Eclipse Mitigation Maneuver Strategy for Earth–Moon 3:2 Resonant Orbits

Abstract

Resonant orbits (ROs) within the Earth–Moon system exhibit periodic motion characterized by a simple integer relationship with the lunar period. These orbits hold significant applications for space situational awareness and the design of transfer orbits in cislunar space. However, due to the expansive scale and extended periods associated with Earth–Moon ROs, the resulting eclipse durations often exceed spacecraft power and thermal system capabilities. This paper conducts an analysis of shadow distribution and eclipse durations within the Earth–Moon 3:2 resonant orbit family. The investigation utilizes relative velocity with respect to the lunar shadow, highlighting the imperative need for eclipse mitigation. The study then proceeds to derive eclipse mitigation methods, specifically employing cross-track maneuver and phasing maneuver techniques. These strategies are applied to mitigate and avoid both Earth and lunar shadows, considering different initial solar phases and eclipse occurrence locations. Simulation results conducted within the circular restricted three-body problem (CRTBP) indicate that the eclipse mitigation strategy via cross-track maneuver incurs a lower $\Delta V$ cost, remaining within the range of 20 m/s. Furthermore, the stability of the resonant orbits post cross-track maneuver is examined. The study presents the maximum allowable cross-track maneuver $\Delta V$s which maintain orbit stable in five years for the entire resonant orbit family. Ultimately, the paper recommends a 2-hour maximum allowable eclipse duration for spacecraft in Earth–Moon 3:2 resonant orbits, offering valuable insights for the future design of missions within these orbits. The eclipse mitigation and avoidance strategies, along with the analytical methods proposed herein, can be readily extended to other resonant orbit families.

1 Introduction

Currently, the majority of space missions are concentrated in terrestrial space, particularly within Geostationary Earth Orbit (GEO). However, with the surge in deep space exploration, space-faring nations, such as the U.S. Space Force (USSF) and the Air Force Research Laboratory (AFRL), have expanded their activities to Extended Geostationary Orbit (XGEO). This shift has led to a series of space situation awareness (SSA) projects focusing on the increasingly active cislunar space [1, 2], necessitating a more profound exploration and understanding of cislunar space.

Resonant orbits (ROs) are a concept originating from astrophysics, denoting a simple integer relationship between the frequencies or periods of celestial bodies [3]. This phenomenon is prevalent in the solar system, with examples like the 8:13 resonance between Earth and Venus around the Sun, and the mean motion resonance of Jupiter-Saturn at about 5:2 [4].

Earth–Moon ROs find diverse applications in transfer orbit design and space situational awareness [5, 6]. Sadaka explored ROs as transfer trajectories corresponding to the planetary period, extending the applicability of transfers with ROs based on homoclinic connection, further delving into the vicinity of ROs [7]. Vaquero et al. addressed the Earth–Moon RO transfer problem by constructing the invariant manifold associated with the RO as a transfer trajectory using the Poincaré section [8]. Liang et al. studied the feasibility of allocating reusable cargoes in a quasi-periodic p:q resonant cycler orbit to construct a cislunar in-orbit infrastructure [9]. Given that the total cislunar space volume is approximately 1000 times that within the GEO radius, ROs have demonstrated extensive coverage of cislunar space. Frueh et al. investigated the feasibility of monitoring cislunar space with the Earth–Moon 2:1 RO constellation, proving its ability to achieve full coverage within 20 revolutions [10]. Fahrner et al. explored the satellite distribution architecture based on revisit rate and coverage, opting for Earth–Moon 3:1 ROs [11].

In cislunar space, the Earth shadow can endure durations exceeding 6 hr, and the Lunar shadow may surpass 15 hr, posing challenges for most mission spacecraft power systems [12, 13]. Consequently, eclipse avoidance has been considered during the mission design phase to alleviate the burden on electric power.

Typically, two approaches are employed: initial phase selection and maneuver avoidance. Many researchers have studied the avoidance of eclipse in various orbits for deep space missions. The Wilkinson Microwave Anisotropy Probe (WMAP), launched in 2001 to a Lissajous orbit about the Sun–Earth L2 Lagrangian point, executed an advanced phasing maneuver of 8.2 m/s on September 5, 2007, resulting in six years without the Earth shadow [14]. Canalias et al. further investigated the Lissajous orbit eclipse avoidance strategy based on invariant manifolds, proposing the Effective Phase Plane Method for a less costly eclipse avoidance strategy [15]. Ye et al. analyzed the eclipse of TianQin candidate orbits between 2034 and 2039, discovering that a 1:8 Earth–Moon RO could significantly avoid eclipse by tuning preliminary orbital radii and initial phases [16].

Recent research has explored eclipse avoidance of periodic orbits in cislunar space. Unlike the Sun–Earth system, the influence of lunar shadow should be further taken into account, particularly in periodic orbits near the Moon, which may experience long-duration lunar shadows. Zimovan et al. studied eclipse avoidance in the Near Rectilinear Halo Orbit (NRHO), NASA’s Lunar Gateway Program targeting orbit, achieving a long-time eclipse-free orbit by adjusting the geometry of ROs near NRHO [17]. Tang et al. analyzed the eclipse avoidance of the Queqiao Relay Satellite at a halo orbit around the Earth–Moon L2 Lagrangian point during a 3-year mission, finding that increasing the orbital radius effectively shortens the duration of a single eclipse [18]. Chikazawa et al. further discussed the EQUULEUS CubeSat at the Earth–Moon L2 halo orbit and MMX at 3:4 quasi-satellite orbits (QSOs) around the Martian Moon Phobos, revealing that specific orbit injection points and resonant radio could optimize orbital eclipse characteristics [19]. Sun Yang et al. conducted research on the eclipse characteristics of Earth–Moon Distant Retrograde Orbits (DROs), proposing a search strategy of the multi-objective genetic algorithm combining the time domain and phase space to avoid solar eclipse during the mission period, testing the robustness of the optimized model against dynamical model error using Monte Carlo simulation [20].

The Earth–Moon ROs, characterized by large scale and long periods and lying in the lunar orbital plane, are susceptible to occlusion by the Earth or the Moon, resulting in prolonged eclipses. Therefore, this paper aims to achieve eclipse avoidance or reduce the maximum single eclipse duration through maneuvering, providing a reference for scenarios where shadows are located in the same plane as the ROs.

Earth–Moon 3:2 ROs have demonstrated their capability for long-term cislunar surveillance, complementing the surveillance range of DROs, and providing navigation support for missions in cislunar space. This paper investigates eclipse mitigation for the planar Earth–Moon 3:2 RO family, designing orbital maneuvers to achieve low-cost avoidance strategies for the Earth and lunar shadows, respectively. It accordingly makes recommendations for the selection of the maximum allowable eclipse time for spacecraft electric power. In this paper, an overview of ROs and their dynamics is presented in Sect. 2; the distributions and durations of the Earth and lunar shadows are simulated and analyzed in Sect. 3. Sections 4 and 5 discuss the Earth and lunar shadow avoidance strategies, respectively, and propose eclipse mitigation strategies. Section 6 evaluates orbital stability after the eclipse mitigation maneuver, and Sect. 7 analyzes the spacecraft under different selections of the maximum allowable eclipse duration. In Sect. 8, the methods discussed in the above work are applied to high-fidelity models, demonstrating the feasibility of eclipse mitigation strategies. The analysis methods presented herein can be generalized to eclipse mitigation studies of other RO families.

2 Dynamics

This section presents the basic theories of ROs, including the model of the Circular Restricted Three-Body Problem (CRTBP) and orbital stability theory.

2.1 Circular Restricted Three-Body Problem

The ROs explored in this paper is based on the CRTBP. A CRTBP is an autonomous dynamic model that approximates the dynamics of the Earth–Moon system. Szebehely described this well-known model in details [21] and its definition is briefly reviewed here. In the CRTBP, the primary and secondary bodies are assumed to follow circular orbits about their mutual barycenter, whereas the spacecraft mass is assumed to be negligible compared with these two masses.

The dynamics are modeled in a rotating frame $C-XYZ$, where C represents the barycenter of the Earth–Moon system, the X axis is directed from the Earth to the Moon, the Z axis is directed along the angular momentum vectors of the primaries, and the Y axis is defined to complete the right-handed coordinate frame. The spacecraft state, ${\varvec{r}} = {[x,y,z]^\mathrm{{T}}}$ and ${\varvec{v}}={{[{\dot{x}},{\dot{y}},{\dot{z}}]}^{\text {T}}}$, is chosen as a Cartesian representation of the spacecraft position and velocity in the synodic frame. This synodic frame is shown schematically in Fig. 1.

Fig. 1

CRTBP model

The CRTBP is commonly normalized such that the system mass, angular velocity, and distance between the primaries are all equal to one. The distance from the barycenter to the primary and the distance from the barycenter to the secondary then become

$$\begin{aligned} 1-\mu =\dfrac{{{m}_{1}}}{{{m}_{1}}+{{m}_{2}}}\text {, }\mu =\dfrac{{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}} \end{aligned}$$

(1)

We then define distances $r_1$ and $r_2$ as the distances from the spacecraft to the primary and secondary bodies, respectively. These become

$$\begin{aligned} {{r}_{1}}= & \sqrt{{{(x+\mu )}^{2}}+{{y}^{2}}+{{z}^{2}}} \end{aligned}$$

(2)

$$\begin{aligned} {{r}_{2}}= & \sqrt{{{(x+\mu -1)}^{2}}+{{y}^{2}}+{{z}^{2}}} \end{aligned}$$

(3)

Using these definitions, the equations of motion for the CRTBP can be expressed as

$$\begin{aligned} \left\{ \begin{array}{l} \ddot{x} - 2\dot{y} = x - \dfrac{{(1 - \mu )(x + \mu )}}{{{r_1}^3}} - \dfrac{{\mu (x + \mu - 1)}}{{{r_2}^3}}\\ \ddot{y} + 2\dot{x} = y - \dfrac{{(1 - \mu )y}}{{{r_1}^3}} - \dfrac{{\mu y}}{{{r_2}^3}}\\ \ddot{z} = - \dfrac{{(1 - \mu )z}}{{{r_1}^3}} - \dfrac{{\mu z}}{{{r_2}^3}} \end{array} \right. \end{aligned}$$

(4)

The values that every unit characterize after the normalization in the CRTBP are shown in Table 1.

Table 1 Earth–Moon CRTBP parameters

One energy-like constant of the motion does exist in the rotating-frame formulation of the CR3BP. This scalar, known as the Jacobi constant, JC, provides significant insight into the dynamical behavior in the CR3BP as

$$\begin{aligned} J = {x^2} + {y^2} + \dfrac{{2(1 - \mu )}}{{{r_1}}} + \dfrac{{2\mu }}{{{r_2}}} - ({\dot{x}^2} + {\dot{y}^2} + {\dot{z}^2}) \end{aligned}$$

(5)

2.2 Resonant Orbit

Consider two objects A and B of arbitrary mass and describe their possible resonant motion relations. In the two-body Kepler motion, the motion period of object A is q and the motion period of object B is p, so the orbital resonance is defined by the parameter p : q. In the case of the Earth–Moon system, object A represents the Moon and object B represents the spacecraft [22]. In this paper, the spacecraft at Earth–Moon 3:2 ROs completed three orbits around the Earth, while the Moon completed two cycles around the Earth.

The initial guess of the ROs in the CRTBP is presented from the two-body problem. The most straightforward approach to generate a planar RO in the two-body model is to select a set of initial parameters at perigee. If the set of initial parameters is selected at periapsis, then the initial velocity points entirely along the y-direction. According to this definition, the initial state of Earth–Moon p : q RO in J2000 can be determined from the following expressions for the selected orbital elements [3]:

$$\begin{aligned} {R_0} = \dfrac{{a\left( {1 - {e^2}} \right) }}{{1 + e\cos \theta }},\quad \mathrm{ }{V_0} = \sqrt{2{\mu _e}\left( {\dfrac{1}{{{R_0}}} - \dfrac{1}{{2a}}} \right) } \end{aligned}$$

(6)

where e is the eccentricity, and the semimajor axis a can be expressed as

$$\begin{aligned} a = {\left[ {{\mu _e}{{\left( {\dfrac{{q{T_{moon}}}}{{2\pi p}}} \right) }^2}} \right] ^{{1 / 3}}} \end{aligned}$$

(7)

Figure 2a shows the Earth–Moon 3:2 ROs in the two-body model which are calculated with the eccentricity e from 0 to 0.6. Based on the initial states provided from the two-body problem, the corresponding RO in the CRTBP can be easily obtained through multi-shooting method [23]. Thereafter, a single-parameter continuation method is used to generate families of the Earth–Moon 3:2 ROs as Fig. 2b.

Fig. 2

Earth–Moon 3:2 resonant orbit family

2.3 Orbital Stability

Stability indices provide a useful measure of orbital stability. This metric is defined as

$$\begin{aligned} {\nu _i} = \dfrac{1}{2}\left( {\left\| {{\lambda _i}} \right\| + \dfrac{1}{{\left\| {{\lambda _i}} \right\| }}} \right) \end{aligned}$$

(8)

where $\lambda _i$ represents the eigenvalues of the monodromy matrix, ${\varvec{M}}$. The monodromy matrix is defined as the state transition matrix (STM) at precisely one full period along a periodic obit:

$$\begin{aligned} {\varvec{M}} = \varvec{\Phi } \left( {t + T,t} \right) = \dfrac{{\partial {{\varvec{X}}_{t + T}}}}{{\partial {{\varvec{X}}_t}}} \end{aligned}$$

(9)

where ${\varvec{X}}_t$ indicates the state at epoch t, and T represents the orbital period in the CRTBP. The periodic orbit is linearly stable when all stability indices are equal to one [24].

For the CRTBP, An characteristic equation for the nontrivial (not equal to 1) eigenvalue has the following form [25]:

$$\begin{aligned} P\left( \lambda \right) = {\lambda ^4} + \alpha {\lambda ^3} + \beta {\lambda ^2} + \alpha \lambda + 1 \end{aligned}$$

(10)

where the parameters $\alpha$ and $\beta$ can be computed via monodromy matrix:

$$\begin{aligned} \alpha= & 2 - Tr\left( {\varvec{M}} \right) \end{aligned}$$

(11)

$$\begin{aligned} \beta= & \dfrac{1}{2}\left( {{\alpha ^2} + 2 - Tr\left( {{{\varvec{M}}^2}} \right) } \right) \end{aligned}$$

(12)

The parameters $\alpha$ and $\beta$ are plotted for each member of a family of periodic orbits on a Broucke stability diagram resulting in a curve that illustrates the evolution of the eigenstructure across the family. The Broucke stability diagram in Fig. 3 offers insight into the eigenvalue configuration in the complex plane.

Fig. 3

Broucke diagram of Earth–Moon 3:2 orbit family

Fig. 4

The distribution of quasi-periodic orbit near Earth–Moon 3:2 resonant orbits

Figure 3 shows that the orbits with Jacobian constant from 2.765 to 2.843 (with position in Earth–Moon line frome 0.328 LU to 0.382 LU) are unstable. Note that a quasi-periodic orbit exists outside the orbital plane when the eigenvalue $\lambda _i$ has the form of $\cos (\theta ) \pm \sin {\theta }$. The number of quasi-periodic orbits of the orbital family can be obtained by observing the eigenvectors of the monodromy matrix ${\varvec{M}}$ in Fig. 4. Figure 4a shows the eigenvalues of the monodromy matrix corresponding to quasi-periodic orbits, where the three dimensional quasi-periodic orbits (out of the orbital plane) are plotted in blue, and the red scatter points indicate the planar orbits.

The number of the quasi-periodic orbital types around Earth–Moon 3:2 ROs are given in Fig. 4b. The horizontal axis of the position of the ROs at the Earth-moon line in CRTBP represents the different orbits in the RO family. It can be found that there are three dimensional quasi-periodic orbits around most of the orbits in Earth–Moon 3:2 RO family.

3 Eclipse Propertise of Resonant Orbits

In this section, the shadow model applied in eclipse mitigation is determined firstly, and then the Earth shadow and lunar shadow are analyzed through celestial rotating frame and relative velocity of shadow respectively.

3.1 Shadow Model

Due to the large scale of cislunar space, long-duration eclipse exists in lots of orbits in this area. The eclipse shadows are divided into Earth shadow and moon shadow according to the obstacle body. Srivastava et al. compared the impact of different lunar shadow models including the cylindrical model, the conical model, and a further refined solar intensity modelon LEO and GEO, contrasting results with STK simulations [26]. Subsequently, Nugnes further considered an oblate ellipsoid of rotation as occulting body and a conical shadow, refining the cumulative error of the models [27].

The conical shadow model is a traditional method for evaluating the eclipses, as shown in Fig. 5a. The conic shadow area is divided into umbra and penumbra. According to the geometric relationship, the angles between the tangent line and center-center line of the obstacle and the Sun are expressed as

$$\begin{aligned} {f_1} = \arcsin \left( {\dfrac{{{R_S} + {R_B}}}{\rho }} \right) , \quad \mathrm{ }{f_2} = \arcsin \left( {\dfrac{{{R_S} - {R_B}}}{\rho }} \right) \end{aligned}$$

(13)

where $R_s$ represents the radius of the Sun, $R_B$ represents the radius of the obstacle body, $\rho$ indicates the distance between these two bodies, and the boundary of umbra and penumbra is symbolized by $f_1$ and $f_2$. The radius of the umbra and penumbra can be written as

$$\begin{aligned} {R_{umbra}}= & {R_B}\cos {f_2} - \tan {f_2}\left( {d - {R_B}\sin {f_2}} \right) \end{aligned}$$

(14)

$$\begin{aligned} {R_{penumbra}}= & {R_B}\cos {f_1} + \tan {f_1}\left( {d + {R_B}\sin {f_1}} \right) \end{aligned}$$

(15)

Generally, spacecraft in umbral area is completely obscured by the obstacle body, while that in penumbral area is partly obscured by the obstacle body, receiving slight sunlight. Due to the large amplitude of the 3:2 ROs, the conservative assumption that the power system in penumbra area cannot supply regular operation of the spacecraft is inconsistent with the actual solar intensity in cislunar space.

Fig. 5

General shadow models

In this paper, 50$\%$ of the penumbral area is considered as the reference shadow. Here the 50$\%$ of the penumbral area is defined as the conical region where the radius of variable cross-section equals the average of the radii for penumbral cross-sectional circle and the umbral cross-sectional circle. The goal of eclipse mitigation is that the single duration of the reference shadow does not exceed the preset maximum allowable shadow duration of the power system $T_{eclipse}$. The radius of the shadow is close to that of the cylindrical shadow model (the difference from the radius of the obstacle body is less than 1$\%$). Therefore, the shadow model is further simplified, and the cylindrical model is used as an approximation in the subsequent study, as shown in Fig. 5b.

There is a variable angle between ROs and the ecliptic plane and the lunar orbital plane. In this paper, the most conservative case of maximizing the eclipse duration is considered, that is, the resonant orbital plane, the lunar orbital plane and the ecliptic plane coincide.

3.2 Distribution of Earth Shadow

In order to facilitate the observation of the distribution of the Earth shadow, the ROs are transformed to the Sun–Earth synodic frame with the origin at the center of the Earth. Figure 6 shows the orientation of the Sun with respect to the Earth–Moon system. The Earth and the moon rotates clockwise around their barycentre, and the solar phase angle is defined as the angle between the position vector of the Sun relative to Earth–Moon barycentre and the Earth–Moon line. Due to the large distance from the Sun to the Earth–Moon system, the azimuths of the Sun with respect to the Earth–Moon Line approximately the solar phase angle $\phi$.

Fig. 6

The position of the Sun in Earth–Moon synodic frame

The solar phase angle at the initial epoch is set to 0, the coordinate system changes with eclipse, and the orbit family is shown in Fig. 7. The gray area represents the penumbra, and the black area represents the umbra region, with the Sun in the negative X-axis direction.

Fig. 7

Resonant orbits in Sun–Earth synodic frame

The period of the RO family rotating around the Earth is much smaller than that of the Earth revolution around the Sun, as a result that the phase of the Earth shadow under the Earth inertial frame is approximately constant for one period (about 18 days). In order to identify the occurrences of lunar shadows, we assigned numbers to three perigees of the 3:2 resonant orbits in the Earth–Moon synodic coordinate, as illustrated in Fig. 8. The perigee along the Earth–Moon line is designated as Perigee-1, and the perigees in the second and third quadrants are respectively denoted as Perigee-2 and Perigee-3. Selecting Perigee-1 in Fig. 8 as the initial state, Fig. 9 shows the duration of the Earth shadow while the eclipse occurrs in different true anomaly (TA) of the 3:2 RO family, with contour units in minutes, and the white area denotes the eclipse duration of less than 2 hr.

Fig. 8

Perigees of Earth–Moon 3:2 RO family in Earth–Moon synodic frame

Fig. 9

Duration of the Earth shadow in different true anomaly for 3:2 RO family

As can be seen from Fig. 9, for the orbits with the location of Perigee-1 greater than 0.4587 LU, the eclipse occurs for more than 2 hr duration at any phase. For orbits with the location of Perigee-1 less than 0.4587 LU, eclipses occurring only near perigee last less than 2 hr (due to large orbital velocities) and can last up to 6 hr. The above result shows that the ROs are difficult to avoid the Earth shadow through the selection of the initial orbital phase or the phasing maneuver.

3.3 Distribution of Lunar Shadow

Compared with the Earth shadow, the influence of the long-duration lunar shadow is more significant. In the Earth–Moon synodic frame, the Sun rotates clockwise around the barycentre of the Earth–Moon system, making the lunar shadow and the spacecraft move in the same direction near Perigee-1, and this “accompanying flight” brings about a long-duration lunar shadow.

The ROs are transformed from Earth–Moon synodic frame to the Sun–Moon synodic frame. Figure 10 shows the transitioned trajectories of 3:2 ROs with the initial solar phase angle of 0$\circ$. As the initial solar phase angle varies counterclockwise, the spacecraft in ROs rotates clockwise around the Moon, and the geometric shape remains unchanged.

Fig. 10

Resonant orbits in Sun–Moon synodic frame

In Sun–Moon synodic frame, the duration of lunar shadow can be approximated as the time of passing the section of lunar shadow with the in-track component of the velocity vector. Because of the invariance of its orbital shape, we can equate the changes in the initial solar phase angle to the shadow rotating counterclockwise around the moon, obtaining the duration of shadow occurring in different positions:

$$\begin{aligned} {T_{eclipse}} = \dfrac{{2R}}{v_{rel}}, \quad v_{rel}=\dfrac{{\left\| {{\varvec{v}} \times {\varvec{r}}} \right\| }}{\left\| {\varvec{r}}\right\| } \end{aligned}$$

(16)

where R represents the radius of the section of lunar shadow, $v_{rel}$ is defined as the velocity relative to the lunar shadow, ${\varvec{r}}$ and ${\varvec{v}}$ represent the position vector and velocity vector of spacecraft in Sun–Moon synodic frame respectively. As an example, the $v_{rel}$ for the 3:2 RO with the period of 55.34 days varies with different epoch (flight time) as shown in Fig. 11

Fig. 11

Velocity relative to the lunar shadow for RO with 55.34-day period

Fig. 12

Velocity relative to the lunar shadow for 3:2 RO family

In terms of the performance of the spacecraft power system, We can define the boundary velocity relative to the lunar shadow as the velocity just right enables the spacecraft pass the lunar shadow with the duration not exceeding the maximum allowable time. For example, the boundary velocity relative to the lunar shadow for maximum allowable time of 2 hr ($T_{eclipse}$=2 hr) can be calculated as

$$\begin{aligned} {v_{rel}} = \dfrac{{2{R_{Moon}}}}{{{T_{eclipse}}}} = 482.8\mathrm{{ m/s}} \end{aligned}$$

(17)

When the value of $v_{rel}$ is less than the boundary velocity relative to the lunar shadow, that is, below the yellow dashed line in Fig. 11, the duration of the lunar shadow occurring in this region exceeds 2 hr. The velocity relative to the lunar shadow at different epoch of the RO family are shown in Fig. 12. The variation of velocity relative to the lunar shadow for part of ROs are given in Fig. 13, with arrows indicating the position where the longest-duration lunar shadow occurs. Observing Figs. 12 and 13, it can be found that the long-duration lunar shadows are mainly concentrated near Perigee-1, while there are no long-duration lunar shadow in the region between Perigee-2 and Perigee-3.

Fig. 13

Distribution of the velocity relative to the lunar shadow for 3:2 ROs in Earth–Moon synodic frame

The distribution of the long-duration lunar shadows for 3:2 RO family under different conditions is obtained by traversing the different initial solar phase angles as Fig. 14. In order to obtain a complete lunar shadow catalog and avoid the situation that the long-duration lunar shadow crosses the initial state, the initial state is set at the perigee in the second quadrant. Since the lunar shadow moves at a similar speed with the spacecraft near Perigee-1, the maximum duration of lunar shadow exceeds 30 hr, which is difficult for the power and thermal control system of spacecraft to withstand.

Fig. 14

Duration of the lunar shadow in different initial solar phase angle for 3:2 RO family

Table 2 Characteristics of Eclipse for 3:2 RO

Table 2 summerizes the distribution characteristics of Earth and lunar shadows in 3:2 ROs. Eclipses occurs in all orbits of the RO family, and the maximum duration of eclipses far exceeds the allowable shadow duration about 2–4 hr for general spacecraft, which requires that the impact of eclipses must be considered in mission design.

4 Eclipse Mitigation Strategy for Earth Shadow

Due to the slow variation of the solar angular motion in inertial space (with an annual period), the phase of the Earth shadow (true anomaly) remains approximately constant, rendering phasing maneuvers ineffective in significantly avoiding eclipses. This chapter explores strategies for mitigating Earth shadow encounters, deriving methods for changing apsis and cross-track maneuver strategy. These techniques are applied to 3:2 ROs in the Earth–Moon system.

4.1 Eclipse Mitigation Strategy by Changing Apsis

According to Fig. 7, it can be observed that the ROs, in the Sun–Earth synodic frame, are approximately an elliptical orbit with a slowly changing apsis. The eclipse mitigation strategy of changing apsis involves altering the orientation of the orbital apsis in inertial space, causing a eclipse to occur at a position with a smaller true anomaly. This leverages the higher orbital velocity to swiftly traverse the shadow region [28].

Fig. 15

The schematic diagram of the eclipse mitigation strategy by changing apsis

In the two-body model, single-impulse apsis adjustment is illustrated in Fig. 15. The left side depicts the target orbit, while the right side shows the initial orbit, with the apsis changing maneuver occurring at the intersection of the two elliptical orbits. The true anomaly of the Earth shadow is denoted as $f_e$, and the true anomaly of the eclipse on the target orbit is denoted as $f_e'$. The changing angle of apsis-adjustment is defined as $\Delta \omega = f_e - f_e'$, where $f_e'$ can be expressed through the maximum allowable shadow duration $T_{eclipse}$ as:

$$\begin{aligned} f_e' = \left\{ \begin{array}{ll} \arccos \left( {\dfrac{{\sqrt{\mu }{T_{eclipse}} - 2\sqrt{a\left( {1 - {e^2}} \right) } R}}{{\sqrt{\mu }{T_{eclipse}}e}}} \right) ,& {f_e} < \pi \\ 2\pi - \arccos \left( {\dfrac{{\sqrt{\mu }{T_{eclipse}} - 2\sqrt{a\left( {1 - {e^2}} \right) } R}}{{\sqrt{\mu }{T_{eclipse}}e}}} \right) ,& {f_e} \ge \pi \end{array} \right. \end{aligned}$$

(18)

The true anomaly for applying the apsis-adjustment maneuver to the spacecraft is given by $f = \pi - \dfrac{\Delta \omega }{2}$, and the flight path angle $\theta$ is expressed as

$$\begin{aligned} \sin \theta = \dfrac{1}{v}\sqrt{\dfrac{\mu }{p}} e\sin f \end{aligned}$$

(19)

then the magnitude of the maneuver is represented by

$$\begin{aligned} \Delta v = 2v\sin \theta = 2e\sqrt{\dfrac{\mu }{{a\left( {1 - {e^2}} \right) }}} \sin \left( {\dfrac{{\Delta \omega }}{2}} \right) \end{aligned}$$

(20)

Note that this method is applicable to orbits where there are positions that eclipses occurring in with durations shorter than $T_{\text {eclipse}}$. For orbits with relatively high perigees (low eccentricity), their maximum orbital velocity is still insufficient to achieve a passage through the Earth’s shadow within the specified time, making the apsis-adjustment strategy ineffective.

The assumed $T_{\text {eclipse}}$ for power and thermal control systems of spacecraft is set to 2 hr. The results of the eclipse mitigation impulse calculations for eclipses at various positions along the RO family are presented in Fig. 16. This method is not suitable for orbits where the location of Perigee-1 is greater than 0.4587 LU (corresponding to Jacobi constant greater than 2.93).

Fig. 16

Results of the eclipse mitigation strategy by changing apsis

As depicted in Fig. 16, the cost of the apsis-adjustment maneuver is substantial, sharply escalating from the boundary true anomaly of the non-maneuver region. For eclipses requiring mitigation maneuver, there are virtually no impulses below 100 m/s. The maneuver magnitude can reach up to 1.4 km/s, When the eclipse occurs at a true anomaly of 180$\circ$. Evidently, the fuel consumption associated with this eclipse mitigation strategy is unacceptable for space missions in 3:2 ROs.

4.2 Eclipse Mitigation Strategy via Cross-Track Maneuver

The nominal RO family assumed in this study is positioned initially on the lunar orbital plane. Through cross-track maneuvers, the orbital inclination can be altered, allowing for mitigation of the Earth shadow. The principle of cross-track maneuver of the Earth shadow mitigation is illustrated in Fig. 17.

Fig. 17

The schematic diagram of the eclipse mitigation strategy via cross-track maneuver

The nominal orbit intersects the center of the shadow section with a radius of R. By applying a cross-track maneuver, the orbit introduces an orbital normal offset, denoted as z in Fig. 17b, at the true anomaly where the spacecraft is in shadow. This adjustment results in a reduction of the distance $s_{pass}$ that the spacecraft traverses through the shadow to a distance within $T_{eclipse}$. $v_{\perp }$ represents the in-cross component of the orbital velocity at the true anomaly corresponding to the occurrence of the eclipse. As indicated by the Gauss’s variational equations (GVEs) [29], it is known that the cross-track maneuver has no impact on the orbital elements a, e, and f. Therefore, the the in-cross component of the orbital velocity remains unchanged before and after the maneuver [30]:

$$\begin{aligned} {v_ \bot } = \sqrt{\dfrac{\mu }{{a\left( {1 - {e^2}} \right) }}} \left( {1 + e\cos {f_e}} \right) \end{aligned}$$

(21)

The change in orbital elements due to a unit cross-track velocity impulse is given by

$$\begin{aligned} \left\{ \begin{array}{l} \delta i = \dfrac{{r\cos \left( {\omega + {f_m}} \right) }}{{n{a^2}\sqrt{1 - {e^2}} }}\\ \delta \Omega = \dfrac{{r\sin \left( {\omega + {f_m}} \right) }}{{n{a^2}\sqrt{1 - {e^2}} \sin i}} \end{array} \right. \end{aligned}$$

(22)

where $f_m$ is the true anomaly at the maneuver epoch. Based on this, the expression for the change in orbital normal due to a unit normal velocity impulse at the phase of eclipse $f_e$ can be written as

$$\begin{aligned} \delta z= & r\left( {\sin {f_e}\delta i - \cos {f_e}\sin i\delta \Omega } \right) \nonumber \\= & \dfrac{{{a^{{3 / 2}}}{{\left( {1 - {e^2}} \right) }^{{3 / 2}}}\left( {\sin \left( {{f_e}} \right) \cos \left( {{f_m}} \right) - \cos \left( {{f_e}} \right) \sin \left( {{f_m}} \right) } \right) }}{{\sqrt{\mu }\left( {1 + e\cos {f_e}} \right) \left( {1 + e\cos {f_m}} \right) }} \end{aligned}$$

(23)

The change required in resonant orbital normal can be obtained from Fig. 17b:

$$\begin{aligned} z = \sqrt{{R^2} - {{\left( {{ ^{{s_{pass}}} \!/ _{2}}} \right) }^2}} \end{aligned}$$

(24)

where $s_{pass} = v_{\perp } T_{eclipse}$. In conclusion, the magnitude of the velocity impulse for the cross-track maneuver is given by

$$\begin{aligned} v= & \dfrac{z}{{\delta z}}\nonumber \\= & \sqrt{\mu \left( { - \cos {{\left( {{f_e}} \right) }^2}{e^2}\mu {T_{eclipse}} - 2\cos {f_e}e\mu T_{eclipse}^2 - 4{R^2}a{e^2} + 4{R^2}a - \mu T_{eclipse}^2} \right) }\nonumber \\ & \left( {1 + e\cos {f_e}} \right) \left( {1 + e\cos {f_m}} \right) /{2{a^2}{{\left( {1 - {e^2}} \right) }^2}\sin \left( {{f_e} - {f_m}} \right) } \end{aligned}$$

(25)

According to Eq. (25), given the nominal orbit and $T_{eclipse}$, the magnitude of the velocity impulse for the cross-track maneuver is solely dependent on the true anomaly of eclipse $f_e$ and the true anomaly at the epoch of the maneuver $f_m$. Taking a RO with a semi-major axis of 2.9 $\times 10^5$ km and an eccentricity of 0.5 as an example, for eclipses occurring at different true anomalies, varying all possible maneuver locations reveals the optimal maneuver phase with the minimum fuel consumption cost. Assuming $T_{eclipse}$ is equal to 2 hr, as shown in Fig. 18, the aforementioned RO incurs the maximum maneuver cost when a eclipse occurs at a true anomaly of 180$\circ$, amounting to 16 m/s. The optimal maneuver position varies linearly with the true anomaly of eclipse. The region beyond the black dashed line indicates areas where there is no need for eclipse mitigation due to shadow duration being less than $T_{eclipse}$.

Fig. 18

The result of the cross-track eclipse mitigation maneuver for the resonant orbit with $a=2.9\times 10^5$ km and $e=0.5$

Assuming $T_{eclipse}$ is 2 hr, the eclipse mitigation results of the cross-track impulses for Earth shadows at various positions in the RO family are depicted in Fig. 19. For ROs with lower perigee heights, the overall eclipse mitigation maneuver is less than 16 m/s. As the perigee height increases, resulting in a decrease in orbital velocity, the required change of the position in normal direction becomes larger, reaching a maximum of no more than 20 m/s.

Observing the variation in the surface plot with respect to true anomaly, for ROs with lower perigee height, the cost of maneuver decreases as the distance from the 180$^\circ$ phase of eclipse increases; on ROs with perigee heights greater than 0.55 LU, the magnitude of impulse achieves maxima not only at eclipse TA around 180$\circ$ but also at eclipse TA in vicinity of the perigee.

Fig. 19

Results of the eclipse mitigation strategy via cross-track maneuver for Earth shadow

In summary, the cost of the eclipse mitigation strategy via cross-track maneuver is significantly reduced compared to that by changing apsis. For a spacecraft with $T_{eclipse}$ set at 2 hr, the eclipse mitigation impulse remains within 20 m/s, which results the broader scope of application. Moreover, in addition to the substantial difference in maneuver impulse, the optimal maneuver phases for both eclipse mitigation strategies are very close, exhibiting a ramp type pattern between 100$^\circ$ and 300$^\circ$, as shown in Figs. 16a and 19a.

5 Eclipse Mitigation Strategy for Lunar Shadow

Due to the fact that the 3:2 RO family are not precisely resonant with the synodic period about 29.5 days in a high-fidelity model, the occurring locations of lunar shadow in these ROs may drift along the trajectory. Consequently, in formulating avoidance strategies for lunar shadow, it is necessary to consider different solar phase angles and account for the conservative scenarios where the initial phase selection cannot avoid the lunar shadow. This chapter discusses two avoidance strategies - phasing avoidance and cross-track maneuver avoidance. The effectiveness of both approaches will be evaluated and compared.

5.1 Eclipse Avoidance Strategy via Phasing maneuver

Phasing avoidance is one of the commonly employed methods for spacecraft to avoid prolonged eclipses. It involves applying a phasing maneuver near the perigee, causing an increase or decrease in the period of the RO. This adjustment allows the spacecraft to avoid the eclipse in the original phase where eclipse occur.

Fig. 20

Distribution of the lunar shadow with different initial solar phase angle for 3:2 RO family in $T_{CRTBP}$

Figure 20 illustrates the scenarios of lunar shadow for the RO family at different initial solar phase angles. The initial epoch is defined when the spacecraft is at Perigee-2 in Fig. 8. The propagation time in Fig. 20 corresponds to the orbital period $T_{CRTBP}$-the flight time for the RO to undergo three perigee passages and return to the initial position in CRTBP. The contour values represent the number of the lunar shadow over 2 hr in $T_{CRTBP}$, which indicate that the range of solar phase angle for maximum shadow duration over 2 hr in low-perigee (below 0.59 LU) orbits is narrow, while that in high-perigee (over 0.59 LU) orbits becomes larger.

Since the lunar angular velocity relative to the Sun is approximately constant, the variation in solar phase angle can be equivalently considered as an epoch change. The change in single-loop orbital period $T_{single}$- the flight time from one perigee to the next perigee, induced by the phasing maneuver, can be viewed as the variation in solar phase angle when the spacecraft reaches a specific position, as depicted in Fig. 21. For every change of $1^{\circ }$ in solar phase angle, the corresponding $\Delta T_{single}$ is 1.97 hr.

Fig. 21

The schematic diagram of the phasing avoidance strategy

Due to the concentration of extended lunar shadows around Perigee-1, we set the position of phasing maneuver at Perigee-2. The phasing goal is to position the spacecraft within a solar phase angle interval that ensures an absence of long-duration lunar shadows (> 2 hr), as indicated by the blank region in Fig. 21. When applying a pulse opposite to the velocity direction at the perigee, the post-phasing orbital period $T_{single}$ decreases, leading to an equivalent higher initial solar phase angle; conversely, applying a pulse along the perigee velocity direction results in an increase of $T_{single}$ and an equivalent lower initial solar phase angle.

Fig. 22

The schematic diagram of the $\Delta T_{single}$ phasing maneuver in Earth–Moon synodic frame

The change of orbital period $\Delta T_{single}$ is defined as the difference between the time it takes for a maneuver applied in the perigee velocity direction to reach the next perigee along the line connecting to the Earth center and the flight time between two consecutive perigees on the nominal orbit, as illustrated in Fig. 22. To provide a reference for the design of phasing maneuvers, Fig. 23 depicts the mapping between $\Delta V$ and $\Delta T_{single}$. The two parameters seem to approximately follow a linear relationship, with each 1 m/s velocity increment corresponding to a $\Delta T_{single}$ of approximately 2.5 hr.

Fig. 23

The relationship of the phasing maneuver and the change of period (unit hours)

An additional analysis for the relationship between $\Delta T_{single}$ and $\Delta V$ at Perigee-2 is presented. The orbit in a two-body system exhibits the following characteristics [30]:

$$\begin{aligned} h=\sqrt{\mu p}=\sqrt{\mu a (1-e^2)} \end{aligned}$$

(26)

The above equation is the expression for angular momentum, where a and e represent the semi-major axis and eccentricity of the ROs respectively. Using the relationship of the eccentricity and the radius of periapsis $e=1-\frac{r_p}{a}$, the velocity of the S/C at the perigee becomes

$$\begin{aligned} v_p= \dfrac{h}{r_p}=\sqrt{\mu a \left( \dfrac{1}{r_p^2}-\left( \dfrac{1}{r_p}-\dfrac{1}{a}\right) ^{2}\right) } \end{aligned}$$

(27)

The semi-major axis can be written as

$$\begin{aligned} a= \dfrac{\mu r_p}{2 \mu -r_p v_p^{2}} \end{aligned}$$

(28)

We can then substitute Eq. (28) into the formula of one orbital period:

$$\begin{aligned} T_{single}= \dfrac{2 \pi }{\sqrt{\mu }}a^{\frac{3}{2}} =2 \pi \mu \left( \dfrac{r_p}{2 \mu -r_p {v_p}^2}\right) ^{\frac{3}{2}} \end{aligned}$$

(29)

The partial derivative of $T_{single}$ with respect to the velocity of the perigee $v_p$ is obtained:

$$\begin{aligned} \dfrac{\textrm{d} T_{single}}{\textrm{d} v_p} =\dfrac{6 \pi \mu {r_p}^{\frac{3}{2}} v_p}{\left( 2 \mu -r_p {v_p}^{2}\right) ^{\frac{3}{2}}} \end{aligned}$$

(30)

We have depicted the trend of $\frac{\textrm{d} T_{single}}{\textrm{d} v_p}$ with respect to changes in velocity of the perigee, as shown in Fig. 24. Since the Eq. (30) is derived in two-body model, it exhibits some disparity from $\Delta T_{single}$ in CRTBP model. It can be observed that within a narrow range of variation in perigee velocity, $\frac{\textrm{d} T_{single}}{\textrm{d} v_p}$ remains relatively constant for the same orbit at different velocities of perigee. This conforms to the expected linear relationship between $\Delta T_{single}$ and $\Delta v_p$, as approximated in Fig. 23.

Fig. 24

The relationship of $\dfrac{\textrm{d} T_{single}}{\textrm{d} v_p}$ and the phasing maneuver in two-body model

Based on the mapping relationship between $\Delta V$ and $\Delta T_{single}$, the phasing maneuver impulses for the ROs at different initial solar phase angles can be computed, as depicted in Fig. 25a. For ROs with lower perigee heights, the required $\Delta V$ are within 10 m/s. The real magnitude of phasing maneuver in high-fidelity model can be obtained easily by exploring points around the phasing maneuver calculated in two-body model.

Fig. 25

Results of the avoidance strategy with phasing maneuver

Note that after applying the phasing maneuver near the perigee, the RO deviates from the nominal orbit. To bring the RO back, a trajectory correction near the next perigee is required. To mitigate deviations introduced by the two-body model, a differential correction method is employed under the CRTBP to refine the phasing maneuver, as illustrated in Fig. 25b.

5.2 Eclipse Mitigation Strategy via Cross-Track Maneuver

The cross-track maneuver eclipse mitigation strategy of the lunar shadow is analogous to that of the Earth shadow. Referring to Fig. 12, which illustrates the relative velocity to the lunar shadow $v_{rel}$ on a RO, for regions where $v_{rel}$ is below the boundary velocity relative to the lunar shadow, an optimal cross-track maneuver impulse is sought by traversing all phases, as depicted in Fig. 26.

Fig. 26

Results of the eclipse mitigation strategy with cross-track maneuver for lunar shadow

Table 3 Eclipse mitigation maneuvers for 3:2 RO family

The different flight times in Fig. 26 represent various positions of eclipses on one period of ROs in CRTBP. The overall magnitudes of the cross-track maneuvers are within 7 m/s, and the costs are lower than that of the phasing maneuver at the perigee. It can be found that the eclipse mitigation strategy via cross-track maneuver exhibits significant advantages, especially for ROs with lower eccentricity.

It is evident that the regions requiring eclipse avoidance correspond to areas with small $v_{rel}$, as shown in Fig. 12. By contrasting the cost of eclipse avoidance across different ROs, it is observed that the RO with the perigee in Earth–Moon line of 0.567 LU incurs the highest cost, which corresponds to the maximum shadow duration in Fig. 12. This is attributed to the elevation of the perigee in Earth–Moon 3:2 RO family, resulting in a gradual decrease in the velocity of perigee. Consequently, the linear velocity of lunar eclipse motion near the perigee also gradually diminishes. Both factors reach an extreme at the RO with the perigee in Earth–Moon line of 0.567 LU, resulting in the longest shadow duration compared to other ROs.

The characteristics of the eclipse mitigation strategies discussed in Sects. 4 and 5 are summarized in Table 3.

6 Orbital Stability after The Cross-Track Maneuver

Sections 4 and 5 discussed various eclipse mitigation strategies for Earth and lunar shadow. In terms of fuel consumption, cross-track maneuvering exhibits a distinct advantage, enabling eclipse mitigation within 20 m/s. However, due to the presence of third-body perturbations in cislunar space, cross-track maneuvers may cause the spacecraft to deviate significantly from the nominal orbit, potentially even leaving the Earth–Moon system. The orbital stability after a cross-track maneuver needs to be evaluated.

Fig. 27

Stability indices for 3:2 RO family after cross-track maneuvers

Applying cross-track maneuvers $\Delta V$ of 0 to 20 m/s at Perigee-1 in Fig. 8, with the facts that the monodromy matrices are the same wherever the initial states are in a periodic orbit in CRTBP, the stability index of the new orbits are calculated. The stability indices for most orbits are nearly equal to one; a few orbits with higher perigee heights exhibit stability indices greater than 1, as shown in Fig. 27. Only orbits that do not deviate from the cislunar space within a 10-year flight time are considered in the calculations. The majority of the deep blue region represents orbits with stability indices equal to or close to 1, while the blank regions in Fig. 27 represent orbits that diverge after propagation of 10 years. In general, unstable orbits are primarily concentrated in orbits with higher perigee heights and Jacobi constants in the range of 2.765 to 2.843.

However, due to the highly chaotic nature of the dynamics in cislunar space, small deviations may lead to substantial deviations over long-term propagation. Propagating the new orbits in CRTBP resulting from applying cross-track maneuvers to the RO family for a period of 5 years, even for orbits with stability indices close to 1, there is a possibility of leaving the cislunar space, as illustrated in Fig. 28.

Fig. 28

Trajectories for a 3:2 resonant orbit after the cross-track maneuvers in Earth–Moon synodic frame

To further provide reference for the magnitude of cross-track maneuver, the maximum allowable cross-track maneuver impulses for which the 3:2 RO family keep stable (The S/C remains within the cislunar space for five years after the maneuver, with a constraint that the variation in its orbital period shall not exceed 0.1 times the nominal orbital period) are depicted in Fig. 29 at all initial solar phases. Comparing Fig. 29 with Figs. 3, 4 and 27, The three pronounced minima of the maximum allowable normal maneuver values correspond to ROs with unstable manifolds, ROs lacking out-of-plane component of eigenvectors, and ROs with high stability indices (low eccentricity). If ROs in these region are designated as mission orbits, then the eclipse mitigation can be achieved through a nonzero orbital inclination, or through phasing maneuvers.

Fig. 29

Maximum allowable cross-track maneuvers for 3:2 RO family

Additionally, due to the autonomy of the CRTBP, there is little correlation between the orbital stability after applying cross-track maneuvers and the phasing maneuvers. For orbits feasible for cross-track maneuvers, it is possible to traverse all phases to find the optimal impulse.

7 Evaluation of The Maximum Allowable Shadow Duration

The eclipse prediction and mitigation analyses in the previous chapters were based on a maximum allowable shadow duration $T_{eclipse}$ of 2 hr. The key factor constraining $T_{eclipse}$ of the spacecraft is the performance of the power system [31]. In practical spacecraft mission planning, there is a requirement to strike a balance between the cost of configuring spacecraft batteries and the cost of eclipse mitigation, which provides a reference for the selection of spacecraft batteries.

Fig. 30

Maximum maneuver impulses in different $T_{eclipse}$ for Earth shadow mitigation

Fig. 31

Distribution of the velocity relative to the lunar shadow for 3:2 ROs

By traversing maximum allowable shadow duration at intervals from 1 to 3 hr, the costs of the eclipse mitigation maneuvers were evaluated for each $T_{eclipse}$. Figure 30 illustrates the scenario for different $T_{eclipse}$, depicting the maximum avoidance maneuver values for the 3:2 RO family under various true anomalies of the Earth shadows. Observing the slope of the distribution surface for the maximum maneuver impulse, it is noted that the maneuver cost undergoes a dramatic change when $T_{eclipse}$ is less than 2 hr, decreasing rapidly from over 45 m/s to 20 m/s; When $T_{eclipse}$ exceeds 2 hr, the maneuver cost decreases more slowly.

Using the method described in Sect. 3, an analysis of lunar shadow exceeding the specified duration for different $T_{eclipse}$ is conducted, as shown in Fig. 31. For $T_{eclipse}$ ranging from 1 to 1.5 hr, the cost of lunar shadow phasing avoidance is relatively high, while for $T_{eclipse}$ of 1.8–3 hr, the distribution of lunar shadow shows minor variations. The cost is lower for orbits with high eccentricity, but for orbits with low eccentricity, the cost of phasing maneuver remains challenging to accommodate.

Fig. 32

Maximum maneuver impulses in different $T_{eclipse}$ for lunar shadow mitigation

Figure 32 illustrates the scenarios for different $T_{eclipse}$, depicting the maximum cross-track maneuver for the RO family under various lunar shadow phase angles. Observing the trend of the maximum impulse polyline, for ROs with higher eccentricity, the cost of maneuver undergoes a dramatic change when $T_{eclipse}$ is less than 2 hr, decreasing rapidly from over 20 m/s to less than 10 m/s; when $T_{eclipse}$ exceeds 2 hr, the maneuver cost decreases more slowly.

Above all, from the perspective for the cost of the eclipse mitigation maneuver, setting the maximum allowable shadow duration to 2 hr enables relatively low fuel consumption, ensuring reasonable demands on the power system.

8 Eclipse Mitigation in High-Fidelity Model

To further demonstrate the feasibility of eclipse mitigation strategies, an analysis of eclipse mitigation in Earth–Moon 3:2 ROs with high-fidelity model is conducted.

The high-fidelity model incorporates gravitational forces from Earth, Moon, and Sun, as well as perturbations due to planets’ oblateness, atmospheric drag and solar radiation pressure. Here we study two applications of strategy via cross-track maneuver. The ROs of these two scenarios are depicted in Fig. 33. Firstly, we consider Scenario-1 of an Earth–Moon 3:2 RO with the orbital plane close to the lunar orbital plane. The initial epoch is set at October 10, 2025, 00:00:00, while the initial state of S/C is chosen in Earth–Moon line as Table 4.

Fig. 33

Earth–Moon 3:2 resonant orbits in high-fidelity model

Table 4 Initial state of Scenario-1

In the Earth–Moon synodic frame, the initial solar phase angle is 135$\circ$. As illustrated in Fig. 34a, the eclipse summary in the next four years reveals five occurrences of eclipses lasting more than 2 hr. The cross-track maneuver strategy is implemented to mitigate these eclipses. Utilizing the mitigation strategy outlined in Sects. 4 and 5, the actual cost of the eclipse mitigation maneuvers is calculated at theoretically determined maneuver positions, based on the true anomalies where the eclipses occur, as summarized in Table 5.

Fig. 34

Distribution of eclipses for Scenario 1

Table 5 Eclipse mitigation maneuvers for Scenario 1

The eclipse summary after eclipse mitigation is depicted in Fig. 34b, exhibiting that there are no eclipses continuing more than 2 hr within the next four years. Note that the change required in resonant orbital normal direction may differ from the planar case because of the inclination of the lunar orbit in high-fidelity model. It is well-known that there is an approximate 5$\circ$ angle between the ecliptic plane and the lunar orbital plane. The duration of eclipses in orbits is significantly influenced by this inclination due to the conical shape of the umbra shadow. Therefore, in practical missions, the cost of eclipse mitigation maneuver is computed based on the actual conical shadow section. The calculation result of Eq. (24) needs to be further subtracted by the distance of deviation of S/C from the center of the shadow section in the orbital normal.

We then take an Earth–Moon 3:2 RO with orbital plane having a relatively large inclination with respect to the ecliptic plane into account. The initial epoch is set at October 10, 2025, 00:00:00, while the initial state of S/C is chosen in Earth–Moon line as Table 6.

Table 6 Initial state of Scenario-2

The eclipse summary of the RO for the next 4 years is illustrated in Fig. 35a, showing that there are 5 occurrences of eclipses lasting more than 2 hr within the next four years. The cross-track maneuver strategy is employed to mitigate the eclipses. Similar to Scenario 1, the eclipse mitigation maneuvers are listed in Table 7.

Fig. 35

Distribution of eclipses for Scenario 2

Table 7 Eclipse mitigation maneuvers for Scenario 2

In some particularly harsh conditions, eclipse mitigation may result in subsequent eclipse that lasted initially less than 2 hr exceeding this threshold. Consequently, the frequency of mitigation maneuvers increases. It can be observed that as the orbital inclination relative to the lunar orbital plane becomes larger, the cost of eclipse mitigation significantly decreases (the third mitigation maneuver of 22 m/s to address the side effect of avoiding two long-duration eclipses with one maneuver). In practical missions, the cost of eclipse mitigation may benefit from slightly increasing the inclination of ROs relative to the lunar orbital plane.

In conclusion, the cross-track maneuver strategy is applicable to high-fidelity model, offering optimal maneuver epoch and impulse for eclipse mitigation in actual missions involving ROs. A correction is required for the shadow section in Eq. (25) due to the inclination of the shadow cone, which leads to differences between the actual maneuver cost and the theoretical cost with the coplanar assumption outlined in Sects. 4 and 5.

9 Conclusions

This study investigates the distribution of solar eclipses for the planar Earth–Moon 3:2 RO family. For orbits with lower eccentricity, the Earth shadow duration exceeds 2 hr for the eclipses occuring in any phase. Lunar shadow predominantly occurs around the Earth–Moon line, spanning two orbital periods $T_{single}$ of the ROs near the perigee, with durations exceeding 24 hr, necessitating mitigation or avoidance maneuvers. Therefore, the paper discusses the results of the eclipse mitigation via cross-track maneuver and phasing maneuver for Earth and lunar shadows, respectively in different true anomalies or initial solar phase angles. The findings reveal that cross-track maneuvers incur smaller costs (within 20 m/s), and further analysis is performed on the stability of resonant orbits after cross-track maneuvers. Finally, a recommended value of 2 hr is suggested for the maximum allowable shadow duration for spacecraft on 3:2 ROs. The analysis methods and eclipse mitigation strategies presented are verified feasible in the high-fidelity model, which can be extended to other RO families.

Due to the absence of spatial quasi-periodic orbits for a few orbits in 3:2 RO family, applying cross-track maneuvers may result in unstable manifold, causing the trajectories to depart the cislunar space after a period of propagation. For mission orbit selection, efforts should be made to avoid these unstable orbits, or eclipse avoidance through phasing maneuver can be implemented. Future research could focus on further studying eclipse mitigation and avoidance methods for these specific orbits and higher fidelity model.