Two-maneuver indirect contingency return from a low lunar orbit

Yunfei LI, Xiaosheng XIN, Xiyun HOU,*

a School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China

b Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210023, China

c Beijing Institute of Tracking and Telecommunication Technology, Beijing 100094, China

KEYWORDS Moon;Low lunar orbit;Contingency return;Third-body maneuver;Gravity assist

Abstract The problem of contingency return from the low lunar orbit is studied.A novel twomaneuver indirect return strategy is proposed.By effectively using the Earth’s gravity to change the orbital plane of the transfer orbit, the second maneuver in the well-known three-maneuver return strategy can be removed, so the total delta-v is reduced.Compared with the singlemaneuver direct return,our strategy has the advantage in that the re-entry epoch for the minimum delta-v cost can be advanced in time, with a minimum delta-v value similar to that of the direct return.The most obvious difference between our strategy and the traditional single- or multiplemaneuver strategies is that the complete transfer orbit is a patch between a two-body conic orbit and a three-body orbit instead of two conic orbits.Our strategy can serve as a useful option for contingency return from a low lunar orbit, especially when the delta-v constraint is stringent for a direct return and the contingency epoch is far away from the return window.

1.Introduction

For a manned lunar mission, risks during the mission’s entire lifetime should be considered.There are unexpected situations that the mission has to be aborted abnormally and a contingency return to the Earth is needed.In the era of Apollo missions, the contingency problem has already been extensively studied.Hyle et al.1studied the abort planning according to two types of flight parts: powered-flight parts and coast parts.The powered-flight parts include launch, translunar injection, lunar orbit insertion, and trans-Earth injection.The coast parts include Earth parking orbit, trans-lunar coast arc, trans-Earth coast arc, and lunar cycling orbit.Focusing on different flight parts,researchers carried out analysis.For example, Eggleston and McGowan studied the contingency orbit in the launch phase.2Kelly and Adornato3studied the single-maneuver direct return during the translunar phase in the two-body model.Hyle et al.classified the lunar orbit insertion failure into three types--escape trajectory,unstable or impacting ellipse, and stable or non-impacting ellipse.For the stable or non-impacting ellipse, an abort of the mission may be unnecessary.1Miller and Baker4studied the abort problem during the lunar landing phase.For the well-known failure of the Apollo 13, some abort trajectories targeting different landing sites were designed after a maneuver two hours later following the pericynthion passage,5which can serve as an example of studies on abort trajectories in the trans-Earth phase.In a more recent review paper, Xi et al.6classified the contingencies into seven levels.Only in level A-1, A-2, B-1 and B-2 contingencies, the mission is required to be aborted, and only in level A-1 and B-1 contingencies, the crew are required to be sent back to the Earth immediately.

With a resurgence of interest in the Moon’s exploration and especially manned missions in the new millennium, some recent new studies on this topic appear.There are several differences of the new studies when compared with the old ones.

(1) Due to more powerful computation ability and extensive use of optimization algorithms in spacecraft orbits,7–10the first difference is that more realistic force models and optimization algorithms are employed to find more realistic abort trajectories.For example, the multiplemaneuver direct contingency return in the trans-lunar phase was reformulated and solved with a high accuracy force model based on optimization algorithms by Lu et al.11An efficient algorithm was proposed by Senent12to design single-maneuver abort contingency return trajectories with or without lunar fly based on a high accuracy force model.

(2) The second difference is that contingency return from polar orbits or high-latitude regions of the lunar surface are considered.For early lunar missions, the free-return orbit is usually used, which limits the latitude on the lunar surface that an astronaut can visit.13,14Although the hybrid transfer orbit can remove this limit, early crewed lunar missions usually landed on low-latitude regions.With the discovery of water resources on lunar polar regions, current crewed missions usually focus on high-latitude regions.This poses new problems to contingency return.Murtazin et al.15proposed to first maneuver from a polar orbit to a high elliptic orbit and then wait on the orbit until the return window opens.Using the three-maneuver strategy, which is to be mentioned in the following, Lu et al.16studied the return orbit design for lunar high-latitude missions.

(3) The third difference is that contingency orbit design is generalized to different mission scenarios.For example,for the unmanned LADEE mission, Genova17considered the lunar orbit insertion failure,and proposed ways to design contingency orbits back to the Moon.Condon et al.18studied the contingency return to the Earth from the Earth-Moon L1 point.Jesick19studied the direct and the lunar fly-by contingency return orbit from the L2 halo orbit to the Earth.Williams and Condon20studied the contingency trajectory design options in the Asteroid Redirect Crewed Mission that sends astronauts to a distant retrograde orbit around the Moon in which the captured asteroid sample resides.Komar and Tartabini21studied the abort modes in the ascent,descent,and entry phases for the crewed Mars mission.Recently, with people’s increasing interest in utilizing the whole cislunar space but not just the Moon,22–24different contingency scenarios can be expected in the future.

A comprehensive treatment of the contingency orbit design during all phases of a manned lunar mission is beyond the scope of a scientific paper.In this study, we only focus on the lunar cycling phase.That is to say, the spacecraft already moves on a Low Lunar Orbit (LLO).At a contingency epoch prior to the scheduled return epoch, it has to enter the trans-Earth phase from the LLO.Due to constraints from the relative geometry between the Earth and the Moon, a spacecraft moving on an LLO has only two return windows in one month, so that the maneuver at the Trans-Earth Insertion(TEI) point is tangential and the minimum.25An unscheduled contingency epoch out of the return window usually needs a larger non-tangential TEI maneuver.Usually,the further away the contingency epoch is from the return window, the larger the TEI maneuver is.

For a trans-Earth orbit lasting a few days, the trajectory inside the Moon’s Sphere of Influence(SOI)is usually a hyperbolic orbit.The well-known patched two-body method produces a good approximation of the hyperbolic orbit’s outbound escape velocity vector vo∞at infinity25.As a result,by neglecting the Earth’s gravity, the trans-Earth orbit design inside the Moon’s SOI is equivalent to the design of a hyperbolic escape orbit from the LLO, with an escape velocity the same as vo∞.At a contingency return epoch,the vector vo∞usually does not lie in the LLO’s orbit plane,so one single tangential maneuver is impossible to fulfill the transfer.There are several ways to adjust the inclination difference between the escape velocity vector vo∞and the transfer orbit’s orbital plane.

(1)The first way is to use a single non-tangential maneuver at the LLO to target the escape velocity vector vo∞.The method to minimize the single maneuver has already been studied by previous researchers.For example, Gunther26provided an analytic approximation of the minimum value of the single maneuver in the two-body model.Shen et al.27used the multi-start algorithm to search for the optimal singlemaneuver direct return in a more realistic force model.Li et al.28proposed an analytical method to generate the single-maneuver direct return for high-latitude missions.One remark is that targeting an escape velocity vector is not only useful for the Moon-to-Earth transfer, but also useful in designing interplanetary transfer trajectories departing from the Earth.For example, Deerwester et al.studied the escape from a circular Earth parking orbit.29Zhang et al.30studied the similar problem from an elliptic orbit.Duan and Liu31studied the transfer from the Earth to near-Earth asteroids.

(2)The second way is to use a two-maneuver strategy to fulfill the transfer, with an additional maneuver at infinity to adjust the difference between the vo∞direction and the transfer orbit plane.The preliminary idea was proposed by Penzo when studying the transfer orbit back to the Earth from a polar lunar orbit,32and was further studied by Gunther26who found that the place for the second maneuver should be at infinity for the minimum cost.Of course,in practice the second maneuver has to be within the Moon’s SOI,usually at the apolune point of a highly elliptic orbit.33Later, Gobetz and Doll34showed that the two-maneuver strategy is never simultaneously better than the single- and the three- maneuver transfer.

(3) An improved three-maneuver strategy was also proposed.35In such a strategy, a first tangential maneuver is used to increase the apolune height,a second maneuver is then used at the apolune point to adjust the orbital plane so that thevector is in the new orbital plane,and a last tangential maneuver is performed when the spacecraft is at the perilune point again.Gerbracht and Penzo36extended the strategy to the departure from an elliptic orbit.Edelbaum37compared the three-maneuver and the four-maneuver strategy, and found that the three-maneuver strategy costs less if the amount of orbit inclination change is smaller than a critical value.Recently, the same three-maneuver problem was also studied by Gavrikova and Golubev,38and Jones and Ocampo.39

As we have mentioned,the escape velocity vector vo∞is generated from the Earth-centered conic orbit by neglecting the Moon’s gravity(see Section 2.1 and Fig.1).It serves as the target vector of the above single- or multiple- maneuver escape conic orbit centered at the Moon.The patched transfer orbit(between the Earth-centered conic orbit and the Mooncentered conic orbit)should be improved in order to get a real transfer orbit in the complete model of the Earth-Moon system.Ocampo and Saudemont40proposed a way to get a better initial estimate in the complete model than the patched conic orbits.They integrated the Earth-centered conic orbit backward from the re-entry point and the Moon-centered conic orbit forward from the TEI point,both in the complete model.A mismatch between the two integrated orbits appeared at the Moon’s SOI.A correction to both the re-entry point and the TEI point was made according to the mismatch to get a better estimate than the patched orbit.In some other studies, the authors simply used the patched conic orbits as the initial guess and numerically refined them to get the results in high accuracy force models, and used optimization algorithms to further reduce the Δv cost.41,42

All the above analyses are in the framework of the twobody problem.The resulting transfer orbit is a patched one between two parts: the Moon-centered single- or multiplemaneuver conic orbit, and the Earth-centered conic orbit.However,the trans-Earth orbit is actually a three-body trajectory, which is simultaneously influenced by the Earth and the Moon.Even inside the Moon’s SOI,when the spacecraft is far away from the Moon, influence from the Earth’s gravity cannot be neglected.In the restricted three-body problem, previous studies already showed that orbital plane of orbits around the secondary (Moon in our case) can be obviously influenced by the gravity of the primary (Earth in our case).43Researchers call this mechanism as the third-body maneuver,and use it to effectively change the orbital plane of an orbiter around the secondary or the first primary.For example,in the Hill model, Villac and Scheeres44studied the problem of orbit inclination change of an orbiter around the secondary.They compared the cost of the one-maneuver strategy with that of the third-body maneuver, and found that the latter always costs less when the orbit inclination change is larger than 40°.Capdevila and Howell45made a systematic study on transfer orbits among low Earth orbit, Distant Retrograde Orbit(DRO), triangular libration point orbit, and Nearly Rectilinear Halo Orbit (NRHO).They found that the lunar gravity can be used to effectively change the inclination of the transfer orbit between the DRO and the NRHO.Circi et al.46studied the geostationary transfer orbits by using the Moon’s gravity assist.They found that compared with Hohmann transfers,lunar assisted transfers are more economical for any inclination greater than the Moon’s orbit inclination.Trofimov et al.47studied the lunar landing problem through several example NRHOs with or without an intermediate low-perilune orbit.Their studies show that orbit inclination of the resulting lowperilune orbit covers a range of values,and this range changes with the selenocentric distance of the departure point on the NRHO.Wang et al.48also studied the transfers between NRHOs and DROs as Capdevila and Howell did.They found that exterior transfers always had relatively lower costs than interior ones,but generally required longer transfer times than the interior ones.Zhang et al.49studied the transfer from planar DROs to Earth orbits.They found that the cost of transfer to a polar Earth orbit from the planar DROs can be saved by using the lunar gravity assist.

Fig.1 Patched two-body problem for trans-Earth orbit.

The same third-body maneuver mechanism can also be used in the contingency return orbit design.In this paper, we propose a novel way to design the contingency return orbit by separating it into two patched phases.In the first phase, a tangential maneuver at some point of the LLO is used to increase the apolune height to an extent that the orbit can be obviously influenced by the Earth’s gravity.When the spacecraft is back to the new perilune point again, the Earth’s gravity perturbs the transfer orbit’s plane to an extent that it is closer to the escape velocity vector vo∞.The second phase starts from the new perilune point of the first phase, and ends at the re-entry point, with a second maneuver at the new perilune point(the patch point)to patch the two phases.In the following studies,we call this type of transfer as two-maneuver indirect return,in contrast with the single-maneuver direct return.As far as the authors know, no similar previous studies have been published.There are two remarks that we need to make.The first remark is that the most obvious difference between our study and previous studies is that the first phase of the complete transfer orbit (from the LLO to the patch point) is a three-body orbit instead of a Moon-centered conic orbit.The second phase of the complete transfer orbit (from the patch point to the re-entry point) however is still approximately a conic orbit centered at the Earth.As a result, we patch a three-body orbit with a conic orbit instead of patching two conic orbits as all previous studies did.The second remark is that our two-maneuver strategy is qualitatively different from the two-maneuver strategy introduced above, which is based on two-body dynamics and the second maneuver is at the Moon’s SOI.

For practical contingency orbit design,there are many constraints that need to be considered.For example:

(1) The fuel budget.In case of a contingency event,different choices of the propulsion system may provide different fuel budget.Take the Apollo 13 as an example.5The option of using the service module without the lunar module attached to it can provide a total delta-v Δv more than 3168 m/s.The option of using the lunar module can provide a total Δv of 608 m/s(or 1472 m/s)with(or without)the service module attached to it.Energy is not the only criterion when making the final choice,and many factors should be considered.Sometimes, choosing an option is closely involved with the energy requirement of the designed contingency orbit.

(2) The flight time budget.This depends on status of the hardware, software, and health of the crew.There are emergency situations that the return should be finished within a short time.Also take the Apollo 13 as an example.5Rupture of the liquid oxygen tank requires the team to send the crew back as soon as possible.One of the reasons for the final choice of keeping the lunar module and service module together is because the lunar module has sufficient lifetime support (～140 h)and this gives more freedom to the design of contingency return orbit.

(3) The communication condition.In case that the automatic navigation system fails and the crew cannot control the spacecraft manually, the control of the spacecraft should be performed by ground stations.In such a case, the key points (such as the place to execute maneuvers) of the designed contingency return orbit should be visible to ground stations.

(4) The attitude of the spacecraft.In case that the spacecraft’s attitude cannot be ideally adjusted,there are limitations to the thrust direction which should be considered in the design of the contingency return orbit.Also,errors of the thrust direction exist due to errors in the spacecraft’s attitude.A large Δv at some point may produce future errors beyond the controllability of the remaining fuel budget.This situation should be avoided when designing the contingency return orbit.5

(5) The re-entry angle and the landing site.For crewed missions, the re-entry angle at the re-entry point should be within a narrow corridor.Also, the sub-satellite point should be able to pass through the landing site.These are basic constraints when designing the contingency return orbit.In our work, we have taken them into consideration.

A comprehensive treatment of various constraints is beyond the scope of this paper,and is not the theme of the current study.For the theoretical analysis in the current work,the main constraints are the fuel and the flight time (or equivalently the return epoch).As being pointed out by Xi et al.,6it is a tradeoff between flight time and energy.The first priority is to bring the crew back to the Earth as soon as possible while satisfying the fuel budget.If starting from the same TEI epoch,our indirect return needs to spend a few more days in the first phase,so it generally takes a longer time from the TEI point to the re-entry point than the direct return.However, if starting from a TEI epoch different from that of the direct return,the indirect return is allowed to have approximately the same re-entry epoch as the direct return.Our studies find that the reentry epoch for the minimum Δv cost can be advanced when compared with the direct return.For the example case in this paper,we find that the epoch can be advanced by about 2 days.In case that we have a limited fuel budget (e.g., close to the minimum Δv), the indirect return can serve as a good option because it can bring the crew back earlier than the direct return at this fuel budget.China is going to start its manned lunar missions.Study in this work can serve as a reference for mission designers to make decisions if contingencies do happen during the lunar cycling phase.Besides,it can be simply taken as an alternative way of sending a probe back to the Earth if it is more economical than the direct return.

The remaining part of the paper is organized as follows.Section 2 introduces the geometry constraint of the trans-Earth orbit.Section 3 makes some preparations for the current study.Section 4 studies the single-maneuver direct contingency return.Section 5 studies the two-maneuver indirect contingency return and compares the results with those of the direct return.Section 6 concludes the study.

2.Problem statement

2.1.Patched two-body model

The well-known patched two-body model is widely used when designing direct trans-lunar or trans-Earth orbits.Details on this model can be found in textbooks.22Briefly speaking,three steps are involved(taking the trans-Earth orbit as an example):

Step 1.Neglect the Moon’s gravity and design the trans-Earth orbit from the Moon’s position to the re-entry point in the two-body problem composed by the spacecraft and the Earth, and then get the relative velocity vo∞of the spacecraft with respect to the Moon at the Moon’s position (see Fig.1(a)) from the designed orbit.We call the two-body orbit shown in the left frame as the Earth-leg.

Step 2.Use the relative velocity vo∞of a probe departing from the Moon as an approximation of the velocity at infinity,and design the hyperbolic orbit inside the Moon’s SOI in the two-body problem composed by the spacecraft and the Moon(see Fig.1(b)).We call the hyperbolic orbit inside the Moon’s SOI as the Moon-leg.With a prescribed angular momentum vector h, we can completely determine the hyperbolic orbit geometry inside the Moon’s SOI.More specific, we have the following relations

where μMis the gravitational constant of the Moon, and iMand ΩMare orbit inclination and ascending node longitude of the hyperbolic orbit, respectively.The angular momentum vector h should be in the plane which is orthogonal to the vector vo∞.It should also have an intersection angle of iMwith the z axis, which indicates that it lies on the cone with an angle of iMwith respect to the z axis.Fig.1(b) shows the geometry.Generally,there are only two vectors h1and h2simultaneously satisfying the two constraints.As a result, there are only two hyperbolic orbits with a prescribed orbit inclination iMand an asymptote direction parallel with the vo∞vector.The longitude of the ascending node ΩMis determined correspondingly by Eq.(1).

Step 3.The patched orbit is then numerically refined using the full force model by considering the boundary conditions at the TEI point and the re-entry point (see Section 4.2).Experience shows that the numerically refined transfer orbit is usually very close to the patched orbit for the transfer orbits lasting only several days.

2.2.Return window

A practical mission scenario is that the spacecraft waits on an LLO with a fixed orbit inclination ilunarand a fixed ascending node longitude Ωlunar, and chooses an optimal epoch to enter the trans-Earth phase.To find this epoch,we can set the transfer orbit inclination iMequal the LLO’s orbit inclination ilunarand design the trans-Earth orbit.The best TEI epoch is then determined by the condition that

In such a condition, a tangential maneuver can be applied at the TEI point, as shown in Fig.2(b).From Fig.1(b), we know that ΩMis determined by iMand the escaping velocity vector vo∞.Since we fix iM= ilunar, the only thing we can do is to choose the vector vo∞to satisfy the condition in Eq.(2).Known from Fig.1(a), the vector vo∞is determined by the Earth-leg which depends on the TEI epoch and the Time of Flight (TOF) of the Earth-leg.Suppose we fix the TOF, then we can only adjust the TEI epoch to satisfy Eq.(2).An obvious fact from Fig.1(a) is that the vector vo∞r(nóng)otates with the Moon’s position in space.Since at each TEI epoch, we have two angular momentum vectors h1and h2(see Fig.1(b)), so we have two values of the hyperbolic orbit’s ascending node longitude.As a result, in one orbital period of the Moon, we have twice the chance that Eq.(2)is satisfied.At other epochs,a larger non-tangential maneuver at the TEI point is needed to send the spacecraft from the LLO to the trans-Earth orbit, as illustrated in Fig.2(a).In this study, we call the epoch satisfying the constraint by Eq.(2) as the return window.

3.Preliminaries

3.1.Force model

Besides the particle gravity of the Earth, the Moon, and the Sun, a more realistic force model for the trans-Earth orbit design should include the following perturbation forces: (A)Particle gravity of major celestial bodies besides the Earth,the Moon and the Sun;(B)Non-spherical gravity of the Earth and the Moon;(C) Perturbation of the Earth’s tides;(D) Solar Radiation Pressure (SRP);(E) General relativity effects.

Fig.2 Geometry between LLO’s orbit plane and hyperbolic transfer orbit plane.The hyperbolic orbit plane nearly coincides with the LLO’s orbit plane.

In the following,we call the force model including only the particle gravity of the Earth, the Moon, and the Sun as the ephemerides model,50and the model including the above perturbations as the complete model.In the complete model,computing the non-spherical gravity and the transformation matrix from Earth’s body-fixed frame to the International Celestial Reference System (ICRS) takes a lot of time.Since we need to extensively search the space for the optimal results,we make some simplifications to the force model used.Actually, for direct trans-Earth orbits lasting only a few days,effects from the perturbations are very small.Considering the computation cost, we choose to use the ephemerides model, even when propagating the LLO.This leads to errors of the propagated LLO, but generally does not influence the conclusions.

3.2.Coordinate

A well-known fact about the Moon’s rotational motion is that it librates with respect to its orbit.Meanwhile, its orbit about the Earth precesses in the ecliptic frame with a period of about 18.6 years.This leads to the fact that the Moon’s true equator oscillates in space,with an amplitude much larger than that of the Earth.To properly describe the orbital elements of an LLO, we need to choose a proper ‘inertial’reference frame.In our study, we choose this reference frame to coincide with the Moon’s Principal Axis(PA)frame at the beginning of each year.For example, the inertial frame coinciding with the Moon’s PA frame of 2025:01:01:00: 00:00.000 is chosen to describe the orbit of the LLO if we want to design the trans-Earth orbit in the year 2025.In the following, we call this frame as the Moon’s equator frame.

3.3.Example LLO

Table 1 shows the example LLO used in the study.The orbital elements are given in the Moon’s inertial reference frame.For readers to check the results, its position and speed in the Moon-centered ICRS are also given in Table 1.

4.Single-maneuver direct return

This section is devoted to the single-maneuver direct contingency return orbit design.It is firstly designed in the patched two-body model, and is then numerically refined in the ephemerides model.We neglect the design of the Earth-leg because it is well explained in textbooks.

4.1.Moon-leg:two types of hyperbolic orbits

For the direct return,we need to choose one point on the LLO as the TEI point.The hyperbolic orbit inside the Moon’s SOI should start from the chosen TEI point and have an escape velocity vector the same as vo∞when it leaves the Moon’s SOI.As already shown in Fig.1(a),the outbound escape velocity vector vo∞is determined by the Earth-leg.Since we require the hyperbolic orbit to pass exactly the TEI point in this case,the design problem is different from the one shown in Fig.1(b), which requires the hyperbolic orbit’s inclination to be a prescribed value.The geometry of the hyperbolic orbit in this case is displayed in Fig.3.

In Fig.3,the angle f is the true anomaly of the TEI point on the hyperbolic orbit.It is a function of rTEIand the orbit eccentricity e.The angle δ is the angle between the eccentricity vector e and the vector -vo∞which satisfies

For a fixed value of rTEI,Eq.(5)can be solved by choosing a proper value of the orbit eccentricity e.In the following, the type of hyperbolic orbit in the left frame of Fig.3 is denoted as Type I, and the type in the right frame of Fig.3 is denoted asNote:*The initial epoch is taken as the contingency epoch.That is to say, an emergency happens at this epoch, and we have to design contingency return orbits starting from an TEI epoch after the contingency epoch.Type II.Once we solve the orbit eccentricity, we get the value of f along with the orbital plane from its normal direction n,then we can simply get the velocity vector at the TEI point.

Table 1 Initial condition of the example LLO used in this study.

Fig.3 Two types of hyperbolic trajectory geometry inside the Moon’s SOI.

4.2.Numerical refinement

The patched orbit designed in the patched two-body model should be numerically refined in the ephemerides model.The TEI point is on the LLO which changes with the time.Once the TEI epoch tTEIis chosen, the TEI point is chosen accordingly by propagating the LLO from the initial epoch to the epoch tTEI.Denote the departure maneuver at the TEI point as Δv = [Δvx, Δvy, Δvz]T.Position and velocity vectors at the re-entry point are determined by the four variables (tTEI,Δvx, Δvy, Δvz) and the transfer time ΔT.At the re-entry point,the following three constraints should be satisfied

where the subscript ‘re-entry’indicates that the value is taken at the re-entry point,aeis the radius of the Earth,hre-entryis the height of the re-entry point, ire-entryis the re-entry inclination,and θre-entryis the re-entry angle.We consider a re-entry point at the southern hemisphere, with hre-entryfixed as 120 km, θre-entryfixed as - 5.8°, and ire-entryfixed as 45°.We add another constraint requiring the sub-satellite point to pass through the landing site.In this study, the landing site is chosen as the Siziwang Banner in China (111°.68E, 41°.37 N).This constraint along with the three constraints in Eq.(7)determine the values of Δvx,Δvy,Δvzand ΔT,once the TEI epoch tTEIis chosen.Starting from the initial guess of Δvx, Δvy, Δvz, and ΔT provided by the patched two-body model, these constraints can be satisfied by fixing tTEIand numerically refining the values of Δvx,Δvy,Δvz,and ΔT.The numerical refinement process is trivial, and the details are omitted here.

4.3.Simulation results

Starting from the initial epoch t0,the spacecraft starts to move on the LLO.For a chosen TEI epoch tTEI>t0,the position of the TEI point can be obtained by integrating the LLO starting from t0to tTEI.A maneuver is executed to send the spacecraft from the LLO to the trans-Earth orbit.Fig.4(a)shows the Δv cost of the Type I (blue) and the Type II (red) transfer orbits for different values of tTEI.The initial epoch of the LLO is given in Table 1.From this frame, the following conclusions can be made:

(1) The Δv cost changes with the epoch tTEI, with a period the same as the LLO’s orbit period.This is because the relative geometry of the LLO and the Earth-Moon system changes in one LLO’s orbit period.This fact means that we can always choose a proper epoch tTEIduring one LLO’s orbit period to make the Δv cost the minimum.We denote the minimum value of the maneuver in one LLO’s orbit period as Δvmin,lp.

(2) A longer‘period’of 24 h exits in these curves due to the constraint from the landing site.When designing the trans-Earth orbits, we require their sub-satellite point after the re-entry point to pass through the landing site(in our case the Siziwang Banner in China).Meanwhile,we also restrict the TOF to be within 4–5 days.As a result, due to Earth’s rotational motion, this ‘24 hperiod’appears in the Δv cost.

(3) For this example, as the epoch tTEIgradually approaches 116 h, the value of Δvmin,lpgradually reduces.This is because that the return window (see Fig.2(b))is being approached.For this specific example,the minimum value of Δvmin,lpis about 801.09 m/s.

(4) Also,the minimum value of Δvmin,lpfor the Type I transfer is smaller than that for the Type II transfer.This fact can be easily understood from Fig.3.The minimal value of Δvmin,lpcorresponds to a near-tangential maneuver at the perilune of the hyperbolic orbit, which obviously belongs to the Type I transfer.This fact suggests us that we only need to consider Type I trans-Earth orbits.

Fig.4 Δv cost of single-maneuver direct return.The TOF is restricted to be 4–5 days.

Compared with the TEI epoch,what is more important for a manned contingency return is the re-entry epoch.Fig.4(b)shows the total Δv cost with respect to the re-entry epoch.The re-entry epoch seems to cluster in the figure.This is due to the fact that we require all trans-Earth orbits’sub-satellite point to pass through the landing site (Siziwang Banner in China).Due to the Earth’s 24-h rotation period, the re-entry epoch clusters in the figure,and the gap between different clusters is about one day.On the other hand,different TEI epochs within one LLO’s orbit period have a negligible effect on the re-entry epoch, i.e., the width of each cluster is much smaller compared with the gap.According to Fig.4(b), the re-entry epoch with the minimum Δv cost is around 236 h for the example case in our study.

5.Two-maneuver indirect return

5.1.Third-body maneuver mechanism

The problem with direct contingency return is that the LLO’s orbit plane does not coincide with the trans-Earth hyperbolic orbit plane at the contingency epoch.A preliminary idea of the multiple-maneuver return strategy is to first change the LLO’s orbit plane to coincide with the trans-Earth hyperbolic orbit plane (or the vo∞direction), and then use a tangential maneuver to enter the trans-Earth orbit.As we have already mentioned,a three-maneuver strategy is proposed by previous researchers to effectively change the orbital plane.The algorithm is displayed in Fig.5(a).A first tangential maneuver Δv1is used to increase the apolune height, then an extra small maneuver Δvmidis used at the apolune to change the orbit inclination,and finally the second tangential maneuver Δv2is used to send the spacecraft to the trans-Earth orbit.This is a common technique to reduce the Δv cost when a large orbit inclination change is needed.

Since the transfer orbit is in the three-body model composed of the Earth, the Moon, and the spacecraft, we can use the well-known third-body maneuver mechanism to remove the mid-course maneuver Δvmid.That is to say, we can use the Earth’s gravity to change the orbit inclination of the spacecraft, instead of using the maneuver Δvmid.An illustration is shown in Fig.5(b).Since the orbit is greatly perturbed by the Earth, the new perilune height is different from the original LLO’s height.The whole contingency return orbit is separated into two phases.The first phase is from the TEI point to the new perilune point, and the second phase is the trans-Earth phase which connects the new perilune point and the re-entry point.

Using the initial data given in Table 1, Fig.6 shows the orbit inclination i and the longitude of the ascending node Ω when the spacecraft first comes back at the new perilune point.Curves are corresponding to Δv1= 580, 600, 620 m/s, respectively.The orbit colliding with the Moon or escaping are not shown.The orbital elements are given in the Moon’s equator frame.Comparing the orbital elements with the initial orbital elements in Table 1,we know that the orbital plane is changed by the Earth’s gravity.The amount of change is small when Δv1is not very large, but can be extremely large when Δv1is close to the escape velocity of the spacecraft.Judging from Fig.6,we know that the Earth’s gravity is effective in changing the orbital plane of orbits around the Moon.

Fig.5 Illustration of three-maneuver contingency return strategy and the third-body maneuver strategy.For the three-maneuver strategy,a first tangential maneuver is used to raise the height of the orbit apolune,followed by a mid-course maneuver to change the orbit plane at the apolune,and then a second tangential maneuver to send the spacecraft to the trans-Earth orbit at the perilune.For the thirdbody maneuver strategy,the mid-course maneuver in the left frame is no longer necessary.The magenta curve in the right frame is the first phase of the indirect return.

Fig.6 Orbit inclination and longitude of ascending node when the spacecraft first comes back at the perilune point.The abscissa is the TEI epoch since the initial epoch.A different epoch means a different point on the LLO.

5.2.Orbit design strategy

In this section,we propose a way to design the indirect contingency return orbit.We restrict the first maneuver Δv1to be tangential.As a result, once the TEI point on the LLO is chosen,the direction of Δv1is fixed.We use two free parameters to characterize the TEI’s position and speed.The first parameter is the TEI epoch tTEI(characterized by the difference from the initial epoch t0),and the second parameter is the magnitude of the first maneuver Δv1.

(1) For the first parameter,we know that the total Δv cost is periodic in one LLO’s orbit period.Since we need a TEI epoch as soon as possible for a contingency return, we only need to constrain the tTEIin one LLO’s orbit period.

(2) For the second parameter, we know that it cannot be small because the apolune point should be high enough to be obviously influenced by the Earth’s gravity.On the other hand,it cannot be too large because the spacecraft has to return to the perilune point again instead of escaping from the Moon.As a result, we limit its value to be close to but smaller than the escape speed of the spacecraft.

Once the epoch tTEIis chosen,the position of the TEI point is fixed.Once the value of Δv1is chosen, the speed of the TEI point is fixed.We can start integrating the first phase of the contingency return orbit and find the new perilune point.Once the new perilune point is obtained,we can compute the second phase of the contingency return orbit using the algorithm described in Section 4.The second maneuver Δv2at the new perilune point is usually not tangential.Since we only have two variables,we can use a simple grid search of the two variables.We record the total Δv cost for each feasible solution,and pick up the one with the minimum Δv cost as the contingency return orbit.

5.3.Simulation results

Fig.7(a) shows the contour map of the total Δv cost vs tTEIand Δv1.We require tre-entry-tpatch?[4,5] days.The map indicates that generally, in one LLO’s orbit period, the minimum of the Δv cost exists.The value of the first tangential maneuver Δv1for the minimum Δv cost is close to 600 m/s in this example.

Fig.7(b) shows the corresponding Δv-tre-entryresults.For readers to compare with the results of direct return, we also plot in the right frame the Δv-tre-entrycurves of the direct return.Judging from this figure, we can know that compared with that of the direct return, the epoch for the minimum Δv cost of the indirect return is obviously advanced, and is advanced by about two days in this example case.This means that in the case of a limited fuel budget, the crew can be brought back to Earth about 2 d earlier.

Table 2 shows the key parameters of some indirect contingency return orbits(labelled as‘Indirect-*’).These orbits have local minimum Δv cost at the corresponding re-entry epoch within 1 day.For comparison, we also show the minimum Δv cost of direct return orbits with about the same re-entry epoch (labelled as ‘Direct-*’in Table 2).From the data in Table 2, the difference of the direct return and the indirect return with about the same re-entry epoch is obvious:

(1) In the direct return, the spacecraft has to wait on the LLO for a better trans-Earth geometry,so tTEIis larger.In the indirect return, the spacecraft adjusts its orbital plane in the extra first phase for a better trans-Earth geometry.

(2) For the orbital plane to be obviously influenced by the Earth’s gravity, a large first tangential maneuver Δv1is necessary.This means the apolune height of the first phase is high, so tpatch-tTEIis large.As a result, in order to shorten the re-entry epoch, a smaller TEI epoch tTEIis necessary.

Fig.7 Left frame shows the contour map of total Δv vs tTEI and Δv1 for indirect return.Different colors indicate total Δv in the unit of m/s.Right frame shows the corresponding Δv vs tre-entry curves.Also shown in the right frame are the Δv vs tre-entry curves for direct contingency return orbits shown in Fig.4(b).

Table 2 *Key parameters of contingency return orbits.

Fig.8 Illustration of the difference between direct and indirect return.Single-maneuver direct return is represented as blue lines,and two-maneuver indirect return is represented as red lines.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this paper).

Fig.8 intuitively shows the difference between the two strategies.The red lines correspond to the indirect return,and the blue lines correspond to the direct return.

As an example of the indirect return,Fig.9 shows the contingency return orbit corresponding to the ‘Indirect-2′one in Table 2.The left frame shows the trajectory in the Earthcentered ICRS, and the right frame shows the trajectory in the Moon-centered ICRS.For readers to see the details within the Moon’s SOI,only part of the second phase is shown in the right frame.

5.4.Discussion

Fig.9 Example indirect contingency return orbit, corresponding to the ‘Indirect-2′ orbit in Table 2.

Fig.10 Δv-tre-entry curves of indirect return(blue)and direct return(red)for two extra example cases.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Obviously, when the contingency epoch is close to the return window, a near-tangential maneuver can directly send the spacecraft to the trans-Earth orbit.In such a case, using the indirect return is not recommended, because the first phase of the indirect return takes some extra time.On the other hand,if the contingency epoch is far away from the return window,the advantage of the indirect return is obvious according to the above results.An intuitive idea is that this advantage is more obvious if the contingency epoch is further away from the return window.Fig.10 shows such two examples.The initial condition of the LLO is the same as that in Table 1.However,the initial epoch is 1 day earlier for the left example,and 2 days earlier for the right example.This means the contingency epoch(chosen as the initial epoch in our study) is 1 day and 2 days further away from the return window.Comparing Fig.10 with Fig.7, the advancement of the minimum Δv re-entry epoch is more obvious in Fig.10.For the left frame, the advancement is about 3 days, and for the right frame, the advancement is about 3–4 days.This supports the conclusion that the advantage of the indirect return is more obvious for the contingency epochs far away from the return window.

We have to remark that the shape of the first phase of the trans-Earth orbit which is inside the Moon’s SOI(Fig.9(b))is different from that of the three-impulse orbit designed by the three-maneuver strategy40–42.Since the the extra midmaneuver in the three-maneuver strategy is saved in our case,the orbit is smooth in the Moon’s SOI.On the other hand,the three-maneuver orbit is not smooth even after optimization due to the mid-course maneuver which changes the orbit plane.42Moreover, the minimum Δv value of our twomaneuver return strategy is about the same as the minimum Δv value of the direct return,i.e.,a single tangential maneuver.This is an obvious fact from Fig.10 and the right frame of Fig.7.In one word,compared with the direct return,our strategy has the advantage in that it can significantly advance the return epoch with the minimum Δv cost.Compared with the three-maneuver return strategy,our strategy has the advantage in that it can save energy.

6.Conclusions

Design of the contingency return orbit from an LLO is studied in this paper.For comparison, the single-maneuver direct return is firstly studied in the ephemerides model.The Δv cost is large if the contingency epoch is far away from the return window and an early return is required.Then, a twomaneuver indirect return strategy which divides the whole return orbit into two phases is proposed.The first phase uses the Earth’s gravity to change the transfer orbit plane with respect to the Moon, and the second phase connects the new perilune point of the first phase with the re-entry point.Our studies show that the re-entry epoch for the minimum Δv cost of the two-maneuver indirect return can be significantly advanced when compared with that of the single-maneuver direct return, especially when the contingency epoch is far away from the return window.As a result, the indirect return can serve as an option of the contingency return orbit when the fuel budget is stringent.One remark is that the same indirect transfer strategy can be used to design trans-lunar orbits,which may significantly increase the width of the launch window, and we are still working on this.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This study was supported by the National Natural Science Foundation of China (No.12233003).