Analysis of Near-Polar and Near-Circular Periodic Orbits Around the Moon with J₂, C₂₂ and Third-Body Perturbations

Abstract

In the Moon–Earth elliptic restricted three-body problem, near-polar and near-circular lunar-type periodic orbits are numerically continued from Keplerian circular orbits using Broyden’s method with line search. The Hamiltonian system, expressed in Cartesian coordinates, is treated via the symplectic scaling method. The radii of the initial Keplerian circular orbits are then scaled and normalized. For cases in which the integer ratios $\{j/k\}$ of the mean motions between the inner and outer orbits are within the range $[9,150]$, some periodic orbits of the elliptic restricted three-body problem are investigated. For the middle-altitude cases with $j/k\in [38,70]$, the perturbations due to $J_2$ and $C_{22}$ are incorporated, and some new near-polar periodic orbits are computed. The orbital dynamics of these near-polar, near-circular periodic orbits are well characterized by the first-order double-averaged system in the Poincaré–Delaunay elements. Linear stability is assessed through characteristic multipliers derived from the fundamental solution matrix of the linear varational system. Stability indices are computed for both the near-polar and planar near-circular periodic orbits across the range $j/k\in [9,50]$.

1. Introduction

Nowadays, with the increasing demand for lunar exploration and advancements in aerospace technology, the cislunar space is becoming a new frontier for human activity. In order to carry out long-term scientific research missions near the Moon and save fuel to place space stations, it is necessary to use stable and approximately stable orbits, such as distant retrograde orbits (DROs), near-rectlinear halo orbits (NRHOs) and near-polar frozen orbits [1]. It is common to first study the periodic orbits of the restricted three-body problem (RTBP) and then extend the solutions to a generalized time-periodic model and lastly to a high-fidelity ephemeris model [2]. It is important to study periodic orbits as they help us to understand real motions, and there is a long history in this area of study. Since the pioneering work of H. Poincaré, periodic orbits have attracted the attention of many mathematicians, astronomers, celestial mechanicians, and so on [3,4].

The family of near-polar and near-circular periodic orbits is interesting. According to Poincaré’s classification, these periodic orbits belong to the third-type of the first kind. Theoretically, these periodic orbits are very close to one primary. The periodic orbits can be computed by solving the periodicity conditions, in which Broyden’s method with line search is used [5]. Via numerical analysis, Xu and Song [6] found that these orbits can be of high altitude, and some other interesting phenomena were found. The eccentricity and the inclination of such a periodic orbit vary periodically and satisfy the Lidov–Kozai effect [7]. For certain values of the eccentricity of the outer orbit and ratios of the mean motion resonances between the inner and outer orbits, the periodic orbits can be approximately linearly stable. This confirms that high-altitude and near-circular polar frozen orbits are suitable for the placement of a lunar station [8]. However, Xu and Song [6] did not provide sufficient detail regarding the near-polar and near-circular lunar-type periodic orbits. For further study, it is logical to study these long-period periodic orbits in a more accurate gravity field model [9].

The motion of a lunar orbiter with an altitude in the range (2000, 16,000) km is mainly affected by the gravitation of the Moon–Earth system. The Moon–Earth system can be simplified to a two-body system, while the Moon can be approximated as a triaxial ellipsoid, as the main non-spherical perturbations come from the $J_2,C_{22}$ terms. The zonal coefficient $J_2$ represents the size of the oblateness, and the sectorial coefficient $C_{22}$ measures the equatorial ellipticity. The obliquity of the ecliptic plane and the Moon’s path is neglected. There is an interesting phenomenon named tidal locking, where the Moon’s rotation period and the revolution period are nearly equal, and the longest axis always passes the Earth. For lower-altitude lunar orbits, Lara et al. [10] studied the long-term dynamics of near-polar frozen orbits by reducing a 50-degree zonal model with the third-body effect of the Earth. Saedeleer [11] explained the Hamiltonian system of a lunar orbiter in the Moon–Earth circular RTBP with $J_2,C_{22}$ perturbations and studied the averaged system via the Lie–Deprit method. With the same model, Nie and Gurfil [12] studied lunar frozen orbits in the first-order double-averaged system in Delaunay elements via Zeipel’s method. For frozen orbits, the slow mean orbital elements remain nearly fixed such that the costs of the orbital corrections are reduced.

With the $J_2,C_{22}$ perturbations and several orders of the Legendre expansions of third-body perturbations, Carvalho et al. [13] studied frozen orbits and the critical inclinations of lunar satellites by analyzing the double-averaged system. Considering the third-degree gravity harmonics of the Moon, Tzirti et al. [14] investigated the Poincaré sections, the Fast Lyapunov Indicator Maps, and some families of periodic orbits. Tzirti et al. [15] studied the secular dynamics of low-altitude lunar orbiters with high-degree gravity models via frequency analysis and investigated the eccentricity–inclination space. Considering a lunar orbiter perturbed by the $J_2,J_3,J_4$ terms, El-Salam and El-Bar [16] investigated families of frozen orbits. In Sirwah et al. [17], the perturbing function was considered up to the seventh zonal harmonic and the third-body perturbation of the Earth in an elliptic inclined orbit. Then, they numerically studied frozen orbits with the arguments of the pericenter at $\pi /2,3\pi /2$. An efficient approach based on the grid search, parallelization and evolution strategy was introduced to compute periodic orbits in Dena et al. [18]. Franz and Russell [19] introduced a database on the symmetric periodic orbits near the Moon in the model of the circular RTBP via grid search and an unsupervised learning clustering algorithm. Legnaro and Efthymiopoulos [20] distinguished three types of lunar orbits through their altitudes and studied the secular dynamics, especially the secular resonances and the eccentricity growth, of lunar satellites according to different models.

In Section 2, we provide a new approach to the numerical continuation of the near-polar and near-circular periodic orbits of the elliptic RTBP and give some numerical examples of lunar-type periodic orbits. In Section 3, we study the existence and stability of the near-polar and near-circular periodic orbits in the elliptic RTBP with $J_2,C_{22}$ perturbations. Some numerical examples are also given. Finally, Section 5 discusses and concludes this work.

2. Elliptic RTBP

2.1. Scaled Hamiltonian System

Against the background of the motion of a lunar orbiter in the cislunar space, we study the periodic orbits around the smaller primary in the elliptic RTBP. Let ${\mathtt{P}}_1$ and ${\mathtt{P}}_2$ represent two mass points. The relative orbit from ${\mathtt{P}}_1$ to ${\mathtt{P}}_2$ is a Keplerian orbit, with the semi-major axis $a_p$, the eccentricity $e_p$, the mean motion $n_p$, and the time of periapsis passage ${\tau }_0$. Choose the units such that the gravitational constant $\mathtt{G}=1$, the distance unit $a_p=1$, the total masses $m_1+m_2=1$, and the mean motion $n_p=1$. Let $\mu =m_2/(m_1+m_2)<1/2$. Set the motion plane of primaries as the reference plane. The direction of the major axis from ${\mathtt{P}}_1$ to ${\mathtt{P}}_2$ is set as the $q_1$-axis. Set ${\mathtt{P}}_2$ as the origin and establish the right-handed Cartesian coordinate system ${\mathtt{P}}_2-q_1{q}_2{q}_3$. The initial time is set as ${\tau }_0=0$ or ${\tau }_0=\pi$.

In the coordinate system ${\mathtt{P}}_2-q_1{q}_2{q}_3$, the position of ${\mathtt{P}}_1$ is denoted as ${\mathit{X}}_p\in {\mathbb{R}}^3$, which is also a solution of the planar Kepler problem

$$\begin{array}{c}\hfill {\ddot{\mathit{X}}}_p=-{\mathit{X}}_p·{\parallel {\mathit{X}}_p\parallel }^{-3}.\end{array}$$

Here, $\parallel ·\parallel$ represents the Euclidean distance norm, and the time is t. We have $\parallel {\mathit{X}}_p\parallel =1-e_p\cos E_p$. Denote $E_p$ as the eccentric anomaly. The vector ${\mathit{X}}_p$ is

$$\begin{array}{c}\hfill {\mathit{X}}_p={\left(X_1,X_2,0\right)}^{\mathrm{T}}={\left(\cos E_p-e_p,\sqrt{1-e_p^2}\sin E_p,0\right)}^{\mathrm{T}},\end{array}$$

where the upper $\mathrm{T}$ represents a transpose. The position of the infinitesimal body is $\mathit{u}\in {\mathbb{R}}^3$, and its conjugate momentum is $\mathit{v}=\dot{\mathit{u}}$. The Hamiltonian dynamical system of this problem is

$$\begin{array}{c}\hfill {\mathcal{H}}^{\mathtt{P}\mathtt{2}}={\displaystyle \frac{1}{2}}{\parallel \mathit{v}\parallel }^2-{\displaystyle \frac{\mu }{\parallel \mathit{u}\parallel }}-{\displaystyle \frac{1-\mu }{\parallel \mathit{u}-{\mathit{X}}_p\parallel }}+(1-\mu ){\displaystyle \frac{{\mathit{u}}^{\mathrm{T}}{\mathit{X}}_p}{\parallel {\mathit{X}}_p{\parallel }^3}}.\end{array}$$

(1)

The canonical differential equation system is $\dot{\mathit{u}}=\partial {\mathcal{H}}^{\mathtt{P}\mathtt{2}}/\partial \mathit{v}$, $\dot{\mathit{v}}=-\partial {\mathcal{H}}^{\mathtt{P}\mathtt{2}}/\partial \mathit{u}$. The second-order differential equation system can be written as

$$\begin{array}{c}\hfill \ddot{\mathit{u}}=-\mu {\displaystyle \frac{\mathit{u}}{{\parallel \mathit{u}\parallel }^3}}-(1-\mu ){\displaystyle \frac{\mathit{u}-{\mathit{X}}_p}{{\parallel \mathit{u}-{\mathit{X}}_p\parallel }^3}}-(1-\mu ){\displaystyle \frac{{\mathit{X}}_p}{\parallel {\mathit{X}}_p{\parallel }^3}}.\end{array}$$

(2)

When $\mu$ is very small, it is not convenient to obtain the numerical solution, as the orbit scale is too small.

The orbit of the infinitesimal body is called the inner orbit, and the orbit of the relative orbit of ${\mathtt{P}}_1$ is the outer orbit. Denote the orbital elements of the inner orbit as $a_s$, $e_s$, $i_s$, ${\mathsf{\Omega }}_s$, ${\omega }_s$, ℓ_s, where ℓ represents the mean anomaly. According to the symplectic scaling method, the variables can be scaled as

$$\begin{array}{c}\hfill \mathit{u}={\epsilon }^2{\mu }^{1/3}\xi ,\hspace{1em}\mathit{v}={\epsilon }^{-1}{\mu }^{1/3}\eta ,\hspace{1em}t={\epsilon }^{3}s,\end{array}$$

(3)

where s is the new time, and the small parameter $\epsilon$ represents the closeness of the infinitesimal body to primary ${\mathtt{P}}_2$. The new Hamiltonian is

$$\begin{array}{cc}\hfill {\hat{\mathcal{H}}}^{\mathtt{P}\mathtt{2}}(\xi ,\eta ,s)=& {\epsilon }^2{\mu }^{-2/3}{\mathcal{H}}^{\mathtt{P}\mathtt{2}}(\mathit{u},\mathit{v},t)={\displaystyle \frac{{\parallel \eta \parallel }^2}{2}}-{\displaystyle \frac{1}{\parallel \xi \parallel }}\hfill \\ \hfill \hspace{1em}& -{\epsilon }^2{\mu }^{-2/3}(1-\mu )\left({\displaystyle \frac{1}{\parallel {\epsilon }^2{\mu }^{1/3}\xi -{\mathit{X}}_p\left(s\right)\parallel }}-{\displaystyle \frac{{\epsilon }^2{\mu }^{1/3}{\xi }^{\mathrm{T}}{\mathit{X}}_p}{\parallel {\mathit{X}}_p{\parallel }^3}}\right).\hfill \end{array}$$

(4)

The new differential equation system becomes

$$\begin{array}{cc}\hfill \xi ″={\displaystyle \frac{{\mathrm{d}}^2\xi }{\mathrm{d}{s}^2}}=& -{\displaystyle \frac{\xi }{{\parallel \xi \parallel }^3}}-{\epsilon }^6(1-\mu ){\displaystyle \frac{\xi }{{\parallel \mathit{u}-{\mathit{X}}_p\parallel }^3}}\hfill \\ \hfill \hspace{1em}& +{\epsilon }^4(1-\mu ){\mu }^{-1/3}{\mathit{X}}_p\left({\displaystyle \frac{1}{{\parallel \mathit{u}-{\mathit{X}}_p\parallel }^3}}-{\displaystyle \frac{1}{{\parallel {\mathit{X}}_p\parallel }^3}}\right),\hfill \end{array}$$

(5)

where ${\xi }^{\prime }=\eta$, $\mathit{u}={\epsilon }^2{\mu }^{1/3}\xi$, and the prime represents the derivative about the scaled time s. Let ${\hat{a}}_s$ be the scaled variable and $a_s={\epsilon }^2{\mu }^{1/3}{\hat{a}}_s$. We have $n_s^2{a}_s^3=\mu$, and $n_s^2{\epsilon }^6\mu {\hat{a}}_s=\mu$, so ${\hat{n}}_s={\epsilon }^3{n}_s$. It is convenient to set ${\hat{n}}_s=1$ if ${\epsilon }^3=n_p/n_s$. Then, we obtain ${\hat{n}}_p={\epsilon }^3$. The Kepler equation for the outer orbit satisfies

$$\begin{array}{c}\hfill E_p-e_p\sin E_p=n_{p}t={\hat{n}}_{p}s.\end{array}$$

The numerical solution of Equation (5) can be calculated by integration.

2.2. Symmetry and Periodicity

The scaled Hamiltonian system $\hat{\mathcal{H}}$ maintains the same symmetries as the original Hamiltonian system ${\mathcal{H}}^{\mathtt{P}\mathtt{2}}$. One time-reversing symmetry ${\mathcal{R}}_1$ with respect to the ${\xi }_1$-axis is recalled.

$$\begin{array}{c}\hfill {\mathcal{R}}_1:({\xi }_1,{\xi }_2,{\xi }_3,{\eta }_1,{\eta }_2,{\eta }_3,s)\to ({\xi }_1,-{\xi }_2,-{\xi }_3,-{\eta }_1,{\eta }_2,{\eta }_3,-s).\end{array}$$

There exist periodic third-body perturbations in the lunar-type orbits in the elliptic RTBP. The desired periodic orbits can be continued from the two uncoupled Kepler orbits. The ratio of the mean motions of the uncoupled inner and outer Kepler orbits is set to be ${\hat{n}}_s/{\hat{n}}_p=n_s/n_p={\epsilon }^3$, and ${\epsilon }^3$ should be a small rational number. ${\epsilon }^3$ is set to equal $k/j$, where $k,j\in \mathbb{N}$ and $k\ll j$. This means that the inner orbit revolves in j circles, while the outer orbit revolves in k circles. In order to understand the ${\mathcal{R}}_1$-symmetric periodic solution, a proposition is summarized as follows.

Lemma 1

(Cors et al. [21]). For the Hamiltonian system (1) of the elliptic RTBP, the Lagrangian set ${\mathcal{L}}_1$ with respect to ${\mathcal{R}}_1$-symmetry is

$$\begin{array}{c}\hfill {\mathcal{L}}_1=\{(\mathbf{u},\mathbf{v},t):u_2=u_3=v_1=0,\hspace{1em}t=0\hspace{1em}mod \pi \}.\end{array}$$

If a solution $Z(Z_1,t)$ satisfies $Z_1\in {\mathcal{L}}_1$ and $Z(Z_1,k\pi )\in {\mathcal{L}}_1$ with $k\in \mathbb{N}$, then the solution is ${\mathcal{R}}_1$-symmetric and periodic with a period $T=2k\pi$. For the Hamiltonian system (4), the corresponding Lagrangian set ${\mathcal{L}}_1^{\left(1\right)}$ is

$$\begin{array}{c}\hfill {\mathcal{L}}_1^{\left(1\right)}=\{(\xi ,\eta ,s):{\xi }_2={\xi }_3={\eta }_1=0,s=0\hspace{1em}mod \pi \},\end{array}$$

and the scaled periodic solution has a period $\hat{T}=2j\pi$ with $j\in \mathbb{N}$. The ratio of the mean motions of the outer and inner orbits is $k/j$. The periodicity conditions used in this paper can be written as

$$\begin{array}{c}\hfill {\hat{Z}}_1\in {\mathcal{L}}_1^{\left(1\right)},\hspace{1em}\hat{Z}({\hat{Z}}_1,j\pi )\in {\mathcal{L}}_1^{\left(1\right)}.\end{array}$$

(6)

According to the description in Xu and Song [6], there exist both near-polar and planar ${\mathcal{R}}_1$-symmetric near-circular lunar-type periodic orbits in the elliptic RTBP. However, Xu and Song [6] did not give sufficient detail regarding the numerical continuation of such periodic orbits. The difficulty lies in the fact that the accumulated integration errors are relatively large when the infinitesimal body is very close to primary ${\mathtt{P}}_2$ and the integration time is long. For the Hamiltonian system (1), the initial values to be continued can be written as

$$\begin{array}{c}\hfill Z_0={\left(u_1^0,u_2^0,u_3^0,v_1^0,v_2^0,v_3^0\right)}^{\mathrm{T}}={\left(\pm {\mu }^{1/3}{(k/j)}^{2/3},0,0,0,0,\pm {\mu }^{1/3}{(k/j)}^{-1/3}\right)}^{\mathrm{T}}.\end{array}$$

The scale of the lunar-type orbits is small but the velocities are relatively large. For the Hamiltonian system (4), the initial values to be continued can be written as

$$\begin{array}{c}\hfill {\hat{Z}}_0={\left({\xi }_1^0,{\xi }_2^0,{\xi }_3^0,{\eta }_1^0,{\eta }_2^0,{\eta }_3^0\right)}^{\mathrm{T}}={\left(\pm 1,0,0,0,0,\pm 1\right)}^{\mathrm{T}}.\end{array}$$

In this paper, the scaled variables are used, and more numerical continuation results can be achieved. The continuation scenario is still based on the combination of the periodicity conditions and Broyden’s method with line search. This means that ${\hat{Z}}_0$ is expected to be continued to

$$\begin{array}{c}\hfill {\hat{Z}}_1={\left(\pm 1+{\delta }_1,0,0,0,{\delta }_2,\pm 1+{\delta }_3;s_0\right)}^{\mathrm{T}},\hspace{1em}{s}_0=0,j\pi ,\end{array}$$

such that

$$\begin{array}{c}\hfill {\xi }_2({\delta }_1,{\delta }_2,{\delta }_3,s_{T/2})=0,\hspace{1em}{\xi }_3({\delta }_1,{\delta }_2,{\delta }_3,s_{T/2})=0,\hspace{1em}{\eta }_1({\delta }_1,{\delta }_2,{\delta }_3,s_{T/2})=0,\end{array}$$

where ${\delta }_1,{\delta }_2,{\delta }_3\in \mathbb{R}$ are small quantities, and $s_{T/2}=s_0+j\pi$.

2.3. Some Periodic Orbits

The numerical solutions of Equation (5) are calculated via the variable step-size Runge–Kutta 7-8 routine with double precision. The routines of Broyden’s method are referred to in Press, et al. [5], and the precision is guaranteed at ${10}^{-8}$. Thus, the periodic orbits can maintain the precision of ${10}^{-8}$. Let ${\mathtt{P}}_1,{\mathtt{P}}_2$ represent the Earth and the Moon, respectively. Let $\mu =0.0121505843947$, $e_p=0.0549$. Set $j/k$ as the ratio of the mean motions between the uncoupled inner and outer orbits. The real length of the semi-major axis of the outer orbit is about A = 328,900.5597 km. The real length of the semi-major axis of the inner orbit is about $A{a}_0$ km with $a_0={\mu }^{1/3}{(k/j)}^{2/3}$. The altitude is defined as the difference in $A{a}_0$ and the real length of the longer equatorial semi-major axis $1738.1$ km. The high-altitude zone is defined in the range of approximately 5000 km and 20,000 km. In this zone, the third-body perturbation of the Earth is dominant. In the theoretical proof of the existence of the near-polar and near-circular periodic orbits in the elliptic RTBP, the small parameter ${\epsilon }^3=k/j$ is supposed to be small enough. One question is how small ${\epsilon }^3$ must be so that the periodic orbits can be calculated. Such periodic orbits are numerically investigated with ${\epsilon }^3$ in the range $9\le j/k\le 150$. In Figure 1 and Figure 2, it is found that the argument of pericenter $\omega$ rotates and oscillates. Figure 3 depicts the case $j/k=37$. All these orbital elements are very symmetric. The details of the figures are shown in the image captions. Some more initial values for the periodic orbits are shown in

Table 1. Some initial values $({\xi }_1,0,0,0,{\eta }_2,{\eta }_3)$ of the near-polar, near-circular, lunar-type periodic orbits in the elliptic RTBP. The types are classified by the signs of $({\xi }_1,{\eta }_3,\cos E_p)$.

$\mathit{j}/\mathit{k}$, Type	${\mathit{\xi }}_1$	${\mathit{\eta }}_2$	${\mathit{\eta }}_3$
9/1, $+\pm +$	0.99620440178	−0.06082772318	±1.0157184687
9/1, $-\pm +$	−0.99470649817	0.06185840160	±1.0154002218
10/1, $+\pm -$	0.99910153226	−0.050852737	±1.0072154827
10/1, $-\pm -$	−0.99837950690	0.0506041258	±1.0087412525
16/1, $+\pm +$	0.99925242695	−0.035922494	±1.0043641526
16/1, $-\pm +$	−0.99851083910	0.0360488668	±1.0047376373
36/1, $+\pm -$	1.00010727626	−0.0146998968	±1.0003869346
36/1, $-\pm -$	−0.99974967250	0.0147026686	±1.0007698831
37/1, $+\pm +$	0.99999430950	−0.0157684742	±1.0006628074
37/1, $-\pm +$	−0.99962561652	0.0157786232	±1.0009929915
50/1, $+\pm +$	1.0000454998	−1.16852281 $\times {10}^{-2}$	±1.0003121582
50/1, $+\pm +$	−0.999748470093	1.169015346 $\times {10}^{-2}$	±1.000591975
50/1, $+\pm -$	1.00010809345	−1.05990235 $\times {10}^{-2}$	±1.00014917507
50/1, $-\pm -$	−0.99981879968	1.060132171 $\times {10}^{-2}$	±1.00044900515
120/1, $+\pm +$	1.000063875	−4.87551 $\times {10}^{-3}$	±0.999997510
120/1, $--+$	−0.9999007299785	4.8763318183 $\times {10}^{-3}$	−1.000158994073
150/1, $+++$	1.00005889302967	−3.900799586228 $\times {10}^{-3}$	0.99998031895716
150/1, $--+$	−0.99991304641419	3.225120713516 $\times {10}^{-3}$	−1.0001276778590

. The types of initial values are defined by the signs of the values of ${\xi }_1,{\eta }_3$, and $\cos E_p$.

3. Elliptic RTBP with J₂, C₂₂ Perturbations

3.1. The Model and Numerical Experiment

The usual orbital elements are the semi-major axis a, the eccentricity e, the inclination i, the longitude of the ascending node $\mathsf{\Omega }$, the argument of the periapsis $\omega$, and the mean anomaly M. The eccentric anomaly is denoted as $E(e,M)$, and the true anomaly is denoted as $f(e,M)$. If only the perturbations of $J_2,C_{22}$ are added to the RTBP, the application of this model is for the case of an altitude in the range of $(2000,5000)$ km. The non-spherical perturbing function is usually expressed by orbital elements. It is necessary to transform the perturbing function into the form of Cartesian coordinates for the convenience of the computation of the near-circular periodic orbits. For a massless artificial satellite in the Moon-centered inertial coordinate frame, the position is denoted as $\mathit{u}$, and the conjugate momentum is $\mathit{v}=\dot{\mathit{u}}$. The Hamiltonian system of a lunar orbiter can be written in the form of perturbations of the RTBP,

$$\begin{array}{c}\hfill {\mathcal{H}}^{\mathtt{P}\mathtt{2}\mathtt{JC}}={\mathcal{H}}^{\mathtt{P}\mathtt{2}}(\mathit{u},\mathit{v},t)+{\mathcal{H}}_{J2}-{\mathcal{H}}_{C22},\end{array}$$

(7)

where

$$\begin{array}{cc}\hfill \hspace{1em}& {\mathcal{H}}_{J2}=J_2{\displaystyle \frac{\mu a_m^2}{r^3}}{\mathcal{P}}_2\left(\sin \phi \right),\hspace{1em}{\mathcal{H}}_{C22}=C_{22}{\displaystyle \frac{\mu a_m^2}{r^3}}{\mathcal{P}}_{22}\left(\sin \phi \right)·\cos 2(\mathsf{\Omega }-f_p+\psi ),\hfill \\ \hfill \hspace{1em}& a_m\approx 1738.1/328900.5597,\hspace{1em}{J}_2\approx 2.0322356\times \hspace{1em}{10}^{-4},\hspace{1em}{C}_{22}\approx 2.2381388\times \hspace{1em}{10}^{-5},\hfill \\ \hfill \hspace{1em}& r=\parallel \mathit{u}\parallel ,\hspace{1em}{n}_p=1,\hspace{1em}\sin \phi ={\displaystyle \frac{u_3}{r}},\hspace{1em}{\mathcal{P}}_2\left(\sin \phi \right)={\displaystyle \frac{3u_3^2}{2r^2}}-{\displaystyle \frac{1}{2}},\hfill \\ \hfill \hspace{1em}& \cos \phi \cos \psi =\cos (f+\omega ),\hspace{1em}\cos \phi \sin \psi =\sin (f+\omega )\cos i,\hfill \\ \hfill \hspace{1em}& {\mathcal{P}}_{22}\left(\sin \phi \right)=3{\cos }^2\phi ,\hspace{1em}\cos 2(\mathsf{\Omega }-f_p+\psi )=2{\cos }^2(\mathsf{\Omega }-f_p+\psi )-1,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \hspace{1em}& {\mathcal{P}}_{22}\left(\sin \phi \right)\cos 2(\mathsf{\Omega }-f_p+\psi )=6{\left[\cos \phi \cos (\mathsf{\Omega }-f_p+\psi )\right]}^2-3{\cos }^2\phi \hfill \\ \hfill \hspace{1em}& =6{\{\cos \phi ·[\cos (\mathsf{\Omega }-f_p)\cos \psi -\sin (\mathsf{\Omega }-f_p)\sin \psi ]\}}^2-3{\cos }^2\phi )\hfill \\ \hfill \hspace{1em}& =6{[\cos (f+\omega )\cos (\mathsf{\Omega }-f_p)-\sin (f+\omega )\cos i\sin (\mathsf{\Omega }-f_p)]}^2-3{\cos }^2\phi \hfill \\ \hfill \hspace{1em}& ={\displaystyle \frac{6}{r^2}}{(u_1\cos f_p+u_2\sin f_p)}^2-3{\cos }^2\phi .\hfill \end{array}$$

In order to better understand the angles $\phi$ and $\psi$, Figure 4 is provided. In the lunar inertial coordinate frame $P_2-q_1{q}_2{q}_3$, the longitude of the ascending node of the infinitesimal satellite is $\mathsf{\Omega }$, and the latitude is $\phi$. The prime meridian is set at the direction of the longest semi-major axis of the ellipsoid. The longitude of the satellite is $\mathsf{\Omega }-f_p+\psi$ in the rotating frame $P_2-x_1{x}_2{x}_3$. The Earth is always on the $x_1$-axis. Moreover, $f_p$ is denoted as the true anomaly of the outer orbit, ${\mathcal{P}}_2(·)$ is the second-order Legendre polynomial, and ${\mathcal{P}}_{22}(·)$ is the unnormalized associative Legendre polynomial.

With the help of symplectic scaling,

$$\begin{array}{c}\hfill \mathit{u}\to {\epsilon }^2{\mu }^{1/3}\xi ,\hspace{1em}\mathit{v}\to {\epsilon }^{-1}{\mu }^{1/3}\eta ,\hspace{1em}t\to {\epsilon }^{3}s,\hspace{1em}{J}_2\to {\epsilon }^6{\tilde{J}}_2,\hspace{1em}{C}_{22}\to {\epsilon }^6{\tilde{C}}_{22},\end{array}$$

and the Legendre polynomial expansion [6], the Hamiltonian (7) becomes

$$\begin{array}{cc}\hfill {\tilde{\mathcal{H}}}^{\mathtt{P}\mathtt{2}\mathtt{JC}}& ={\hat{\mathcal{H}}}^{\mathtt{P}\mathtt{2}}(\xi ,\eta ,s)+{\epsilon }^6({\hat{\mathcal{H}}}_{J2}-{\hat{\mathcal{H}}}_{C22})\hfill \\ \hfill \hspace{1em}& ={\hat{\mathcal{H}}}_0^{\mathtt{P}\mathtt{2}}+{\epsilon }^6({\hat{\mathcal{H}}}_1+{\hat{\mathcal{H}}}_{J2}-{\hat{\mathcal{H}}}_{C22})+\mathcal{O}\left({\epsilon }^8\right),\hfill \end{array}$$

(8)

where

$$\begin{array}{cc}\hfill \hspace{1em}& {\hat{\mathcal{H}}}_1=-{\displaystyle \frac{(1-\mu )r^2}{\parallel {\mathit{X}}_p{\parallel }^3}}{\mathcal{P}}_2\left(\cos \theta \right),\hspace{1em}r=\parallel \xi \parallel ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \hspace{1em}& {\hat{\mathcal{H}}}_{J2}={\tilde{a}}_m^2{\tilde{J}}_2\left({\displaystyle \frac{3{\xi }_3^2}{2r^5}}-{\displaystyle \frac{1}{2r^3}}\right),\hfill \\ \hfill \hspace{1em}& {\hat{\mathcal{H}}}_{C22}={\tilde{a}}_m^2{\tilde{C}}_{22}\left({\displaystyle \frac{6{\tilde{x}}_1^2}{r^5}}-{\displaystyle \frac{3}{r^3}}+{\displaystyle \frac{3{\xi }_3^2}{r^5}}\right),\hfill \\ \hfill \hspace{1em}& \cos \theta ={\displaystyle \frac{\xi }{\parallel \xi \parallel }}·{\displaystyle \frac{{\mathit{X}}_p}{\parallel {\mathit{X}}_p\parallel }}={\displaystyle \frac{1}{r}}({\xi }_1\cos f_p+{\xi }_2\sin f_p)={\displaystyle \frac{{\tilde{x}}_1}{r}},\hfill \\ \hfill \hspace{1em}& {\tilde{x}}_1={\xi }_1\cos f_p+{\xi }_2\sin f_p,\hfill \\ \hfill \hspace{1em}& {\tilde{a}}_m={\epsilon }^{-2}{\mu }^{-1/3}{a}_m.\hfill \end{array}$$

The relations between the scaled rectangular coordinates and the osculating orbital elements are

$$\begin{array}{c}\hfill \left\{\begin{array}{c}r=a(1-e^2)/(1+e\cos f),\hfill \\ u_1=r\cos (f+\omega )\cos \mathsf{\Omega }-r\sin (f+\omega )\cos i\sin \mathsf{\Omega },\hfill \\ u_2=r\cos (f+\omega )\sin \mathsf{\Omega }+r\sin (f+\omega )\cos i\cos \mathsf{\Omega },\hfill \\ u_3=r\sin (f+\omega )\sin i,\hfill \end{array}\right.\end{array}$$

Here, ${\tilde{x}}_1=r\cos \theta$, and $\cos \theta$ can be expressed as

$$\begin{array}{cc}\hfill \cos \theta =& \cos (f+\omega )\cos (\mathsf{\Omega }-f_p)-\cos i\sin (f+\omega )\sin (\mathsf{\Omega }-f_p)\hfill \\ \hfill =& (I_1+I_2)\cos (f+\omega )\cos (\mathsf{\Omega }-f_p)-(I_1-I_2)\sin (f+\omega )\sin (\mathsf{\Omega }-f_p)\hfill \\ \hfill =& I_1\cos (f+\omega +\mathsf{\Omega }-f_p)+I_2\cos (f+\omega -\mathsf{\Omega }+f_p),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill I_1={\displaystyle \frac{1}{2}}\left(1+\cos i\right),\hspace{1em}{I}_2={\displaystyle \frac{1}{2}}\left(1-\cos i\right).\end{array}$$

According to knowledge of complex variable functions, the real triangular functions can be replaced by complex exponential functions, such that the powers of the real triangular functions can be easily calculated. By this method, we obtain

$$\begin{array}{cc}\hfill {\cos }^2\theta =& {\displaystyle \frac{I_1^2}{2}}\cos 2(f+\omega +\mathsf{\Omega }-f_p)+{\displaystyle \frac{I_2^2}{2}}\cos 2(f+\omega -\mathsf{\Omega }+f_p)+{\displaystyle \frac{I_1^2+I_2^2}{2}}\hfill \\ \hfill \hspace{1em}& +I_1{I}_2\left[\cos 2(f+\omega )+\cos 2(\mathsf{\Omega }-f_p)\right].\hfill \end{array}$$

Now, it is convenient to apply the Hamiltonian system (8) both for the numerical computation of the periodic orbits and for the analysis of the first-order perturbed system. Through a numerical experiment, it is found that $\omega$ is near 0 when $\cos E_p=-1$ and $\omega$ is near $\pi$ when $\cos E_p=1$. An example is given in Figure 5. Some more initial values of these frozen periodic orbits can be found in

Table 2. Some initial values of the near-polar, near-circular, lunar frozen periodic orbits in the elliptic RTBP perturbed by the $J2,C22$ terms.

$\mathit{j}/\mathit{k}$, Type 1	${\mathit{\xi }}_1$	${\mathit{\eta }}_2$	${\mathit{\eta }}_3$
38, $+++$	0.999996415501457	−1.53265760125584 $\times {10}^{-2}$	1.00063234441343
38, $+-+$	0.999996323959347	−1.53265221246967 $\times {10}^{-2}$	−1.00063243652928
38, $-++$	−0.999631537537576	1.53360715507054 $\times {10}^{-2}$	1.00096137315743
38, $--+$	−0.999631572957102	1.53361607836104 $\times {10}^{-2}$	−1.00096133644985
38, $++-$	1.00010457611908	−1.39262375517334 $\times {10}^{-2}$	1.00034505669525
38, $+--$	1.00010457613606	−1.39262375145222 $\times {10}^{-2}$	−1.00034505667884
38, $-+-$	−0.999755336066306	1.39291300677261 $\times {10}^{-2}$	1.00071623453464
38, $---$	−0.999755336071655	1.39291298616095 $\times {10}^{-2}$	−1.00071623453208
50, $+++$	1.00004036237187	−1.16399095055137 $\times {10}^{-2}$	1.00032604091855
50, $+-+$	1.00004036442821	−1.16399055608615 $\times {10}^{-2}$	−1.00032603891154
50, $-++$	−0.999736185844210	1.16448719424375 $\times {10}^{-2}$	1.00061301131761
50, $--+$	−0.999736185751775	1.16448718613465 $\times {10}^{-2}$	−1.00061301141084
50, $++-$	1.00010324877502	−1.06014270499141 $\times {10}^{-2}$	1.00016221064751
50, $+--$	1.00010324742691	−1.06015693670615 $\times {10}^{-2}$	−1.00016221050752
50, $-+-$	−0.999806261225514	1.06039245231929 $\times {10}^{-2}$	1.00046973982848
50, $---$	−0.999806260045887	1.06039241923682 $\times {10}^{-2}$	−1.00046974101071
50, $+++\left(1\right)$	1.00004772495397	−1.16437280686876 $\times {10}^{-2}$	1.00031865811095
50, $-++\left(1\right)$	−0.999727871097841	1.16482584577867 $\times {10}^{-2}$	1.00062129340388
50, $--+\left(1\right)$	−0.999727871210650	1.16482585774023 $\times {10}^{-2}$	−1.0006212932898344
50, $++-\left(1\right)$	1.000047724953971	−1.16437280686877 $\times {10}^{-2}$	1.00031865811095
50, $+--\left(1\right)$	1.000047723820043	−1.16437303795379 $\times {10}^{-2}$	−1.00031865921609
60, $+++$	1.00005445614547	−9.69443727684011 $\times {10}^{-3}$	1.00020411562384
60, $+-+$	1.00005445666738	−9.69443733030811 $\times {10}^{-3}$	−1.00020411510206
60, $-++$	−0.999780858404711	9.69780610332384 $\times {10}^{-3}$	1.00046714423176
60, $--+$	−0.999780830682264	9.69781696395454 $\times {10}^{-3}$	−1.00046717183446
60, $++-$	1.00009836808750	−8.86151811506304 $\times {10}^{-3}$	1.00009091857796
60, $+--$	1.00009836626972	−8.86152667714532 $\times {10}^{-3}$	−1.00009092031881
60, $-+-$	−0.999829057478298	8.86344960938257 $\times {10}^{-3}$	1.00036671317776
60, $---$	−0.999829058314773	8.86344660447464 $\times {10}^{-3}$	−1.00036671236851
70, $+++$	1.00006122089056	−8.32258582949308 $\times {10}^{-3}$	1.00013342805500
70, $+-+$	1.00006122070046	−8.32258595263936 $\times {10}^{-3}$	−1.00013342824388
70, $-++$	−0.999807148471916	8.32518964348363 $\times {10}^{-3}$	1.00038049962717
70, $--+$	−0.999807149512763	8.32518602724036 $\times {10}^{-3}$	−1.00038049861647
70, $++-$	1.00009370579491	−7.65010043854636 $\times {10}^{-3}$	1.00005034663127
70, $+--$	1.00009370571585	−7.65010142502368 $\times {10}^{-3}$	−1.00005034670266
70, $-+-$	−0.999842137241648	7.65183065190509 $\times {10}^{-3}$	1.00030621538970
70, $---$	−0.999842137612930	7.65182240120627 $\times {10}^{-3}$	−1.00030621508136
70, $-+-\left(1\right)$	−0.99980224406757	8.326726523174 $\times {10}^{-3}$	1.000385394761

¹ The types are defined by the signs of $\mathrm{\xi }$₁, η₃ = $\stackrel{.}{\mathrm{\xi }}$₃, cos E_p. For one type, there may be more than one set of initial values.

. It is interesting to explain this phenomenon in the first-order double-averaged system.

3.2. First-Order Averaged System

The scaled canonical Delaunay elements are introduced in order to apply the Von Zeipel transform and obtain the first-order averaged system.

$$\begin{array}{c}\hfill \ell =M,\hspace{1em}g=\omega ,\hspace{1em}h=\mathsf{\Omega },\hspace{1em}L=\sqrt{a},\hspace{1em}G=L\sqrt{1-e^2},\hspace{1em}H=G\cos i.\end{array}$$

The complete expansion of the perturbed system in the Delaunay elements is difficult, so the mean anomaly ℓ is usually contained in $E\left(e\right(L,G),M)$ and $f\left(e\right(L,G),M)$. Hansen coefficients are usually taken in use in the elliptic expansion [3],

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{r}{a}}\right)}^{n}exp\mathrm{i}mf=\sum _{k=-\infty }^{\infty }{X}_k^{n,m}\left(e\right)exp\mathrm{i}kM.\end{array}$$

The symbols above the Hamiltonian notations in (8) are removed, and we let $r_2=\parallel {\mathit{X}}_p\parallel$.

After the elimination of the fast variable M, we have

$$\begin{array}{cc}\hfill {\overline{\mathcal{H}}}_1& =-{\displaystyle \frac{(1-\mu )a^2}{r_2^3}}\{{\displaystyle \frac{15}{4}}{e}_1^2\left[I_1^2\cos 2(\omega +\mathsf{\Omega }-f_p)+I_2^2\cos 2(\omega -\mathsf{\Omega }+f_p)+I_1{I}_2\cos 2\omega \right]\hfill \\ \hfill \hspace{1em}& +\left(1+{\displaystyle \frac{3}{2}}{e}^2\right)\left[{\displaystyle \frac{3}{4}}(I_1^2+I_2^2)+{\displaystyle \frac{3}{2}}{I}_1{I}_2\cos 2(\mathsf{\Omega }-f_p)-{\displaystyle \frac{1}{2}}\right]\}.\hfill \\ \hfill {\overline{\mathcal{H}}}_{J2}& ={\displaystyle \frac{{\tilde{J}}_2{\tilde{a}}_m^2}{L^3{G}^3}}\left({\displaystyle \frac{3}{4}}{\sin }^{2}i-{\displaystyle \frac{1}{2}}\right),\hspace{1em}{\overline{\mathcal{H}}}_{C22}=-{\displaystyle \frac{3{\tilde{C}}_{22}{\tilde{a}}_m^2}{2L^3{G}^3}}·{\sin }^{2}i·\cos 2(\mathsf{\Omega }-f_p).\hfill \end{array}$$

Considering that the second fast variable is $f_p$, the terms above can be averaged again as

$$\begin{array}{cc}\hfill {\stackrel{=}{\mathcal{H}}}_1& =-{\displaystyle \frac{(1-\mu )L^4}{L_2^3{G}_2^3}}\left\{{\displaystyle \frac{15}{16}}\left(1-{\displaystyle \frac{G^2}{L^2}}\right)\left(1-{\displaystyle \frac{H^2}{G^2}}\right)\cos 2g+{\displaystyle \frac{1}{16}}\left(5-{\displaystyle \frac{3G^2}{L^2}}\right)\left({\displaystyle \frac{3H^2}{G^2}}-1\right)\right\},\hfill \\ \hfill {\stackrel{=}{\mathcal{H}}}_{J2}& ={\overline{\mathcal{H}}}_{J2}={\displaystyle \frac{{\tilde{J}}_2{\tilde{a}}_m^2}{L^3{G}^3}}\left({\displaystyle \frac{1}{4}}-{\displaystyle \frac{3H^2}{4G^2}}\right),\hspace{1em}{\stackrel{=}{\mathcal{H}}}_{C22}=0,\hspace{1em}{L}_2=1,\hspace{1em}{G}_2=\sqrt{1-e_p^2}.\hfill \end{array}$$

The canonical differential equations of the first-order averaged system are

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}\ell }{\mathrm{d}s}}& ={\displaystyle \frac{1}{L^3}}+{\epsilon }^6{\displaystyle \frac{\partial ({\stackrel{=}{\mathcal{H}}}_1+{\stackrel{=}{\mathcal{H}}}_{J2})}{\partial L}},\hspace{1em}{\displaystyle \frac{\mathrm{d}g}{\mathrm{d}s}}={\epsilon }^6{\displaystyle \frac{\partial ({\stackrel{=}{\mathcal{H}}}_1+{\stackrel{=}{\mathcal{H}}}_{J2})}{\partial G}},\hspace{1em}{\displaystyle \frac{\mathrm{d}h}{\mathrm{d}s}}={\epsilon }^6{\displaystyle \frac{\partial ({\stackrel{=}{\mathcal{H}}}_1+{\stackrel{=}{\mathcal{H}}}_{J2})}{\partial H}},\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}L}{\mathrm{d}s}}& =0,\hspace{1em}{\displaystyle \frac{\mathrm{d}G}{\mathrm{d}s}}=-{\displaystyle \frac{15(1-\mu )L^4}{8G_2^3}}{e}^2{\sin }^{2}i·\sin 2g,\hspace{1em}{\displaystyle \frac{\mathrm{d}H}{\mathrm{d}s}}=0.\hfill \end{array}$$

(9)

As a further step, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}\ell }{\mathrm{d}s}}=& {\displaystyle \frac{1}{L^3}}-{\epsilon }^6{\displaystyle \frac{(1-\mu )L^3}{G_2^3}}\left[{\displaystyle \frac{15}{8}}(1+e^2){\sin }^{2}i·\cos 2g+{\displaystyle \frac{1}{8}}(7+3e^2)(3{\cos }^{2}i-1)\right]\hfill \\ \hfill \hspace{1em}& -{\epsilon }^6{\displaystyle \frac{3{\tilde{J}}_2{\tilde{a}}_m^2}{4L^4{G}^3}}(1-3{\cos }^{2}i),\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}g}{\mathrm{d}s}}=& -{\epsilon }^6{\displaystyle \frac{3(1-\mu )L^4}{8G_2^{3}G}}\left[5(-{\sin }^{2}i+e^2)\cos 2g+(1-e^2-5{\cos }^{2}i)\right]\hfill \\ \hfill \hspace{1em}& -{\epsilon }^6{\displaystyle \frac{3{\tilde{J}}_2{\tilde{a}}_m^2}{4L^3{G}^4}}(1-3{\cos }^{2}i)+{\epsilon }^6{\displaystyle \frac{3{\tilde{J}}_2{\tilde{a}}_m^2}{2L^3{G}^4}}{\cos }^{2}i,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}h}{\mathrm{d}s}}=& -{\epsilon }^6{\displaystyle \frac{3(1-\mu )L^4}{8G_2^{3}G}}(-5e^2\cos 2g+2+3e^2)\cos i-{\epsilon }^6{\displaystyle \frac{3{\tilde{J}}_2{\tilde{a}}_m^2}{2L^3{G}^4}}\cos i.\hfill \end{array}$$

In the near-polar and near-circular frozen orbits, the orbital elements a, e, i, $\mathsf{\Omega }$ are almost constant, with small periodic amplitudes. Substituting the initial unperturbed orbital elements $a=1$, $e=0$, and $i={\displaystyle \frac{\pi }{2}}$ into the differential equation about $\mathrm{d}g/\mathrm{d}s$, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}g}{\mathrm{d}s}}=k_1\cos 2g-k_2\approx {\displaystyle \frac{15(1-\mu )}{8G_2^3}}{\epsilon }^6\cos 2g-{\displaystyle \frac{3(1-\mu )}{8G_2^3}}{\epsilon }^6-0.75J_2{\tilde{a}}_m^2.\end{array}$$

(10)

For $38\le {\epsilon }^{-3}\le 70$, we have $3.798\times {10}^{-4}<k_1<1.288\times \hspace{1em}{10}^{-3}$, $J_2{\tilde{a}}_m^2\sim {10}^{-7}$, and $0<k_2<k_1$. Equation (10) has stable and unstable fixed points, which are

$$\begin{array}{c}\hfill g=k\pi \pm {\displaystyle \frac{1}{2}}\arccos {\displaystyle \frac{k_2}{k_1}}\approx k\pi \pm \arccos {\displaystyle \frac{1}{5}},\hspace{1em}k\in \mathbb{Z}.\end{array}$$

However, from Figure 5 and Figure 6, we can see that it is not effective to theoretically study the near-polar and near-circular orbits by Delaunay elements in the first-order double-averaged system.

In order to understand the periodic behavior of these frozen orbits, we resort to Poincaré–Delaunay elements, as they are effective for near-circular and near-polar orbits.

$$\begin{array}{c}\hfill \left\{\begin{array}{ccccccccc}{Q}_1& =& \ell +g,& Q_2& =& -\sqrt{2(L-G)}\sin g,& Q_3& =& h,\\ P_1& =& L,& P_2& =& \sqrt{2(L-G)}\cos g,& P_3& =& H.\end{array}\right.\end{array}$$

The first-order double-averaged system with the Poincaré–Delaunay elements can be written as

$$\begin{array}{c}\hfill {\mathcal{K}}_f=-{\displaystyle \frac{1}{2P_1^2}}+{\epsilon }^6\left(-c_1{\mathcal{K}}_1+c_2{\mathcal{K}}_2\right),\hspace{1em}{c}_1={\displaystyle \frac{(1-\mu )}{16G_2^3}},\hspace{1em}{c}_2={\displaystyle \frac{{\tilde{J}}_2{\tilde{a}}_m^2}{4}},\end{array}$$

(11)

where

$$\begin{array}{cc}\hfill {\mathcal{K}}_1& =P_1^4(15{\mathcal{A}}_1{\mathcal{A}}_2+{\mathcal{A}}_3),\hspace{1em}{\mathcal{K}}_2={\displaystyle \frac{8}{P_1^3{G}_{two}^3}}-{\displaystyle \frac{96P_3^2}{P_1^3{G}_{two}^5}},\hfill \\ \hfill {\mathcal{A}}_1& =1-{\displaystyle \frac{G_{two}^2}{4P_1^2}}-{\displaystyle \frac{4P_3^2}{G_{two}^2}}+{\displaystyle \frac{P_3^2}{P_1^2}},\hspace{1em}{\mathcal{A}}_2={\displaystyle \frac{2P_2^2}{P_2^2+Q_2^2}}-1,\hfill \\ \hfill {\mathcal{A}}_3& =-5+{\displaystyle \frac{3G_{two}^2}{4P_1^2}}+{\displaystyle \frac{60P_3^2}{G_{two}^2}}-{\displaystyle \frac{9P_3^2}{P_1^2}},\hspace{1em}{G}_{two}=2P_1^2-P_2^2-Q_2^2=2G.\hfill \end{array}$$

The canonical differential equations can be calculated with the help of symbolic computation software like wxMaxima:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{Q}_1}{\mathrm{d}s}}& ={\displaystyle \frac{\partial {\mathcal{K}}_f}{\partial P_1}}={\displaystyle \frac{1}{P_1^3}}+{\epsilon }^6\left(-c_1{\displaystyle \frac{\partial {\mathcal{K}}_1}{\partial P_1}}+c_2{\displaystyle \frac{\partial {\mathcal{K}}_2}{\partial P_1}}\right),\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}{Q}_i}{\mathrm{d}s}}& ={\displaystyle \frac{\partial {\mathcal{K}}_f}{\partial P_i}}={\epsilon }^6\left(-c_1{\displaystyle \frac{\partial {\mathcal{K}}_1}{\partial P_i}}+c_2{\displaystyle \frac{\partial {\mathcal{K}}_2}{\partial P_i}}\right),\hspace{1em}i=2,3,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}{P}_2}{\mathrm{d}s}}& =-{\displaystyle \frac{\partial {\mathcal{K}}_f}{\partial Q_2}}=-{\epsilon }^6\left(-c_1{\displaystyle \frac{\partial {\mathcal{K}}_1}{\partial Q_2}}+c_2{\displaystyle \frac{\partial {\mathcal{K}}_2}{\partial Q_2}}\right),\hspace{1em}{\displaystyle \frac{\mathrm{d}{P}_j}{\mathrm{d}s}}={\displaystyle \frac{\partial {\mathcal{K}}_f}{\partial Q_j}}=0,\hspace{1em}j=1,3.\hfill \end{array}$$

The key parts of the differential equations of the first-order double-averaged Hamiltonian system ${\mathcal{K}}_f$ (11) are as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{Q}_1}{\mathrm{d}s}}=& {\displaystyle \frac{1}{P_1^3}}-{\epsilon }^6{\displaystyle \frac{3c_2}{P_1^4}}\left({\displaystyle \frac{8}{G_{two}^2}}-{\displaystyle \frac{96P_3^2}{G_{two}^5}}\right)+{\epsilon }^6{\displaystyle \frac{c_2}{P_1^3}}\left({\displaystyle \frac{960P_3^2}{G_{two}^6}}-{\displaystyle \frac{32}{G_{two}^3}}\right)\hfill \\ \hfill \hspace{1em}& -{\epsilon }^{6}4c_1{P}_1^3(15{\mathcal{A}}_1{\mathcal{A}}_2+{\mathcal{A}}_3)-{\epsilon }^6{c}_1{P}_1^4[15{\mathcal{A}}_2·({\displaystyle \frac{G_{two}^2}{2P_1^3}}+{\displaystyle \frac{16P_3^2}{G_{two}^3}}\hfill \\ \hfill \hspace{1em}& -{\displaystyle \frac{G_{two}}{P_1^2}}-{\displaystyle \frac{2P_3^2}{P_1^3}})-{\displaystyle \frac{3G_{two}^2}{2P_1^3}}-{\displaystyle \frac{240P_3^2}{G_{two}^3}}+{\displaystyle \frac{3G_{two}}{P_1^2}}+{\displaystyle \frac{18P_3^2}{P_1^3}}].\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{Q}_2}{\mathrm{d}s}}=& {\epsilon }^6{\displaystyle \frac{c_2}{P_1^3}}\left({\displaystyle \frac{32}{G_{two}^3}}-{\displaystyle \frac{960P_3^2}{G_{two}^6}}\right)P_2-{\epsilon }^6{c}_1{P}_1^4[15\left({\displaystyle \frac{G_{two}}{P_1^2}}-{\displaystyle \frac{16P_3^2}{G_{two}^3}}\right){\mathcal{A}}_2{P}_2\hfill \\ \hfill \hspace{1em}& +\left({\displaystyle \frac{240P_2{P}_3^2}{G_{two}^3}}-{\displaystyle \frac{60Q_2^2{\mathcal{A}}_1}{{(P_2^2+Q_2^2)}^2}}-{\displaystyle \frac{3G_{two}}{P_1^2}}\right)P_2].\hfill \end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}{Q}_3}{\mathrm{d}s}}=-{\epsilon }^6{\displaystyle \frac{192c_2{P}_3}{P_1^3{G}_{two}^5}}-{\epsilon }^6{c}_1{P}_1^4\left[15{\mathcal{A}}_2\left({\displaystyle \frac{2P_3}{P_1^2}}-{\displaystyle \frac{8P_3}{G_{two}^2}}\right)+{\displaystyle \frac{120P_3}{G_{two}^2}}-{\displaystyle \frac{18P_3}{P_1^2}}\right].\end{array}$$

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}{P}_2}{\mathrm{d}s}}=& -{\epsilon }^6{\displaystyle \frac{c_2}{P_1^3}}\left({\displaystyle \frac{32}{G_{two}^3}}-{\displaystyle \frac{960P_3^2}{G_{two}^6}}\right)Q_2+{\epsilon }^6{c}_1{P}_1^4[15\left({\displaystyle \frac{G_{two}}{P_1^2}}-{\displaystyle \frac{16P_3^2}{G_{two}^3}}\right){\mathcal{A}}_2{Q}_2\hfill \\ \hfill \hspace{1em}& +\left({\displaystyle \frac{240P_3^2}{G_{two}^3}}-{\displaystyle \frac{60P_2^2{\mathcal{A}}_1}{{(P_2^2+Q_2^2)}^2}}-{\displaystyle \frac{3G_{two}}{P_1^2}}\right)Q_2].\hfill \end{array}$$

In the first-order double-averaged system, $P_1$ and $P_3$ are constants. Set $P_1=1$ and $P_3=0$, and we have $Q_3\equiv Q_3\left(0\right)$. The differential equations about $Q_2,P_2$ can be simplified as

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}s}}\left(\begin{array}{c}{Q}_2\\ P_2\end{array}\right)={\epsilon }^6\left\{{\displaystyle \frac{32c_2}{G_{two}^3}}-c_1\left[15{\mathcal{A}}_2{G}_{two}-246+18(P_2^2+Q_2^2)\right]\right\}\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\begin{array}{c}{Q}_2\\ P_2\end{array}\right).$$

Suppose that $Q_2\left(0\right)=P_2\left(0\right)=0$; then, $Q_2\left(t\right)=P_2\left(t\right)=0$. We have ${\displaystyle \frac{\mathrm{d}{Q}_1}{\mathrm{d}s}}=1+{\epsilon }^6(8c_1-10c_2)$. Thus, there exist polar-type and circular periodic orbits in the first-order double-averaged system.

Corollary 1

(Xu and Song [6]). A class of symmetric, near-polar, near-circular lunar-type periodic orbits exists in the spatial elliptic RTBP with perturbations of the $J_2$ and $C_{22}$ terms. The model is described by the Hamiltonian (7).

4. Linear Stability

The Hamiltonian system ${\hat{\mathcal{H}}}^{\mathtt{P}\mathtt{2}\mathtt{JC}}$ is

$$\begin{array}{cc}\hfill {\hat{\mathcal{H}}}^{\mathtt{P}\mathtt{2}\mathtt{JC}}=& {\displaystyle \frac{\parallel \eta \parallel }{2}}-{\displaystyle \frac{1}{\parallel \xi \parallel }}-{\epsilon }^2{\mu }^{-2/3}(1-\mu )\left({\displaystyle \frac{1}{r_3}}-{\displaystyle \frac{{\epsilon }^2{\mu }^{1/3}{\xi }^{\mathrm{T}}{\mathit{X}}_p}{r_2^3}}\right)\hfill \\ \hfill \hspace{1em}& +J_2{\tilde{a}}_m^2\left({\displaystyle \frac{3{\xi }_3^2}{2r^5}}-{\displaystyle \frac{1}{2r^3}}\right)-C_{22}{\tilde{a}}_m^2\left({\displaystyle \frac{6{\tilde{x}}_1^2}{r^5}}-{\displaystyle \frac{3}{r^3}}+{\displaystyle \frac{3{\xi }_3^2}{r^5}}\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \hspace{1em}& r=\parallel \xi \parallel ,\hspace{1em}{r}_2=\parallel {\mathit{X}}_p\left(s\right)\parallel ,\hspace{1em}{r}_3=\parallel {\epsilon }^2{\mu }^{1/3}\xi -{\mathit{X}}_p\parallel ,\hfill \\ \hfill \hspace{1em}& {\tilde{x}}_1={\xi }_1\cos f_p+{\xi }_2\sin f_p,\hspace{1em}{f}_p=f_p\left(E_p(n_{p}s,e_p),e_p\right).\hfill \end{array}$$

The second-order differential equations are

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\mathrm{d}}^2\xi }{\mathrm{d}{s}^2}}=& -{\displaystyle \frac{\xi }{r^3}}-{\epsilon }^6(1-\mu ){\displaystyle \frac{\xi }{r_3^3}}+{\epsilon }^4(1-\mu ){\mu }^{-1/3}{\mathit{X}}_p\left({\displaystyle \frac{1}{r_3^3}}-{\displaystyle \frac{1}{r_2^3}}\right)-J_2{\tilde{a}}_m^2{\mathit{V}}_1+C_{22}{\tilde{a}}_m^2{\mathit{V}}_2,\hfill \end{array}$$

where ${\tilde{a}}_m={\epsilon }^{-2}{\mu }^{-1/3}{a}_m$, and

$$\begin{array}{c}\hfill {\mathit{V}}_1={\displaystyle \frac{\partial }{\partial \xi }}\left({\displaystyle \frac{3{\xi }_3^2}{2r^5}}-{\displaystyle \frac{1}{2r^2}}\right),\hspace{1em}{\mathit{V}}_2={\displaystyle \frac{\partial }{\partial \xi }}\left({\displaystyle \frac{6{\tilde{x}}_1^2}{r^5}}-{\displaystyle \frac{3}{r^3}}+{\displaystyle \frac{3{\xi }_3^2}{r^5}}\right).\end{array}$$

The canonical differential equations of the Hamiltonian system ${\hat{\mathcal{H}}}^{\mathtt{P}\mathtt{2}\mathtt{JC}}$ are

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}\xi }{\mathrm{d}s}}=\eta ,\hspace{1em}{\displaystyle \frac{\mathrm{d}\eta }{\mathrm{d}s}}=-{\displaystyle \frac{\partial {\hat{\mathcal{H}}}^{\mathtt{P}\mathtt{2}\mathtt{JC}}}{\partial \xi }}={\displaystyle \frac{{\mathrm{d}}^2\xi }{\mathrm{d}{s}^2}}.\end{array}$$

(12)

The solution is expressed as $\xi =\xi (s,{\xi }_0,{\eta }_0)$, $\eta =\eta (s,{\xi }_0,{\eta }_0)$. The fundamental solution matrix of the linear variational equations is written as

$$\begin{array}{c}\hfill \mathsf{\Phi }=\mathsf{\Phi }(s,{\xi }_0,{\eta }_0)=\left(\begin{array}{cc}{\displaystyle \frac{\partial \xi }{\partial {\xi }_0}}& {\displaystyle \frac{\partial \xi }{\partial {\eta }_0}}\\ {\displaystyle \frac{\partial \eta }{\partial {\xi }_0}}& {\displaystyle \frac{\partial \eta }{\partial {\eta }_0}}\end{array}\right).\end{array}$$

The system of the linear variational equations is

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}\mathsf{\Phi }}{\mathrm{d}s}}=\left(\begin{array}{cc}{\mathit{O}}_3& {\mathit{I}}_3\\ \mathit{B}& {\mathit{O}}_3\end{array}\right)\mathsf{\Phi },\hspace{1em}\mathit{B}=-{\displaystyle \frac{{\partial }^2{\hat{\mathcal{H}}}^{\mathtt{P}\mathtt{2}\mathtt{JC}}}{\partial \xi \partial {\xi }_0}},\end{array}$$

(13)

where ${\mathit{I}}_3$ is a $3\times 3$ identical matrix, ${\mathit{O}}_3$ is a $3\times 3$ zero matrix, and

$$\begin{array}{c}\hfill \mathit{B}={\mathit{B}}_1+{\epsilon }^6(1-\mu ){\mathit{B}}_2-{\epsilon }^6(1-\mu ){\mathit{B}}_3-J_2{\tilde{a}}_m^2{\mathit{B}}_4+C_{22}{\tilde{a}}_m^2{\mathit{B}}_5,\end{array}$$

$$\begin{array}{c}\hfill {\mathit{B}}_1=\left(\begin{array}{ccc}{b}_{11}& b_{12}& b_{13}\\ b_{12}& b_{22}& b_{23}\\ b_{13}& b_{23}& b_{33}\end{array}\right),\hspace{1em}{b}_{ii}=-{\displaystyle \frac{1}{r^3}}+{\displaystyle \frac{3{\xi }_i^2}{r^5}},\hspace{1em}{b}_{ij}={\displaystyle \frac{3{\xi }_i{\xi }_j}{r^5}},\end{array}$$

$$\begin{array}{c}\hfill {\mathit{B}}_2=\left(\begin{array}{ccc}{\hat{b}}_{11}& {\hat{b}}_{12}& {\hat{b}}_{13}\\ {\hat{b}}_{12}& {\hat{b}}_{22}& {\hat{b}}_{23}\\ {\hat{b}}_{13}& {\hat{b}}_{23}& {\hat{b}}_{33}\end{array}\right),\hspace{1em}{\hat{b}}_{ii}=-{\displaystyle \frac{1}{r_3^3}}+{\displaystyle \frac{3{\mathit{u}}_i({\mathit{u}}_i-X_i)}{r_3^5}},\hspace{1em}{\hat{b}}_{ij}={\displaystyle \frac{3{\mathit{u}}_i({\mathit{u}}_j-X_j)}{r^5}},\end{array}$$

$$\begin{array}{c}\hfill {\mathit{B}}_3=\left(\begin{array}{ccc}{\tilde{b}}_{11}& {\tilde{b}}_{12}& {\tilde{b}}_{13}\\ {\tilde{b}}_{12}& {\tilde{b}}_{22}& {\tilde{b}}_{23}\\ {\tilde{b}}_{13}& {\tilde{b}}_{23}& {\tilde{b}}_{33}\end{array}\right),\hspace{1em}{\tilde{b}}_{ij}={\displaystyle \frac{3X_i({\mathit{u}}_j-X_j)}{r_3^5}},\end{array}$$

$$\begin{array}{cc}\hfill {\mathit{B}}_4& =\left(\begin{array}{ccc}{\overline{b}}_{11}& {\overline{b}}_{12}& {\overline{b}}_{13}\\ {\overline{b}}_{12}& {\overline{b}}_{22}& {\overline{b}}_{23}\\ {\overline{b}}_{13}& {\overline{b}}_{23}& {\overline{b}}_{33}\end{array}\right),\hspace{1em}{\overline{b}}_{ii}={\displaystyle \frac{2}{r_3^5}}-{\displaystyle \frac{15{\xi }_3^2}{2r^7}}-{\displaystyle \frac{12{\xi }_i^2}{r^8}}+{\displaystyle \frac{105{\xi }_i^2{\xi }_3^2}{2r^9}},\hfill \\ \hfill {\overline{b}}_{12}& ={\displaystyle \frac{105{\xi }_1{\xi }_2{\xi }_3^2}{2r^9}}-{\displaystyle \frac{12{\xi }_1{\xi }_2}{r^8}},\hspace{1em}{\overline{b}}_{i3}=-{\displaystyle \frac{15{\xi }_i{\xi }_3}{2r^7}}-{\displaystyle \frac{12{\xi }_i{\xi }_3}{r^8}}+{\displaystyle \frac{105{\xi }_i^2{\xi }_3^2}{2r^9}},\hfill \\ \hfill {\overline{b}}_{33}& ={\displaystyle \frac{3}{r_3^5}}+{\displaystyle \frac{2}{r_3^6}}-{\displaystyle \frac{75{\xi }_3^2}{2r^7}}-{\displaystyle \frac{12{\xi }_3^2}{r^8}}+{\displaystyle \frac{105{\xi }_3^4}{2r^9}},\hspace{1em}\hspace{1em}i=1,2,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathit{B}}_5& =\left(\begin{array}{ccc}{\check{b}}_{11}& {\check{b}}_{12}& {\check{b}}_{13}\\ {\check{b}}_{12}& {\check{b}}_{22}& {\check{b}}_{23}\\ {\check{b}}_{13}& {\check{b}}_{23}& {\check{b}}_{33}\end{array}\right),\hspace{1em}{\check{b}}_i={\displaystyle \frac{9}{r^5}}-{\displaystyle \frac{15{\xi }_3^2+30{\tilde{x}}_1^2+45{\xi }_i^2}{r^7}}+{\displaystyle \frac{105{\xi }_i^2{\xi }_3^2+210{\xi }_i^2{\tilde{x}}_1^2}{r^9}},\hfill \\ \hfill {\check{b}}_{11}& ={\check{b}}_1+{\displaystyle \frac{12{\cos }^2{f}_p}{r^5}}-{\displaystyle \frac{120{\tilde{x}}_1{\xi }_1\cos f_p}{r^7}},\hspace{1em}{\check{b}}_{22}={\check{b}}_2+{\displaystyle \frac{12{\sin }^2{f}_p}{r^5}}-{\displaystyle \frac{120{\tilde{x}}_1{\xi }_2\sin f_p}{r^7}},\hfill \\ \hfill {\check{b}}_{12}& ={\displaystyle \frac{6\sin 2f_p}{r^5}}-{\displaystyle \frac{60{\tilde{x}}_1({\xi }_1\sin f_p+{\xi }_2\cos f_p)+45{\xi }_1{\xi }_2}{r^7}}+{\displaystyle \frac{105{\xi }_1{\xi }_2{\xi }_3^2+210{\tilde{x}}_1^2{\xi }_1{\xi }_2}{r^9}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\check{b}}_{i3}& =-{\displaystyle \frac{60{\tilde{x}}_1{\xi }_3\cos f_p+75{\xi }_i{\xi }_3}{r^7}}+{\displaystyle \frac{105{\xi }_i{\xi }_3^2+210{\tilde{x}}_1^2{\xi }_i{\xi }_3}{r^9}},\hspace{1em}i=1,2,\hfill \\ \hfill {\check{b}}_{33}& ={\displaystyle \frac{15}{r^5}}-{\displaystyle \frac{120{\xi }_3^2+30{\tilde{x}}_1^2}{r^7}}+{\displaystyle \frac{105{\xi }_3^4+210{\tilde{x}}_1^2{\xi }_3^2}{r^9}}.\hfill \end{array}$$

The initial values for Equation (13) are the $6\times 6$ identical matrix. Equations (12) and (13) should be integrated. The scaled time for the numerical integration is the period $T_{jk}=2j\pi /k$ of a periodic orbit. However, the complete information of a symmetric periodic orbit can be obtained via the integration of a half period.

The monodromy matrix can be calculated via the following formula [6]:

$$\begin{array}{c}\hfill \mathsf{\Phi }\left(2j\pi \right)=\mathsf{\Phi }\left(j\pi \right){\mathsf{\Gamma }}_1{\mathsf{\Phi }}^{-1}\left(j\pi \right){\mathsf{\Gamma }}_1,\hspace{1em}{\mathsf{\Gamma }}_1=\mathbf{diag}\{1,-1,-1,-1,1,1\}.\end{array}$$

There are three pairs of conjugate eigenvalues for the matrix $\mathsf{\Phi }\left(2j\pi \right)$. If one periodic orbit is linearly stable, the eigenvalues are in the unit circle. If not, at least one eigenvalue will be far away from the unit circle. One index $\rho$ to describe the stability is the summation of the moduli of the eigenvalues. The eigenvalues of $\mathsf{\Phi }\left(2j\pi \right)$ are also called characteristic multipliers. To provide a few examples, the summation of the moduli of the multipliers can be calculated directly from the eigenvalues of $\mathsf{\Phi }\left(2j\pi \right)$. With $J_2,C_{22}$ and the third-body perturbations, the stability index $\rho$ for the periodic orbit of the $(j/k=38,+++)$ type is about 24, and it is 114 for the case of $(j/k=50,+++)$, 590 for the case of $(j/k=60,+++)$, and 3778 for the case of $(j/k=70,+++)$. In the elliptic RTBP, the linear stability index is about 7 for the case of $(j/k=9,+++)$. This reveals that it saves fuel for high-altitude orbits.

5. Discussion and Conclusions

In the Hamiltonian system (7), the planar symmetric near-circular periodic orbits can be calculated. The initial solution can be set as

$$\begin{array}{c}\hfill {\hat{Z}}_2=(\pm 1+{\delta }_1,0,0,0,\pm 1+{\delta }_2,0),\hspace{1em}{s}_0=0,j\pi ,\end{array}$$

and the periodicity conditions are

$$\begin{array}{c}\hfill {\xi }_2({\delta }_1,{\delta }_2,s_0+j\pi )=0,\hspace{1em}{\eta }_1({\delta }_1,{\delta }_2,s_0+j\pi )=0,\end{array}$$

where ${\delta }_1,{\delta }_2\in \mathbb{R}$. In order to study the linear stability of the planar periodic orbits, we can select the $4\times 4$ submatrix of the monodromy matrix and calculate the characteristic multipliers. The linear stability can also be studied within the framework of the three-dimensional system. We use $\rho$ as the summation of the moduli of the six characteristic multipliers. These planar periodic orbits can also be classified via the types $(j/k,\pm ,\pm ,\pm )$, where the signs come from ${\xi }_1,{\eta }_2$ and $\cos E_p$. Figure 7 shows the linear stability indices $\rho$ of the near-circular periodic orbits. The left subgraph is for the near-polar orbits, and the right subgraph is for the planar orbits. For the near-circular periodic orbits with an altitude in the range (2713.8, 15,737.7) km, the high-altitude orbits are more linearly stable. However, it is necessary to consider more perturbations to check the rules.

For high-altitude orbits, the perturbation from the Sun becomes larger and cannot be neglected. The Sun can be supposed to move on a circular orbit around the mass center of the Earth–Moon system. This is the elliptic–circular double two-body model for the Earth–Moon–Sun system [22]. There can exist near-circular periodic orbits in the elliptic–circular Earth–Moon–Sun restricted four-body problem with symmetric non-spherical perturbations. The existence and stability of periodic orbits in the new model will be considered later. Furthermore, the lunar-type periodic orbits in the quasi-bicircular model of the Moon–Earth–Sun system can also be studied. The asymmetric periodic orbits, quasi-periodic orbits, transition orbits, analytical solutions, and the evolution of the orbits are interesting topics for the future. The formulas of this paper can also be adjusted to study the orbits around a natural satellite of other planets.

This paper provides a wealth information about the near-polar, near-circular, lunar-type periodic orbits in the Moon–Earth elliptic RTBP with $J_2,C_{22}$ and third-body perturbations. Some periodic orbits are calculated and the orbital dynamics are well explained by the first-order double-averaged system. The symplectic scaling technique is introduced, and the small parameter ${\epsilon }^3=k/j$ represents a small ratio of the mean resonances between the inner orbit of the infinitesimal body and the outer orbit of the relative motion of the Earth. The linear stability index is introduced and the linear variational equations are calculated. The scaled Hamiltonian system can also be applied to the study of planar lunar orbits. For the Hamiltonian system (4) of the elliptic RTBP, we provide some initial values of the near-polar and near-circular symmetric periodic orbits in Table 1. Some figures can be seen in Figure 1, Figure 2, Figure 3 and Figure 4. For the Hamiltonian system (7) with $J_2$ and $C_{22}$ perturbations, some initial values of such symmetric periodic orbits are listed in Table 2. Figure 5 and Figure 6 are given as examples. Lastly, the linear stability indices of both the near-polar and planar near-circular periodic orbits are investigated in Figure 7.