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AAS 21-234

EXPLORATION OF DEIMOS AND PHOBOS

LEVERAGING RESONANT ORBITS

David Canales*

, Maaninee Gupta†

, Beom Park‡

, and Kathleen C. Howell§

While the interest in future missions devoted to Phobos and Deimos increases,

missions that explore both moons are expensive in terms of maneuver capabil-

ities partly due to low-energy transfer options that may not be readily available.

The proposed approach in this investigation includes Mars-Deimos resonant orbits

that offer repeated Deimos ﬂybys as well as access to libration point orbits in the

Phobos vicinity. A strategy to select the candidate orbits is discussed and associ-

ated costs are analyzed, both for impulsive and low-thrust propulsion capabilities,

within the context of the coupled spatial circular restricted three body problem.

The trajectory concepts are then validated in a higher-ﬁdelity ephemeris model.

INTRODUCTION

The future of space exploration includes a focus on Mars and its moons, Phobos and Deimos. Al-

though not generally a primary objective, observations of these two moons as part of various Mars

missions have occurred since the 1990’s.1One example is the Phobos 2 mission,1where contact

was lost when the spacecraft was approaching the surface of Phobos. However, both moons are now

considered potentially key destinations in support of the Martian exploration program including, for

example, telecommunications capabilities, radiation protection, and infrastructure for transporta-

tion and operations.2, 3 Recognizing the importance of both moons, a number of missions to explore

Phobos and Deimos have been proposed by both ESA and NASA, including PHOOTPRINT,4PAN-

DORA,5and PADME.6In addition, JAXA’s MMX,7planned for launch in 2024, incorporates the

return of a sample from Phobos to Earth.8While some of the proposed mission scenarios are based

on different scientiﬁc objectives, many are focused on exploring only one of the two moons.

Given the small masses of both Martian moons, constructing low-propellant transfers is challeng-

ing, particularly for capture into a science orbit in the vicinity of either of the two moons. In contrast

to other multi-moon planetary systems such as Jupiter or Uranus, the gravitational inﬂuence of the

Martian moons is so small that the moons do not signiﬁcantly inﬂuence the path of a spacecraft (s/c)

when traversing the Martian vicinity. Consequently, capture at either moon is challenging using the

classical two-body problem (2BP). With the objective of performing in-situ observation of Phobos,

previous authors have proposed different alternatives based on Lambert arcs to arrive at Phobos and

*Ph.D. Candidate, School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907;

dcanales@purdue.edu

†Ph.D. Student, School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907;

gupta208@purdue.edu

‡MSc. Student, School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907;

park1103@purdue.edu

§Hsu Lo Distinguished Professor of Aeronautics and Astronautics, School of Aeronautics and Astronautics, Purdue Uni-

versity, West Lafayette, IN 47907; howell@purdue.edu

1

to return to Earth.9Likewise, some low-thrust analysis has also been introduced for the exploration

of both moons.10 Another commonly proposed approach is a three-staged Mars orbit insertion

towards an orbit similar to that of Deimos or Phobos, with Mars at its focus.11, 12 Additionally,

Phobos/Deimos resonant gravity assists have also been exploited to explore both moons.13

Capture options in the vicinity of Phobos and Deimos are potentially accomplished by leverag-

ing multi-body dynamics techniques to understand the motion in the vicinity of the moons. Since

Phobos is frequently the primary target for a number of the previous missions, the motion in its

vicinity is explored by various authors. Prior research results are available that explore potential

science orbits under the gravitational inﬂuence of both Mars and Phobos. In particular, retrograde

orbits around Phobos that exist in the Mars-Phobos circular restricted three body problem (CR3BP)

are computed and assessed for their stability by Canalias et al.,14 Chen et al.,15 as well as Wallace

et al.16 Periodic orbits are also computed within the context of the Mars-Phobos elliptic restricted

three body problem (ER3BP), incorporating the irregular gravitational ﬁeld of Phobos, by Zamaro

et al.,17 and Wang et al.18 These previous works demonstrate that the inclusion of both the Mars

and Phobos gravity ﬁelds is a key component in the computation of Phobos science orbits that could

potentially be transitioned into a higher-ﬁdelity model for an actual mission scenario.

The complex construction of transfers between moons in a multi-body environment is the focus

of numerous previous investigations. It is demonstrated by various authors that low-energy transfers

between two distinct moons orbiting a common planet can be successfully achieved using invariant

manifolds.19–21 While most of these studies assume that the moons are in co-planar orbits, strategies

to produce transfers between spatial periodic orbits corresponding to two different moons located

in their true orbital planes are also available.21 Such schemes are most often demonstrated in multi-

moon systems where the masses of the moons and their relative distances enable these transfers,

e.g., the Jupiter or Uranus systems. However, due to the small masses of Phobos and Deimos,

the invariant manifold trajectories emanating from orbits near these moons do not intersect in con-

ﬁguration space. Thus, it is difﬁcult to plan potential exploration missions targeting both moons

using libration point orbits; typical transfers from one moon to the other are generally expensive

in terms of propellant. Furthermore, the gravitational inﬂuence of the moons on the s/c is so small

that capture scenarios that rely solely on resonant gravity assists via Tisserand graphs22,23 present

signiﬁcant challenges without extra maneuvers (∆Vs).

The current investigation proposes an alternative mission scenario that includes Mars-Deimos

resonant orbits (ROs) for exploring both moons with some reduced propellant costs. The exploration

via Mars-Deimos resonant orbits is twofold: (i) repeated Deimos ﬂybys, and (ii) relatively low

cost access to Phobos science orbits. The resonant orbits that fulﬁll these requirements are, thus,

introduced. Resonant orbits computed in the Mars-Deimos CR3BP offer periodic Deimos ﬂybys.24

For this analysis, Phobos science orbits are assumed to be periodic libration point orbits within

the context of the Mars-Phobos CR3BP. Mars-Deimos resonant orbits that allow for an intersection

with the invariant manifolds arriving into the Phobos science orbits are presented as candidates given

their low ∆Vs. Both moons are modeled in their true orbital planes at a given epoch. While prior

work relevant to this investigation focuses on the s/c arriving at an orbit with the same properties

as the orbit of the target moon, in this current work, the Phobos science orbit is assumed to be a

Lyapunov orbit. Thus, this investigation is focused on selecting candidate resonant orbits in the

Mars-Deimos CR3BP that offer repeated Deimos ﬂybys, and assessing the costs for transfers into

the Phobos science orbit.

The dynamical models to fulﬁll the objectives of this investigation are ﬁrst introduced, followed

2

by the assumptions for arrival to the Martian system. Then, a background on resonant orbits com-

puted in the Mars-Deimos CR3BP is presented. The methodology employed for the selection of

candidate resonant orbits that provide access to Phobos and Deimos is introduced, and an impulsive

as well as a low-thrust analysis of the transfers is completed. The transfers involving resonant orbits

that offer repeated Deimos ﬂybys and offer access to Phobos are then analyzed in a higher-ﬁdelity

ephemeris model. Finally, some concluding remarks are offered.

DYNAMICAL MODELS

The Martian system is a multi-body environment with two moons. Despite the fact that the masses

of Phobos and Deimos are small, a s/c traversing the system is still subject to the gravitational ac-

celerations due to several bodies simultaneously. For the purposes of this investigation, the CR3BP

is leveraged for computing resonant orbits in the Mars-Deimos system, and for computing periodic

orbits in the vicinity of L2in the Mars-Phobos system. Additionally, two variants of the CR3BP are

introduced: a patched 2BP-CR3BP model to locate transfers between resonant orbits and the vicin-

ity of Phobos, and a CR3BP with a low-thrust model (CR3BP-LT) for further analysis. Transfers

leveraging these models are then validated in a higher-ﬁdelity ephemeris model, accommodating

both impulsive and low-thrust propulsion.

Circular Restricted Three-Body Problem

The CR3BP builds upon insights from the two-body model, while also incorporating some of

the complexities of the N-body model. For the purposes of this investigation, the three bodies that

comprise the model includes Mars, either one of its moons, and the s/c. The s/c is assumed to pos-

sess inﬁnitesimal mass relative to Mars and its moons. Additionally, it is assumed that Mars and

Phobos/Deimos move in circular orbits relative to their mutual barycenter. Considering the eccen-

tricities of the orbits of Phobos and Deimos around Mars, equal to 0.0151 and 0.0002, respectively,

this assumption for circular moon orbits allows the CR3BP to serve as a reasonable approximation

of the true dynamics governed by Mars and either of the moons. The orbital planes of the moons are

deﬁned by the appropriate epoch in the Mars-centered Ecliptic J2000 frame, but the moons move

in their respective circular orbits in their recognized orbital planes. The speciﬁcs of the CR3BP are

formulated within the context of the Mars-Phobos or Mars-Deimos systems. Note that the moon

refers to either Phobos or Deimos.

A rotating frame, represented as ˆx-ˆy-ˆz, is deﬁned such that it rotates with the motion of Mars and

its moon; the origin of this frame is located at the Mars-moon system barycenter. It is convenient

to nondimensionalize the equations of motion for the s/c in the CR3BP, resulting in an autonomous

system that provides greater insight into a range of design problems. The following quantities are

deﬁned: the characteristic length, deﬁned as the distance between Mars and the moon, and the

characteristic time, determined such that the nondimensional mean motion of the two bodies is

equal to one. Additionally, the nondimensional mass ratio is deﬁned as µ=mP2/(mP1+mP2),

where mP1and mP2denote the masses of Mars and its moon, respectively. Note that the mass

ratios of the Mars-Phobos and Mars-Deimos CR3BP systems are µMP = 1.6548 ×10−8and

µMD = 0.2245 ×10−8, respectively. The nondimensional position vector locating the s/c relative

to the system barycenter is deﬁned as ¯rs/c =xˆx+yˆy+zˆz, where the overbars indicate vector

quantities. The nondimensional equations of motion for the s/c then, expressed in terms of the

3

rotating frame, take the form,

¨x−2 ˙y=∂U ∗

∂x ,¨y+ 2 ˙x=∂U ∗

∂y ,¨z=∂U ∗

∂z (1)

where the dots indicate the derivative with respect to nondimensional time. Here, the pseudo-

potential function is computed as U∗=1−µ

d+µ

s+(x2+y2)

2, where the distance between Mars and

the s/c is d=p(x+µ)2+y2+z2, and the distance between the moon and the s/c is evaluated as

s=p(x−1 + µ)2+y2+z2. There exists a constant of the motion, i.e., an energy-like quantity

formulated in the rotating frame, denoted the Jacobi constant, computed as JC = 2U∗−( ˙x2+ ˙y2+

˙z2). Additionally, ﬁve equilibrium solutions exist in the CR3BP, also termed the libration points.

Three of the libration points, namely L1,L2,L3, are collinear and lie along the rotating ˆx-axis.

The remaining two points, L4and L5, form equilateral triangles with Mars and the moon, in the

primary plane of motion as viewed in the rotating frame. Knowledge of the locations of the libration

points allows insight into the system dynamics relative to these solutions. In the vicinity of the

libration points, periodic solutions exist, such as the planar Lyapunov and the spatial halo orbits. The

computation of one periodic solution in the CR3BP guarantees the existence of a family of periodic

solutions. The process of natural parameter continuation is employed to generate families of such

solutions that possess similar characteristics. Relevant to this investigation, the L2Lyapunov and

L2halo orbit families in the Mars-Phobos system are computed and further explored to offer access

to Phobos. Given periodic orbits in the CR3BP, the knowledge of the ﬂow structures associated

with such solutions is leveraged for trajectory design. In particular, stable and unstable manifolds

associated with periodic orbits are computed and exploited in the search for connections between

different periodic solutions.

Patched 2BP-CR3BP Model The spatial patched 2BP-CR3BP model21 approximates trajecto-

ries modeled in either the CR3BP or the 2BP depending on the location of the s/c in the system.

Within a certain threshold distance from the moon, the trajectory is modeled with the CR3BP. At

such a distance, a sphere of inﬂuence (SoI) is deﬁned as a sphere surrounding the moon where the

gravitational inﬂuence of the moon is considerably high. In this investigation, this distance, or the

radius of the SoI, is computed at the location along the ˆxaxis where the ratio of the gravitational

accelerations due to the two primaries is equal to 10−5, or xSoI = ¨rmoon/¨rM ars = 5 ·10−5.xSoI ,

denotes the ratio of acceleration, where ¨rmoon and ¨rMars correspond to the gravitational accelera-

tions due to the moon and Mars, respectively. When the trajectories pass beyond the SoI, the motion

is approximated as Keplerian with a focus at the larger primary and uniquely determined by the os-

culating orbital elements: semi-major axis (a) , eccentricity (e), right ascension of the ascending

node (Ω), inclination (i), argument of periapsis (ω) and true anomaly (θ). For example, assume

a trajectory is desired that arrives into the vicinity of Phobos. From an orbit in the Mars-Phobos

system, the state is propagated backwards in time in the CR3BP towards the Phobos SoI, where

the state is instantaneously deﬁned as a back-propagated Keplerian orbit in the inertial frame, with

Mars at its focus (see Figure 1 for a schematic). Simultaneously, there exists a Keplerian orbit ap-

proaching from Deimos. An analytical exploration of the potential intersection of the two conics

is possible since trajectories departing Deimos and arriving at Phobos are blended into the Ecliptic

J2000 Mars-centered inertial frame. A strategy based in the spatial patched 2BP-CR3BP algorithm

then blends the arcs into an end-to-end transfer as detailed by Canales et al.,21 incorporating maneu-

vers as necessary. Given the relatively small masses of both Deimos and Phobos, results are readily

transitioned to the higher-ﬁdelity ephemeris model as well.

4

Figure 1. Scheme representing the blending of two different systems in the Ecliptic

J2000 planet-centered inertial frame according to the patched 2BP-CR3BP model.

CR3BP-LT Model For the analysis of the low-thrust engine model within the context of the

CR3BP, an additional acceleration term is added to the right side of Eq. (1). This term is ¯

T /m,

where ¯

Tis the low-thrust force and mis the mass of the s/c. Since malso depends on time, it is

incorporated into Eq. (1) as follows,

¨x−2 ˙y= +∂U∗

∂x +Tx

m,¨y+ ˙x= +∂U∗

∂y +Ty

m,¨z=∂U ∗

∂z +Tz

m,˙m=−T

g0Isp

(2)

where the low-thrust force components acting along the ˆx, ˆy, ˆzdirections are Tx, Ty, Tz, respec-

tively. The magnitude of the thrust vector, ¯

T=T, is between 0and Tmax. The value of the

speciﬁc impulse, Isp, is assumed to be constant and, together with the standard gravity acceleration

g0= 9.80665m/s2, these parameters determine the mass ﬂow rate ˙m.

Higher-Fidelity Ephemeris Model

The higher-ﬁdelity ephemeris model represents the motion of a s/c subject to multiple gravita-

tional accelerations. The motion of a s/c of mass miis modeled relative to a central body of mass

mq. Located relative to the central body, Nperturbing bodies of mass mjfor j= 1, . . . , N, are

also incorporated in the model. In this model, all the bodies are assumed to be centrobaric point

masses. This implementation relies on the SPICE libraries25 supplied by the Ancillary Data Services

of NASA’s Navigation and Ancillary Information Facility (NAIF), through which precise positions

and velocities are retrieved. The dimensional acceleration acting on the s/c in a model with N

perturbing bodies is then expressed as,

¨

¯rq−i=−G(mi+mq)

r3

q−i

¯rq−i+G

N

X

j=1

mj ¯ri−j

r3

i−j

−¯rq−j

r3

q−j!+¯

T

m,(3)

where idenotes the s/c, qdenotes the central body, and jcorresponds to the perturbing bodies.

The gravitational constant is denoted by G, and the dots in the equation represent derivatives with

respect to dimensional time. Note that ¯ro−frepresents the position vector of body ‘f’ relative to

body ‘o’, where the subscripts oand fare i,qor j, as apparent in Eq. (3). In this investigation,

the transfers obtained using the CR3BP models are validated in a higher-ﬁdelity ephemeris model

that includes the s/c, Mars, Sun, Phobos, Deimos and Jupiter. For the low-thrust engine model, the

acceleration from the low-thrust force, ¯

T /m, is incorporated as well.

ARRIVAL INTO THE MARTIAN SYSTEM

The main goal of this investigation is an approach that yields access to Deimos and Phobos

given their orbits as represented in Table 1. Of course, any trajectory to Mars is constrained by the

5

departure state from the Earth vicinity and an optimal arrival to the Mars or the moon science orbit

cannot be separated from the Earth departure. Thus, given the extensive prior analysis of the Earth-

to-Mars transfer problem, impulsive transfers to the Martian system are beyond the scope of this

investigation, although it is noted that a three-stage Mars orbit insertion strategy is a typical arrival

approach.11, 12 However, some assumptions regarding low-thrust propulsion can be leveraged to

simplify the arrival state into the Martian system.

Table 1. Orbital data of Phobos and Deimos obtained from the SPICE database25 in the Ecliptic

J2000 reference frame. Last accessed 08/05/2020. ais semi-major axis, Pis the orbital period, eis the

eccentricity, iis the inclination, and Ωis the right ascension of the ascending node (RAAN). iand Ω

are computed with respect to the Mars J2000 equatorial plane.

a[km] P[hour] e[nd] i[◦]Ω[◦]

Phobos 9,377.82 7.65 0.01482 1.05 131.71

Deimos 23,459.61 30.29 0.00019 2.44 260.12

Interplanetary low-thrust trajectories are a function of interdependent parameters associated with

the mission architecture, engine speciﬁcations and time of ﬂight. The arrival state of the s/c to

the Martian system is also dependent on these parameters. The Earth departure options include a

dedicated launcher with a considerable characteristic energy (C3) with respect to the Earth, and a

launch as a secondary payload on other interplanetary/geocentric missions; these various options

govern the initial state of the s/c and, thus, render a broad range for the optimal state for arrival

to the Martian system. For this investigation, the arrival state is conveniently deﬁned by utilizing

the history of the energy-like Jacobi constant in the Sun-Mars system, JCSM , of the low-thrust

Earth-to-Mars trajectories. A sample scenario is selected and the resulting trajectory is generated

using the higher-ﬁdelity ephemeris model, as shown in Figure 2(a). The trajectory departs from the

vicinity of the Earth at a location 4×105km away from the Earth and arrives at a Mars-centered

circular orbit in the Sun-Mars J2000 ecliptic plane with an orbital radius equal to that of Deimos.

Figure 2(b) illustrates the history of the Jacobi constant in the Sun-Mars CR3BP system, JCS M ,

corresponding to the transfer trajectory. Although this trajectory illustrates one speciﬁc scenario, the

time history for JCSM in the low-thrust Earth-to-Mars transfer trajectories must generally exhibit

the characteristics in Figure 2(b). First, the JCSM along the Earth-to-Mars transfer arc must be

lower than that of the Sun-Mars L1libration point to enable the s/c to enter the Martian system.

Additionally, the JCSM for a destination orbit near Mars in the Sun-Mars system must be higher

than that of the Sun-Mars L1,JCSM ,L1, to ensure capture by the second primary. Therefore, all

candidate orbits possess a JCSM value greater than J CS M,L1. Contrary to impulsive engines, low-

thrust engines change the velocity and energy of the s/c continuously. Thus, the JCSM of the states

of the s/c along the transfer trajectory must be lower than JCSM,L1while approaching the Martian

system, and it must be continuously raised to the ﬁnal JCSM of the destination orbit. Without any

loss of generality, it is assumed that the tentative arrival state in the Martian system is associated

with a JCSM smaller than J CSM,L1. In this investigation, the J CS M at the tentative arrival state

is assumed to be 3.00018, a value that is slightly lower than JCS M,L1= 3.00020 to ensure that

the trajectory passes through the L1libration point gateway. The energy difference between this

value and the JCSM of the various candidate science orbits is a useful indicator of the associated

propellant costs, as the Sun and Mars are the dominant gravitational bodies upon arrival, and the

Jacobi constant is adjusted only by the low-thrust maneuvers. This assumption is analogous to ﬁxing

the arrival state such that v∞= 0, consistent with the Mars two-body model, which is a common

practice when generating preliminary trajectories.26

6

(a) Sun-centered inertial frame view (b) JCS M history

Figure 2. A sample low-thrust Earth-to-Mars transfer trajectory and energy

(J CSM ) history. Departure on 08/31/24, arrival on 09/27/25.

The speciﬁcations of the s/c are selected to be consistent with the values that appear in previous

investigations. Upon the arrival across all resonant orbits, the mass of the s/c is assumed to be

m0= 180 kg, a realistic value when the s/c departs from an Earth geostationary transfer orbit

as a secondary payload on the EELV Secondary Payload Adapter (ESPA) ring27 with a low-thrust

engine. Then, the propellant costs between the different resonant orbits and the Phobos science

orbit are measured as the consumed mass from this consistent value. The model for the low-thrust

engine assumes a maximum thrust level of Tmax = 60 mN , and a speciﬁc impulse such that

Isp = 3000 s. These values are comparable to those in prior investigations for Earth-Mars transfers

with ballistic escape and low-thrust capture.28 Both sets of parameters produce similar levels of

maximum acceleration (3.3×10−4m/s2vs. 2.5×10−4m/s2) with the same speciﬁc impulse

value. Despite the assumptions regarding the arrival state and the s/c speciﬁcations, the results from

this investigation are extendable.

RESONANT ORBITS BACKGROUND

The proposed mission scenario entails that the s/c completes repeated Deimos ﬂybys, while also

providing access to Phobos. To that end, resonant orbits are computed and assessed for their ap-

plicability to this investigation. Resonant orbits have an extensive history for planetary ﬂybys, and

the invariant manifolds of unstable resonant orbits are applied frequently towards transfer trajectory

design. Additionally, mission designs leveraging the inherent stability of some resonant orbits are

conceptualized and successfully applied to long-term mission scenarios. For example, the Interstel-

lar Boundary Explorer (IBEX), originally launched in 2008 into a highly elliptical orbit around the

Earth, was later transferred into a spatial 3 : 1 resonant orbit, thereby guaranteeing long-term stabil-

ity.29 Following, in 2018, the Transiting Exoplanet Survey Satellite (TESS) was launched directly

into an operationally stable spatial 2 : 1 resonant orbit.30 This investigation exploits the stability of

resonant orbits, coupled with their distinctive repeating geometry, as visualized in a rotating frame.

Prior to computing resonant orbits in the Mars-Deimos CR3BP system, resonant orbits are com-

puted in the two-body model such that the s/c orbiting Mars is in resonance with Deimos. In the

two-body model, the relationship between the orbital periods of the s/c and Deimos, which are de-

noted as Ts/c and TDrespectively, is expressed as p/q =TD/Ts/c. The integer prepresents the

7

number of revolutions completed by the s/c around Mars, and qrepresents the number of revolutions

completed by Deimos in the same time interval. The s/c is then in p:qresonance with Deimos. For

instance, a s/c in 3 : 4 resonance with Deimos around Mars completes three revolutions in the time

that Deimos requires to orbit Mars four times. Visualizing this orbit in the Mars-Deimos rotating

frame reveals the characteristic loops associated with resonant orbits. The number of loops that

appear in this frame corresponds to the value of pin the associated p:qresonance ratio.

The two-body resonant orbits are then transitioned to the Mars-Deimos CR3BP model, noting

the following distinction: due to the additional gravitational forces in this model, the resulting res-

onant orbit no longer possesses a perfect integer resonance ratio with Deimos. Therefore, in the

Mars-Deimos CR3BP system, the p:qresonance ratio implies that the s/c completes prevolutions

in approximately the time that Deimos completes qrevolutions around Mars. Additionally, the

transitioned orbit is no longer precisely periodic and thus, differential corrections techniques are

employed to produce the analogous periodic resonant orbit in the CR3BP. Utilizing the initial con-

ditions corresponding to the two-body resonant orbit, a single-shooting targeting scheme is applied

to converge an equivalent resonance in the CR3BP.31

For this investigation, a variety of resonant orbits in the Mars-Deimos system are examined,

including the planar 2 : 1,2 : 3,3 : 2,3 : 4,3 : 5,4 : 3,5 : 3, and 5 : 4 orbit families. Spatial res-

onances belonging to the 2 : 3,3 : 2,3 : 4,4 : 1, and 4 : 3 resonant orbit families are evaluated as

well. Figures 3 and 4 highlight the distinctive geometries possessed by various planar and spatial

orbits corresponding to different resonance ratios in the Mars-Deimos system. For clarity concern-

ing the robustness of the expected geometries for these resonant orbits, individual orbits are also

propagated in the higher-ﬁdelity ephemeris model. Speciﬁcally, the orbits are transitioned from the

Mars-Deimos CR3BP model to the Mars-Deimos-Phobos-Sun-Jupiter ephemeris model. With the

epoch as 11/22/2020 22:00:00, the initial states corresponding to various resonances are propagated

for approximately 30 days. As a demonstrative example, the results from propagating a 2 : 1 and

a3 : 2 resonant orbit are illustrated in Figure 5. As is apparent, the orbits maintain their geometry

without any signiﬁcant deviations, and remain bounded to the base-orbit computed in the CR3BP.

Thus, these stable orbits are considered as Deimos science orbit candidates.

(a) 2 : 1 resonant orbits. (b) 3 : 2 resonant orbits.

Figure 3. Representative members from the planar 2 : 1 and 3 : 2 resonant orbit

families in the Mars-Deimos CR3BP system. The magenta arcs are the candidates

2 : 1A and 3 : 2B that are further investigated.

8

(a) 5 : 4 resonant orbits. (b) 3 : 4 resonant orbits.

Figure 4. Representative members from the planar 5 : 4 and spatial 3 : 4 resonant

orbit families in the Mars-Deimos CR3BP system. The magenta arc is the candidate

5 : 4B orbit that is further analyzed in Figure 14.

(a) 2 : 1 resonant orbit. (b) 3 : 2 resonant orbit.

Figure 5. Propagation of a 2 : 1 and a 3 : 2 resonant orbit for 30 days. Model: higher-

ﬁdelity ephemeris model. Epoch: 11/12/2020 22:00:00.

RESONANT ORBITS WITH ACCESS TO PHOBOS AND DEIMOS

Given the cost and challenges of sending a s/c from Earth to the moons of Mars, it is convenient

to ﬁnd orbits in the Martian system that aid with the exploration of both Phobos and Deimos. Given

the advantages that resonant orbits offer in the Mars-Deimos system, these orbits are considered to

link arrival states into the Martian system to the desired science orbits in the Phobos vicinity. With

this objective in mind, the most important requirement for the selection of the desirable resonant

orbits is identifying orbits that permit access to Phobos and, in particular, the Phobos science orbits.

Thus, the target Phobos science orbits are initially deﬁned. Given the small variation in the Mars-

Phobos Jacobi constant (JCM P ) along the L2families for both the Lyapunov and halo periodic

orbits in the Mars-Phobos CR3BP system, the arrival ∆Vat any orbit along either family is similar,

since JCM P indicates the energy level of the science orbits. For example, representative members

from the family of the L2Lyapunov and halo orbits in the Mars-Phobos rotating frame are plotted

in Figure 6, where each orbit is colored according to the nondimensional value of its JCM P . An

L2Lyapunov orbit possessing a Jacobi constant value equal to 3.000023 is selected as the Phobos

science orbit, as highlighted in Figure 6(a). From this Lyapunov orbit, a stable manifold trajectory

9

is propagated in reverse time towards the Phobos SoI, where it becomes a back-propagated arrival

conic. Then, given that the resonant orbit and the arrival conic are not in the same plane, the patched

2BP-CR3BP model is used to construct spatial intersections with the resonant orbit.21 Although the

selected Lyapunov orbit is planar, the associated costs are extendable to a spatial halo orbit given

the small variations in JCMP .

(a) Mars-Phobos L2Lyapunov family. (b) Mars-Phobos L2Halo family.

Figure 6. L2Lyapunov and halo orbits families in Mars-Phobos CR3BP system.

In black, the arrival L2Lyapunov periodic orbit with JCM P = 3.000023 is high-

lighted. Phobos is assumed to be perfectly spherical in this investigation and appears

in black.

To compute impulsive transfers from any resonant orbits to the arrival moon vicinity, the Moon-

to-Moon Analytical Transfer method (MMAT)21 is leveraged. One of the main challenges in con-

structing transfers between two moons is determining the conditions such that they can occur and

the relative phase between moons in their respective planes is one of the key elements. Employing

the MMAT approach, an intersection in conﬁguration space between the given resonant orbit and the

arrival conic is determined if available. As a result, for a given departure angle relative to Deimos,

the location of Phobos at arrival is delivered such that the intersection between the resonant orbit

and arrival conic is guaranteed. At the initial time of the transfer, the s/c is assumed to be located at

a desired point along the Mars-Deimos resonant orbit. Given that the resonant orbit is computed for

the Mars-Deimos CR3BP, its argument of periapsis is selected such that the apoapsis of the resonant

orbit is directed towards Deimos at instant 1. Thus, the argument of periapsis of the resonant orbit

is related to the true anomaly of Deimos in its orbit at the origin of the transfer to Phobos, θ0Dei ,

which is measured from the Deimos orbit’s right ascension of the ascending node (RAAN). Recall

that the orbit of Deimos is approximated as circular. Note that instant 1 refers to the moment that

the transfer from the resonant orbit towards the vicinity of Phobos originates. Figure 7 illustrates

the orientation of the resonant orbit with respect to the orbital plane of Deimos for different θ0Dei .

Two different examples are illustrated with the two angles θ0Dei,1and θ0Dei,2in Figure 7. Note that,

in the schematic, the s/c is assumed to depart towards Phobos at the apoapsis of the resonant orbit.

The objective is to determine whether a transfer between the resonant orbit and the Phobos vicin-

ity is available. The true anomaly of Deimos in its orbit at instant 1, θ0Dei , is considered ﬁxed. Note

that the departure conic can be deﬁned as either the same resonant orbit approximated in the Mars-

2BP or an intermediate conic that joins the resonant orbit with the arrival conic. Both options are

extensively explained below. Spatial intersections between the departure and arrival conics occur

10

Figure 7. Scheme that represents the location of the s/c and the resonant orbit with

respect to Deimos depending on θ0Dei . Then, θ0Dei ,1and θ0Dei ,2represent two dif-

ferent sample epochs at instant 1 assuming that the s/c departs towards Phobos at the

apoapsis of the resonant orbit.

if and only if the following analytical condition is satisﬁed. The condition must be fulﬁlled such

that the arrival conic is re-phased, or re-oriented, to deliver a spatial intersection between the arrival

conic and the departure conic:21

aa(1 −ea)≤ad(1 −e2

d)

1 + edcos(θdInt +nπ)≤aa(1 + ea),being n= 0,1,(4)

where aaand adare the semi-major axes of the arrival conic and the resonant orbit, respectively; ea

and edare the eccentricities of the arrival conic and the resonant orbit, respectively; and the angle

θdInt or θdI nt +πcorrespond to the true anomaly of the departure conic when it intersects with the

arrival plane, measured from the argument of periapsis, ωd. Recall that the angle θdI nt is generally

deﬁned by the values of the inclination, i, and the right ascension of the ascending node angle, Ω,

for the departure and arrival planes, as well as the argument of periapsis for the resonant orbit, ωd.

Note that ωddepends upon the initial epoch at the origination of the transfer with respect to Deimos,

θ0Dei . If the inequality constraint in Eq. (4) is satisﬁed, the unique phase for Phobos that yields such

a conﬁguration is produced. Then, the process is repeated for every value of θ0Dei over an entire

period of Deimos in its orbit, i.e., all available phases for Deimos in its orbit measured from Ωover

an entire period. The conﬁguration of the two moons, or the relative orientation between Phobos and

Deimos, that provides the minimum analytical ∆Vis determined. The total time of ﬂight, tT O F ,

is also evaluated. Given that the MMAT method is applied to generate transfers between resonant

orbits and the vicinity of Phobos, the transfers are divided into two different types depending on the

conic arc used to assess Eq. (4): (1) direct transfers from the resonant orbit to the arrival conic, and

(2) two-burn transfers, that incorporate an intermediate arc to link the resonant orbit with the arrival

conic, as represented in Figure 8.

Case 1: Direct Transfers

As is evident in Figure 9, the higher the value of the Mars-Deimos Jacobi constant, JCM D , cor-

responding to the resonant orbit in its associated orbit family, the lower the minimum ∆Vobtained

in the feasibility analysis for a direct transfer to Phobos. As an example, Figure 9 illustrates the

evolution of the ∆Vfor a direct transfer to Phobos for the 3 : 2 and the 5 : 4 planar families as a

function of the JCM D of the resonant orbit. Therefore, Eq. (4) is evaluated across the resonant

11

Figure 8. Two types of scenarios for the construction of transfers between resonant

orbits and a periodic orbit in the vicinity of L2in the Mars-Phobos system.

orbit family for every θ0Dei of Deimos in its orbit. If Eq. (4) is satisﬁed, then the minimum ∆V

conﬁguration between Phobos and Deimos is produced. Then, the orbit with the maximum JCMD

value that supplies access to Phobos is selected. As a result, the candidates for resonant orbits in the

Mars-Deimos system are reduced to members from the planar 3 : 2,3 : 4, and the 5 : 4 families, and

orbits from the spatial 3 : 4 resonant orbits. Figure 10(a) illustrates an example of a direct transfer

from the spatial 3 : 4B resonant orbit. These results are summarized in the ﬁrst four rows of Table

2 under Case 1. A s/c in these resonant orbits can access Phobos with only one maneuver, hence

the arc is denoted as a direct transfer. Additionally, the orbits provide the opportunity to conduct

ﬂybys of Phobos, given their proximity to its orbit. Recall that the numbers 1-4, which denote the

locations of the two moons at instants along the transfer as represented in Figure 10(a), are deﬁned

in Figure 8. Also, for reference, these instants are deﬁned in the same way as Canales et al.21

(a) Evolution in the 3:2 resonant orbit family. (b) Evolution in the 5:4 resonant orbit family.

Figure 9. Evolution of the ∆Vmagnitude with respect to JCM D for a direct transfer

to Phobos for the 3 : 2 and the 5 : 4 planar families.

Case 2: Two-Burn Transfers

The selected resonant orbits for Case 1 do not offer close Deimos ﬂybys. To reach Deimos, an

intermediate conic arc is incorporated, as represented by the orange arc in Figure 8, to bridge the gap

from the resonant orbits to Phobos while guaranteeing that close Deimos ﬂybys are available. Thus,

the semi-major axis of the resonant orbit, ad, is adjusted such that the ﬂybys with Deimos occur

at relatively closer distances. Then, to decrease the ∆Vfor the extra maneuver, this intermediate

conic arc departs from the apoapsis of the resonant orbit and targets a periapsis that equates to that

of the arrival conic. Note that this strategy does not correspond to a Hohmann transfer since the

12

(a) Case 1: Transfer from 3 : 4B resonant spatial orbit.

∆Vtot = 653.4m/s, tT OF = 5.41 days.

(b) Case 2: Transfer from 5 : 4B resonant planar orbit.

∆Vtot = 643.9m/s, tT OF = 43.74 hours.

Figure 10. Sample candidate RO transfers to a Mars-Phobos L2Lyapunov orbit

(Ecliptic J2000 Mars centered inertial frame). Model: 2BP-CR3BP patched model.

Table 2. Candidate Resonant Orbits (ROs) belonging to Case 1 (direct transfer, Figure 10(a)), Case 2

(two-burn transfer, Figure 10(b))

Candidate ROs x0[nd] z0[nd] ˙y0[nd] Period [nd] ∆V[m/s] J CM D

Case 1

3:2A 1.1200 N/A -0.4305 12.5664 480 2.8547

3:4A 0.4005 N/A 1.6410 25.1327 630 2.4609

3:4B 0.3991 0.0230 1.6448 25.1327 650 2.4569

5:4A 1.3199 N/A -0.7244 25.1327 530 2.7328

Case 2

3:2B 1.0010 N/A -0.1718 12.5663 528 2.9705

2:1A 0.9982 N/A -0.3530 6.2832 580 2.8753

2:1B 1.0010 N/A -0.3602 6.2832 582 2.8702

5:4B 1.0010 N/A -0.0858 25.1324 643 2.9926

intermediate and the arrival conics are not contained in the same plane. Yet, given its similarity

to a Hohmann transfer, this intermediate arc connecting the resonant orbit and the arrival conic

corresponds to a closer minimum ∆Vmaneuver for this Case 2. The Keplerian elements of the

intermediate arc are now used to evaluate Eq. (4), instead of the resonant orbit orbital elements.

This case requires two separate maneuvers, as illustrated by red dots in Figure 8, but the total ∆V

remains comparable to Case 1. Despite the additional ∆V, the new selected resonant orbits possess

higher JCM D in their respective families, which results in a lower cost to transfer to Phobos. Along

the family of resonant orbits, the orbits that include close Deimos ﬂybys and also satisfy Eq. (4) are

added to Table 2 under Case 2. As it was demonstrated in the resonant orbits background section, all

the candidate orbits for both cases offer long term stability when transitioned into a higher-ﬁdelity

ephemeris model, which is generally desired to meet the scientiﬁc requirements. A sample Case 2

transfer from the 5 : 4B planar resonant orbit is illustrated in Figure 10(b). Recall that each instant

identiﬁed in Figure 10(b) is deﬁned in Figure 8.

LOW-THRUST ANALYSIS OF THE SELECTED RESONANT ORBITS

The costs associated with each candidate resonant orbit, summarized in Table 2, are now tran-

sitioned to a low-thrust engine model. Since the thrust level is low, it requires many revolutions

around Mars, or a spiral-down arc, to achieve the energy change associated with the transfers.

These spiral-down arcs are decomposed into two phases as represented in the schematic in Figure

13

11: spiral-down (A) and spiral-down (B). The ﬁrst phase of the arc connects the tentative arrival

state, associated with a ﬁxed Sun-Mars Jacobi constant value, JCS M = 3.00018, to each resonant

orbit. The second phase of arc corresponds to the transfer from each resonant orbit to the sample

Phobos science orbit.

Spiral-Down (A): Anti-Velocity Steering Law

An anti-velocity steering law in Sun-Mars CR3BP is employed to generate spiral-down (A),

where the s/c is thrusting with the maximum thrust magnitude in the direction opposite to the ro-

tating velocity. Thus, the low-thrust term in Eq. (2) is evaluated as follows: ¯

T=−Tmax ¯v

v, where

¯vdenotes the rotating velocity of the s/c. This steering law offers a minimum time of ﬂight for a

given difference in JC, and is a useful reference value for the cost associated with a speciﬁed time

of ﬂight and propellant consumption, corresponding to arrival at each resonant orbit. From each

resonant orbit, the trajectories are propagated in reverse time with the anti-velocity steering law

until the value of the Jacobi constant reaches JCSM = 3.00018 < J CS M,L1, or when the energy is

sufﬁcient to open the L1gateway. Although this strategy offers a limited control over the targeted

quantity, it can be coupled with a differential corrections scheme to produce feasible and optimized

trajectories from the Earth vicinity to the candidate resonant orbits. Figure 12 illustrates a low-thrust

arc connected to a heliocentric leg that enters the Martian system through the Sun-Mars L1gateway.

These orbits serve as an initial guess to generate an end-to-end trajectory, as in Figure 2(a). Thus,

the anti-velocity steering law provides useful estimates of the costs associated with arriving at the

resonant orbits given the difference in the energy of the s/c along a heliocentric path and the energy

of the s/c in the Mars-centered resonant orbits. The corresponding costs for spiral-down (A) are

included in Table 3.

Figure 11. Two spiral-down strategies schematic Figure 12. Spiral-down (A) example

Spiral-Down (B): Q-law + Direct Collocation

Other options from each resonant orbit to the Phobos science orbit include an arc with several

revolutions, deﬁned as the spiral-down option (B) in Figure 11. Since the resonant orbits are linearly

stable as well as periodic, the spiral-down (B) is is solved independently from the spiral-down (A).

The main distinction between the (A) and (B) arcs is an additional boundary constraint on the ﬁnal

state along the arc that coincides with the state along the stable manifold of the Phobos science

orbit, as represented by the blue line segment in Figure 8. An algorithm capable of incorporating

multiple revolutions as well as rendezvous is required, as the state along the manifold changes with

the epoch. To that end, a methodology that combines Q-law32 and direct collocation33 is developed.

14

The history of state and control along the spiral-down (B) arc generated by Q-law is introduced as

an initial guess for the direct collocation process, by which the rendezvous is achieved and also the

trajectory is optimized. Both Q-law and direct collocation are brieﬂy summarized, followed by a

step-by-step description of the interface between the algorithms.

Q-law. A Q-law control strategy utilizes a candidate Lyapunov function, Q, to quantify the dis-

tance from the osculating orbital elements to the target orbital elements. The dynamics of the s/c

is represented in the Gauss’s form of variational equations, by which the time derivatives of the

osculating orbital elements are represented as functions of the osculating orbital elements as well

as additional forces besides the gravitational force from the central body, Mars in this investigation.

While the thrust is assumed to be at the maximum value, the control history of the two thrusting

angles is computed at each moment to maximize the decrease of Q, which is also labelled the dis-

tance quotient. One of the two thrusting angles is measured in the osculating orbital plane with

respect to the circumferential direction, assumed to be positive away from the central gravitational

body. The other angle is measured from the osculating orbital plane, positive in the direction of the

osculating angular momentum. These two angles fully deﬁne the three-dimensional thrust vector

in the osculating radial, circumferential and angular momentum directions. Although Q-law, in its

simplest form, is efﬁcient in generating a possible transfer between two orbits, it fails to target the

fast-variable (true anomaly), implying that it cannot pinpoint the exact location along the destination

orbit where the s/c arrives at. Moreover, the time of ﬂight along a trajectory generated with Q-law

is unknown a priori; thus, the Q-law algorithm by itself cannot handle a rendezvous problem. This

poses a problem since in this analysis, for the s/c to arrive at the Phobos science orbit or to encounter

Phobos, both the true anomaly of the s/c along the Phobos science orbit at the arrival and the time

of ﬂight for the spiral-down arc should be speciﬁed. This investigation overcomes this challenge by

leveraging direct collocation, and the trajectory generated with Q-law only serves as an initial guess

and remains a preliminary transfer solution.

Direct Collocation. The direct collocation algorithm used in this investigation is based on the

one implemented and extensively explained by Pritchett.33 A collocation scheme discretizes a con-

tinuous trajectory into ssegments. While the scheme supports different dynamical models, the

higher-ﬁdelity ephemeris model (Equation (3)) is employed and approximated as a polynomial of

degree n. Formulated with the Legendre-Gauss-Lobatto node placement scheme, the collocation

algorithm is equivalent to an implicit Runge-Kutta integration with (2n−2)th order of accuracy.34

In this investigation, n= 7, thus, the dynamics are approximated with 7th order polynomials, and

are equal to an implicit Runge-Kutta integration of 12th order. While classical orbital elements or

modiﬁed equinoctial elements are potentially available as the state variables, Cartesian position and

velocity are employed here. The control variables consist of the magnitude and the direction of the

thrust, and are assumed to be constant over a segment. In this investigation, this constant thrust

direction is represented in the osculating radial, circumferential and angular momentum directions

that are not ﬁxed in the inertial frame. As realistic missions utilize the turn-and-hold strategy for

the thrust vector in the inertial directions, it is noted that an extra step may be required to convert

the current solutions into a more realistic scenario. The problem formulation also incorporates the

following boundary constraints: the s/c originates from a selected resonant orbit at its apoapsis, and

the s/c arrives along the stable manifold of the Phobos science orbit at the SoI for Phobos. Then, the

feasible state and control history is produced by employing a differential corrections process, and

passed to a direct optimization algorithm to solve for mass-optimal trajectories. This approach of

pairing collocation and direct optimization is denoted direct collocation.

15

Interfacing Q-Law and Direct Collocation. Recall that the main challenge with a Q-law control

approach for the application of the spiral-down (B) scheme is that it fails to target the true anomaly

of the s/c upon its arrival at the Phobos science orbit. To address that deﬁciency, the results from

Q-law are now passed to a direct collocation algorithm that not only accommodates the Phobos

rendezvous, but also serves as an optimizer. The algorithm that interfaces the Q-law control history

and the direct collocation targeting is described with the following steps:

1. The ﬁnal state, deﬁned as the state along the stable manifold associated with the Phobos

science orbit when it crosses the SoI for Phobos, is converted into the osculating orbital

elements in the Mars-2BP at an estimated ﬁnal epoch, JDf,est . These elements serve as the

target variables for the Q-law guidance process. The position of this ﬁnal state in a Mars-

centered inertial frame is represented as the yellow circle in Figure 13(a).

2. A preliminary transfer from a candidate resonant orbit to the target variables is generated with

the Q-law algorithm. As a result, the state and control history for this transfer is constructed,

where the position history is represented as the purple arc in Figure 13(a).

3. Since Q-law fails to target the true anomaly, the location at the end of the preliminary transfer

(red circle in Figure 13(a)) does not coincide with the anticipated ﬁnal state. This discrepancy

in positions is accommodated by shifting the yellow circle closer to the red circle by selecting

a different ﬁnal epoch. This new ﬁnal epoch is determined via the equation,

JDf,new = arg min

JDf

|¯rQf −¯rf(J Df)|(5)

Here, ¯rdenotes a position vector of the s/c with respect to Mars in the Mars-centered inertial

frame, ¯rMars−s/c, following Equation (3), but represented as ¯rfor simplicity. The subscripts

fand Qf correspond to the ﬁnal state along the manifold to be targeted (yellow circle in

Figure 13(a)) and the ﬁnal location along the preliminary transfer generated with the Q-law

strategy (red circle in Figure 13(a)), respectively. Note that ¯rfdepends on the ﬁnal epoch,

JDf, since the epoch determines the location of Phobos as well as the location along the

manifold in the Mars-centered inertial frame. On the contrary, ¯rQf is constant over different

epochs, which is a valid assumption since the osculating orbital elements of Phobos around

Mars do not change substantially. Then, Eq. (5) is equivalent to determining a new ﬁnal epoch

that minimizes the distance between the ﬁnal state at the ﬁnal epoch, ¯rf(JDf), represented as

the yellow circle, and the ﬁnal location (the red circle) along the original trajectory generated

with Q-law, ¯rQf , represented as the red circle.

4. The difference between the estimated and the new ﬁnal epoch is deﬁned as ∆JDf=JDf,new−

JDf,est . An intermediate Mars-centered conic arc, with a period equal to ∆J Df, is intro-

duced. Then, the semi-major axis corresponding to this period is computed as ainter =

(GmMars∆J D2

f)1/3, where GmMars = 4.282 ×104km3/s2is the gravitational parameter

of Mars. One revolution of this intermediate conic, i.e., the blue arc in Figure 13(a), is in-

serted into the state history of the state generated by Q-law. As the s/c spends more time on

this conic, the distance between ¯rQf and ¯rfis minimized.

5. The state and control histories, including the intermediate ballistic conic, is discretized to

form an initial guess for the direct collocation scheme, as illustrated in Figure 13(b). Subse-

quently, the initial guess is computed for a feasible solution where the trajectory is continuous

16

along all segments while satisfying the boundary conditions. The feasible solution is then op-

timized for propellant consumption. An example of an optimized solution is plotted in Figure

14(b).

(a) Preliminary transfer generated with Q-law (b) Discretized state and control for direct collocation

Figure 13. Interfacing Q-law and direct collocation for generating spiral-down (B)

The above steps comprise the interface between the Q-law control and the direct collocation targeter,

and succeed in acquiring optimized transfers from each resonant orbit to the Phobos science orbit.

The associated costs are included in Table 3. Note that ∆Vcorresponds to the equivalent ∆V

computed as ∆V=Ispg0log (m0/(m0−∆m)).

Table 3. Low-Thrust Results: Q-law + direct collocation (*: optimized)

Candidate Orbits Spiral-down (A) Spiral-down (B) (A) + (B)

∆m[kg] ∆V[m/s] TOF [days] ∆m[kg] ∆V[m/s] TOF [days] ∆m[kg] ∆V[m/s] TOF [days]

Case 1

3:2 6.06 1217 34.37 4.58* 954* 30.67 10.63 2171 65.03

3:4 4.69 938 26.62 9.38 1969 53.67 14.07 2907 80.29

3:4 4.79 961 27.19 10.21 2157 58.43 15.00 3118 85.62

5:4 5.49 1101 31.17 3.44* 711* 26.76 8.93 1813 57.93

Case 2

3:2 6.00 1205 34.07 3.62* 752* 25.98 9.63 1957 60.06

2:1 7.10 1430 40.29 4.58* 961* 31.50 11.68 2391 71.79

2:1 7.10 1431 40.31 4.43* 928* 31.46 11.53 2359 71.77

5:4 5.47 1096 31.02 3.35* 693* 25.48 8.82 1789 56.49

CONCLUDING REMARKS

This investigation proposes a trajectory design strategy for observation of both Deimos and Pho-

bos within the context of the CR3BP. Firstly, it is demonstrated that resonant orbits computed in the

Mars-Deimos CR3BP model offer long term stability in a higher-ﬁdelity ephemeris model, gener-

ally desirable to meet scientiﬁc requirements. Thus, such resonant trajectories are a good resource

for periodic observation of Deimos, given that a s/c in this orbit repeatedly encounters around the

moon. Additionally, it is demonstrated that by employing the MMAT method, it is possible to de-

duce trajectories departing Deimos that yield a connection to transfer to the vicinity of Phobos. In

the context of this application, resonant orbits are promising candidates that grant access to Phobos,

given their observational ﬂybys of both moons but at the expense of a low ∆V. As a result, an

assessment of various resonant orbits is accomplished seeking access to CR3BP periodic orbits in

17

the vicinity of L2of the Mars-Phobos system. Using MMAT, a relationship is established between

the variation of JCMD along a resonant orbit family and the cost to access Phobos. It is observed

that the resonant orbit in its family with a higher value of JCM D , the lower the cost to access the

vicinity of Phobos. Once the resonant orbits that guarantee lower ∆Vs for impulsive transfers are

identiﬁed, a Q-law strategy is interfaced in a scheme to transition the results to the low-thrust model

and, thus, construct transfers from the resonant orbit towards the Phobos vicinity using low-thrust.

The low-thrust algorithm addresses the fact that Q-Law does not target the true anomaly for the

arrival at Phobos by adding an intermediate conic. This modiﬁcation not only aids convergence in a

higher-ﬁdelity ephemeris model, but also results in an optimized transfer. As a result of the analysis,

the 5 : 4B resonant orbit serves as a suitable candidate for access from outside the Martian system;

this orbit also offers frequent access to Deimos and more efﬁcient access to Phobos compared to the

other resonant orbits. To the extent that impulsive analysis is concerned, the resulting transfer (Fig-

ure 10(b)) is obtained in the patched 2BP-CR3BP model and then, using this transfer as an initial

guess, it is transitioned to a higher-ﬁdelity ephemeris model (Figure 14). The optimal low-thrust

trajectory that connects the 5 : 4B resonant orbit with the Phobos vicinity is also illustrated in Figure

14. It is noted that, given the small mass of Phobos, the JCM P variation to ensure capture around

the moon is sufﬁciently small that the same transfer from a resonant orbit is applicable for capture

into different orbits in the Phobos vicinity: both planar and spatial periodic orbits around L2, or a

quasi-satellite orbit around the moon. This generalization is applicable due to the low ∆Vrequired

to target instant 4 (deﬁned in Figure 8) in the impulsive analysis: less than 10 m/s is required to

transfer from the arrival arc to the selected Lyapunov orbit in a higher-ﬁdelity ephemeris model,

reﬂected in Figure 14 (g). Further reﬁnement of these trajectories can also incorporate the moon

harmonics due to their irregular shapes, e.g., Phobos.

ACKNOWLEDGEMENTS

The ﬁrst three authors have equally contributed to the work. Assistance from colleagues in the

Multi-Body Dynamics Research group at Purdue University is appreciated, as is the support from the

Purdue University School of Aeronautics and Astronautics and College of Engineering, including

access to the Rune and Barbara Eliasen Visualization Laboratory.

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(a) Mars centered inertial frame (MARSIAU) (b) Mars centered inertial frame (MARSIAU)

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(e) Mars-Phobos rotating frame: zoomed in (f) Mars-Phobos rotating frame: zoomed in

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19

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