝜎𝑘𝑡subscript𝜎𝑖𝑡subscript𝜎𝑗𝑡\alpha_{k}(\sigma_{k}(t),\sigma_{i}(t),\sigma_{j}(t))italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) , italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) , italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ), respectively.
其中 i𝑖iitalic_ij𝑗jitalic_j , 和 k𝑘kitalic_k 取值 1、2 或 3 和 ijk𝑖𝑗𝑘i\neq j\neq kitalic_i ≠ italic_j ≠ italic_k 。将方程 (24) 代入方程。(28)-(30) 中,对于这三个运动指示符,可以分别获得 、 和 vij(σi(t),σj(t))subscript𝑣𝑖𝑗subscript𝜎𝑖𝑡subscript𝜎𝑗𝑡v_{ij}(\sigma_{i}(t),\sigma_{j}(t))italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) , italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ) αk(σk(t),σi(t),σj(t))subscript𝛼𝑘subscript𝜎𝑘𝑡subscript𝜎𝑖𝑡subscript𝜎𝑗𝑡\alpha_{k}(\sigma_{k}(t),\sigma_{i}(t),\sigma_{j}(t))italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) , italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) , italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ) 、 的 Lij(σi(t),σj(t))subscript𝐿𝑖𝑗subscript𝜎𝑖𝑡subscript𝜎𝑗𝑡L_{ij}(\sigma_{i}(t),\sigma_{j}(t))italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) , italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ) 显式表达式。

To maintain the constellation as an equilateral triangle, i.e. L12(t)=L13(t)=L23(t)subscript𝐿12𝑡subscript𝐿13𝑡subscript𝐿23𝑡L_{12}(t)=L_{13}(t)=L_{23}(t)italic_L start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT ( italic_t ) = italic_L start_POSTSUBSCRIPT 13 end_POSTSUBSCRIPT ( italic_t ) = italic_L start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT ( italic_t ), the orbits of the three satellites need to be purposefully designed. One intuitive orbit design involves circular orbits for the satellites in the point-mass gravitational field of the Earth:
为了保持星座为等边三角形 L12(t)=L13(t)=L23(t)subscript𝐿12𝑡subscript𝐿13𝑡subscript𝐿23𝑡L_{12}(t)=L_{13}(t)=L_{23}(t)italic_L start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT ( italic_t ) = italic_L start_POSTSUBSCRIPT 13 end_POSTSUBSCRIPT ( italic_t ) = italic_L start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT ( italic_t ) ,即三颗卫星的轨道需要有目的地设计。一种直观的轨道设计涉及卫星在地球点质量引力场中的圆形轨道:

e1(t)=e2(t)=e3(t)=0,subscript𝑒1𝑡subscript𝑒2𝑡subscript𝑒3𝑡0\displaystyle e_{1}(t)=e_{2}(t)=e_{3}(t)=0,italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) = italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) = italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) = 0 , (31)

while ensuring that they share the same orbit size, lie in the same orbital plane, and are phased 120 degrees apart from each other:
同时确保它们具有相同的轨道大小,位于相同的轨道平面上,并且彼此相距 120 度:

{a1(t)=a2(t)=a3(t),i1(t)=i2(t)=i3(t),Ω1(t)=Ω2(t)=Ω3(t),λk(t)=2π3(k1)+λ1(t).casessubscript𝑎1𝑡subscript𝑎2𝑡subscript𝑎3𝑡subscript𝑖1𝑡subscript𝑖2𝑡subscript𝑖3𝑡subscriptΩ1𝑡subscriptΩ2𝑡subscriptΩ3𝑡subscript𝜆𝑘𝑡2𝜋3𝑘1subscript𝜆1𝑡\displaystyle\left\{\begin{array}[]{l}a_{1}(t)=a_{2}(t)=a_{3}(t),\vskip 3.0pt % plus 1.0pt minus 1.0pt\\ i_{1}(t)=i_{2}(t)=i_{3}(t),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \Omega_{1}(t)=\Omega_{2}(t)=\Omega_{3}(t),\vskip 3.0pt plus 1.0pt minus 1.0pt% \\ \lambda_{k}(t)=\frac{2\pi}{3}(k-1)+\lambda_{1}(t).\end{array}\right.{ start_ARRAY start_ROW start_CELL italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) = italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) = italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) , end_CELL end_ROW start_ROW start_CELL italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) = italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) = italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) , end_CELL end_ROW start_ROW start_CELL roman_Ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) = roman_Ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) = roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) , end_CELL end_ROW start_ROW start_CELL italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) = divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_k - 1 ) + italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) . end_CELL end_ROW end_ARRAY (36)

The above requirements on the inter-satellite parameters can be achieved in the two-body problem, if the initial orbital elements σ0ksubscript𝜎0𝑘\sigma_{0k}italic_σ start_POSTSUBSCRIPT 0 italic_k end_POSTSUBSCRIPT in Eq. (26) are set to
如果方程(26)中的初始轨道单元 σ0ksubscript𝜎0𝑘\sigma_{0k}italic_σ start_POSTSUBSCRIPT 0 italic_k end_POSTSUBSCRIPT 设置为

σ0k=σo+δι2π3(k1),subscript𝜎0𝑘subscript𝜎osubscript𝛿𝜄2𝜋3𝑘1\displaystyle\sigma_{0k}=\sigma_{\text{o}}+\delta_{\iota}\frac{2\pi}{3}(k-1),italic_σ start_POSTSUBSCRIPT 0 italic_k end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_k - 1 ) , (37)

where the parameters with subscript “o” are the nominal ones of the TianQin constellation. For instance, these values can be chosen as ao=105subscript𝑎osuperscript105a_{\text{o}}=10^{5}italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT km, eo=0subscript𝑒o0e_{\text{o}}=0italic_e start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = 0, and io=94.7subscript𝑖osuperscript94.7i_{\text{o}}=94.7^{\circ}italic_i start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = 94.7 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT, Ωo=210.4subscriptΩosuperscript210.4\Omega_{\text{o}}=210.4^{\circ}roman_Ω start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = 210.4 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT, respectively establishing the orbit size and orienting the orbital plane perpendicular to J0806 [7, 24]. The initial value λosubscript𝜆o\lambda_{\text{o}}italic_λ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT associated with the orbit phase is typically selected to be any value within the range of 0superscript00^{\circ}0 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT to 120superscript120120^{\circ}120 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT, or it may be specifically designated to avoid Moon eclipses [37].
其中带下标 “o” 的参数是天琴座的名义参数。例如,这些值可以选择为 ao=105subscript𝑎osuperscript105a_{\text{o}}=10^{5}italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = 10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT km、 eo=0subscript𝑒o0e_{\text{o}}=0italic_e start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = 0 、 和 io=94.7subscript𝑖osuperscript94.7i_{\text{o}}=94.7^{\circ}italic_i start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = 94.7 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPTΩo=210.4subscriptΩosuperscript210.4\Omega_{\text{o}}=210.4^{\circ}roman_Ω start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = 210.4 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT ,分别建立轨道大小和垂直于 J0806 的轨道平面 [724]。与轨道相位相关的初始值 λosubscript𝜆o\lambda_{\text{o}}italic_λ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT 通常选择为 0superscript00^{\circ}0 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT120superscript120120^{\circ}120 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT 范围内的任何值,或者可以专门指定以避免月食 [37]。

To analyze additional nominal orbit design allowing for e0𝑒0e\neq 0italic_e ≠ 0 and quantify the impact of eccentricity on the three indicators, the constraint specified by Eq. (31) is relaxed. Subsequently, employing only Eq. (36) or Eq. (37) (for σ{a,i,Ω,λ}𝜎𝑎𝑖Ω𝜆\sigma\in\{a,i,\Omega,\lambda\}italic_σ ∈ { italic_a , italic_i , roman_Ω , italic_λ }), the variations of these indicators in the two-body problem, up to the first order of e𝑒eitalic_e, can be expressed as
为了分析允许 e0𝑒0e\neq 0italic_e ≠ 0 和量化偏心率对三个指标影响的额外标称轨道设计,式(31)规定的约束是放宽的。随后,仅使用方程(36)或方程(37)(for σ{a,i,Ω,λ}𝜎𝑎𝑖Ω𝜆\sigma\in\{a,i,\Omega,\lambda\}italic_σ ∈ { italic_a , italic_i , roman_Ω , italic_λ } ),这些指标在两体问题中的变化,直到 e𝑒eitalic_e 的一阶,可以表示为

Lijkepl(t)=superscriptsubscript𝐿𝑖𝑗kepl𝑡absent\displaystyle L_{ij}^{\text{kepl}}(t)=italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT kepl end_POSTSUPERSCRIPT ( italic_t ) = 3ao+72ao[eisin(Mi+β)\displaystyle~{}\sqrt{3}a_{\text{o}}+\frac{\sqrt{7}}{2}a_{\text{o}}[-e_{i}\sin% \left(M_{i}+\beta\right)square-root start_ARG 3 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT + divide start_ARG square-root start_ARG 7 end_ARG end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ - italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_sin ( italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_β )
+ejsin(Mjβ)],\displaystyle+e_{j}\sin\left(M_{j}-\beta\right)],+ italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_sin ( italic_M start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_β ) ] , (38)
vijkepl(t)=superscriptsubscript𝑣𝑖𝑗kepl𝑡absent\displaystyle v_{ij}^{\text{kepl}}(t)=italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT kepl end_POSTSUPERSCRIPT ( italic_t ) = 0+72aono[eicos(Mi+β)\displaystyle~{}0+\frac{\sqrt{7}}{2}a_{\text{o}}n_{\text{o}}[-e_{i}\cos\left(M% _{i}+\beta\right)0 + divide start_ARG square-root start_ARG 7 end_ARG end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ - italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_cos ( italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_β )
+ejcos(Mjβ)],\displaystyle+e_{j}\cos\left(M_{j}-\beta\right)],+ italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_cos ( italic_M start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_β ) ] , (39)
αkkepl(t)=superscriptsubscript𝛼𝑘kepl𝑡absent\displaystyle\alpha_{k}^{\text{kepl}}(t)=italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT kepl end_POSTSUPERSCRIPT ( italic_t ) = π3+73ekcosMksinβ𝜋373subscript𝑒𝑘subscript𝑀𝑘𝛽\displaystyle~{}\frac{\pi}{3}+\frac{\sqrt{7}}{3}e_{k}\cos M_{k}\sin\betadivide start_ARG italic_π end_ARG start_ARG 3 end_ARG + divide start_ARG square-root start_ARG 7 end_ARG end_ARG start_ARG 3 end_ARG italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_cos italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_sin italic_β
76ei[sin(Miβ)+2sin(Mi+β)]76subscript𝑒𝑖delimited-[]subscript𝑀𝑖𝛽2subscript𝑀𝑖𝛽\displaystyle-\frac{\sqrt{7}}{6}e_{i}[\sin\left(M_{i}-\beta\right)+2\sin\left(% M_{i}+\beta\right)]- divide start_ARG square-root start_ARG 7 end_ARG end_ARG start_ARG 6 end_ARG italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT [ roman_sin ( italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_β ) + 2 roman_sin ( italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_β ) ]
+76ej[2sin(Mjβ)+sin(Mj+β)],76subscript𝑒𝑗delimited-[]2subscript𝑀𝑗𝛽subscript𝑀𝑗𝛽\displaystyle+\frac{\sqrt{7}}{6}e_{j}[2\sin\left(M_{j}-\beta\right)+\sin\left(% M_{j}+\beta\right)],+ divide start_ARG square-root start_ARG 7 end_ARG end_ARG start_ARG 6 end_ARG italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT [ 2 roman_sin ( italic_M start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - italic_β ) + roman_sin ( italic_M start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + italic_β ) ] , (40)

where Mi(t)=2π3(i1)+λ1(t)ωisubscript𝑀𝑖𝑡2𝜋3𝑖1subscript𝜆1𝑡subscript𝜔𝑖M_{i}(t)=\frac{2\pi}{3}(i-1)+\lambda_{1}(t)-\omega_{i}italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) = divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_i - 1 ) + italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) - italic_ω start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and β:=arccos27assign𝛽27\beta:=\arccos\frac{2}{\sqrt{7}}italic_β := roman_arccos divide start_ARG 2 end_ARG start_ARG square-root start_ARG 7 end_ARG end_ARG, with the indices i𝑖iitalic_i, j𝑗jitalic_j, and k𝑘kitalic_k using cyclic indexing (i,j,k=1231𝑖𝑗𝑘1231i,\,j,\,k=1\to 2\to 3\to 1italic_i , italic_j , italic_k = 1 → 2 → 3 → 1). If we further set e1=e2=e3eo0subscript𝑒1subscript𝑒2subscript𝑒3subscript𝑒o0e_{1}=e_{2}=e_{3}\equiv e_{\text{o}}\neq 0italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ≡ italic_e start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ≠ 0 and ω1=ω2=ω3subscript𝜔1subscript𝜔2subscript𝜔3\omega_{1}=\omega_{2}=\omega_{3}italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, then it follows that
其中 Mi(t)=2π3(i1)+λ1(t)ωisubscript𝑀𝑖𝑡2𝜋3𝑖1subscript𝜆1𝑡subscript𝜔𝑖M_{i}(t)=\frac{2\pi}{3}(i-1)+\lambda_{1}(t)-\omega_{i}italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) = divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_i - 1 ) + italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) - italic_ω start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPTβ:=arccos27assign𝛽27\beta:=\arccos\frac{2}{\sqrt{7}}italic_β := roman_arccos divide start_ARG 2 end_ARG start_ARG square-root start_ARG 7 end_ARG end_ARG ,带有索引 i𝑖iitalic_ij𝑗jitalic_j ,并使用 k𝑘kitalic_k 循环索引 ( i,j,k=1231𝑖𝑗𝑘1231i,\,j,\,k=1\to 2\to 3\to 1italic_i , italic_j , italic_k = 1 → 2 → 3 → 1 )。如果我们进一步设置 e1=e2=e3eo0subscript𝑒1subscript𝑒2subscript𝑒3subscript𝑒o0e_{1}=e_{2}=e_{3}\equiv e_{\text{o}}\neq 0italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ≡ italic_e start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ≠ 0ω1=ω2=ω3subscript𝜔1subscript𝜔2subscript𝜔3\omega_{1}=\omega_{2}=\omega_{3}italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,则

Lijkepl-A(t)=3ao32aoeocosMk,superscriptsubscript𝐿𝑖𝑗kepl-A𝑡3subscript𝑎o32subscript𝑎osubscript𝑒osubscript𝑀𝑘\displaystyle L_{ij}^{\text{kepl-A}}(t)=\sqrt{3}a_{\text{o}}-\frac{\sqrt{3}}{2% }a_{\text{o}}e_{\text{o}}\cos M_{k},italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT kepl-A end_POSTSUPERSCRIPT ( italic_t ) = square-root start_ARG 3 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT - divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT o end_POSTSUBSCRIPT roman_cos italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (41)
vijkepl-A(t)=0+32aonoeosinMk,superscriptsubscript𝑣𝑖𝑗kepl-A𝑡032subscript𝑎osubscript𝑛osubscript𝑒osubscript𝑀𝑘\displaystyle v_{ij}^{\text{kepl-A}}(t)=0+\frac{\sqrt{3}}{2}a_{\text{o}}n_{% \text{o}}e_{\text{o}}\sin M_{k},italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT kepl-A end_POSTSUPERSCRIPT ( italic_t ) = 0 + divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT o end_POSTSUBSCRIPT roman_sin italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (42)
αkkepl-A(t)=π332eocosMk,superscriptsubscript𝛼𝑘kepl-A𝑡𝜋332subscript𝑒osubscript𝑀𝑘\displaystyle\alpha_{k}^{\text{kepl-A}}(t)=\frac{\pi}{3}-\frac{\sqrt{3}}{2}e_{% \text{o}}\cos M_{k},italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT kepl-A end_POSTSUPERSCRIPT ( italic_t ) = divide start_ARG italic_π end_ARG start_ARG 3 end_ARG - divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 2 end_ARG italic_e start_POSTSUBSCRIPT o end_POSTSUBSCRIPT roman_cos italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (43)

where Mk(t)=2π3(k1)+M1(t)subscript𝑀𝑘𝑡2𝜋3𝑘1subscript𝑀1𝑡M_{k}(t)=\frac{2\pi}{3}(k-1)+M_{1}(t)italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) = divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_k - 1 ) + italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ). Equations (41)-(43) indicate that, at the zeroth order of e𝑒eitalic_e, the three TianQin satellites can form a constant equilateral triangle. However, when accounting for eccentricity, as observed in perturbed orbits, the constellation’s evolution deviates from the ideal equilateral triangle, exhibiting periodic variations.
其中 Mk(t)=2π3(k1)+M1(t)subscript𝑀𝑘𝑡2𝜋3𝑘1subscript𝑀1𝑡M_{k}(t)=\frac{2\pi}{3}(k-1)+M_{1}(t)italic_M start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) = divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_k - 1 ) + italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) . 方程 (41)-(43) 表明,在 的零阶 e𝑒eitalic_e 处,三颗天琴卫星可以形成一个恒定的等边三角形。然而,当考虑到在扰动轨道上观察到的偏心率时,星座的演化偏离了理想的等边三角形,表现出周期性变化。

The close-to-circular orbits, as inspired by Eqs. (31) and (36) for Keplerian orbits, are currently employed in TianQin orbit studies (see, e.g., [21, 24, 13, 38, 39, 40, 41, 42]). It is worth noting that, to obtain the nominal equilateral triangle configuration, there is another option: elliptical frozen orbits. From Eqs. (38)-(40), if e1=e2=e3eo0subscript𝑒1subscript𝑒2subscript𝑒3subscript𝑒o0e_{1}=e_{2}=e_{3}\equiv e_{\text{o}}\neq 0italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ≡ italic_e start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ≠ 0, and M1=M2=M3subscript𝑀1subscript𝑀2subscript𝑀3M_{1}=M_{2}=M_{3}italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, namely ωk=ω1+2π3(k1)subscript𝜔𝑘subscript𝜔12𝜋3𝑘1\omega_{k}=\omega_{1}+\frac{2\pi}{3}(k-1)italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_k - 1 ), then Lijkepl-B(t)3ao3aoeocosE1(t)superscriptsubscript𝐿𝑖𝑗kepl-B𝑡3subscript𝑎o3subscript𝑎osubscript𝑒osubscript𝐸1𝑡L_{ij}^{\text{kepl-B}}(t)\equiv\sqrt{3}a_{\text{o}}-\sqrt{3}a_{\text{o}}e_{% \text{o}}\cos E_{1}(t)italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT kepl-B end_POSTSUPERSCRIPT ( italic_t ) ≡ square-root start_ARG 3 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT - square-root start_ARG 3 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT o end_POSTSUBSCRIPT roman_cos italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ), representing an equilateral-triangle constellation with three arm-lengths that vary synchronously. Preliminary numerical simulation results show that the long-term stability of the constellation, based on this design, is not as favorable as that of the close-to-circular orbits. Furthermore, the impact of this design on other aspects of the mission, such as point-ahead angle variations associated with the finite speed of light, requires further assessment. In this paper, we focus on the study of a three-satellite constellation with close-to-circular orbits.
受 Eqs 启发的接近圆形的轨道。(31) 和 (36) 目前用于天琴轨道研究(参见 [2124133839404142])。值得注意的是,要获得名义上的等边三角形构型,还有另一种选择:椭圆冻结轨道。来自 Eqs.(38)-(40),如果 e1=e2=e3eo0subscript𝑒1subscript𝑒2subscript𝑒3subscript𝑒o0e_{1}=e_{2}=e_{3}\equiv e_{\text{o}}\neq 0italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ≡ italic_e start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ≠ 0 ,和 M1=M2=M3subscript𝑀1subscript𝑀2subscript𝑀3M_{1}=M_{2}=M_{3}italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,即 ωk=ω1+2π3(k1)subscript𝜔𝑘subscript𝜔12𝜋3𝑘1\omega_{k}=\omega_{1}+\frac{2\pi}{3}(k-1)italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_k - 1 ) ,则 Lijkepl-B(t)3ao3aoeocosE1(t)superscriptsubscript𝐿𝑖𝑗kepl-B𝑡3subscript𝑎o3subscript𝑎osubscript𝑒osubscript𝐸1𝑡L_{ij}^{\text{kepl-B}}(t)\equiv\sqrt{3}a_{\text{o}}-\sqrt{3}a_{\text{o}}e_{% \text{o}}\cos E_{1}(t)italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT kepl-B end_POSTSUPERSCRIPT ( italic_t ) ≡ square-root start_ARG 3 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT - square-root start_ARG 3 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_e start_POSTSUBSCRIPT o end_POSTSUBSCRIPT roman_cos italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) ,代表一个等边三角形星座,具有三个同步变化的臂长。初步数值模拟结果表明,基于此设计的星座的长期稳定性不如接近圆形的轨道。此外,这种设计对任务其他方面的影响,例如与有限光速相关的前方角度变化,需要进一步评估。在本文中,我们专注于研究具有接近圆形轨道的三卫星星座。

III Effects of lunisolar perturbations on TianQin constellation
、阴阳扰动对天秦星座的影响

The TianQin constellation is subject not only to the central gravitational attraction but also to gravitational perturbations. These perturbations can distort the carefully designed equilateral-triangle configuration. To gain a more accurate understanding of the TianQin constellation’s variations, it is crucial to account for these gravitational perturbations.
天琴座不仅受中心引力的影响,还受引力扰动的影响。这些扰动会扭曲精心设计的等边三角形构型。为了更准确地了解天琴座的变化,考虑这些引力扰动至关重要。

The primary perturbations originate from the Moon and the Sun, with magnitudes of approximately 4×1044superscript1044\times 10^{-4}4 × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT and 2×1042superscript1042\times 10^{-4}2 × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT, respectively [33]. In this section, we collectively address the point mass effects of these two perturbing bodies. Furthermore, we also incorporate the secular perturbation arising from the third most significant perturbation, Earth’s J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT perturbation, which has a magnitude of 6×1066superscript1066\times 10^{-6}6 × 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT. Other perturbations, e.g., the higher-degree non-spherical gravity fields of the Earth, have a minor impact on satellite positions and constellation stability. As illustrated in Fig. 2, these perturbations lead to deviations of approximately 3.3 km in satellite positions, ±2.2plus-or-minus2.2\pm 2.2± 2.2 km in arm-lengths, ±0.0020plus-or-minus0.0020\pm 0.0020± 0.0020 m/s in relative velocities, and ±0.0012plus-or-minussuperscript0.0012\pm 0.0012^{\circ}± 0.0012 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT in breathing angles over a 5-year period. In contrast to Refs. [21, 22, 23, 36, 16, 32], the perturbation solutions developed in this study offer explicit expressions with improved precision, enabling a more precise description of the distinctive 105superscript10510^{5}10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT km-radius orbits relevant to space-based GW detection.
主要扰动起源于月球和太阳,震等分别约为 4×1044superscript1044\times 10^{-4}4 × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT2×1042superscript1042\times 10^{-4}2 × 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT [33]。在本节中,我们将共同讨论这两个扰动体的点质量效应。此外,我们还考虑了由第三个最显著的扰动,即地球 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 的扰动引起的长期扰动,其震级为 6×1066superscript1066\times 10^{-6}6 × 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT 。其他扰动,例如地球的更高程度的非球形重力场,对卫星位置和星座稳定性影响较小。如图所示。2,这些扰动导致卫星位置偏差约为 3.3 km,臂长偏差约为 km, ±2.2plus-or-minus2.2\pm 2.2± 2.2 ±0.0020plus-or-minus0.0020\pm 0.0020± 0.0020 相对速度偏差为 m/s, ±0.0012plus-or-minussuperscript0.0012\pm 0.0012^{\circ}± 0.0012 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT 呼吸角偏差为 5 年。与 Refs 相反。 [212223361632],本研究开发的扰动解决方案提供了明确的表达式,精度更高,能够更精确地描述与天基GW探测相关的不同 105superscript10510^{5}10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT km半径轨道。

Figure 2: Time evolution of deviations between the simplified (SNM) and high-precision (HPNM) numerical models for satellite positions and three indicators. SNM incorporates the point-mass gravity fields of the Earth, Moon, and Sun, as well as the Earth’s J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. HPNM additionally considers the higher-degree non-spherical gravity fields of the Earth and the point-mass gravity fields of other planets (for further details, see Appendix B.3). In these plots, red corresponds to SC1, v23subscript𝑣23v_{23}italic_v start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT, L23subscript𝐿23L_{23}italic_L start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT, or α1subscript𝛼1\alpha_{1}italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT; green represents SC2, v31subscript𝑣31v_{31}italic_v start_POSTSUBSCRIPT 31 end_POSTSUBSCRIPT, L31subscript𝐿31L_{31}italic_L start_POSTSUBSCRIPT 31 end_POSTSUBSCRIPT, or α2subscript𝛼2\alpha_{2}italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT; and blue indicates SC3, v12subscript𝑣12v_{12}italic_v start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT, L12subscript𝐿12L_{12}italic_L start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT, or α3subscript𝛼3\alpha_{3}italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. The initial orbital elements used are the same as those presented in Table 6.
图 2: 卫星位置和三个指标的简化 (SNM) 和高精度 (HPNM) 数值模型之间偏差的时间演变。SNM 包含地球、月球和太阳的点质量重力场,以及地球的 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT .HPNM 还考虑了地球的更高阶非球形重力场和其他行星的点质量重力场(有关详细信息,请参阅附录 B.3)。在这些图中,红色对应于 SC1、 v23subscript𝑣23v_{23}italic_v start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPTL23subscript𝐿23L_{23}italic_L start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPTα1subscript𝛼1\alpha_{1}italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ;绿色代表 SC2、 v31subscript𝑣31v_{31}italic_v start_POSTSUBSCRIPT 31 end_POSTSUBSCRIPTL31subscript𝐿31L_{31}italic_L start_POSTSUBSCRIPT 31 end_POSTSUBSCRIPTα2subscript𝛼2\alpha_{2}italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ;蓝色表示 SC3、 v12subscript𝑣12v_{12}italic_v start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPTL12subscript𝐿12L_{12}italic_L start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPTα3subscript𝛼3\alpha_{3}italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 。使用的初始轨道元件与表 6 中所示的相同。

III.1 Dynamic model
三.1 动态模型

The gravitational potential U𝑈Uitalic_U acting on a satellite can be expressed as
作用在卫星上的引力势 U𝑈Uitalic_U 可以表示为

U=U0+,U0=μr,formulae-sequence𝑈subscript𝑈0subscript𝑈0𝜇𝑟\displaystyle U=U_{0}+\mathcal{R},\qquad U_{0}=\frac{\mu}{r},italic_U = italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + caligraphic_R , italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = divide start_ARG italic_μ end_ARG start_ARG italic_r end_ARG , (44)

where U0subscript𝑈0U_{0}italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the gravitational potential of a pointlike Earth, and \mathcal{R}caligraphic_R represents a perturbative potential describing the satellite’s perturbed motion. Under the influence of \mathcal{R}caligraphic_R, the evolution of the satellite’s orbital elements is governed by Lagrange’s planetary equations [43],
其中 U0subscript𝑈0U_{0}italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT 是点状地球的引力势,表示 \mathcal{R}caligraphic_R 描述卫星扰动运动的扰动势。在 \mathcal{R}caligraphic_R 的影响下,卫星轨道元件的演化受拉格朗日行星方程 [43] 的控制,

dadt=𝑑𝑎𝑑𝑡absent\displaystyle\frac{da}{dt}=divide start_ARG italic_d italic_a end_ARG start_ARG italic_d italic_t end_ARG = 2naλ,2𝑛𝑎𝜆\displaystyle~{}\frac{2}{na}\frac{\partial\mathcal{R}}{\partial\lambda},divide start_ARG 2 end_ARG start_ARG italic_n italic_a end_ARG divide start_ARG ∂ caligraphic_R end_ARG start_ARG ∂ italic_λ end_ARG , (45)
didt=𝑑𝑖𝑑𝑡absent\displaystyle\frac{di}{dt}=divide start_ARG italic_d italic_i end_ARG start_ARG italic_d italic_t end_ARG = 1na2eˇsini[cosi(ξηηξ+λ)\displaystyle~{}\frac{1}{na^{2}\check{e}\sin i}\left[\cos i\left(\xi\frac{% \partial\mathcal{R}}{\partial\eta}-\eta\frac{\partial\mathcal{R}}{\partial\xi}% +\frac{\partial\mathcal{R}}{\partial\lambda}\right)\right.divide start_ARG 1 end_ARG start_ARG italic_n italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT overroman_ˇ start_ARG italic_e end_ARG roman_sin italic_i end_ARG [ roman_cos italic_i ( italic_ξ divide start_ARG ∂ caligraphic_R end_ARG start_ARG ∂ italic_η end_ARG - italic_η divide start_ARG ∂ caligraphic_R end_ARG start_ARG ∂ italic_ξ end_ARG + divide start_ARG ∂ caligraphic_R end_ARG start_ARG ∂ italic_λ end_ARG )
Ω],\displaystyle\left.-\frac{\partial\mathcal{R}}{\partial\Omega}\right],- divide start_ARG ∂ caligraphic_R end_ARG start_ARG ∂ roman_Ω end_ARG ] , (46)
dΩdt=𝑑Ω𝑑𝑡absent\displaystyle\frac{d\Omega}{dt}=divide start_ARG italic_d roman_Ω end_ARG start_ARG italic_d italic_t end_ARG = 1na2eˇsinii,1𝑛superscript𝑎2ˇ𝑒𝑖𝑖\displaystyle~{}\frac{1}{na^{2}\check{e}\sin i}\frac{\partial\mathcal{R}}{% \partial i},divide start_ARG 1 end_ARG start_ARG italic_n italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT overroman_ˇ start_ARG italic_e end_ARG roman_sin italic_i end_ARG divide start_ARG ∂ caligraphic_R end_ARG start_ARG ∂ italic_i end_ARG , (47)
dξdt=𝑑𝜉𝑑𝑡absent\displaystyle\frac{d\xi}{dt}=divide start_ARG italic_d italic_ξ end_ARG start_ARG italic_d italic_t end_ARG = eˇna2ηξeˇna2(1+eˇ)λ+ηcosidΩdt,ˇ𝑒𝑛superscript𝑎2𝜂𝜉ˇ𝑒𝑛superscript𝑎21ˇ𝑒𝜆𝜂𝑖𝑑Ω𝑑𝑡\displaystyle-\frac{\check{e}}{na^{2}}\frac{\partial\mathcal{R}}{\partial\eta}% -\xi\frac{\check{e}}{na^{2}(1+\check{e})}\frac{\partial\mathcal{R}}{\partial% \lambda}+\eta\cos i\frac{d\Omega}{dt},- divide start_ARG overroman_ˇ start_ARG italic_e end_ARG end_ARG start_ARG italic_n italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG divide start_ARG ∂ caligraphic_R end_ARG start_ARG ∂ italic_η end_ARG - italic_ξ divide start_ARG overroman_ˇ start_ARG italic_e end_ARG end_ARG start_ARG italic_n italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 + overroman_ˇ start_ARG italic_e end_ARG ) end_ARG divide start_ARG ∂ caligraphic_R end_ARG start_ARG ∂ italic_λ end_ARG + italic_η roman_cos italic_i divide start_ARG italic_d roman_Ω end_ARG start_ARG italic_d italic_t end_ARG , (48)
dηdt=𝑑𝜂𝑑𝑡absent\displaystyle\frac{d\eta}{dt}=divide start_ARG italic_d italic_η end_ARG start_ARG italic_d italic_t end_ARG = eˇna2ξηeˇna2(1+eˇ)λξcosidΩdt,ˇ𝑒𝑛superscript𝑎2𝜉𝜂ˇ𝑒𝑛superscript𝑎21ˇ𝑒𝜆𝜉𝑖𝑑Ω𝑑𝑡\displaystyle~{}\frac{\check{e}}{na^{2}}\frac{\partial\mathcal{R}}{\partial\xi% }-\eta\frac{\check{e}}{na^{2}(1+\check{e})}\frac{\partial\mathcal{R}}{\partial% \lambda}-\xi\cos i\frac{d\Omega}{dt},divide start_ARG overroman_ˇ start_ARG italic_e end_ARG end_ARG start_ARG italic_n italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG divide start_ARG ∂ caligraphic_R end_ARG start_ARG ∂ italic_ξ end_ARG - italic_η divide start_ARG overroman_ˇ start_ARG italic_e end_ARG end_ARG start_ARG italic_n italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 + overroman_ˇ start_ARG italic_e end_ARG ) end_ARG divide start_ARG ∂ caligraphic_R end_ARG start_ARG ∂ italic_λ end_ARG - italic_ξ roman_cos italic_i divide start_ARG italic_d roman_Ω end_ARG start_ARG italic_d italic_t end_ARG , (49)
dλdt=𝑑𝜆𝑑𝑡absent\displaystyle\frac{d\lambda}{dt}=divide start_ARG italic_d italic_λ end_ARG start_ARG italic_d italic_t end_ARG = n2naa+eˇna2(1+eˇ)(ξξ+ηη)𝑛2𝑛𝑎𝑎ˇ𝑒𝑛superscript𝑎21ˇ𝑒𝜉𝜉𝜂𝜂\displaystyle~{}n-\frac{2}{na}\frac{\partial\mathcal{R}}{\partial a}+\frac{% \check{e}}{na^{2}(1+\check{e})}\left(\xi\frac{\partial\mathcal{R}}{\partial\xi% }+\eta\frac{\partial\mathcal{R}}{\partial\eta}\right)italic_n - divide start_ARG 2 end_ARG start_ARG italic_n italic_a end_ARG divide start_ARG ∂ caligraphic_R end_ARG start_ARG ∂ italic_a end_ARG + divide start_ARG overroman_ˇ start_ARG italic_e end_ARG end_ARG start_ARG italic_n italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 + overroman_ˇ start_ARG italic_e end_ARG ) end_ARG ( italic_ξ divide start_ARG ∂ caligraphic_R end_ARG start_ARG ∂ italic_ξ end_ARG + italic_η divide start_ARG ∂ caligraphic_R end_ARG start_ARG ∂ italic_η end_ARG )
cosidΩdt,𝑖𝑑Ω𝑑𝑡\displaystyle-\cos i\frac{d\Omega}{dt},- roman_cos italic_i divide start_ARG italic_d roman_Ω end_ARG start_ARG italic_d italic_t end_ARG , (50)

where eˇ:=1e2assignˇ𝑒1superscript𝑒2\check{e}:=\sqrt{1-e^{2}}overroman_ˇ start_ARG italic_e end_ARG := square-root start_ARG 1 - italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG, and new variables ξ𝜉\xiitalic_ξ and η𝜂\etaitalic_η are introduced,
其中 eˇ:=1e2assignˇ𝑒1superscript𝑒2\check{e}:=\sqrt{1-e^{2}}overroman_ˇ start_ARG italic_e end_ARG := square-root start_ARG 1 - italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , 和 新变量 ξ𝜉\xiitalic_ξη𝜂\etaitalic_η 被引入,

ξ:=ecosω,η:=esinω,formulae-sequenceassign𝜉𝑒𝜔assign𝜂𝑒𝜔\displaystyle\xi:=e\cos\omega,\qquad\eta:=e\sin\omega,italic_ξ := italic_e roman_cos italic_ω , italic_η := italic_e roman_sin italic_ω , (51)

to avoid the singularity at e=0𝑒0e=0italic_e = 0. When =00\mathcal{R}=0caligraphic_R = 0, the solutions to Eqs. (45)-(50) revert to the Keplerian case discussed in Sec. II.1.
为了避免 处 e=0𝑒0e=0italic_e = 0 的奇点。当 =00\mathcal{R}=0caligraphic_R = 0 , 方程的解 程.(45)-(50) 回到第 II.1 节讨论的 Keplerian 案。

For TianQin orbits with an orbital radius of 105superscript10510^{5}10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT km, the perturbative potential \mathcal{R}caligraphic_R predominantly encompasses the perturbation effects arising from the Sun, Moon, and Earth’s J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT term, as expressed in the following expressions:
对于轨道半径为 105superscript10510^{5}10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT km 的天琴轨道,扰动势 \mathcal{R}caligraphic_R 主要包括由太阳、月亮和地球 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 项引起的扰动效应,如以下表达式所示:

=s+m+J2subscriptssubscriptmsubscriptsubscript𝐽2\displaystyle\mathcal{R}=\mathcal{R}_{\text{s}}+\mathcal{R}_{\text{m}}+% \mathcal{R}_{J_{2}}caligraphic_R = caligraphic_R start_POSTSUBSCRIPT s end_POSTSUBSCRIPT + caligraphic_R start_POSTSUBSCRIPT m end_POSTSUBSCRIPT + caligraphic_R start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT (52)

with [33]
[33]

s=μ2r2r233cos2ψ212,subscriptssubscript𝜇2superscript𝑟2superscriptsubscript𝑟233superscript2subscript𝜓212\displaystyle\mathcal{R}_{\text{s}}=\frac{\mu_{2}r^{2}}{r_{2}^{3}}\frac{3\cos^% {2}\psi_{2}-1}{2},caligraphic_R start_POSTSUBSCRIPT s end_POSTSUBSCRIPT = divide start_ARG italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG divide start_ARG 3 roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - 1 end_ARG start_ARG 2 end_ARG , (53)
m=μ3r3N=2𝒩(rr3)NPN(cosψ3),subscriptmsubscript𝜇3subscript𝑟3superscriptsubscript𝑁2𝒩superscript𝑟subscript𝑟3𝑁subscript𝑃𝑁subscript𝜓3\displaystyle\mathcal{R}_{\text{m}}=\frac{\mu_{3}}{r_{3}}\sum_{N=2}^{\mathcal{% N}}\left(\frac{r}{r_{3}}\right)^{N}P_{N}(\cos\psi_{3}),caligraphic_R start_POSTSUBSCRIPT m end_POSTSUBSCRIPT = divide start_ARG italic_μ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_N = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ( divide start_ARG italic_r end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( roman_cos italic_ψ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (54)
J2=μRe2r3J23sin2φ12,subscriptsubscript𝐽2𝜇superscriptsubscript𝑅e2superscript𝑟3subscript𝐽23superscript2𝜑12\displaystyle\mathcal{R}_{\!J_{2}}=-\frac{\mu R_{\text{e}}^{2}}{r^{3}}J_{2}% \frac{3\sin^{2}\varphi-1}{2},caligraphic_R start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = - divide start_ARG italic_μ italic_R start_POSTSUBSCRIPT e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 3 roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_φ - 1 end_ARG start_ARG 2 end_ARG , (55)

where μ2=GMssubscript𝜇2𝐺subscript𝑀s\mu_{2}=GM_{\text{s}}italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_G italic_M start_POSTSUBSCRIPT s end_POSTSUBSCRIPT and μ3=GMmsubscript𝜇3𝐺subscript𝑀m\mu_{3}=GM_{\text{m}}italic_μ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_G italic_M start_POSTSUBSCRIPT m end_POSTSUBSCRIPT are the gravitational constants of the Sun and the Moon, respectively. r2subscript𝑟2r_{2}italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and r3subscript𝑟3r_{3}italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT denote the geocentric distances of the Sun and the Moon. Moreover, PN(x)subscript𝑃𝑁𝑥P_{N}(x)italic_P start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_x ) is the Legendre polynomial of degree N𝑁Nitalic_N, with 𝒩=6𝒩6\mathcal{N}=6caligraphic_N = 6 signifying the truncation degree. The derivation of Eq. (54) is presented in Appendix B.1, suggesting that employing Legendre polynomial expansions is more advantageous than the original square root form (Eq. (117)) for solving Lagrange’s equations. Additionally, Resubscript𝑅eR_{\text{e}}italic_R start_POSTSUBSCRIPT e end_POSTSUBSCRIPT stands for the equatorial radius of the Earth, and J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT represents the second zonal harmonic coefficient. Furthermore, ψ2subscript𝜓2\psi_{2}italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is the angular separation of the Sun and the satellite as observed from the Earth’s center,
其中 μ2=GMssubscript𝜇2𝐺subscript𝑀s\mu_{2}=GM_{\text{s}}italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_G italic_M start_POSTSUBSCRIPT s end_POSTSUBSCRIPTμ3=GMmsubscript𝜇3𝐺subscript𝑀m\mu_{3}=GM_{\text{m}}italic_μ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_G italic_M start_POSTSUBSCRIPT m end_POSTSUBSCRIPT 分别是太阳和月亮的引力常数。 r2subscript𝑟2r_{2}italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPTr3subscript𝑟3r_{3}italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 表示太阳和月亮的地心距离。此外, PN(x)subscript𝑃𝑁𝑥P_{N}(x)italic_P start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_x ) 是度 N𝑁Nitalic_N 的勒让德多项式,表示 𝒩=6𝒩6\mathcal{N}=6caligraphic_N = 6 截断度。方程(54)的推导见附录 B.1,表明采用勒让德多项式展开比原始平方根形式(方程(117))更有利于求解拉格朗日方程。此外, Resubscript𝑅eR_{\text{e}}italic_R start_POSTSUBSCRIPT e end_POSTSUBSCRIPT 表示地球的赤道半径,并 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 表示第二纬向谐波系数。 此外, ψ2subscript𝜓2\psi_{2}italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 是从地心观测到的太阳和卫星的角度分离,

cosψ2=𝐫^𝟐𝐫^,subscript𝜓2subscript^𝐫2^𝐫\displaystyle\cos\psi_{2}=\mathbf{\hat{r}_{2}}\cdot\mathbf{\hat{r}},roman_cos italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = over^ start_ARG bold_r end_ARG start_POSTSUBSCRIPT bold_2 end_POSTSUBSCRIPT ⋅ over^ start_ARG bold_r end_ARG , (56)

where 𝐫^𝟐subscript^𝐫2\mathbf{\hat{r}_{2}}over^ start_ARG bold_r end_ARG start_POSTSUBSCRIPT bold_2 end_POSTSUBSCRIPT and 𝐫^^𝐫\mathbf{\hat{r}}over^ start_ARG bold_r end_ARG denote the unit position vectors of the Sun and the satellite, respectively,
其中 𝐫^𝟐subscript^𝐫2\mathbf{\hat{r}_{2}}over^ start_ARG bold_r end_ARG start_POSTSUBSCRIPT bold_2 end_POSTSUBSCRIPT𝐫^^𝐫\mathbf{\hat{r}}over^ start_ARG bold_r end_ARG 分别表示太阳和卫星的单位位置向量,

𝐫^𝟐=[cosu2sinu20],𝐫^=Rz(Ω)Rx(i)Rz(ω)[cosνsinν0],formulae-sequencesubscript^𝐫2matrixsubscriptu2subscriptu20^𝐫subscript𝑅𝑧Ωsubscript𝑅𝑥𝑖subscript𝑅𝑧𝜔matrix𝜈𝜈0\displaystyle\mathbf{\hat{r}_{2}}=\begin{bmatrix}\cos\text{u}_{2}\\ \sin\text{u}_{2}\\ 0\end{bmatrix},\quad\mathbf{\hat{r}}=R_{z}(-\Omega)R_{x}(-i)R_{z}(-\omega)% \begin{bmatrix}\cos\nu\\ \sin\nu\\ 0\end{bmatrix},over^ start_ARG bold_r end_ARG start_POSTSUBSCRIPT bold_2 end_POSTSUBSCRIPT = [ start_ARG start_ROW start_CELL roman_cos u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL roman_sin u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW end_ARG ] , over^ start_ARG bold_r end_ARG = italic_R start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( - roman_Ω ) italic_R start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( - italic_i ) italic_R start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( - italic_ω ) [ start_ARG start_ROW start_CELL roman_cos italic_ν end_CELL end_ROW start_ROW start_CELL roman_sin italic_ν end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW end_ARG ] , (57)

with u2:=Ω2+ω2+ν2assignsubscriptu2subscriptΩ2subscript𝜔2subscript𝜈2\text{u}_{2}:=\Omega_{2}+\omega_{2}+\nu_{2}u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := roman_Ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT representing the Sun’s ecliptic longitude. Similarly, ψ3subscript𝜓3\psi_{3}italic_ψ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT is given by
,代表 u2:=Ω2+ω2+ν2assignsubscriptu2subscriptΩ2subscript𝜔2subscript𝜈2\text{u}_{2}:=\Omega_{2}+\omega_{2}+\nu_{2}u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := roman_Ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 太阳的黄道经度。 同样, ψ3subscript𝜓3\psi_{3}italic_ψ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT

cosψ3=𝐫^𝟑𝐫^,subscript𝜓3subscript^𝐫3^𝐫\displaystyle\cos\psi_{3}=\mathbf{\hat{r}_{3}}\cdot\mathbf{\hat{r}},roman_cos italic_ψ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = over^ start_ARG bold_r end_ARG start_POSTSUBSCRIPT bold_3 end_POSTSUBSCRIPT ⋅ over^ start_ARG bold_r end_ARG , (58)

with 𝐫^𝟑subscript^𝐫3\mathbf{\hat{r}_{3}}over^ start_ARG bold_r end_ARG start_POSTSUBSCRIPT bold_3 end_POSTSUBSCRIPT being the Moon’s unit position vector,
𝐫^𝟑subscript^𝐫3\mathbf{\hat{r}_{3}}over^ start_ARG bold_r end_ARG start_POSTSUBSCRIPT bold_3 end_POSTSUBSCRIPT 月球的单位位置向量,

𝐫^𝟑=Rz(Ω3)Rx(i3)Rz(u3)[100],subscript^𝐫3subscript𝑅𝑧subscriptΩ3subscript𝑅𝑥subscript𝑖3subscript𝑅𝑧subscriptu3matrix100\displaystyle\mathbf{\hat{r}_{3}}=R_{z}(-\Omega_{3})R_{x}(-i_{3})R_{z}(-\text{% u}_{3})\begin{bmatrix}1\\ 0\\ 0\end{bmatrix},over^ start_ARG bold_r end_ARG start_POSTSUBSCRIPT bold_3 end_POSTSUBSCRIPT = italic_R start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( - roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_R start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( - italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_R start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( - u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) [ start_ARG start_ROW start_CELL 1 end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW end_ARG ] , (59)

where Ω3subscriptΩ3\Omega_{3}roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, i3subscript𝑖3i_{3}italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, and u3:=ω3+ν3assignsubscriptu3subscript𝜔3subscript𝜈3\text{u}_{3}:=\omega_{3}+\nu_{3}u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT := italic_ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_ν start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT correspond to the Moon’s longitude of ascending node, inclination, and latitude argument, respectively. φ𝜑\varphiitalic_φ of Eq. (55) signifies the satellite’s geocentric latitude in the Earth-fixed coordinate system,
其中 Ω3subscriptΩ3\Omega_{3}roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPTi3subscript𝑖3i_{3}italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , 和 u3:=ω3+ν3assignsubscriptu3subscript𝜔3subscript𝜈3\text{u}_{3}:=\omega_{3}+\nu_{3}u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT := italic_ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_ν start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 分别对应于月球的升交点经度、倾角和纬度参数。 φ𝜑\varphiitalic_φ 方程(55)表示卫星在地球固定坐标系中的地心纬度,

sinφ=sinisin(ω+ν).𝜑𝑖𝜔𝜈\displaystyle\sin\varphi=\sin i\sin(\omega+\nu).roman_sin italic_φ = roman_sin italic_i roman_sin ( italic_ω + italic_ν ) . (60)

Substituting Eqs. (56)-(60) into (53)-(55), \mathcal{R}caligraphic_R is formulated as a function of the satellite’s orbital elements and those of the Sun and the Moon, (a,e,i,Ω,ω,ν;σ23)𝑎𝑒𝑖Ω𝜔𝜈subscript𝜎23\mathcal{R}(a,e,i,\Omega,\omega,\nu;\sigma_{23})caligraphic_R ( italic_a , italic_e , italic_i , roman_Ω , italic_ω , italic_ν ; italic_σ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT ), where σ23={r2,u2,i3,Ω3,r3,u3}subscript𝜎23subscript𝑟2subscriptu2subscript𝑖3subscriptΩ3subscript𝑟3subscriptu3\sigma_{23}=\{r_{2},\text{u}_{2},i_{3},\Omega_{3},r_{3},\text{u}_{3}\}italic_σ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT = { italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT }. Consequently, the singularity-free form of the potential, (a,ξ,η,i,Ω,λ;σ23)𝑎𝜉𝜂𝑖Ω𝜆subscript𝜎23\mathcal{R}(a,\xi,\eta,i,\Omega,\lambda;\sigma_{23})caligraphic_R ( italic_a , italic_ξ , italic_η , italic_i , roman_Ω , italic_λ ; italic_σ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT ), can be obtained. Moreover, it is important to note that Eq. (60) is formulated in the equatorial coordinate system. For a unified description of the influence of all three perturbations on satellite orbits, including both solar and lunar perturbations, the transformation to the ecliptic coordinate system must be taken into account (see Appendix C.3 for more details).
代入 eqs.(56)-(60) 转换为 (53)-(55), \mathcal{R}caligraphic_R 表示为卫星轨道元件与太阳和月亮轨道元件的函数, (a,e,i,Ω,ω,ν;σ23)𝑎𝑒𝑖Ω𝜔𝜈subscript𝜎23\mathcal{R}(a,e,i,\Omega,\omega,\nu;\sigma_{23})caligraphic_R ( italic_a , italic_e , italic_i , roman_Ω , italic_ω , italic_ν ; italic_σ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT ) 其中 σ23={r2,u2,i3,Ω3,r3,u3}subscript𝜎23subscript𝑟2subscriptu2subscript𝑖3subscriptΩ3subscript𝑟3subscriptu3\sigma_{23}=\{r_{2},\text{u}_{2},i_{3},\Omega_{3},r_{3},\text{u}_{3}\}italic_σ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT = { italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT } 。因此,可以获得势能 (a,ξ,η,i,Ω,λ;σ23)𝑎𝜉𝜂𝑖Ω𝜆subscript𝜎23\mathcal{R}(a,\xi,\eta,i,\Omega,\lambda;\sigma_{23})caligraphic_R ( italic_a , italic_ξ , italic_η , italic_i , roman_Ω , italic_λ ; italic_σ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT ) 的无奇点形式 。此外,需要注意的是,方程(60)是在赤道坐标系中表示的。要统一描述所有三种扰动对卫星轨道的影响,包括太阳和月球的扰动,必须考虑到黄道坐标系的变换(更多细节见附录 C.3)。

III.2 Motion of the Sun and Moon
三.2 太阳和月亮的运动

To solve the Lagrange equations, the coordinates of the Sun and the Moon, relative to r2subscript𝑟2r_{2}italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, u2subscriptu2\text{u}_{2}u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, i3subscript𝑖3i_{3}italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, Ω3subscriptΩ3\Omega_{3}roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, r3subscript𝑟3r_{3}italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, and u3subscriptu3\text{u}_{3}u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, are required. While high-precision coordinates are available through numerical integration, such as the JPL ephemerides DE430 [44], they are less suitable for analytical purposes. References like [35, 45] offer analytical formulas with reduced precision, providing geocentric solar coordinates based on a simplified, unperturbed motion of the Earth around the Sun and expressed using appropriate mean orbital elements. In contrast, the Moon’s motion, influenced by strong solar and terrestrial perturbations, is described through linear terms corresponding to its long-term precessing elliptical orbit and numerous trigonometric terms capturing periodic variations.
要求解拉格朗日方程,需要太阳和月亮相对于 r2subscript𝑟2r_{2}italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPTu2subscriptu2\text{u}_{2}u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT i3subscript𝑖3i_{3}italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Ω3subscriptΩ3\Omega_{3}roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT r3subscript𝑟3r_{3}italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPTu3subscriptu3\text{u}_{3}u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 的坐标。虽然可以通过数值积分获得高精度坐标,例如 JPL 星历仪 DE430 [44],但它们不太适合用于分析目的。像 [3545] 这样的参考文献提供了精度降低的解析公式,根据地球围绕太阳的简化、不受干扰的运动提供地心太阳坐标,并使用适当的平均轨道元素表示。相比之下,月球的运动受强烈的太阳和地球扰动的影响,通过对应于其长期进动椭圆轨道的线性项和捕捉周期性变化的众多三角项来描述。

Generally, higher precision in these analytical coordinates results in more complex expressions, rendering the analytical solutions of the Lagrange equations challenging. In this study, for a balance between solvability and precision, essential components of these coordinates are retained, and fitting is applied using JPL ephemerides data [46] from around 2035 to reduce discrepancies in the positions of the Sun and the Moon.
通常,这些解析坐标的精度越高,表达式就越复杂,这使得拉格朗日方程的解析解具有挑战性。在这项研究中,为了在可解性和精度之间取得平衡,保留了这些坐标的基本组成部分,并使用2035年左右的JPL星历数据[46]进行拟合,以减少太阳和月亮位置的差异。

The apparent motion of the Sun around the Earth is approximated as a circular orbit on the ecliptic plane with a one-sidereal-year period,
太阳绕地球的视运动近似为黄道平面上的圆形轨道,周期为单侧实年,

r2(t)r¯2,similar-to-or-equalssubscript𝑟2𝑡subscript¯𝑟2\displaystyle r_{2}(t)\simeq\overline{r}_{2},italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) ≃ over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , (61)
u2(t)u2(t)=n2t+u20,similar-to-or-equalssubscriptu2𝑡subscript𝑢2𝑡subscript𝑛2𝑡subscript𝑢subscript20\displaystyle\text{u}_{2}(t)\simeq u_{2}(t)=n_{2}\,t+u_{2_{0}},u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) ≃ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) = italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_t + italic_u start_POSTSUBSCRIPT 2 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , (62)

where r¯2subscript¯𝑟2\overline{r}_{2}over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is the mean Sun-Earth distance, and u2subscript𝑢2u_{2}italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT represents the mean longitude of the Sun. Moreover, n2=2π/(one sidereal year)subscript𝑛22𝜋one sidereal yearn_{2}=2\pi/(\text{one sidereal year})italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 2 italic_π / ( one sidereal year ) is the mean motion, and u20subscript𝑢subscript20u_{2_{0}}italic_u start_POSTSUBSCRIPT 2 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT denotes the initial phase of the Sun’s orbit. The specific parameter values can be found in Table 1. The Moon’s orbit is considered as an inclined and elliptical precessing orbit,
其中 r¯2subscript¯𝑟2\overline{r}_{2}over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 是太阳与地球的平均距离, u2subscript𝑢2u_{2}italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 代表太阳的平均经度。此外, n2=2π/(one sidereal year)subscript𝑛22𝜋one sidereal yearn_{2}=2\pi/(\text{one sidereal year})italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 2 italic_π / ( one sidereal year ) 是平均运动,表示 u20subscript𝑢subscript20u_{2_{0}}italic_u start_POSTSUBSCRIPT 2 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT 太阳轨道的初始阶段。具体参数值见表1。月球的轨道被认为是一个倾斜和椭圆形的前部轨道,

i3(t)i¯3,similar-to-or-equalssubscript𝑖3𝑡subscript¯𝑖3\displaystyle i_{3}(t)\simeq\overline{i}_{3},italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) ≃ over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , (63)
Ω3(t)Ω¯3(t),similar-to-or-equalssubscriptΩ3𝑡subscript¯Ω3𝑡\displaystyle\Omega_{3}(t)\simeq\overline{\Omega}_{3}(t),roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) ≃ over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) , (64)
r3(t)r¯3+r3AcosM¯3(t),similar-to-or-equalssubscript𝑟3𝑡subscript¯𝑟3superscriptsubscript𝑟3𝐴subscript¯𝑀3𝑡\displaystyle r_{3}(t)\simeq\overline{r}_{3}+r_{3}^{A}\cos\overline{M}_{3}(t),italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) ≃ over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT roman_cos over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) , (65)
u3(t)u3(t)+u3AsinM¯3(t),similar-to-or-equalssubscriptu3𝑡subscript𝑢3𝑡superscriptsubscript𝑢3𝐴subscript¯𝑀3𝑡\displaystyle\text{u}_{3}(t)\simeq u_{3}(t)+u_{3}^{A}\sin\overline{M}_{3}(t),u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) ≃ italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) + italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT roman_sin over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) , (66)

with 

Ω¯3(t)=nΩ3t+Ω30,subscript¯Ω3𝑡subscript𝑛subscriptΩ3𝑡subscriptΩsubscript30\displaystyle\overline{\Omega}_{3}(t)=n_{\Omega_{3}}t+\Omega_{3_{0}},over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) = italic_n start_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_t + roman_Ω start_POSTSUBSCRIPT 3 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , (67)
M¯3(t)=nM3t+M30,subscript¯𝑀3𝑡subscript𝑛subscript𝑀3𝑡subscript𝑀subscript30\displaystyle\overline{M}_{3}(t)=n_{\!M_{3}}t+M_{3_{0}},over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) = italic_n start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_t + italic_M start_POSTSUBSCRIPT 3 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , (68)
u3(t)=n3t+u30,subscript𝑢3𝑡subscript𝑛3𝑡subscript𝑢subscript30\displaystyle u_{3}(t)=n_{3}\,t+u_{3_{0}},italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) = italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_t + italic_u start_POSTSUBSCRIPT 3 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , (69)

where i¯3subscript¯𝑖3\overline{i}_{3}over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT is the mean inclination of the Moon’s orbit, Ω¯3subscript¯Ω3\overline{\Omega}_{3}over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT represents the secular variation in the Moon’s longitude of ascending node, r¯3subscript¯𝑟3\overline{r}_{3}over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT denotes the mean Earth-Moon distance, and u3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT signifies the secular variation in the argument of latitude. Equations (65) and (66) include trigonometric corrections, related to the Moon’s mean anomaly M¯3subscript¯𝑀3\overline{M}_{3}over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, aimed at more accurately describing the Moon’s motion in the radial and transverse directions. The periods of variation for Ω¯3subscript¯Ω3\overline{\Omega}_{3}over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, M¯3subscript¯𝑀3\overline{M}_{3}over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, and u3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT are approximately 18.6 years, 27.55 days (anomalistic month), and 27.21 days (draconic month), respectively.
其中 i¯3subscript¯𝑖3\overline{i}_{3}over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 是月球轨道的平均倾角, Ω¯3subscript¯Ω3\overline{\Omega}_{3}over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 表示月球升交点经度的长期变化, r¯3subscript¯𝑟3\overline{r}_{3}over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 表示平均地月距离,并 u3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 表示纬度论证中的长期变化。方程 (65) 和 (66) 包括与月球平均距平相关的三角校正 M¯3subscript¯𝑀3\overline{M}_{3}over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,旨在更准确地描述月球在径向和横向上的运动。 Ω¯3subscript¯Ω3\overline{\Omega}_{3}over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPTM¯3subscript¯𝑀3\overline{M}_{3}over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPTu3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 的变化周期分别约为 18.6 年、27.55 天(异常月)和 27.21 天(龙月)。

Table 1: Parameter settings for the motion of the Sun and Moon. The subscript “0” indicates values taken at the epoch 1 January, 2034, 00:00:00 UTC.
表 1: 太阳和月亮运动的参数设置。下标“0”表示在 2034 年 1 月 1 日 00:00:00 UTC 纪元获取的值。
Symbols 符号 Parameters 参数 Values 
r¯2subscript¯𝑟2\overline{r}_{2}over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT Mean Sun-Earth distance 平均日地距离 1.496 191×108km1.496191superscript108km1.496\,191\times 10^{8}\,\text{km}1.496 191 × 10 start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT km
n2subscript𝑛2n_{2}italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT Mean motion of the Earth
地球的平均运动
2π/365.2564days2𝜋365.2564days2\pi/365.2564\,\text{days}2 italic_π / 365.2564 days
u20subscript𝑢subscript20u_{2_{0}}italic_u start_POSTSUBSCRIPT 2 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT Sun initial phase Sun 初始阶段 280.251superscript280.251280.251^{\circ}280.251 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT
i¯3subscript¯𝑖3\overline{i}_{3}over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Mean lunar orbit inclination
平均月球轨道倾角
5.162superscript5.1625.162^{\circ}5.162 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT
nΩ3subscript𝑛subscriptΩ3n_{\Omega_{3}}italic_n start_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT Rate of change of Ω¯3subscript¯Ω3\overline{\Omega}_{3}over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
变化 Ω¯3subscript¯Ω3\overline{\Omega}_{3}over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
2π/18.6years2𝜋18.6years2\pi/18.6\,\text{years}2 italic_π / 18.6 years
Ω30subscriptΩsubscript30\Omega_{3_{0}}roman_Ω start_POSTSUBSCRIPT 3 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT Initial phase of Ω¯3subscript¯Ω3\overline{\Omega}_{3}over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
的起步阶段 Ω¯3subscript¯Ω3\overline{\Omega}_{3}over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
186.988superscript186.988186.988^{\circ}186.988 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT
r¯3subscript¯𝑟3\overline{r}_{3}over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Mean Earth-Moon distance 平均地月距离 384 151km384151km384\,151\,\text{km}384 151 km
r3Asuperscriptsubscript𝑟3𝐴r_{3}^{A}italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT Amplitude of correction term
校正项的振幅
20 905km20905km-20\,905\,\text{km}- 20 905 km
nM3subscript𝑛subscript𝑀3n_{\!M_{3}}italic_n start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT Rate of change of M¯3subscript¯𝑀3\overline{M}_{3}over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
变化 M¯3subscript¯𝑀3\overline{M}_{3}over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
2π/27.55days2𝜋27.55days2\pi/27.55\,\text{days}2 italic_π / 27.55 days
M30subscript𝑀subscript30M_{3_{0}}italic_M start_POSTSUBSCRIPT 3 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT Initial phase of M¯3subscript¯𝑀3\overline{M}_{3}over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
的初始阶段 M¯3subscript¯𝑀3\overline{M}_{3}over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
22.578superscript22.57822.578^{\circ}22.578 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT
n3subscript𝑛3n_{3}italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Rate of change of u3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
变化 u3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
2π/27.21days2𝜋27.21days2\pi/27.21\,\text{days}2 italic_π / 27.21 days
u30subscript𝑢subscript30u_{3_{0}}italic_u start_POSTSUBSCRIPT 3 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT Initial phase of u3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
的初始阶段 u3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
221.970superscript221.970221.970^{\circ}221.970 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT
u3Asuperscriptsubscript𝑢3𝐴u_{3}^{A}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT Amplitude of correction term
校正项的振幅
6.289superscript6.2896.289^{\circ}6.289 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT

III.3 Lunisolar perturbations on the TianQin satellites
三.3 天琴卫星上的阴阳扰动

Let σ=[aiΩξηλ]T𝜎superscriptmatrix𝑎𝑖Ω𝜉𝜂𝜆𝑇\sigma=\begin{bmatrix}a&i&\Omega&\xi&\eta&\lambda\end{bmatrix}^{T}italic_σ = [ start_ARG start_ROW start_CELL italic_a end_CELL start_CELL italic_i end_CELL start_CELL roman_Ω end_CELL start_CELL italic_ξ end_CELL start_CELL italic_η end_CELL start_CELL italic_λ end_CELL end_ROW end_ARG ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT; then, the Lagrange perturbation equations (45)-(50) can be reformulated as
σ=[aiΩξηλ]T𝜎superscriptmatrix𝑎𝑖Ω𝜉𝜂𝜆𝑇\sigma=\begin{bmatrix}a&i&\Omega&\xi&\eta&\lambda\end{bmatrix}^{T}italic_σ = [ start_ARG start_ROW start_CELL italic_a end_CELL start_CELL italic_i end_CELL start_CELL roman_Ω end_CELL start_CELL italic_ξ end_CELL start_CELL italic_η end_CELL start_CELL italic_λ end_CELL end_ROW end_ARG ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ;那么,拉格朗日扰动方程 (45)-(50) 可以重新表述为

dσdt=f0(a)+f1(σ,t,ε),𝑑𝜎𝑑𝑡subscript𝑓0𝑎subscript𝑓1𝜎𝑡𝜀\displaystyle\frac{d\sigma}{dt}=f_{0}(a)+f_{1}(\sigma,t,\varepsilon),divide start_ARG italic_d italic_σ end_ARG start_ARG italic_d italic_t end_ARG = italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a ) + italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_σ , italic_t , italic_ε ) , (70)

where the functions f0subscript𝑓0f_{0}italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and f1subscript𝑓1f_{1}italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT are both 6-dimensional vector functions,
其中函数 f0subscript𝑓0f_{0}italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPTf1subscript𝑓1f_{1}italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT 都是 6 维向量函数

f0(a)=διn,δι=[000001]T,formulae-sequencesubscript𝑓0𝑎subscript𝛿𝜄𝑛subscript𝛿𝜄superscriptmatrix000001𝑇\displaystyle f_{0}(a)=\delta_{\iota}n,\qquad\delta_{\iota}=\begin{bmatrix}0&0% &0&0&0&1\end{bmatrix}^{T},italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_a ) = italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT italic_n , italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT = [ start_ARG start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL end_ROW end_ARG ] start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT , (71)
(f1)ζ=𝒪(ε),ζ=1,2,,6,formulae-sequencesubscriptsubscript𝑓1𝜁𝒪𝜀𝜁126\displaystyle(f_{1})_{\zeta}=\mathcal{O}(\varepsilon),\qquad\zeta=1,2,\cdots,6,( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT = caligraphic_O ( italic_ε ) , italic_ζ = 1 , 2 , ⋯ , 6 , (72)

and ε1much-less-than𝜀1\varepsilon\ll 1italic_ε ≪ 1 is a small parameter. Since the perturbing forces are significantly weaker than the Earth’s central gravitational attraction, the solution of Eq. (70) is assumed to be
and ε1much-less-than𝜀1\varepsilon\ll 1italic_ε ≪ 1 是一个小参数。由于扰动力明显弱于地球的中心引力,因此假设方程(70)的解为

σ(t)=σ(0)(t)+σ(1)(t).𝜎𝑡superscript𝜎0𝑡superscript𝜎1𝑡\displaystyle\sigma(t)=\sigma^{(0)}(t)+\sigma^{(1)}(t).italic_σ ( italic_t ) = italic_σ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_t ) + italic_σ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) . (73)

Here, σ(0)(t)superscript𝜎0𝑡\sigma^{(0)}(t)italic_σ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_t ) represents the unperturbed Keplerian orbit (as shown in Eq. (26)),
这里, σ(0)(t)superscript𝜎0𝑡\sigma^{(0)}(t)italic_σ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_t ) 表示未受扰动的开普勒轨道(如方程(26)所示),

σ(0)(t)=σ0+διn(tt0),superscript𝜎0𝑡subscript𝜎0subscript𝛿𝜄𝑛𝑡subscript𝑡0\displaystyle\sigma^{(0)}(t)=\sigma_{0}+\delta_{\iota}n(t-t_{0}),italic_σ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_t ) = italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT italic_n ( italic_t - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , (74)

and σ(1)(t)superscript𝜎1𝑡\sigma^{(1)}(t)italic_σ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) is the perturbation solution,
并且是 σ(1)(t)superscript𝜎1𝑡\sigma^{(1)}(t)italic_σ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) 扰动解,

σ(1)(t)=(σc(t)+σl[c](t)+σl[l](t)+σs(t))|t0t.superscript𝜎1𝑡evaluated-atsubscript𝜎𝑐𝑡subscript𝜎𝑙delimited-[]𝑐𝑡subscript𝜎𝑙delimited-[]𝑙𝑡subscript𝜎𝑠𝑡subscript𝑡0𝑡\displaystyle\sigma^{(1)}(t)=\bigl{(}\sigma_{c}(t)+\sigma_{l[c]}(t)+\sigma_{l[% l]}(t)+\sigma_{s}(t)\bigr{)}\!\large{|}_{t_{0}}^{t}.italic_σ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) = ( italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t ) + italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) + italic_σ start_POSTSUBSCRIPT italic_l [ italic_l ] end_POSTSUBSCRIPT ( italic_t ) + italic_σ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_t ) ) | start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT . (75)

In Eq. (75), σ(1)(t)superscript𝜎1𝑡\sigma^{(1)}(t)italic_σ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) is decomposed into four parts, distinguished by the unique time scales of orbital variations induced by perturbations: the secular term σcsubscript𝜎𝑐\sigma_{c}italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, special long-period term σl[c]subscript𝜎𝑙delimited-[]𝑐\sigma_{l[c]}italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT, general long-period term σl[l]subscript𝜎𝑙delimited-[]𝑙\sigma_{l[l]}italic_σ start_POSTSUBSCRIPT italic_l [ italic_l ] end_POSTSUBSCRIPT, and short-period term σssubscript𝜎𝑠\sigma_{s}italic_σ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT. σcsubscript𝜎𝑐\sigma_{c}italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT signifies the linear change over time, while σl[c]subscript𝜎𝑙delimited-[]𝑐\sigma_{l[c]}italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT, σl[l]subscript𝜎𝑙delimited-[]𝑙\sigma_{l[l]}italic_σ start_POSTSUBSCRIPT italic_l [ italic_l ] end_POSTSUBSCRIPT, and σssubscript𝜎𝑠\sigma_{s}italic_σ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT are associated with periodic variations. These variations are linked to, for instance, Ω3subscriptΩ3\Omega_{3}roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT with an 18.6-year period, u3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT with a 27.21-day period, and λ𝜆\lambdaitalic_λ with a 3.64-day period. The explicit expression for σ(1)(t)superscript𝜎1𝑡\sigma^{(1)}(t)italic_σ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) can be derived using perturbation methods to solve Eq. (70).
在方程(75)中, σ(1)(t)superscript𝜎1𝑡\sigma^{(1)}(t)italic_σ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) 被分解为四个部分,其特点是由扰动引起的轨道变化的独特时间尺度:长期项 σcsubscript𝜎𝑐\sigma_{c}italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT 、特殊长周期项 σl[c]subscript𝜎𝑙delimited-[]𝑐\sigma_{l[c]}italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT 、一般长期项 σl[l]subscript𝜎𝑙delimited-[]𝑙\sigma_{l[l]}italic_σ start_POSTSUBSCRIPT italic_l [ italic_l ] end_POSTSUBSCRIPT 和短期项 σssubscript𝜎𝑠\sigma_{s}italic_σ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPTσcsubscript𝜎𝑐\sigma_{c}italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT 表示随时间的线性变化,而 σl[c]subscript𝜎𝑙delimited-[]𝑐\sigma_{l[c]}italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPTσl[l]subscript𝜎𝑙delimited-[]𝑙\sigma_{l[l]}italic_σ start_POSTSUBSCRIPT italic_l [ italic_l ] end_POSTSUBSCRIPT 和 与 σssubscript𝜎𝑠\sigma_{s}italic_σ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT 周期性变化相关联。例如, Ω3subscriptΩ3\Omega_{3}roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 这些变化与18.6年、27.21天 λ𝜆\lambdaitalic_λ 和3.64天的周期 u3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 有关。可以使用扰动方法求解方程(70)来推导的 σ(1)(t)superscript𝜎1𝑡\sigma^{(1)}(t)italic_σ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) 显式表达式。

Table 2: The components of the analytical solution σ(t)𝜎𝑡\sigma(t)italic_σ ( italic_t ) for the TianQin orbit describing the perturbing effects from the Sun, Moon, and Earth’s J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. The symbol “--” denotes that terms do not appear separately due to the joint effects of solar and lunar perturbations. “similar-to\sim” indicates neglected contributions, considering that the J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT perturbation induces a negligible eccentricity variation of 106superscript10610^{-6}10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT. Terms like σJ2ssuperscriptsubscript𝜎subscript𝐽2𝑠\sigma_{J_{2}}^{s}italic_σ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT are also ignored, and σJ2l[l]=σJ2l[c]=0superscriptsubscript𝜎subscript𝐽2𝑙delimited-[]𝑙superscriptsubscript𝜎subscript𝐽2𝑙delimited-[]𝑐0\sigma_{J_{2}}^{l[l]}=\sigma_{J_{2}}^{l[c]}=0italic_σ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT = italic_σ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT = 0. Explicit expressions for each component are detailed in Appendix C.
表 2: 天琴轨道的解析解 σ(t)𝜎𝑡\sigma(t)italic_σ ( italic_t ) 的分量,描述了太阳、月亮和地球的 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 扰动效应。符号 “ -- ” 表示由于太阳和月球扰动的共同影响,术语不会单独出现。“ similar-to\sim ” 表示被忽略的贡献,考虑到 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 扰动引起的 的偏心率变化可以忽略不计。 106superscript10610^{-6}10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT 术语 like σJ2ssuperscriptsubscript𝜎subscript𝐽2𝑠\sigma_{J_{2}}^{s}italic_σ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT 也被忽略,和 σJ2l[l]=σJ2l[c]=0superscriptsubscript𝜎subscript𝐽2𝑙delimited-[]𝑙superscriptsubscript𝜎subscript𝐽2𝑙delimited-[]𝑐0\sigma_{J_{2}}^{l[l]}=\sigma_{J_{2}}^{l[c]}=0italic_σ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT = italic_σ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT = 0 .附录 C 中详细介绍了每个组件的显式表达式。
σ(t)𝜎𝑡\sigma(t)italic_σ ( italic_t ) σ0+διn¯tsubscript𝜎0subscript𝛿𝜄¯𝑛𝑡\sigma_{0}+\delta_{\iota}\overline{n}\,titalic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT over¯ start_ARG italic_n end_ARG italic_t σc(t)subscript𝜎𝑐𝑡\sigma_{c}(t)italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t ) σl[c](t)subscript𝜎𝑙delimited-[]𝑐𝑡\sigma_{l[c]}(t)italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) σl[l](t)subscript𝜎𝑙delimited-[]𝑙𝑡\sigma_{l[l]}(t)italic_σ start_POSTSUBSCRIPT italic_l [ italic_l ] end_POSTSUBSCRIPT ( italic_t ) σs(t)subscript𝜎𝑠𝑡\sigma_{s}(t)italic_σ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_t )
s m J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT s m s m s m
a(t)𝑎𝑡a(t)italic_a ( italic_t ) a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT 0 0 0 0 0 0 0 asssuperscriptsubscript𝑎s𝑠a_{\text{s}}^{s}italic_a start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT amssuperscriptsubscript𝑎m𝑠a_{\text{m}}^{s}italic_a start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT
i(t)𝑖𝑡i(t)italic_i ( italic_t ) i0subscript𝑖0i_{0}italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT 0 0 iJ2csuperscriptsubscript𝑖subscript𝐽2𝑐i_{J_{2}}^{c}italic_i start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT 0 iml[c]superscriptsubscript𝑖m𝑙delimited-[]𝑐i_{\text{m}}^{l[c]}italic_i start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT isl[l]superscriptsubscript𝑖s𝑙delimited-[]𝑙i_{\text{s}}^{l[l]}italic_i start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT iml[l]superscriptsubscript𝑖m𝑙delimited-[]𝑙i_{\text{m}}^{l[l]}italic_i start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT isssuperscriptsubscript𝑖s𝑠i_{\text{s}}^{s}italic_i start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT imssuperscriptsubscript𝑖m𝑠i_{\text{m}}^{s}italic_i start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT
ξ(t)𝜉𝑡\xi(t)italic_ξ ( italic_t ) ξ0subscript𝜉0\xi_{0}italic_ξ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT -- ξcsubscript𝜉𝑐\xi_{c}italic_ξ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT similar-to\sim -- ξl[c]subscript𝜉𝑙delimited-[]𝑐\xi_{l[c]}italic_ξ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT 0 ξml[l]superscriptsubscript𝜉m𝑙delimited-[]𝑙\xi_{\text{m}}^{l[l]}italic_ξ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ξsssuperscriptsubscript𝜉s𝑠\xi_{\text{s}}^{s}italic_ξ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ξmssuperscriptsubscript𝜉m𝑠\xi_{\text{m}}^{s}italic_ξ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT
η(t)𝜂𝑡\eta(t)italic_η ( italic_t ) η0subscript𝜂0\eta_{0}italic_η start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT -- ηcsubscript𝜂𝑐\eta_{c}italic_η start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT similar-to\sim -- ηl[c]subscript𝜂𝑙delimited-[]𝑐\eta_{l[c]}italic_η start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT 0 ηml[l]superscriptsubscript𝜂m𝑙delimited-[]𝑙\eta_{\text{m}}^{l[l]}italic_η start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ηsssuperscriptsubscript𝜂s𝑠\eta_{\text{s}}^{s}italic_η start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ηmssuperscriptsubscript𝜂m𝑠\eta_{\text{m}}^{s}italic_η start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT
Ω(t)Ω𝑡\Omega(t)roman_Ω ( italic_t ) Ω0subscriptΩ0\Omega_{0}roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ΩscsuperscriptsubscriptΩs𝑐\Omega_{\text{s}}^{c}roman_Ω start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ΩmcsuperscriptsubscriptΩm𝑐\Omega_{\text{m}}^{c}roman_Ω start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ΩJ2csuperscriptsubscriptΩsubscript𝐽2𝑐\Omega_{J_{2}}^{c}roman_Ω start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT 0 Ωml[c]superscriptsubscriptΩm𝑙delimited-[]𝑐\Omega_{\text{m}}^{l[c]}roman_Ω start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT Ωsl[l]superscriptsubscriptΩs𝑙delimited-[]𝑙\Omega_{\text{s}}^{l[l]}roman_Ω start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT Ωml[l]superscriptsubscriptΩm𝑙delimited-[]𝑙\Omega_{\text{m}}^{l[l]}roman_Ω start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ΩsssuperscriptsubscriptΩs𝑠\Omega_{\text{s}}^{s}roman_Ω start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ΩmssuperscriptsubscriptΩm𝑠\Omega_{\text{m}}^{s}roman_Ω start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT
λ(t)𝜆𝑡\lambda(t)italic_λ ( italic_t ) λ0+n¯tsubscript𝜆0¯𝑛𝑡\lambda_{0}+\overline{n}\,titalic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + over¯ start_ARG italic_n end_ARG italic_t λscsuperscriptsubscript𝜆s𝑐\lambda_{\text{s}}^{c}italic_λ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT λmcsuperscriptsubscript𝜆m𝑐\lambda_{\text{m}}^{c}italic_λ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT λJ2csuperscriptsubscript𝜆subscript𝐽2𝑐\lambda_{J_{2}}^{c}italic_λ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT 0 λml[c]superscriptsubscript𝜆m𝑙delimited-[]𝑐\lambda_{\text{m}}^{l[c]}italic_λ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT λsl[l]superscriptsubscript𝜆s𝑙delimited-[]𝑙\lambda_{\text{s}}^{l[l]}italic_λ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT λml[l]superscriptsubscript𝜆m𝑙delimited-[]𝑙\lambda_{\text{m}}^{l[l]}italic_λ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT λsssuperscriptsubscript𝜆s𝑠\lambda_{\text{s}}^{s}italic_λ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT λmssuperscriptsubscript𝜆m𝑠\lambda_{\text{m}}^{s}italic_λ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT

To enhance the precision of the analytical solution, we employ a perturbation method known as the mean element method [34, 35], which uses the mean orbital elements σ¯(t)¯𝜎𝑡\overline{\sigma}(t)over¯ start_ARG italic_σ end_ARG ( italic_t ) as a reference solution (defined in Eq. (122)), rather than the Keplerian orbit σ(0)(t)superscript𝜎0𝑡\sigma^{(0)}(t)italic_σ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_t ). Additionally, for simplicity, only terms up to the first order of eccentricity in the solution are considered. For more derivation details, one can refer to Appendix B. The components of the perturbation solution are outlined in Table 2, with the average value n¯=μa¯3¯𝑛𝜇superscript¯𝑎3\overline{n}=\sqrt{\frac{\mu}{\overline{a}^{3}}}over¯ start_ARG italic_n end_ARG = square-root start_ARG divide start_ARG italic_μ end_ARG start_ARG over¯ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG end_ARG replacing n𝑛nitalic_n in σ(0)(t)superscript𝜎0𝑡\sigma^{(0)}(t)italic_σ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_t ). Detailed expressions for the terms in Table 2 can be found in Appendix C. Moreover, the effectiveness of the analytical solution σ(t)𝜎𝑡\sigma(t)italic_σ ( italic_t ) is evaluated through a comparison with high-fidelity numerical orbit simulations (see Appendix B.3). For the TianQin orbit, the 5-year average deviation in position is approximately 87 km.
为了提高解析解的精度,我们采用了一种称为均元法 [3435] 的微扰方法,该方法使用均值轨道元 σ¯(t)¯𝜎𝑡\overline{\sigma}(t)over¯ start_ARG italic_σ end_ARG ( italic_t ) 作为参考解(在方程(122)中定义),而不是开普勒轨道 σ(0)(t)superscript𝜎0𝑡\sigma^{(0)}(t)italic_σ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_t ) 。此外,为简单起见,仅考虑解中偏心率的一阶项。有关更多推导的详细信息,可以参考附录 B。表 2 概述了扰动解的组成部分,平均值 n¯=μa¯3¯𝑛𝜇superscript¯𝑎3\overline{n}=\sqrt{\frac{\mu}{\overline{a}^{3}}}over¯ start_ARG italic_n end_ARG = square-root start_ARG divide start_ARG italic_μ end_ARG start_ARG over¯ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG end_ARG 替换为 n𝑛nitalic_n σ(0)(t)superscript𝜎0𝑡\sigma^{(0)}(t)italic_σ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_t ) 。表 2 中术语的详细表达式可在附录 C 中找到。此外,通过与高保真数值轨道模拟的比较来评估解析解 σ(t)𝜎𝑡\sigma(t)italic_σ ( italic_t ) 的有效性(见附录 B.3)。对于天琴轨道,5年的平均位置偏差约为87公里。

Table 2 illustrates the effects of gravitational perturbations on the orbital elements of the TianQin satellite. Variation in a𝑎aitalic_a are solely induced by short-period perturbation. However, the other five orbital elements are also influenced by both secular and long-period perturbations, particularly the secular one, leading to cumulative change. In the case of i𝑖iitalic_i, its secular variation are not due to lunisolar perturbations (which would occur when considering second-order eccentricity [47]), but instead result from J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT perturbation, tied to a coordinate transformation involving the obliquity ϵitalic-ϵ\epsilonitalic_ϵ (see Eq. (182)). As for ΩΩ\Omegaroman_Ω, λ𝜆\lambdaitalic_λ, ξ𝜉\xiitalic_ξ, and η𝜂\etaitalic_η, their secular variations are predominantly driven by lunar and solar perturbations.
2 说明了引力扰动对天琴卫星轨道元件的影响。的变化 a𝑎aitalic_a 完全是由短周期扰动引起的。然而,其他五种轨道元素也受到长期和长期扰动的影响,尤其是长期扰动,导致累积变化。在 i𝑖iitalic_i 的情况下,它的长期变化不是由于阴阳扰动(在考虑二阶偏心率时会发生 [47]),而是由 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 扰动引起的,与涉及倾斜度的坐标变换有关 ϵitalic-ϵ\epsilonitalic_ϵ (见方程(182))。至于 ΩΩ\Omegaroman_Ωλ𝜆\lambdaitalic_λξ𝜉\xiitalic_ξη𝜂\etaitalic_η ,它们的长期变化主要是由月球和太阳的扰动驱动的。

The two elements, ΩΩ\Omegaroman_Ω and i𝑖iitalic_i, determine the orientation of the orbital plane. As indicated in Eqs. (137) and (156), the secular variation of ΩΩ\Omegaroman_Ω is dependent on the satellite’s mean semimajor axis a¯¯𝑎\overline{a}over¯ start_ARG italic_a end_ARG and mean inclination i¯¯𝑖\overline{i}over¯ start_ARG italic_i end_ARG, which implies that ΩΩ\Omegaroman_Ω experiences negligible precession when i90similar-to𝑖superscript90i\sim 90^{\circ}italic_i ∼ 90 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT. Similarly, the secular variation of i𝑖iitalic_i for the TianQin satellite is also minimal (<0.1absentsuperscript0.1<0.1^{\circ}< 0.1 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT in five years). As a result, the orientation of the TianQin detector plane remains nearly constant, changing by less than 2.6superscript2.62.6^{\circ}2.6 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT over five years. This is in stark contrast to LISA orbits [4], where the plane undergoes a full 360-degree rotation annually.
这两个单元 ΩΩ\Omegaroman_Ωi𝑖iitalic_i 确定轨道平面的方向。如 Eqs 所示。(137) 和 (156) 中,的长期 ΩΩ\Omegaroman_Ω 变化取决于卫星的平均长半轴 a¯¯𝑎\overline{a}over¯ start_ARG italic_a end_ARG 和平均倾角 i¯¯𝑖\overline{i}over¯ start_ARG italic_i end_ARG ,这意味着 ΩΩ\Omegaroman_Ω 当 时经历的岁差可以忽略不计。 i90similar-to𝑖superscript90i\sim 90^{\circ}italic_i ∼ 90 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT 同样,天琴卫星的长期变化 i𝑖iitalic_i 也很小( <0.1absentsuperscript0.1<0.1^{\circ}< 0.1 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT 五年内)。因此,天琴探测器平面的方向几乎保持不变,变化时间不到 2.6superscript2.62.6^{\circ}2.6 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT 五年。这与 LISA 轨道 [4] 形成鲜明对比,LISA 轨道上的飞机每年都会进行完整的 360 度旋转。

For the periodic variations, their periods are linked to the motions of the satellite, the Moon, and the Sun. Especially, for the short-period variation σs(t)subscript𝜎𝑠𝑡\sigma_{s}(t)italic_σ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_t ), the arguments of trigonometric terms take the form of κλ+pu3+qθ3𝜅𝜆𝑝subscript𝑢3𝑞subscript𝜃3\kappa\,\lambda+p\,u_{3}+q\,\theta_{3}italic_κ italic_λ + italic_p italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT or κλ+pU2𝜅𝜆𝑝subscript𝑈2\kappa\,\lambda+p\,U_{2}italic_κ italic_λ + italic_p italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (see, e.g., Eqs. (174) and (146)), which indicates that orbital variations occur at multiples of the satellite’s orbital frequency and are modulated by the motions of the Moon and the Sun. This insight aids in understanding the perturbing effects of the Moon’s and the Sun’s gravitational fields on the TianQin inter-satellite range acceleration noise (cf. Fig. 3 in Ref. [38]).
对于周期性变化,它们的周期与卫星、月亮和太阳的运动有关。特别是,对于短周期变化 σs(t)subscript𝜎𝑠𝑡\sigma_{s}(t)italic_σ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_t ) ,三角项的参数采用 κλ+pu3+qθ3𝜅𝜆𝑝subscript𝑢3𝑞subscript𝜃3\kappa\,\lambda+p\,u_{3}+q\,\theta_{3}italic_κ italic_λ + italic_p italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT or κλ+pU2𝜅𝜆𝑝subscript𝑈2\kappa\,\lambda+p\,U_{2}italic_κ italic_λ + italic_p italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 的形式(参见,例如,方程。(174) 和 (146)),这表明轨道变化发生在卫星轨道频率的倍数处,并受到月球和太阳运动的调制。这种洞察力有助于理解月球和太阳引力场对天琴星间距离加速度噪声的扰动影响(参见参考文献 [38] 中的图 3)。

Moreover, σs(t)subscript𝜎𝑠𝑡\sigma_{s}(t)italic_σ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_t ) is correlated with the orbit phase λk(t)subscript𝜆𝑘𝑡\lambda_{k}(t)italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) of SCk𝑘kitalic_k, with a 120-degree phase difference among the three satellites, indicating that short-period perturbation will influence the relative motion between satellites. However, it can be demonstrated that the other three components of Eq. (75) may have little impact on the relative motion. Furthermore, an ideal equilateral-triangle constellation requires zero eccentricity, which is unlikely to hold, as shown by the perturbation solutions ξk(1)(t)superscriptsubscript𝜉𝑘1𝑡\xi_{k}^{(1)}(t)italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) and ηk(1)(t)superscriptsubscript𝜂𝑘1𝑡\eta_{k}^{(1)}(t)italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ). Consequently, the presence of these components underscores the potential to significantly disturb the nominal TianQin triangle constellation.
此外, σs(t)subscript𝜎𝑠𝑡\sigma_{s}(t)italic_σ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_t ) 与 SC k𝑘kitalic_k 的轨道相位 λk(t)subscript𝜆𝑘𝑡\lambda_{k}(t)italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) 相关,三颗卫星之间的相位差为 120 度,表明短周期扰动会影响卫星之间的相对运动。然而,可以证明方程(75)的其他三个分量可能对相对运动影响很小。此外,理想的等边三角形星座需要零偏心率,这不太可能成立,如扰动解 ξk(1)(t)superscriptsubscript𝜉𝑘1𝑡\xi_{k}^{(1)}(t)italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t )ηk(1)(t)superscriptsubscript𝜂𝑘1𝑡\eta_{k}^{(1)}(t)italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) 所示。因此,这些组件的存在强调了显著扰乱名义上的天琴三角形星座的可能性。

III.4 Perturbed motion of the TianQin constellation
III.4 天琴星座的扰动运动

Equations (73)-(75) describe the variations in orbital elements for SCk𝑘kitalic_k (k𝑘kitalic_k = 1, 2, 3) under the influence of lunisolar perturbations and Earth’s J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT perturbation in the geocentric ecliptic coordinate system,
方程(73)-(75)描述了在月球太阳扰动和地球 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 扰动在地心黄道坐标系的影响下SC( k𝑘kitalic_k = 1, 2, 3)的轨道元素 k𝑘kitalic_k 的变化,

σk(t)=subscript𝜎𝑘𝑡absent\displaystyle\sigma_{k}(t)=italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) = σ0k+διn¯k(tt0)subscript𝜎0𝑘subscript𝛿𝜄subscript¯𝑛𝑘𝑡subscript𝑡0\displaystyle~{}\sigma_{0k}+\delta_{\iota}\,\overline{n}_{k}(t-t_{0})italic_σ start_POSTSUBSCRIPT 0 italic_k end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )
+(σkc(t)+σkl[c](t)+σkl[l](t)+σks(t))|t0t,evaluated-atsuperscriptsubscript𝜎𝑘𝑐𝑡superscriptsubscript𝜎𝑘𝑙delimited-[]𝑐𝑡superscriptsubscript𝜎𝑘𝑙delimited-[]𝑙𝑡superscriptsubscript𝜎𝑘𝑠𝑡subscript𝑡0𝑡\displaystyle+\bigl{(}\sigma_{k}^{c}(t)+\sigma_{k}^{l[c]}(t)+\sigma_{k}^{l[l]}% (t)+\sigma_{k}^{s}(t)\bigr{)}\!\large{|}_{t_{0}}^{t},+ ( italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ) + italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) + italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ( italic_t ) + italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t ) ) | start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT , (76)

with their explicit expressions detailed in Table 2 and Appendix C. Substituting Eq. (76) into Eq. (24) and using Eq. (51), the position vector 𝐫ksubscript𝐫𝑘\mathbf{r}_{k}bold_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and velocity vector 𝐫˙ksubscript˙𝐫𝑘\mathbf{\dot{r}}_{k}over˙ start_ARG bold_r end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT can be obtained. Then, employing the definitions in Eqs. (28)-(30), one can derive analytical expressions for the constellation’s three kinematic indicators, Lij(t)subscript𝐿𝑖𝑗𝑡L_{ij}(t)italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ), vij(t)subscript𝑣𝑖𝑗𝑡v_{ij}(t)italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ), and αk(t)subscript𝛼𝑘𝑡\alpha_{k}(t)italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ). The time evolution of these three quantities is illustrated in Fig. 3 for a set of simulated TianQin orbits, comparing both analytical and numerical models (see Appendix B.4 for more details).
2和附录C中详细说明了其明确的表达方式。将式(76)代入式(24)中,使用式(51), 𝐫˙ksubscript˙𝐫𝑘\mathbf{\dot{r}}_{k}over˙ start_ARG bold_r end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 可以得到位置矢量 𝐫ksubscript𝐫𝑘\mathbf{r}_{k}bold_r start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 和速度矢量。然后,采用方程中的定义。(28)-(30),可以推导出星座的三个运动学指标 Lij(t)subscript𝐿𝑖𝑗𝑡L_{ij}(t)italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t )vij(t)subscript𝑣𝑖𝑗𝑡v_{ij}(t)italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t )αk(t)subscript𝛼𝑘𝑡\alpha_{k}(t)italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) 的解析表达式。这三个量的时间演化如图所示。图3为一组模拟的天琴轨道,比较了分析和数值模型(详见附录B.4)。

Variations in the triangular constellation can be decomposed into two parts,
三角形星座的变化可以分解为两部分,

Lij(t)=subscript𝐿𝑖𝑗𝑡absent\displaystyle L_{ij}(t)=italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) = Lo+δLij(t),subscript𝐿o𝛿subscript𝐿𝑖𝑗𝑡\displaystyle~{}L_{\text{o}}+\delta L_{ij}(t),italic_L start_POSTSUBSCRIPT o end_POSTSUBSCRIPT + italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) , (77)
vij(t)=subscript𝑣𝑖𝑗𝑡absent\displaystyle v_{ij}(t)=italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) = vo+δvij(t),subscript𝑣o𝛿subscript𝑣𝑖𝑗𝑡\displaystyle~{}v_{\text{o}}+\delta v_{ij}(t),italic_v start_POSTSUBSCRIPT o end_POSTSUBSCRIPT + italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) , (78)
αk(t)=subscript𝛼𝑘𝑡absent\displaystyle\alpha_{k}(t)=italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) = αo+δαk(t),subscript𝛼o𝛿subscript𝛼𝑘𝑡\displaystyle~{}\alpha_{\text{o}}+\delta\alpha_{k}(t),italic_α start_POSTSUBSCRIPT o end_POSTSUBSCRIPT + italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) , (79)

where Lo=3×105subscript𝐿o3superscript105L_{\text{o}}=\sqrt{3}\times 10^{5}italic_L start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = square-root start_ARG 3 end_ARG × 10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT km, vo=0subscript𝑣o0v_{\text{o}}=0italic_v start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = 0 m/s, and αo=π3subscript𝛼o𝜋3\alpha_{\text{o}}=\frac{\pi}{3}italic_α start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = divide start_ARG italic_π end_ARG start_ARG 3 end_ARG represent the desired equilateral-triangle configuration. Conversely, δLij(t)𝛿subscript𝐿𝑖𝑗𝑡\delta L_{ij}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ), δvij(t)𝛿subscript𝑣𝑖𝑗𝑡\delta v_{ij}(t)italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ), and δαk(t)𝛿subscript𝛼𝑘𝑡\delta\alpha_{k}(t)italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) signify distortions from the ideal configuration. As previously mentioned, the magnitude of these distortions significantly impacts GW detection missions, including TDI data processing and the design of instruments such as phase meters and telescopes. It is crucial to minimize these distortions as much as possible.
其中 Lo=3×105subscript𝐿o3superscript105L_{\text{o}}=\sqrt{3}\times 10^{5}italic_L start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = square-root start_ARG 3 end_ARG × 10 start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT km、 vo=0subscript𝑣o0v_{\text{o}}=0italic_v start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = 0 m/s 和 αo=π3subscript𝛼o𝜋3\alpha_{\text{o}}=\frac{\pi}{3}italic_α start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = divide start_ARG italic_π end_ARG start_ARG 3 end_ARG 表示所需的等边三角形配置。反之, δLij(t)𝛿subscript𝐿𝑖𝑗𝑡\delta L_{ij}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) 、、 δvij(t)𝛿subscript𝑣𝑖𝑗𝑡\delta v_{ij}(t)italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t )δαk(t)𝛿subscript𝛼𝑘𝑡\delta\alpha_{k}(t)italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) 表示与理想配置的扭曲。如前所述,这些扭曲的程度极大地影响了GW探测任务,包括TDI数据处理和相位计和望远镜等仪器的设计。尽可能减少这些失真至关重要。

These distortions are all zero when the three satellites are solely influenced by the Earth’s point mass and satisfy the conditions (31) and (36). However, these conditions no longer hold when accounting for gravitational perturbations, which result in variations in ek(t)subscript𝑒𝑘𝑡e_{k}(t)italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) and the inclusion of non-synchronous short-period terms σks(t)superscriptsubscript𝜎𝑘𝑠𝑡\sigma_{k}^{s}(t)italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t ) in σk(t)subscript𝜎𝑘𝑡\sigma_{k}(t)italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ). To achieve an equilateral triangle, σk(t)subscript𝜎𝑘𝑡\sigma_{k}(t)italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) in Eqs. (31) and (36), instead, can be substituted with σkref(t)superscriptsubscript𝜎𝑘ref𝑡\sigma_{k}^{\text{ref}}(t)italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ):
当三颗卫星仅受地球点质量的影响并满足条件 (31) 和 (36) 时,这些失真都为零。然而,在考虑引力扰动时,这些条件不再成立,引力扰动导致 中的变化 ek(t)subscript𝑒𝑘𝑡e_{k}(t)italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) 和不同步的短周期项 σks(t)superscriptsubscript𝜎𝑘𝑠𝑡\sigma_{k}^{s}(t)italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t ) 的包含 σk(t)subscript𝜎𝑘𝑡\sigma_{k}(t)italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) 。为了实现等边三角形, σk(t)subscript𝜎𝑘𝑡\sigma_{k}(t)italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) 在 Eqs.(31) 和 (36) 可以替换为 σkref(t)superscriptsubscript𝜎𝑘ref𝑡\sigma_{k}^{\text{ref}}(t)italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t )

{e1ref(t)=e2ref(t)=e3ref(t)eo=0,a1ref(t)=a2ref(t)=a3ref(t)ao=Lo3,i1ref(t)=i2ref(t)=i3ref(t)io(t),Ω1ref(t)=Ω2ref(t)=Ω3ref(t)Ωo(t),λkref(t)2π3(k1)+λo(t),casessuperscriptsubscript𝑒1ref𝑡superscriptsubscript𝑒2ref𝑡superscriptsubscript𝑒3ref𝑡subscript𝑒o0superscriptsubscript𝑎1ref𝑡superscriptsubscript𝑎2ref𝑡superscriptsubscript𝑎3ref𝑡subscript𝑎osubscript𝐿o3superscriptsubscript𝑖1ref𝑡superscriptsubscript𝑖2ref𝑡superscriptsubscript𝑖3ref𝑡subscript𝑖o𝑡superscriptsubscriptΩ1ref𝑡superscriptsubscriptΩ2ref𝑡superscriptsubscriptΩ3ref𝑡subscriptΩo𝑡superscriptsubscript𝜆𝑘ref𝑡2𝜋3𝑘1subscript𝜆o𝑡\displaystyle\left\{\begin{array}[]{l}e_{1}^{\text{ref}}(t)=e_{2}^{\text{ref}}% (t)=e_{3}^{\text{ref}}(t)\equiv e_{\text{o}}=0,\vskip 3.0pt plus 1.0pt minus 1% .0pt\\ a_{1}^{\text{ref}}(t)=a_{2}^{\text{ref}}(t)=a_{3}^{\text{ref}}(t)\equiv a_{% \text{o}}=\frac{L_{\text{o}}}{\sqrt{3}},\vskip 3.0pt plus 1.0pt minus 1.0pt\\ i_{1}^{\text{ref}}(t)=i_{2}^{\text{ref}}(t)=i_{3}^{\text{ref}}(t)\equiv i_{% \text{o}}(t),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \Omega_{1}^{\text{ref}}(t)=\Omega_{2}^{\text{ref}}(t)=\Omega_{3}^{\text{ref}}(% t)\equiv\Omega_{\text{o}}(t),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \lambda_{k}^{\text{ref}}(t)\equiv\frac{2\pi}{3}(k-1)+\lambda_{\text{o}}(t),% \end{array}\right.{ start_ARRAY start_ROW start_CELL italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ) = italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ) = italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ) ≡ italic_e start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = 0 , end_CELL end_ROW start_ROW start_CELL italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ) = italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ) = italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ) ≡ italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = divide start_ARG italic_L start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG start_ARG square-root start_ARG 3 end_ARG end_ARG , end_CELL end_ROW start_ROW start_CELL italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ) = italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ) = italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ) ≡ italic_i start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) , end_CELL end_ROW start_ROW start_CELL roman_Ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ) = roman_Ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ) = roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ) ≡ roman_Ω start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) , end_CELL end_ROW start_ROW start_CELL italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ) ≡ divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_k - 1 ) + italic_λ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) , end_CELL end_ROW end_ARRAY (85)

where 哪里

σkref(t):={0,σ=e,σk(t)σks(t),σ{a,i,Ω,λ},assignsuperscriptsubscript𝜎𝑘ref𝑡cases0𝜎𝑒subscript𝜎𝑘𝑡superscriptsubscript𝜎𝑘𝑠𝑡𝜎𝑎𝑖Ω𝜆\sigma_{k}^{\text{ref}}(t):=\left\{\begin{array}[]{ll}0,&\sigma=e,\vskip 3.0pt% plus 1.0pt minus 1.0pt\\ \sigma_{k}(t)-\sigma_{k}^{s}(t),&\sigma\in\{a,i,\Omega,\lambda\},\end{array}\right.italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ref end_POSTSUPERSCRIPT ( italic_t ) := { start_ARRAY start_ROW start_CELL 0 , end_CELL start_CELL italic_σ = italic_e , end_CELL end_ROW start_ROW start_CELL italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) - italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t ) , end_CELL start_CELL italic_σ ∈ { italic_a , italic_i , roman_Ω , italic_λ } , end_CELL end_ROW end_ARRAY (86)

and σo(t)subscript𝜎o𝑡\sigma_{\text{o}}(t)italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) serves as the reference orbit for the synchronous motion of the three satellites. Utilizing Eqs. (76), (85), and (86), the form of σo(t)subscript𝜎o𝑡\sigma_{\text{o}}(t)italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) for σ{a,i,Ω,λ}𝜎𝑎𝑖Ω𝜆\sigma\in\{a,i,\Omega,\lambda\}italic_σ ∈ { italic_a , italic_i , roman_Ω , italic_λ } is
σo(t)subscript𝜎o𝑡\sigma_{\text{o}}(t)italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) 作为三颗卫星同步运动的参考轨道。利用方程。(76)、(85)和(86),for σ{a,i,Ω,λ}𝜎𝑎𝑖Ω𝜆\sigma\in\{a,i,\Omega,\lambda\}italic_σ ∈ { italic_a , italic_i , roman_Ω , italic_λ } 的形式 σo(t)subscript𝜎o𝑡\sigma_{\text{o}}(t)italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t )

σo(t)=σ¯0o+διn¯o(tt0)+Δσoc(t)+Δσol[c](t)+σol[l](t),subscript𝜎o𝑡subscript¯𝜎0osubscript𝛿𝜄subscript¯𝑛o𝑡subscript𝑡0Δsuperscriptsubscript𝜎o𝑐𝑡Δsuperscriptsubscript𝜎o𝑙delimited-[]𝑐𝑡superscriptsubscript𝜎o𝑙delimited-[]𝑙𝑡\displaystyle\sigma_{\text{o}}(t)=\overline{\sigma}_{0\text{o}}+\delta_{\iota}% \overline{n}_{\text{o}}(t-t_{0})+\Delta\sigma_{\text{o}}^{c}(t)+\Delta\sigma_{% \text{o}}^{l[c]}(t)+\sigma_{\text{o}}^{l[l]}(t),italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) = over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 o end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + roman_Δ italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ) + roman_Δ italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) + italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ( italic_t ) , (87)

with σ¯0o:=σ0k[σol[l](t0)+σks(t0)]δι2π3(k1)=const.assignsubscript¯𝜎0osubscript𝜎0𝑘delimited-[]superscriptsubscript𝜎o𝑙delimited-[]𝑙subscript𝑡0superscriptsubscript𝜎𝑘𝑠subscript𝑡0subscript𝛿𝜄2𝜋3𝑘1const\overline{\sigma}_{0\text{o}}:=\sigma_{0k}-[\sigma_{\text{o}}^{l[l]}(t_{0})+% \sigma_{k}^{s}(t_{0})]-\delta_{\iota}\frac{2\pi}{3}(k-1)=\text{const}.over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 o end_POSTSUBSCRIPT := italic_σ start_POSTSUBSCRIPT 0 italic_k end_POSTSUBSCRIPT - [ italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ] - italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_k - 1 ) = const ., Δσoc(t):=σoc(t)σoc(t0)assignΔsuperscriptsubscript𝜎o𝑐𝑡superscriptsubscript𝜎o𝑐𝑡superscriptsubscript𝜎o𝑐subscript𝑡0\Delta\sigma_{\text{o}}^{c}(t):=\sigma_{\text{o}}^{c}(t)-\sigma_{\text{o}}^{c}% (t_{0})roman_Δ italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ) := italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ) - italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ), and Δσol[c](t):=σol[c](t)σol[c](t0)assignΔsuperscriptsubscript𝜎o𝑙delimited-[]𝑐𝑡superscriptsubscript𝜎o𝑙delimited-[]𝑐𝑡superscriptsubscript𝜎o𝑙delimited-[]𝑐subscript𝑡0\Delta\sigma_{\text{o}}^{l[c]}(t):=\sigma_{\text{o}}^{l[c]}(t)-\sigma_{\text{o% }}^{l[c]}(t_{0})roman_Δ italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) := italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) - italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). In other words, the three satellites move along the same virtual circular orbit σo(t)subscript𝜎o𝑡\sigma_{\text{o}}(t)italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) with secular and long-period variations, while maintaining a 120-degree phase difference, forming a rotating, precessing equilateral-triangle constellation.
其中 σ¯0o:=σ0k[σol[l](t0)+σks(t0)]δι2π3(k1)=const.assignsubscript¯𝜎0osubscript𝜎0𝑘delimited-[]superscriptsubscript𝜎o𝑙delimited-[]𝑙subscript𝑡0superscriptsubscript𝜎𝑘𝑠subscript𝑡0subscript𝛿𝜄2𝜋3𝑘1const\overline{\sigma}_{0\text{o}}:=\sigma_{0k}-[\sigma_{\text{o}}^{l[l]}(t_{0})+% \sigma_{k}^{s}(t_{0})]-\delta_{\iota}\frac{2\pi}{3}(k-1)=\text{const}.over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 o end_POSTSUBSCRIPT := italic_σ start_POSTSUBSCRIPT 0 italic_k end_POSTSUBSCRIPT - [ italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ] - italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_k - 1 ) = const .Δσoc(t):=σoc(t)σoc(t0)assignΔsuperscriptsubscript𝜎o𝑐𝑡superscriptsubscript𝜎o𝑐𝑡superscriptsubscript𝜎o𝑐subscript𝑡0\Delta\sigma_{\text{o}}^{c}(t):=\sigma_{\text{o}}^{c}(t)-\sigma_{\text{o}}^{c}% (t_{0})roman_Δ italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ) := italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ) - italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )Δσol[c](t):=σol[c](t)σol[c](t0)assignΔsuperscriptsubscript𝜎o𝑙delimited-[]𝑐𝑡superscriptsubscript𝜎o𝑙delimited-[]𝑐𝑡superscriptsubscript𝜎o𝑙delimited-[]𝑐subscript𝑡0\Delta\sigma_{\text{o}}^{l[c]}(t):=\sigma_{\text{o}}^{l[c]}(t)-\sigma_{\text{o% }}^{l[c]}(t_{0})roman_Δ italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) := italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) - italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) 。换句话说,这三颗卫星沿着同一个虚拟圆形轨道移动 σo(t)subscript𝜎o𝑡\sigma_{\text{o}}(t)italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) ,具有长期和长期变化,同时保持 120 度的相位差,形成一个旋转的进动等边三角形星座。

Correspondingly, δLij(t)𝛿subscript𝐿𝑖𝑗𝑡\delta L_{ij}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ), δvij(t)𝛿subscript𝑣𝑖𝑗𝑡\delta v_{ij}(t)italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ), and δαk(t)𝛿subscript𝛼𝑘𝑡\delta\alpha_{k}(t)italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) result from discrepancies δσk(t)𝛿subscript𝜎𝑘𝑡\delta\sigma_{k}(t)italic_δ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ):
相应地, δLij(t)𝛿subscript𝐿𝑖𝑗𝑡\delta L_{ij}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t )δvij(t)𝛿subscript𝑣𝑖𝑗𝑡\delta v_{ij}(t)italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) , 和 δαk(t)𝛿subscript𝛼𝑘𝑡\delta\alpha_{k}(t)italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) 则由差异产生 δσk(t)𝛿subscript𝜎𝑘𝑡\delta\sigma_{k}(t)italic_δ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t )

δσk(t):=σk(t)σok(t),assign𝛿subscript𝜎𝑘𝑡subscript𝜎𝑘𝑡subscript𝜎o𝑘𝑡\displaystyle\delta\sigma_{k}(t):=\sigma_{k}(t)-\sigma_{\text{o}k}(t),italic_δ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) := italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) - italic_σ start_POSTSUBSCRIPT o italic_k end_POSTSUBSCRIPT ( italic_t ) , (88)

between the real orbits σk(t)subscript𝜎𝑘𝑡\sigma_{k}(t)italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) and the reference orbits σok(t)=σo(t)+δι2π3(k1)subscript𝜎o𝑘𝑡subscript𝜎o𝑡subscript𝛿𝜄2𝜋3𝑘1\sigma_{\text{o}k}(t)=\sigma_{\text{o}}(t)+\delta_{\iota}\frac{2\pi}{3}(k-1)italic_σ start_POSTSUBSCRIPT o italic_k end_POSTSUBSCRIPT ( italic_t ) = italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) + italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_k - 1 ). By expanding Lij(t)subscript𝐿𝑖𝑗𝑡L_{ij}(t)italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ), vij(t)subscript𝑣𝑖𝑗𝑡v_{ij}(t)italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ), and αk(t)subscript𝛼𝑘𝑡\alpha_{k}(t)italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) into a Taylor series along σok(t)subscript𝜎o𝑘𝑡\sigma_{\text{o}k}(t)italic_σ start_POSTSUBSCRIPT o italic_k end_POSTSUBSCRIPT ( italic_t ), the triangle distortions caused by δσk(t)𝛿subscript𝜎𝑘𝑡\delta\sigma_{k}(t)italic_δ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) can be obtained. For the arm-length distortion, we have
在实轨道 σk(t)subscript𝜎𝑘𝑡\sigma_{k}(t)italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) 和参考轨道之间 σok(t)=σo(t)+δι2π3(k1)subscript𝜎o𝑘𝑡subscript𝜎o𝑡subscript𝛿𝜄2𝜋3𝑘1\sigma_{\text{o}k}(t)=\sigma_{\text{o}}(t)+\delta_{\iota}\frac{2\pi}{3}(k-1)italic_σ start_POSTSUBSCRIPT o italic_k end_POSTSUBSCRIPT ( italic_t ) = italic_σ start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) + italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_k - 1 ) 。 通过将 、 vij(t)subscript𝑣𝑖𝑗𝑡v_{ij}(t)italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t )αk(t)subscript𝛼𝑘𝑡\alpha_{k}(t)italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) 沿 σok(t)subscript𝜎o𝑘𝑡\sigma_{\text{o}k}(t)italic_σ start_POSTSUBSCRIPT o italic_k end_POSTSUBSCRIPT ( italic_t ) 展开 Lij(t)subscript𝐿𝑖𝑗𝑡L_{ij}(t)italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) 为泰勒级数,可以得到 引起的 δσk(t)𝛿subscript𝜎𝑘𝑡\delta\sigma_{k}(t)italic_δ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) 三角形畸变。对于臂长畸变,我们有

δLij(t)=𝛿subscript𝐿𝑖𝑗𝑡absent\displaystyle\delta L_{ij}(t)=italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) = Lij(t)3aosubscript𝐿𝑖𝑗𝑡3subscript𝑎o\displaystyle~{}L_{ij}(t)-\sqrt{3}a_{\text{o}}italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) - square-root start_ARG 3 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT
=\displaystyle== 32[δai(t)+δaj(t)]+12aoδλji(t)+12aocosio(t)δΩji(t)32delimited-[]𝛿subscript𝑎𝑖𝑡𝛿subscript𝑎𝑗𝑡12subscript𝑎o𝛿subscript𝜆𝑗𝑖𝑡12subscript𝑎osubscript𝑖o𝑡𝛿subscriptΩ𝑗𝑖𝑡\displaystyle~{}\frac{\sqrt{3}}{2}[{\delta a}_{i}(t)+{\delta a}_{j}(t)]+\frac{% 1}{2}a_{\text{o}}\delta\lambda_{ji}(t)+\frac{1}{2}a_{\text{o}}\cos i_{\text{o}% }(t)\,\delta\Omega_{{ji}}(t)divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 2 end_ARG [ italic_δ italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) + italic_δ italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ] + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_δ italic_λ start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT roman_cos italic_i start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) italic_δ roman_Ω start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t )
+72ao[sinMoj(t)δej(t)sinMoi+(t)δei(t)]+𝒪(δσ(t)2),72subscript𝑎odelimited-[]superscriptsubscript𝑀o𝑗𝑡𝛿subscript𝑒𝑗𝑡superscriptsubscript𝑀o𝑖𝑡𝛿subscript𝑒𝑖𝑡𝒪𝛿𝜎superscript𝑡2\displaystyle+\frac{\sqrt{7}}{2}a_{\text{o}}[\sin M_{\text{o}j}^{-}(t)\,{% \delta e}_{j}(t)-\sin M_{\text{o}i}^{+}(t)\,{\delta e}_{i}(t)]+\mathcal{O}(% \delta\sigma(t)^{2}),+ divide start_ARG square-root start_ARG 7 end_ARG end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ roman_sin italic_M start_POSTSUBSCRIPT o italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_t ) italic_δ italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) - roman_sin italic_M start_POSTSUBSCRIPT o italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_t ) italic_δ italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ] + caligraphic_O ( italic_δ italic_σ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , (89)

where 哪里

δσji(t):=δσj(t)δσi(t),assign𝛿subscript𝜎𝑗𝑖𝑡𝛿subscript𝜎𝑗𝑡𝛿subscript𝜎𝑖𝑡\displaystyle\delta\sigma_{ji}(t):=\delta\sigma_{j}(t)-\delta\sigma_{i}(t),italic_δ italic_σ start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t ) := italic_δ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) - italic_δ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) , (90)
Mok±(t):=Mok(t)±β,Mok(t)=λok(t)ωk(t),formulae-sequenceassignsuperscriptsubscript𝑀o𝑘plus-or-minus𝑡plus-or-minussubscript𝑀o𝑘𝑡𝛽subscript𝑀o𝑘𝑡subscript𝜆o𝑘𝑡subscript𝜔𝑘𝑡\displaystyle M_{\text{o}k}^{\pm}(t):=M_{\text{o}k}(t)\pm\beta,\qquad M_{\text% {o}k}(t)=\lambda_{\text{o}k}(t)-\omega_{k}(t),italic_M start_POSTSUBSCRIPT o italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT ( italic_t ) := italic_M start_POSTSUBSCRIPT o italic_k end_POSTSUBSCRIPT ( italic_t ) ± italic_β , italic_M start_POSTSUBSCRIPT o italic_k end_POSTSUBSCRIPT ( italic_t ) = italic_λ start_POSTSUBSCRIPT o italic_k end_POSTSUBSCRIPT ( italic_t ) - italic_ω start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) , (91)

and the indices i𝑖iitalic_i, j𝑗jitalic_j, and k𝑘kitalic_k follow a cyclic permutation (i,j,k=1231𝑖𝑗𝑘1231i,\,j,\,k=1\to 2\to 3\to 1italic_i , italic_j , italic_k = 1 → 2 → 3 → 1). It can be seen from Eq. (89) that, up to (δσ)1superscript𝛿𝜎1(\delta\sigma)^{1}( italic_δ italic_σ ) start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT order, δLij(t)𝛿subscript𝐿𝑖𝑗𝑡\delta L_{ij}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) is unaffected by inclination deviations δii𝛿subscript𝑖𝑖\delta i_{i}italic_δ italic_i start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and δij𝛿subscript𝑖𝑗\delta i_{j}italic_δ italic_i start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT. Additionally, deviations in ΩΩ\Omegaroman_Ω have minimal influence on δLij(t)𝛿subscript𝐿𝑖𝑗𝑡\delta L_{ij}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) due to the approximately 90superscript9090^{\circ}90 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT inclinations of TianQin orbits, which render the constellation stability insensitive to orbital plane deviations. Combining Eqs. (90), (88), (76), and (87), and defining io(t)=i¯o+ioϵ(t)subscript𝑖o𝑡subscript¯𝑖osubscriptsuperscript𝑖italic-ϵo𝑡i_{\text{o}}(t)=\overline{i}_{\text{o}}+i^{\epsilon}_{\text{o}}(t)italic_i start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) = over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT + italic_i start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ), where i¯osubscript¯𝑖o\overline{i}_{\text{o}}over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT is the average, the right-hand side of Eq. (89) can be categorized into distinct types:
和索引 i𝑖iitalic_ij𝑗jitalic_jk𝑘kitalic_k 遵循循环排列 ( i,j,k=1231𝑖𝑗𝑘1231i,\,j,\,k=1\to 2\to 3\to 1italic_i , italic_j , italic_k = 1 → 2 → 3 → 1 )。从方程(89)可以看出,在阶 (δσ)1superscript𝛿𝜎1(\delta\sigma)^{1}( italic_δ italic_σ ) start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT 次下, δLij(t)𝛿subscript𝐿𝑖𝑗𝑡\delta L_{ij}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) 不受倾角偏差和 δij𝛿subscript𝑖𝑗\delta i_{j}italic_δ italic_i start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT 的影响 δii𝛿subscript𝑖𝑖\delta i_{i}italic_δ italic_i start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT 。此外,由于天琴轨道的近似 90superscript9090^{\circ}90 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT 倾角,星 ΩΩ\Omegaroman_Ω 座的偏差对 δLij(t)𝛿subscript𝐿𝑖𝑗𝑡\delta L_{ij}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) 轨道平面偏差的影响很小,这使得星座稳定性对轨道平面偏差不敏感。结合方程。(90)、(88)、(76)和(87),定义 io(t)=i¯o+ioϵ(t)subscript𝑖o𝑡subscript¯𝑖osubscriptsuperscript𝑖italic-ϵo𝑡i_{\text{o}}(t)=\overline{i}_{\text{o}}+i^{\epsilon}_{\text{o}}(t)italic_i start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) = over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT + italic_i start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) ,其中 i¯osubscript¯𝑖o\overline{i}_{\text{o}}over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT 是平均值,方程(89)的右侧可以分为不同的类型:

δLij(t)=𝛿subscript𝐿𝑖𝑗𝑡absent\displaystyle\delta L_{ij}(t)=italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) = δLijdrift(t)+δLijbias+δLijfluc(t)+𝒪(ioϵ(t)1)δΩji(t)+𝒪(δσ(t)2),𝛿superscriptsubscript𝐿𝑖𝑗drift𝑡𝛿superscriptsubscript𝐿𝑖𝑗bias𝛿superscriptsubscript𝐿𝑖𝑗fluc𝑡𝒪subscriptsuperscript𝑖italic-ϵosuperscript𝑡1𝛿subscriptΩ𝑗𝑖𝑡𝒪𝛿𝜎superscript𝑡2\displaystyle~{}\delta L_{ij}^{\text{drift}}(t)+\delta L_{ij}^{\text{bias}}+% \delta L_{ij}^{\text{fluc}}(t)+\mathcal{O}(i^{\epsilon}_{\text{o}}(t)^{1})% \delta\Omega_{{ji}}(t)+\mathcal{O}(\delta\sigma(t)^{2}),italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT drift end_POSTSUPERSCRIPT ( italic_t ) + italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT bias end_POSTSUPERSCRIPT + italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT fluc end_POSTSUPERSCRIPT ( italic_t ) + caligraphic_O ( italic_i start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) italic_δ roman_Ω start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t ) + caligraphic_O ( italic_δ italic_σ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , (92)

with 

δLijdrift(t)=𝛿superscriptsubscript𝐿𝑖𝑗drift𝑡absent\displaystyle\delta L_{ij}^{\text{drift}}(t)=italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT drift end_POSTSUPERSCRIPT ( italic_t ) = 12aoδn¯ji(tt0)+12ao[δΔλjic(t)+δΔλjil[c](t)]+12aocosi¯o[δΔΩjic(t)+δΔΩjil[c](t)],12subscript𝑎o𝛿subscript¯𝑛𝑗𝑖𝑡subscript𝑡012subscript𝑎odelimited-[]subscript𝛿Δsuperscriptsubscript𝜆𝑗𝑖𝑐𝑡subscript𝛿Δsuperscriptsubscript𝜆𝑗𝑖𝑙delimited-[]𝑐𝑡12subscript𝑎osubscript¯𝑖odelimited-[]subscript𝛿ΔsuperscriptsubscriptΩ𝑗𝑖𝑐𝑡subscript𝛿ΔsuperscriptsubscriptΩ𝑗𝑖𝑙delimited-[]𝑐𝑡\displaystyle~{}\frac{1}{2}a_{\text{o}}\delta\overline{n}_{ji}(t-t_{0})+\frac{% 1}{2}a_{\text{o}}[\delta_{\!\Delta}\lambda_{ji}^{c}(t)+\delta_{\!\Delta}% \lambda_{ji}^{l[c]}(t)]+\frac{1}{2}a_{\text{o}}\cos\overline{i}_{\text{o}}\,[% \delta_{\!\Delta}\Omega_{ji}^{c}(t)+\delta_{\!\Delta}\Omega_{ji}^{l[c]}(t)],divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_δ over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ) + italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) ] + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT roman_cos over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ) + italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) ] , (93)
δLijbias=𝛿superscriptsubscript𝐿𝑖𝑗biasabsent\displaystyle\delta L_{ij}^{\text{bias}}=italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT bias end_POSTSUPERSCRIPT = 32(δa¯0i+δa¯0j)+12aoδλ¯0ji+12aocosi¯oδΩ¯0ji,32𝛿subscript¯𝑎0𝑖𝛿subscript¯𝑎0𝑗12subscript𝑎o𝛿subscript¯𝜆0𝑗𝑖12subscript𝑎osubscript¯𝑖o𝛿subscript¯Ω0𝑗𝑖\displaystyle~{}\frac{\sqrt{3}}{2}(\delta\overline{a}_{0i}+\delta\overline{a}_% {0j})+\frac{1}{2}a_{\text{o}}\delta\overline{\lambda}_{0{ji}}+\frac{1}{2}a_{% \text{o}}\cos\overline{i}_{\text{o}}\,\delta\overline{\Omega}_{0{ji}},divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 2 end_ARG ( italic_δ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 italic_i end_POSTSUBSCRIPT + italic_δ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 italic_j end_POSTSUBSCRIPT ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_δ over¯ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 0 italic_j italic_i end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT roman_cos over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_δ over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 0 italic_j italic_i end_POSTSUBSCRIPT , (94)
δLijfluc(t)=𝛿superscriptsubscript𝐿𝑖𝑗fluc𝑡absent\displaystyle\delta L_{ij}^{\text{fluc}}(t)=italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT fluc end_POSTSUPERSCRIPT ( italic_t ) = 72ao[sinMoj(t)ej(t)sinMoi+(t)ei(t)]+32[ais(t)+ajs(t)]72subscript𝑎odelimited-[]superscriptsubscript𝑀o𝑗𝑡subscript𝑒𝑗𝑡superscriptsubscript𝑀o𝑖𝑡subscript𝑒𝑖𝑡32delimited-[]subscriptsuperscript𝑎𝑠𝑖𝑡subscriptsuperscript𝑎𝑠𝑗𝑡\displaystyle~{}\frac{\sqrt{7}}{2}a_{\text{o}}[\sin M_{\text{o}j}^{-}(t)\,e_{j% }(t)-\sin M_{\text{o}i}^{+}(t)\,e_{i}(t)]+\frac{\sqrt{3}}{2}[a^{s}_{i}(t)+a^{s% }_{j}(t)]divide start_ARG square-root start_ARG 7 end_ARG end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ roman_sin italic_M start_POSTSUBSCRIPT o italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) - roman_sin italic_M start_POSTSUBSCRIPT o italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ] + divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 2 end_ARG [ italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) + italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ]
+12ao[δλjil[l](t)+δλjis(t)]+12aocosi¯o[δΩjil[l](t)+δΩjis(t)],12subscript𝑎odelimited-[]𝛿superscriptsubscript𝜆𝑗𝑖𝑙delimited-[]𝑙𝑡𝛿subscriptsuperscript𝜆𝑠𝑗𝑖𝑡12subscript𝑎osubscript¯𝑖odelimited-[]𝛿superscriptsubscriptΩ𝑗𝑖𝑙delimited-[]𝑙𝑡𝛿subscriptsuperscriptΩ𝑠𝑗𝑖𝑡\displaystyle+\frac{1}{2}a_{\text{o}}[\delta\lambda_{ji}^{l[l]}(t)+\delta% \lambda^{s}_{ji}(t)]+\frac{1}{2}a_{\text{o}}\cos\overline{i}_{\text{o}}\,[% \delta\Omega_{ji}^{l[l]}(t)+\delta\Omega^{s}_{ji}(t)],+ divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ italic_δ italic_λ start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ( italic_t ) + italic_δ italic_λ start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t ) ] + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT roman_cos over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ italic_δ roman_Ω start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ( italic_t ) + italic_δ roman_Ω start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t ) ] , (95)

where δn¯ji=n¯jn¯i𝛿subscript¯𝑛𝑗𝑖subscript¯𝑛𝑗subscript¯𝑛𝑖\delta\overline{n}_{ji}=\overline{n}_{j}-\overline{n}_{i}italic_δ over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT = over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, 其中 δn¯ji=n¯jn¯i𝛿subscript¯𝑛𝑗𝑖subscript¯𝑛𝑗subscript¯𝑛𝑖\delta\overline{n}_{ji}=\overline{n}_{j}-\overline{n}_{i}italic_δ over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT = over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT

δΔσij(t):=δσij(t)δσij(t0),assignsubscript𝛿Δsubscript𝜎𝑖𝑗𝑡𝛿subscript𝜎𝑖𝑗𝑡𝛿subscript𝜎𝑖𝑗subscript𝑡0\displaystyle\delta_{\!\Delta}\sigma_{ij}(t):=\delta\sigma_{ij}(t)-\delta% \sigma_{ij}(t_{0}),italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) := italic_δ italic_σ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) - italic_δ italic_σ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , (96)

δa¯0i=a¯0iao𝛿subscript¯𝑎0𝑖subscript¯𝑎0𝑖subscript𝑎o\delta\overline{a}_{0{i}}=\overline{a}_{0{i}}-a_{\text{o}}italic_δ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 italic_i end_POSTSUBSCRIPT = over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 italic_i end_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT, δλ¯0ji=λ¯0jλ¯0i2π3(ji)𝛿subscript¯𝜆0𝑗𝑖subscript¯𝜆0𝑗subscript¯𝜆0𝑖2𝜋3𝑗𝑖\delta\overline{\lambda}_{0{ji}}=\overline{\lambda}_{0{j}}-\overline{\lambda}_% {0{i}}-\frac{2\pi}{3}(j-i)italic_δ over¯ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 0 italic_j italic_i end_POSTSUBSCRIPT = over¯ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 0 italic_j end_POSTSUBSCRIPT - over¯ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 0 italic_i end_POSTSUBSCRIPT - divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_j - italic_i ), and so on. Equation (92) illustrates that arm-length distortion manifests in three possible types: linear drift δLijdrift(t)𝛿superscriptsubscript𝐿𝑖𝑗drift𝑡\delta L_{ij}^{\text{drift}}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT drift end_POSTSUPERSCRIPT ( italic_t ), constant bias δLijbias𝛿superscriptsubscript𝐿𝑖𝑗bias\delta L_{ij}^{\text{bias}}italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT bias end_POSTSUPERSCRIPT, and periodic fluctuation δLijfluc(t)𝛿superscriptsubscript𝐿𝑖𝑗fluc𝑡\delta L_{ij}^{\text{fluc}}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT fluc end_POSTSUPERSCRIPT ( italic_t ). Specifically, (1) δLijdrift(t)𝛿superscriptsubscript𝐿𝑖𝑗drift𝑡\delta L_{ij}^{\text{drift}}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT drift end_POSTSUPERSCRIPT ( italic_t ) consists of five components of inter-satellite deviations, including δn¯ji𝛿subscript¯𝑛𝑗𝑖\delta\overline{n}_{ji}italic_δ over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT, δΔλjicsubscript𝛿Δsuperscriptsubscript𝜆𝑗𝑖𝑐\delta_{\!\Delta}\lambda_{ji}^{c}italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT, δΔΩjicsubscript𝛿ΔsuperscriptsubscriptΩ𝑗𝑖𝑐\delta_{\!\Delta}\Omega_{ji}^{c}italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT, δΔλjil[c]subscript𝛿Δsuperscriptsubscript𝜆𝑗𝑖𝑙delimited-[]𝑐\delta_{\!\Delta}\lambda_{ji}^{l[c]}italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT, and δΔΩjil[c]subscript𝛿ΔsuperscriptsubscriptΩ𝑗𝑖𝑙delimited-[]𝑐\delta_{\!\Delta}\Omega_{ji}^{l[c]}italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT; (2) δLijbias𝛿superscriptsubscript𝐿𝑖𝑗bias\delta L_{ij}^{\text{bias}}italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT bias end_POSTSUPERSCRIPT comprises initial mean deviations, δa¯0i𝛿subscript¯𝑎0𝑖\delta\overline{a}_{0i}italic_δ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 italic_i end_POSTSUBSCRIPT, δa¯0j𝛿subscript¯𝑎0𝑗\delta\overline{a}_{0j}italic_δ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 italic_j end_POSTSUBSCRIPT, δλ¯0ji𝛿subscript¯𝜆0𝑗𝑖\delta\overline{\lambda}_{0{ji}}italic_δ over¯ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 0 italic_j italic_i end_POSTSUBSCRIPT, and δΩ¯0ji𝛿subscript¯Ω0𝑗𝑖\delta\overline{\Omega}_{0ji}italic_δ over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 0 italic_j italic_i end_POSTSUBSCRIPT; and (3) δLijfluc(t)𝛿superscriptsubscript𝐿𝑖𝑗fluc𝑡\delta L_{ij}^{\text{fluc}}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT fluc end_POSTSUPERSCRIPT ( italic_t ) is linked to eccentricity variations ei(t)subscript𝑒𝑖𝑡e_{i}(t)italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) and ej(t)subscript𝑒𝑗𝑡e_{j}(t)italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) and short-period variations in semi-major axis ais(t)subscriptsuperscript𝑎𝑠𝑖𝑡a^{s}_{i}(t)italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) and ajs(t)subscriptsuperscript𝑎𝑠𝑗𝑡a^{s}_{j}(t)italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ), along with inter-satellite periodic deviations in λ𝜆\lambdaitalic_λ and ΩΩ\Omegaroman_Ω. Among these types, the drift, which progressively increases over time, emerges as the predominant factor affecting the stability of the constellation.
δa¯0i=a¯0iao𝛿subscript¯𝑎0𝑖subscript¯𝑎0𝑖subscript𝑎o\delta\overline{a}_{0{i}}=\overline{a}_{0{i}}-a_{\text{o}}italic_δ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 italic_i end_POSTSUBSCRIPT = over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 italic_i end_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPTδλ¯0ji=λ¯0jλ¯0i2π3(ji)𝛿subscript¯𝜆0𝑗𝑖subscript¯𝜆0𝑗subscript¯𝜆0𝑖2𝜋3𝑗𝑖\delta\overline{\lambda}_{0{ji}}=\overline{\lambda}_{0{j}}-\overline{\lambda}_% {0{i}}-\frac{2\pi}{3}(j-i)italic_δ over¯ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 0 italic_j italic_i end_POSTSUBSCRIPT = over¯ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 0 italic_j end_POSTSUBSCRIPT - over¯ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 0 italic_i end_POSTSUBSCRIPT - divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_j - italic_i ) 、 等。 方程 (92) 说明了臂长失真表现为三种可能的类型: 线性漂移 δLijdrift(t)𝛿superscriptsubscript𝐿𝑖𝑗drift𝑡\delta L_{ij}^{\text{drift}}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT drift end_POSTSUPERSCRIPT ( italic_t ) 、 恒定偏置 δLijbias𝛿superscriptsubscript𝐿𝑖𝑗bias\delta L_{ij}^{\text{bias}}italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT bias end_POSTSUPERSCRIPT 和 周期性波动 δLijfluc(t)𝛿superscriptsubscript𝐿𝑖𝑗fluc𝑡\delta L_{ij}^{\text{fluc}}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT fluc end_POSTSUPERSCRIPT ( italic_t ) 。具体来说,(1) δLijdrift(t)𝛿superscriptsubscript𝐿𝑖𝑗drift𝑡\delta L_{ij}^{\text{drift}}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT drift end_POSTSUPERSCRIPT ( italic_t ) 由卫星间偏差的五个分量组成,包括 δn¯ji𝛿subscript¯𝑛𝑗𝑖\delta\overline{n}_{ji}italic_δ over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT 、 、 δΔλjicsubscript𝛿Δsuperscriptsubscript𝜆𝑗𝑖𝑐\delta_{\!\Delta}\lambda_{ji}^{c}italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT δΔΩjicsubscript𝛿ΔsuperscriptsubscriptΩ𝑗𝑖𝑐\delta_{\!\Delta}\Omega_{ji}^{c}italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPTδΔλjil[c]subscript𝛿Δsuperscriptsubscript𝜆𝑗𝑖𝑙delimited-[]𝑐\delta_{\!\Delta}\lambda_{ji}^{l[c]}italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPTδΔΩjil[c]subscript𝛿ΔsuperscriptsubscriptΩ𝑗𝑖𝑙delimited-[]𝑐\delta_{\!\Delta}\Omega_{ji}^{l[c]}italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ;(2) δLijbias𝛿superscriptsubscript𝐿𝑖𝑗bias\delta L_{ij}^{\text{bias}}italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT bias end_POSTSUPERSCRIPT 包括初始平均偏差 δa¯0i𝛿subscript¯𝑎0𝑖\delta\overline{a}_{0i}italic_δ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 italic_i end_POSTSUBSCRIPT 、 、 δa¯0j𝛿subscript¯𝑎0𝑗\delta\overline{a}_{0j}italic_δ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 italic_j end_POSTSUBSCRIPT δλ¯0ji𝛿subscript¯𝜆0𝑗𝑖\delta\overline{\lambda}_{0{ji}}italic_δ over¯ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 0 italic_j italic_i end_POSTSUBSCRIPTδΩ¯0ji𝛿subscript¯Ω0𝑗𝑖\delta\overline{\Omega}_{0ji}italic_δ over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 0 italic_j italic_i end_POSTSUBSCRIPT ;(3) δLijfluc(t)𝛿superscriptsubscript𝐿𝑖𝑗fluc𝑡\delta L_{ij}^{\text{fluc}}(t)italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT fluc end_POSTSUPERSCRIPT ( italic_t ) 与半长轴 ais(t)subscriptsuperscript𝑎𝑠𝑖𝑡a^{s}_{i}(t)italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t )ajs(t)subscriptsuperscript𝑎𝑠𝑗𝑡a^{s}_{j}(t)italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) 的偏心率变化 ei(t)subscript𝑒𝑖𝑡e_{i}(t)italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t )ej(t)subscript𝑒𝑗𝑡e_{j}(t)italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) 短周期变化以及 和 ΩΩ\Omegaroman_Ωλ𝜆\lambdaitalic_λ 卫星间周期性偏差有关 。在这些类型中,随着时间的推移逐渐增加的漂移成为影响星座稳定性的主要因素。

Regarding the relative velocity, one has
关于相对速度,有

δvij(t)=𝛿subscript𝑣𝑖𝑗𝑡absent\displaystyle\delta v_{ij}(t)=italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) = vij(t)0subscript𝑣𝑖𝑗𝑡0\displaystyle~{}v_{ij}(t)-0italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) - 0
=\displaystyle== 34voao[δaj(t)δai(t)]+72vo[cosMoj(t)δej(t)cosMoi+(t)δei(t)]+𝒪(δσ(t)2),34subscript𝑣osubscript𝑎odelimited-[]𝛿subscript𝑎𝑗𝑡𝛿subscript𝑎𝑖𝑡72subscript𝑣odelimited-[]superscriptsubscript𝑀o𝑗𝑡𝛿subscript𝑒𝑗𝑡superscriptsubscript𝑀o𝑖𝑡𝛿subscript𝑒𝑖𝑡𝒪𝛿𝜎superscript𝑡2\displaystyle-\frac{3}{4}\frac{v_{\text{o}}}{a_{\text{o}}}[{\delta a}_{j}(t)-{% \delta a}_{i}(t)]+\frac{\sqrt{7}}{2}v_{\text{o}}[\cos M_{\text{o}j}^{-}(t)\,{% \delta e}_{j}(t)-\cos M_{\text{o}i}^{+}(t)\,{\delta e}_{i}(t)]+\mathcal{O}(% \delta\sigma(t)^{2}),- divide start_ARG 3 end_ARG start_ARG 4 end_ARG divide start_ARG italic_v start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG start_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG [ italic_δ italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) - italic_δ italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ] + divide start_ARG square-root start_ARG 7 end_ARG end_ARG start_ARG 2 end_ARG italic_v start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ roman_cos italic_M start_POSTSUBSCRIPT o italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_t ) italic_δ italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) - roman_cos italic_M start_POSTSUBSCRIPT o italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_t ) italic_δ italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ] + caligraphic_O ( italic_δ italic_σ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , (97)
=\displaystyle== δvijbias+δvijfluc(t)+𝒪(δσ(t)2),𝛿superscriptsubscript𝑣𝑖𝑗bias𝛿superscriptsubscript𝑣𝑖𝑗fluc𝑡𝒪𝛿𝜎superscript𝑡2\displaystyle~{}\delta v_{ij}^{\text{bias}}+\delta v_{ij}^{\text{fluc}}(t)+% \mathcal{O}(\delta\sigma(t)^{2}),italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT bias end_POSTSUPERSCRIPT + italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT fluc end_POSTSUPERSCRIPT ( italic_t ) + caligraphic_O ( italic_δ italic_σ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , (98)

where vo:=μaoassignsubscript𝑣o𝜇subscript𝑎ov_{\text{o}}:=\sqrt{\frac{\mu}{a_{\text{o}}}}italic_v start_POSTSUBSCRIPT o end_POSTSUBSCRIPT := square-root start_ARG divide start_ARG italic_μ end_ARG start_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG end_ARG, and 其中 vo:=μaoassignsubscript𝑣o𝜇subscript𝑎ov_{\text{o}}:=\sqrt{\frac{\mu}{a_{\text{o}}}}italic_v start_POSTSUBSCRIPT o end_POSTSUBSCRIPT := square-root start_ARG divide start_ARG italic_μ end_ARG start_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG end_ARG , 和

δvijbias=𝛿superscriptsubscript𝑣𝑖𝑗biasabsent\displaystyle\delta v_{ij}^{\text{bias}}=italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT bias end_POSTSUPERSCRIPT = 34voao(δa¯0jδa¯0i),34subscript𝑣osubscript𝑎o𝛿subscript¯𝑎0𝑗𝛿subscript¯𝑎0𝑖\displaystyle-\frac{3}{4}\frac{v_{\text{o}}}{a_{\text{o}}}(\delta\overline{a}_% {0j}-\delta\overline{a}_{0i}),- divide start_ARG 3 end_ARG start_ARG 4 end_ARG divide start_ARG italic_v start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG start_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG ( italic_δ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 italic_j end_POSTSUBSCRIPT - italic_δ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 italic_i end_POSTSUBSCRIPT ) , (99)
δvijfluc(t)=𝛿superscriptsubscript𝑣𝑖𝑗fluc𝑡absent\displaystyle\delta v_{ij}^{\text{fluc}}(t)=italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT fluc end_POSTSUPERSCRIPT ( italic_t ) = 34voao[ajs(t)ais(t)]+72vo[cosMoj(t)ej(t)cosMoi+(t)ei(t)].34subscript𝑣osubscript𝑎odelimited-[]subscriptsuperscript𝑎𝑠𝑗𝑡subscriptsuperscript𝑎𝑠𝑖𝑡72subscript𝑣odelimited-[]superscriptsubscript𝑀o𝑗𝑡subscript𝑒𝑗𝑡superscriptsubscript𝑀o𝑖𝑡subscript𝑒𝑖𝑡\displaystyle-\frac{3}{4}\frac{v_{\text{o}}}{a_{\text{o}}}[a^{s}_{j}(t)-a^{s}_% {i}(t)]+\frac{\sqrt{7}}{2}v_{\text{o}}[\cos M_{\text{o}j}^{-}(t)\,{e}_{j}(t)-% \cos M_{\text{o}i}^{+}(t)\,{e}_{i}(t)].- divide start_ARG 3 end_ARG start_ARG 4 end_ARG divide start_ARG italic_v start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG start_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG [ italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) - italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ] + divide start_ARG square-root start_ARG 7 end_ARG end_ARG start_ARG 2 end_ARG italic_v start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ roman_cos italic_M start_POSTSUBSCRIPT o italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) - roman_cos italic_M start_POSTSUBSCRIPT o italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ] . (100)

Equation (98) illustrates that there is little long-term variation in relative velocity, δvijdrift(t)0similar-to-or-equals𝛿superscriptsubscript𝑣𝑖𝑗drift𝑡0\delta v_{ij}^{\text{drift}}(t)\simeq 0italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT drift end_POSTSUPERSCRIPT ( italic_t ) ≃ 0, consistent with numerical simulation results (cf. Fig. 10 in Ref. [41]). Additionally, the breathing angle within the TianQin triangle experiences three types of distortion akin to those observed in arm-length:
方程 (98) 表明相对速度的长期变化很小, δvijdrift(t)0similar-to-or-equals𝛿superscriptsubscript𝑣𝑖𝑗drift𝑡0\delta v_{ij}^{\text{drift}}(t)\simeq 0italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT drift end_POSTSUPERSCRIPT ( italic_t ) ≃ 0 这与数值模拟结果一致(参见参考文献 [41] 中的图 10)。此外,天琴三角内的呼吸角经历了三种类型的扭曲,类似于在手臂长度中观察到的扭曲:

δαk(t)=𝛿subscript𝛼𝑘𝑡absent\displaystyle\delta\alpha_{k}(t)=italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) = αk(t)π3subscript𝛼𝑘𝑡𝜋3\displaystyle~{}\alpha_{k}(t)-\frac{\pi}{3}italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) - divide start_ARG italic_π end_ARG start_ARG 3 end_ARG
=\displaystyle== 123ao[δai(t)+δaj(t)2δak(t)]+12δλji(t)+12cosio(t)δΩji(t)123subscript𝑎odelimited-[]𝛿subscript𝑎𝑖𝑡𝛿subscript𝑎𝑗𝑡2𝛿subscript𝑎𝑘𝑡12𝛿subscript𝜆𝑗𝑖𝑡12subscript𝑖o𝑡𝛿subscriptΩ𝑗𝑖𝑡\displaystyle~{}\frac{1}{2\sqrt{3}a_{\text{o}}}[{\delta a}_{i}(t)+{\delta a}_{% j}(t)-2{\delta a}_{k}(t)]+\frac{1}{2}\delta\lambda_{ji}(t)+\frac{1}{2}\cos i_{% \text{o}}(t)\,\delta\Omega_{{ji}}(t)divide start_ARG 1 end_ARG start_ARG 2 square-root start_ARG 3 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG [ italic_δ italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) + italic_δ italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) - 2 italic_δ italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) ] + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_δ italic_λ start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_cos italic_i start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) italic_δ roman_Ω start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t )
+76[fek(t)δek(t)+fei(t)δei(t)+fej(t)δej(t)]+𝒪(δσ(t)2)76delimited-[]superscriptsubscript𝑓𝑒𝑘𝑡𝛿subscript𝑒𝑘𝑡superscriptsubscript𝑓𝑒𝑖𝑡𝛿subscript𝑒𝑖𝑡superscriptsubscript𝑓𝑒𝑗𝑡𝛿subscript𝑒𝑗𝑡𝒪𝛿𝜎superscript𝑡2\displaystyle+\frac{\sqrt{7}}{6}[f_{e}^{k}(t){\delta e}_{k}(t)+f_{e}^{i}(t){% \delta e}_{i}(t)+f_{e}^{j}(t){\delta e}_{j}(t)]+\mathcal{O}(\delta\sigma(t)^{2})+ divide start_ARG square-root start_ARG 7 end_ARG end_ARG start_ARG 6 end_ARG [ italic_f start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_t ) italic_δ italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) + italic_f start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t ) italic_δ italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) + italic_f start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ( italic_t ) italic_δ italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ] + caligraphic_O ( italic_δ italic_σ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) (101)
=\displaystyle== δαkdrift(t)+δαkbias+δαkfluc(t)+𝒪(ioϵ(t)1)δΩji(t)+𝒪(δσ(t)2),𝛿superscriptsubscript𝛼𝑘drift𝑡𝛿superscriptsubscript𝛼𝑘bias𝛿superscriptsubscript𝛼𝑘fluc𝑡𝒪subscriptsuperscript𝑖italic-ϵosuperscript𝑡1𝛿subscriptΩ𝑗𝑖𝑡𝒪𝛿𝜎superscript𝑡2\displaystyle~{}\delta\alpha_{k}^{\text{drift}}(t)+\delta\alpha_{k}^{\text{% bias}}+\delta\alpha_{k}^{\text{fluc}}(t)+\mathcal{O}(i^{\epsilon}_{\text{o}}(t% )^{1})\delta\Omega_{{ji}}(t)+\mathcal{O}(\delta\sigma(t)^{2}),italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT drift end_POSTSUPERSCRIPT ( italic_t ) + italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT bias end_POSTSUPERSCRIPT + italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT fluc end_POSTSUPERSCRIPT ( italic_t ) + caligraphic_O ( italic_i start_POSTSUPERSCRIPT italic_ϵ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT o end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) italic_δ roman_Ω start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t ) + caligraphic_O ( italic_δ italic_σ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , (102)

with 

δαkdrift(t)=𝛿superscriptsubscript𝛼𝑘drift𝑡absent\displaystyle\delta\alpha_{k}^{\text{drift}}(t)=italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT drift end_POSTSUPERSCRIPT ( italic_t ) = 12δn¯ji(tt0)+12[δΔλjic(t)+δΔλjil[c](t)]+12cosi¯o[δΔΩjic(t)+δΔΩjil[c](t)],12𝛿subscript¯𝑛𝑗𝑖𝑡subscript𝑡012delimited-[]subscript𝛿Δsuperscriptsubscript𝜆𝑗𝑖𝑐𝑡subscript𝛿Δsuperscriptsubscript𝜆𝑗𝑖𝑙delimited-[]𝑐𝑡12subscript¯𝑖odelimited-[]subscript𝛿ΔsuperscriptsubscriptΩ𝑗𝑖𝑐𝑡subscript𝛿ΔsuperscriptsubscriptΩ𝑗𝑖𝑙delimited-[]𝑐𝑡\displaystyle~{}\frac{1}{2}\delta\overline{n}_{ji}(t-t_{0})+\frac{1}{2}[\delta% _{\!\Delta}\lambda_{ji}^{c}(t)+\delta_{\!\Delta}\lambda_{ji}^{l[c]}(t)]+\frac{% 1}{2}\cos\overline{i}_{\text{o}}\,[\delta_{\!\Delta}\Omega_{ji}^{c}(t)+\delta_% {\!\Delta}\Omega_{ji}^{l[c]}(t)],divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_δ over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ) + italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) ] + divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_cos over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ) + italic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) ] , (103)
δαkbias=𝛿superscriptsubscript𝛼𝑘biasabsent\displaystyle\delta\alpha_{k}^{\text{bias}}=italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT bias end_POSTSUPERSCRIPT = 123ao(δa¯0ik+δa¯0jk)+12δλ¯0ji+12cosi¯oδΩ¯0ji,123subscript𝑎o𝛿subscript¯𝑎0𝑖𝑘𝛿subscript¯𝑎0𝑗𝑘12𝛿subscript¯𝜆0𝑗𝑖12subscript¯𝑖o𝛿subscript¯Ω0𝑗𝑖\displaystyle~{}\frac{1}{2\sqrt{3}a_{\text{o}}}(\delta\overline{a}_{0ik}+% \delta\overline{a}_{0jk})+\frac{1}{2}\delta\overline{\lambda}_{0{ji}}+\frac{1}% {2}\cos\overline{i}_{\text{o}}\,\delta\overline{\Omega}_{0{ji}},divide start_ARG 1 end_ARG start_ARG 2 square-root start_ARG 3 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG ( italic_δ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 italic_i italic_k end_POSTSUBSCRIPT + italic_δ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 0 italic_j italic_k end_POSTSUBSCRIPT ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_δ over¯ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 0 italic_j italic_i end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_cos over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_δ over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 0 italic_j italic_i end_POSTSUBSCRIPT , (104)
δαkfluc(t)=𝛿superscriptsubscript𝛼𝑘fluc𝑡absent\displaystyle\delta\alpha_{k}^{\text{fluc}}(t)=italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT fluc end_POSTSUPERSCRIPT ( italic_t ) = 76[fek(t)ek(t)+fei(t)ei(t)+fej(t)ej(t)]+123ao[δaiks(t)+δajks(t)]76delimited-[]superscriptsubscript𝑓𝑒𝑘𝑡subscript𝑒𝑘𝑡superscriptsubscript𝑓𝑒𝑖𝑡subscript𝑒𝑖𝑡superscriptsubscript𝑓𝑒𝑗𝑡subscript𝑒𝑗𝑡123subscript𝑎odelimited-[]𝛿subscriptsuperscript𝑎𝑠𝑖𝑘𝑡𝛿subscriptsuperscript𝑎𝑠𝑗𝑘𝑡\displaystyle~{}\frac{\sqrt{7}}{6}[f_{e}^{k}(t)e_{k}(t)+f_{e}^{i}(t)e_{i}(t)+f% _{e}^{j}(t)e_{j}(t)]+\frac{1}{2\sqrt{3}a_{\text{o}}}[\delta a^{s}_{ik}(t)+% \delta a^{s}_{jk}(t)]divide start_ARG square-root start_ARG 7 end_ARG end_ARG start_ARG 6 end_ARG [ italic_f start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) + italic_f start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) + italic_f start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ] + divide start_ARG 1 end_ARG start_ARG 2 square-root start_ARG 3 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG [ italic_δ italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT ( italic_t ) + italic_δ italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT ( italic_t ) ]
+12[δλjil[l](t)+δλjis(t)]+12cosi¯o[δΩjil[l](t)+δΩjis(t)],12delimited-[]𝛿superscriptsubscript𝜆𝑗𝑖𝑙delimited-[]𝑙𝑡𝛿subscriptsuperscript𝜆𝑠𝑗𝑖𝑡12subscript¯𝑖odelimited-[]𝛿superscriptsubscriptΩ𝑗𝑖𝑙delimited-[]𝑙𝑡𝛿subscriptsuperscriptΩ𝑠𝑗𝑖𝑡\displaystyle+\frac{1}{2}[\delta\lambda_{ji}^{l[l]}(t)+\delta\lambda^{s}_{ji}(% t)]+\frac{1}{2}\cos\overline{i}_{\text{o}}\,[\delta\Omega_{ji}^{l[l]}(t)+% \delta\Omega^{s}_{ji}(t)],+ divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_δ italic_λ start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ( italic_t ) + italic_δ italic_λ start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t ) ] + divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_cos over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ italic_δ roman_Ω start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ( italic_t ) + italic_δ roman_Ω start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t ) ] , (105)

where 哪里

fek(t)=2cosMok(t)sinβ,fei(t)=sinMoi(t)2sinMoi+(t),fej(t)=2sinMoj(t)+sinMoj+(t).formulae-sequencesuperscriptsubscript𝑓𝑒𝑘𝑡2subscript𝑀o𝑘𝑡𝛽formulae-sequencesuperscriptsubscript𝑓𝑒𝑖𝑡superscriptsubscript𝑀o𝑖𝑡2superscriptsubscript𝑀o𝑖𝑡superscriptsubscript𝑓𝑒𝑗𝑡2superscriptsubscript𝑀o𝑗𝑡superscriptsubscript𝑀o𝑗𝑡\displaystyle f_{e}^{k}(t)=2\cos M_{\text{o}k}(t)\sin\beta,\qquad f_{e}^{i}(t)% =-\sin M_{\text{o}i}^{-}(t)-2\sin M_{\text{o}i}^{+}(t),\qquad f_{e}^{j}(t)=2% \sin M_{\text{o}j}^{-}(t)+\sin M_{\text{o}j}^{+}(t).italic_f start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_t ) = 2 roman_cos italic_M start_POSTSUBSCRIPT o italic_k end_POSTSUBSCRIPT ( italic_t ) roman_sin italic_β , italic_f start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t ) = - roman_sin italic_M start_POSTSUBSCRIPT o italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_t ) - 2 roman_sin italic_M start_POSTSUBSCRIPT o italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_t ) , italic_f start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ( italic_t ) = 2 roman_sin italic_M start_POSTSUBSCRIPT o italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_t ) + roman_sin italic_M start_POSTSUBSCRIPT o italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_t ) . (106)

Equations (104) and (105) show that bias and fluctuation in the breathing angle, as observed from SCk𝑘kitalic_k, are associated with deviations in a𝑎aitalic_a and e𝑒eitalic_e of all three satellites. However, concerning λ𝜆\lambdaitalic_λ and ΩΩ\Omegaroman_Ω, they are exclusively linked to the relative differences between the other two satellites.
方程 (104) 和 (105) 表明,从 SC k𝑘kitalic_k 观察到的呼吸角度的偏差和波动与所有三个卫星的 a𝑎aitalic_a e𝑒eitalic_e 偏差有关。然而,关于 λ𝜆\lambdaitalic_λΩΩ\Omegaroman_Ω ,它们完全与其他两颗卫星之间的相对差异有关。

The variations in the three types, drift, bias, and fluctuation, all impact the constellation’s stability, necessitating optimization. The drift, associated with δn¯ji=3no2aoδa¯ji𝛿subscript¯𝑛𝑗𝑖3subscript𝑛o2subscript𝑎o𝛿subscript¯𝑎𝑗𝑖\delta\overline{n}_{ji}=-\frac{3n_{\text{o}}}{2a_{\text{o}}}{\delta\bar{a}}_{{% ji}}italic_δ over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT = - divide start_ARG 3 italic_n start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG start_ARG 2 italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG italic_δ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT, can be significantly mitigated by aligning the mean semi-major axes a¯joptim=a¯ioptimsuperscriptsubscript¯𝑎𝑗optimsuperscriptsubscript¯𝑎𝑖optim\overline{a}_{j}^{\,\text{optim}}=\overline{a}_{i}^{\,\text{optim}}over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT = over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT. More generally, one can see from, e.g., Eqs. (103)-(105), that terms within the drift, bias, and long-period fluctuation, are contingent on the mean or initial mean values of parameters a𝑎aitalic_a, i𝑖iitalic_i, ΩΩ\Omegaroman_Ω, and λ𝜆\lambdaitalic_λ. Additionally, the fluctuation is also correlated with eccentricities, which exhibit secular variations and serve as the primary factor influencing the amplitude of the fluctuation. Thus, the optimization of constellation variations can be achieved by imposing the following conditions:
漂移、偏置和波动这三种类型的变化都会影响星座的稳定性,因此需要进行优化。与 δn¯ji=3no2aoδa¯ji𝛿subscript¯𝑛𝑗𝑖3subscript𝑛o2subscript𝑎o𝛿subscript¯𝑎𝑗𝑖\delta\overline{n}_{ji}=-\frac{3n_{\text{o}}}{2a_{\text{o}}}{\delta\bar{a}}_{{% ji}}italic_δ over¯ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT = - divide start_ARG 3 italic_n start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG start_ARG 2 italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG italic_δ over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT 关联的漂移可以通过对齐平均半长轴 a¯joptim=a¯ioptimsuperscriptsubscript¯𝑎𝑗optimsuperscriptsubscript¯𝑎𝑖optim\overline{a}_{j}^{\,\text{optim}}=\overline{a}_{i}^{\,\text{optim}}over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT = over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT 来显著缓解。更一般地说,人们可以从 Eqs 等中看到。(103)-(105),即漂移、偏差和长期波动内的项取决于参数 a𝑎aitalic_ai𝑖iitalic_iΩΩ\Omegaroman_Ωλ𝜆\lambdaitalic_λ 的平均值或初始平均值。此外,波动还与偏心率相关,偏心率表现出长期变化,是影响波动幅度的主要因素。因此,可以通过施加以下条件来实现星座变化的优化:

{a¯1optim=a¯2optim=a¯3optim,i¯1optim=i¯2optim=i¯3optim,Ω¯1optim=Ω¯2optim=Ω¯3optim,λ¯koptim=2π3(k1)+λ¯1optim,ekoptim(t)0.casessuperscriptsubscript¯𝑎1optimsuperscriptsubscript¯𝑎2optimsuperscriptsubscript¯𝑎3optimsuperscriptsubscript¯𝑖1optimsuperscriptsubscript¯𝑖2optimsuperscriptsubscript¯𝑖3optimsuperscriptsubscript¯Ω1optimsuperscriptsubscript¯Ω2optimsuperscriptsubscript¯Ω3optimsuperscriptsubscript¯𝜆𝑘optim2𝜋3𝑘1superscriptsubscript¯𝜆1optimsimilar-to-or-equalssuperscriptsubscript𝑒𝑘optim𝑡0\displaystyle\left\{\begin{array}[]{l}\overline{a}_{1}^{\,\text{optim}}=% \overline{a}_{2}^{\,\text{optim}}=\overline{a}_{3}^{\,\text{optim}},\\ \overline{i}_{1}^{\,\text{optim}}=\overline{i}_{2}^{\,\text{optim}}=\overline{% i}_{3}^{\,\text{optim}},\\ \overline{\Omega}_{1}^{\,\text{optim}}=\overline{\Omega}_{2}^{\,\text{optim}}=% \overline{\Omega}_{3}^{\,\text{optim}},\\ \overline{\lambda}_{k}^{\,\text{optim}}=\frac{2\pi}{3}(k-1)+\overline{\lambda}% _{1}^{\,\text{optim}},\\ e_{k}^{\text{optim}}(t)\simeq 0.\end{array}\right.{ start_ARRAY start_ROW start_CELL over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT = over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT = over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT = over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT = over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT = over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT = over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL over¯ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT = divide start_ARG 2 italic_π end_ARG start_ARG 3 end_ARG ( italic_k - 1 ) + over¯ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT ( italic_t ) ≃ 0 . end_CELL end_ROW end_ARRAY (112)

Further, the optimized indicators, up to the leading order, are
此外,优化后的指标,直到前导顺序,都是

δLijoptim(t)=𝛿superscriptsubscript𝐿𝑖𝑗optim𝑡absent\displaystyle\delta L_{ij}^{\text{optim}}(t)=italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT ( italic_t ) = 72ao[sinMoj(t)ej(t)sinMoi+(t)ei(t)]+32[ais(t)+ajs(t)]+12aoδλjis(t)+12aocosi¯oδΩjis(t),72subscript𝑎odelimited-[]superscriptsubscript𝑀o𝑗𝑡subscript𝑒𝑗𝑡superscriptsubscript𝑀o𝑖𝑡subscript𝑒𝑖𝑡32delimited-[]subscriptsuperscript𝑎𝑠𝑖𝑡subscriptsuperscript𝑎𝑠𝑗𝑡12subscript𝑎o𝛿subscriptsuperscript𝜆𝑠𝑗𝑖𝑡12subscript𝑎osubscript¯𝑖o𝛿subscriptsuperscriptΩ𝑠𝑗𝑖𝑡\displaystyle~{}\frac{\sqrt{7}}{2}a_{\text{o}}[\sin M_{\text{o}j}^{-}(t)\,e_{j% }(t)-\sin M_{\text{o}i}^{+}(t)\,e_{i}(t)]+\frac{\sqrt{3}}{2}[a^{s}_{i}(t)+a^{s% }_{j}(t)]+\frac{1}{2}a_{\text{o}}\delta\lambda^{s}_{ji}(t)+\frac{1}{2}a_{\text% {o}}\cos\overline{i}_{\text{o}}\,\delta\Omega^{s}_{ji}(t),divide start_ARG square-root start_ARG 7 end_ARG end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ roman_sin italic_M start_POSTSUBSCRIPT o italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) - roman_sin italic_M start_POSTSUBSCRIPT o italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ] + divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 2 end_ARG [ italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) + italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ] + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_δ italic_λ start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT roman_cos over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_δ roman_Ω start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t ) , (113)
δvijoptim(t)=𝛿superscriptsubscript𝑣𝑖𝑗optim𝑡absent\displaystyle\delta v_{ij}^{\text{optim}}(t)=italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT ( italic_t ) = 72vo[cosMoj(t)ej(t)cosMoi+(t)ei(t)]34voao[ajs(t)ais(t)],72subscript𝑣odelimited-[]superscriptsubscript𝑀o𝑗𝑡subscript𝑒𝑗𝑡superscriptsubscript𝑀o𝑖𝑡subscript𝑒𝑖𝑡34subscript𝑣osubscript𝑎odelimited-[]subscriptsuperscript𝑎𝑠𝑗𝑡subscriptsuperscript𝑎𝑠𝑖𝑡\displaystyle~{}\frac{\sqrt{7}}{2}v_{\text{o}}[\cos M_{\text{o}j}^{-}(t)\,{e}_% {j}(t)-\cos M_{\text{o}i}^{+}(t)\,{e}_{i}(t)]-\frac{3}{4}\frac{v_{\text{o}}}{a% _{\text{o}}}[a^{s}_{j}(t)-a^{s}_{i}(t)],divide start_ARG square-root start_ARG 7 end_ARG end_ARG start_ARG 2 end_ARG italic_v start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ roman_cos italic_M start_POSTSUBSCRIPT o italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) - roman_cos italic_M start_POSTSUBSCRIPT o italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ] - divide start_ARG 3 end_ARG start_ARG 4 end_ARG divide start_ARG italic_v start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG start_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG [ italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) - italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) ] , (114)
δαkoptim(t)=𝛿superscriptsubscript𝛼𝑘optim𝑡absent\displaystyle\delta\alpha_{k}^{\text{optim}}(t)=italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT ( italic_t ) = 76[fek(t)ek(t)+fei(t)ei(t)+fej(t)ej(t)]+123ao[δaiks(t)+δajks(t)]+12δλjis(t)+12cosi¯oδΩjis(t),76delimited-[]superscriptsubscript𝑓𝑒𝑘𝑡subscript𝑒𝑘𝑡superscriptsubscript𝑓𝑒𝑖𝑡subscript𝑒𝑖𝑡superscriptsubscript𝑓𝑒𝑗𝑡subscript𝑒𝑗𝑡123subscript𝑎odelimited-[]𝛿subscriptsuperscript𝑎𝑠𝑖𝑘𝑡𝛿subscriptsuperscript𝑎𝑠𝑗𝑘𝑡12𝛿subscriptsuperscript𝜆𝑠𝑗𝑖𝑡12subscript¯𝑖o𝛿subscriptsuperscriptΩ𝑠𝑗𝑖𝑡\displaystyle~{}\frac{\sqrt{7}}{6}[f_{e}^{k}(t)e_{k}(t)+f_{e}^{i}(t)e_{i}(t)+f% _{e}^{j}(t)e_{j}(t)]+\frac{1}{2\sqrt{3}a_{\text{o}}}[\delta a^{s}_{ik}(t)+% \delta a^{s}_{jk}(t)]+\frac{1}{2}\delta\lambda^{s}_{ji}(t)+\frac{1}{2}\cos% \overline{i}_{\text{o}}\,\delta\Omega^{s}_{ji}(t),divide start_ARG square-root start_ARG 7 end_ARG end_ARG start_ARG 6 end_ARG [ italic_f start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) + italic_f start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) + italic_f start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ( italic_t ) italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) ] + divide start_ARG 1 end_ARG start_ARG 2 square-root start_ARG 3 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT end_ARG [ italic_δ italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT ( italic_t ) + italic_δ italic_a start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT ( italic_t ) ] + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_δ italic_λ start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_cos over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_δ roman_Ω start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_i end_POSTSUBSCRIPT ( italic_t ) , (115)

with right-hand functions adopting the conditions given by Eq. (112), i.e., ao=a¯koptimsubscript𝑎osuperscriptsubscript¯𝑎𝑘optima_{\text{o}}=\overline{a}_{k}^{\,\text{optim}}italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT, i¯o=i¯koptimsubscript¯𝑖osuperscriptsubscript¯𝑖𝑘optim\overline{i}_{\text{o}}=\overline{i}_{k}^{\,\text{optim}}over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT, ek(t)=ekoptim(t)subscript𝑒𝑘𝑡subscriptsuperscript𝑒optim𝑘𝑡e_{k}(t)=e^{\text{optim}}_{k}(t)italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) = italic_e start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ), etc. Equations (113)-(115) reveal the optimized TianQin triangle as intrinsic fluctuation variations induced by perturbations, with amplitude dependent on eccentricities and short-period variations in other elements. Notably, conditions in Eq. (112) correspond to the optimal stable configuration, and therefore can provide useful guidelines for numerical optimization and orbit control.
右手函数采用式(112)给出的条件,即、 ao=a¯koptimsubscript𝑎osuperscriptsubscript¯𝑎𝑘optima_{\text{o}}=\overline{a}_{k}^{\,\text{optim}}italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = over¯ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPTi¯o=i¯koptimsubscript¯𝑖osuperscriptsubscript¯𝑖𝑘optim\overline{i}_{\text{o}}=\overline{i}_{k}^{\,\text{optim}}over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT o end_POSTSUBSCRIPT = over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT ek(t)=ekoptim(t)subscript𝑒𝑘𝑡subscriptsuperscript𝑒optim𝑘𝑡e_{k}(t)=e^{\text{optim}}_{k}(t)italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) = italic_e start_POSTSUPERSCRIPT optim end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) 等。方程(113)-(115)揭示了优化后的天琴三角是由扰动引起的内在涨落变化,其振幅取决于偏心率和其他元素的短周期变化。值得注意的是,方程(112)中的条件对应于最优稳定配置,因此可以为数值优化和轨道控制提供有用的指导。

IV Concluding Remarks
、结束语

Detecting GWs with TianQin requires a stable three-satellite constellation, configured as closely to an equilateral triangle as possible. In high Earth orbits, gravitational perturbations, especially from lunar and solar influences, can distort this configuration. To quantify the impact, we have developed an analytical model delineating the effects of lunar and solar point masses on the TianQin constellation. This model provides expressions for three kinematic indicators, including arm-lengths, relative velocities, and breathing angles, derived from the first-order perturbation solution for satellite orbital elements.
用天琴探测 GW 需要一个稳定的三星星座,配置尽可能靠近等边三角形。在高地球轨道上,引力扰动,尤其是来自月球和太阳影响的引力扰动,可以扭曲这种配置。为了量化影响,我们开发了一个分析模型,描述了月球和太阳点质量对天琴星座的影响。该模型提供了三个运动学指标的表达式,包括臂长、相对速度和呼吸角,这些指标来自卫星轨道元件的一阶扰动解。

The analysis of these indicators has revealed that gravitational perturbations induce secular, long-period, and short-period variations in satellite orbital elements, leading to relative motion between satellites and distortions in the constellation. These distortions appear as three distinct types, i.e., linear drift, bias, and fluctuation. The drift, progressively increasing over time, is a primary destabilizing factor affecting arm-lengths and breathing angles but has almost no impact on relative velocities. To alleviate design constraints on onboard scientific payloads, these three distortions have been further optimized. It is demonstrated that both drift and bias can be eliminated, and fluctuation amplitude reduced, if the three orbits adhere to the following constraints on average: synchronized orbital periods, aligned orbital planes, equally spaced phases, and minimized eccentricities. The expressions for the optimized indicators are presented, revealing that the optimized TianQin constellation displays only fluctuation with amplitude dependent on eccentricities and short-period variations in other elements.
对这些指标的分析表明,引力扰动会引起卫星轨道元件的长期、长期和短期变化,从而导致卫星之间的相对运动和星座的扭曲。这些失真表现为三种不同的类型,即线性漂移、偏置和波动。漂移随着时间的推移逐渐增加,是影响臂长和呼吸角度的主要不稳定因素,但对相对速度几乎没有影响。为了减轻对机载科学有效载荷的设计限制,这三种失真得到了进一步优化。结果表明,如果三个轨道平均遵守以下约束条件,则可以消除漂移和偏置,并减小波动幅度:同步轨道周期、对齐的轨道平面、等距相位和最小化偏心率。给出了优化指标的表达式,揭示了优化后的天琴星座仅显示振幅取决于偏心率和其他元素的短周期变化的波动。

These results can provide valuable insights and guidelines for enhancing the stability of the GW observatory constellation, such as in numerical optimization and orbit control. For future works, this model will be extended to incorporate the influence of initial orbit errors [22, 48, 49, 50]. The perturbation solution can be applied to explore the dynamics of TianQin satellite eccentricity, closely linked to the constellation stability. Potential applications in celestial mechanics, especially for high-inclination TianQin-like orbits subject to the Kozai–Lidov effect [51, 52], may arise. Further discussions are deferred to future work.
这些结果可以为增强 GW 天文台星座的稳定性提供有价值的见解和指导,例如在数值优化和轨道控制方面。在未来的工作中,该模型将被扩展以纳入初始轨道误差的影响 [22484950]。该微扰解可用于探究天琴卫星偏心率的动力学特性,与星座稳定性密切相关。在天体力学中的潜在应用,特别是受 Kozai-Lidov 效应影响的高倾角天琴样轨道 [5152],可能会出现。进一步的讨论推迟到未来的工作中。

Acknowledgements. 确认。
The authors thank Jianwei Mei, Yunhe Meng, Defeng Gu, Liang-Cheng Tu, Cheng-Gang Shao, Yan Wang, and Jun Luo for helpful discussions and comments. Special thanks to the anonymous referee for valuable suggestions and comments. X. Z. is supported by the National Key R&D Program of China (Grant Nos. 2020YFC2201202 and 2022YFC2204600), NSFC (Grant No. 12373116), and Fundamental Research Funds for the Central Universities, Sun Yat-sen University (Grant No. 23xkjc001).
作者感谢 Jianwei Mei、Yunhe Meng、Defeng Gu、Liang-Cheng Tu、Cheng-Gang Shao、Yan Wang 和 Jun Luo 的有益讨论和评论。特别感谢匿名审稿人的宝贵建议和评论。X. Z. 得到了国家重点研发计划(批准号 2020YFC2201202 和 2022YFC2204600)、国家自然科学基金(批准号 12373116)和中央高校基本研究基金(批准号 23xkjc001)的支持。

Appendix A Table of Symbols
附录 A 符号表

Table LABEL:table:symbols below lists the main symbols used in the paper and their meanings for quick look-ups.
下面的表格 LABEL:table:symbols 列出了论文中使用的主要符号及其含义,以便快速查找。

Table 3: List of symbols and their meanings.
表 3: 符号列表及其含义。
Symbols 符号 Meanings 意义
t𝑡titalic_t Time 时间
t0subscript𝑡0t_{0}italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT Reference epoch 参考纪元
UTC Coordinated Universal Time
协调世界时
a𝑎aitalic_a Semimajor axis 半长轴
e𝑒eitalic_e Orbital eccentricity 轨道偏心率
i𝑖iitalic_i Orbital inclination 轨道倾角
ΩΩ\Omegaroman_Ω Longitude of ascending node
升序节点的经度
ω𝜔\omegaitalic_ω Argument of perigee 近地点的论点
ν𝜈\nuitalic_ν True anomaly 真正的异常
M𝑀Mitalic_M Mean anomaly 平均异常
E𝐸Eitalic_E Eccentric anomaly 偏心异常
u(=ω+ν)annotateduabsent𝜔𝜈\text{u}\,(=\omega+\nu)u ( = italic_ω + italic_ν ) Argument of latitude 纬度参数
λ𝜆\lambdaitalic_λ Defined as (ω+M)𝜔𝑀(\omega+M)( italic_ω + italic_M ) 定义为 (ω+M)𝜔𝑀(\omega+M)( italic_ω + italic_M )
ξ(=ecosω)annotated𝜉absent𝑒𝜔\xi\,(=e\cos\omega)italic_ξ ( = italic_e roman_cos italic_ω ) Singularity-free variable for eccentricity
偏心率的无奇点变量
η(=esinω)annotated𝜂absent𝑒𝜔\eta\,(=e\sin\omega)italic_η ( = italic_e roman_sin italic_ω ) Singularity-free variable for eccentricity
偏心率的无奇点变量
σ𝜎\sigmaitalic_σ General representation of orbital elements
轨道元素的一般表示
σ0subscript𝜎0\sigma_{0}italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT Initial orbital elements 初始轨道元件
Subscript e 下标 e Refers to the Earth 指地球
Subscript s 下标 s Refers to the Sun 指太阳
Subscript m 下标 m Refers to the Moon 指月亮
Subscript J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 下标 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT Refers to the Earth’s oblateness J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT
指地球的扁圆度 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT
Subscript 0 下标 0 Denoting initial values or zeroth order
表示初始值或零阶
Subscript 1 下标 1 Denoting SC1 or first order
表示 SC1 或一阶
Subscript 2 下标 2 Denoting the Sun, SC2 or second order
表示太阳、SC2 或二阶
Subscript 3 下标 3 Denoting the Moon or SC3
表示月亮或 SC3
Subscript o 下标 o Denoting nominal values 表示标称值
Notation c𝑐citalic_c 表示法 c𝑐citalic_c Denoting secular variation
表示长期变化
Notation l𝑙litalic_l 表示法 l𝑙litalic_l Denoting long-period variation
表示长周期变化
Notation l[c]𝑙delimited-[]𝑐l[c]italic_l [ italic_c ] 表示法 l[c]𝑙delimited-[]𝑐l[c]italic_l [ italic_c ] Denoting special long-period variation
表示特殊的长周期变化
Notation l[l]𝑙delimited-[]𝑙l[l]italic_l [ italic_l ] 表示法 l[l]𝑙delimited-[]𝑙l[l]italic_l [ italic_l ] Denoting general long-period variation
表示一般的长周期变化
Notation s𝑠sitalic_s 表示法 s𝑠sitalic_s Denoting short-period variation
表示短期变化
i𝑖iitalic_i, j𝑗jitalic_j, k𝑘kitalic_k
i𝑖iitalic_ij𝑗jitalic_jk𝑘kitalic_k
Represent satellites and take values of 1, 2, 3
表示卫星并取值 1、2、3
κ𝜅\kappaitalic_κ, p𝑝pitalic_p, q𝑞qitalic_q
κ𝜅\kappaitalic_κp𝑝pitalic_pq𝑞qitalic_q
Represent components of the perturbation solution
表示扰动解的分量
y˙˙𝑦\dot{y}over˙ start_ARG italic_y end_ARG Time derivative of y𝑦yitalic_y
的时间 y𝑦yitalic_y 导数
y¯¯𝑦\overline{y}over¯ start_ARG italic_y end_ARG Mean orbital elements or average values
平均轨道元素或平均值
ΔyΔ𝑦\Delta yroman_Δ italic_y Difference (See Eq. (87))
差异(见方程 (87))
δy𝛿𝑦\delta yitalic_δ italic_y Change relative to the nominal value yosubscript𝑦oy_{\text{o}}italic_y start_POSTSUBSCRIPT o end_POSTSUBSCRIPT (See Eqs. (77), (88), and (90))
相对于标称值 yosubscript𝑦oy_{\text{o}}italic_y start_POSTSUBSCRIPT o end_POSTSUBSCRIPT 的变化(参见方程。(77)、(88) 和 (90))
δΔysubscript𝛿Δ𝑦\delta_{\!\Delta}yitalic_δ start_POSTSUBSCRIPT roman_Δ end_POSTSUBSCRIPT italic_y See Eq. (96)
见方程 (96
μ(=GMe)annotated𝜇absent𝐺subscript𝑀e\mu\,(=GM_{\text{e}})italic_μ ( = italic_G italic_M start_POSTSUBSCRIPT e end_POSTSUBSCRIPT ) Gravitational constant of the Earth
地球的引力常数
μ2(=GMs)annotatedsubscript𝜇2absent𝐺subscript𝑀s\mu_{2}\,(=GM_{\text{s}})italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( = italic_G italic_M start_POSTSUBSCRIPT s end_POSTSUBSCRIPT ) Gravitational constant of the Sun
太阳的引力常数
μ3(=GMm)annotatedsubscript𝜇3absent𝐺subscript𝑀m\mu_{3}\,(=GM_{\text{m}})italic_μ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( = italic_G italic_M start_POSTSUBSCRIPT m end_POSTSUBSCRIPT ) Gravitational constant of the Moon
月球的引力常数
Resubscript𝑅eR_{\text{e}}italic_R start_POSTSUBSCRIPT e end_POSTSUBSCRIPT Equatorial radius of the Earth
地球的赤道半径
J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT Second zonal harmonic coefficient
二次区域谐波系数
ϵitalic-ϵ\epsilonitalic_ϵ Obliquity of the ecliptic
黄道的倾角
U𝑈Uitalic_U Gravitational potential 重力势
U0subscript𝑈0U_{0}italic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT Central gravitational potential
中心引力势
\mathcal{R}caligraphic_R Perturbative potential 扰动电位
PN(x)subscript𝑃𝑁𝑥P_{N}(x)italic_P start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( italic_x ) Legendre polynomial of degree N𝑁Nitalic_N
度的勒让德多项式 N𝑁Nitalic_N
𝒩𝒩\mathcal{N}caligraphic_N Truncation degree 截断度
Rx(γ)subscript𝑅𝑥𝛾R_{x}(\gamma)italic_R start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_γ ), Rz(γ)subscript𝑅𝑧𝛾R_{z}(\gamma)italic_R start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_γ )
Rx(γ)subscript𝑅𝑥𝛾R_{x}(\gamma)italic_R start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_γ )Rz(γ)subscript𝑅𝑧𝛾R_{z}(\gamma)italic_R start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_γ )
Rotation matrices about the x𝑥xitalic_x and z𝑧zitalic_z axes by angle γ𝛾\gammaitalic_γ
按 角度 γ𝛾\gammaitalic_γx𝑥xitalic_xz𝑧zitalic_z 轴的旋转矩阵
f0subscript𝑓0f_{0}italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, f1subscript𝑓1f_{1}italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT
f0subscript𝑓0f_{0}italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPTf1subscript𝑓1f_{1}italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT
Zeroth and first order of the function f𝑓fitalic_f
函数 f𝑓fitalic_f 的零阶和一阶
διsubscript𝛿𝜄\delta_{\iota}italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT Takes values 0 or 1 (see Eqs. (27) and (71))
取值 0 或 1(参见 Eqs。(27) 和 (71))
r𝑟ritalic_r Geocentric satellite distance
地心卫星距离
𝐫𝐫\mathbf{r}bold_r Geocentric satellite position vector
地心卫星位置矢量
𝐫˙˙𝐫\mathbf{\dot{r}}over˙ start_ARG bold_r end_ARG Geocentric satellite velocity vector
地心卫星速度矢量
𝐫^^𝐫\mathbf{\hat{r}}over^ start_ARG bold_r end_ARG, 𝐫^𝟐subscript^𝐫2\mathbf{\hat{r}_{2}}over^ start_ARG bold_r end_ARG start_POSTSUBSCRIPT bold_2 end_POSTSUBSCRIPT, 𝐫^𝟑subscript^𝐫3\mathbf{\hat{r}_{3}}over^ start_ARG bold_r end_ARG start_POSTSUBSCRIPT bold_3 end_POSTSUBSCRIPT
𝐫^^𝐫\mathbf{\hat{r}}over^ start_ARG bold_r end_ARG𝐫^𝟐subscript^𝐫2\mathbf{\hat{r}_{2}}over^ start_ARG bold_r end_ARG start_POSTSUBSCRIPT bold_2 end_POSTSUBSCRIPT𝐫^𝟑subscript^𝐫3\mathbf{\hat{r}_{3}}over^ start_ARG bold_r end_ARG start_POSTSUBSCRIPT bold_3 end_POSTSUBSCRIPT
Unit vectors for satellite, the Sun, and the Moon, respectively
分别用于卫星、太阳和月亮的单位向量
ψ2,ψ3subscript𝜓2subscript𝜓3\psi_{2},\psi_{3}italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_ψ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Geocentric angles between satellite and the Sun, and the Moon
卫星与太阳和月球之间的地心角
φ𝜑\varphiitalic_φ Geocentric latitude in Earth-fixed coordinate system
地球固定坐标系中的地心纬度
r2subscript𝑟2r_{2}italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT Sun-Earth distance 日地距离
r¯2subscript¯𝑟2\overline{r}_{2}over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT Mean Sun-Earth distance 平均日地距离
u2subscriptu2\text{u}_{2}u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT Ecliptic longitude of the Sun
太阳的黄道经度
u2subscript𝑢2u_{2}italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT Mean longitude of the Sun
太阳的平均经度
U2subscript𝑈2U_{2}italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT Defined as (u2Ω¯)subscript𝑢2¯Ω(u_{2}-\overline{\Omega})( italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - over¯ start_ARG roman_Ω end_ARG ) 定义为 (u2Ω¯)subscript𝑢2¯Ω(u_{2}-\overline{\Omega})( italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - over¯ start_ARG roman_Ω end_ARG )
r3subscript𝑟3r_{3}italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Earth-Moon distance 地月距离
r¯3subscript¯𝑟3\overline{r}_{3}over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Mean Earth-Moon distance 平均地月距离
i¯3subscript¯𝑖3\overline{i}_{3}over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Mean inclination of the Moon’s orbit
月球轨道的平均倾角
Ω¯3subscript¯Ω3\overline{\Omega}_{3}over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Secular variation in the Moon’s longitude of ascending node
月球上升交点经度的长期变化
M¯3subscript¯𝑀3\overline{M}_{3}over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Secular variation in the Moon’s mean anomaly
月球平均距平的长期变化
u3subscriptu3\text{u}_{3}u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Latitude argument of the Moon
月球的纬度论点
u3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Secular variation in the Moon’s latitude argument
月球纬度论点的长期变化
Δu3Δsubscript𝑢3\Delta u_{3}roman_Δ italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Defined as (u3M¯3)subscript𝑢3subscript¯𝑀3(u_{3}-\overline{M}_{3})( italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )
定义为 (u3M¯3)subscript𝑢3subscript¯𝑀3(u_{3}-\overline{M}_{3})( italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )
θ3subscript𝜃3\theta_{3}italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Defined as (Ω¯Ω¯3)¯Ωsubscript¯Ω3(\overline{\Omega}-\overline{\Omega}_{3})( over¯ start_ARG roman_Ω end_ARG - over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )
定义为 (Ω¯Ω¯3)¯Ωsubscript¯Ω3(\overline{\Omega}-\overline{\Omega}_{3})( over¯ start_ARG roman_Ω end_ARG - over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )
n𝑛nitalic_n Mean motion of satellite with an orbit period of 3.64 days
轨道周期为3.64天的卫星平均运动
nΩsuperscriptsubscript𝑛Ωn_{\Omega}^{\prime}italic_n start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT Rate of change of ΩΩ\Omegaroman_Ω
变化 ΩΩ\Omegaroman_Ω
nλsuperscriptsubscript𝑛𝜆n_{\lambda}^{\prime}italic_n start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT Rate of change of λ𝜆\lambdaitalic_λ
变化 λ𝜆\lambdaitalic_λ
n2subscript𝑛2n_{2}italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT Mean motion of the Earth with a period of 365.2564 days
地球的平均运动周期为 365.2564 天
n3subscript𝑛3n_{3}italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Rate of change of u3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT with a period of 27.21 days (draconic month)
u3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 周期为 27.21 天的变化率 (draconic month)
nM3subscript𝑛subscript𝑀3n_{\!M_{3}}italic_n start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT Rate of change of M¯3subscript¯𝑀3\overline{M}_{3}over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT with a period of 27.55 days (anomalistic month)
周期为 27.55 天(异常月份)的变化 M¯3subscript¯𝑀3\overline{M}_{3}over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
nΩ3subscript𝑛subscriptΩ3n_{\Omega_{3}}italic_n start_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT Rate of change of Ω¯3subscript¯Ω3\overline{\Omega}_{3}over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT with a period of 18.6 years
18.6 年的变化 Ω¯3subscript¯Ω3\overline{\Omega}_{3}over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
Δn3(=n3nM3)annotatedΔsubscript𝑛3absentsubscript𝑛3subscript𝑛subscript𝑀3\Delta n_{3}\,(=n_{3}-n_{\!M_{3}})roman_Δ italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( = italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_n start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) Rate of change of Δu3Δsubscript𝑢3\Delta u_{3}roman_Δ italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT with a period of 6.0 years
6.0 年期间的变化 Δu3Δsubscript𝑢3\Delta u_{3}roman_Δ italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
nθ3(=nΩnΩ3)annotatedsubscript𝑛subscript𝜃3absentsuperscriptsubscript𝑛Ωsubscript𝑛subscriptΩ3n_{\theta_{3}}\,(=n_{\Omega}^{\prime}-n_{\Omega_{3}})italic_n start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( = italic_n start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_n start_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) Rate of change of θ3subscript𝜃3\theta_{3}italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
变化 θ3subscript𝜃3\theta_{3}italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
nU2(=n2nΩ)annotatedsubscript𝑛subscript𝑈2absentsubscript𝑛2superscriptsubscript𝑛Ωn_{U_{2}}\,(=n_{2}-n_{\Omega}^{\prime})italic_n start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( = italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_n start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) Rate of change of U2subscript𝑈2U_{2}italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT
变化 U2subscript𝑈2U_{2}italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT
Lijsubscript𝐿𝑖𝑗L_{ij}italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT Arm-length formed by SCi𝑖iitalic_i and SCj𝑗jitalic_j
由 SC i𝑖iitalic_i 和 SC j𝑗jitalic_j 形成的臂长
vijsubscript𝑣𝑖𝑗v_{ij}italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT Relative line-of-sight velocity (rate of change of Lijsubscript𝐿𝑖𝑗L_{ij}italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT)
相对视距速度 (变化率 Lijsubscript𝐿𝑖𝑗L_{ij}italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT
αksubscript𝛼𝑘\alpha_{k}italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT Breathing angle at SCk𝑘kitalic_k
SC k𝑘kitalic_k 处的呼吸角度
Losubscript𝐿oL_{\text{o}}italic_L start_POSTSUBSCRIPT o end_POSTSUBSCRIPT, vosubscript𝑣ov_{\text{o}}italic_v start_POSTSUBSCRIPT o end_POSTSUBSCRIPT, αosubscript𝛼o\alpha_{\text{o}}italic_α start_POSTSUBSCRIPT o end_POSTSUBSCRIPT
Losubscript𝐿oL_{\text{o}}italic_L start_POSTSUBSCRIPT o end_POSTSUBSCRIPTvosubscript𝑣ov_{\text{o}}italic_v start_POSTSUBSCRIPT o end_POSTSUBSCRIPTαosubscript𝛼o\alpha_{\text{o}}italic_α start_POSTSUBSCRIPT o end_POSTSUBSCRIPT
Nominal values of arm-length, relative velocity, and breathing angle
臂长、相对速度和呼吸角的标称值
f[q]σ(N)(i),f[p,q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁delimited-[]𝑞𝑖subscriptsuperscript𝑓𝜎𝑁𝑝𝑞𝑖f^{\sigma(N)}_{[q]}(i),f^{\sigma(N)}_{[p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ) , italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ) Inclination functions in the lunar perturbation solution
月球扰动解中的倾角函数

Appendix B Model derivation and verification
附录 B 模型推导和验证

B.1 Derivation and motivation of Eq. (54)
B.1 方程 (54) 的推导和动机

For lunar point-mass perturbation, the perturbative potential msubscriptm\mathcal{R}_{\text{m}}caligraphic_R start_POSTSUBSCRIPT m end_POSTSUBSCRIPT is represented as [35]
对于月球点质量扰动,扰动势 msubscriptm\mathcal{R}_{\text{m}}caligraphic_R start_POSTSUBSCRIPT m end_POSTSUBSCRIPT 表示为 [35]

msubscriptm\displaystyle\mathcal{R}_{\text{m}}caligraphic_R start_POSTSUBSCRIPT m end_POSTSUBSCRIPT =μ3(1|𝐫𝐫𝟑|𝐫𝟑r33𝐫)absentsubscript𝜇31𝐫subscript𝐫3subscript𝐫3superscriptsubscript𝑟33𝐫\displaystyle=\mu_{3}\left(\frac{1}{|\mathbf{r}-\mathbf{r_{3}}|}-\frac{\mathbf% {r_{3}}}{r_{3}^{3}}\cdot\mathbf{r}\right)= italic_μ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG | bold_r - bold_r start_POSTSUBSCRIPT bold_3 end_POSTSUBSCRIPT | end_ARG - divide start_ARG bold_r start_POSTSUBSCRIPT bold_3 end_POSTSUBSCRIPT end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG ⋅ bold_r ) (116)
=μ3(1r22rr3cosψ3+r32rr32cosψ3),absentsubscript𝜇31superscript𝑟22𝑟subscript𝑟3subscript𝜓3superscriptsubscript𝑟32𝑟superscriptsubscript𝑟32subscript𝜓3\displaystyle=\mu_{3}\left(\frac{1}{\sqrt{{r}^{2}-2\,r\,r_{3}\cos\psi_{3}+r_{3% }^{2}}}-\frac{r}{r_{3}^{2}}\cos\psi_{3}\right),= italic_μ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_r italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_cos italic_ψ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG - divide start_ARG italic_r end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG roman_cos italic_ψ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (117)

where r=a(1e2)1+ecosν𝑟𝑎1superscript𝑒21𝑒𝜈r=\frac{a(1-e^{2})}{1+e\cos\nu}italic_r = divide start_ARG italic_a ( 1 - italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 1 + italic_e roman_cos italic_ν end_ARG, r3=a3(1e32)1+e3cosν3subscript𝑟3subscript𝑎31superscriptsubscript𝑒321subscript𝑒3subscript𝜈3r_{3}=\frac{a_{3}(1-e_{3}^{2})}{1+e_{3}\cos\nu_{3}}italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = divide start_ARG italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( 1 - italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 1 + italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_cos italic_ν start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG, and cosψ3=𝐫^𝟑𝐫^subscript𝜓3subscript^𝐫3^𝐫\cos\psi_{3}=\mathbf{\hat{r}_{3}}\cdot\mathbf{\hat{r}}roman_cos italic_ψ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = over^ start_ARG bold_r end_ARG start_POSTSUBSCRIPT bold_3 end_POSTSUBSCRIPT ⋅ over^ start_ARG bold_r end_ARG. msubscriptm\mathcal{R}_{\text{m}}caligraphic_R start_POSTSUBSCRIPT m end_POSTSUBSCRIPT, described by Eq. (117), exhibits a square root form, introducing challenges in solving the Lagrange perturbation equations. This complexity can be circumvented by expressing 1|𝐫𝐫3|1𝐫subscript𝐫3\frac{1}{|\mathbf{r}-\mathbf{r}_{3}|}divide start_ARG 1 end_ARG start_ARG | bold_r - bold_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | end_ARG as an expansion of Legendre polynomials:
其中 r=a(1e2)1+ecosν𝑟𝑎1superscript𝑒21𝑒𝜈r=\frac{a(1-e^{2})}{1+e\cos\nu}italic_r = divide start_ARG italic_a ( 1 - italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 1 + italic_e roman_cos italic_ν end_ARGr3=a3(1e32)1+e3cosν3subscript𝑟3subscript𝑎31superscriptsubscript𝑒321subscript𝑒3subscript𝜈3r_{3}=\frac{a_{3}(1-e_{3}^{2})}{1+e_{3}\cos\nu_{3}}italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = divide start_ARG italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( 1 - italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 1 + italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_cos italic_ν start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG , 和 cosψ3=𝐫^𝟑𝐫^subscript𝜓3subscript^𝐫3^𝐫\cos\psi_{3}=\mathbf{\hat{r}_{3}}\cdot\mathbf{\hat{r}}roman_cos italic_ψ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = over^ start_ARG bold_r end_ARG start_POSTSUBSCRIPT bold_3 end_POSTSUBSCRIPT ⋅ over^ start_ARG bold_r end_ARG . msubscriptm\mathcal{R}_{\text{m}}caligraphic_R start_POSTSUBSCRIPT m end_POSTSUBSCRIPT ,由方程(117)描述,表现出平方根形式,给求解拉格朗日微扰方程带来了挑战。这种复杂性可以通过表示 1|𝐫𝐫3|1𝐫subscript𝐫3\frac{1}{|\mathbf{r}-\mathbf{r}_{3}|}divide start_ARG 1 end_ARG start_ARG | bold_r - bold_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | end_ARG 为勒让德多项式的扩展来规避:

1|𝐫𝐫𝟑|1𝐫subscript𝐫3\displaystyle\frac{1}{|\mathbf{r}-\mathbf{r_{3}}|}divide start_ARG 1 end_ARG start_ARG | bold_r - bold_r start_POSTSUBSCRIPT bold_3 end_POSTSUBSCRIPT | end_ARG =1r3[12(rr3)cosψ3+(rr3)2]1/2absent1subscript𝑟3superscriptdelimited-[]12𝑟subscript𝑟3subscript𝜓3superscript𝑟subscript𝑟3212\displaystyle=\frac{1}{r_{3}}\left[1-2\left(\frac{r}{r_{3}}\right)\cos\psi_{3}% +\left(\frac{r}{r_{3}}\right)^{2}\right]^{-1/2}= divide start_ARG 1 end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG [ 1 - 2 ( divide start_ARG italic_r end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ) roman_cos italic_ψ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( divide start_ARG italic_r end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT (118)
=1r3N=0(rr3)NPN(cosψ3).absent1subscript𝑟3superscriptsubscript𝑁0superscript𝑟subscript𝑟3𝑁subscript𝑃𝑁subscript𝜓3\displaystyle=\frac{1}{r_{3}}\sum_{N=0}^{\infty}\left(\frac{r}{r_{3}}\right)^{% N}P_{N}(\cos\psi_{3}).= divide start_ARG 1 end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_N = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( divide start_ARG italic_r end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( roman_cos italic_ψ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) . (119)

Further substituting Eq. (119) into Eq. (117) and removing the first term 1r31subscript𝑟3\frac{1}{r_{3}}divide start_ARG 1 end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG (as σ(1r3)=0𝜎1subscript𝑟30\frac{\partial}{\partial\sigma}(\frac{1}{r_{3}})=0divide start_ARG ∂ end_ARG start_ARG ∂ italic_σ end_ARG ( divide start_ARG 1 end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ) = 0 after substitution into Lagrange’s equations) yields
进一步将方程(119)代入方程(117)并删除第一项 1r31subscript𝑟3\frac{1}{r_{3}}divide start_ARG 1 end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG (如 σ(1r3)=0𝜎1subscript𝑟30\frac{\partial}{\partial\sigma}(\frac{1}{r_{3}})=0divide start_ARG ∂ end_ARG start_ARG ∂ italic_σ end_ARG ( divide start_ARG 1 end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ) = 0 代入拉格朗日方程后)得到

m=μ3r3N=2(rr3)NPN(cosψ3).subscriptmsubscript𝜇3subscript𝑟3superscriptsubscript𝑁2superscript𝑟subscript𝑟3𝑁subscript𝑃𝑁subscript𝜓3\displaystyle\mathcal{R}_{\text{m}}=\frac{\mu_{3}}{r_{3}}\sum_{N=2}^{\infty}% \left(\frac{r}{r_{3}}\right)^{N}P_{N}(\cos\psi_{3}).caligraphic_R start_POSTSUBSCRIPT m end_POSTSUBSCRIPT = divide start_ARG italic_μ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_N = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( divide start_ARG italic_r end_ARG start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( roman_cos italic_ψ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) . (120)

This formulation proves more advantageous for solving the Lagrange equations than Eq. (117). Based on estimated magnitudes and validation through numerical simulations, we set the maximum degree of Legendre polynomials in msubscriptm\mathcal{R}_{\text{m}}caligraphic_R start_POSTSUBSCRIPT m end_POSTSUBSCRIPT at N=6𝑁6N=6italic_N = 6. Additionally, the solar potential ssubscripts\mathcal{R}_{\text{s}}caligraphic_R start_POSTSUBSCRIPT s end_POSTSUBSCRIPT resembles Eq. (120), with the maximum degree set at N=2𝑁2N=2italic_N = 2.
事实证明,这个公式比方程 (117) 更有利于求解拉格朗日方程。根据估计的大小和通过数值模拟的验证,我们将勒让德多项式的最大次数设置为 msubscriptm\mathcal{R}_{\text{m}}caligraphic_R start_POSTSUBSCRIPT m end_POSTSUBSCRIPTN=6𝑁6N=6italic_N = 6 此外,太阳势 ssubscripts\mathcal{R}_{\text{s}}caligraphic_R start_POSTSUBSCRIPT s end_POSTSUBSCRIPT 类似于方程(120),最大度数设置为 N=2𝑁2N=2italic_N = 2

B.2 Derivation of Eqs. (73)-(75)
乙.2 方程的推导。(73)-(75

The analytical expressions for Eqs. (73)–(75) can be derived by applying perturbation methods to solve Eq. (70). To enhance the accuracy of the analytical solution, it is more advantageous to use the mean orbital elements σ¯(t)¯𝜎𝑡\overline{\sigma}(t)over¯ start_ARG italic_σ end_ARG ( italic_t ) [34, 35], corresponding to a long-term precessing elliptical orbit, rather than the Keplerian orbit σ(0)(t)superscript𝜎0𝑡\sigma^{(0)}(t)italic_σ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_t ) as the reference solution. Consequently, the perturbation solution’s form (73) is reformulated as
Eqs 的解析表达式。(73)–(75) 可以通过应用微扰方法求解方程 (70) 来推导。为了提高解析解的准确性,使用对应于长期进动椭圆轨道的平均轨道单元 σ¯(t)¯𝜎𝑡\overline{\sigma}(t)over¯ start_ARG italic_σ end_ARG ( italic_t ) [3435] 而不是开普勒轨道 σ(0)(t)superscript𝜎0𝑡\sigma^{(0)}(t)italic_σ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_t ) 作为参考解更有利。因此,扰动解的形式 (73) 被重新表述为

σ(t)=σ¯(t)+[σl[l](t)+σs(t)]𝜎𝑡¯𝜎𝑡delimited-[]subscript𝜎𝑙delimited-[]𝑙𝑡subscript𝜎𝑠𝑡\displaystyle\sigma(t)=\overline{\sigma}(t)+[\sigma_{l[l]}(t)+\sigma_{s}(t)]italic_σ ( italic_t ) = over¯ start_ARG italic_σ end_ARG ( italic_t ) + [ italic_σ start_POSTSUBSCRIPT italic_l [ italic_l ] end_POSTSUBSCRIPT ( italic_t ) + italic_σ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_t ) ] (121)

with 

σ¯(t)=σ¯(0)(t)+Δσc(t)+Δσl[c](t),¯𝜎𝑡superscript¯𝜎0𝑡Δsubscript𝜎𝑐𝑡Δsubscript𝜎𝑙delimited-[]𝑐𝑡\displaystyle\overline{\sigma}(t)=\overline{\sigma}^{(0)}(t)+\Delta\sigma_{c}(% t)+\Delta\sigma_{l[c]}(t),over¯ start_ARG italic_σ end_ARG ( italic_t ) = over¯ start_ARG italic_σ end_ARG start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_t ) + roman_Δ italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t ) + roman_Δ italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) , (122)

and 

σ¯(0)(t)=σ¯0+διn¯(tt0),superscript¯𝜎0𝑡subscript¯𝜎0subscript𝛿𝜄¯𝑛𝑡subscript𝑡0\displaystyle\overline{\sigma}^{(0)}(t)=\overline{\sigma}_{0}+\delta_{\iota}% \overline{n}(t-t_{0}),over¯ start_ARG italic_σ end_ARG start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_t ) = over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT over¯ start_ARG italic_n end_ARG ( italic_t - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , (123)
σ¯0=σ0[σl[l](t0)+σs(t0)],subscript¯𝜎0subscript𝜎0delimited-[]subscript𝜎𝑙delimited-[]𝑙subscript𝑡0subscript𝜎𝑠subscript𝑡0\displaystyle\overline{\sigma}_{0}=\sigma_{0}-[\sigma_{l[l]}(t_{0})+\sigma_{s}% (t_{0})],over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - [ italic_σ start_POSTSUBSCRIPT italic_l [ italic_l ] end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_σ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ] , (124)

where σ¯(0)superscript¯𝜎0\overline{\sigma}^{(0)}over¯ start_ARG italic_σ end_ARG start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT represents the unperturbed secular variations, σ¯0subscript¯𝜎0\overline{\sigma}_{0}over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the initial mean elements, Δσc(t):=σc(t)σc(t0)assignΔsubscript𝜎𝑐𝑡subscript𝜎𝑐𝑡subscript𝜎𝑐subscript𝑡0\Delta\sigma_{c}(t):=\sigma_{c}(t)-\sigma_{c}(t_{0})roman_Δ italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t ) := italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t ) - italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ), and Δσl[c](t):=σl[c](t)σl[c](t0)assignΔsubscript𝜎𝑙delimited-[]𝑐𝑡subscript𝜎𝑙delimited-[]𝑐𝑡subscript𝜎𝑙delimited-[]𝑐subscript𝑡0\Delta\sigma_{l[c]}(t):=\sigma_{l[c]}(t)-\sigma_{l[c]}(t_{0})roman_Δ italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) := italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) - italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ). Notably, σl[c](t)subscript𝜎𝑙delimited-[]𝑐𝑡\sigma_{l[c]}(t)italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) is incorporated into σ¯(t)¯𝜎𝑡\overline{\sigma}(t)over¯ start_ARG italic_σ end_ARG ( italic_t ), considering its short-term behavior akin to secular variation.
其中 σ¯(0)superscript¯𝜎0\overline{\sigma}^{(0)}over¯ start_ARG italic_σ end_ARG start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT 表示未受扰动的长期变化, σ¯0subscript¯𝜎0\overline{\sigma}_{0}over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT 是初始均值元素 Δσc(t):=σc(t)σc(t0)assignΔsubscript𝜎𝑐𝑡subscript𝜎𝑐𝑡subscript𝜎𝑐subscript𝑡0\Delta\sigma_{c}(t):=\sigma_{c}(t)-\sigma_{c}(t_{0})roman_Δ italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t ) := italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t ) - italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ,和 Δσl[c](t):=σl[c](t)σl[c](t0)assignΔsubscript𝜎𝑙delimited-[]𝑐𝑡subscript𝜎𝑙delimited-[]𝑐𝑡subscript𝜎𝑙delimited-[]𝑐subscript𝑡0\Delta\sigma_{l[c]}(t):=\sigma_{l[c]}(t)-\sigma_{l[c]}(t_{0})roman_Δ italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) := italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) - italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) 。值得注意的是, σl[c](t)subscript𝜎𝑙delimited-[]𝑐𝑡\sigma_{l[c]}(t)italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) 考虑到其类似于长期变化的短期行为,它被纳入 σ¯(t)¯𝜎𝑡\overline{\sigma}(t)over¯ start_ARG italic_σ end_ARG ( italic_t )

In relation to the left-side partitioning of Eq. (70) concerning σ(t)𝜎𝑡\sigma(t)italic_σ ( italic_t ), the function f1subscript𝑓1f_{1}italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT on the right is similarly decomposed into
关于方程(70)的左侧划分 σ(t)𝜎𝑡\sigma(t)italic_σ ( italic_t ) ,右侧的函数 f1subscript𝑓1f_{1}italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT 同样被分解为

f1=f1c+f1l[c]+f1l[l]+f1s.subscript𝑓1subscript𝑓1𝑐subscript𝑓1𝑙delimited-[]𝑐subscript𝑓1𝑙delimited-[]𝑙subscript𝑓1𝑠\displaystyle f_{1}=f_{1c}+f_{1l[c]}+f_{1l[l]}+f_{1s}.italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT 1 italic_c end_POSTSUBSCRIPT + italic_f start_POSTSUBSCRIPT 1 italic_l [ italic_c ] end_POSTSUBSCRIPT + italic_f start_POSTSUBSCRIPT 1 italic_l [ italic_l ] end_POSTSUBSCRIPT + italic_f start_POSTSUBSCRIPT 1 italic_s end_POSTSUBSCRIPT . (125)

f1csubscript𝑓1𝑐f_{1c}italic_f start_POSTSUBSCRIPT 1 italic_c end_POSTSUBSCRIPT depends solely on a¯¯𝑎\overline{a}over¯ start_ARG italic_a end_ARG, ξ¯¯𝜉\overline{\xi}over¯ start_ARG italic_ξ end_ARG, η¯¯𝜂\overline{\eta}over¯ start_ARG italic_η end_ARG, and i¯¯𝑖\overline{i}over¯ start_ARG italic_i end_ARG. Both f1l[c]subscript𝑓1𝑙delimited-[]𝑐f_{1l[c]}italic_f start_POSTSUBSCRIPT 1 italic_l [ italic_c ] end_POSTSUBSCRIPT and f1l[l]subscript𝑓1𝑙delimited-[]𝑙f_{1l[l]}italic_f start_POSTSUBSCRIPT 1 italic_l [ italic_l ] end_POSTSUBSCRIPT involve trigonometric functions with arguments related to slow variables, such as Ω3subscriptΩ3\Omega_{3}roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT with an 18.6-year period and u3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT with a 27.21-day period, while f1ssubscript𝑓1𝑠f_{1s}italic_f start_POSTSUBSCRIPT 1 italic_s end_POSTSUBSCRIPT incorporates the fast variable λ𝜆\lambdaitalic_λ, which has a 3.64-day period, as the argument. The decomposition in Eq. (125) is achieved through averaging [53, 47, 16], where, for instance, f1ssubscript𝑓1𝑠f_{1s}italic_f start_POSTSUBSCRIPT 1 italic_s end_POSTSUBSCRIPT is obtained via
f1csubscript𝑓1𝑐f_{1c}italic_f start_POSTSUBSCRIPT 1 italic_c end_POSTSUBSCRIPT 仅依赖于 a¯¯𝑎\overline{a}over¯ start_ARG italic_a end_ARGξ¯¯𝜉\overline{\xi}over¯ start_ARG italic_ξ end_ARGη¯¯𝜂\overline{\eta}over¯ start_ARG italic_η end_ARGi¯¯𝑖\overline{i}over¯ start_ARG italic_i end_ARG 。和 f1l[l]subscript𝑓1𝑙delimited-[]𝑙f_{1l[l]}italic_f start_POSTSUBSCRIPT 1 italic_l [ italic_l ] end_POSTSUBSCRIPTf1l[c]subscript𝑓1𝑙delimited-[]𝑐f_{1l[c]}italic_f start_POSTSUBSCRIPT 1 italic_l [ italic_c ] end_POSTSUBSCRIPT 涉及三角函数,其参数与慢变量相关,例如 Ω3subscriptΩ3\Omega_{3}roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 18.6 年周期和 u3subscript𝑢3u_{3}italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 27.21 天周期,同时 f1ssubscript𝑓1𝑠f_{1s}italic_f start_POSTSUBSCRIPT 1 italic_s end_POSTSUBSCRIPT 包含快速变量 λ𝜆\lambdaitalic_λ ,其周期为 3.64 天,作为参数。方程(125)中的分解是通过平均[534716]实现的,例如 f1ssubscript𝑓1𝑠f_{1s}italic_f start_POSTSUBSCRIPT 1 italic_s end_POSTSUBSCRIPT ,通过求法得到

f1s=f1f1λ,f1λ:=12π02πf1𝑑λ.formulae-sequencesubscript𝑓1𝑠subscript𝑓1subscriptdelimited-⟨⟩subscript𝑓1𝜆assignsubscriptdelimited-⟨⟩subscript𝑓1𝜆12𝜋superscriptsubscript02𝜋subscript𝑓1differential-d𝜆\displaystyle f_{1s}=f_{1}-\left\langle f_{1}\right\rangle_{\lambda},\qquad% \left\langle f_{1}\right\rangle_{\lambda}:=\frac{1}{2\pi}\int_{0}^{2\pi}f_{1}% \,d\lambda.italic_f start_POSTSUBSCRIPT 1 italic_s end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - ⟨ italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT , ⟨ italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT := divide start_ARG 1 end_ARG start_ARG 2 italic_π end_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 italic_π end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_d italic_λ . (126)

A similar averaging over slow variables is applied to derive f1csubscript𝑓1𝑐f_{1c}italic_f start_POSTSUBSCRIPT 1 italic_c end_POSTSUBSCRIPT, fl[c]subscript𝑓𝑙delimited-[]𝑐f_{l[c]}italic_f start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT, and fl[l]subscript𝑓𝑙delimited-[]𝑙f_{l[l]}italic_f start_POSTSUBSCRIPT italic_l [ italic_l ] end_POSTSUBSCRIPT. Moreover, given that the TianQin orbits are nearly circular with e¯0.0005similar-to-or-equals¯𝑒0.0005\overline{e}\simeq 0.0005over¯ start_ARG italic_e end_ARG ≃ 0.0005 [24], the terms on the right side of Eq. (125) consider only the leading-order effects of eccentricity for simplicity.
对慢速变量进行类似的平均应用于派生 f1csubscript𝑓1𝑐f_{1c}italic_f start_POSTSUBSCRIPT 1 italic_c end_POSTSUBSCRIPTfl[c]subscript𝑓𝑙delimited-[]𝑐f_{l[c]}italic_f start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPTfl[l]subscript𝑓𝑙delimited-[]𝑙f_{l[l]}italic_f start_POSTSUBSCRIPT italic_l [ italic_l ] end_POSTSUBSCRIPT 。此外,鉴于天琴轨道与 [24] 几乎是圆形 e¯0.0005similar-to-or-equals¯𝑒0.0005\overline{e}\simeq 0.0005over¯ start_ARG italic_e end_ARG ≃ 0.0005 的,为简单起见,方程(125)右侧的项只考虑了偏心率的超前序效应。

By inserting the formal solution (121) into both sides of Eq. (70) and conducting a Taylor expansion around σ¯(t)¯𝜎𝑡\overline{\sigma}(t)over¯ start_ARG italic_σ end_ARG ( italic_t ), the comparison of coefficients for the same powers (ε0superscript𝜀0\varepsilon^{0}italic_ε start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, ε1superscript𝜀1\varepsilon^{1}italic_ε start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT, ε2superscript𝜀2\varepsilon^{2}italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, \cdots) yields [35]
通过将形式解 (121) 代入方程 (70) 的两侧并围绕 σ¯(t)¯𝜎𝑡\overline{\sigma}(t)over¯ start_ARG italic_σ end_ARG ( italic_t ) 进行泰勒展开,相同幂 ( ε0superscript𝜀0\varepsilon^{0}italic_ε start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPTε1superscript𝜀1\varepsilon^{1}italic_ε start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPTε2superscript𝜀2\varepsilon^{2}italic_ε start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT \cdots , ) 的系数比较得到 [35]

σ¯(0)(t)=t0t[f0]a¯𝑑t=σ¯0+διn¯(tt0),superscript¯𝜎0𝑡subscriptsuperscript𝑡subscript𝑡0subscriptdelimited-[]subscript𝑓0¯𝑎differential-d𝑡subscript¯𝜎0subscript𝛿𝜄¯𝑛𝑡subscript𝑡0\displaystyle\overline{\sigma}^{(0)}(t)=\int^{t}_{t_{0}}[f_{0}]_{\overline{a}}% \,dt=\overline{\sigma}_{0}+\delta_{\iota}\overline{n}(t-t_{0}),over¯ start_ARG italic_σ end_ARG start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ( italic_t ) = ∫ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT over¯ start_ARG italic_a end_ARG end_POSTSUBSCRIPT italic_d italic_t = over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT over¯ start_ARG italic_n end_ARG ( italic_t - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , (127)
σc(1)(t)=t[f1c]σ¯𝑑t,superscriptsubscript𝜎𝑐1𝑡superscript𝑡subscriptdelimited-[]subscript𝑓1𝑐¯𝜎differential-d𝑡\displaystyle\sigma_{c}^{(1)}(t)=\int^{t}[f_{1c}]_{\overline{\sigma}}\,dt,italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) = ∫ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT [ italic_f start_POSTSUBSCRIPT 1 italic_c end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT over¯ start_ARG italic_σ end_ARG end_POSTSUBSCRIPT italic_d italic_t , (128)
σl[c](1)(t)=t[f1l[c]]σ¯𝑑t,superscriptsubscript𝜎𝑙delimited-[]𝑐1𝑡superscript𝑡subscriptdelimited-[]subscript𝑓1𝑙delimited-[]𝑐¯𝜎differential-d𝑡\displaystyle\sigma_{l[c]}^{(1)}(t)=\int^{t}[f_{1l[c]}]_{\overline{\sigma}}\,dt,italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) = ∫ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT [ italic_f start_POSTSUBSCRIPT 1 italic_l [ italic_c ] end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT over¯ start_ARG italic_σ end_ARG end_POSTSUBSCRIPT italic_d italic_t , (129)
σl[l](1)(t)=t[f1l[l]]σ¯𝑑t,superscriptsubscript𝜎𝑙delimited-[]𝑙1𝑡superscript𝑡subscriptdelimited-[]subscript𝑓1𝑙delimited-[]𝑙¯𝜎differential-d𝑡\displaystyle\sigma_{l[l]}^{(1)}(t)=\int^{t}[f_{1l[l]}]_{\overline{\sigma}}\,dt,italic_σ start_POSTSUBSCRIPT italic_l [ italic_l ] end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) = ∫ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT [ italic_f start_POSTSUBSCRIPT 1 italic_l [ italic_l ] end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT over¯ start_ARG italic_σ end_ARG end_POSTSUBSCRIPT italic_d italic_t , (130)
σs(1)(t)=t[f0aas(1)+f1s]σ¯𝑑t,superscriptsubscript𝜎𝑠1𝑡superscript𝑡subscriptdelimited-[]subscript𝑓0𝑎superscriptsubscript𝑎𝑠1subscript𝑓1𝑠¯𝜎differential-d𝑡\displaystyle\sigma_{s}^{(1)}(t)=\int^{t}[\frac{\partial f_{0}}{\partial a}a_{% s}^{(1)}+f_{1s}]_{\overline{\sigma}}\,dt,italic_σ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) = ∫ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT [ divide start_ARG ∂ italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_a end_ARG italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT + italic_f start_POSTSUBSCRIPT 1 italic_s end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT over¯ start_ARG italic_σ end_ARG end_POSTSUBSCRIPT italic_d italic_t , (131)
σc(2)(t)=t[122f0a2[as(1)]c2\displaystyle\sigma_{c}^{(2)}(t)=\int^{t}\biggl{[}\frac{1}{2}\frac{\partial^{2% }\!f_{0}}{\partial a^{2}}[a_{s}^{(1)}]^{2}_{c}italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ( italic_t ) = ∫ start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT [ divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG [ italic_a start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT
+(j=16f1σj[σs(1)+σl[l](1)]j)c]σ¯dt,\displaystyle\qquad\quad\;\;\,\,+\biggl{(}\sum_{j=1}^{6}\frac{\partial f_{1}}{% \partial\sigma_{j}}[\sigma_{s}^{(1)}+\sigma_{l[l]}^{(1)}]_{j}\biggr{)}_{\!\!c}% \biggr{]}_{\!\overline{\sigma}}\,dt,+ ( ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT divide start_ARG ∂ italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG [ italic_σ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT + italic_σ start_POSTSUBSCRIPT italic_l [ italic_l ] end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ] start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT over¯ start_ARG italic_σ end_ARG end_POSTSUBSCRIPT italic_d italic_t , (132)
.\displaystyle\cdots.⋯ .

The superscript in parentheses denotes the order of the perturbation solution; in this paper, we focus on the first-order solution. Utilizing Eqs. (128)-(131), we derive explicit expressions for the four components of σ(1)(t)superscript𝜎1𝑡\sigma^{(1)}(t)italic_σ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ), presented in Appendix C. Particularly, for ξcsubscript𝜉𝑐\xi_{c}italic_ξ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, ξl[c]subscript𝜉𝑙delimited-[]𝑐\xi_{l[c]}italic_ξ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT, ηcsubscript𝜂𝑐\eta_{c}italic_η start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, and ηl[c]subscript𝜂𝑙delimited-[]𝑐\eta_{l[c]}italic_η start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT, it is more reasonable to directly solve the oscillation equations they satisfy (see Eqs. (191) and (192)) [35]; detailed derivations are provided in Appendix C.4.
括号中的上标表示扰动解的顺序;在本文中,我们重点介绍一阶解。利用方程。(128)-(131) 中,我们推导出了附录 C 中所示的 σ(1)(t)superscript𝜎1𝑡\sigma^{(1)}(t)italic_σ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) 的四个组成部分的显式表达式。特别是,对于 ξcsubscript𝜉𝑐\xi_{c}italic_ξ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT 、 、 ξl[c]subscript𝜉𝑙delimited-[]𝑐\xi_{l[c]}italic_ξ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ηcsubscript𝜂𝑐\eta_{c}italic_η start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPTηl[c]subscript𝜂𝑙delimited-[]𝑐\eta_{l[c]}italic_η start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ,直接求解它们满足的振荡方程更合理(参见 Eqs。(191) 和 (192)) [35];附录 C.4 提供了详细的推导。

Note that in Eqs. (127)-(132), σ𝜎\sigmaitalic_σ on the right-hand side all take the form σ¯(t)¯𝜎𝑡\overline{\sigma}(t)over¯ start_ARG italic_σ end_ARG ( italic_t ) defined in Eq. (122). For i𝑖iitalic_i, ΩΩ\Omegaroman_Ω, and λ𝜆\lambdaitalic_λ, they are embedded in the trigonometric functions of f1subscript𝑓1f_{1}italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. To enable integrable solutions, σl[c](t)subscript𝜎𝑙delimited-[]𝑐𝑡\sigma_{l[c]}(t)italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) in σ¯(t)¯𝜎𝑡\overline{\sigma}(t)over¯ start_ARG italic_σ end_ARG ( italic_t ) is approximated as a linear term with a rate of change n~σsubscript~𝑛𝜎\widetilde{n}_{\sigma}over~ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT,
请注意,在 Eqs.(127)-(132), σ𝜎\sigmaitalic_σ 右侧都采用方程 (122) 中定义的形式 σ¯(t)¯𝜎𝑡\overline{\sigma}(t)over¯ start_ARG italic_σ end_ARG ( italic_t ) 。对于 i𝑖iitalic_iΩΩ\Omegaroman_Ωλ𝜆\lambdaitalic_λ ,它们嵌入在 的 f1subscript𝑓1f_{1}italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT 三角函数中。为了实现可积解, σl[c](t)subscript𝜎𝑙delimited-[]𝑐𝑡\sigma_{l[c]}(t)italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) in σ¯(t)¯𝜎𝑡\overline{\sigma}(t)over¯ start_ARG italic_σ end_ARG ( italic_t ) 近似为具有变化 n~σsubscript~𝑛𝜎\widetilde{n}_{\sigma}over~ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT 率 的线性项 ,

n~σ=Δσl[c](t)ττ,f(t)τ:=1τt0t0+τf(t)𝑑t,formulae-sequencesubscript~𝑛𝜎subscriptdelimited-⟨⟩Δsubscript𝜎𝑙delimited-[]𝑐𝑡𝜏𝜏assignsubscriptdelimited-⟨⟩𝑓𝑡𝜏1𝜏superscriptsubscriptsubscript𝑡0subscript𝑡0𝜏𝑓𝑡differential-d𝑡\displaystyle\widetilde{n}_{\sigma}=\frac{\left\langle\Delta\sigma_{l[c]}(t)% \right\rangle_{\tau}}{\tau},\qquad\left\langle f(t)\right\rangle_{\tau}:=\frac% {1}{\tau}\int_{t_{0}}^{t_{0}+\tau}f(t)\,dt,over~ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT = divide start_ARG ⟨ roman_Δ italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) ⟩ start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT end_ARG start_ARG italic_τ end_ARG , ⟨ italic_f ( italic_t ) ⟩ start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT := divide start_ARG 1 end_ARG start_ARG italic_τ end_ARG ∫ start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_τ end_POSTSUPERSCRIPT italic_f ( italic_t ) italic_d italic_t , (133)

where τ𝜏\tauitalic_τ denotes the duration, implying
其中 τ𝜏\tauitalic_τ 表示持续时间,表示

σ¯(t){σ¯0+nσ(tt0),σ{Ω,λ},i(t)τ,σ=i,similar-to-or-equals¯𝜎𝑡casessubscript¯𝜎0superscriptsubscript𝑛𝜎𝑡subscript𝑡0𝜎Ω𝜆subscriptdelimited-⟨⟩𝑖𝑡𝜏𝜎𝑖\overline{\sigma}(t)\simeq\left\{\begin{array}[]{ll}\overline{\sigma}_{0}+n_{% \sigma}^{\prime}(t-t_{0}),&\sigma\in\{\Omega,\lambda\},\vskip 3.0pt plus 1.0pt% minus 1.0pt\\ \left\langle i(t)\right\rangle_{\tau},&\sigma=i,\\ \end{array}\right.over¯ start_ARG italic_σ end_ARG ( italic_t ) ≃ { start_ARRAY start_ROW start_CELL over¯ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , end_CELL start_CELL italic_σ ∈ { roman_Ω , italic_λ } , end_CELL end_ROW start_ROW start_CELL ⟨ italic_i ( italic_t ) ⟩ start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT , end_CELL start_CELL italic_σ = italic_i , end_CELL end_ROW end_ARRAY (134)

with 

nσ:=διn¯+nσ+n~σ.assignsuperscriptsubscript𝑛𝜎subscript𝛿𝜄¯𝑛subscript𝑛𝜎subscript~𝑛𝜎\displaystyle n_{\sigma}^{\prime}:=\delta_{\iota}\overline{n}+n_{\sigma}+% \widetilde{n}_{\sigma}.italic_n start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := italic_δ start_POSTSUBSCRIPT italic_ι end_POSTSUBSCRIPT over¯ start_ARG italic_n end_ARG + italic_n start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT + over~ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT . (135)

Here, nσsubscript𝑛𝜎n_{\sigma}italic_n start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT represents the rate of change for σc(t)subscript𝜎𝑐𝑡\sigma_{c}(t)italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t ), given by nσ=nσs+nσm+nσJ2subscript𝑛𝜎subscript𝑛𝜎ssubscript𝑛𝜎msubscript𝑛subscript𝜎subscript𝐽2n_{\sigma}=n_{\sigma\text{s}}+n_{\sigma\text{m}}+n_{\sigma\!_{J_{2}}}italic_n start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT italic_σ s end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT italic_σ m end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT. Additionally, i¯(t)¯𝑖𝑡\overline{i}(t)over¯ start_ARG italic_i end_ARG ( italic_t ) is approximated as the mean value i(t)τsubscriptdelimited-⟨⟩𝑖𝑡𝜏\left\langle i(t)\right\rangle_{\tau}⟨ italic_i ( italic_t ) ⟩ start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT, due to its small secular variation.
其中, nσsubscript𝑛𝜎n_{\sigma}italic_n start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT 表示 σc(t)subscript𝜎𝑐𝑡\sigma_{c}(t)italic_σ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t ) 的变化率 ,由 给出。 nσ=nσs+nσm+nσJ2subscript𝑛𝜎subscript𝑛𝜎ssubscript𝑛𝜎msubscript𝑛subscript𝜎subscript𝐽2n_{\sigma}=n_{\sigma\text{s}}+n_{\sigma\text{m}}+n_{\sigma\!_{J_{2}}}italic_n start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT italic_σ s end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT italic_σ m end_POSTSUBSCRIPT + italic_n start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT 此外, 由于 其长期变化较小, i¯(t)¯𝑖𝑡\overline{i}(t)over¯ start_ARG italic_i end_ARG ( italic_t ) 因此近似为 平均值 i(t)τsubscriptdelimited-⟨⟩𝑖𝑡𝜏\left\langle i(t)\right\rangle_{\tau}⟨ italic_i ( italic_t ) ⟩ start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT

B.3 Verification of Eqs. (73)-(75)
乙.3 方程验证。(73)-(75

To validate the derived analytical solution for satellite orbits, we conduct high-precision numerical orbit simulations using the NASA General Mission Analysis Tool (GMAT) [54]. The force models, consistent with those in Ref. [24], include the point-mass gravity fields of the Moon, Sun, and solar system planets (the ephemeris DE421), a 10×10101010\times 1010 × 10 spherical-harmonic model of the Earth’s gravity field (JGM-3), and the first-order relativistic correction. Non-gravitational perturbations, such as solar radiation pressure, are omitted as the satellites are drag-free controlled. Additionally, an adaptive step, ninth-order Runge-Kutta integrator with eighth-order error control (RungeKutta89) is employed, with the maximum integration step size set to 45 minutes. Initial orbital elements for the test orbits are detailed in Table 4. Orbit-1 corresponds to the nominal orbit of the TianQin satellite. In addition, three cases with different inclinations are considered to facilitate a more comprehensive validation, considering that the inclination is a crucial parameter in the analytical solution.
为了验证卫星轨道的推导解析解,我们使用 NASA 通用任务分析工具 (GMAT) [54] 进行了高精度数值轨道模拟。力模型与参考文献[24]中的模型一致,包括月球、太阳和太阳系行星的点质量重力场(星历表DE421)、地球引力场的 10×10101010\times 1010 × 10 球谐模型(JGM-3)和一阶相对论校正。由于卫星是无阻力控制的,因此省略了非引力扰动,例如太阳辐射压力。此外,采用自适应步长、具有八阶误差控制的九阶 Runge-Kutta 积分器 (RungeKutta89),最大积分步长设置为 45 分钟。表 4 中详细说明了测试轨道的初始轨道元件。Orbit-1 对应于天琴卫星的标称轨道。此外,三种具有不同倾向的情况是 考虑到倾斜度是解析解中的关键参数,因此有助于进行更全面的验证。

Table 4: Initial orbital elements of the test orbits in the J2000-based Earth-centered ecliptic coordinate system at the epoch 1 Jan, 2034, 00:00:00 UTC.
表 4: 在 2034 年 1 月 1 日 00:00:00 UTC 纪元的基于 J2000 的地心黄道坐标系中测试轨道的初始轨道元素。
Test orbits 测试轨道 a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT (km)  a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT (公里) e0subscript𝑒0e_{0}italic_e start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT i0subscript𝑖0i_{0}italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ()
i0subscript𝑖0i_{0}italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT
Ω0subscriptΩ0\Omega_{0}roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ()
Ω0subscriptΩ0\Omega_{0}roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT
ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ()
ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT
ν0subscript𝜈0\nu_{0}italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ()
ν0subscript𝜈0\nu_{0}italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT
Orbit-1 轨道-1 100 000 0 94.7 210.4 0 60
Orbit-2 轨道 2 100 000 0 65.0 210.4 0 60
Orbit-3 轨道 3 100 000 0 35.0 210.4 0 60
Orbit-4 轨道 4 100 000 0 05.0 210.4 0 60

The comparison between analytical and numerical orbits reveals the errors Δσ(t):=σana(t)σnum(t)assignΔ𝜎𝑡subscript𝜎ana𝑡subscript𝜎num𝑡\Delta\sigma(t):=\sigma_{\text{ana}}(t)-\sigma_{\text{num}}(t)roman_Δ italic_σ ( italic_t ) := italic_σ start_POSTSUBSCRIPT ana end_POSTSUBSCRIPT ( italic_t ) - italic_σ start_POSTSUBSCRIPT num end_POSTSUBSCRIPT ( italic_t ) (σ{a,e,i,Ω,ω,λ}𝜎𝑎𝑒𝑖Ω𝜔𝜆\sigma\in\{a,e,i,\Omega,\omega,\lambda\}italic_σ ∈ { italic_a , italic_e , italic_i , roman_Ω , italic_ω , italic_λ }). Statistical results, shown in Table 5, demonstrate that analytical expressions for e𝑒eitalic_e, i𝑖iitalic_i, ΩΩ\Omegaroman_Ω, and λ𝜆\lambdaitalic_λ are in good agreement with numerical simulations, with the relative mean deviation of e𝑒eitalic_e being less than 6%percent66\%6 %, and long-term deviations for i𝑖iitalic_i, ΩΩ\Omegaroman_Ω, and λ𝜆\lambdaitalic_λ being small. In addition, there are relatively large errors in a𝑎aitalic_a and ω𝜔\omegaitalic_ω, with the latter having a minor influence on the constellation stability (see, e.g., Eq. (89)). Table 5 also includes a comparison of satellite positions, denoted as |Δ𝐫(t)|:=|𝐫ana(t)𝐫num(t)|assignΔ𝐫𝑡subscript𝐫ana𝑡subscript𝐫num𝑡|\Delta\mathbf{r}(t)|:=|\mathbf{r}_{\text{ana}}(t)-\mathbf{r}_{\text{num}}(t)|| roman_Δ bold_r ( italic_t ) | := | bold_r start_POSTSUBSCRIPT ana end_POSTSUBSCRIPT ( italic_t ) - bold_r start_POSTSUBSCRIPT num end_POSTSUBSCRIPT ( italic_t ) |. For the TianQin orbit, the average and maximum deviations over a 5-year period are approximately 87 km and 210 km, respectively.
解析轨道和数值轨道之间的比较揭示了误差 Δσ(t):=σana(t)σnum(t)assignΔ𝜎𝑡subscript𝜎ana𝑡subscript𝜎num𝑡\Delta\sigma(t):=\sigma_{\text{ana}}(t)-\sigma_{\text{num}}(t)roman_Δ italic_σ ( italic_t ) := italic_σ start_POSTSUBSCRIPT ana end_POSTSUBSCRIPT ( italic_t ) - italic_σ start_POSTSUBSCRIPT num end_POSTSUBSCRIPT ( italic_t )σ{a,e,i,Ω,ω,λ}𝜎𝑎𝑒𝑖Ω𝜔𝜆\sigma\in\{a,e,i,\Omega,\omega,\lambda\}italic_σ ∈ { italic_a , italic_e , italic_i , roman_Ω , italic_ω , italic_λ } )。表 5 所示的统计结果表明,、、 和 i𝑖iitalic_i ΩΩ\Omegaroman_Ω λ𝜆\lambdaitalic_λe𝑒eitalic_e 解析表达式与数值模拟非常一致,相对平均偏差 e𝑒eitalic_e 小于 6%percent66\%6 % ,而 i𝑖iitalic_iΩΩ\Omegaroman_Ωλ𝜆\lambdaitalic_λ 的长期偏差很小。此外,在 和 ω𝜔\omegaitalic_ωa𝑎aitalic_a 还存在相对较大的误差,后者对星座稳定性的影响很小(参见方程(89))。表 5 还包括卫星位置的比较,表示为 |Δ𝐫(t)|:=|𝐫ana(t)𝐫num(t)|assignΔ𝐫𝑡subscript𝐫ana𝑡subscript𝐫num𝑡|\Delta\mathbf{r}(t)|:=|\mathbf{r}_{\text{ana}}(t)-\mathbf{r}_{\text{num}}(t)|| roman_Δ bold_r ( italic_t ) | := | bold_r start_POSTSUBSCRIPT ana end_POSTSUBSCRIPT ( italic_t ) - bold_r start_POSTSUBSCRIPT num end_POSTSUBSCRIPT ( italic_t ) | 。对于天琴轨道,5 年期间的平均和最大偏差分别约为 87 公里和 210 公里。

Table 5: Comparison of the analytical solution for satellite orbits with numerical simulations over 5 years, indicating mean errors (and maximum errors). To address the secular variation of ΩΩ\Omegaroman_Ω, the second-order solution Ωc(2)(t)superscriptsubscriptΩ𝑐2𝑡\Omega_{c}^{(2)}(t)roman_Ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ( italic_t ) obtained from Eq. (132) has been incorporated into the analytical solution. When propagating λ𝜆\lambdaitalic_λ and, consequently, 𝐫𝐫\mathbf{r}bold_r using Eqs. (73)-(75), the mean semimajor axes of the numerical orbits were employed.
表 5: 卫星轨道的解析解与 5 年内的数值模拟的比较,表明平均误差(和最大误差)。为了解决 ΩΩ\Omegaroman_Ω 的长期变化,从方程(132)得到的二阶解 Ωc(2)(t)superscriptsubscriptΩ𝑐2𝑡\Omega_{c}^{(2)}(t)roman_Ω start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ( italic_t ) 已被纳入解析解中。当传播 λ𝜆\lambdaitalic_λ 并因此 𝐫𝐫\mathbf{r}bold_r 使用 Eqs.(73)-(75),采用了数值轨道的平均半长轴。
Test orbits 测试轨道 ΔaΔ𝑎\Delta aroman_Δ italic_a (km)  ΔaΔ𝑎\Delta aroman_Δ italic_a (公里) Δe/eΔ𝑒𝑒\Delta e/eroman_Δ italic_e / italic_e (%)  Δe/eΔ𝑒𝑒\Delta e/eroman_Δ italic_e / italic_e (%) ΔiΔ𝑖\Delta iroman_Δ italic_i ()
ΔiΔ𝑖\Delta iroman_Δ italic_i
ΔΩΔΩ\Delta\Omegaroman_Δ roman_Ω ()
ΔΩΔΩ\Delta\Omegaroman_Δ roman_Ω
ΔωΔ𝜔\Delta\omegaroman_Δ italic_ω ()
ΔωΔ𝜔\Delta\omegaroman_Δ italic_ω
ΔλΔ𝜆\Delta\lambdaroman_Δ italic_λ ()
ΔλΔ𝜆\Delta\lambdaroman_Δ italic_λ
|Δ𝐫|Δ𝐫|\Delta\mathbf{r}|| roman_Δ bold_r | (km)  |Δ𝐫|Δ𝐫|\Delta\mathbf{r}|| roman_Δ bold_r | (公里)
Orbit-1 轨道-1 --1.4 (--16.1)
-- 1.4 ( -- 16.1)
+++5.7 (+++7.3)
+++ 5.7 ( +++ 7.3)
--0.00 (--0.02)
-- 0.00 ( -- 0.02)
--0.00 (--0.01)
-- 0.00 ( -- 0.01)
--1.1 (+++29.9)
-- 1.1 ( +++ 29.9)
--0.05 (--0.11)
-- 0.05 ( -- 0.11)
087 (210)
087 (210)
Orbit-2 轨道 2 +++0.4 (--15.5)
+++ 0.4 ( -- 15.5)
+++1.2 (+++3.0)
+++ 1.2 ( +++ 3.0 )
+++0.01 (+++0.06)
+++ 0,01 ( +++ 0,06)
+++0.08 (+++0.12)
+++ 0,08 ( +++ 0,12)
--1.1 (+++70.9)
-- 1.1 ( +++ 70.9)
+++0.01 (--0.10)
+++ 0,01 ( -- 0,10)
126 (240)
Orbit-3 轨道 3 +++1.9 (+++16.2)
+++ 1.9 ( +++ 16.2 英寸)
--4.4 (--8.5)
-- 4.4 ( -- 8.5)
+++0.04 (+++0.14)
+++ 0.04 ( +++ 0.14)
+++0.08 (+++0.14)
+++ 0,08 ( +++ 0,14)
--1.6 (--44.4)
-- 1.6 ( -- 44.4)
+++0.08 (+++0.18)
+++ 0,08 ( +++ 0,18)
291 (502)
Orbit-4 轨道 4 +++2.2 (+++18.2)
+++ 2.2 ( +++ 18.2 页)
--1.0 (+++4.9)
-- 1.0 ( +++ 4.9)
--0.02 (--0.05)
-- 0,02 ( -- 0,05)
--0.38 (--0.96)
-- 0.38 ( -- 0.96)
+++7.3 (+++19.9)
+++ 7.3 ( +++ 19.9)
+++0.42 (+++1.02)
+++ 0.42 ( +++ 1.02)
161 (426)

For future improvements, potential dominant sources causing the aforementioned errors are briefly outlined as follows. Firstly, simplified analytical coordinates for the Sun and Moon (see Sec. III.2) were utilized, instead of higher-precision ones with multiple trigonometric corrections [35, 45]. Secondly, smaller perturbative effects, including those from other planets in the solar system and the nonspherical gravitational field of the Sun and Moon, were omitted in Eq. (52). Lastly, the second-order solution was lacking, and the next-leading-order eccentricity effect was neglected in Eq. (125), etc.
为了将来的改进,导致上述错误的潜在主要来源简要概述如下。首先,使用了太阳和月亮的简化解析坐标(见第III.2节),而不是具有多次三角校正的高精度坐标[35,45]。 其次,方程(52)中省略了较小的扰动效应,包括来自太阳系中其他行星的扰动效应以及太阳和月亮的非球形引力场。最后,在方程(125)中忽略了二阶解,忽略了次导阶偏心率效应。

B.4 Model verification for the three indicators
乙.4 3个指标的模型验证

Appendix B.3 verifies the analytical solution for satellite orbits, focusing on individual satellites. Additionally, this subsection presents the verification of the analytical expressions for the kinematic indicators of the three-satellite constellation: Lij(t)subscript𝐿𝑖𝑗𝑡L_{ij}(t)italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ), vij(t)subscript𝑣𝑖𝑗𝑡v_{ij}(t)italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ), and αk(t)subscript𝛼𝑘𝑡\alpha_{k}(t)italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ), derived from Eq. (76) (or Eqs. (73)-(75)).
附录 B.3 验证了卫星轨道的分析解,重点关注单个卫星。此外,本小节还介绍了对三卫星星座运动学指标的分析表达式的验证: Lij(t)subscript𝐿𝑖𝑗𝑡L_{ij}(t)italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t )vij(t)subscript𝑣𝑖𝑗𝑡v_{ij}(t)italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_t ) 、和 αk(t)subscript𝛼𝑘𝑡\alpha_{k}(t)italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_t ) ,源自方程(76)(或方程。(73)-(75))。

The time evolution of these three indicators in both analytical and numerical models is plotted in the left panel of Fig. 3 for a representative set of initial orbital elements provided in Table 6. In the numerical model, the considered perturbations, integrator, and step size align with those detailed in Appendix B.3. The right panel illustrates the time evolution of the deviations between the analytical and numerical models for these three quantities. Figure 3 indicates that the analytical model can effectively capture the long-term variations in the indicators, while noticeable periodic deviations exist. Numerical simulation results suggest that these deviations primarily arise from approximations in the Sun and Moon analytical dynamical model. By employing higher-precision models for solar and lunar motion [35, 45], incorporating numerous trigonometric correction terms in the Sun’s ecliptic longitude u2subscriptu2\text{u}_{2}u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and the Moon’s latitude argument u3subscriptu3\text{u}_{3}u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, these deviations can be effectively reduced. On the other hand, this enhanced complexity presents challenges in analytically solving the Lagrange equations, as u2subscriptu2\text{u}_{2}u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and u3subscriptu3\text{u}_{3}u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT themselves involve trigonometric functions (see Eqs. (57) and (59)). Additionally, beyond the orbits specified in Table 6, the analytical model has been validated on two additional sets: nominal orbits (with SC1’s initial elements matching those of Orbit-1 in Table 4) and optimized orbits (refer to Table 3 in [24]), yielding consistent results.
这三个指标在分析模型和数值模型中的时间演变如图左侧面板所示。3 表示表6中提供的一组具有代表性的初始轨道元素。在数值模型中,考虑的扰动、积分器和步长与附录 B.3 中详述的一致。右图显示了这三个量的分析模型和数值模型之间偏差的时间演变。图3表明,分析模型可以有效地捕捉指标的长期变化,同时存在明显的周期性偏差。数值模拟结果表明,这些偏差主要源于太阳和月亮解析动力学模型中的近似值。通过采用更高精度的太阳和月球运动模型[35,45],在太阳的黄道经度 u2subscriptu2\text{u}_{2}u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 和月球的纬度参数 u3subscriptu3\text{u}_{3}u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 中结合许多三角校正项,可以有效地减少这些偏差。另一方面,这种增加的复杂性给解析求解拉格朗日方程带来了挑战,因为 u2subscriptu2\text{u}_{2}u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPTu3subscriptu3\text{u}_{3}u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 格朗日方程本身涉及三角函数(参见方程。(57)和(59))。 此外,除了表 6 中规定的轨道之外,分析模型还在另外两组上进行了验证:标称轨道(SC1 的初始元件与表 4 中 Orbit-1 的初始元件相匹配)和优化轨道(参见 [24] 中的表 3),产生了一致的结果。

Figure 3: Time evolution of three kinematic indicators for the analytical model compared to the numerical model. The left panel displays subplots illustrating variations in arm-lengths (Lijsubscript𝐿𝑖𝑗L_{ij}italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT), relative velocities (vijsubscript𝑣𝑖𝑗v_{ij}italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT), and breathing angles (αksubscript𝛼𝑘\alpha_{k}italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT), respectively. Colors in each subplot represent the numerical model (blue for L12subscript𝐿12L_{12}italic_L start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT, v12subscript𝑣12v_{12}italic_v start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT, or α3subscript𝛼3\alpha_{3}italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT; red for L23subscript𝐿23L_{23}italic_L start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT, v23subscript𝑣23v_{23}italic_v start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT, or α1subscript𝛼1\alpha_{1}italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT; green for L31subscript𝐿31L_{31}italic_L start_POSTSUBSCRIPT 31 end_POSTSUBSCRIPT, v31subscript𝑣31v_{31}italic_v start_POSTSUBSCRIPT 31 end_POSTSUBSCRIPT, or α2subscript𝛼2\alpha_{2}italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT), while black denotes the analytical model (only v12subscript𝑣12v_{12}italic_v start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT is plotted for clarity in relative velocities). The right panel illustrates the temporal evolution of discrepancies between the analytical and numerical models for these three indicators.
图 3: 与数值模型相比,解析模型的三个运动学指标的时间演变。左侧面板显示子图,分别说明了臂长 ( Lijsubscript𝐿𝑖𝑗L_{ij}italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT )、相对速度 ( vijsubscript𝑣𝑖𝑗v_{ij}italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) 和呼吸角度 ( αksubscript𝛼𝑘\alpha_{k}italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) 的变化。每个子图中的颜色代表数值模型(蓝色代表 、 或 ;红色代表 L12subscript𝐿12L_{12}italic_L start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT L23subscript𝐿23L_{23}italic_L start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPTv23subscript𝑣23v_{23}italic_v start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPTα1subscript𝛼1\alpha_{1}italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ;绿色代表 L31subscript𝐿31L_{31}italic_L start_POSTSUBSCRIPT 31 end_POSTSUBSCRIPT 、 或 v31subscript𝑣31v_{31}italic_v start_POSTSUBSCRIPT 31 end_POSTSUBSCRIPT α2subscript𝛼2\alpha_{2}italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ),而黑色表示解析模型(仅 v12subscript𝑣12v_{12}italic_v start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT 绘制为了相对速度的清晰度)。 α3subscript𝛼3\alpha_{3}italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT v12subscript𝑣12v_{12}italic_v start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT 右图说明了这三个指标的解析模型和数值模型之间差异的时间演变。
Table 6: Initial orbital elements for simulated TianQin orbits in the J2000-based Earth-centered ecliptic coordinate system at the epoch of 22 May, 2034, 12:00:00 UTC. These initial elements deviate by approximately 1 km, 105superscript10510^{-5}10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT, 0.2, 0.2, 0.1, and 0.1 in a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, e0subscript𝑒0e_{0}italic_e start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, i0subscript𝑖0i_{0}italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, Ω0subscriptΩ0\Omega_{0}roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and ν0subscript𝜈0\nu_{0}italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, respectively, from the optimized orbits (cf. Table 3 in [24]). The subsequent orbital evolution is illustrated in Fig. 3.
表 6: 2034 年 5 月 22 日 12:00:00 UTC 纪元,基于 J2000 的地心黄道坐标系中模拟天琴轨道的初始轨道元素。这些初始元素在优化轨道上 a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT e0subscript𝑒0e_{0}italic_e start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT i0subscript𝑖0i_{0}italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT Ω0subscriptΩ0\Omega_{0}roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ν0subscript𝜈0\nu_{0}italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT 分别偏离了大约 1 km、 105superscript10510^{-5}10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT 、0.2、0.2、0.1 和 0.1(参见 [24] 中的表 3)。随后的轨道演化如图 1 所示。3.
a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT (km)  a0subscript𝑎0a_{0}italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT (公里) e0subscript𝑒0e_{0}italic_e start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT i0subscript𝑖0i_{0}italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ()
i0subscript𝑖0i_{0}italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT
SC1 099 996.572 323
电话 099 996.572 323
0.000 440 94.897 997
SC2 100 010.400 095 0.000 010 94.904 363
SC3 099 992.041 899
电话 099 992.041 899
0.000 296 94.509 747
Ω0subscriptΩ0\Omega_{0}roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ()
Ω0subscriptΩ0\Omega_{0}roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT
ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ()
ω0subscript𝜔0\omega_{0}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT
ν0subscript𝜈0\nu_{0}italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ()
ν0subscript𝜈0\nu_{0}italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT
SC1 210.645 892 358.724 463 061.429 603
电话 061.429 603
SC2 210.240 199 359.900 000 180.130 706
SC3 210.644 582 359.901 624 299.812 164

Furthermore, the expressions (89), (97), and (101), derived from the series expansion of the three indicators, have been verified. The results suggest that to achieve a deviation magnitude similar to that before the series expansion, the second-order term δLij(δλ(t)2)=34aoδλi(t)δλj(t)38ao[δλi(t)2+δλj(t)2]𝛿subscript𝐿𝑖𝑗𝛿𝜆superscript𝑡234subscript𝑎o𝛿subscript𝜆𝑖𝑡𝛿subscript𝜆𝑗𝑡38subscript𝑎odelimited-[]𝛿subscript𝜆𝑖superscript𝑡2𝛿subscript𝜆𝑗superscript𝑡2\delta L_{ij}(\delta\lambda(t)^{2})=\frac{\sqrt{3}}{4}a_{\text{o}}\delta% \lambda_{i}(t)\delta\lambda_{j}(t)-\frac{\sqrt{3}}{8}a_{\text{o}}[\delta% \lambda_{i}(t)^{2}+\delta\lambda_{j}(t)^{2}]italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_δ italic_λ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 4 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_δ italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) italic_δ italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) - divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 8 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ italic_δ italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_δ italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] in the arm-length, where δλ(t)2δn2t2similar-to𝛿𝜆superscript𝑡2𝛿superscript𝑛2superscript𝑡2\delta\lambda(t)^{2}\sim\delta n^{2}\,t^{2}italic_δ italic_λ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∼ italic_δ italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT rapidly increases with time, needs to be taken into account. Notably, for the relative velocity and breathing angle, these second-order terms are both zero, δvij(δλ(t)2)=δαk(δλ(t)2)=0𝛿subscript𝑣𝑖𝑗𝛿𝜆superscript𝑡2𝛿subscript𝛼𝑘𝛿𝜆superscript𝑡20\delta v_{ij}(\delta\lambda(t)^{2})=\delta\alpha_{k}(\delta\lambda(t)^{2})=0italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_δ italic_λ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_δ italic_λ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = 0.
此外,从三个指标的级数扩展中得出的表达式 (89)、(97) 和 (101) 也得到了验证。结果表明,为了获得与级数展开前相似的偏差幅度,需要考虑臂长中的二阶项,该 δλ(t)2δn2t2similar-to𝛿𝜆superscript𝑡2𝛿superscript𝑛2superscript𝑡2\delta\lambda(t)^{2}\sim\delta n^{2}\,t^{2}italic_δ italic_λ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∼ italic_δ italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPTδLij(δλ(t)2)=34aoδλi(t)δλj(t)38ao[δλi(t)2+δλj(t)2]𝛿subscript𝐿𝑖𝑗𝛿𝜆superscript𝑡234subscript𝑎o𝛿subscript𝜆𝑖𝑡𝛿subscript𝜆𝑗𝑡38subscript𝑎odelimited-[]𝛿subscript𝜆𝑖superscript𝑡2𝛿subscript𝜆𝑗superscript𝑡2\delta L_{ij}(\delta\lambda(t)^{2})=\frac{\sqrt{3}}{4}a_{\text{o}}\delta% \lambda_{i}(t)\delta\lambda_{j}(t)-\frac{\sqrt{3}}{8}a_{\text{o}}[\delta% \lambda_{i}(t)^{2}+\delta\lambda_{j}(t)^{2}]italic_δ italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_δ italic_λ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 4 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT italic_δ italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) italic_δ italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) - divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 8 end_ARG italic_a start_POSTSUBSCRIPT o end_POSTSUBSCRIPT [ italic_δ italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_δ italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] 随时间迅速增加。值得注意的是,对于相对速度和呼吸角,这些二阶项都是零 。 δvij(δλ(t)2)=δαk(δλ(t)2)=0𝛿subscript𝑣𝑖𝑗𝛿𝜆superscript𝑡2𝛿subscript𝛼𝑘𝛿𝜆superscript𝑡20\delta v_{ij}(\delta\lambda(t)^{2})=\delta\alpha_{k}(\delta\lambda(t)^{2})=0italic_δ italic_v start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ( italic_δ italic_λ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = italic_δ italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_δ italic_λ ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = 0

Appendix C Explicit expressions for terms in Eq. (75)
附录 C 方程 (75) 中项的显式表达式

In this section, explicit expressions for each term of σ(1)(t)superscript𝜎1𝑡\sigma^{(1)}(t)italic_σ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ), as listed in Table 2, are presented. These expressions, categorized by the perturbations of the Sun, Moon, and Earth’s J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, are detailed in C.1, C.2, and C.3. Appendix C.4 provides the perturbation solutions for jointly solved ξ𝜉\xiitalic_ξ and η𝜂\etaitalic_η: ξcsubscript𝜉𝑐\xi_{c}italic_ξ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, ηcsubscript𝜂𝑐\eta_{c}italic_η start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, ξl[c]subscript𝜉𝑙delimited-[]𝑐\xi_{l[c]}italic_ξ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT, and ηl[c]subscript𝜂𝑙delimited-[]𝑐\eta_{l[c]}italic_η start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT, considering both solar and lunar perturbations. Note that in the subsequent expressions, the orbital elements a𝑎aitalic_a, ΩΩ\Omegaroman_Ω, λ𝜆\lambdaitalic_λ, and i𝑖iitalic_i, take the mean value a¯¯𝑎\overline{a}over¯ start_ARG italic_a end_ARG or the form defined by Eq. (134). Similarly, the Moon’s orbit inclination i3subscript𝑖3i_{3}italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT represents i¯3subscript¯𝑖3\overline{i}_{3}over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT.
在本节中,显示了表 2 中所列的每个术语 σ(1)(t)superscript𝜎1𝑡\sigma^{(1)}(t)italic_σ start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_t ) 的显式表达式。这些表达式按太阳、月亮和地球 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 的扰动分类,在 C.1C.2C.3 中有详细说明。附录 C.4 提供了联合解决 ξ𝜉\xiitalic_ξ 的 AND η𝜂\etaitalic_η 的扰动解: ξcsubscript𝜉𝑐\xi_{c}italic_ξ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT 、、 ηcsubscript𝜂𝑐\eta_{c}italic_η start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ξl[c]subscript𝜉𝑙delimited-[]𝑐\xi_{l[c]}italic_ξ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPTηl[c]subscript𝜂𝑙delimited-[]𝑐\eta_{l[c]}italic_η start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ,同时考虑了太阳和月球的扰动。请注意,在后续表达式中,轨道元素 a𝑎aitalic_aΩΩ\Omegaroman_Ωλ𝜆\lambdaitalic_λi𝑖iitalic_i 采用平均值 a¯¯𝑎\overline{a}over¯ start_ARG italic_a end_ARG 或方程 (134) 定义的形式。同样,月球的轨道倾角 i3subscript𝑖3i_{3}italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT 表示 i¯3subscript¯𝑖3\overline{i}_{3}over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT

C.1 Solar perturbation solution σs(t)subscript𝜎s𝑡\sigma_{\text{s}}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT ( italic_t )
C.1 太阳能扰动解决方案 σs(t)subscript𝜎s𝑡\sigma_{\text{s}}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT ( italic_t )

The secular variation σsc(t)superscriptsubscript𝜎s𝑐𝑡\sigma_{\text{s}}^{c}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ), long-period variation σsl(t)superscriptsubscript𝜎s𝑙𝑡\sigma_{\text{s}}^{l}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT ( italic_t ), and short-period variation σss(t)superscriptsubscript𝜎s𝑠𝑡\sigma_{\text{s}}^{s}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t ) within σs(t)subscript𝜎s𝑡\sigma_{\text{s}}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT ( italic_t ) are presented as follows:
内部的长期变化 σsc(t)superscriptsubscript𝜎s𝑐𝑡\sigma_{\text{s}}^{c}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ) 、长期变化 σsl(t)superscriptsubscript𝜎s𝑙𝑡\sigma_{\text{s}}^{l}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT ( italic_t ) 和短期变化 σss(t)superscriptsubscript𝜎s𝑠𝑡\sigma_{\text{s}}^{s}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t ) 表示 σs(t)subscript𝜎s𝑡\sigma_{\text{s}}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT ( italic_t ) 如下:

(1) Secular terms with the form σsc(t)superscriptsubscript𝜎s𝑐𝑡\sigma_{\text{s}}^{c}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ):
(1) 形式 σsc(t)superscriptsubscript𝜎s𝑐𝑡\sigma_{\text{s}}^{c}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ) 为 :

asc=superscriptsubscript𝑎s𝑐absent\displaystyle a_{\text{s}}^{c}=italic_a start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = isc=0,superscriptsubscript𝑖s𝑐0\displaystyle~{}i_{\text{s}}^{c}=0,italic_i start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = 0 , (136)
Ωsc=superscriptsubscriptΩs𝑐absent\displaystyle\Omega_{\text{s}}^{c}=roman_Ω start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = nΩst,nΩs:=34cscosi,assignsubscript𝑛Ωs𝑡subscript𝑛Ωs34subscript𝑐s𝑖\displaystyle~{}n_{\Omega{\text{s}}}\,t,\qquad n_{\Omega{\text{s}}}:=-\frac{3}% {4}c_{\text{s}}\cos i,italic_n start_POSTSUBSCRIPT roman_Ω s end_POSTSUBSCRIPT italic_t , italic_n start_POSTSUBSCRIPT roman_Ω s end_POSTSUBSCRIPT := - divide start_ARG 3 end_ARG start_ARG 4 end_ARG italic_c start_POSTSUBSCRIPT s end_POSTSUBSCRIPT roman_cos italic_i , (137)
λsc=superscriptsubscript𝜆s𝑐absent\displaystyle\lambda_{\text{s}}^{c}=italic_λ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = nλst,nλs:=18cs(13cos2i),assignsubscript𝑛𝜆s𝑡subscript𝑛𝜆s18subscript𝑐s132𝑖\displaystyle~{}n_{\lambda\text{s}}\,t,\qquad n_{\lambda\text{s}}:=\frac{1}{8}% c_{\text{s}}(1-3\cos 2i),italic_n start_POSTSUBSCRIPT italic_λ s end_POSTSUBSCRIPT italic_t , italic_n start_POSTSUBSCRIPT italic_λ s end_POSTSUBSCRIPT := divide start_ARG 1 end_ARG start_ARG 8 end_ARG italic_c start_POSTSUBSCRIPT s end_POSTSUBSCRIPT ( 1 - 3 roman_cos 2 italic_i ) , (138)

where cs:=μ2n¯r¯23assignsubscript𝑐ssubscript𝜇2¯𝑛superscriptsubscript¯𝑟23c_{\text{s}}:=\frac{\mu_{2}}{\overline{n}\,\overline{r}_{2}^{3}}italic_c start_POSTSUBSCRIPT s end_POSTSUBSCRIPT := divide start_ARG italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG over¯ start_ARG italic_n end_ARG over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG. 其中 cs:=μ2n¯r¯23assignsubscript𝑐ssubscript𝜇2¯𝑛superscriptsubscript¯𝑟23c_{\text{s}}:=\frac{\mu_{2}}{\overline{n}\,\overline{r}_{2}^{3}}italic_c start_POSTSUBSCRIPT s end_POSTSUBSCRIPT := divide start_ARG italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG over¯ start_ARG italic_n end_ARG over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG .

(2) Long-period terms with the form σsl(t)superscriptsubscript𝜎s𝑙𝑡\sigma_{\text{s}}^{l}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT ( italic_t ):
(2) 格式为 σsl(t)superscriptsubscript𝜎s𝑙𝑡\sigma_{\text{s}}^{l}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT ( italic_t )

asl[l]=superscriptsubscript𝑎s𝑙delimited-[]𝑙absent\displaystyle a_{\text{s}}^{l[l]}=italic_a start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT = ξsl[l]=ηsl[l]=0,superscriptsubscript𝜉s𝑙delimited-[]𝑙superscriptsubscript𝜂s𝑙delimited-[]𝑙0\displaystyle~{}\xi_{\text{s}}^{l[l]}=\eta_{\text{s}}^{l[l]}=0,italic_ξ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT = italic_η start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT = 0 , (139)
isl[l]=superscriptsubscript𝑖s𝑙delimited-[]𝑙absent\displaystyle i_{\text{s}}^{l[l]}=italic_i start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT = 38csnU2sinicos2U2,38subscript𝑐ssubscript𝑛subscript𝑈2𝑖2subscript𝑈2\displaystyle~{}\frac{3}{8}\frac{c_{\text{s}}}{n_{U_{2}}}\sin i\cos 2U_{2},divide start_ARG 3 end_ARG start_ARG 8 end_ARG divide start_ARG italic_c start_POSTSUBSCRIPT s end_POSTSUBSCRIPT end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG roman_sin italic_i roman_cos 2 italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , (140)
Ωsl[l]=superscriptsubscriptΩs𝑙delimited-[]𝑙absent\displaystyle\Omega_{\text{s}}^{l[l]}=roman_Ω start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT = 38csnU2cosisin2U2,38subscript𝑐ssubscript𝑛subscript𝑈2𝑖2subscript𝑈2\displaystyle~{}\frac{3}{8}\frac{c_{\text{s}}}{n_{U_{2}}}\cos i\sin 2U_{2},divide start_ARG 3 end_ARG start_ARG 8 end_ARG divide start_ARG italic_c start_POSTSUBSCRIPT s end_POSTSUBSCRIPT end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG roman_cos italic_i roman_sin 2 italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , (141)
λsl[l]=superscriptsubscript𝜆s𝑙delimited-[]𝑙absent\displaystyle\lambda_{\text{s}}^{l[l]}=italic_λ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT = 316csnU2(cos2i3)sin2U2,316subscript𝑐ssubscript𝑛subscript𝑈22𝑖32subscript𝑈2\displaystyle~{}\frac{3}{16}\frac{c_{\text{s}}}{n_{U_{2}}}(\cos 2i-3)\sin 2U_{% 2},divide start_ARG 3 end_ARG start_ARG 16 end_ARG divide start_ARG italic_c start_POSTSUBSCRIPT s end_POSTSUBSCRIPT end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG ( roman_cos 2 italic_i - 3 ) roman_sin 2 italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , (142)

with 

U2(t):=u2(t)Ω¯(t)=nU2(tt0)+U20,assignsubscript𝑈2𝑡subscript𝑢2𝑡¯Ω𝑡subscript𝑛subscript𝑈2𝑡subscript𝑡0subscript𝑈subscript20\displaystyle U_{2}(t):=u_{2}(t)-\overline{\Omega}(t)=n_{U_{2}}(t-t_{0})+U_{2_% {0}},italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) := italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) - over¯ start_ARG roman_Ω end_ARG ( italic_t ) = italic_n start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_t - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_U start_POSTSUBSCRIPT 2 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , (143)
nU2=n2nΩ,U20=u20Ω¯0,formulae-sequencesubscript𝑛subscript𝑈2subscript𝑛2superscriptsubscript𝑛Ωsubscript𝑈subscript20subscript𝑢subscript20subscript¯Ω0\displaystyle n_{U_{2}}=n_{2}-n_{\Omega}^{\prime},\qquad U_{2_{0}}=u_{2_{0}}-% \overline{\Omega}_{0},italic_n start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_n start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_U start_POSTSUBSCRIPT 2 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_u start_POSTSUBSCRIPT 2 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT - over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , (144)

where nΩsuperscriptsubscript𝑛Ωn_{\Omega}^{\prime}italic_n start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is the rate of change of Ω¯(t)¯Ω𝑡\overline{\Omega}(t)over¯ start_ARG roman_Ω end_ARG ( italic_t ) as defined in Eq. (134). These expressions reveal that solar perturbation induces general long-period variations in satellite orbital elements with an annual period tied to the solar apparent motion n2subscript𝑛2n_{2}italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. The magnitudes of these variations are governed by a¯¯𝑎\overline{a}over¯ start_ARG italic_a end_ARG and i¯¯𝑖\overline{i}over¯ start_ARG italic_i end_ARG. ξsl[l]=ηsl[l]=0superscriptsubscript𝜉s𝑙delimited-[]𝑙superscriptsubscript𝜂s𝑙delimited-[]𝑙0\xi_{\text{s}}^{l[l]}=\eta_{\text{s}}^{l[l]}=0italic_ξ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT = italic_η start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT = 0 arises from considering only the leading order N=2𝑁2N=2italic_N = 2 within PN(cosψ2)subscript𝑃𝑁subscript𝜓2P_{N}(\cos\psi_{2})italic_P start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( roman_cos italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) (refer to Eq. (53)); when N3𝑁3N\geq 3italic_N ≥ 3, both ξ𝜉\xiitalic_ξ and η𝜂\etaitalic_η will exhibit periodic variations, as observed in the case of lunar perturbation (see Eqs. (168) and (169)).
其中 nΩsuperscriptsubscript𝑛Ωn_{\Omega}^{\prime}italic_n start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT 是方程(134)中定义的 的变化 Ω¯(t)¯Ω𝑡\overline{\Omega}(t)over¯ start_ARG roman_Ω end_ARG ( italic_t ) 率。这些表达式表明,太阳扰动会引起卫星轨道元素的一般长周期变化,其年周期与太阳视运动 n2subscript𝑛2n_{2}italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 有关。这些变化的大小由 a¯¯𝑎\overline{a}over¯ start_ARG italic_a end_ARGi¯¯𝑖\overline{i}over¯ start_ARG italic_i end_ARG 控制。 ξsl[l]=ηsl[l]=0superscriptsubscript𝜉s𝑙delimited-[]𝑙superscriptsubscript𝜂s𝑙delimited-[]𝑙0\xi_{\text{s}}^{l[l]}=\eta_{\text{s}}^{l[l]}=0italic_ξ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT = italic_η start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT = 0 产生于仅考虑 ( PN(cosψ2)subscript𝑃𝑁subscript𝜓2P_{N}(\cos\psi_{2})italic_P start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ( roman_cos italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) 参见方程 (53))中的前导顺序 N=2𝑁2N=2italic_N = 2 );当 N3𝑁3N\geq 3italic_N ≥ 3 时,两者 和 ξ𝜉\xiitalic_ξ η𝜂\etaitalic_η 都将表现出周期性变化,如在月球扰动的情况下观察到的那样(参见方程。(168) 和 (169))。

There are no special long-period variations in a𝑎aitalic_a, i𝑖iitalic_i, ΩΩ\Omegaroman_Ω, and λ𝜆\lambdaitalic_λ,
a𝑎aitalic_a 在 、 、 i𝑖iitalic_i ΩΩ\Omegaroman_Ωλ𝜆\lambdaitalic_λ 中没有特殊的长周期变化

σsl[c]0,forσ{a,i,Ω,λ}.formulae-sequencesuperscriptsubscript𝜎s𝑙delimited-[]𝑐0for𝜎𝑎𝑖Ω𝜆\displaystyle\sigma_{\text{s}}^{l[c]}\equiv 0,\quad\text{for}\ \sigma\in\{a,i,% \Omega,\lambda\}.italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ≡ 0 , for italic_σ ∈ { italic_a , italic_i , roman_Ω , italic_λ } . (145)

For ξ𝜉\xiitalic_ξ and η𝜂\etaitalic_η, they exhibit special long-period variations coupled with lunar perturbation, as indicated in Eqs. (204) and (205).
对于 ξ𝜉\xiitalic_ξη𝜂\etaitalic_η ,它们表现出特殊的长周期变化和月球扰动,如方程所示。(204)和(205)。

(3) Short-period terms with the form σss(t)superscriptsubscript𝜎s𝑠𝑡\sigma_{\text{s}}^{s}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t ):
(3) 短期期限,表格 σss(t)superscriptsubscript𝜎s𝑠𝑡\sigma_{\text{s}}^{s}(t)italic_σ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t ) 如下:

asssuperscriptsubscript𝑎s𝑠\displaystyle a_{\text{s}}^{s}italic_a start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT =p=2(Δp=2)2a×h(2,p)acos(2λ+pU2),absentsuperscriptsubscript𝑝2Δ𝑝22𝑎subscriptsuperscript𝑎2𝑝2𝜆𝑝subscript𝑈2\displaystyle=\sum_{\begin{subarray}{c}p=-2\\ (\Delta p=2)\end{subarray}}^{2}a\times h^{a}_{(2,\,p)}\cos(2\,\lambda+p\,U_{2}),= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p = - 2 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a × italic_h start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( 2 , italic_p ) end_POSTSUBSCRIPT roman_cos ( 2 italic_λ + italic_p italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , (146)
ξsssuperscriptsubscript𝜉s𝑠\displaystyle\xi_{\text{s}}^{s}italic_ξ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT =κ=1(Δκ=2)3p=2(Δp=2)2h(κ,p)ξcos(κλ+pU2),absentsuperscriptsubscript𝜅1Δ𝜅23superscriptsubscript𝑝2Δ𝑝22subscriptsuperscript𝜉𝜅𝑝𝜅𝜆𝑝subscript𝑈2\displaystyle=\sum_{\begin{subarray}{c}\kappa=1\\ (\Delta\kappa=2)\end{subarray}}^{3}\sum_{\begin{subarray}{c}p=-2\\ (\Delta p=2)\end{subarray}}^{2}h^{\xi}_{(\kappa,\,p)}\cos(\kappa\,\lambda+p\,U% _{2}),= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_κ = 1 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_κ = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p = - 2 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_h start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_κ , italic_p ) end_POSTSUBSCRIPT roman_cos ( italic_κ italic_λ + italic_p italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , (147)
ηsssuperscriptsubscript𝜂s𝑠\displaystyle\eta_{\text{s}}^{s}italic_η start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT =κ=1(Δκ=2)3p=2(Δp=2)2h(κ,p)ηsin(κλ+pU2),absentsuperscriptsubscript𝜅1Δ𝜅23superscriptsubscript𝑝2Δ𝑝22subscriptsuperscript𝜂𝜅𝑝𝜅𝜆𝑝subscript𝑈2\displaystyle=\sum_{\begin{subarray}{c}\kappa=1\\ (\Delta\kappa=2)\end{subarray}}^{3}\sum_{\begin{subarray}{c}p=-2\\ (\Delta p=2)\end{subarray}}^{2}h^{\eta}_{(\kappa,\,p)}\sin(\kappa\,\lambda+p\,% U_{2}),= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_κ = 1 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_κ = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p = - 2 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_h start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_κ , italic_p ) end_POSTSUBSCRIPT roman_sin ( italic_κ italic_λ + italic_p italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , (148)
isssuperscriptsubscript𝑖s𝑠\displaystyle i_{\text{s}}^{s}italic_i start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT =p=2(Δp=2)2h(2,p)icos(2λ+pU2),absentsuperscriptsubscript𝑝2Δ𝑝22subscriptsuperscript𝑖2𝑝2𝜆𝑝subscript𝑈2\displaystyle=\sum_{\begin{subarray}{c}p=-2\\ (\Delta p=2)\end{subarray}}^{2}h^{i}_{(2,\,p)}\cos(2\,\lambda+p\,U_{2}),= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p = - 2 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_h start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( 2 , italic_p ) end_POSTSUBSCRIPT roman_cos ( 2 italic_λ + italic_p italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , (149)
ΩsssuperscriptsubscriptΩs𝑠\displaystyle\Omega_{\text{s}}^{s}roman_Ω start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT =p=2(Δp=2)2h(2,p)Ωsin(2λ+pU2),absentsuperscriptsubscript𝑝2Δ𝑝22subscriptsuperscriptΩ2𝑝2𝜆𝑝subscript𝑈2\displaystyle=\sum_{\begin{subarray}{c}p=-2\\ (\Delta p=2)\end{subarray}}^{2}h^{\Omega}_{(2,\,p)}\sin(2\,\lambda+p\,U_{2}),= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p = - 2 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_h start_POSTSUPERSCRIPT roman_Ω end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( 2 , italic_p ) end_POSTSUBSCRIPT roman_sin ( 2 italic_λ + italic_p italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , (150)
λsssuperscriptsubscript𝜆s𝑠\displaystyle\lambda_{\text{s}}^{s}italic_λ start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT =p=2(Δp=2)2h(2,p)λsin(2λ+pU2),absentsuperscriptsubscript𝑝2Δ𝑝22subscriptsuperscript𝜆2𝑝2𝜆𝑝subscript𝑈2\displaystyle=\sum_{\begin{subarray}{c}p=-2\\ (\Delta p=2)\end{subarray}}^{2}h^{\lambda}_{(2,\,p)}\sin(2\,\lambda+p\,U_{2}),= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p = - 2 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_h start_POSTSUPERSCRIPT italic_λ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( 2 , italic_p ) end_POSTSUBSCRIPT roman_sin ( 2 italic_λ + italic_p italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , (151)

with 

h(κ,p)σ:=csκnλ+pnU2×h[κ,p]σ(i)assignsubscriptsuperscript𝜎𝜅𝑝subscript𝑐s𝜅superscriptsubscript𝑛𝜆𝑝subscript𝑛subscript𝑈2subscriptsuperscript𝜎𝜅𝑝𝑖\displaystyle h^{\sigma}_{(\kappa,\,p)}:=\frac{c_{\text{s}}}{\kappa\,n_{% \lambda}^{\prime}+p\,n_{U_{2}}}\times h^{\sigma}_{[\kappa,\,p]}(i)italic_h start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_κ , italic_p ) end_POSTSUBSCRIPT := divide start_ARG italic_c start_POSTSUBSCRIPT s end_POSTSUBSCRIPT end_ARG start_ARG italic_κ italic_n start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + italic_p italic_n start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG × italic_h start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p ] end_POSTSUBSCRIPT ( italic_i ) (152)

for σ{a,ξ,η,i,Ω}𝜎𝑎𝜉𝜂𝑖Ω\sigma\in\{a,\xi,\eta,i,\Omega\}italic_σ ∈ { italic_a , italic_ξ , italic_η , italic_i , roman_Ω }, and for σ{a,ξ,η,i,Ω}𝜎𝑎𝜉𝜂𝑖Ω\sigma\in\{a,\xi,\eta,i,\Omega\}italic_σ ∈ { italic_a , italic_ξ , italic_η , italic_i , roman_Ω }

h(κ,p)λ:=3csn×h[κ,p]λ[a](i)(κnλ+pnU2)2+cs×h[κ,p]λ[λ](i)κnλ+pnU2,assignsubscriptsuperscript𝜆𝜅𝑝3subscript𝑐s𝑛subscriptsuperscript𝜆delimited-[]𝑎𝜅𝑝𝑖superscript𝜅superscriptsubscript𝑛𝜆𝑝subscript𝑛subscript𝑈22subscript𝑐ssubscriptsuperscript𝜆delimited-[]𝜆𝜅𝑝𝑖𝜅superscriptsubscript𝑛𝜆𝑝subscript𝑛subscript𝑈2\displaystyle h^{\lambda}_{(\kappa,\,p)}:=\frac{3\,c_{\text{s}}\,n\times h^{% \lambda[a]}_{[\kappa,\,p]}(i)}{(\kappa\,n_{\lambda}^{\prime}+p\,n_{U_{2}})^{2}% }+\frac{c_{\text{s}}\times h^{\lambda[\lambda]}_{[\kappa,\,p]}(i)}{\kappa\,n_{% \lambda}^{\prime}+p\,n_{U_{2}}},italic_h start_POSTSUPERSCRIPT italic_λ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_κ , italic_p ) end_POSTSUBSCRIPT := divide start_ARG 3 italic_c start_POSTSUBSCRIPT s end_POSTSUBSCRIPT italic_n × italic_h start_POSTSUPERSCRIPT italic_λ [ italic_a ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p ] end_POSTSUBSCRIPT ( italic_i ) end_ARG start_ARG ( italic_κ italic_n start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + italic_p italic_n start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + divide start_ARG italic_c start_POSTSUBSCRIPT s end_POSTSUBSCRIPT × italic_h start_POSTSUPERSCRIPT italic_λ [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p ] end_POSTSUBSCRIPT ( italic_i ) end_ARG start_ARG italic_κ italic_n start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + italic_p italic_n start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG , (153)

where nλsuperscriptsubscript𝑛𝜆n_{\lambda}^{\prime}italic_n start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT denotes the rate of change of λ¯(t)¯𝜆𝑡\overline{\lambda}(t)over¯ start_ARG italic_λ end_ARG ( italic_t ) as defined in Eq. (134). In the specific case of solar perturbation alone, nλ=n¯+nλssuperscriptsubscript𝑛𝜆¯𝑛subscript𝑛𝜆sn_{\lambda}^{\prime}=\overline{n}+n_{\lambda\text{s}}italic_n start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = over¯ start_ARG italic_n end_ARG + italic_n start_POSTSUBSCRIPT italic_λ s end_POSTSUBSCRIPT. The terms on the right side of Eq. (153) correspond to the integrals of the two terms in Eq. (131). The explicit forms of h[κ,p]σ(i)subscriptsuperscript𝜎𝜅𝑝𝑖h^{\sigma}_{[\kappa,\,p]}(i)italic_h start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p ] end_POSTSUBSCRIPT ( italic_i ) are given by
其中 nλsuperscriptsubscript𝑛𝜆n_{\lambda}^{\prime}italic_n start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT 表示方程 (134) 中定义的 的变化 λ¯(t)¯𝜆𝑡\overline{\lambda}(t)over¯ start_ARG italic_λ end_ARG ( italic_t ) 率。在单独的太阳扰动的特定情况下, nλ=n¯+nλssuperscriptsubscript𝑛𝜆¯𝑛subscript𝑛𝜆sn_{\lambda}^{\prime}=\overline{n}+n_{\lambda\text{s}}italic_n start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = over¯ start_ARG italic_n end_ARG + italic_n start_POSTSUBSCRIPT italic_λ s end_POSTSUBSCRIPT .方程(153)右侧的项对应于方程(131)中两项的积分。的 h[κ,p]σ(i)subscriptsuperscript𝜎𝜅𝑝𝑖h^{\sigma}_{[\kappa,\,p]}(i)italic_h start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p ] end_POSTSUBSCRIPT ( italic_i ) 显式形式由下式给出

h[2,0]a=32sin2i,h[2,±2]a=38(34cosi+cos2i),formulae-sequencesubscriptsuperscript𝑎2032superscript2𝑖subscriptsuperscript𝑎2plus-or-minus238minus-or-plus34𝑖2𝑖\displaystyle h^{a}_{[2,0]}=\frac{3}{2}\sin^{2}i,\qquad h^{a}_{[2,\pm 2]}=% \frac{3}{8}(3\mp 4\cos i+\cos 2i),italic_h start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 2 end_ARG roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_i , italic_h start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 8 end_ARG ( 3 ∓ 4 roman_cos italic_i + roman_cos 2 italic_i ) ,
h[2,0]i=38sin2i,h[2,±2]i=316(±2sinisin2i),formulae-sequencesubscriptsuperscript𝑖20382𝑖subscriptsuperscript𝑖2plus-or-minus2316plus-or-minus2𝑖2𝑖\displaystyle h^{i}_{[2,0]}=\frac{3}{8}\sin 2i,\qquad h^{i}_{[2,\pm 2]}=\frac{% 3}{16}(\pm 2\sin i-\sin 2i),italic_h start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 8 end_ARG roman_sin 2 italic_i , italic_h start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 16 end_ARG ( ± 2 roman_sin italic_i - roman_sin 2 italic_i ) ,
h[2,0]Ω=34cosi,h[2,±2]Ω=38(±1cosi),formulae-sequencesubscriptsuperscriptΩ2034𝑖subscriptsuperscriptΩ2plus-or-minus238plus-or-minus1𝑖\displaystyle h^{\Omega}_{[2,0]}=\frac{3}{4}\cos i,\qquad h^{\Omega}_{[2,\pm 2% ]}=\frac{3}{8}(\pm 1-\cos i),italic_h start_POSTSUPERSCRIPT roman_Ω end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 4 end_ARG roman_cos italic_i , italic_h start_POSTSUPERSCRIPT roman_Ω end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 8 end_ARG ( ± 1 - roman_cos italic_i ) ,
h[2,0]λ[a]=34sin2i,h[2,±2]λ[a]=316(34cosi+cos2i),formulae-sequencesubscriptsuperscript𝜆delimited-[]𝑎2034superscript2𝑖subscriptsuperscript𝜆delimited-[]𝑎2plus-or-minus2316minus-or-plus34𝑖2𝑖\displaystyle h^{\lambda[a]}_{[2,0]}=-\frac{3}{4}\sin^{2}i,\qquad h^{\lambda[a% ]}_{[2,\pm 2]}=-\frac{3}{16}(3\mp 4\cos i+\cos 2i),italic_h start_POSTSUPERSCRIPT italic_λ [ italic_a ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 4 end_ARG roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_i , italic_h start_POSTSUPERSCRIPT italic_λ [ italic_a ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 16 end_ARG ( 3 ∓ 4 roman_cos italic_i + roman_cos 2 italic_i ) ,
h[2,0]λ[λ]=38(3cos2i),subscriptsuperscript𝜆delimited-[]𝜆203832𝑖\displaystyle h^{\lambda[\lambda]}_{[2,0]}=-\frac{3}{8}(3-\cos 2i),italic_h start_POSTSUPERSCRIPT italic_λ [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 8 end_ARG ( 3 - roman_cos 2 italic_i ) ,
h[2,±2]λ[λ]=316(56cosi+cos2i),subscriptsuperscript𝜆delimited-[]𝜆2plus-or-minus2316minus-or-plus56𝑖2𝑖\displaystyle h^{\lambda[\lambda]}_{[2,\pm 2]}=-\frac{3}{16}(5\mp 6\cos i+\cos 2% i),italic_h start_POSTSUPERSCRIPT italic_λ [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 16 end_ARG ( 5 ∓ 6 roman_cos italic_i + roman_cos 2 italic_i ) , (154)
h[1,0]ξ=116(715cos2i),subscriptsuperscript𝜉101167152𝑖\displaystyle h^{\xi}_{[1,0]}=\frac{1}{16}(7-15\cos 2i),italic_h start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 16 end_ARG ( 7 - 15 roman_cos 2 italic_i ) ,
h[1,±2]ξ=332(712cosi+5cos2i),subscriptsuperscript𝜉1plus-or-minus2332minus-or-plus712𝑖52𝑖\displaystyle h^{\xi}_{[1,\pm 2]}=\frac{3}{32}(7\mp 12\cos i+5\cos 2i),italic_h start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( 7 ∓ 12 roman_cos italic_i + 5 roman_cos 2 italic_i ) ,
h[3,0]ξ=38sin2i,h[3,±2]ξ=332(34cosi+cos2i),formulae-sequencesubscriptsuperscript𝜉3038superscript2𝑖subscriptsuperscript𝜉3plus-or-minus2332minus-or-plus34𝑖2𝑖\displaystyle h^{\xi}_{[3,0]}=\frac{3}{8}\sin^{2}i,\qquad h^{\xi}_{[3,\pm 2]}=% \frac{3}{32}(3\mp 4\cos i+\cos 2i),italic_h start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 8 end_ARG roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_i , italic_h start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( 3 ∓ 4 roman_cos italic_i + roman_cos 2 italic_i ) ,
h[1,0]η=116(113cos2i),subscriptsuperscript𝜂101161132𝑖\displaystyle h^{\eta}_{[1,0]}=-\frac{1}{16}(11-3\cos 2i),italic_h start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 0 ] end_POSTSUBSCRIPT = - divide start_ARG 1 end_ARG start_ARG 16 end_ARG ( 11 - 3 roman_cos 2 italic_i ) ,
h[1,±2]η=332(1112cosi+cos2i),subscriptsuperscript𝜂1plus-or-minus2332minus-or-plus1112𝑖2𝑖\displaystyle h^{\eta}_{[1,\pm 2]}=-\frac{3}{32}(11\mp 12\cos i+\cos 2i),italic_h start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( 11 ∓ 12 roman_cos italic_i + roman_cos 2 italic_i ) ,
h[3,0]η=38sin2i,h[3,±2]η=332(34cosi+cos2i).formulae-sequencesubscriptsuperscript𝜂3038superscript2𝑖subscriptsuperscript𝜂3plus-or-minus2332minus-or-plus34𝑖2𝑖\displaystyle h^{\eta}_{[3,0]}=\frac{3}{8}\sin^{2}i,\qquad h^{\eta}_{[3,\pm 2]% }=\frac{3}{32}(3\mp 4\cos i+\cos 2i).italic_h start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 8 end_ARG roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_i , italic_h start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( 3 ∓ 4 roman_cos italic_i + roman_cos 2 italic_i ) .

C.2 Lunar perturbation solution σm(t)subscript𝜎m𝑡\sigma_{\text{m}}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT ( italic_t )
C.2 月球扰动解 σm(t)subscript𝜎m𝑡\sigma_{\text{m}}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT ( italic_t )

The secular variation σmc(t)superscriptsubscript𝜎m𝑐𝑡\sigma_{\text{m}}^{c}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ), special long-period variation σml[c](t)superscriptsubscript𝜎m𝑙delimited-[]𝑐𝑡\sigma_{\text{m}}^{l[c]}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ), general long-period variation σml[l](t)superscriptsubscript𝜎m𝑙delimited-[]𝑙𝑡\sigma_{\text{m}}^{l[l]}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ( italic_t ), and short-period variation σms(t)superscriptsubscript𝜎m𝑠𝑡\sigma_{\text{m}}^{s}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t ) within σm(t)subscript𝜎m𝑡\sigma_{\text{m}}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT ( italic_t ) are shown as follows:
长期变化 σmc(t)superscriptsubscript𝜎m𝑐𝑡\sigma_{\text{m}}^{c}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ) 、特殊长期变化 σml[c](t)superscriptsubscript𝜎m𝑙delimited-[]𝑐𝑡\sigma_{\text{m}}^{l[c]}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) 、一般长期变化 σml[l](t)superscriptsubscript𝜎m𝑙delimited-[]𝑙𝑡\sigma_{\text{m}}^{l[l]}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ( italic_t ) 和内部 σm(t)subscript𝜎m𝑡\sigma_{\text{m}}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT ( italic_t ) 的短期变化 σms(t)superscriptsubscript𝜎m𝑠𝑡\sigma_{\text{m}}^{s}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t ) 如下所示:

(1) Secular terms with the form σmc(t)superscriptsubscript𝜎m𝑐𝑡\sigma_{\text{m}}^{c}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ):
(1) 形式 σmc(t)superscriptsubscript𝜎m𝑐𝑡\sigma_{\text{m}}^{c}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ( italic_t ) 为 :

amc=superscriptsubscript𝑎m𝑐absent\displaystyle a_{\text{m}}^{c}=italic_a start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = imc=0,superscriptsubscript𝑖m𝑐0\displaystyle~{}i_{\text{m}}^{c}=0,italic_i start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = 0 , (155)
Ωmc=superscriptsubscriptΩm𝑐absent\displaystyle\Omega_{\text{m}}^{c}=roman_Ω start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = nΩmt,subscript𝑛Ωm𝑡\displaystyle~{}n_{\Omega{\text{m}}}\,t,italic_n start_POSTSUBSCRIPT roman_Ω m end_POSTSUBSCRIPT italic_t , (156)
λmc=superscriptsubscript𝜆m𝑐absent\displaystyle\lambda_{\text{m}}^{c}=italic_λ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = nλmt,subscript𝑛𝜆m𝑡\displaystyle~{}n_{\lambda\text{m}}\,t,italic_n start_POSTSUBSCRIPT italic_λ m end_POSTSUBSCRIPT italic_t , (157)

where 哪里

nΩm:=assignsubscript𝑛Ωmabsent\displaystyle n_{\Omega{\text{m}}}:=italic_n start_POSTSUBSCRIPT roman_Ω m end_POSTSUBSCRIPT := 3cm(2)16cosi(1+3cos2i3)45cm(4)327683superscriptsubscript𝑐m216𝑖132subscript𝑖345superscriptsubscript𝑐m432768\displaystyle-\frac{3\,c_{\text{m}}^{(2)}}{16}\cos i\,(1+3\cos 2i_{3})-\frac{4% 5\,c_{\text{m}}^{(4)}}{32768}- divide start_ARG 3 italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT end_ARG start_ARG 16 end_ARG roman_cos italic_i ( 1 + 3 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) - divide start_ARG 45 italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 4 ) end_POSTSUPERSCRIPT end_ARG start_ARG 32768 end_ARG
×(9cosi+7cos3i)(9+20cos2i3+35cos4i3)absent9𝑖73𝑖9202subscript𝑖3354subscript𝑖3\displaystyle\times(9\cos i+7\cos 3i)(9+20\cos 2i_{3}+35\cos 4i_{3})× ( 9 roman_cos italic_i + 7 roman_cos 3 italic_i ) ( 9 + 20 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 35 roman_cos 4 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )
525cm(6)16777216(50cosi+45cos3i+33cos5i)525superscriptsubscript𝑐m61677721650𝑖453𝑖335𝑖\displaystyle-\frac{525\,c_{\text{m}}^{(6)}}{16777216}(50\cos i+45\cos 3i+33% \cos 5i)- divide start_ARG 525 italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 6 ) end_POSTSUPERSCRIPT end_ARG start_ARG 16777216 end_ARG ( 50 roman_cos italic_i + 45 roman_cos 3 italic_i + 33 roman_cos 5 italic_i )
×(50+105cos2i3+126cos4i3+231cos6i3),absent501052subscript𝑖31264subscript𝑖32316subscript𝑖3\displaystyle\times(50+105\cos 2i_{3}+126\cos 4i_{3}+231\cos 6i_{3}),× ( 50 + 105 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 126 roman_cos 4 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 231 roman_cos 6 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (158)
nλm:=assignsubscript𝑛𝜆mabsent\displaystyle n_{\lambda\text{m}}:=italic_n start_POSTSUBSCRIPT italic_λ m end_POSTSUBSCRIPT := 132cm(2)(13cos2i)(1+3cos2i3)+9cm(4)65536132superscriptsubscript𝑐m2132𝑖132subscript𝑖39superscriptsubscript𝑐m465536\displaystyle~{}\frac{1}{32}c_{\text{m}}^{(2)}(1-3\cos 2i)(1+3\cos 2i_{3})+% \frac{9\,c_{\text{m}}^{(4)}}{65536}divide start_ARG 1 end_ARG start_ARG 32 end_ARG italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ( 1 - 3 roman_cos 2 italic_i ) ( 1 + 3 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) + divide start_ARG 9 italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 4 ) end_POSTSUPERSCRIPT end_ARG start_ARG 65536 end_ARG
×(27+40cos2i35cos4i)(9+20cos2i3\displaystyle\times(27+40\cos 2i-35\cos 4i)(9+20\cos 2i_{3}× ( 27 + 40 roman_cos 2 italic_i - 35 roman_cos 4 italic_i ) ( 9 + 20 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+35cos4i3)+75cm(6)33554432(250+455cos2i\displaystyle+35\cos 4i_{3})+\frac{75\,c_{\text{m}}^{(6)}}{33554432}(250+455% \cos 2i+ 35 roman_cos 4 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) + divide start_ARG 75 italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 6 ) end_POSTSUPERSCRIPT end_ARG start_ARG 33554432 end_ARG ( 250 + 455 roman_cos 2 italic_i
+294cos4i231cos6i)(50+105cos2i3\displaystyle+294\cos 4i-231\cos 6i)(50+105\cos 2i_{3}+ 294 roman_cos 4 italic_i - 231 roman_cos 6 italic_i ) ( 50 + 105 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+126cos4i3+231cos6i3),\displaystyle+126\cos 4i_{3}+231\cos 6i_{3}),+ 126 roman_cos 4 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 231 roman_cos 6 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (159)

and cm(N):=μ3n¯r¯23(a¯r¯2)N2assignsuperscriptsubscript𝑐m𝑁subscript𝜇3¯𝑛superscriptsubscript¯𝑟23superscript¯𝑎subscript¯𝑟2𝑁2c_{\text{m}}^{(N)}:=\frac{\mu_{3}}{\overline{n}\,\overline{r}_{2}^{3}}(\frac{% \overline{a}}{\overline{r}_{2}})^{N-2}italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_N ) end_POSTSUPERSCRIPT := divide start_ARG italic_μ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG over¯ start_ARG italic_n end_ARG over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG ( divide start_ARG over¯ start_ARG italic_a end_ARG end_ARG start_ARG over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_N - 2 end_POSTSUPERSCRIPT. cm(N):=μ3n¯r¯23(a¯r¯2)N2assignsuperscriptsubscript𝑐m𝑁subscript𝜇3¯𝑛superscriptsubscript¯𝑟23superscript¯𝑎subscript¯𝑟2𝑁2c_{\text{m}}^{(N)}:=\frac{\mu_{3}}{\overline{n}\,\overline{r}_{2}^{3}}(\frac{% \overline{a}}{\overline{r}_{2}})^{N-2}italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_N ) end_POSTSUPERSCRIPT := divide start_ARG italic_μ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG over¯ start_ARG italic_n end_ARG over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG ( divide start_ARG over¯ start_ARG italic_a end_ARG end_ARG start_ARG over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT italic_N - 2 end_POSTSUPERSCRIPT .

(2) Special long-period terms with the form σml[c](t)superscriptsubscript𝜎m𝑙delimited-[]𝑐𝑡\sigma_{\text{m}}^{l[c]}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ):
(2) 特殊长期条款,格式 σml[c](t)superscriptsubscript𝜎m𝑙delimited-[]𝑐𝑡\sigma_{\text{m}}^{l[c]}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) 为:

aml[c]superscriptsubscript𝑎m𝑙delimited-[]𝑐\displaystyle a_{\text{m}}^{l[c]}italic_a start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT =0,absent0\displaystyle=0,= 0 , (160)
iml[c]superscriptsubscript𝑖m𝑙delimited-[]𝑐\displaystyle i_{\text{m}}^{l[c]}italic_i start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT =N=2(ΔN=2)𝒩q=1Nf(q)i(N)cosqθ3,absentsuperscriptsubscript𝑁2Δ𝑁2𝒩superscriptsubscript𝑞1𝑁subscriptsuperscript𝑓𝑖𝑁𝑞𝑞subscript𝜃3\displaystyle=\sum_{\begin{subarray}{c}N=2\\ (\Delta N=2)\end{subarray}}^{\mathcal{N}}\sum_{q=1}^{N}f^{i(N)}_{(q)}\cos q\,% \theta_{3},= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_N = 2 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_N = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_q = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_i ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT roman_cos italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , (161)
Ωml[c]superscriptsubscriptΩm𝑙delimited-[]𝑐\displaystyle\Omega_{\text{m}}^{l[c]}roman_Ω start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT =N=2(ΔN=2)𝒩q=1Nf(q)Ω(N)sinqθ3,absentsuperscriptsubscript𝑁2Δ𝑁2𝒩superscriptsubscript𝑞1𝑁subscriptsuperscript𝑓Ω𝑁𝑞𝑞subscript𝜃3\displaystyle=\sum_{\begin{subarray}{c}N=2\\ (\Delta N=2)\end{subarray}}^{\mathcal{N}}\sum_{q=1}^{N}f^{\Omega(N)}_{(q)}\sin q% \,\theta_{3},= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_N = 2 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_N = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_q = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT roman_Ω ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT roman_sin italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , (162)
λml[c]superscriptsubscript𝜆m𝑙delimited-[]𝑐\displaystyle\lambda_{\text{m}}^{l[c]}italic_λ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT =N=2(ΔN=2)𝒩q=1Nf(q)λ(N)sinqθ3,absentsuperscriptsubscript𝑁2Δ𝑁2𝒩superscriptsubscript𝑞1𝑁subscriptsuperscript𝑓𝜆𝑁𝑞𝑞subscript𝜃3\displaystyle=\sum_{\begin{subarray}{c}N=2\\ (\Delta N=2)\end{subarray}}^{\mathcal{N}}\sum_{q=1}^{N}f^{\lambda(N)}_{(q)}% \sin q\,\theta_{3},= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_N = 2 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_N = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_q = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_λ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT roman_sin italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , (163)

with 

f(q)σ(N):=cm(N)qnθ3×f[q]σ(N)(i),assignsubscriptsuperscript𝑓𝜎𝑁𝑞superscriptsubscript𝑐m𝑁𝑞subscript𝑛subscript𝜃3subscriptsuperscript𝑓𝜎𝑁delimited-[]𝑞𝑖\displaystyle f^{\sigma(N)}_{(q)}:=\frac{c_{\text{m}}^{(N)}}{q\,n_{\theta_{3}}% }\times f^{\sigma(N)}_{[q]}(i),italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT := divide start_ARG italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_N ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_q italic_n start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG × italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ) , (164)

and 

θ3(t):=Ω¯(t)Ω¯3(t)=nθ3(tt0)+θ30,assignsubscript𝜃3𝑡¯Ω𝑡subscript¯Ω3𝑡subscript𝑛subscript𝜃3𝑡subscript𝑡0subscript𝜃subscript30\displaystyle\theta_{3}(t):=\overline{\Omega}(t)-\overline{\Omega}_{3}(t)=n_{% \theta_{3}}(t-t_{0})+\theta_{3_{0}},italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) := over¯ start_ARG roman_Ω end_ARG ( italic_t ) - over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) = italic_n start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_t - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + italic_θ start_POSTSUBSCRIPT 3 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , (165)
nθ3=nΩnΩ3,θ30=Ω¯0Ω30.formulae-sequencesubscript𝑛subscript𝜃3superscriptsubscript𝑛Ωsubscript𝑛subscriptΩ3subscript𝜃subscript30subscript¯Ω0subscriptΩsubscript30\displaystyle n_{\theta_{3}}=n_{\Omega}^{\prime}-n_{\Omega_{3}},\qquad\theta_{% 3_{0}}=\overline{\Omega}_{0}-\Omega_{3_{0}}.italic_n start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - italic_n start_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_θ start_POSTSUBSCRIPT 3 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = over¯ start_ARG roman_Ω end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - roman_Ω start_POSTSUBSCRIPT 3 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT . (166)

For explicit forms of f[q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁delimited-[]𝑞𝑖f^{\sigma(N)}_{[q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ), see Appendix D.1.
有关 的 f[q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁delimited-[]𝑞𝑖f^{\sigma(N)}_{[q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ) 明确形式,请参阅附录 D.1

(3) General long-period terms with the form σml[l](t)superscriptsubscript𝜎m𝑙delimited-[]𝑙𝑡\sigma_{\text{m}}^{l[l]}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ( italic_t ):
(3) 一般长期条款,格式 σml[l](t)superscriptsubscript𝜎m𝑙delimited-[]𝑙𝑡\sigma_{\text{m}}^{l[l]}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ( italic_t ) 为:

aml[l]superscriptsubscript𝑎m𝑙delimited-[]𝑙\displaystyle a_{\text{m}}^{l[l]}italic_a start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT =0,absent0\displaystyle=0,= 0 , (167)

and 

ξml[l]superscriptsubscript𝜉m𝑙delimited-[]𝑙\displaystyle\xi_{\text{m}}^{l[l]}italic_ξ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT =N=3(ΔN=2)𝒩p>0(Δp=2)Nq=NNf(p,q)ξ(N)cos(pu3+qθ3),absentsuperscriptsubscript𝑁3Δ𝑁2𝒩superscriptsubscript𝑝0Δ𝑝2𝑁superscriptsubscript𝑞𝑁𝑁subscriptsuperscript𝑓𝜉𝑁𝑝𝑞𝑝subscript𝑢3𝑞subscript𝜃3\displaystyle=\sum_{\begin{subarray}{c}N=3\\ (\Delta N=2)\end{subarray}}^{\mathcal{N}}\sum_{\begin{subarray}{c}p>0\\ (\Delta p=2)\end{subarray}}^{N}\sum_{q=-N}^{N}f^{\xi(N)}_{(p,\,q)}\cos(p\,u_{3% }+q\,\theta_{3}),= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_N = 3 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_N = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p > 0 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_q = - italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_ξ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_p , italic_q ) end_POSTSUBSCRIPT roman_cos ( italic_p italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (168)
ηml[l]superscriptsubscript𝜂m𝑙delimited-[]𝑙\displaystyle\eta_{\text{m}}^{l[l]}italic_η start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT =N=3(ΔN=2)𝒩p>0(Δp=2)Nq=N(q0)Nf(p,q)η(N)sin(pu3+qθ3),absentsuperscriptsubscript𝑁3Δ𝑁2𝒩superscriptsubscript𝑝0Δ𝑝2𝑁superscriptsubscript𝑞𝑁𝑞0𝑁subscriptsuperscript𝑓𝜂𝑁𝑝𝑞𝑝subscript𝑢3𝑞subscript𝜃3\displaystyle=\sum_{\begin{subarray}{c}N=3\\ (\Delta N=2)\end{subarray}}^{\mathcal{N}}\sum_{\begin{subarray}{c}p>0\\ (\Delta p=2)\end{subarray}}^{N}\sum_{\begin{subarray}{c}q=-N\\ (q\neq 0)\end{subarray}}^{N}f^{\eta(N)}_{(p,\,q)}\sin(p\,u_{3}+q\,\theta_{3}),= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_N = 3 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_N = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p > 0 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_q = - italic_N end_CELL end_ROW start_ROW start_CELL ( italic_q ≠ 0 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_η ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_p , italic_q ) end_POSTSUBSCRIPT roman_sin ( italic_p italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (169)
iml[l]superscriptsubscript𝑖m𝑙delimited-[]𝑙\displaystyle i_{\text{m}}^{l[l]}italic_i start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT =N=2(ΔN=2)𝒩p>0(Δp=2)Nq=N(q0)Nf(p,q)i(N)cos(pu3+qθ3),absentsuperscriptsubscript𝑁2Δ𝑁2𝒩superscriptsubscript𝑝0Δ𝑝2𝑁superscriptsubscript𝑞𝑁𝑞0𝑁subscriptsuperscript𝑓𝑖𝑁𝑝𝑞𝑝subscript𝑢3𝑞subscript𝜃3\displaystyle=\sum_{\begin{subarray}{c}N=2\\ (\Delta N=2)\end{subarray}}^{\mathcal{N}}\sum_{\begin{subarray}{c}p>0\\ (\Delta p=2)\end{subarray}}^{N}\sum_{\begin{subarray}{c}q=-N\\ (q\neq 0)\end{subarray}}^{N}f^{i(N)}_{(p,\,q)}\cos(p\,u_{3}+q\,\theta_{3}),= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_N = 2 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_N = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p > 0 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_q = - italic_N end_CELL end_ROW start_ROW start_CELL ( italic_q ≠ 0 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_i ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_p , italic_q ) end_POSTSUBSCRIPT roman_cos ( italic_p italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (170)
Ωml[l]superscriptsubscriptΩm𝑙delimited-[]𝑙\displaystyle\Omega_{\text{m}}^{l[l]}roman_Ω start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT =N=2(ΔN=2)𝒩p>0(Δp=2)Nq=NNf(p,q)Ω(N)sin(pu3+qθ3),absentsuperscriptsubscript𝑁2Δ𝑁2𝒩superscriptsubscript𝑝0Δ𝑝2𝑁superscriptsubscript𝑞𝑁𝑁subscriptsuperscript𝑓Ω𝑁𝑝𝑞𝑝subscript𝑢3𝑞subscript𝜃3\displaystyle=\sum_{\begin{subarray}{c}N=2\\ (\Delta N=2)\end{subarray}}^{\mathcal{N}}\sum_{\begin{subarray}{c}p>0\\ (\Delta p=2)\end{subarray}}^{N}\sum_{q=-N}^{N}f^{\Omega(N)}_{(p,\,q)}\sin(p\,u% _{3}+q\,\theta_{3}),= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_N = 2 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_N = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p > 0 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_q = - italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT roman_Ω ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_p , italic_q ) end_POSTSUBSCRIPT roman_sin ( italic_p italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (171)
λml[l]superscriptsubscript𝜆m𝑙delimited-[]𝑙\displaystyle\lambda_{\text{m}}^{l[l]}italic_λ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT =N=2(ΔN=2)𝒩p>0(Δp=2)Nq=NNf(p,q)λ(N)sin(pu3+qθ3),absentsuperscriptsubscript𝑁2Δ𝑁2𝒩superscriptsubscript𝑝0Δ𝑝2𝑁superscriptsubscript𝑞𝑁𝑁subscriptsuperscript𝑓𝜆𝑁𝑝𝑞𝑝subscript𝑢3𝑞subscript𝜃3\displaystyle=\sum_{\begin{subarray}{c}N=2\\ (\Delta N=2)\end{subarray}}^{\mathcal{N}}\sum_{\begin{subarray}{c}p>0\\ (\Delta p=2)\end{subarray}}^{N}\sum_{q=-N}^{N}f^{\lambda(N)}_{(p,\,q)}\sin(p\,% u_{3}+q\,\theta_{3}),= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_N = 2 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_N = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p > 0 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_q = - italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_λ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_p , italic_q ) end_POSTSUBSCRIPT roman_sin ( italic_p italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (172)

with 

f(p,q)σ(N):=cm(N)pn3+qnθ3×f[p,q]σ(N)(i).assignsubscriptsuperscript𝑓𝜎𝑁𝑝𝑞superscriptsubscript𝑐m𝑁𝑝subscript𝑛3𝑞subscript𝑛subscript𝜃3subscriptsuperscript𝑓𝜎𝑁𝑝𝑞𝑖\displaystyle f^{\sigma(N)}_{(p,\,q)}:=\frac{c_{\text{m}}^{(N)}}{p\,n_{3}+q\,n% _{\theta_{3}}}\times f^{\sigma(N)}_{[p,\,q]}(i).italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_p , italic_q ) end_POSTSUBSCRIPT := divide start_ARG italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_N ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_p italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_n start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG × italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ) . (173)

For explicit forms of f[p,q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁𝑝𝑞𝑖f^{\sigma(N)}_{[p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ), see Appendix D.2.
有关 的 f[p,q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁𝑝𝑞𝑖f^{\sigma(N)}_{[p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ) 明确形式,请参阅附录 D.2

(4) Short-period terms with the form σms(t)superscriptsubscript𝜎m𝑠𝑡\sigma_{\text{m}}^{s}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t ):
(4) 短期条款,格式 σms(t)superscriptsubscript𝜎m𝑠𝑡\sigma_{\text{m}}^{s}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t ) 为:

amssuperscriptsubscript𝑎m𝑠\displaystyle a_{\text{m}}^{s}italic_a start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT =N=2𝒩κ>0(Δκ=2)Np=N(Δp=2)Nq=NNaf(κ,p,q)a(N)cosΛ,absentsuperscriptsubscript𝑁2𝒩superscriptsubscript𝜅0Δ𝜅2𝑁superscriptsubscript𝑝𝑁Δ𝑝2𝑁superscriptsubscript𝑞𝑁𝑁𝑎subscriptsuperscript𝑓𝑎𝑁𝜅𝑝𝑞Λ\displaystyle=~{}\sum_{N=2}^{\mathcal{N}}\sum_{\begin{subarray}{c}\kappa>0\\ (\Delta\kappa=2)\end{subarray}}^{N}\sum_{\begin{subarray}{c}p=-N\\ (\Delta p=2)\end{subarray}}^{N}\sum_{q=-N}^{N}a\,f^{a(N)}_{(\kappa,\,p,\,q)}% \cos\Lambda,= ∑ start_POSTSUBSCRIPT italic_N = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_κ > 0 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_κ = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p = - italic_N end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_q = - italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_a italic_f start_POSTSUPERSCRIPT italic_a ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_κ , italic_p , italic_q ) end_POSTSUBSCRIPT roman_cos roman_Λ , (174)
ξmssuperscriptsubscript𝜉m𝑠\displaystyle\xi_{\text{m}}^{s}italic_ξ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT =N=2𝒩κ>0(Δκ=2)N+1p=N(Δp=2)Nq=NNf(κ,p,q)ξ(N)cosΛ,absentsuperscriptsubscript𝑁2𝒩superscriptsubscript𝜅0Δ𝜅2𝑁1superscriptsubscript𝑝𝑁Δ𝑝2𝑁superscriptsubscript𝑞𝑁𝑁subscriptsuperscript𝑓𝜉𝑁𝜅𝑝𝑞Λ\displaystyle=\sum_{N=2}^{\mathcal{N}}\sum_{\begin{subarray}{c}\kappa>0\\ (\Delta\kappa=2)\end{subarray}}^{N+1}\sum_{\begin{subarray}{c}p=-N\\ (\Delta p=2)\end{subarray}}^{N}\sum_{q=-N}^{N}f^{\xi(N)}_{(\kappa,\,p,\,q)}% \cos\Lambda,= ∑ start_POSTSUBSCRIPT italic_N = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_κ > 0 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_κ = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N + 1 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p = - italic_N end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_q = - italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_ξ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_κ , italic_p , italic_q ) end_POSTSUBSCRIPT roman_cos roman_Λ , (175)
ηmssuperscriptsubscript𝜂m𝑠\displaystyle\eta_{\text{m}}^{s}italic_η start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT =N=2𝒩κ>0(Δκ=2)N+1p=N(Δp=2)Nq=NNf(κ,p,q)η(N)sinΛ,absentsuperscriptsubscript𝑁2𝒩superscriptsubscript𝜅0Δ𝜅2𝑁1superscriptsubscript𝑝𝑁Δ𝑝2𝑁superscriptsubscript𝑞𝑁𝑁subscriptsuperscript𝑓𝜂𝑁𝜅𝑝𝑞Λ\displaystyle=\sum_{N=2}^{\mathcal{N}}\sum_{\begin{subarray}{c}\kappa>0\\ (\Delta\kappa=2)\end{subarray}}^{N+1}\sum_{\begin{subarray}{c}p=-N\\ (\Delta p=2)\end{subarray}}^{N}\sum_{q=-N}^{N}f^{\eta(N)}_{(\kappa,\,p,\,q)}% \sin\Lambda,= ∑ start_POSTSUBSCRIPT italic_N = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_κ > 0 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_κ = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N + 1 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p = - italic_N end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_q = - italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_η ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_κ , italic_p , italic_q ) end_POSTSUBSCRIPT roman_sin roman_Λ , (176)
imssuperscriptsubscript𝑖m𝑠\displaystyle i_{\text{m}}^{s}italic_i start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT =N=2𝒩κ>0(Δκ=2)Np=N(Δp=2)Nq=NNf(κ,p,q)i(N)cosΛ,absentsuperscriptsubscript𝑁2𝒩superscriptsubscript𝜅0Δ𝜅2𝑁superscriptsubscript𝑝𝑁Δ𝑝2𝑁superscriptsubscript𝑞𝑁𝑁subscriptsuperscript𝑓𝑖𝑁𝜅𝑝𝑞Λ\displaystyle=\sum_{N=2}^{\mathcal{N}}\sum_{\begin{subarray}{c}\kappa>0\\ (\Delta\kappa=2)\end{subarray}}^{N}\sum_{\begin{subarray}{c}p=-N\\ (\Delta p=2)\end{subarray}}^{N}\sum_{q=-N}^{N}f^{i(N)}_{(\kappa,\,p,\,q)}\cos\Lambda,= ∑ start_POSTSUBSCRIPT italic_N = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_κ > 0 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_κ = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p = - italic_N end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_q = - italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_i ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_κ , italic_p , italic_q ) end_POSTSUBSCRIPT roman_cos roman_Λ , (177)
ΩmssuperscriptsubscriptΩm𝑠\displaystyle\Omega_{\text{m}}^{s}roman_Ω start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT =N=2𝒩κ>0(Δκ=2)Np=N(Δp=2)Nq=NNf(κ,p,q)Ω(N)sinΛ,absentsuperscriptsubscript𝑁2𝒩superscriptsubscript𝜅0Δ𝜅2𝑁superscriptsubscript𝑝𝑁Δ𝑝2𝑁superscriptsubscript𝑞𝑁𝑁subscriptsuperscript𝑓Ω𝑁𝜅𝑝𝑞Λ\displaystyle=\sum_{N=2}^{\mathcal{N}}\sum_{\begin{subarray}{c}\kappa>0\\ (\Delta\kappa=2)\end{subarray}}^{N}\sum_{\begin{subarray}{c}p=-N\\ (\Delta p=2)\end{subarray}}^{N}\sum_{q=-N}^{N}f^{\Omega(N)}_{(\kappa,\,p,\,q)}% \sin\Lambda,= ∑ start_POSTSUBSCRIPT italic_N = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_κ > 0 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_κ = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p = - italic_N end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_q = - italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT roman_Ω ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_κ , italic_p , italic_q ) end_POSTSUBSCRIPT roman_sin roman_Λ , (178)
λmssuperscriptsubscript𝜆m𝑠\displaystyle\lambda_{\text{m}}^{s}italic_λ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT =N=2𝒩κ>0(Δκ=2)Np=N(Δp=2)Nq=NNf(κ,p,q)λ(N)sinΛ,absentsuperscriptsubscript𝑁2𝒩superscriptsubscript𝜅0Δ𝜅2𝑁superscriptsubscript𝑝𝑁Δ𝑝2𝑁superscriptsubscript𝑞𝑁𝑁subscriptsuperscript𝑓𝜆𝑁𝜅𝑝𝑞Λ\displaystyle=\sum_{N=2}^{\mathcal{N}}\sum_{\begin{subarray}{c}\kappa>0\\ (\Delta\kappa=2)\end{subarray}}^{N}\sum_{\begin{subarray}{c}p=-N\\ (\Delta p=2)\end{subarray}}^{N}\sum_{q=-N}^{N}f^{\lambda(N)}_{(\kappa,\,p,\,q)% }\sin\Lambda,= ∑ start_POSTSUBSCRIPT italic_N = 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_κ > 0 end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_κ = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_p = - italic_N end_CELL end_ROW start_ROW start_CELL ( roman_Δ italic_p = 2 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_q = - italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_λ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_κ , italic_p , italic_q ) end_POSTSUBSCRIPT roman_sin roman_Λ , (179)

with Λ:=κλ+pu3+qθ3assignΛ𝜅𝜆𝑝subscript𝑢3𝑞subscript𝜃3\Lambda:=\kappa\,\lambda+p\,u_{3}+q\,\theta_{3}roman_Λ := italic_κ italic_λ + italic_p italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, 其中 Λ:=κλ+pu3+qθ3assignΛ𝜅𝜆𝑝subscript𝑢3𝑞subscript𝜃3\Lambda:=\kappa\,\lambda+p\,u_{3}+q\,\theta_{3}roman_Λ := italic_κ italic_λ + italic_p italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT

f(κ,p,q)σ(N):=cm(N)κnλ+pn3+qnθ3×f[κ,p,q]σ(N)(i)assignsubscriptsuperscript𝑓𝜎𝑁𝜅𝑝𝑞superscriptsubscript𝑐m𝑁𝜅superscriptsubscript𝑛𝜆𝑝subscript𝑛3𝑞subscript𝑛subscript𝜃3subscriptsuperscript𝑓𝜎𝑁𝜅𝑝𝑞𝑖\displaystyle f^{\sigma(N)}_{(\kappa,\,p,\,q)}:=\frac{c_{\text{m}}^{(N)}}{% \kappa\,n_{\lambda}^{\prime}+p\,n_{3}+q\,n_{\theta_{3}}}\times f^{\sigma(N)}_{% [\kappa,\,p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_κ , italic_p , italic_q ) end_POSTSUBSCRIPT := divide start_ARG italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_N ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_κ italic_n start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + italic_p italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_n start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG × italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ) (180)

for σ{a,ξ,η,i,Ω}𝜎𝑎𝜉𝜂𝑖Ω\sigma\in\{a,\xi,\eta,i,\Omega\}italic_σ ∈ { italic_a , italic_ξ , italic_η , italic_i , roman_Ω }, and for σ{a,ξ,η,i,Ω}𝜎𝑎𝜉𝜂𝑖Ω\sigma\in\{a,\xi,\eta,i,\Omega\}italic_σ ∈ { italic_a , italic_ξ , italic_η , italic_i , roman_Ω }

f(κ,p,q)λ(N):=assignsubscriptsuperscript𝑓𝜆𝑁𝜅𝑝𝑞absent\displaystyle f^{\lambda(N)}_{(\kappa,\,p,\,q)}:=italic_f start_POSTSUPERSCRIPT italic_λ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_κ , italic_p , italic_q ) end_POSTSUBSCRIPT := 3cm(N)n(κnλ+pn3+qnθ3)2×f[κ,p,q]λ(N)[a](i)3superscriptsubscript𝑐m𝑁𝑛superscript𝜅superscriptsubscript𝑛𝜆𝑝subscript𝑛3𝑞subscript𝑛subscript𝜃32subscriptsuperscript𝑓𝜆𝑁delimited-[]𝑎𝜅𝑝𝑞𝑖\displaystyle~{}\frac{3\,c_{\text{m}}^{(N)}\,n}{(\kappa\,n_{\lambda}^{\prime}+% p\,n_{3}+q\,n_{\theta_{3}})^{2}}\times f^{\lambda(N)[a]}_{[\kappa,\,p,\,q]}(i)divide start_ARG 3 italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_N ) end_POSTSUPERSCRIPT italic_n end_ARG start_ARG ( italic_κ italic_n start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + italic_p italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_n start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG × italic_f start_POSTSUPERSCRIPT italic_λ ( italic_N ) [ italic_a ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i )
+cm(N)κnλ+pn3+qnθ3×f[κ,p,q]λ(N)[λ](i).superscriptsubscript𝑐m𝑁𝜅superscriptsubscript𝑛𝜆𝑝subscript𝑛3𝑞subscript𝑛subscript𝜃3subscriptsuperscript𝑓𝜆𝑁delimited-[]𝜆𝜅𝑝𝑞𝑖\displaystyle+\frac{c_{\text{m}}^{(N)}}{\kappa\,n_{\lambda}^{\prime}+p\,n_{3}+% q\,n_{\theta_{3}}}\times f^{\lambda(N)[\lambda]}_{[\kappa,\,p,\,q]}(i).+ divide start_ARG italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_N ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_κ italic_n start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + italic_p italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_n start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG × italic_f start_POSTSUPERSCRIPT italic_λ ( italic_N ) [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ) . (181)

For explicit forms of f[κ,p,q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁𝜅𝑝𝑞𝑖f^{\sigma(N)}_{[\kappa,\,p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ), see Appendix D.3.
有关 f[κ,p,q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁𝜅𝑝𝑞𝑖f^{\sigma(N)}_{[\kappa,\,p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ) 的显式形式,请参阅附录 D.3

C.3 Earth’s J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT perturbation solution σJ2(t)subscript𝜎subscript𝐽2𝑡\sigma_{\!J_{2}}(t)italic_σ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_t )
C.3 地球的 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 扰动解 σJ2(t)subscript𝜎subscript𝐽2𝑡\sigma_{\!J_{2}}(t)italic_σ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_t )

The Earth’s J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT perturbation solution, σJ2(eq)subscript𝜎subscript𝐽2(eq)\sigma_{\!J_{2}\text{(eq)}}italic_σ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (eq) end_POSTSUBSCRIPT in the geocentric equatorial coordinate system, can be derived straightforwardly from Eqs. (55) and (128)-(131). For consistency with the solar and lunar perturbation solutions, the ecliptic representation σJ2subscript𝜎subscript𝐽2\sigma_{\!J_{2}}italic_σ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is essential. Given that the J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT perturbation is significantly smaller than lunisolar perturbations, this study focuses on the secular components, including iJ2csuperscriptsubscript𝑖𝐽2𝑐i_{J2}^{c}italic_i start_POSTSUBSCRIPT italic_J 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT, ΩJ2csuperscriptsubscriptΩ𝐽2𝑐\Omega_{J2}^{c}roman_Ω start_POSTSUBSCRIPT italic_J 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT, and λJ2csuperscriptsubscript𝜆𝐽2𝑐\lambda_{J2}^{c}italic_λ start_POSTSUBSCRIPT italic_J 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT. Employing spherical trigonometry [35, 33] and variational method, we have
在地心赤道坐标系 σJ2(eq)subscript𝜎subscript𝐽2(eq)\sigma_{\!J_{2}\text{(eq)}}italic_σ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (eq) end_POSTSUBSCRIPT 中,地球的 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 扰动解可以直接从 Eqs 推导出来。(55) 和 (128)-(131)。为了与太阳和月球扰动解保持一致,黄道表示 σJ2subscript𝜎subscript𝐽2\sigma_{\!J_{2}}italic_σ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT 是必不可少的。 J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 鉴于扰动明显小于阴阳扰动,本研究侧重于长期分量,包括 iJ2csuperscriptsubscript𝑖𝐽2𝑐i_{J2}^{c}italic_i start_POSTSUBSCRIPT italic_J 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPTΩJ2csuperscriptsubscriptΩ𝐽2𝑐\Omega_{J2}^{c}roman_Ω start_POSTSUBSCRIPT italic_J 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPTλJ2csuperscriptsubscript𝜆𝐽2𝑐\lambda_{J2}^{c}italic_λ start_POSTSUBSCRIPT italic_J 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT 。采用球面三角学 [3533] 和变分方法,我们有

iJ2csuperscriptsubscript𝑖subscript𝐽2𝑐\displaystyle i_{J_{2}}^{c}italic_i start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT =sinΩsinϵΩJ2(eq)c=niJ2t,absentΩitalic-ϵsuperscriptsubscriptΩsubscript𝐽2(eq)𝑐subscript𝑛subscript𝑖subscript𝐽2𝑡\displaystyle=\sin\Omega\sin\epsilon\,\Omega_{J_{2}\text{(eq)}}^{c}=n_{i_{J_{2% }}}t,= roman_sin roman_Ω roman_sin italic_ϵ roman_Ω start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (eq) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = italic_n start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_t , (182)
ΩJ2csuperscriptsubscriptΩsubscript𝐽2𝑐\displaystyle\Omega_{J_{2}}^{c}roman_Ω start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT =(cosϵ+coticosΩsinϵ)ΩJ2(eq)c=nΩJ2t,absentitalic-ϵ𝑖Ωitalic-ϵsuperscriptsubscriptΩsubscript𝐽2(eq)𝑐subscript𝑛subscriptΩsubscript𝐽2𝑡\displaystyle=(\cos\epsilon+\cot i\cos\Omega\sin\epsilon)\,\Omega_{J_{2}\text{% (eq)}}^{c}=n_{\Omega_{J_{2}}}t,= ( roman_cos italic_ϵ + roman_cot italic_i roman_cos roman_Ω roman_sin italic_ϵ ) roman_Ω start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (eq) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = italic_n start_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_t , (183)
λJ2csuperscriptsubscript𝜆subscript𝐽2𝑐\displaystyle\lambda_{J_{2}}^{c}italic_λ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT =λJ2(eq)ccosΩsinisinϵΩJ2(eq)c=nλJ2t,absentsuperscriptsubscript𝜆subscript𝐽2(eq)𝑐Ω𝑖italic-ϵsuperscriptsubscriptΩsubscript𝐽2(eq)𝑐subscript𝑛subscript𝜆subscript𝐽2𝑡\displaystyle=\lambda_{J_{2}\text{(eq)}}^{c}-\frac{\cos\Omega}{\sin i}\sin% \epsilon\,\Omega_{J_{2}\text{(eq)}}^{c}=n_{\lambda_{J_{2}}}\,t,= italic_λ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (eq) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT - divide start_ARG roman_cos roman_Ω end_ARG start_ARG roman_sin italic_i end_ARG roman_sin italic_ϵ roman_Ω start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (eq) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = italic_n start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_t , (184)

with 

niJ2subscript𝑛subscript𝑖subscript𝐽2\displaystyle n_{i_{J_{2}}}italic_n start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT :=32cJ2sinΩsinϵcosieq,assignabsent32subscript𝑐subscript𝐽2Ωitalic-ϵsubscript𝑖eq\displaystyle:=-\frac{3}{2}c_{J_{2}}\sin\Omega\sin\epsilon\cos i_{\text{eq}},:= - divide start_ARG 3 end_ARG start_ARG 2 end_ARG italic_c start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_sin roman_Ω roman_sin italic_ϵ roman_cos italic_i start_POSTSUBSCRIPT eq end_POSTSUBSCRIPT , (185)
nΩJ2subscript𝑛subscriptΩsubscript𝐽2\displaystyle n_{\Omega_{J_{2}}}italic_n start_POSTSUBSCRIPT roman_Ω start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT :=32cJ2(cosϵ+coticosΩsinϵ)cosieq,assignabsent32subscript𝑐subscript𝐽2italic-ϵ𝑖Ωitalic-ϵsubscript𝑖eq\displaystyle:=-\frac{3}{2}c_{J_{2}}(\cos\epsilon+\cot i\cos\Omega\sin\epsilon% )\cos i_{\text{eq}},:= - divide start_ARG 3 end_ARG start_ARG 2 end_ARG italic_c start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( roman_cos italic_ϵ + roman_cot italic_i roman_cos roman_Ω roman_sin italic_ϵ ) roman_cos italic_i start_POSTSUBSCRIPT eq end_POSTSUBSCRIPT , (186)
nλJ2subscript𝑛subscript𝜆subscript𝐽2\displaystyle n_{\lambda_{J_{2}}}italic_n start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT :=32cJ2(1+2cos2ieq)+32cJ2cosieqcosΩsinisinϵ,assignabsent32subscript𝑐subscript𝐽2122subscript𝑖eq32subscript𝑐subscript𝐽2subscript𝑖eqΩ𝑖italic-ϵ\displaystyle:=\frac{3}{2}c_{\small{J_{2}}}(1+2\cos 2i_{\text{eq}})+\frac{3}{2% }c_{J_{2}}\cos i_{\text{eq}}\frac{\cos\Omega}{\sin i}\sin\epsilon,:= divide start_ARG 3 end_ARG start_ARG 2 end_ARG italic_c start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( 1 + 2 roman_cos 2 italic_i start_POSTSUBSCRIPT eq end_POSTSUBSCRIPT ) + divide start_ARG 3 end_ARG start_ARG 2 end_ARG italic_c start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_cos italic_i start_POSTSUBSCRIPT eq end_POSTSUBSCRIPT divide start_ARG roman_cos roman_Ω end_ARG start_ARG roman_sin italic_i end_ARG roman_sin italic_ϵ , (187)

where ϵ=23.439 291italic-ϵsuperscript23.439291\epsilon=23.439\,291^{\circ}italic_ϵ = 23.439 291 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT is the obliquity of the ecliptic, cJ2:=μJ2Re2a¯5n¯assignsubscript𝑐subscript𝐽2𝜇subscript𝐽2superscriptsubscript𝑅e2superscript¯𝑎5¯𝑛c_{\small{J_{2}}}:=\frac{\mu J_{2}R_{\text{e}}^{2}}{\overline{a}^{5}\,% \overline{n}}italic_c start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT := divide start_ARG italic_μ italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG over¯ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT over¯ start_ARG italic_n end_ARG end_ARG, and
其中 ϵ=23.439 291italic-ϵsuperscript23.439291\epsilon=23.439\,291^{\circ}italic_ϵ = 23.439 291 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT 是黄道的倾角, cJ2:=μJ2Re2a¯5n¯assignsubscript𝑐subscript𝐽2𝜇subscript𝐽2superscriptsubscript𝑅e2superscript¯𝑎5¯𝑛c_{\small{J_{2}}}:=\frac{\mu J_{2}R_{\text{e}}^{2}}{\overline{a}^{5}\,% \overline{n}}italic_c start_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT := divide start_ARG italic_μ italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG over¯ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT over¯ start_ARG italic_n end_ARG end_ARG

cosieq=cosicosϵsinicosΩsinϵ.subscript𝑖eq𝑖italic-ϵ𝑖Ωitalic-ϵ\displaystyle\cos i_{\text{eq}}=\cos i\cos\epsilon-\sin i\cos\Omega\sin\epsilon.roman_cos italic_i start_POSTSUBSCRIPT eq end_POSTSUBSCRIPT = roman_cos italic_i roman_cos italic_ϵ - roman_sin italic_i roman_cos roman_Ω roman_sin italic_ϵ . (188)

C.4 Expressions for ξcsubscript𝜉𝑐\xi_{c}italic_ξ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, ηcsubscript𝜂𝑐\eta_{c}italic_η start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, ξl[c]subscript𝜉𝑙delimited-[]𝑐\xi_{l[c]}italic_ξ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT, and ηl[c]subscript𝜂𝑙delimited-[]𝑐\eta_{l[c]}italic_η start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT
C.4 ξcsubscript𝜉𝑐\xi_{c}italic_ξ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPTηcsubscript𝜂𝑐\eta_{c}italic_η start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPTξl[c]subscript𝜉𝑙delimited-[]𝑐\xi_{l[c]}italic_ξ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPTηl[c]subscript𝜂𝑙delimited-[]𝑐\eta_{l[c]}italic_η start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT 的表达式

As pointed out in Appendix B, deriving the “secular” perturbation solutions ξnewc(t)subscriptsuperscript𝜉𝑐new𝑡\xi^{c}_{\text{new}}(t)italic_ξ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT new end_POSTSUBSCRIPT ( italic_t ) and ηnewc(t)subscriptsuperscript𝜂𝑐new𝑡\eta^{c}_{\text{new}}(t)italic_η start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT new end_POSTSUBSCRIPT ( italic_t ),
如附录 B 中所指出的,推导出“长期”扰动解 ξnewc(t)subscriptsuperscript𝜉𝑐new𝑡\xi^{c}_{\text{new}}(t)italic_ξ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT new end_POSTSUBSCRIPT ( italic_t )ηnewc(t)subscriptsuperscript𝜂𝑐new𝑡\eta^{c}_{\text{new}}(t)italic_η start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT new end_POSTSUBSCRIPT ( italic_t )

ξnewc(t)subscriptsuperscript𝜉𝑐new𝑡\displaystyle\xi^{c}_{\text{new}}(t)italic_ξ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT new end_POSTSUBSCRIPT ( italic_t ) :=ξc(t)+ξl[c](t),assignabsentsubscript𝜉𝑐𝑡subscript𝜉𝑙delimited-[]𝑐𝑡\displaystyle:=\xi_{c}(t)+\xi_{l[c]}(t),:= italic_ξ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t ) + italic_ξ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) , (189)
ηnewc(t)subscriptsuperscript𝜂𝑐new𝑡\displaystyle\eta^{c}_{\text{new}}(t)italic_η start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT new end_POSTSUBSCRIPT ( italic_t ) :=ηc(t)+ηl[c](t),assignabsentsubscript𝜂𝑐𝑡subscript𝜂𝑙delimited-[]𝑐𝑡\displaystyle:=\eta_{c}(t)+\eta_{l[c]}(t),:= italic_η start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t ) + italic_η start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) , (190)

involves solving the oscillation equations:
涉及求解振荡方程:

dξnewc(t)dtdsubscriptsuperscript𝜉𝑐new𝑡d𝑡\displaystyle\frac{\mathrm{d}\xi^{c}_{\text{new}}(t)}{\mathrm{d}t}divide start_ARG roman_d italic_ξ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT new end_POSTSUBSCRIPT ( italic_t ) end_ARG start_ARG roman_d italic_t end_ARG =hξηnewc(t)+f1ξl[c](t),absentsubscript𝜉subscriptsuperscript𝜂𝑐new𝑡superscriptsubscript𝑓1𝜉𝑙delimited-[]𝑐𝑡\displaystyle=h_{\xi}\,\eta^{c}_{\text{new}}(t)+f_{1\xi}^{l[c]}(t),= italic_h start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT italic_η start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT new end_POSTSUBSCRIPT ( italic_t ) + italic_f start_POSTSUBSCRIPT 1 italic_ξ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) , (191)
dηnewc(t)dtdsubscriptsuperscript𝜂𝑐new𝑡d𝑡\displaystyle\frac{\mathrm{d}\eta^{c}_{\text{new}}(t)}{\mathrm{d}t}divide start_ARG roman_d italic_η start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT new end_POSTSUBSCRIPT ( italic_t ) end_ARG start_ARG roman_d italic_t end_ARG =hηξnewc(t)+f1ηl[c](t).absentsubscript𝜂subscriptsuperscript𝜉𝑐new𝑡superscriptsubscript𝑓1𝜂𝑙delimited-[]𝑐𝑡\displaystyle=h_{\eta}\,\xi^{c}_{\text{new}}(t)+f_{1\eta}^{l[c]}(t).= italic_h start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT italic_ξ start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT new end_POSTSUBSCRIPT ( italic_t ) + italic_f start_POSTSUBSCRIPT 1 italic_η end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) . (192)

Due to the minor eccentricity variations (similar-to\sim106superscript10610^{-6}10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT) induced by the J2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT perturbation, only the effects of the lunisolar perturbations are taken into account, resulting in expressions for hξsubscript𝜉h_{\xi}italic_h start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT and hηsubscript𝜂h_{\eta}italic_h start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT given by
由于扰动引起的微小偏心率变化 ( similar-to\sim 106superscript10610^{-6}10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT ),仅考虑阴阳扰动的影响,因此表达式 for hξsubscript𝜉h_{\xi}italic_h start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPThηsubscript𝜂h_{\eta}italic_h start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPTJ2subscript𝐽2J_{2}italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT

hξhsξ+hmξ,hηhsη+hmη,formulae-sequencesimilar-to-or-equalssubscript𝜉superscriptsubscripts𝜉superscriptsubscriptm𝜉similar-to-or-equalssubscript𝜂superscriptsubscripts𝜂superscriptsubscriptm𝜂\displaystyle h_{\xi}\simeq h_{\text{s}}^{\xi}+h_{\text{m}}^{\xi},\qquad h_{% \eta}\simeq h_{\text{s}}^{\eta}+h_{\text{m}}^{\eta},italic_h start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT ≃ italic_h start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT + italic_h start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT , italic_h start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ≃ italic_h start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT + italic_h start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT , (193)

with 

hsξ=38cs(15cos2i),hsη=32cs,formulae-sequencesuperscriptsubscripts𝜉38subscript𝑐s152𝑖superscriptsubscripts𝜂32subscript𝑐s\displaystyle h_{\text{s}}^{\xi}=\frac{3}{8}c_{\text{s}}(1-5\cos 2i),\qquad h_% {\text{s}}^{\eta}=\frac{3}{2}c_{\text{s}},italic_h start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT = divide start_ARG 3 end_ARG start_ARG 8 end_ARG italic_c start_POSTSUBSCRIPT s end_POSTSUBSCRIPT ( 1 - 5 roman_cos 2 italic_i ) , italic_h start_POSTSUBSCRIPT s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT = divide start_ARG 3 end_ARG start_ARG 2 end_ARG italic_c start_POSTSUBSCRIPT s end_POSTSUBSCRIPT , (194)

and 

hmξ=superscriptsubscriptm𝜉absent\displaystyle h_{\text{m}}^{\xi}=italic_h start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT = 3cm(2)32(15cos2i)(1+3cos2i3)45cm(4)65536(3\displaystyle~{}\frac{3\,c_{\text{m}}^{(2)}}{32}(1-5\cos 2i)(1+3\cos 2i_{3})-% \frac{45\,c_{\text{m}}^{(4)}}{65536}(3divide start_ARG 3 italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT end_ARG start_ARG 32 end_ARG ( 1 - 5 roman_cos 2 italic_i ) ( 1 + 3 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) - divide start_ARG 45 italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 4 ) end_POSTSUPERSCRIPT end_ARG start_ARG 65536 end_ARG ( 3
+12cos2i+49cos4i)(9+20cos2i3+35cos4i3)\displaystyle+12\cos 2i+49\cos 4i)(9+20\cos 2i_{3}+35\cos 4i_{3})+ 12 roman_cos 2 italic_i + 49 roman_cos 4 italic_i ) ( 9 + 20 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 35 roman_cos 4 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )
525cm(6)33554432(30+71cos2i+114cos4i+297cos6i)525superscriptsubscript𝑐m63355443230712𝑖1144𝑖2976𝑖\displaystyle-\frac{525\,c_{\text{m}}^{(6)}}{33554432}(30+71\cos 2i+114\cos 4i% +297\cos 6i)- divide start_ARG 525 italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 6 ) end_POSTSUPERSCRIPT end_ARG start_ARG 33554432 end_ARG ( 30 + 71 roman_cos 2 italic_i + 114 roman_cos 4 italic_i + 297 roman_cos 6 italic_i )
×(50+105cos2i3+126cos4i3+231cos6i3),absent501052subscript𝑖31264subscript𝑖32316subscript𝑖3\displaystyle\times(50+105\cos 2i_{3}+126\cos 4i_{3}+231\cos 6i_{3}),× ( 50 + 105 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 126 roman_cos 4 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 231 roman_cos 6 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
hmη=superscriptsubscriptm𝜂absent\displaystyle h_{\text{m}}^{\eta}=italic_h start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT = 3cm(2)8(1+3cos2i3)3superscriptsubscript𝑐m28132subscript𝑖3\displaystyle~{}\frac{3\,c_{\text{m}}^{(2)}}{8}(1+3\cos 2i_{3})divide start_ARG 3 italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT end_ARG start_ARG 8 end_ARG ( 1 + 3 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )
+45cm(4)8192(3+5cos2i)(9+20cos2i3+35cos4i3)45superscriptsubscript𝑐m48192352𝑖9202subscript𝑖3354subscript𝑖3\displaystyle+\frac{45\,c_{\text{m}}^{(4)}}{8192}(3+5\cos 2i)(9+20\cos 2i_{3}+% 35\cos 4i_{3})+ divide start_ARG 45 italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 4 ) end_POSTSUPERSCRIPT end_ARG start_ARG 8192 end_ARG ( 3 + 5 roman_cos 2 italic_i ) ( 9 + 20 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 35 roman_cos 4 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )
+525cm(6)4194304(15+28cos2i+21cos4i)525superscriptsubscript𝑐m6419430415282𝑖214𝑖\displaystyle+\frac{525\,c_{\text{m}}^{(6)}}{4194304}(15+28\cos 2i+21\cos 4i)+ divide start_ARG 525 italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 6 ) end_POSTSUPERSCRIPT end_ARG start_ARG 4194304 end_ARG ( 15 + 28 roman_cos 2 italic_i + 21 roman_cos 4 italic_i )
×(50+105cos2i3+126cos4i3+231cos6i3).absent501052subscript𝑖31264subscript𝑖32316subscript𝑖3\displaystyle\times(50+105\cos 2i_{3}+126\cos 4i_{3}+231\cos 6i_{3}).× ( 50 + 105 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 126 roman_cos 4 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 231 roman_cos 6 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) . (195)

Moreover, the expressions for f1ξl[c](t)superscriptsubscript𝑓1𝜉𝑙delimited-[]𝑐𝑡f_{1\xi}^{l[c]}(t)italic_f start_POSTSUBSCRIPT 1 italic_ξ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) and f1ηl[c](t)superscriptsubscript𝑓1𝜂𝑙delimited-[]𝑐𝑡f_{1\eta}^{l[c]}(t)italic_f start_POSTSUBSCRIPT 1 italic_η end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) are

f1ξl[c]superscriptsubscript𝑓1𝜉𝑙delimited-[]𝑐\displaystyle f_{1\xi}^{l[c]}italic_f start_POSTSUBSCRIPT 1 italic_ξ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT =q=55f(q)ξsin(Δu3+qθ3),absentsuperscriptsubscript𝑞55subscriptsuperscript𝑓𝜉𝑞Δsubscript𝑢3𝑞subscript𝜃3\displaystyle=\sum_{q=-5}^{5}-f^{\xi}_{(q)}\sin(\Delta u_{3}+q\,\theta_{3}),= ∑ start_POSTSUBSCRIPT italic_q = - 5 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT - italic_f start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT roman_sin ( roman_Δ italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (196)
f1ηl[c]superscriptsubscript𝑓1𝜂𝑙delimited-[]𝑐\displaystyle f_{1\eta}^{l[c]}italic_f start_POSTSUBSCRIPT 1 italic_η end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT =q=5(q0)5f(q)ηcos(Δu3+qθ3),absentsuperscriptsubscript𝑞5𝑞05subscriptsuperscript𝑓𝜂𝑞Δsubscript𝑢3𝑞subscript𝜃3\displaystyle=\sum_{\begin{subarray}{c}q=-5\\ (q\neq 0)\end{subarray}}^{5}f^{\eta}_{(q)}\cos(\Delta u_{3}+q\,\theta_{3}),= ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_q = - 5 end_CELL end_ROW start_ROW start_CELL ( italic_q ≠ 0 ) end_CELL end_ROW end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT roman_cos ( roman_Δ italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (197)

with

f(q)ξsubscriptsuperscript𝑓𝜉𝑞\displaystyle f^{\xi}_{(q)}italic_f start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT :=cm(3)(4r3ε+u3A)f[q]ξ(3)(i)+cm(5)(6r3ε+u3A)f[q]ξ(5)(i),assignabsentsuperscriptsubscript𝑐m34superscriptsubscript𝑟3𝜀superscriptsubscript𝑢3𝐴subscriptsuperscript𝑓𝜉3delimited-[]𝑞𝑖superscriptsubscript𝑐m56superscriptsubscript𝑟3𝜀superscriptsubscript𝑢3𝐴subscriptsuperscript𝑓𝜉5delimited-[]𝑞𝑖\displaystyle:=c_{\text{m}}^{(3)}(4\,r_{3}^{\varepsilon}+u_{3}^{A})f^{\xi(3)}_% {[q]}(i)+c_{\text{m}}^{(5)}(6\,r_{3}^{\varepsilon}+u_{3}^{A})f^{\xi(5)}_{[q]}(% i),:= italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT ( 4 italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT + italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ) + italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 5 ) end_POSTSUPERSCRIPT ( 6 italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT + italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) italic_f start_POSTSUPERSCRIPT italic_ξ ( 5 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ) , (198)
f(q)ηsubscriptsuperscript𝑓𝜂𝑞\displaystyle f^{\eta}_{(q)}italic_f start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT :=cm(3)(4r3ε+u3A)f[q]η(3)(i)+cm(5)(6r3ε+u3A)f[q]η(5)(i),assignabsentsuperscriptsubscript𝑐m34superscriptsubscript𝑟3𝜀superscriptsubscript𝑢3𝐴subscriptsuperscript𝑓𝜂3delimited-[]𝑞𝑖superscriptsubscript𝑐m56superscriptsubscript𝑟3𝜀superscriptsubscript𝑢3𝐴subscriptsuperscript𝑓𝜂5delimited-[]𝑞𝑖\displaystyle:=c_{\text{m}}^{(3)}(4\,r_{3}^{\varepsilon}+u_{3}^{A})f^{\eta(3)}% _{[q]}(i)+c_{\text{m}}^{(5)}(6\,r_{3}^{\varepsilon}+u_{3}^{A})f^{\eta(5)}_{[q]% }(i),:= italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT ( 4 italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT + italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ) + italic_c start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 5 ) end_POSTSUPERSCRIPT ( 6 italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT + italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) italic_f start_POSTSUPERSCRIPT italic_η ( 5 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ) , (199)

and

Δu3(t):=u3(t)M¯3(t)=Δn3(tt0)+Δu30,assignΔsubscript𝑢3𝑡subscript𝑢3𝑡subscript¯𝑀3𝑡Δsubscript𝑛3𝑡subscript𝑡0Δsubscript𝑢subscript30\displaystyle\Delta u_{3}(t):=u_{3}(t)-\overline{M}_{3}(t)=\Delta n_{3}\,(t-t_% {0})+\Delta u_{3_{0}},roman_Δ italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) := italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) - over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) = roman_Δ italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t - italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) + roman_Δ italic_u start_POSTSUBSCRIPT 3 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , (200)
Δn3=n3nM3,Δu30=u30M30,formulae-sequenceΔsubscript𝑛3subscript𝑛3subscript𝑛subscript𝑀3Δsubscript𝑢subscript30subscript𝑢subscript30subscript𝑀subscript30\displaystyle\Delta n_{3}=n_{3}-n_{\!M_{3}},\qquad\Delta u_{3_{0}}=u_{3_{0}}-M% _{3_{0}},roman_Δ italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_n start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , roman_Δ italic_u start_POSTSUBSCRIPT 3 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_u start_POSTSUBSCRIPT 3 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_M start_POSTSUBSCRIPT 3 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , (201)

where r3ε:=r3Ar¯3assignsuperscriptsubscript𝑟3𝜀superscriptsubscript𝑟3𝐴subscript¯𝑟3r_{3}^{\varepsilon}:=\frac{r_{3}^{A}}{\overline{r}_{3}}italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT := divide start_ARG italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_ARG start_ARG over¯ start_ARG italic_r end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG, and f(0)η=0subscriptsuperscript𝑓𝜂00f^{\eta}_{(0)}=0italic_f start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( 0 ) end_POSTSUBSCRIPT = 0. Further, solving Eqs. (191) and (192) yields

ξc(t)=c1cosh(hξhηt)+c2hξhηsinh(hξhηt),subscript𝜉𝑐𝑡subscript𝑐1subscript𝜉subscript𝜂𝑡subscript𝑐2subscript𝜉subscript𝜂subscript𝜉subscript𝜂𝑡\displaystyle\xi_{c}(t)=c_{1}\cosh(\sqrt{h_{\xi}}\sqrt{h_{\eta}}t)+c_{2}\frac{% \sqrt{h_{\xi}}}{\sqrt{h_{\eta}}}\sinh(\sqrt{h_{\xi}}\sqrt{h_{\eta}}t),italic_ξ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t ) = italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cosh ( square-root start_ARG italic_h start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT end_ARG square-root start_ARG italic_h start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT end_ARG italic_t ) + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG square-root start_ARG italic_h start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT end_ARG end_ARG start_ARG square-root start_ARG italic_h start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT end_ARG end_ARG roman_sinh ( square-root start_ARG italic_h start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT end_ARG square-root start_ARG italic_h start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT end_ARG italic_t ) , (202)
ηc(t)=c2cosh(hξhηt)+c1hηhξsinh(hξhηt),subscript𝜂𝑐𝑡subscript𝑐2subscript𝜉subscript𝜂𝑡subscript𝑐1subscript𝜂subscript𝜉subscript𝜉subscript𝜂𝑡\displaystyle\eta_{c}(t)=c_{2}\cosh(\sqrt{h_{\xi}}\sqrt{h_{\eta}}t)+c_{1}\frac% {\sqrt{h_{\eta}}}{\sqrt{h_{\xi}}}\sinh(\sqrt{h_{\xi}}\sqrt{h_{\eta}}t),italic_η start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_t ) = italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_cosh ( square-root start_ARG italic_h start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT end_ARG square-root start_ARG italic_h start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT end_ARG italic_t ) + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT divide start_ARG square-root start_ARG italic_h start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT end_ARG end_ARG start_ARG square-root start_ARG italic_h start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT end_ARG end_ARG roman_sinh ( square-root start_ARG italic_h start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT end_ARG square-root start_ARG italic_h start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT end_ARG italic_t ) , (203)
ξl[c](t)=q=55f(q)ξ(Δn3+qnθ3)hξf(q)η(Δn3+qnθ3)2+hξhηcosΓ,subscript𝜉𝑙delimited-[]𝑐𝑡superscriptsubscript𝑞55subscriptsuperscript𝑓𝜉𝑞Δsubscript𝑛3𝑞subscript𝑛subscript𝜃3subscript𝜉subscriptsuperscript𝑓𝜂𝑞superscriptΔsubscript𝑛3𝑞subscript𝑛subscript𝜃32subscript𝜉subscript𝜂Γ\displaystyle\xi_{l[c]}(t)=\sum_{q=-5}^{5}\frac{f^{\xi}_{(q)}(\Delta n_{3}+q\,% n_{\theta_{3}})-h_{\xi}f^{\eta}_{(q)}}{(\Delta n_{3}+q\,n_{\theta_{3}})^{2}+h_% {\xi}h_{\eta}}\cos\Gamma,italic_ξ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) = ∑ start_POSTSUBSCRIPT italic_q = - 5 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT divide start_ARG italic_f start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT ( roman_Δ italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_n start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) - italic_h start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT end_ARG start_ARG ( roman_Δ italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_n start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_h start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT end_ARG roman_cos roman_Γ , (204)
ηl[c](t)=q=55f(q)η(Δn3+qnθ3)+hηf(q)ξ(Δn3+qnθ3)2+hξhηsinΓ,subscript𝜂𝑙delimited-[]𝑐𝑡superscriptsubscript𝑞55subscriptsuperscript𝑓𝜂𝑞Δsubscript𝑛3𝑞subscript𝑛subscript𝜃3subscript𝜂subscriptsuperscript𝑓𝜉𝑞superscriptΔsubscript𝑛3𝑞subscript𝑛subscript𝜃32subscript𝜉subscript𝜂Γ\displaystyle\eta_{l[c]}(t)=\sum_{q=-5}^{5}\frac{f^{\eta}_{(q)}(\Delta n_{3}+q% \,n_{\theta_{3}})+h_{\eta}f^{\xi}_{(q)}}{(\Delta n_{3}+q\,n_{\theta_{3}})^{2}+% h_{\xi}h_{\eta}}\sin\Gamma,italic_η start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) = ∑ start_POSTSUBSCRIPT italic_q = - 5 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT divide start_ARG italic_f start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT ( roman_Δ italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_n start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) + italic_h start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT end_ARG start_ARG ( roman_Δ italic_n start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_n start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_h start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT end_ARG roman_sin roman_Γ , (205)

where

c1=ξ¯0ξl[c](t0),c2=\displaystyle c_{1}=\overline{\xi}_{\!0}-\xi_{l[c]}(t_{0}),\qquad c_{2}=italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = over¯ start_ARG italic_ξ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_ξ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = η¯0ηl[c](t0),subscript¯𝜂0subscript𝜂𝑙delimited-[]𝑐subscript𝑡0\displaystyle~{}\overline{\eta}_{\!0}-\eta_{l[c]}(t_{0}),over¯ start_ARG italic_η end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_η start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) , (206)

and Γ:=Δu3+qθ3assignΓΔsubscript𝑢3𝑞subscript𝜃3\Gamma:=\Delta u_{3}+q\,\theta_{3}roman_Γ := roman_Δ italic_u start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_q italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. Equations (204) and (205) indicate that solar perturbation alone induces no special long-period variations in ξ𝜉\xiitalic_ξ and η𝜂\etaitalic_η, since the terms f(q)ξsubscriptsuperscript𝑓𝜉𝑞f^{\xi}_{(q)}italic_f start_POSTSUPERSCRIPT italic_ξ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT and f(q)ηsubscriptsuperscript𝑓𝜂𝑞f^{\eta}_{(q)}italic_f start_POSTSUPERSCRIPT italic_η end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( italic_q ) end_POSTSUBSCRIPT are exclusively associated with lunar perturbation. Additionally, it is worth noting that introducing σl[c](t)subscript𝜎𝑙delimited-[]𝑐𝑡\sigma_{l[c]}(t)italic_σ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t ) in the reference solution (122), as well as in Eqs. (189) and (190), is crucial. Without this term, a significant increase in analytical solution errors would occur, leading to the disappearance of terms related to ξl[c](t0)subscript𝜉𝑙delimited-[]𝑐subscript𝑡0\xi_{l[c]}(t_{0})italic_ξ start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and ηl[c](t0)subscript𝜂𝑙delimited-[]𝑐subscript𝑡0\eta_{l[c]}(t_{0})italic_η start_POSTSUBSCRIPT italic_l [ italic_c ] end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) in Eq. (206), as well as the terms associated with hξsubscript𝜉h_{\xi}italic_h start_POSTSUBSCRIPT italic_ξ end_POSTSUBSCRIPT and hηsubscript𝜂h_{\eta}italic_h start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT in Eqs. (204) and (205).

Appendix D Explicit forms of inclination functions

The explicit forms of inclination functions within the lunar perturbation solution are shown below. These encompass f[q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁delimited-[]𝑞𝑖f^{\sigma(N)}_{[q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ) in Eqs. (164), (198), and (199) for the special long-period terms, f[p,q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁𝑝𝑞𝑖f^{\sigma(N)}_{[p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ) in Eq. (173) for the general long-period terms, and f[κ,p,q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁𝜅𝑝𝑞𝑖f^{\sigma(N)}_{[\kappa,\,p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ) in Eqs. (180) and (181) for the short-period terms. For brevity, only the leading-order inclination functions with N=2𝑁2N=2italic_N = 2 or N=3𝑁3N=3italic_N = 3 are presented (cf. Table 7). Inclination functions for other orders can be derived using the methods outlined in Appendix B. Note that i𝑖iitalic_i and i3subscript𝑖3i_{3}italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT in these functions represent mean values i¯¯𝑖\overline{i}over¯ start_ARG italic_i end_ARG and i¯3subscript¯𝑖3\overline{i}_{3}over¯ start_ARG italic_i end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, respectively.

Table 7: Relationship between the inclination functions and the Legendre polynomial degree N𝑁Nitalic_N.
σ𝜎\sigmaitalic_σ f[q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁delimited-[]𝑞𝑖f^{\sigma(N)}_{[q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ) f[p,q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁𝑝𝑞𝑖f^{\sigma(N)}_{[p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ) f[κ,p,q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁𝜅𝑝𝑞𝑖f^{\sigma(N)}_{[\kappa,\,p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i )
a𝑎aitalic_a -- -- N=2,3,4,𝑁234N=2,3,4,\cdotsitalic_N = 2 , 3 , 4 , ⋯
ξ𝜉\xiitalic_ξ, η𝜂\etaitalic_η N=3,5,𝑁35N=3,5,\cdotsitalic_N = 3 , 5 , ⋯ N=3,5,𝑁35N=3,5,\cdotsitalic_N = 3 , 5 , ⋯ N=2,3,4,𝑁234N=2,3,4,\cdotsitalic_N = 2 , 3 , 4 , ⋯
i𝑖iitalic_i, ΩΩ\Omegaroman_Ω, λ𝜆\lambdaitalic_λ N=2,4,𝑁24N=2,4,\cdotsitalic_N = 2 , 4 , ⋯ N=2,4,𝑁24N=2,4,\cdotsitalic_N = 2 , 4 , ⋯ N=2,3,4,𝑁234N=2,3,4,\cdotsitalic_N = 2 , 3 , 4 , ⋯

D.1 Inclination functions for special long-period terms

The inclination functions f[q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁delimited-[]𝑞𝑖f^{\sigma(N)}_{[q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ), associated with the special long-period terms σml[c](t)superscriptsubscript𝜎m𝑙delimited-[]𝑐𝑡\sigma_{\text{m}}^{l[c]}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_c ] end_POSTSUPERSCRIPT ( italic_t ) in Eqs. (204)-(205) and (161)-(163), are listed below in terms of the orbital elements ξ𝜉\xiitalic_ξ, η𝜂\etaitalic_η, i𝑖iitalic_i, ΩΩ\Omegaroman_Ω, and λ𝜆\lambdaitalic_λ:

(1) Inclination functions with the form f[q]ξ(3)(i)subscriptsuperscript𝑓𝜉3delimited-[]𝑞𝑖f^{\xi(3)}_{[q]}(i)italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ):

f[-3]ξ(3)=22532768(cosicos3i)(34cos2i3+cos4i3)1cosi3,subscriptsuperscript𝑓𝜉3delimited-[]-322532768𝑖3𝑖342subscript𝑖34subscript𝑖31subscript𝑖3\displaystyle\displaystyle f^{\xi(3)}_{[\raisebox{0.0pt}{-}3]}=\frac{225}{3276% 8}\frac{(\cos i-\cos 3i)(3-4\cos 2i_{3}+\cos 4i_{3})}{1-\cos i_{3}},italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ - 3 ] end_POSTSUBSCRIPT = divide start_ARG 225 end_ARG start_ARG 32768 end_ARG divide start_ARG ( roman_cos italic_i - roman_cos 3 italic_i ) ( 3 - 4 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 4 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG start_ARG 1 - roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ,
f[0]ξ(3)=454096(sini+5sin3i)(sini3+5sin3i3),subscriptsuperscript𝑓𝜉3delimited-[]0454096𝑖53𝑖subscript𝑖353subscript𝑖3\displaystyle\displaystyle f^{\xi(3)}_{[0]}=\frac{45}{4096}(\sin i+5\sin 3i)(% \sin i_{3}+5\sin 3i_{3}),italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 0 ] end_POSTSUBSCRIPT = divide start_ARG 45 end_ARG start_ARG 4096 end_ARG ( roman_sin italic_i + 5 roman_sin 3 italic_i ) ( roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 5 roman_sin 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[±1]ξ(3)=1516384(cosi+15cos3i)(±6cosi3\displaystyle\displaystyle f^{\xi(3)}_{[\pm 1]}=-\frac{15}{16384}(\cos i+15% \cos 3i)(\pm 6-\cos i_{3}italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 15 end_ARG start_ARG 16384 end_ARG ( roman_cos italic_i + 15 roman_cos 3 italic_i ) ( ± 6 - roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
±10cos2i315cos3i3),\displaystyle\displaystyle\phantom{f^{\xi(3)}_{[\pm 1]}=}\pm 10\cos 2i_{3}-15% \cos 3i_{3}),± 10 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 15 roman_cos 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[±2]ξ(3)=758192(sini3sin3i)(sini3±4sin2i3\displaystyle\displaystyle f^{\xi(3)}_{[\pm 2]}=\frac{75}{8192}(\sin i-3\sin 3% i)(\sin i_{3}\pm 4\sin 2i_{3}italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 75 end_ARG start_ARG 8192 end_ARG ( roman_sin italic_i - 3 roman_sin 3 italic_i ) ( roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± 4 roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
3sin3i3),\displaystyle\displaystyle\phantom{f^{\xi(3)}_{[\pm 2]}=}-3\sin 3i_{3}),- 3 roman_sin 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3]ξ(3)=22516384(cosicos3i)(2cosi32cos2i3\displaystyle\displaystyle f^{\xi(3)}_{[3]}=-\frac{225}{16384}(\cos i-\cos 3i)% (2-\cos i_{3}-2\cos 2i_{3}italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 ] end_POSTSUBSCRIPT = - divide start_ARG 225 end_ARG start_ARG 16384 end_ARG ( roman_cos italic_i - roman_cos 3 italic_i ) ( 2 - roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 2 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+cos3i3).\displaystyle\displaystyle\phantom{f^{\xi(3)}_{[3]}=}+\cos 3i_{3}).+ roman_cos 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) . (207)

(2) Inclination functions with the form f[q]η(3)(i)subscriptsuperscript𝑓𝜂3delimited-[]𝑞𝑖f^{\eta(3)}_{[q]}(i)italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ):
(2) 形式 f[q]η(3)(i)subscriptsuperscript𝑓𝜂3delimited-[]𝑞𝑖f^{\eta(3)}_{[q]}(i)italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ) 为的倾斜函数 :

f[-3]η(3)=22516384(1cos2i)(34cos2i3+cos4i3)1cosi3,subscriptsuperscript𝑓𝜂3delimited-[]-32251638412𝑖342subscript𝑖34subscript𝑖31subscript𝑖3\displaystyle\displaystyle f^{\eta(3)}_{[\raisebox{0.0pt}{-}3]}=\frac{225}{163% 84}\frac{(1-\cos 2i)(3-4\cos 2i_{3}+\cos 4i_{3})}{1-\cos i_{3}},italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ - 3 ] end_POSTSUBSCRIPT = divide start_ARG 225 end_ARG start_ARG 16384 end_ARG divide start_ARG ( 1 - roman_cos 2 italic_i ) ( 3 - 4 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 4 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG start_ARG 1 - roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ,
f[±1]η(3)=158192(3+5cos2i)(6cosi3+10cos2i3\displaystyle\displaystyle f^{\eta(3)}_{[\pm 1]}=\frac{15}{8192}(3+5\cos 2i)(6% \mp\cos i_{3}+10\cos 2i_{3}italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ ± 1 ] end_POSTSUBSCRIPT = divide start_ARG 15 end_ARG start_ARG 8192 end_ARG ( 3 + 5 roman_cos 2 italic_i ) ( 6 ∓ roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 10 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
15cos3i3),\displaystyle\displaystyle\phantom{f^{\eta(3)}_{[\pm 1]}=}\mp 15\cos 3i_{3}),∓ 15 roman_cos 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (208)
f[±2]η(3)=752048sin2i(±sini3+4sin2i33sin3i3),subscriptsuperscript𝑓𝜂3delimited-[]plus-or-minus27520482𝑖minus-or-plusplus-or-minussubscript𝑖342subscript𝑖333subscript𝑖3\displaystyle\displaystyle f^{\eta(3)}_{[\pm 2]}=\frac{75}{2048}\sin 2i(\pm% \sin i_{3}+4\sin 2i_{3}\mp 3\sin 3i_{3}),italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 75 end_ARG start_ARG 2048 end_ARG roman_sin 2 italic_i ( ± roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 4 roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∓ 3 roman_sin 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3]η(3)=2258192(1cos2i)(2cosi32cos2i3+cos3i3).subscriptsuperscript𝑓𝜂3delimited-[]3225819212𝑖2subscript𝑖322subscript𝑖33subscript𝑖3\displaystyle\displaystyle f^{\eta(3)}_{[3]}=\frac{225}{8192}(1-\cos 2i)(2-% \cos i_{3}-2\cos 2i_{3}+\cos 3i_{3}).italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 ] end_POSTSUBSCRIPT = divide start_ARG 225 end_ARG start_ARG 8192 end_ARG ( 1 - roman_cos 2 italic_i ) ( 2 - roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 2 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) .

(3) Inclination functions with the form f[q]i(2)(i)subscriptsuperscript𝑓𝑖2delimited-[]𝑞𝑖f^{i(2)}_{[q]}(i)italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ):
(3) 形式 f[q]i(2)(i)subscriptsuperscript𝑓𝑖2delimited-[]𝑞𝑖f^{i(2)}_{[q]}(i)italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ) 为的倾斜函数 :

f[1]i(2)=38cosisin2i3,subscriptsuperscript𝑓𝑖2delimited-[]138𝑖2subscript𝑖3\displaystyle f^{i(2)}_{[1]}=-\frac{3}{8}\cos i\sin 2i_{3},italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 8 end_ARG roman_cos italic_i roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,
f[2]i(2)=316(1cos2i3)sini.subscriptsuperscript𝑓𝑖2delimited-[]231612subscript𝑖3𝑖\displaystyle f^{i(2)}_{[2]}=-\frac{3}{16}(1-\cos 2i_{3})\sin i.italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 16 end_ARG ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_sin italic_i . (209)

(4) Inclination functions with the form f[q]Ω(2)(i)subscriptsuperscript𝑓Ω2delimited-[]𝑞𝑖f^{\Omega(2)}_{[q]}(i)italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ):

f[1]Ω(2)=38cos2icscisin2i3,subscriptsuperscript𝑓Ω2delimited-[]1382𝑖𝑖2subscript𝑖3\displaystyle f^{\Omega(2)}_{[1]}=\frac{3}{8}\cos 2i\csc i\sin 2i_{3},italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 8 end_ARG roman_cos 2 italic_i roman_csc italic_i roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,
f[2]Ω(2)=316cosi(1cos2i3).subscriptsuperscript𝑓Ω2delimited-[]2316𝑖12subscript𝑖3\displaystyle f^{\Omega(2)}_{[2]}=\frac{3}{16}\cos i(1-\cos 2i_{3}).italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 16 end_ARG roman_cos italic_i ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) . (210)

(5) Inclination functions with the form f[q]λ(2)(i)subscriptsuperscript𝑓𝜆2delimited-[]𝑞𝑖f^{\lambda(2)}_{[q]}(i)italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_q ] end_POSTSUBSCRIPT ( italic_i ):

f[1]λ(2)=316(3cosicos3i)cscisin2i3,subscriptsuperscript𝑓𝜆2delimited-[]13163𝑖3𝑖𝑖2subscript𝑖3\displaystyle f^{\lambda(2)}_{[1]}=-\frac{3}{16}(3\cos i-\cos 3i)\csc i\sin 2i% _{3},italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 16 end_ARG ( 3 roman_cos italic_i - roman_cos 3 italic_i ) roman_csc italic_i roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,
f[2]λ(2)=332(3cos2i)(1cos2i3).subscriptsuperscript𝑓𝜆2delimited-[]233232𝑖12subscript𝑖3\displaystyle f^{\lambda(2)}_{[2]}=-\frac{3}{32}(3-\cos 2i)(1-\cos 2i_{3}).italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( 3 - roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) . (211)

D.2 Inclination functions for general long-period terms

The inclination functions f[p,q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁𝑝𝑞𝑖f^{\sigma(N)}_{[p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ) for the general long-period terms σml[l](t)superscriptsubscript𝜎m𝑙delimited-[]𝑙𝑡\sigma_{\text{m}}^{l[l]}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l [ italic_l ] end_POSTSUPERSCRIPT ( italic_t ) in Eqs. (168)-(172) are listed in the order of ξ𝜉\xiitalic_ξ, η𝜂\etaitalic_η, i𝑖iitalic_i, ΩΩ\Omegaroman_Ω, and λ𝜆\lambdaitalic_λ, as follows:

(1) Inclination functions with the form f[p,q]ξ(3)(i)subscriptsuperscript𝑓𝜉3𝑝𝑞𝑖f^{\xi(3)}_{[p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ):

f[1,-3]ξ(3)=22516384(cosicos3i)(34cos2i3+cos4i3)1cosi3,subscriptsuperscript𝑓𝜉31-322516384𝑖3𝑖342subscript𝑖34subscript𝑖31subscript𝑖3\displaystyle\displaystyle f^{\xi(3)}_{[1,\raisebox{0.0pt}{-}3]}=-\frac{225}{1% 6384}\frac{(\cos i-\cos 3i)(3-4\cos 2i_{3}+\cos 4i_{3})}{1-\cos i_{3}},italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , - 3 ] end_POSTSUBSCRIPT = - divide start_ARG 225 end_ARG start_ARG 16384 end_ARG divide start_ARG ( roman_cos italic_i - roman_cos 3 italic_i ) ( 3 - 4 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 4 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG start_ARG 1 - roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ,
f[1,0]ξ(3)=452048(sini+5sin3i)(sini3+5sin3i3),subscriptsuperscript𝑓𝜉310452048𝑖53𝑖subscript𝑖353subscript𝑖3\displaystyle\displaystyle f^{\xi(3)}_{[1,0]}=-\frac{45}{2048}(\sin i+5\sin 3i% )(\sin i_{3}+5\sin 3i_{3}),italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 0 ] end_POSTSUBSCRIPT = - divide start_ARG 45 end_ARG start_ARG 2048 end_ARG ( roman_sin italic_i + 5 roman_sin 3 italic_i ) ( roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 5 roman_sin 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,±1]ξ(3)=158192(cosi+15cos3i)(±6cosi3±10cos2i3\displaystyle\displaystyle f^{\xi(3)}_{[1,\pm 1]}=\frac{15}{8192}(\cos i+15% \cos 3i)(\pm 6-\cos i_{3}\pm 10\cos 2i_{3}italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , ± 1 ] end_POSTSUBSCRIPT = divide start_ARG 15 end_ARG start_ARG 8192 end_ARG ( roman_cos italic_i + 15 roman_cos 3 italic_i ) ( ± 6 - roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± 10 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
15cos3i3),\displaystyle\displaystyle\phantom{f^{\xi(3)}_{[1,\pm 1]}=}-15\cos 3i_{3}),- 15 roman_cos 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,±2]ξ(3)=754096(sini3sin3i)(sini3±4sin2i3\displaystyle\displaystyle f^{\xi(3)}_{[1,\pm 2]}=-\frac{75}{4096}(\sin i-3% \sin 3i)(\sin i_{3}\pm 4\sin 2i_{3}italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 75 end_ARG start_ARG 4096 end_ARG ( roman_sin italic_i - 3 roman_sin 3 italic_i ) ( roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± 4 roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
3sin3i3),\displaystyle\displaystyle\phantom{f^{\xi(3)}_{[1,\pm 2]}=}-3\sin 3i_{3}),- 3 roman_sin 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,3]ξ(3)=2258192(cosicos3i)(2cosi32cos2i3\displaystyle\displaystyle f^{\xi(3)}_{[1,3]}=\frac{225}{8192}(\cos i-\cos 3i)% (2-\cos i_{3}-2\cos 2i_{3}italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 3 ] end_POSTSUBSCRIPT = divide start_ARG 225 end_ARG start_ARG 8192 end_ARG ( roman_cos italic_i - roman_cos 3 italic_i ) ( 2 - roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 2 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+cos3i3),\displaystyle\displaystyle\phantom{f^{\xi(3)}_{[1,3]}=}+\cos 3i_{3}),+ roman_cos 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,-2]ξ(3)=75(3sin3isini)(10sini35sin3i3+sin5i3)8192(34cosi3+cos2i3),subscriptsuperscript𝑓𝜉33-27533𝑖𝑖10subscript𝑖353subscript𝑖35subscript𝑖3819234subscript𝑖32subscript𝑖3\displaystyle\displaystyle f^{\xi(3)}_{[3,\raisebox{0.0pt}{-}2]}=\frac{75(3% \sin 3i-\sin i)(10\sin i_{3}-5\sin 3i_{3}+\sin 5i_{3})}{8192(3-4\cos i_{3}+% \cos 2i_{3})},italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , - 2 ] end_POSTSUBSCRIPT = divide start_ARG 75 ( 3 roman_sin 3 italic_i - roman_sin italic_i ) ( 10 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 5 roman_sin 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_sin 5 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG start_ARG 8192 ( 3 - 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG ,
f[3,0]ξ(3)=752048(sini+5sin3i)(3sini3sin3i3),subscriptsuperscript𝑓𝜉330752048𝑖53𝑖3subscript𝑖33subscript𝑖3\displaystyle\displaystyle f^{\xi(3)}_{[3,0]}=-\frac{75}{2048}(\sin i+5\sin 3i% )(3\sin i_{3}-\sin 3i_{3}),italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 0 ] end_POSTSUBSCRIPT = - divide start_ARG 75 end_ARG start_ARG 2048 end_ARG ( roman_sin italic_i + 5 roman_sin 3 italic_i ) ( 3 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - roman_sin 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,±1]ξ(3)=758192(cosi+15cos3i)(±2cosi32cos2i3\displaystyle\displaystyle f^{\xi(3)}_{[3,\pm 1]}=\frac{75}{8192}(\cos i+15% \cos 3i)(\pm 2-\cos i_{3}\mp 2\cos 2i_{3}italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , ± 1 ] end_POSTSUBSCRIPT = divide start_ARG 75 end_ARG start_ARG 8192 end_ARG ( roman_cos italic_i + 15 roman_cos 3 italic_i ) ( ± 2 - roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∓ 2 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+cos3i3),\displaystyle\displaystyle\phantom{f^{\xi(3)}_{[3,\pm 1]}=}+\cos 3i_{3}),+ roman_cos 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,2]ξ(3)=754096(3sin3isini)(5sini34sin2i3+sin3i3),subscriptsuperscript𝑓𝜉33275409633𝑖𝑖5subscript𝑖342subscript𝑖33subscript𝑖3\displaystyle\displaystyle f^{\xi(3)}_{[3,2]}=\frac{75}{4096}(3\sin 3i-\sin i)% (5\sin i_{3}-4\sin 2i_{3}+\sin 3i_{3}),italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 2 ] end_POSTSUBSCRIPT = divide start_ARG 75 end_ARG start_ARG 4096 end_ARG ( 3 roman_sin 3 italic_i - roman_sin italic_i ) ( 5 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 4 roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_sin 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,±3]ξ(3)=758192(cosicos3i)(±1015cosi3±6cos2i3\displaystyle\displaystyle f^{\xi(3)}_{[3,\pm 3]}=\frac{75}{8192}(\cos i-\cos 3% i)(\pm 10-15\cos i_{3}\pm 6\cos 2i_{3}italic_f start_POSTSUPERSCRIPT italic_ξ ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , ± 3 ] end_POSTSUBSCRIPT = divide start_ARG 75 end_ARG start_ARG 8192 end_ARG ( roman_cos italic_i - roman_cos 3 italic_i ) ( ± 10 - 15 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± 6 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
cos3i3).\displaystyle\displaystyle\phantom{f^{\xi(3)}_{[3,\pm 3]}=}-\cos 3i_{3}).- roman_cos 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) . (212)

(2) Inclination functions with the form f[p,q]η(3)(i)subscriptsuperscript𝑓𝜂3𝑝𝑞𝑖f^{\eta(3)}_{[p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ):

f[1,-3]η(3)=2258192(1cos2i)(34cos2i3+cos4i3)1cosi3,subscriptsuperscript𝑓𝜂31-3225819212𝑖342subscript𝑖34subscript𝑖31subscript𝑖3\displaystyle\displaystyle f^{\eta(3)}_{[1,\raisebox{0.0pt}{-}3]}=-\frac{225}{% 8192}\frac{(1-\cos 2i)(3-4\cos 2i_{3}+\cos 4i_{3})}{1-\cos i_{3}},italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , - 3 ] end_POSTSUBSCRIPT = - divide start_ARG 225 end_ARG start_ARG 8192 end_ARG divide start_ARG ( 1 - roman_cos 2 italic_i ) ( 3 - 4 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 4 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG start_ARG 1 - roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ,
f[1,±1]η(3)=154096(3+5cos2i)(6cosi3+10cos2i3\displaystyle\displaystyle f^{\eta(3)}_{[1,\pm 1]}=-\frac{15}{4096}(3+5\cos 2i% )(6\mp\cos i_{3}+10\cos 2i_{3}italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 15 end_ARG start_ARG 4096 end_ARG ( 3 + 5 roman_cos 2 italic_i ) ( 6 ∓ roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 10 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
15cos3i3),\displaystyle\displaystyle\phantom{f^{\eta(3)}_{[1,\pm 1]}=}\mp 15\cos 3i_{3}),∓ 15 roman_cos 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,±2]η(3)=751024sin2i(±sini3+4sin2i33sin3i3),subscriptsuperscript𝑓𝜂31plus-or-minus27510242𝑖minus-or-plusplus-or-minussubscript𝑖342subscript𝑖333subscript𝑖3\displaystyle\displaystyle f^{\eta(3)}_{[1,\pm 2]}=-\frac{75}{1024}\sin 2i(\pm% \sin i_{3}+4\sin 2i_{3}\mp 3\sin 3i_{3}),italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 75 end_ARG start_ARG 1024 end_ARG roman_sin 2 italic_i ( ± roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 4 roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∓ 3 roman_sin 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,3]η(3)=2254096(cos2i1)(2cosi32cos2i3+cos3i3),subscriptsuperscript𝑓𝜂31322540962𝑖12subscript𝑖322subscript𝑖33subscript𝑖3\displaystyle\displaystyle f^{\eta(3)}_{[1,3]}=\frac{225}{4096}(\cos 2i-1)(2-% \cos i_{3}-2\cos 2i_{3}+\cos 3i_{3}),italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 3 ] end_POSTSUBSCRIPT = divide start_ARG 225 end_ARG start_ARG 4096 end_ARG ( roman_cos 2 italic_i - 1 ) ( 2 - roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 2 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,±1]η(3)=754096(3+5cos2i)(2cosi32cos2i3\displaystyle\displaystyle f^{\eta(3)}_{[3,\pm 1]}=-\frac{75}{4096}(3+5\cos 2i% )(2\mp\cos i_{3}-2\cos 2i_{3}italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 75 end_ARG start_ARG 4096 end_ARG ( 3 + 5 roman_cos 2 italic_i ) ( 2 ∓ roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 2 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
±cos3i3),\displaystyle\displaystyle\phantom{f^{\eta(3)}_{[3,\pm 1]}=}\pm\cos 3i_{3}),± roman_cos 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,±2]η(3)=751024sin2i(±5sini34sin2i3±sin3i3),subscriptsuperscript𝑓𝜂33plus-or-minus27510242𝑖plus-or-minusplus-or-minus5subscript𝑖342subscript𝑖33subscript𝑖3\displaystyle\displaystyle f^{\eta(3)}_{[3,\pm 2]}=-\frac{75}{1024}\sin 2i(\pm 5% \sin i_{3}-4\sin 2i_{3}\pm\sin 3i_{3}),italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 75 end_ARG start_ARG 1024 end_ARG roman_sin 2 italic_i ( ± 5 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 4 roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± roman_sin 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,±3]η(3)=754096(1cos2i)(1015cosi3+6cos2i3\displaystyle\displaystyle f^{\eta(3)}_{[3,\pm 3]}=-\frac{75}{4096}(1-\cos 2i)% (10\mp 15\cos i_{3}+6\cos 2i_{3}italic_f start_POSTSUPERSCRIPT italic_η ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , ± 3 ] end_POSTSUBSCRIPT = - divide start_ARG 75 end_ARG start_ARG 4096 end_ARG ( 1 - roman_cos 2 italic_i ) ( 10 ∓ 15 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 6 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
cos3i3).\displaystyle\displaystyle\phantom{f^{\eta(3)}_{[3,\pm 3]}=}\mp\cos 3i_{3}).∓ roman_cos 3 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) . (213)

(3) Inclination functions with the form f[p,q]i(2)(i)subscriptsuperscript𝑓𝑖2𝑝𝑞𝑖f^{i(2)}_{[p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ):

f[2,±1]i(2)=316cosi(2sini3sin2i3),subscriptsuperscript𝑓𝑖22plus-or-minus1316𝑖minus-or-plus2subscript𝑖32subscript𝑖3\displaystyle f^{i(2)}_{[2,\pm 1]}=-\frac{3}{16}\cos i(2\sin i_{3}\mp\sin 2i_{% 3}),italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 16 end_ARG roman_cos italic_i ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∓ roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,±2]i(2)=332(±34cosi3±cos2i3)sini.subscriptsuperscript𝑓𝑖22plus-or-minus2332plus-or-minusplus-or-minus34subscript𝑖32subscript𝑖3𝑖\displaystyle f^{i(2)}_{[2,\pm 2]}=-\frac{3}{32}(\pm 3-4\cos i_{3}\pm\cos 2i_{% 3})\sin i.italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( ± 3 - 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_sin italic_i . (214)

(4) Inclination functions with the form f[p,q]Ω(2)(i)subscriptsuperscript𝑓Ω2𝑝𝑞𝑖f^{\Omega(2)}_{[p,\,q]}(i)italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ):

f[2,0]Ω(2)=916cosi(1cos2i3),subscriptsuperscript𝑓Ω220916𝑖12subscript𝑖3\displaystyle f^{\Omega(2)}_{[2,0]}=-\frac{9}{16}\cos i(1-\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 ] end_POSTSUBSCRIPT = - divide start_ARG 9 end_ARG start_ARG 16 end_ARG roman_cos italic_i ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,±1]Ω(2)=316cos2icsci(±2sini3sin2i3),subscriptsuperscript𝑓Ω22plus-or-minus13162𝑖𝑖plus-or-minus2subscript𝑖32subscript𝑖3\displaystyle f^{\Omega(2)}_{[2,\pm 1]}=\frac{3}{16}\cos 2i\csc i(\pm 2\sin i_% {3}-\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , ± 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 16 end_ARG roman_cos 2 italic_i roman_csc italic_i ( ± 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,±2]Ω(2)=332cosi(34cosi3+cos2i3).subscriptsuperscript𝑓Ω22plus-or-minus2332𝑖minus-or-plus34subscript𝑖32subscript𝑖3\displaystyle f^{\Omega(2)}_{[2,\pm 2]}=\frac{3}{32}\cos i(3\mp 4\cos i_{3}+% \cos 2i_{3}).italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 32 end_ARG roman_cos italic_i ( 3 ∓ 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) . (215)

(5) Inclination functions with the form f[p,q]λ(2)(i)subscriptsuperscript𝑓𝜆2𝑝𝑞𝑖f^{\lambda(2)}_{[p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ):
(5) 形式 f[p,q]λ(2)(i)subscriptsuperscript𝑓𝜆2𝑝𝑞𝑖f^{\lambda(2)}_{[p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ) 为的倾斜函数 :

f[2,0]λ(2)=332(13cos2i)(1cos2i3),subscriptsuperscript𝑓𝜆220332132𝑖12subscript𝑖3\displaystyle f^{\lambda(2)}_{[2,0]}=\frac{3}{32}(1-3\cos 2i)(1-\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( 1 - 3 roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,±1]λ(2)=332(3cosicos3i)csci(±2sini3sin2i3),subscriptsuperscript𝑓𝜆22plus-or-minus13323𝑖3𝑖𝑖plus-or-minus2subscript𝑖32subscript𝑖3\displaystyle f^{\lambda(2)}_{[2,\pm 1]}=-\frac{3}{32}(3\cos i-\cos 3i)\csc i(% \pm 2\sin i_{3}-\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( 3 roman_cos italic_i - roman_cos 3 italic_i ) roman_csc italic_i ( ± 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,±2]λ(2)=364(3cos2i)(34cosi3+cos2i3).subscriptsuperscript𝑓𝜆22plus-or-minus236432𝑖minus-or-plus34subscript𝑖32subscript𝑖3\displaystyle f^{\lambda(2)}_{[2,\pm 2]}=-\frac{3}{64}(3-\cos 2i)(3\mp 4\cos i% _{3}+\cos 2i_{3}).italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 3 - roman_cos 2 italic_i ) ( 3 ∓ 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) . (216)

D.3 Inclination functions for short-period terms

The inclination functions f[κ,p,q]σ(N)(i)subscriptsuperscript𝑓𝜎𝑁𝜅𝑝𝑞𝑖f^{\sigma(N)}_{[\kappa,\,p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_σ ( italic_N ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ) for the short-period terms σms(t)superscriptsubscript𝜎m𝑠𝑡\sigma_{\text{m}}^{s}(t)italic_σ start_POSTSUBSCRIPT m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( italic_t ) in Eqs. (174)-(179) are listed below in the order of a𝑎aitalic_a, ξ𝜉\xiitalic_ξ, η𝜂\etaitalic_η, i𝑖iitalic_i, ΩΩ\Omegaroman_Ω, and λ𝜆\lambdaitalic_λ:

(1) Inclination functions with the form f[κ,p,q]a(2)(i)subscriptsuperscript𝑓𝑎2𝜅𝑝𝑞𝑖f^{a(2)}_{[\kappa,\,p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_a ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ):

f[2,-2,0]a(2)=932(1cos2i)(1cos2i3),subscriptsuperscript𝑓𝑎22-2093212𝑖12subscript𝑖3\displaystyle f^{a(2)}_{[2,\raisebox{0.0pt}{-}2,0]}=\frac{9}{32}(1-\cos 2i)(1-% \cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_a ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 9 end_ARG start_ARG 32 end_ARG ( 1 - roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,-2,±1]a(2)=316(2sini±sin2i)(2sini3±sin2i3),subscriptsuperscript𝑓𝑎22-2plus-or-minus1316plus-or-minus2𝑖2𝑖plus-or-minus2subscript𝑖32subscript𝑖3\displaystyle f^{a(2)}_{[2,\raisebox{0.0pt}{-}2,\pm 1]}=\frac{3}{16}(2\sin i% \pm\sin 2i)(2\sin i_{3}\pm\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_a ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , ± 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 16 end_ARG ( 2 roman_sin italic_i ± roman_sin 2 italic_i ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,-2,±2]a(2)=364(3±4cosi+cos2i)(3±4cosi3+cos2i3),subscriptsuperscript𝑓𝑎22-2plus-or-minus2364plus-or-minus34𝑖2𝑖plus-or-minus34subscript𝑖32subscript𝑖3\displaystyle f^{a(2)}_{[2,\raisebox{0.0pt}{-}2,\pm 2]}=\frac{3}{64}(3\pm 4% \cos i+\cos 2i)(3\pm 4\cos i_{3}+\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_a ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 3 ± 4 roman_cos italic_i + roman_cos 2 italic_i ) ( 3 ± 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,0,0]a(2)=316(1cos2i)(1+3cos2i3),subscriptsuperscript𝑓𝑎220031612𝑖132subscript𝑖3\displaystyle f^{a(2)}_{[2,0,0]}=\frac{3}{16}(1-\cos 2i)(1+3\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_a ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 16 end_ARG ( 1 - roman_cos 2 italic_i ) ( 1 + 3 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,0,±1]a(2)=38(±2sini+sin2i)sin2i3,subscriptsuperscript𝑓𝑎220plus-or-minus138plus-or-minus2𝑖2𝑖2subscript𝑖3\displaystyle f^{a(2)}_{[2,0,\pm 1]}=-\frac{3}{8}(\pm 2\sin i+\sin 2i)\sin 2i_% {3},italic_f start_POSTSUPERSCRIPT italic_a ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 8 end_ARG ( ± 2 roman_sin italic_i + roman_sin 2 italic_i ) roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , (217)
f[2,0,±2]a(2)=332(3±4cosi+cos2i)(1cos2i3),subscriptsuperscript𝑓𝑎220plus-or-minus2332plus-or-minus34𝑖2𝑖12subscript𝑖3\displaystyle f^{a(2)}_{[2,0,\pm 2]}=\frac{3}{32}(3\pm 4\cos i+\cos 2i)(1-\cos 2% i_{3}),italic_f start_POSTSUPERSCRIPT italic_a ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( 3 ± 4 roman_cos italic_i + roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,2,0]a(2)=f[2,-2,0]a(2),subscriptsuperscript𝑓𝑎2220subscriptsuperscript𝑓𝑎22-20\displaystyle f^{a(2)}_{[2,2,0]}=f^{a(2)}_{[2,\raisebox{0.0pt}{-}2,0]},italic_f start_POSTSUPERSCRIPT italic_a ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , 0 ] end_POSTSUBSCRIPT = italic_f start_POSTSUPERSCRIPT italic_a ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , 0 ] end_POSTSUBSCRIPT ,
f[2,2,±1]a(2)=316(2sini±sin2i)(2sini3sin2i3),subscriptsuperscript𝑓𝑎222plus-or-minus1316plus-or-minus2𝑖2𝑖minus-or-plus2subscript𝑖32subscript𝑖3\displaystyle f^{a(2)}_{[2,2,\pm 1]}=-\frac{3}{16}(2\sin i\pm\sin 2i)(2\sin i_% {3}\mp\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_a ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 16 end_ARG ( 2 roman_sin italic_i ± roman_sin 2 italic_i ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∓ roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,2,±2]a(2)=364(3±4cosi+cos2i)(34cosi3+cos2i3).subscriptsuperscript𝑓𝑎222plus-or-minus2364plus-or-minus34𝑖2𝑖minus-or-plus34subscript𝑖32subscript𝑖3\displaystyle f^{a(2)}_{[2,2,\pm 2]}=\frac{3}{64}(3\pm 4\cos i+\cos 2i)(3\mp 4% \cos i_{3}+\cos 2i_{3}).italic_f start_POSTSUPERSCRIPT italic_a ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 3 ± 4 roman_cos italic_i + roman_cos 2 italic_i ) ( 3 ∓ 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) .

(2) Inclination functions with the form f[κ,p,q]ξ(2)(i)subscriptsuperscript𝑓𝜉2𝜅𝑝𝑞𝑖f^{\xi(2)}_{[\kappa,\,p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ):

f[1,-2,0]ξ(2)=3128(715cos2i)(1cos2i3),subscriptsuperscript𝑓𝜉21-2031287152𝑖12subscript𝑖3\displaystyle\displaystyle f^{\xi(2)}_{[1,\raisebox{0.0pt}{-}2,0]}=\frac{3}{12% 8}(7-15\cos 2i)(1-\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , - 2 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 128 end_ARG ( 7 - 15 roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,-2,±1]ξ(2)=364(6sini±5sin2i)(2sini3±sin2i3),subscriptsuperscript𝑓𝜉21-2plus-or-minus1364plus-or-minus6𝑖52𝑖plus-or-minus2subscript𝑖32subscript𝑖3\displaystyle\displaystyle f^{\xi(2)}_{[1,\raisebox{0.0pt}{-}2,\pm 1]}=\frac{3% }{64}(6\sin i\pm 5\sin 2i)(2\sin i_{3}\pm\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , - 2 , ± 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 6 roman_sin italic_i ± 5 roman_sin 2 italic_i ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,-2,±2]ξ(2)=3256(7±12cosi+5cos2i)(3±4cosi3\displaystyle\displaystyle f^{\xi(2)}_{[1,\raisebox{0.0pt}{-}2,\pm 2]}=\frac{3% }{256}(7\pm 12\cos i+5\cos 2i)(3\pm 4\cos i_{3}italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , - 2 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 256 end_ARG ( 7 ± 12 roman_cos italic_i + 5 roman_cos 2 italic_i ) ( 3 ± 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+cos2i3),\displaystyle\displaystyle\phantom{f^{\xi(2)}_{[1,\raisebox{0.0pt}{-}2,\pm 2]}% =}+\cos 2i_{3}),+ roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,0,0]ξ(2)=164(715cos2i)(1+3cos2i3),subscriptsuperscript𝑓𝜉21001647152𝑖132subscript𝑖3\displaystyle\displaystyle f^{\xi(2)}_{[1,0,0]}=\frac{1}{64}(7-15\cos 2i)(1+3% \cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 0 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 64 end_ARG ( 7 - 15 roman_cos 2 italic_i ) ( 1 + 3 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,0,±1]ξ(2)=332(±6sini+5sin2i)sin2i3,subscriptsuperscript𝑓𝜉210plus-or-minus1332plus-or-minus6𝑖52𝑖2subscript𝑖3\displaystyle\displaystyle f^{\xi(2)}_{[1,0,\pm 1]}=-\frac{3}{32}(\pm 6\sin i+% 5\sin 2i)\sin 2i_{3},italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 0 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( ± 6 roman_sin italic_i + 5 roman_sin 2 italic_i ) roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,
f[1,0,±2]ξ(2)=3128(7±12cosi+5cos2i)(1cos2i3),subscriptsuperscript𝑓𝜉210plus-or-minus23128plus-or-minus712𝑖52𝑖12subscript𝑖3\displaystyle\displaystyle f^{\xi(2)}_{[1,0,\pm 2]}=\frac{3}{128}(7\pm 12\cos i% +5\cos 2i)(1-\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 0 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 128 end_ARG ( 7 ± 12 roman_cos italic_i + 5 roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,2,0]ξ(2)=f[1,-2,0]ξ(2),subscriptsuperscript𝑓𝜉2120subscriptsuperscript𝑓𝜉21-20\displaystyle\displaystyle f^{\xi(2)}_{[1,2,0]}=f^{\xi(2)}_{[1,\raisebox{0.0pt% }{-}2,0]},italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 2 , 0 ] end_POSTSUBSCRIPT = italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , - 2 , 0 ] end_POSTSUBSCRIPT ,
f[1,2,±1]ξ(2)=364(6sini±5sin2i)(2sini3sin2i3),subscriptsuperscript𝑓𝜉212plus-or-minus1364plus-or-minus6𝑖52𝑖minus-or-plus2subscript𝑖32subscript𝑖3\displaystyle\displaystyle f^{\xi(2)}_{[1,2,\pm 1]}=-\frac{3}{64}(6\sin i\pm 5% \sin 2i)(2\sin i_{3}\mp\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 2 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 6 roman_sin italic_i ± 5 roman_sin 2 italic_i ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∓ roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,2,±2]ξ(2)=3256(7±12cosi+5cos2i)(34cosi3\displaystyle\displaystyle f^{\xi(2)}_{[1,2,\pm 2]}=\frac{3}{256}(7\pm 12\cos i% +5\cos 2i)(3\mp 4\cos i_{3}italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 2 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 256 end_ARG ( 7 ± 12 roman_cos italic_i + 5 roman_cos 2 italic_i ) ( 3 ∓ 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+cos2i3),\displaystyle\displaystyle\phantom{f^{\xi(2)}_{[1,2,\pm 2]}=}+\cos 2i_{3}),+ roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (218)
f[3,-2,0]ξ(2)=9128(1cos2i)(1cos2i3),subscriptsuperscript𝑓𝜉23-20912812𝑖12subscript𝑖3\displaystyle\displaystyle f^{\xi(2)}_{[3,\raisebox{0.0pt}{-}2,0]}=\frac{9}{12% 8}(1-\cos 2i)(1-\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , - 2 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 9 end_ARG start_ARG 128 end_ARG ( 1 - roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,-2,±1]ξ(2)=364(2sini±sin2i)(2sini3±sin2i3),subscriptsuperscript𝑓𝜉23-2plus-or-minus1364plus-or-minus2𝑖2𝑖plus-or-minus2subscript𝑖32subscript𝑖3\displaystyle\displaystyle f^{\xi(2)}_{[3,\raisebox{0.0pt}{-}2,\pm 1]}=\frac{3% }{64}(2\sin i\pm\sin 2i)(2\sin i_{3}\pm\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , - 2 , ± 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 2 roman_sin italic_i ± roman_sin 2 italic_i ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,-2,±2]ξ(2)=3256(3±4cosi+cos2i)(3±4cosi3\displaystyle\displaystyle f^{\xi(2)}_{[3,\raisebox{0.0pt}{-}2,\pm 2]}=\frac{3% }{256}(3\pm 4\cos i+\cos 2i)(3\pm 4\cos i_{3}italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , - 2 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 256 end_ARG ( 3 ± 4 roman_cos italic_i + roman_cos 2 italic_i ) ( 3 ± 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+cos2i3),\displaystyle\displaystyle\phantom{f^{\xi(2)}_{[3,\raisebox{0.0pt}{-}2,\pm 2]}% =}+\cos 2i_{3}),+ roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,0,0]ξ(2)=364(1cos2i)(1+3cos2i3),subscriptsuperscript𝑓𝜉230036412𝑖132subscript𝑖3\displaystyle\displaystyle f^{\xi(2)}_{[3,0,0]}=\frac{3}{64}(1-\cos 2i)(1+3% \cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 0 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 1 - roman_cos 2 italic_i ) ( 1 + 3 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,0,±1]ξ(2)=332(±2sini+sin2i)sin2i3,subscriptsuperscript𝑓𝜉230plus-or-minus1332plus-or-minus2𝑖2𝑖2subscript𝑖3\displaystyle\displaystyle f^{\xi(2)}_{[3,0,\pm 1]}=-\frac{3}{32}(\pm 2\sin i+% \sin 2i)\sin 2i_{3},italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 0 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( ± 2 roman_sin italic_i + roman_sin 2 italic_i ) roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,
f[3,0,±2]ξ(2)=3128(3±4cosi+cos2i)(1cos2i3),subscriptsuperscript𝑓𝜉230plus-or-minus23128plus-or-minus34𝑖2𝑖12subscript𝑖3\displaystyle\displaystyle f^{\xi(2)}_{[3,0,\pm 2]}=\frac{3}{128}(3\pm 4\cos i% +\cos 2i)(1-\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 0 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 128 end_ARG ( 3 ± 4 roman_cos italic_i + roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,2,0]ξ(2)=f[3,-2,0]ξ(2),subscriptsuperscript𝑓𝜉2320subscriptsuperscript𝑓𝜉23-20\displaystyle\displaystyle f^{\xi(2)}_{[3,2,0]}=f^{\xi(2)}_{[3,\raisebox{0.0pt% }{-}2,0]},italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 2 , 0 ] end_POSTSUBSCRIPT = italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , - 2 , 0 ] end_POSTSUBSCRIPT ,
f[3,2,±1]ξ(2)=364(2sini±sin2i)(2sini3sin2i3),subscriptsuperscript𝑓𝜉232plus-or-minus1364plus-or-minus2𝑖2𝑖minus-or-plus2subscript𝑖32subscript𝑖3\displaystyle\displaystyle f^{\xi(2)}_{[3,2,\pm 1]}=-\frac{3}{64}(2\sin i\pm% \sin 2i)(2\sin i_{3}\mp\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 2 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 2 roman_sin italic_i ± roman_sin 2 italic_i ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∓ roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,2,±2]ξ(2)=3256(3±4cosi+cos2i)(34cosi3+cos2i3).subscriptsuperscript𝑓𝜉232plus-or-minus23256plus-or-minus34𝑖2𝑖minus-or-plus34subscript𝑖32subscript𝑖3\displaystyle\displaystyle f^{\xi(2)}_{[3,2,\pm 2]}=\frac{3}{256}(3\pm 4\cos i% +\cos 2i)(3\mp 4\cos i_{3}+\cos 2i_{3}).italic_f start_POSTSUPERSCRIPT italic_ξ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 2 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 256 end_ARG ( 3 ± 4 roman_cos italic_i + roman_cos 2 italic_i ) ( 3 ∓ 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) .

(3) Inclination functions with the form f[κ,p,q]η(2)(i)subscriptsuperscript𝑓𝜂2𝜅𝑝𝑞𝑖f^{\eta(2)}_{[\kappa,\,p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ):

f[1,-2,0]η(2)=3128(113cos2i)(1cos2i3),subscriptsuperscript𝑓𝜂21-2031281132𝑖12subscript𝑖3\displaystyle\displaystyle f^{\eta(2)}_{[1,\raisebox{0.0pt}{-}2,0]}=-\frac{3}{% 128}(11-3\cos 2i)(1-\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , - 2 , 0 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 128 end_ARG ( 11 - 3 roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,-2,±1]η(2)=364(6sini±sin2i)(2sini3±sin2i3),subscriptsuperscript𝑓𝜂21-2plus-or-minus1364plus-or-minus6𝑖2𝑖plus-or-minus2subscript𝑖32subscript𝑖3\displaystyle\displaystyle f^{\eta(2)}_{[1,\raisebox{0.0pt}{-}2,\pm 1]}=-\frac% {3}{64}(6\sin i\pm\sin 2i)(2\sin i_{3}\pm\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , - 2 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 6 roman_sin italic_i ± roman_sin 2 italic_i ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,-2,±2]η(2)=3256(11±12cosi+cos2i)(3±4cosi3\displaystyle\displaystyle f^{\eta(2)}_{[1,\raisebox{0.0pt}{-}2,\pm 2]}=-\frac% {3}{256}(11\pm 12\cos i+\cos 2i)(3\pm 4\cos i_{3}italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , - 2 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 256 end_ARG ( 11 ± 12 roman_cos italic_i + roman_cos 2 italic_i ) ( 3 ± 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+cos2i3),\displaystyle\displaystyle\phantom{f^{\eta(2)}_{[1,\raisebox{0.0pt}{-}2,\pm 2]% }=}+\cos 2i_{3}),+ roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,0,0]η(2)=164(113cos2i)(1+3cos2i3),subscriptsuperscript𝑓𝜂21001641132𝑖132subscript𝑖3\displaystyle\displaystyle f^{\eta(2)}_{[1,0,0]}=-\frac{1}{64}(11-3\cos 2i)(1+% 3\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 0 , 0 ] end_POSTSUBSCRIPT = - divide start_ARG 1 end_ARG start_ARG 64 end_ARG ( 11 - 3 roman_cos 2 italic_i ) ( 1 + 3 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,0,±1]η(2)=332(±6sini+sin2i)sin2i3,subscriptsuperscript𝑓𝜂210plus-or-minus1332plus-or-minus6𝑖2𝑖2subscript𝑖3\displaystyle\displaystyle f^{\eta(2)}_{[1,0,\pm 1]}=\frac{3}{32}(\pm 6\sin i+% \sin 2i)\sin 2i_{3},italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 0 , ± 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( ± 6 roman_sin italic_i + roman_sin 2 italic_i ) roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,
f[1,0,±2]η(2)=3128(11±12cosi+cos2i)(1cos2i3),subscriptsuperscript𝑓𝜂210plus-or-minus23128plus-or-minus1112𝑖2𝑖12subscript𝑖3\displaystyle\displaystyle f^{\eta(2)}_{[1,0,\pm 2]}=-\frac{3}{128}(11\pm 12% \cos i+\cos 2i)(1-\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 0 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 128 end_ARG ( 11 ± 12 roman_cos italic_i + roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,2,0]η(2)=f[1,-2,0]η(2),subscriptsuperscript𝑓𝜂2120subscriptsuperscript𝑓𝜂21-20\displaystyle\displaystyle f^{\eta(2)}_{[1,2,0]}=f^{\eta(2)}_{[1,\raisebox{0.0% pt}{-}2,0]},italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 2 , 0 ] end_POSTSUBSCRIPT = italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , - 2 , 0 ] end_POSTSUBSCRIPT ,
f[1,2,±1]η(2)=364(6sini±sin2i)(2sini3sin2i3),subscriptsuperscript𝑓𝜂212plus-or-minus1364plus-or-minus6𝑖2𝑖minus-or-plus2subscript𝑖32subscript𝑖3\displaystyle\displaystyle f^{\eta(2)}_{[1,2,\pm 1]}=\frac{3}{64}(6\sin i\pm% \sin 2i)(2\sin i_{3}\mp\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 2 , ± 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 6 roman_sin italic_i ± roman_sin 2 italic_i ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∓ roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[1,2,±2]η(2)=3256(11±12cosi+cos2i)(34cosi3\displaystyle\displaystyle f^{\eta(2)}_{[1,2,\pm 2]}=-\frac{3}{256}(11\pm 12% \cos i+\cos 2i)(3\mp 4\cos i_{3}italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 1 , 2 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 256 end_ARG ( 11 ± 12 roman_cos italic_i + roman_cos 2 italic_i ) ( 3 ∓ 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+cos2i3),\displaystyle\displaystyle\phantom{f^{\eta(2)}_{[1,2,\pm 2]}=}+\cos 2i_{3}),+ roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (219)
f[3,-2,0]η(2)=9128(1cos2i)(1cos2i3),subscriptsuperscript𝑓𝜂23-20912812𝑖12subscript𝑖3\displaystyle\displaystyle f^{\eta(2)}_{[3,\raisebox{0.0pt}{-}2,0]}=\frac{9}{1% 28}(1-\cos 2i)(1-\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , - 2 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 9 end_ARG start_ARG 128 end_ARG ( 1 - roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,-2,±1]η(2)=364(2sini±sin2i)(2sini3±sin2i3),subscriptsuperscript𝑓𝜂23-2plus-or-minus1364plus-or-minus2𝑖2𝑖plus-or-minus2subscript𝑖32subscript𝑖3\displaystyle\displaystyle f^{\eta(2)}_{[3,\raisebox{0.0pt}{-}2,\pm 1]}=\frac{% 3}{64}(2\sin i\pm\sin 2i)(2\sin i_{3}\pm\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , - 2 , ± 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 2 roman_sin italic_i ± roman_sin 2 italic_i ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,-2,±2]η(2)=3256(3±4cosi+cos2i)(3±4cosi3\displaystyle\displaystyle f^{\eta(2)}_{[3,\raisebox{0.0pt}{-}2,\pm 2]}=\frac{% 3}{256}(3\pm 4\cos i+\cos 2i)(3\pm 4\cos i_{3}italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , - 2 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 256 end_ARG ( 3 ± 4 roman_cos italic_i + roman_cos 2 italic_i ) ( 3 ± 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+cos2i3),\displaystyle\displaystyle\phantom{f^{\eta(2)}_{[3,\raisebox{0.0pt}{-}2,\pm 2]% }=}+\cos 2i_{3}),+ roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,0,0]η(2)=364(1cos2i)(1+3cos2i3),subscriptsuperscript𝑓𝜂230036412𝑖132subscript𝑖3\displaystyle\displaystyle f^{\eta(2)}_{[3,0,0]}=\frac{3}{64}(1-\cos 2i)(1+3% \cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 0 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 1 - roman_cos 2 italic_i ) ( 1 + 3 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,0,±1]η(2)=332(±2sini+sin2i)sin2i3,subscriptsuperscript𝑓𝜂230plus-or-minus1332plus-or-minus2𝑖2𝑖2subscript𝑖3\displaystyle\displaystyle f^{\eta(2)}_{[3,0,\pm 1]}=-\frac{3}{32}(\pm 2\sin i% +\sin 2i)\sin 2i_{3},italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 0 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( ± 2 roman_sin italic_i + roman_sin 2 italic_i ) roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,
f[3,0,±2]η(2)=3128(3±4cosi+cos2i)(1cos2i3),subscriptsuperscript𝑓𝜂230plus-or-minus23128plus-or-minus34𝑖2𝑖12subscript𝑖3\displaystyle\displaystyle f^{\eta(2)}_{[3,0,\pm 2]}=\frac{3}{128}(3\pm 4\cos i% +\cos 2i)(1-\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 0 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 128 end_ARG ( 3 ± 4 roman_cos italic_i + roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,2,0]η(2)=f[3,-2,0]η(2),subscriptsuperscript𝑓𝜂2320subscriptsuperscript𝑓𝜂23-20\displaystyle\displaystyle f^{\eta(2)}_{[3,2,0]}=f^{\eta(2)}_{[3,\raisebox{0.0% pt}{-}2,0]},italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 2 , 0 ] end_POSTSUBSCRIPT = italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , - 2 , 0 ] end_POSTSUBSCRIPT ,
f[3,2,±1]η(2)=364(2sini±sin2i)(2sini3sin2i3),subscriptsuperscript𝑓𝜂232plus-or-minus1364plus-or-minus2𝑖2𝑖minus-or-plus2subscript𝑖32subscript𝑖3\displaystyle\displaystyle f^{\eta(2)}_{[3,2,\pm 1]}=-\frac{3}{64}(2\sin i\pm% \sin 2i)(2\sin i_{3}\mp\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 2 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 2 roman_sin italic_i ± roman_sin 2 italic_i ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∓ roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[3,2,±2]η(2)=3256(3±4cosi+cos2i)(34cosi3+cos2i3).subscriptsuperscript𝑓𝜂232plus-or-minus23256plus-or-minus34𝑖2𝑖minus-or-plus34subscript𝑖32subscript𝑖3\displaystyle\displaystyle f^{\eta(2)}_{[3,2,\pm 2]}=\frac{3}{256}(3\pm 4\cos i% +\cos 2i)(3\mp 4\cos i_{3}+\cos 2i_{3}).italic_f start_POSTSUPERSCRIPT italic_η ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 3 , 2 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 256 end_ARG ( 3 ± 4 roman_cos italic_i + roman_cos 2 italic_i ) ( 3 ∓ 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) .

(4) Inclination functions with the form f[κ,p,q]i(2)(i)subscriptsuperscript𝑓𝑖2𝜅𝑝𝑞𝑖f^{i(2)}_{[\kappa,\,p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ):

f[2,-2,0]i(2)=964(1cos2i3)sin2i,subscriptsuperscript𝑓𝑖22-2096412subscript𝑖32𝑖\displaystyle f^{i(2)}_{[2,\raisebox{0.0pt}{-}2,0]}=\frac{9}{64}(1-\cos 2i_{3}% )\sin 2i,italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 9 end_ARG start_ARG 64 end_ARG ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_sin 2 italic_i ,
f[2,-2,±1]i(2)=332(cosi±cos2i)(2sini3±sin2i3),subscriptsuperscript𝑓𝑖22-2plus-or-minus1332plus-or-minus𝑖2𝑖plus-or-minus2subscript𝑖32subscript𝑖3\displaystyle f^{i(2)}_{[2,\raisebox{0.0pt}{-}2,\pm 1]}=\frac{3}{32}(\cos i\pm% \cos 2i)(2\sin i_{3}\pm\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , ± 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( roman_cos italic_i ± roman_cos 2 italic_i ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,-2,±2]i(2)=3128(±3+4cosi3±cos2i3)(2sini±sin2i),subscriptsuperscript𝑓𝑖22-2plus-or-minus23128plus-or-minusplus-or-minus34subscript𝑖32subscript𝑖3plus-or-minus2𝑖2𝑖\displaystyle f^{i(2)}_{[2,\raisebox{0.0pt}{-}2,\pm 2]}=-\frac{3}{128}(\pm 3+4% \cos i_{3}\pm\cos 2i_{3})(2\sin i\pm\sin 2i),italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 128 end_ARG ( ± 3 + 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ( 2 roman_sin italic_i ± roman_sin 2 italic_i ) ,
f[2,0,0]i(2)=332(1+3cos2i3)sin2i,subscriptsuperscript𝑓𝑖2200332132subscript𝑖32𝑖\displaystyle f^{i(2)}_{[2,0,0]}=\frac{3}{32}(1+3\cos 2i_{3})\sin 2i,italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( 1 + 3 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_sin 2 italic_i ,
f[2,0,±1]i(2)=316(±cosi+cos2i)sin2i3,subscriptsuperscript𝑓𝑖220plus-or-minus1316plus-or-minus𝑖2𝑖2subscript𝑖3\displaystyle f^{i(2)}_{[2,0,\pm 1]}=-\frac{3}{16}(\pm\cos i+\cos 2i)\sin 2i_{% 3},italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 16 end_ARG ( ± roman_cos italic_i + roman_cos 2 italic_i ) roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , (220)
f[2,0,±2]i(2)=364(1cos2i3)(±2sini+sin2i),subscriptsuperscript𝑓𝑖220plus-or-minus236412subscript𝑖3plus-or-minus2𝑖2𝑖\displaystyle f^{i(2)}_{[2,0,\pm 2]}=-\frac{3}{64}(1-\cos 2i_{3})(\pm 2\sin i+% \sin 2i),italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ( ± 2 roman_sin italic_i + roman_sin 2 italic_i ) ,
f[2,2,0]i(2)=f[2,-2,0]i(2),subscriptsuperscript𝑓𝑖2220subscriptsuperscript𝑓𝑖22-20\displaystyle f^{i(2)}_{[2,2,0]}=f^{i(2)}_{[2,\raisebox{0.0pt}{-}2,0]},italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , 0 ] end_POSTSUBSCRIPT = italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , 0 ] end_POSTSUBSCRIPT ,
f[2,2,±1]i(2)=332(cosi±cos2i)(2sini3sin2i3),subscriptsuperscript𝑓𝑖222plus-or-minus1332plus-or-minus𝑖2𝑖minus-or-plus2subscript𝑖32subscript𝑖3\displaystyle f^{i(2)}_{[2,2,\pm 1]}=-\frac{3}{32}(\cos i\pm\cos 2i)(2\sin i_{% 3}\mp\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , ± 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( roman_cos italic_i ± roman_cos 2 italic_i ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∓ roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,2,±2]i(2)=3128(±34cosi3±cos2i3)(2sini±sin2i).subscriptsuperscript𝑓𝑖222plus-or-minus23128plus-or-minusplus-or-minus34subscript𝑖32subscript𝑖3plus-or-minus2𝑖2𝑖\displaystyle f^{i(2)}_{[2,2,\pm 2]}=-\frac{3}{128}(\pm 3-4\cos i_{3}\pm\cos 2% i_{3})(2\sin i\pm\sin 2i).italic_f start_POSTSUPERSCRIPT italic_i ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 128 end_ARG ( ± 3 - 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ( 2 roman_sin italic_i ± roman_sin 2 italic_i ) .

(5) Inclination functions with the form f[κ,p,q]Ω(2)(i)subscriptsuperscript𝑓Ω2𝜅𝑝𝑞𝑖f^{\Omega(2)}_{[\kappa,\,p,\,q]}(i)italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ):
(5) 形式 f[κ,p,q]Ω(2)(i)subscriptsuperscript𝑓Ω2𝜅𝑝𝑞𝑖f^{\Omega(2)}_{[\kappa,\,p,\,q]}(i)italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ) 为的倾斜函数 :

f[2,-2,-1]Ω(2)=332seci2sin3i2(2sini3sin2i3),subscriptsuperscript𝑓Ω22-2-1332𝑖23𝑖22subscript𝑖32subscript𝑖3\displaystyle f^{\Omega(2)}_{[2,\raisebox{0.0pt}{-}2,\raisebox{0.0pt}{-}1]}=% \frac{3}{32}\sec\frac{i}{2}\sin\frac{3i}{2}(2\sin i_{3}-\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , - 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 32 end_ARG roman_sec divide start_ARG italic_i end_ARG start_ARG 2 end_ARG roman_sin divide start_ARG 3 italic_i end_ARG start_ARG 2 end_ARG ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,-2,0]Ω(2)=932cosi(1cos2i3),subscriptsuperscript𝑓Ω22-20932𝑖12subscript𝑖3\displaystyle f^{\Omega(2)}_{[2,\raisebox{0.0pt}{-}2,0]}=\frac{9}{32}\cos i(1-% \cos 2i_{3}),italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 9 end_ARG start_ARG 32 end_ARG roman_cos italic_i ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,-2,1]Ω(2)=332cos3i2csci2(2sini3+sin2i3),subscriptsuperscript𝑓Ω22-213323𝑖2𝑖22subscript𝑖32subscript𝑖3\displaystyle f^{\Omega(2)}_{[2,\raisebox{0.0pt}{-}2,1]}=\frac{3}{32}\cos\frac% {3i}{2}\csc\frac{i}{2}(2\sin i_{3}+\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 32 end_ARG roman_cos divide start_ARG 3 italic_i end_ARG start_ARG 2 end_ARG roman_csc divide start_ARG italic_i end_ARG start_ARG 2 end_ARG ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,-2,±2]Ω(2)=364(1±cosi)(±3+4cosi3±cos2i3),subscriptsuperscript𝑓Ω22-2plus-or-minus2364plus-or-minus1𝑖plus-or-minusplus-or-minus34subscript𝑖32subscript𝑖3\displaystyle f^{\Omega(2)}_{[2,\raisebox{0.0pt}{-}2,\pm 2]}=-\frac{3}{64}(1% \pm\cos i)(\pm 3+4\cos i_{3}\pm\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 1 ± roman_cos italic_i ) ( ± 3 + 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,0,-1]Ω(2)=316seci2sin3i2sin2i3,subscriptsuperscript𝑓Ω220-1316𝑖23𝑖22subscript𝑖3\displaystyle f^{\Omega(2)}_{[2,0,\raisebox{0.0pt}{-}1]}=\frac{3}{16}\sec\frac% {i}{2}\sin\frac{3i}{2}\sin 2i_{3},italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , - 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 16 end_ARG roman_sec divide start_ARG italic_i end_ARG start_ARG 2 end_ARG roman_sin divide start_ARG 3 italic_i end_ARG start_ARG 2 end_ARG roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,
f[2,0,0]Ω(2)=316cosi(1+3cos2i3),subscriptsuperscript𝑓Ω2200316𝑖132subscript𝑖3\displaystyle f^{\Omega(2)}_{[2,0,0]}=\frac{3}{16}\cos i(1+3\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , 0 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 16 end_ARG roman_cos italic_i ( 1 + 3 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (221)
f[2,0,1]Ω(2)=316cos3i2csci2sin2i3,subscriptsuperscript𝑓Ω22013163𝑖2𝑖22subscript𝑖3\displaystyle f^{\Omega(2)}_{[2,0,1]}=-\frac{3}{16}\cos\frac{3i}{2}\csc\frac{i% }{2}\sin 2i_{3},italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 16 end_ARG roman_cos divide start_ARG 3 italic_i end_ARG start_ARG 2 end_ARG roman_csc divide start_ARG italic_i end_ARG start_ARG 2 end_ARG roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,
f[2,0,±2]Ω(2)=332(±1+cosi)(1cos2i3),subscriptsuperscript𝑓Ω220plus-or-minus2332plus-or-minus1𝑖12subscript𝑖3\displaystyle f^{\Omega(2)}_{[2,0,\pm 2]}=-\frac{3}{32}(\pm 1+\cos i)(1-\cos 2% i_{3}),italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( ± 1 + roman_cos italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,2,-1]Ω(2)=332seci2sin3i2(2sini3+sin2i3),subscriptsuperscript𝑓Ω222-1332𝑖23𝑖22subscript𝑖32subscript𝑖3\displaystyle f^{\Omega(2)}_{[2,2,\raisebox{0.0pt}{-}1]}=-\frac{3}{32}\sec% \frac{i}{2}\sin\frac{3i}{2}(2\sin i_{3}+\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , - 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 32 end_ARG roman_sec divide start_ARG italic_i end_ARG start_ARG 2 end_ARG roman_sin divide start_ARG 3 italic_i end_ARG start_ARG 2 end_ARG ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,2,0]Ω(2)=f[2,-2,0]Ω(2),subscriptsuperscript𝑓Ω2220subscriptsuperscript𝑓Ω22-20\displaystyle f^{\Omega(2)}_{[2,2,0]}=f^{\Omega(2)}_{[2,\raisebox{0.0pt}{-}2,0% ]},italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , 0 ] end_POSTSUBSCRIPT = italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , 0 ] end_POSTSUBSCRIPT ,
f[2,2,1]Ω(2)=332cos3i2csci2(2sini3sin2i3),subscriptsuperscript𝑓Ω22213323𝑖2𝑖22subscript𝑖32subscript𝑖3\displaystyle f^{\Omega(2)}_{[2,2,1]}=-\frac{3}{32}\cos\frac{3i}{2}\csc\frac{i% }{2}(2\sin i_{3}-\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 32 end_ARG roman_cos divide start_ARG 3 italic_i end_ARG start_ARG 2 end_ARG roman_csc divide start_ARG italic_i end_ARG start_ARG 2 end_ARG ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,2,±2]Ω(2)=364(±1+cosi)(34cosi3+cos2i3).subscriptsuperscript𝑓Ω222plus-or-minus2364plus-or-minus1𝑖minus-or-plus34subscript𝑖32subscript𝑖3\displaystyle f^{\Omega(2)}_{[2,2,\pm 2]}=-\frac{3}{64}(\pm 1+\cos i)(3\mp 4% \cos i_{3}+\cos 2i_{3}).italic_f start_POSTSUPERSCRIPT roman_Ω ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( ± 1 + roman_cos italic_i ) ( 3 ∓ 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) .

(6) Inclination functions with the form f[κ,p,q]λ(2)(i)subscriptsuperscript𝑓𝜆2𝜅𝑝𝑞𝑖f^{\lambda(2)}_{[\kappa,\,p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ):
(6) 形式 f[κ,p,q]λ(2)(i)subscriptsuperscript𝑓𝜆2𝜅𝑝𝑞𝑖f^{\lambda(2)}_{[\kappa,\,p,\,q]}(i)italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ italic_κ , italic_p , italic_q ] end_POSTSUBSCRIPT ( italic_i ) 为的倾斜函数 :

f[2,-2,0]λ(2)[a]=964(1cos2i)(1cos2i3),subscriptsuperscript𝑓𝜆2delimited-[]𝑎2-2096412𝑖12subscript𝑖3\displaystyle\displaystyle f^{\lambda(2)[a]}_{[2,\raisebox{0.0pt}{-}2,0]}=-% \frac{9}{64}(1-\cos 2i)(1-\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_a ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , 0 ] end_POSTSUBSCRIPT = - divide start_ARG 9 end_ARG start_ARG 64 end_ARG ( 1 - roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,-2,±1]λ(2)[a]=332(2sini±sin2i)(2sini3±sin2i3),subscriptsuperscript𝑓𝜆2delimited-[]𝑎2-2plus-or-minus1332plus-or-minus2𝑖2𝑖plus-or-minus2subscript𝑖32subscript𝑖3\displaystyle\displaystyle f^{\lambda(2)[a]}_{[2,\raisebox{0.0pt}{-}2,\pm 1]}=% \frac{-3}{32}(2\sin i\pm\sin 2i)(2\sin i_{3}\pm\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_a ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , ± 1 ] end_POSTSUBSCRIPT = divide start_ARG - 3 end_ARG start_ARG 32 end_ARG ( 2 roman_sin italic_i ± roman_sin 2 italic_i ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,-2,±2]λ(2)[a]=3128(3±4cosi+cos2i)(3±4cosi3\displaystyle\displaystyle f^{\lambda(2)[a]}_{[2,\raisebox{0.0pt}{-}2,\pm 2]}=% -\frac{3}{128}(3\pm 4\cos i+\cos 2i)(3\pm 4\cos i_{3}italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_a ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 128 end_ARG ( 3 ± 4 roman_cos italic_i + roman_cos 2 italic_i ) ( 3 ± 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+cos2i3),\displaystyle\displaystyle\phantom{f^{\lambda(2)[a]}_{[2,\raisebox{0.0pt}{-}2,% \pm 2]}=}+\cos 2i_{3}),+ roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,0,0]λ(2)[a]=332(1cos2i)(1+3cos2i3),subscriptsuperscript𝑓𝜆2delimited-[]𝑎20033212𝑖132subscript𝑖3\displaystyle\displaystyle f^{\lambda(2)[a]}_{[2,0,0]}=-\frac{3}{32}(1-\cos 2i% )(1+3\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_a ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , 0 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( 1 - roman_cos 2 italic_i ) ( 1 + 3 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,0,±1]λ(2)[a]=316(±2sini+sin2i)sin2i3,subscriptsuperscript𝑓𝜆2delimited-[]𝑎20plus-or-minus1316plus-or-minus2𝑖2𝑖2subscript𝑖3\displaystyle\displaystyle f^{\lambda(2)[a]}_{[2,0,\pm 1]}=\frac{3}{16}(\pm 2% \sin i+\sin 2i)\sin 2i_{3},italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_a ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , ± 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 16 end_ARG ( ± 2 roman_sin italic_i + roman_sin 2 italic_i ) roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,
f[2,0,±2]λ(2)[a]=364(3±4cosi+cos2i)(1cos2i3),subscriptsuperscript𝑓𝜆2delimited-[]𝑎20plus-or-minus2364plus-or-minus34𝑖2𝑖12subscript𝑖3\displaystyle\displaystyle f^{\lambda(2)[a]}_{[2,0,\pm 2]}=-\frac{3}{64}(3\pm 4% \cos i+\cos 2i)(1-\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_a ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 3 ± 4 roman_cos italic_i + roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,2,0]λ(2)[a]=f[2,-2,0]λ(2)[a],subscriptsuperscript𝑓𝜆2delimited-[]𝑎220subscriptsuperscript𝑓𝜆2delimited-[]𝑎2-20\displaystyle\displaystyle f^{\lambda(2)[a]}_{[2,2,0]}=f^{\lambda(2)[a]}_{[2,% \raisebox{0.0pt}{-}2,0]},italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_a ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , 0 ] end_POSTSUBSCRIPT = italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_a ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , 0 ] end_POSTSUBSCRIPT ,
f[2,2,±1]λ(2)[a]=332(2sini±sin2i)(2sini3sin2i3),subscriptsuperscript𝑓𝜆2delimited-[]𝑎22plus-or-minus1332plus-or-minus2𝑖2𝑖minus-or-plus2subscript𝑖32subscript𝑖3\displaystyle\displaystyle f^{\lambda(2)[a]}_{[2,2,\pm 1]}=\frac{3}{32}(2\sin i% \pm\sin 2i)(2\sin i_{3}\mp\sin 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_a ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , ± 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( 2 roman_sin italic_i ± roman_sin 2 italic_i ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∓ roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,2,±2]λ(2)[a]=3128(3±4cosi+cos2i)(34cosi3\displaystyle\displaystyle f^{\lambda(2)[a]}_{[2,2,\pm 2]}=-\frac{3}{128}(3\pm 4% \cos i+\cos 2i)(3\mp 4\cos i_{3}italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_a ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 128 end_ARG ( 3 ± 4 roman_cos italic_i + roman_cos 2 italic_i ) ( 3 ∓ 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+cos2i3),\displaystyle\displaystyle\phantom{f^{\lambda(2)[a]}_{[2,2,\pm 2]}=}+\cos 2i_{% 3}),+ roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,-2,-1]λ(2)[λ]=364seci2(5sini2+2sin3i2sin5i2)(2sini3\displaystyle\displaystyle f^{\lambda(2)[\lambda]}_{[2,\raisebox{0.0pt}{-}2,% \raisebox{0.0pt}{-}1]}=-\frac{3}{64}\sec\frac{i}{2}(5\sin\frac{i}{2}+2\sin% \frac{3i}{2}-\sin\frac{5i}{2})(2\sin i_{3}italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , - 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 64 end_ARG roman_sec divide start_ARG italic_i end_ARG start_ARG 2 end_ARG ( 5 roman_sin divide start_ARG italic_i end_ARG start_ARG 2 end_ARG + 2 roman_sin divide start_ARG 3 italic_i end_ARG start_ARG 2 end_ARG - roman_sin divide start_ARG 5 italic_i end_ARG start_ARG 2 end_ARG ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
sin2i3),\displaystyle\displaystyle\phantom{f^{\lambda(2)[\lambda]}_{[2,\raisebox{0.0pt% }{-}2,\raisebox{0.0pt}{-}1]}=}-\sin 2i_{3}),- roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , (222)
f[2,-2,0]λ(2)[λ]=964(3cos2i)(1cos2i3),subscriptsuperscript𝑓𝜆2delimited-[]𝜆2-2096432𝑖12subscript𝑖3\displaystyle\displaystyle f^{\lambda(2)[\lambda]}_{[2,\raisebox{0.0pt}{-}2,0]% }=-\frac{9}{64}(3-\cos 2i)(1-\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , 0 ] end_POSTSUBSCRIPT = - divide start_ARG 9 end_ARG start_ARG 64 end_ARG ( 3 - roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,-2,1]λ(2)[λ]=364csci2(5cosi22cos3i2cos5i2)(2sini3\displaystyle\displaystyle f^{\lambda(2)[\lambda]}_{[2,\raisebox{0.0pt}{-}2,1]% }=-\frac{3}{64}\csc\frac{i}{2}(5\cos\frac{i}{2}-2\cos\frac{3i}{2}-\cos\frac{5i% }{2})(2\sin i_{3}italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 64 end_ARG roman_csc divide start_ARG italic_i end_ARG start_ARG 2 end_ARG ( 5 roman_cos divide start_ARG italic_i end_ARG start_ARG 2 end_ARG - 2 roman_cos divide start_ARG 3 italic_i end_ARG start_ARG 2 end_ARG - roman_cos divide start_ARG 5 italic_i end_ARG start_ARG 2 end_ARG ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+sin2i3),\displaystyle\displaystyle\phantom{f^{\lambda(2)[\lambda]}_{[2,\raisebox{0.0pt% }{-}2,1]}=}+\sin 2i_{3}),+ roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,-2,±2]λ(2)[λ]=3128(5±6cosi+cos2i)(3±4cosi3\displaystyle\displaystyle f^{\lambda(2)[\lambda]}_{[2,\raisebox{0.0pt}{-}2,% \pm 2]}=-\frac{3}{128}(5\pm 6\cos i+\cos 2i)(3\pm 4\cos i_{3}italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 128 end_ARG ( 5 ± 6 roman_cos italic_i + roman_cos 2 italic_i ) ( 3 ± 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+cos2i3),\displaystyle\displaystyle\phantom{f^{\lambda(2)[\lambda]}_{[2,\raisebox{0.0pt% }{-}2,\pm 2]}=}+\cos 2i_{3}),+ roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,0,-1]λ(2)[λ]=332seci2(5sini2+2sin3i2sin5i2)sin2i3,subscriptsuperscript𝑓𝜆2delimited-[]𝜆20-1332𝑖25𝑖223𝑖25𝑖22subscript𝑖3\displaystyle\displaystyle f^{\lambda(2)[\lambda]}_{[2,0,\raisebox{0.0pt}{-}1]% }=-\frac{3}{32}\sec\frac{i}{2}(5\sin\frac{i}{2}+2\sin\frac{3i}{2}-\sin\frac{5i% }{2})\sin 2i_{3},italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , - 1 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 32 end_ARG roman_sec divide start_ARG italic_i end_ARG start_ARG 2 end_ARG ( 5 roman_sin divide start_ARG italic_i end_ARG start_ARG 2 end_ARG + 2 roman_sin divide start_ARG 3 italic_i end_ARG start_ARG 2 end_ARG - roman_sin divide start_ARG 5 italic_i end_ARG start_ARG 2 end_ARG ) roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,
f[2,0,0]λ(2)[λ]=332(3cos2i)(1+3cos2i3),subscriptsuperscript𝑓𝜆2delimited-[]𝜆20033232𝑖132subscript𝑖3\displaystyle\displaystyle f^{\lambda(2)[\lambda]}_{[2,0,0]}=-\frac{3}{32}(3-% \cos 2i)(1+3\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , 0 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 32 end_ARG ( 3 - roman_cos 2 italic_i ) ( 1 + 3 roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,0,1]λ(2)[λ]=332csci2(5cosi22cos3i2cos5i2)sin2i3,subscriptsuperscript𝑓𝜆2delimited-[]𝜆201332𝑖25𝑖223𝑖25𝑖22subscript𝑖3\displaystyle\displaystyle f^{\lambda(2)[\lambda]}_{[2,0,1]}=\frac{3}{32}\csc% \frac{i}{2}(5\cos\frac{i}{2}-2\cos\frac{3i}{2}-\cos\frac{5i}{2})\sin 2i_{3},italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 32 end_ARG roman_csc divide start_ARG italic_i end_ARG start_ARG 2 end_ARG ( 5 roman_cos divide start_ARG italic_i end_ARG start_ARG 2 end_ARG - 2 roman_cos divide start_ARG 3 italic_i end_ARG start_ARG 2 end_ARG - roman_cos divide start_ARG 5 italic_i end_ARG start_ARG 2 end_ARG ) roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,
f[2,0,±2]λ(2)[λ]=364(5±6cosi+cos2i)(1cos2i3),subscriptsuperscript𝑓𝜆2delimited-[]𝜆20plus-or-minus2364plus-or-minus56𝑖2𝑖12subscript𝑖3\displaystyle\displaystyle f^{\lambda(2)[\lambda]}_{[2,0,\pm 2]}=-\frac{3}{64}% (5\pm 6\cos i+\cos 2i)(1-\cos 2i_{3}),italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 0 , ± 2 ] end_POSTSUBSCRIPT = - divide start_ARG 3 end_ARG start_ARG 64 end_ARG ( 5 ± 6 roman_cos italic_i + roman_cos 2 italic_i ) ( 1 - roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,2,-1]λ(2)[λ]=364seci2(5sini2+2sin3i2sin5i2)(2sini3\displaystyle\displaystyle f^{\lambda(2)[\lambda]}_{[2,2,\raisebox{0.0pt}{-}1]% }=\frac{3}{64}\sec\frac{i}{2}(5\sin\frac{i}{2}+2\sin\frac{3i}{2}-\sin\frac{5i}% {2})(2\sin i_{3}italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , - 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 64 end_ARG roman_sec divide start_ARG italic_i end_ARG start_ARG 2 end_ARG ( 5 roman_sin divide start_ARG italic_i end_ARG start_ARG 2 end_ARG + 2 roman_sin divide start_ARG 3 italic_i end_ARG start_ARG 2 end_ARG - roman_sin divide start_ARG 5 italic_i end_ARG start_ARG 2 end_ARG ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
+sin2i3),\displaystyle\displaystyle\phantom{f^{\lambda(2)[\lambda]}_{[2,2,\raisebox{0.0% pt}{-}1]}=}+\sin 2i_{3}),+ roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,2,0]λ(2)[λ]=f[2,-2,0]λ(2)[λ],subscriptsuperscript𝑓𝜆2delimited-[]𝜆220subscriptsuperscript𝑓𝜆2delimited-[]𝜆2-20\displaystyle\displaystyle f^{\lambda(2)[\lambda]}_{[2,2,0]}=f^{\lambda(2)[% \lambda]}_{[2,\raisebox{0.0pt}{-}2,0]},italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , 0 ] end_POSTSUBSCRIPT = italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , - 2 , 0 ] end_POSTSUBSCRIPT ,
f[2,2,1]λ(2)[λ]=364csci2(5cosi22cos3i2cos5i2)(2sini3\displaystyle\displaystyle f^{\lambda(2)[\lambda]}_{[2,2,1]}=\frac{3}{64}\csc% \frac{i}{2}(5\cos\frac{i}{2}-2\cos\frac{3i}{2}-\cos\frac{5i}{2})(2\sin i_{3}italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , 1 ] end_POSTSUBSCRIPT = divide start_ARG 3 end_ARG start_ARG 64 end_ARG roman_csc divide start_ARG italic_i end_ARG start_ARG 2 end_ARG ( 5 roman_cos divide start_ARG italic_i end_ARG start_ARG 2 end_ARG - 2 roman_cos divide start_ARG 3 italic_i end_ARG start_ARG 2 end_ARG - roman_cos divide start_ARG 5 italic_i end_ARG start_ARG 2 end_ARG ) ( 2 roman_sin italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT
sin2i3),\displaystyle\displaystyle\phantom{f^{\lambda(2)[\lambda]}_{[2,2,1]}=}-\sin 2i% _{3}),- roman_sin 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ,
f[2,2,±2]λ(2)[λ]=3128(5±6cosi+cos2i)(34cosi3+cos2i3).subscriptsuperscript𝑓𝜆2delimited-[]𝜆22plus-or-minus23128plus-or-minus56𝑖2𝑖minus-or-plus34subscript𝑖32subscript𝑖3\displaystyle\displaystyle f^{\lambda(2)[\lambda]}_{[2,2,\pm 2]}=\frac{-3}{128% }(5\pm 6\cos i+\cos 2i)(3\mp 4\cos i_{3}+\cos 2i_{3}).italic_f start_POSTSUPERSCRIPT italic_λ ( 2 ) [ italic_λ ] end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ 2 , 2 , ± 2 ] end_POSTSUBSCRIPT = divide start_ARG - 3 end_ARG start_ARG 128 end_ARG ( 5 ± 6 roman_cos italic_i + roman_cos 2 italic_i ) ( 3 ∓ 4 roman_cos italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos 2 italic_i start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) .

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