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
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 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 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。
†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 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 [2, 3]。对于此类天文台,拟议的项目包括 LISA [4, 5]、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 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 [9, 10, 11]。
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 [3, 7]。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) [12, 13, 14]。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[16, 17, 18, 19, 20, 17]。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 [18, 19]。它们还有助于对星间光链路和光传播进行理论研究 [20, 17]。
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 [26, 20, 27, 28, 29, 30, 18] 和太极[17, 31],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 [33, 34, 35, 36],尤其是均元法 [34, 35]。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 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 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
二、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.
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 - plane, where the -axis points toward the perigee. In this system, the satellite’s Cartesian coordinates can be denoted as [35, 33]:
是轨道右手坐标系,原点位于地球质心。卫星的轨道平面与 - 平面相同,其中 -轴指向近地点。在这个系统中,卫星的笛卡尔坐标 可以表示为 [35, 33]:
(7) |
with representing the geocentric radius, the true anomaly, the semimajor axis, the orbital eccentricity, the Earth’s gravitational constant, and the eccentric anomaly. satisfies Kepler’s equation,
代表 地心半径、 真实异常、 长半轴、 轨道偏心率、 地球引力常数和 偏心异常。 满足 Kepler 方程,
(8) |
where denotes the mean anomaly. Specifically, is given by in the two-body problem, with the mean motion and the passing time of the perigee . Equation (8), which is a transcendental equation, can be solved iteratively, resulting in the following expression [21]:
其中 表示均值距平。具体来说, 由 在双体问题中给出,具有近地点的平均运动 和通过时间 。方程 (8) 是一个超越方程,可以迭代求解,得到以下表达式 [21]:
(9) |
By substituting Eq. (9) into Eq. (7), one can obtain the explicit coordinates .
通过将方程 (9) 代入方程 (7),可以得到显式坐标 。
The orbital planes may not be identical for the three TianQin satellites. Thus, the geocentric ecliptic coordinate system is also employed, where the - plane is the ecliptic plane. The -axis points toward the vernal equinox, and the -axis is normal to the ecliptic plane. The coordinates and in this system can be obtained by and 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 ,其中 - 平面是黄道平面。 -轴指向春分, -轴垂直于黄道平面。在这个系统中,坐标 和 可以通过以下变换 [35, 33] 获得 :
(10) | |||
(11) |
where , , and denote the satellite’s longitude of the ascending node, inclination, and argument of perigee, respectively. Additionally, and are the rotation matrices that rotate vectors by an angle about the or axis,
其中 ,、 和 分别表示卫星的升交点经度、倾角和近地点参数。此外, 和 是将向量绕 or 轴旋转一定角度 的旋转矩阵,
(12) | |||
(13) |
Combining Eqs. (7) and (9)-(13), the position vector and velocity vector of SC (, 2, 3), and , are given by
组合方程。(7) 和 (9)-(13),则 SC ( , 2, 3) 和 的位置矢量和速度矢量由下式给出
(24) |
where 哪里
(25) |
, , , and . Define , in Eq. (24) are straightforwardly determined in the two-body problem by
、 和 .定义 , 在方程(24)中直接由下式确定
(26) |
where 哪里
(27) |
Note that Eq. (24) remains valid even when considering gravitational perturbations, with the only change being the replacement of 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(26) 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 , relative line-of-sight velocity between satellites , and breathing angle ,
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 、卫星 之间的相对视线速度和呼吸角 ,
(28) | ||||
(29) | ||||
(30) |
where , , and take values 1, 2, or 3 and . Substituting Eq. (24) into Eqs. (28)-(30), one can obtain the explicit expressions, for these three kinematic indicators, with forms , , and , respectively.
其中 , , 和 取值 1、2 或 3 和 。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 、 的 显式表达式。
To maintain the constellation as an equilateral triangle, i.e. , 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 ,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:
(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:
(36) |
The above requirements on the inter-satellite parameters can be achieved in the two-body problem, if the initial orbital elements in Eq. (26) are set to
如果方程(26)中的初始轨道单元 设置为
(37) |
where the parameters with subscript “o” are the nominal ones of the TianQin constellation. For instance, these values can be chosen as km, , and , , respectively establishing the orbit size and orienting the orbital plane perpendicular to J0806 [7, 24]. The initial value associated with the orbit phase is typically selected to be any value within the range of to , 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 km、 、 和 、 ,分别建立轨道大小和垂直于 J0806 的轨道平面 [7, 24]。与轨道相位相关的初始值 通常选择为 到 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 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 ), the variations of these indicators in the two-body problem, up to the first order of , can be expressed as
为了分析允许 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 ),这些指标在两体问题中的变化,直到 的一阶,可以表示为
(38) |
(39) |
(40) |
where and , with the indices , , and using cyclic indexing (). If we further set and , then it follows that
其中 和 ,带有索引 , ,并使用 循环索引 ( )。如果我们进一步设置 和 ,则
(41) | |||
(42) | |||
(43) |
where .
Equations (41)-(43) indicate that, at the zeroth order of , 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 .
方程 (41)-(43) 表明,在 的零阶 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 , and , namely , then , 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 [21, 24, 13, 38, 39, 40, 41, 42])。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 ,和 ,即 ,则 ,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
三、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 and , 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 perturbation, which has a magnitude of . 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, km in arm-lengths, m/s in relative velocities, and 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 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 和 [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 的扰动引起的长期扰动,其震级为 。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, 相对速度偏差为 m/s, 呼吸角偏差为 5 年。与 Refs 相反。
[21, 22, 23, 36, 16, 32],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 km半径轨道。
III.1 Dynamic model
三.1 动态模型
The gravitational potential acting on a satellite can be expressed as
作用在卫星上的引力势 可以表示为
(44) |
where is the gravitational potential of a pointlike Earth, and represents a perturbative potential describing the satellite’s perturbed motion. Under the influence of , the evolution of the satellite’s orbital elements is governed by Lagrange’s planetary equations [43],
其中 是点状地球的引力势,表示 描述卫星扰动运动的扰动势。在 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,
(45) | ||||
(46) | ||||
(47) | ||||
(48) | ||||
(49) | ||||
(50) |
where , and new variables and are introduced,
其中 , 和 新变量 和 被引入,
(51) |
to avoid the singularity at . When , the solutions to Eqs. (45)-(50) revert to the Keplerian case discussed in Sec. II.1.
为了避免 处 的奇点。当 , 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 km, the perturbative potential predominantly encompasses the perturbation effects arising from the Sun, Moon, and Earth’s term, as expressed in the following expressions:
对于轨道半径为 km 的天琴轨道,扰动势 主要包括由太阳、月亮和地球 项引起的扰动效应,如以下表达式所示:
(52) |
(53) | |||
(54) | |||
(55) |
where and are the gravitational constants of the Sun and the Moon, respectively. and denote the geocentric distances of the Sun and the Moon. Moreover, is the Legendre polynomial of degree , with 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, stands for the equatorial radius of the Earth, and represents the second zonal harmonic coefficient.
Furthermore, is the angular separation of the Sun and the satellite as observed from the Earth’s center,
其中 和 分别是太阳和月亮的引力常数。 并 表示太阳和月亮的地心距离。此外, 是度 的勒让德多项式,表示 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,