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

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.

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$\{X,Y,Z\}$ 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 $X$-$Y$ plane, where the $X$-axis points toward the perigee. In this system, the satellite’s Cartesian coordinates $(X,Y,Z,\dot{X},\dot{Y},\dot{Z})$ can be denoted as [35, 33]:
$\{X,Y,Z\}$ 是轨道右手坐标系，原点位于地球质心。卫星的轨道平面与 $X$ - $Y$ 平面相同，其中 $X$ -轴指向近地点。在这个系统中，卫星的笛卡尔坐标 $(X,Y,Z,\dot{X},\dot{Y},\dot{Z})$ 可以表示为 [35， 33]：

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$\displaystyle\left\{\begin{array}[]{l}X=r\cos\nu=a(\cos E-e),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ Y=r\sin\nu=a\sqrt{1-e^{2}}\sin E,\vskip 3.0pt plus 1.0pt minus 1.0pt\\ Z=0,\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \dot{X}=-\frac{\sqrt{\mu a}}{r}\sin E,\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \dot{Y}=\frac{\sqrt{\mu a}}{r}\sqrt{1-e^{2}}\cos E,\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \dot{Z}=0,\end{array}\right.$

(7)

with $r$ representing the geocentric radius, $\nu$ the true anomaly, $a$ the semimajor axis, $e$ the orbital eccentricity, $\mu=GM_{\text{e}}$ the Earth’s gravitational constant, and $E$ the eccentric anomaly. $E$ satisfies Kepler’s equation,
代表 $r$ 地心半径、 $\nu$ 真实异常、 $a$ 长半轴、 $e$ 轨道偏心率、 $\mu=GM_{\text{e}}$ 地球引力常数和 $E$ 偏心异常。 $E$ 满足 Kepler 方程，

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$\displaystyle E-e\sin E=M,$

(8)

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

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$\displaystyle E=M+e\sin M+e^{2}\cos M\sin M+\mathcal{O}(e^{3}).$

(9)

By substituting Eq. (9) into Eq. (7), one can obtain the explicit coordinates $(X,Y,Z,\dot{X},\dot{Y},\dot{Z})$.
通过将方程 （9） 代入方程 （7），可以得到显式坐标 $(X,Y,Z,\dot{X},\dot{Y},\dot{Z})$ 。

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The orbital planes may not be identical for the three TianQin satellites. Thus, the geocentric ecliptic coordinate system $\{x,y,z\}$ is also employed, where the $x$-$y$ plane is the ecliptic plane. The $x$-axis points toward the vernal equinox, and the $z$-axis is normal to the ecliptic plane. The coordinates $(x,y,z)$ and $(\dot{x},\dot{y},\dot{z})$ in this system can be obtained by $(X,Y,Z)$ and $(\dot{X},\dot{Y},\dot{Z})$ 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 $\{x,y,z\}$ ，其中 $x$ - $y$ 平面是黄道平面。 $x$ -轴指向春分， $z$ -轴垂直于黄道平面。在这个系统中，坐标 $(x,y,z)$ 和 $(\dot{x},\dot{y},\dot{z})$ 可以通过以下变换 [35， 33] 获得 $(X,Y,Z)$ $(\dot{X},\dot{Y},\dot{Z})$ ：

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	$\displaystyle\begin{bmatrix}x\\ y\\ z\end{bmatrix}=R_{z}(-\Omega)R_{x}(-i)R_{z}(-\omega)\begin{bmatrix}X\\ Y\\ Z\end{bmatrix},$		(10)
	$\displaystyle\begin{bmatrix}\dot{x}\\ \dot{y}\\ \dot{z}\end{bmatrix}=R_{z}(-\Omega)R_{x}(-i)R_{z}(-\omega)\begin{bmatrix}\dot{X}\\ \dot{Y}\\ \dot{Z}\end{bmatrix},$		(11)

where $\Omega$, $i$, and $\omega$ denote the satellite’s longitude of the ascending node, inclination, and argument of perigee, respectively. Additionally, $R_{z}(\gamma)$ and $R_{x}(\gamma)$ are the rotation matrices that rotate vectors by an angle $\gamma$ about the $z-$ or $x-$axis,
其中 $\Omega$ ，、 $i$ 和 $\omega$ 分别表示卫星的升交点经度、倾角和近地点参数。此外， $R_{z}(\gamma)$ 和 $R_{x}(\gamma)$ 是将向量绕 $z-$ or $x-$ 轴旋转一定角度 $\gamma$ 的旋转矩阵，

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	$\displaystyle R_{z}(\gamma)=\begin{bmatrix}\cos\gamma&\sin\gamma&0\\ -\sin\gamma&\cos\gamma&0\\ 0&0&1\end{bmatrix},$		(12)
	$\displaystyle R_{x}(\gamma)=\begin{bmatrix}1&0&0\\ 0&\cos\gamma&\sin\gamma\\ 0&-\sin\gamma&\cos\gamma\end{bmatrix}.$		(13)

Combining Eqs. (7) and (9)-(13), the position vector and velocity vector of SC$k$ ($k=1$, 2, 3), $\mathbf{r}_{k}=(x_{k},y_{k},z_{k})$ and $\mathbf{\dot{r}}_{k}=(\dot{x}_{k},\dot{y}_{k},\dot{z}_{k})$, are given by
组合方程。（7） 和 （9）-（13），则 SC $k$ （ $k=1$ ， 2， 3） $\mathbf{r}_{k}=(x_{k},y_{k},z_{k})$ 和 $\mathbf{\dot{r}}_{k}=(\dot{x}_{k},\dot{y}_{k},\dot{z}_{k})$ 的位置矢量和速度矢量由下式给出

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$\displaystyle\left\{\begin{array}[]{l}\displaystyle x_{k}=a_{k}[\cos\Omega_{k}\cos\lambda_{k}-\cos i_{k}\sin\Omega_{k}\sin\lambda_{k}\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \displaystyle\phantom{x_{k}=}+\frac{1}{2}e_{k}(\cos\Omega_{k}\,f_{\text{c1}k}-\cos i_{k}\sin\Omega_{k}\,f_{\text{s1}k})]+\mathcal{O}(e^{2}_{k}),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \displaystyle y_{k}=a_{k}[\sin\Omega_{k}\cos\lambda_{k}+\cos i_{k}\cos\Omega_{k}\sin\lambda_{k}\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \displaystyle\phantom{y_{k}=}+\frac{1}{2}e_{k}(\sin\Omega_{k}\,f_{\text{c1}k}+\cos i_{k}\cos\Omega_{k}\,f_{\text{s1}k})]+\mathcal{O}(e^{2}_{k}),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \displaystyle z_{k}=a_{k}(\sin i_{k}\sin\lambda_{k}+\frac{1}{2}e_{k}\sin i_{k}\,f_{\text{s1}k})+\mathcal{O}(e^{2}_{k}),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \displaystyle\dot{x}_{k}=\frac{\sqrt{\mu}}{\sqrt{a_{k}}}[-\cos\Omega_{k}\sin\lambda_{k}-\cos i_{k}\sin\Omega_{k}\cos\lambda_{k}\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \displaystyle\phantom{\dot{x}_{k}=}+e_{k}(-\cos\Omega_{k}\,f_{\text{s2}k}-\cos i_{k}\sin\Omega_{k}\,f_{\text{c2}k})]+\mathcal{O}(e^{2}_{k}),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \displaystyle\dot{y}_{k}=\frac{\sqrt{\mu}}{\sqrt{a_{k}}}[-\sin\Omega_{k}\sin\lambda_{k}+\cos i_{k}\cos\Omega_{k}\cos\lambda_{k}\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \displaystyle\phantom{\dot{y}_{k}=}+e_{k}(-\sin\Omega_{k}\,f_{\text{s2}k}+\cos i_{k}\cos\Omega_{k}\,f_{\text{c2}k})]+\mathcal{O}(e^{2}_{k}),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \displaystyle\dot{z}_{k}=\frac{\sqrt{\mu}}{\sqrt{a_{k}}}(\sin i_{k}\cos\lambda_{k}+e_{k}\sin i_{k}\,f_{\text{c2}k})+\mathcal{O}(e^{2}_{k}),\end{array}\right.$

(24)

where 哪里

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$\displaystyle\lambda_{k}:=M_{k}+\omega_{k},$

(25)

$f_{\text{c1}k}:=\cos(2\lambda_{k}-\omega_{k})-3\cos\omega_{k}$, $f_{\text{s1}k}:=\sin(2\lambda_{k}-\omega_{k})-3\sin\omega_{k}$, $f_{\text{c2}k}:=\cos(2\lambda_{k}-\omega_{k})$, and $f_{\text{s2}k}:=\sin(2\lambda_{k}-\omega_{k})$. Define $\sigma\in\{a,e,i,\Omega,\omega,\lambda\}$, $\sigma_{k}(t)$ in Eq. (24) are straightforwardly determined in the two-body problem by
$f_{\text{c1}k}:=\cos(2\lambda_{k}-\omega_{k})-3\cos\omega_{k}$ $f_{\text{c2}k}:=\cos(2\lambda_{k}-\omega_{k})$ 、 $f_{\text{s1}k}:=\sin(2\lambda_{k}-\omega_{k})-3\sin\omega_{k}$ 和 $f_{\text{s2}k}:=\sin(2\lambda_{k}-\omega_{k})$ .定义 $\sigma\in\{a,e,i,\Omega,\omega,\lambda\}$ ， $\sigma_{k}(t)$ 在方程（24）中直接由下式确定

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$\displaystyle\sigma_{k}(t)=\sigma_{0k}+\delta_{\iota}n_{k}(t-t_{0}),\qquad\text{for Keplerian orbit},$

(26)

where 哪里

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$$\sigma_{0k}:=\sigma_{k}(t_{0}),\qquad\delta_{\iota}=\left\{\begin{array}[]{ll}1,&\sigma=\lambda,\vskip 3.0pt plus 1.0pt minus 1.0pt\\ 0,&\sigma\neq\lambda.\end{array}\right.$$

(27)

Note that Eq. (24) remains valid even when considering gravitational perturbations, with the only change being the replacement of $\sigma_{k}(t)$ 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（26） $\sigma_{k}(t)$ 中的方程。

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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。

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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 $L_{ij}$, relative line-of-sight velocity between satellites $v_{ij}$, and breathing angle $\alpha_{k}$,
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 $L_{ij}$ 、卫星 $v_{ij}$ 之间的相对视线速度和呼吸角 $\alpha_{k}$ ，

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	$\displaystyle L_{ij}$	$\displaystyle=\left|\mathbf{r}_{i}-\mathbf{r}_{j}\right|,$		(28)
	$\displaystyle v_{ij}$	$\displaystyle=\dot{L}_{ij},$		(29)
	$\displaystyle\alpha_{k}$	$\displaystyle=\arccos\frac{L_{ki}^{2}+L_{kj}^{2}-L_{ij}^{2}}{2L_{ki}L_{kj}},$		(30)

where $i$, $j$, and $k$ take values 1, 2, or 3 and $i\neq j\neq k$. Substituting Eq. (24) into Eqs. (28)-(30), one can obtain the explicit expressions, for these three kinematic indicators, with forms $L_{ij}(\sigma_{i}(t),\sigma_{j}(t))$, $v_{ij}(\sigma_{i}(t),\sigma_{j}(t))$, and $\alpha_{k}(\sigma_{k}(t),\sigma_{i}(t),\sigma_{j}(t))$, respectively.
其中 $i$ ， $j$ ， 和 $k$ 取值 1、2 或 3 和 $i\neq j\neq k$ 。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 $v_{ij}(\sigma_{i}(t),\sigma_{j}(t))$ $\alpha_{k}(\sigma_{k}(t),\sigma_{i}(t),\sigma_{j}(t))$ 、 的 $L_{ij}(\sigma_{i}(t),\sigma_{j}(t))$ 显式表达式。

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To maintain the constellation as an equilateral triangle, i.e. $L_{12}(t)=L_{13}(t)=L_{23}(t)$, 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 $L_{12}(t)=L_{13}(t)=L_{23}(t)$ ，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：

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$\displaystyle e_{1}(t)=e_{2}(t)=e_{3}(t)=0,$

(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：

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$\displaystyle\left\{\begin{array}[]{l}a_{1}(t)=a_{2}(t)=a_{3}(t),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ i_{1}(t)=i_{2}(t)=i_{3}(t),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \Omega_{1}(t)=\Omega_{2}(t)=\Omega_{3}(t),\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \lambda_{k}(t)=\frac{2\pi}{3}(k-1)+\lambda_{1}(t).\end{array}\right.$

(36)

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

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$\displaystyle\sigma_{0k}=\sigma_{\text{o}}+\delta_{\iota}\frac{2\pi}{3}(k-1),$

(37)

where the parameters with subscript “o” are the nominal ones of the TianQin constellation. For instance, these values can be chosen as $a_{\text{o}}=10^{5}$ km, $e_{\text{o}}=0$, and $i_{\text{o}}=94.7^{\circ}$, $\Omega_{\text{o}}=210.4^{\circ}$, respectively establishing the orbit size and orienting the orbital plane perpendicular to J0806 [7, 24]. The initial value $\lambda_{\text{o}}$ associated with the orbit phase is typically selected to be any value within the range of $0^{\circ}$ to $120^{\circ}$, 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 $a_{\text{o}}=10^{5}$ km、 $e_{\text{o}}=0$ 、 和 $i_{\text{o}}=94.7^{\circ}$ 、 $\Omega_{\text{o}}=210.4^{\circ}$ ，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 $\lambda_{\text{o}}$ 通常选择为 $0^{\circ}$ 到 $120^{\circ}$ 范围内的任何值，或者可以专门指定以避免月食 [37]。

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To analyze additional nominal orbit design allowing for $e\neq 0$ 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 $\sigma\in\{a,i,\Omega,\lambda\}$), the variations of these indicators in the two-body problem, up to the first order of $e$, can be expressed as
为了分析允许 $e\neq 0$ 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 $\sigma\in\{a,i,\Omega,\lambda\}$ ），这些指标在两体问题中的变化，直到 $e$ 的一阶，可以表示为

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	$\displaystyle L_{ij}^{\text{kepl}}(t)=$	$\displaystyle~{}\sqrt{3}a_{\text{o}}+\frac{\sqrt{7}}{2}a_{\text{o}}[-e_{i}\sin\left(M_{i}+\beta\right)$
		$\displaystyle+e_{j}\sin\left(M_{j}-\beta\right)],$		(38)

	$\displaystyle v_{ij}^{\text{kepl}}(t)=$	$\displaystyle~{}0+\frac{\sqrt{7}}{2}a_{\text{o}}n_{\text{o}}[-e_{i}\cos\left(M_{i}+\beta\right)$
		$\displaystyle+e_{j}\cos\left(M_{j}-\beta\right)],$		(39)

	$\displaystyle\alpha_{k}^{\text{kepl}}(t)=$	$\displaystyle~{}\frac{\pi}{3}+\frac{\sqrt{7}}{3}e_{k}\cos M_{k}\sin\beta$
		$\displaystyle-\frac{\sqrt{7}}{6}e_{i}[\sin\left(M_{i}-\beta\right)+2\sin\left(M_{i}+\beta\right)]$
		$\displaystyle+\frac{\sqrt{7}}{6}e_{j}[2\sin\left(M_{j}-\beta\right)+\sin\left(M_{j}+\beta\right)],$		(40)

where $M_{i}(t)=\frac{2\pi}{3}(i-1)+\lambda_{1}(t)-\omega_{i}$ and $\beta:=\arccos\frac{2}{\sqrt{7}}$, with the indices $i$, $j$, and $k$ using cyclic indexing ($i,\,j,\,k=1\to 2\to 3\to 1$). If we further set $e_{1}=e_{2}=e_{3}\equiv e_{\text{o}}\neq 0$ and $\omega_{1}=\omega_{2}=\omega_{3}$, then it follows that
其中 $M_{i}(t)=\frac{2\pi}{3}(i-1)+\lambda_{1}(t)-\omega_{i}$ 和 $\beta:=\arccos\frac{2}{\sqrt{7}}$ ，带有索引 $i$ ， $j$ ，并使用 $k$ 循环索引 （ $i,\,j,\,k=1\to 2\to 3\to 1$ ）。如果我们进一步设置 $e_{1}=e_{2}=e_{3}\equiv e_{\text{o}}\neq 0$ 和 $\omega_{1}=\omega_{2}=\omega_{3}$ ，则

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	$\displaystyle L_{ij}^{\text{kepl-A}}(t)=\sqrt{3}a_{\text{o}}-\frac{\sqrt{3}}{2}a_{\text{o}}e_{\text{o}}\cos M_{k},$		(41)
	$\displaystyle v_{ij}^{\text{kepl-A}}(t)=0+\frac{\sqrt{3}}{2}a_{\text{o}}n_{\text{o}}e_{\text{o}}\sin M_{k},$		(42)
	$\displaystyle\alpha_{k}^{\text{kepl-A}}(t)=\frac{\pi}{3}-\frac{\sqrt{3}}{2}e_{\text{o}}\cos M_{k},$		(43)

where $M_{k}(t)=\frac{2\pi}{3}(k-1)+M_{1}(t)$. Equations (41)-(43) indicate that, at the zeroth order of $e$, 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 $M_{k}(t)=\frac{2\pi}{3}(k-1)+M_{1}(t)$ . 方程 （41）-（43） 表明，在 的零阶 $e$ 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。

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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 $e_{1}=e_{2}=e_{3}\equiv e_{\text{o}}\neq 0$, and $M_{1}=M_{2}=M_{3}$, namely $\omega_{k}=\omega_{1}+\frac{2\pi}{3}(k-1)$, then $L_{ij}^{\text{kepl-B}}(t)\equiv\sqrt{3}a_{\text{o}}-\sqrt{3}a_{\text{o}}e_{\text{o}}\cos E_{1}(t)$, 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 [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 $e_{1}=e_{2}=e_{3}\equiv e_{\text{o}}\neq 0$ ，和 $M_{1}=M_{2}=M_{3}$ ，即 $\omega_{k}=\omega_{1}+\frac{2\pi}{3}(k-1)$ ，则 $L_{ij}^{\text{kepl-B}}(t)\equiv\sqrt{3}a_{\text{o}}-\sqrt{3}a_{\text{o}}e_{\text{o}}\cos E_{1}(t)$ ，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。

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