Identification of New Orbits to Enable Future Mission Opportunities forthe Human Exploration of the Martian Moon PhobosI

Mattia Zamaro1,∗, James D. Biggs2,∗

Advanced Space Concepts Laboratory, University of Strathclyde, 75 Montrose Street, G1 1XJ, Glasgow, United Kingdom

Abstract

One of the paramount stepping stones towards NASA’s long-term goal of undertaking human missions toMars is the exploration of the Martian moons. Since a precursor mission to Phobos would be easier thanlanding on Mars itself, NASA is targeting this moon for future exploration, and ESA has also announcedPhootprint as a candidate Phobos sample-and-return mission. Orbital dynamics around small planetarysatellites are particularly complex because many strong perturbations are involved, and the classical circularrestricted three-body problem (R3BP) does not provide an accurate approximation to describe the system’sdynamics. Phobos is a special case, since the combination of a small mass-ratio and length-scale meansthat the sphere-of-influence of the moon moves very close to its surface. Thus, an accurate nonlinearmodel of a spacecraft’s motion in the vicinity of this moon must consider the additional perturbations dueto the orbital eccentricity and the complete gravity field of Phobos, which is far from a spherical-shapedbody, and it is incorporated into an elliptic R3BP using the gravity harmonics series-expansion (ER3BP-GH). In this paper, a showcase of various classes of non-keplerian orbits are identified and a number ofpotential mission applications in the Mars-Phobos system are proposed: these results could be exploited inupcoming unmanned missions targeting the exploration of this Martian moon. These applications include:low-thrust hovering and orbits around Phobos for close-range observations; the dynamical substitutes ofperiodic and quasi-periodic Libration Point Orbits in the ER3BP-GH to enable unique low-cost operationsfor space missions in the proximity of Phobos; their manifold structure for high-performance landing/take-off maneuvers to and from Phobos’ surface and for transfers from and to Martian orbits; Quasi-SatelliteOrbits for long-period station-keeping and maintenance. In particular, these orbits could exploit Phobos’occulting bulk and shadowing wake as a passive radiation shield during future manned flights to Mars toreduce human exposure to radiation, and the latter orbits can be used as an orbital garage, requiring noorbital maintenance, where a spacecraft could make planned pit-stops during a round-trip mission to Mars.

Keywords: Phobos, Mars manned mission, Libration Point orbits, Quasi-Satellite orbits, ArtificialEquilibrium points, radiation shielding2010 MSC: 70F15, 70G60, 70Kxx, 37D05PACS: 91.10.Sp, 95.55.Pe, 96.30.Hf, 95.10.Ce, 45.50.Pk, 98.70.Sa

IThis paper was presented during the 65th InternationalAstronautical Congress, Toronto, Canada.

∗Corresponding authorEmail addresses: mattia.zamaro@strath.ac.uk

(Mattia Zamaro), james.biggs@strath.ac.uk (James D.Biggs)

1+44 (0) 141 574 50252+44 (0) 141 574 5011

List of Acronyms

2B Two-Body2B-P Perturbed Two-Body3B Three-Body2BP Two-Body Problem3BP Three-Body Problem

Preprint submitted to Acta Astronautica December 6, 2015

AEP Artificial Equilibrium PointBCBF Body-Centered Body-FixedCR3BP Circular Restricted Three-Body

ProblemCR3BP-CA Circular Restricted Three-Body

Problem with Constant Acceler-ation

CR3BP-GH Circular Restricted Three-BodyProblem with Gravity Harmonics

ER3BP Elliptic Restricted Three-BodyProblem

ER3BP-GH Elliptic Restricted Three-BodyProblem with Gravity Harmonics

Ef.D. Effective DoseGCR Galactic Cosmic RayGH Gravity HarmonicGNC Guidance, Navigation and Con-

trolIM Invariant ManifoldLP Libration PointLPO Libration Point OrbitMRP Mars Radiation PressureOE Orbital ElementPO Periodic OrbitPRP Phobos Radiation PressureQPO Quasi-Periodic OrbitQSO Quasi-Satellite OrbitR3BP Restricted Three-Body ProblemR4BP Restricted Four-Body ProblemSEP Solar Electric PropulsionSEPE Solar Energetic Particle EventSOI Sphere of InfluenceSRP Solar Radiation PressureSS Sun-SynchronousVDCO Vertical-Displaced Circular Orbit

1. Introduction

Since the discovery of Phobos and Deimos in1877, the two natural satellites of Mars have be-come increasingly interesting astronomical objectsto investigate. Phobos is closer to Mars thanDeimos and almost double its size, but despite this,they are very similar, since they share commonphysical, orbital and geometrical features. Theirorigin is still largely unknown [1, 2], and is cur-rently debated to have been either an asteroid cap-ture by Mars, or coalescence from proto-Mars orSolar System material, or even accretion of mate-rial from Mars ejected from its surface after an im-pact with a previous small body. This puzzle is

supported by the mysterious composition of thesemoons inferred from infrared spectral analysis: dueto their relative low density and high porosity, theycould hide a considerable amount of iced water [2],which is an attractive in-situ resource that couldbe exploited by human missions. In addition, itis speculated that the Martian moons’ rocks couldprovide evidence of alien life [3]. Phobos has someunique characteristics that make it also astrody-namically interesting. The low altitude of its orbitaround Mars has produced speculation on its evo-lution: due to its tidal interaction with Mars, itsaltitude is currently decreasing, which means Pho-bos will eventually crash into Mars or break up intoa planetary ring [4].

Due to its proximity to Mars, Phobos is cur-rently of great interest for future missions to theRed Planet. During its Ministerial Council Meetingof November 2012, ESA confirmed post-2018 mis-sion concepts: the Mars Robotic Exploration Pro-gramme would include a mission (Phootprint) toreturn back to Earth a sample from Phobos [5, 6].Another sample-and-return mission to this moon iscurrently proposed by NASA Innovative AdvancedConcepts team, that will use two CubeSats pro-pelled by a solar sail and joined by a tether mech-anism [7]. In addition, NASA has identified a mis-sion to Phobos as a key milestone to be achievedbefore bringing humans to Mars [8, 9, 10, 11], sincethe absence of atmosphere on Phobos and Deimosmakes landing and take-off easier for a mannedspacecraft than on Mars. Therefore, the Mar-tian moons could be exploited as outposts for as-tronauts: Phobos’ proximity and fast orbital pe-riod can provide a relay for robotic exploration onMars, and protection from space radiation hazardsfor manned spacecraft orbiting Mars (Phobos’ bulkand shadow shielding the spacecraft). At the begin-ning of 2013, with the development of a new roverplatform for the exploration of minor bodies, con-sisting of robotic hedgehogs, it has been reportedthat NASA is taking into consideration a mission(Surveyor) to Phobos as a test-bed for this newtechnology [12].

The purpose of this paper is to present a break-down of different kinds of orbits that could be ex-ploited in future space missions to Phobos. In [13],a collection of different options that a spacecraftcan use to orbit around Phobos is discussed, andtheir properties are preliminarily assessed in theframework of the three-body dynamics. This paperbuilds on this approach by expanding the compu-

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tation to all the available orbits within each classof trajectories, refined in more tailored models ofthe relative dynamics, and by computing a broadrange of their properties for space engineering ap-plications. These results will become useful for theultimate selection of the operative orbits in the mis-sion design loop to plan the exploration of Phobos.

The outline of this paper is as follows. Section 2introduces the reader to the physical environmentconnected to the orbital dynamics and constraintsof a spacecraft in the vicinity of Phobos. The fol-lowing sections 3-6 showcase each of the differentkinds of orbits around this moon, such as: hoveringpoints using Solar Electric Propulsion (SEP); Verti-cal Displaced Circular Orbits with low-thrust; natu-ral Libration Point Orbits and their Invariant Man-ifold trajectories, and their artificial equivalent withconstant low-thrust; Quasi-Satellite Orbits aroundPhobos. Section 7 provides a summary of the differ-ent solutions focusing on their applications in spacemissions to Phobos. Section 8 concludes the papersuggesting their potential usefulness in a real-worldmission scenario.

2. Preliminary Analysis for a Space Missionaround Phobos

In this section we introduce the basic design as-pects of the dynamics and physics of a spacecraftin orbit of Phobos.

2.1. Physical and Astrodynamical Characteristics

The immediately noticeable characteristics ofPhobos are its small size (even smaller than someasteroids) and its irregular shape: in particular thesurface is marked by a dense texture of grooves andby several big craters, one of them, named Stick-ney, is by far the largest and is located on the faceof the moon pointing towards Mars. Phobos has analmost circular and equatorial orbit around Mars,and it rotates with synchronous period and almostzero-tilt with respect to its orbital motion. Thelow altitude of its orbit is lower than that requiredfor a Mars-synchronous rotation: Phobos rises fromthe West more than three times a day as seen fromMars’ surface near its Equator, whereas near thepoles Phobos is never seen, since it is always un-der the horizon. Table 1 presents a summary of thephysical and orbital parameters of Mars and Pho-bos that have been used in the analysis of the orbitsundertaken in this paper.

Table 1: Physical and astrodynamical properties. Thevalues are retrieved from the following sources. Ephemerides:NASA JPL at 25th July 2012 00.00CT (ICRF/J2000.0).Phobos axial tilt θ at date: mean value at epoch 1.08.Gravity models: Mars MGM1025, Phobos [14].

Property Mars Phobosm [kg] 6.42 · 1023 1.07 · 1016

Size [km] R mean sphere R mean ellipsoid3.39 · 103 13.1× 11.1× 9.3

Revolution T 687d 7.65hRotation T 24.6h 7.65hθ [] 25.19 0.30Gravity Field GHs GHsJ2 0.00196 0.105J2,2 0.0000631 0.0147J3 0.0000315 0.00775J4 0.0000154 0.0229Orbital Elements Sun-Ecliptic Mars-Equatoriala [km] 2.28 · 108 9.38 · 103

e 0.0934 0.0156i [] 1.85 1.07

2.2. Relative Dynamics

The general equations of motion of the relativeorbital dynamics that will be used to describe thedifferent kinds of orbits presented in this paper arestated in Eq.1,

q = −aA + aG + aP + aC + aD (1)

aA = aA,T + ω ∧ ω ∧ q + ω ∧ q + 2ω ∧ q (2)

where q is the position of the spacecraft and aA

is the apparent acceleration of the general relativeframe of reference. aA is presented in Eq.2 as afunction of the frame’s translational accelerationaA,T and angular velocity ω with respect to an in-ertial reference; aG is the sum of the gravity ac-celerations of the celestial bodies of interest, eachdefined as the gradient of the gravitational poten-tial uG,⊕ = Gm⊕/||q− q⊕||, where G is the grav-itational constant, m⊕ and q⊕ are the mass andposition of the body ⊕; aP indicates the thrustingacceleration of the propulsion system of the space-craft required for artificial orbits, while for naturalorbits aP = 0. These three terms constitute themodel of the dynamics where the reference signalof the orbit over time q(t) is solved, to be used bythe guidance system in the mission. This motionwill be perturbed in the real world by the distur-bance aD, consisting of the forces not considered inthe model, and by the perturbations on the initialcondition q0, due to the inaccuracies of the navi-gation system; to track the guidance law, such per-

3

turbations need to be counteracted by the station-keeping action aC of the orbital control system ofthe spacecraft, either planned in feedforward or per-formed in feedback.

The study of the dynamics of a spacecraft aboutPhobos is conducted in the first instance with themodel of the classical circular restricted three-bodyproblem (CR3BP) [15], where the two massive bod-ies are Mars (1) and Phobos (2), and the frame ofreference is centered in the Mars-Phobos barycen-ter and aligned with the rotating Hill’s frame of theMars-Phobos orbit, which is considered keplerian,circular, and equatorial. The equations of motion ofthe CR3BP are derived from Eq.1 using a constantvertical ω corresponding to the Mars-Phobos meanmotion, and considering the gravity aG1 and aG2

of the two massive bodies placed in fixed positionin this frame’s choice. Using non-dimensional units,the only parameter of the CR3BP is the mass factorµ, the normalized mass of the secondary body withrespect to the total mass of the primaries, while thesemi-major axis of their orbit provides the lengthunit l3. For the case of Mars and Phobos, two pe-culiarities are evident from the physical parametersof Table 1: the mass parameter of the system is verysmall (µ = 1.66 · 10−8), if compared to other casesstudied so far in the Solar System, and the lengthunit of the system is very small too since the alti-tude of Phobos’ orbit is less than twice the radiusR1 of Mars (R1/l = 36%), an unusual condition fora pair of primaries in our Solar System.

The Hill’s sphere of influence (SOI) is the regionaround each body where the dynamics is dominatedby its own gravity field, and its radius for Phobos is0.17% of the distance from Mars, and consideringthe fact that Phobos is very irregular in shape therelated maximum altitude is only 3.5km, thereforeit is impossible to naturally orbit around Phoboswith a Keplerian motion, as shown in Fig.1. Inparticular the microgravity environment of Phobosis characterized by a keplerian escape velocity at itsmean surface of only 11m/s, which means a humanbeing (or a rover) could auto-inject itself out of thebody with a very small force.

The collapse of the realm of attraction of the sec-ondary body of the CR3BP towards its surface is aresult of the two peculiarities of the Mars-Phobossystem, which has some additional physical andorbital features: not only the revolution of Pho-

3x,y,z axes in all the figures are normalized in l unitsunless otherwise indicated.

Figure 1: Hill’s surface for L2 energy. x-z projection.Phobos mean sphere (dashed line) and ellipsoid (plain line).

Figure 2: Misalignment angle between the equatorialand orbital planes of the Martian moons. Mean valuesdotted.

Figure 3: Differential perturbations analysis. Verticaldotted lines indicate Phobos major size and Hill’s SOI radius.

4

bos around Mars and the rotation around its spinaxis are synchronous, but they are also respectivelyequatorial and zero-tilted [16], therefore Phobos’ at-titude, expressed by the body-centered body-fixedreference frame BCBF, is approximately fixed inthe rotating frame of the CR3BP, and so they dif-fer only by the definition of the Prime Meridian[17, 16]. Before this choice, the actual misalignmentbetween the two frames oscillates between a mini-mum of 0.30 and a maximum of 1.90, and the dy-namics of this libration motion is much slower thanthe time-scale of a mission segment around Phobos(period of 2.26 terrestrial years), as presented inFig.2.

2.3. Orbital Perturbations

An analysis is undertaken to quantify approxi-mately the errors that occur in the Mars-Phobossystem when it is approximated with a CR3BP.The major physical orbital perturbations are dis-tinguished by gravitational and non-gravitationalforces. In the first class, we have the net termthat when added to the basic newtonian point-massforce provides the true gravity pull of a general non-spherical and non-uniform body, which is usuallymodeled with a spherical harmonics series expan-sion known as gravity harmonics (GHs): for a firstanalysis, we consider the dominant term for bothMars and Phobos gravity fields, which is known asJ2 and is related to the oblateness of the body; thesecond type of gravitational perturbation is the ba-sic gravitational term of additional bodies, that inthe framework of the 3BP is referred as a fourthbody perturbation, which is the sum of the bodygravity and its apparent force on the 3BP frame: fora basic analysis, we consider the perturbing bodyin the closest conjunction configuration with Pho-bos. In the second class, we have the pressure dis-turbances of the atmospheric drag and the electro-magnetic radiation: Phobos does not have an at-mosphere, and the atmosphere of Mars is negligibleat Phobos’ altitude, therefore we consider the radi-ation pressures of the Sun (SRP), Mars (MRP, en-closing also the portion of the albedo of the SRP),and Phobos (PRP, with its albedo). These pertur-bations require knowledge of additional technicalparameters of the structure and the subsystems ofthe spacecraft, so for a first analysis we considercommon mean values for them. In addition to thesephysical actions, we must consider the modelingperturbations represented by the approximation of

the dynamics as a circular R3BP, which is the ef-fect of the eccentricity of the Mars-Phobos orbit. Toderive a related physical acceleration value, we con-sider the difference between the acceleration field ofthe CR3BP and the one of the elliptic R3BP of theMars-Phobos system, where ω and the primaries’positions are variable, and we take the maximumover the orbital phases of Phobos, with the refer-ence of the same relative state with respect to Pho-bos.

An important point to consider is that Phobos’orbit around Mars is not keplerian, but it seems tofollow a classical low altitude J2 perturbed orbit.Therefore, the analysis of every orbital perturbationwhich is not due to Phobos (gravity and PRP) inits proximity must be conducted in the frameworkof the relative dynamics with respect to Phobos,considering the resulting differential perturbation.

Since this analysis is undertaken to derive a ba-sic reference for the orbital perturbations, it will beapplied for the simple case of fixed relative points.Also, as we are interested in the dynamics nearPhobos, due to the small size of its SOI, all theperturbations are nearly isotropic, and the onlyvariable for this simple analysis is the radial dis-tance from Phobos along the Mars-Phobos direc-tion (apart from the eccentricity, which has beenevaluated along all three directions). Outcomesof the differential analysis are presented in Fig.3,where the perturbations are shown as a ratio aP/a2

with respect to Phobos’ keplerian gravity term inthe point, and they correspond to [18]. In conclu-sion, the CR3BP does not provide an accurate ap-proximation to describe the Mars-Phobos system’sdynamics: the gravity harmonics and the orbital ec-centricity of Phobos are the main orbital perturba-tions in proximity of the moon, and outside its Hill’sSOI boundary the eccentricity becomes the domi-nant term, with Mars J2 being the second largest.

2.4. Radiation Environment

The microgravity that characterizes the space en-vironment has important implications on the space-craft structure and subsystems design, as well as forhuman crew physiological and biological effects. Inparticular, the ionizing part of the space radiationin the Solar System, which is not shielded by theatmosphere and magnetic field as it happens hereon the Earth surface, is currently considered themost challenging engineering aspect in designing asafe manned mission in deep space [19].

The Sun’s activity is variable and constituted

5

by gradual radiative and particle production andby impulsive particle events, the latter emission iscollectively called Solar Energetic Particle Events(SEPEs): they are high-energy charged particlesand they constitute the hazardous ionizing part forthe organic tissues. The low-energy charged grad-ual particles (mainly protons and electrons) consti-tute a plasma flow, called the solar wind. Its inter-action with the magnetic field of a planet producesthe so-called Magnetosphere: this is a region wherethe charged particles remain trapped by the mag-netic field lines, known as radiation belts, but de-spite they are hazardous when crossed, they providethe natural shield that protects life on the Earth’ssurface (in combination with the ozone layer coun-teracting the UV rays), satellites in LEO, and inparticular the crew of the ISS, from outer space ra-diation. In addition to the Sun, there is a secondvery important radiation source in the Solar Systemknown as Galactic Cosmic Rays (GCRs). These aregradual high-energy charged particles that originatefrom interstellar space.

Focusing now on the Mars-Phobos system, theMars magnetic field is very weak, so no trappedparticles (and related shielding) constitute the radi-ation environment for a mission following the orbitof Phobos, which is similar to a deep space environ-ment at the Sun-Mars distance, constituted by twomain sources: the protons from SEPEs and protonsand alpha particles from GCRs. For applications tofuture manned mission to Phobos’ orbital environ-ment, we conducted an estimation analysis with theopen-source SPENVIS program [20] and its dedi-cated model for Mars MEREM [21]. To derive anapproximated figure of the gross effect of the ra-diation environment (without any shielding effectof the spacecraft structure) to human factors, weconsider the dosimetry quantity called the Effec-tive Dose (Ef.D., whose IS unit is the Sievert, Sv),which represents the amount of energy that the ra-diation deposits in 1kg of the material’s mass, aver-aged through both the incoming radiation and thereference tissue compositions.

The result obtained for the gross radiation haz-ards for a mission in Phobos’ orbit from 2010 to2030 is Ef.D.= 1.9Sv/y, 1.1Sv/y from SEPEs pro-tons and 0.8Sv/y from GCRs protons and alphaparticles. This should now be compared with theestimated allowable dose amount for astronauts,which is based on the recommendations of the Na-tional Council for Radiological Protection (NCRP)and is currently used by both NASA and ESA [22].

Considering the case of a 35-year old astronaut,the figure derived from our analysis, for a Mar-tian orbital segment of one year without any struc-tural shielding, falls inside the range 1.75-2.5Svthat indicates the maximum amount of radiationdose that such human crew could be allowed to ab-sorb throughout the entire mission. Thus, the de-velopment of a strong shielding strategy for crewedmissions is required. An interesting idea that hasrecently gained attention, is that a manned space-craft during a Mars orbital mission segment couldexploit Phobos as a passive radiation shield: stay-ing in its shadowing wake would theoretically coun-teract the gross Ef.D. of the directional part of theSEPEs, while simply remaining close to the moonwill block any incoming isotropic particles (remain-ing SEPEs and GCRs) as much as its bulk coversthe sky. Following this idea, in this paper we inves-tigate possible orbits to be used also for shieldingpurposes about Phobos. However, some additionalalbedo effects (of the GCRs neutrons) could be rel-evant, and also covering the field of view of the Sunhas not been proved to be relevant, as the scatteringeffect of the particles along the Heliosphere’s linesof field is still a current topic of research [23].

2.5. Lighting Conditions

In this subsection we analyze and quantify thelighting conditions around Phobos, in particular wedescribe the shadowing opportunities that could ex-ploit Phobos as a natural shield against the direc-tional solar radiation.

Since we are interested in Phobos’ neighborhood,the analysis is conducted in the CR3BP referenceframe but centered in Phobos, aligned with itsBCBF frame. The kinematics must now consideralso the motion of the Sun in this frame, using arestricted four-body model to evaluate the field ofview of the Sun over time for points around Phobos.In particular, the Mars heliocentric orbit is the sec-ond most elliptic among the Solar System planets,and Phobos’ orbit is equatorial, with the resultingorbital plane inclined with respect to its eclipticplane by Mars’ rotational tilt θM = 25.19. Theresulting Sun adimensionalized position vector inthe CR3BP frame rotates clockwise with an angu-lar velocity equal to the difference between Phobosand Mars revolution rates (dominated by the first),with a fixed declination in the range [−θM , θM ] ac-cording to the seasonal phase of Mars (seasons ofPhobos correspond chronologically with Mars’ ones,so we refer to them without any distinction).

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Figure 4: Sun-Mars eclipse at Phobos. Eclipse times aredefined for a single eclipse during one Phobos’ rotation.

Figure 5: Sun-Phobos eclipse. Light function of the dual-cone model. CR3BP frame, Sun in superior conjunction.

The analysis of the shadowing effects in thissystem is undertaken using eclipse modeling, whichis to derive the zones of light and shadow producedby a shadowing central body ⊕ when illuminatedby a radiating body , described by a scalar lightfunction field L,⊕, ranging from 0 to 1 to expressthe ratio of incident light with respect to the com-plete light case (the shadow function S is the 1-complement of L). The most accurate dual-conemodel in Fig.5 is able to discriminate positions ofcomplete light (L = 1), complete shadow (L = 0)inside a conic wake, and penumbra (L ∈ (0, 1)).When the positions of interest are very close to ⊕and when there is a great difference between the twobodies dimensions, the analysis could be simplifiedto a cylindric model, where the shadowing wake iscylindrical and the transition zones of penumbra

Figure 6: Sun-Phobos eclipse. Mean light function versusdistance from Phobos on the anti-Sun surface of motion.

vanish, so positions are only in complete light orshadow.

The approach we used is to analyze the shadow-ing effect of each couple of bodies. The first caseis the Sun-Mars couple. Here we are interested inthe value of L at the Phobos location. Using thedual-cone model the Phobos SOI is a near and smalldomain in the Mars shadowing wake, therefore theanalysis of this couple is undertaken with the cylin-dric model. The Sun-Mars L1 over time at Phoboshas a small-period variation due to the fast revo-lution of Phobos, and a long-period variation dueto Mars’ revolution; the latter motion inclines theMars shadow cone with respect to the 3BP orbitalplane, such that during winter and summer Pho-bos is constantly in light, without Martian eclipses:due to Mars high eccentricity, seasons are unequal,with the Northern Hemisphere of Mars and Pho-bos experiencing a summer longer than that oc-curring in the South. Fig.4 summarizes the out-comes: the maximum eclipse time at the equinoxesis 54min, corresponding to 12% of Phobos daytime;summer’s total light period is about 164 days (3Martian months), and in the winter this is about110 days (2 months).

The second case is the shadowing effect pro-vided by the Sun-Phobos couple. For this prelimi-nary analysis, the mean spherical shape of Phobosis considered. Like for the Sun-Mars L, the cylin-dric approximation is suitable in proximity of Pho-bos, but now the Sun-Phobos L2 is a time-variant3D field: since this is axially symmetric along theirrotating conjunction line, it is defined on the half-plane whose reference axis is such line. The mean

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Figure 7: Sun-Mars-Phobos eclipse. Anti-Sun daily orbitin spherical coordinates around Phobos over the Martianyear.

integral value L2 along one Phobos’ revolution pe-riod, for a given distance to Phobos, is minimumon the surface of motion where the conjunction linebetween the anti-Sun and Phobos revolves, shortlybecoming 1 in points out of the surface: such mini-mum value rapidly increases with the distance fromPhobos. This is shown in Fig.6: from L = 50% atthe body’s surface, L = 78% at the SOI’s boundary,L = 83% at 2 Phobos radii.

The last shadowing case is the Mars-Phobos cou-ple. Since the radiation of Mars (without thealbedo) is inside the IR spectra, such eclipse analy-sis is neglected because it brings little variation tothe radiation hazards that we aim to reduce (MRPflux at Phobos is two orders of magnitude lowerthan SRP one).

The conclusion of the analysis of the shadowingeffects is now obtained combining the previous sin-gle couples into the system of three massive bodiesof the R4BP, focusing the analysis in Phobos’ neigh-borhood. The Sun-Mars L1 is simply a scalar valuealong the Martian year, while the Sun-Phobos L2

field must now consider the real dynamics of theSun, which is moving in the CR3BP frame of refer-ence (actually we are going to consider the directionof the anti-Sun ′ because it is more immediate torelate it with the position of the shadowing wake ofPhobos). Fig.7 shows a complete orbit of the anti-Sun for every season: the cylindrical shadowingwake of Phobos revolves around its spin axis withthe period of Phobos, and varies its inclination withthe season θ′ ∈ [−θM , θM ] (when the anti-Sun isin the Northern Hemisphere, we are in winter). Be-

sides, Phobos realm is in complete shadow when aMartian eclipse occurs, which is when the anti-Sunis close to the positive direction of the 3BP x-axisframe, enduring from a maximum at the equinoxesto zero during summer and winter.

The approach taken to compute the light func-tion of the R4BP model in proximity of Phobosfollows the following procedure. First we computethe Sun-Phobos mean integral field L2 along onePhobos’ revolution for different seasons. Due tothe axial-symmetry of L, the desired integration ofa 3D field along time is simplified to an uncoupled1D integral in polar coordinates, and it is also sym-metrical along the spin axis. For the particular caseof a cylindrical light function, the analytical solu-tion is derived in Eq.4, where r and ϕ are the radialdistance and declination over the equatorial planeof the point considered [24], and R is Phobos’ ra-dius.

γ = π2 − θ′

θ = ϕ− θ′

α (R/r, θ) = Re

arccos(

cos arcsin(R/r)sin(γ−θ) sin γ −

1tan(γ−θ) tan γ

)(3)

L2 (r, ϕ) = 1− 1π |α (R/r, θ)− α (0, θ)| (4)

Further averaging the daily L2 along the seasons ofthe Martian year is then straightforward.

But before doing this, the second step is toextend the Sun-Phobos daily L2 to take into con-sideration the correction due to the Sun-Mars L1

at the current day previously derived, to obtain theaimed Sun-Mars-Phobos mean 3D light field L12

that models a coupled 3B eclipse. This is far froman easy operation, but since both 2B light fields arecylindrical, and their shadowing wakes have verydifferent dimensions, the instantaneous combinedL12 around Phobos is the logical conjunction ∧ ofthe two single instantaneous light functions, whichresults in their product: L12 = L1 ∧ L2 = L1L2.This allows us to compute the sought mean integralfield L12 over one Phobos’ revolution with only theinformation of the two single daily values L1 andL2, with the deal of introducing a further variableψ, which is the right ascension with respect to Pho-bos of the point analyzed. This is described in Eq.5.L2 is axially symmetric along the spin axis, beingthe same for points with same ϕ: they have thesame profile L2(t), but shifted along time for differ-ent ψ. Therefore the integral of the product L1L2

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Figure 8: Lighting conditions around Phobos. On the top, light field around Phobos in the radial-vertical plane of theCR3BP frame, averaged over a year, a spring equinox month, a summer solstice month. On the bottom, corresponding plotsfunction of the declination, evaluated for some radial distances, show also right ascension dependency, where upper/lowerborder of the filled area is for points in superior/inferior conjunction positions, and black line is for points in quadrature.

produces a different L12, expressed by the correc-tion term ∆t/T which is the percentage of time inone Phobos’ revolution period T such that L1 is inshadow and L2 is in light. This correction will bemaximum for points facing Mars and minimum forthose on the other side as presented in Eq.6, whilethe values in quadrature become closer to the max-imum as far as the radial distance increases.

L12 (r, ϕ, ψ) = 1T

∫ T0L1 (t)L2 (r, ϕ, ψ, t) dt =

= L2 (r, ϕ)− ∆tL1=0∧L2=1

T

(L1, L2, ψ

)(5)

minψL12 = L12 (r, ϕ, π) = L2 (r, ϕ)−min

(S1, 1− S2 (r, ϕ)

)maxψ

L12 = L12 (r, ϕ, 0) = L2 (r, ϕ)−max(0, S1 − S2 (r, ϕ)

)(6)

To interpret the results in Fig.8 we should distin-guish seasonal and yearly averaging. The seasonaltilt inclines the shadow wake, so complete light andone cone of complete shadow appear in the Pho-bos polar regions: the cone’s maximum altitude,using a mean ellipsoidal model for Phobos shape, isof 1.4km. Instead during spring and fall, no com-plete shadow zones are present, and the minimaldaily L at the day of equinoxes is 38% along thesub-Mars meridian onto the surface of the moon

(this without considering its orography and mor-phology). Considering now the annual L, yearlyaveraging drastically increases the light conditions.Due to Mars’ eccentricity, Southern regions experi-ence more shadow time than the upper counterpart;due to the Martian eclipses, points in-between Marsand Phobos and close to the moon experience moreshadow time per annum.

In conclusion, this analysis provides the lightingconditions for a spacecraft orbiting Phobos. Focus-ing on the shadowing opportunities, a fixed observa-tion point in the 3BP frame could provide relevantreduction of the field of view of the Sun for long-period station-keeping only if it is inside the SOIand onto the equatorial plane, and pointing Mars;instead, a shorter period could provide continuousshadowing opportunities for points over the polesduring the solstice seasons, in particular inside theSouthern polar cone during summer. Instead, formiddle seasons the minimum of the light field movestowards lower latitudes, and relevant reduction ofthe Sun field of view is obtained only very close tothe surface. Therefore during equinoctial or longobservation periods, the lighting conditions aroundPhobos are close to experience continuous light, upto 88% due to the unavoidable Martian eclipses.Suitable shadowing exploitation could be obtained

9

Figure 9: Sky occultation around Phobos. Occulta-tion field around Phobos in the radial-vertical plane of theCR3BP frame.

only using orbits that track the daily anti-Sun path,such a vertical displaced circular orbit around thespin axis. We acknowledge that a complete shad-owing is not possible with current technologies, be-cause a spacecraft requires sunlight for the electricalpower generation from the solar arrays, with the so-lar flux decreasing with the distance from the Sun.

2.6. Sky Occultation

In this subsection we consider the possible ex-ploitation of Phobos as a natural shield against theisotropic cosmic rays (SEPEs and GCRs): the ideais that the incoming radiation on a spacecraft islowered proportionally to the filling fraction in thesky of the apparent size of the body’s bulk, as seenby the spacecraft’s location. In astronomy, when abody is totally or partially hidden by the bulk of an-other one that passes between it and the observer,we speak about occultations or transits. Since inthis case the hidden body is the total backgroundsky, in this paper we refer to this action as sky oc-cultation.

A point-like observer sees an object with an ap-parent shape that corresponds to the area that itcovers on a sphere centered on the observer, andradius equal to their distance. In 3D geometry,the area subtends the 2D solid angle Ω on the unitsphere (whose IS unit is the steradian, sr), and thetotal spherical surface has Ω = 4πsr. For a general

body placed at distance d from the observer,

Ω =

∫∫ 2π,π

0,0

M(ϑ, φ) sinϑdϑdφ (7)

where M is the mask function of the body. This isa binary function of the polar and azimuthal spher-ical coordinates ϑ and φ centered on the observer,whose value is 1 or 0 if the related direction from theobserver intersects or not the body: it representsthe apparent shape of the body on the unit sphere.The ratio with 4πsr represents the filling fractionof the body with respect to the background.

The occulting bodies in our case are Mars andPhobos, while the Sun is neglected because it isvery small as seen from Phobos. The approach issimilar to the one undertaken for the lighting con-ditions, defining an occultation function field O⊕which represents the bulk/sky filling fraction of theocculting body ⊕. This analysis is easier becausein the CR3BP frame Mars and Phobos are fixed,and their O does not depend on time. For a firstanalysis, we consider the mean spherical shapes forthe two bodies, of radius R. In this case, the oc-culting body fills a spherical cap on the unit sphere,with apparent angular radius α = arcsin(R/d), andO = 1−cosα

2 , which is spherically-symmetric. Firstwe consider the occultation of Mars, at the Phoboslocation, since the region of interest is small: the re-sult is O1 = 3.4%. Second we consider the occulta-tion of Phobos, which depends only from the radialdistance from the body. This function starts fromO2 = 50% on the surface (astronauts staying insideof a deep crater would be shielded also laterally bythe mountain ridge), and then decreases rapidly:O2 = 13% at the SOI’s boundary, O2 = 7% at 2Phobos radii. The conclusion of the analysis of thesky occultation is obtained combining the previoussingle effects. This requires to determine if the ap-parent shapes of the two bodies’ bulks intersect, andhow much they overlap: such axially-symmetric 3Dfield corresponds to the light function of the Mars-Phobos couple L1,2 that we avoided to compute inthe previous lighting conditions analysis, but we doneed here. In particular, this light function must becomputed with the accurate dual-cone model, sincedue to the proximity of Mars, the shadow cone ver-tex of Phobos is located only at 2.77 Phobos radiiin the anti-Mars direction, therefore its inclinationinside the SOI is not negligible. The resulting 2Bcombined occultation function is O12 = O2+L1,2O1

and it is shown in Fig.9.

10

Figure 10: AEPs of the Mars-Phobos system. Iso-surfaces slices of propulsive acceleration magnitude (loga-rithmic scale).

This analysis highlighted that mild but relevantreduction of the isotropic SEPEs and GCRs by us-ing the bulk of Phobos to occult part of the celestialsphere is obtained inside the SOI of the moon. Be-sides, points on Mars’ side and over the poles expe-rience an additional but small reduction due to theoccultation of Mars. Orbits that remain inside thePhobos’ SOI are therefore suitable to enhance theradiation protection of the spacecraft by exploitingPhobos’ bulk as a passive radiation shield.

3. Hovering Points around Phobos

A simple trajectory for a mission around Phobosis provided by maintaining a fixed position with re-spect to its BCBF frame: due to the small µ, thisis similar to a Martian keplerian orbit close to Pho-bos, analogous to the Trailing/Leading configura-tions used in Formation Flying [25]. The analysisof these trajectories is undertaken adding a con-stant propulsive acceleration aP to the equationsof motion of the CR3BP, which will be referredCR3BP-CA. The aim of the hovering in a point istherefore to counteract the natural acceleration ofthe CR3BP, and thus leading to an Artificial Equi-librium Point (AEP).

aP = aA − aG (8)

Recall from [25] that SEP Hall/ion thrusters op-erate roughly in the medium range of 0.01mN -0.1N , new generation FEEP and colloid thrustersprovide low-thrust down to 1µN scale, and elec-

trothermal propulsion (resistojets and arcjets) sup-plies the higher ranges up to 1N , but the levels ofpropulsive acceleration must be scaled accordinglyto the mass of the spacecraft (100kg for a medium-size interplanetary satellite, 100 times larger fora big manned module). Fig.10 presents the iso-surfaces of the thrusting acceleration level requiredto hover around Phobos: as far as the propulsiongrows, AEPs could be further displaced from thenatural equilibrium points of the CR3BP, which arethe three collinear libration points (LPs) alignedwith the two bodies (two close to the secondary, L1

in inferior and L2 in superior conjunction, and L3 inopposition), and the two equilateral LPs L4 and L5

equidistant from them in the two quadrature config-urations; in particular, the cis/trans couple of LPsL1−2 is located on the boundary of the Hill’s SOI ofthe secondary, at an altitude of 3.5km from Phobos.Despite this proximity, the thrust level required toestablish an AEP displacing a collinear LP is verydemanding. Instead, displacing the equilateral LPsL4−5 is very cheap and effective, arriving close toPhobos along the y-axis still with small values ofthrust; on the other end, establishing AEPs overpolar regions requires high thrust levels.

The next step in the hovering analysis is to lookfor the stability of the AEPs analyzing the lin-earized CR3BP-CA: since the propulsive acceler-ation is constant, the linearized system coincideswith the one of the CR3BP. The 3D stability re-quires the computation of the eigenvalues λ of the6D linearized state-matrix, that contains the 3DHessian matrix Hu of the gravity potential eval-uated at the current AEP. Due to the sparse struc-ture within the state-matrix, it is possible to derivea condensed analytical expression of the three cou-ples of opposite eigenvalues in terms of the threescalar invariants of the hessian matrix (which issymmetric) I1, I2, and I3. To do so, we intro-duce some coefficients: A, B and D are strictly realscalars by definition, instead C is complex. The ex-pression of the three couples of eigenvalues is ex-pressed in a convenient symmetrical form in thecomplex field in Eq.10.

A = I1−23

B =I2+I1+1−Hu3,3

3 −A2

C =3√

2√D2 +B3 +D

D =I3+I2−Hu1,1Hu2,2+H2

u1,2+Hu3,3

2 − 3AB2 −A3

2

(9)

11

Figure 11: AEPs of the Mars-Phobos system. On thetop, planar stability region (in green) around Phobos. Fol-lowing figures show the inner boundary of the 3D stabil-ity region around Phobos (in the last picture, only one halfof the lower hemisphere is represented to visualize Phobosmean sphere).

λ = ±√A+ B

C e(π−θ)i + Ceθi, θ = 0,±2π/3

(10)The linear Lyapunov marginal stability for the

AEP requires all the eigenvalues to be purely imag-inary, which is their squares to be real and neg-ative. To solve these two constraints, we use themagnitude-phase notation for C = |C|eθCi: the firstconstraint is satisfied by one simple relationship be-tween B and C, which is the necessary condition forreal eigenvalues. Using this, Eq.10 could be rewrit-ten in a compact way where the six solutions de-rive from the same definitions of C as a cubic rootof Eq.9. The second stability constraint consists ofone simple inequality, which considers the algebraicroot of C with maximum real part (there is at leastone that is strictly positive-definite).

λ = ±√A+ 2 |C| cos (θC + θ), θ = 0,±2π/3 (11)

B = −|C|2

A+ 2maxRe C < 0(12)

This approach extends the result obtained foronly the planar case of the CR3BP-CA of [26]. Forthe Mars-Phobos system, the 3D stability region ismade of three realms: one central ring, and twosymmetric half hyperbolic coronas placed at veryhigh out-of-plane altitudes. More interesting as pre-sented in Fig.11, the inner surface of the ring isdistorted in proximity to the second massive bodyleaving outside the body’s SOI. The planar sta-bility region in the orbital plane is a thin coronaextending along the Mars-Phobos orbital distance,that comprises the equilateral LPs and cuts off thethree collinear LPs, and in proximity of the sec-ondary body, the inner stability region boundaryis distorted to represent a three-leaves clover: thiscorresponds to the outcome in [27]. Recall thatthis is the linearized stability region, which for thecase of marginal stability encountered by the AEPsdoes not assure stability in the original CR3BP-CA.However, it has been proved by analysis of higherorder terms that the AEPs of the linearized stabilityregion are also stable in the full nonlinear dynamicsapart from singular cases that lie on co-1D domainsof the stability region, where resonance effects arepresent [26].

If we compare the stability region with the equi-thrust curves in the orbital plane, it can be seen

12

that the planar stability boundary is not too farfrom Phobos, starting from 25km (in the petal-headconnection border, requiring 1.9mm/s2), 71km (onthe tip of the top and down leaves, 0.4mm/s2) or81km (along the x-axis, 12mm/s2) and arriving to400km (along the x-axis, where the outer bound-ary lies). Therefore six attractive positions formedium distance observation of Phobos are iden-tified: four minimum-distance AEPs at 25km, andtwo minimum-control AEPs with 0.4mm/s2, all ob-tained by the displacement of the equilateral LPsand affordable by current light electric thrusters.Further trailing/leading orbits around Mars pro-vide attractive cheap, stable, in-light fixed positionswith respect to Phobos at long distances from themoon. All the AEPs available with a low-thrustlevel are not stable near the collinear LPs, andAEPs closer to Phobos, used to maximize the shad-owing time and the sky occultation ratio, or per-form short dedicated operations, are feasible onlywith heavy or multiple thrusters, and must takeinto consideration in the model also the completeinhomogeneous gravity of Phobos: this increasesthe precision, frequency and computational load ofthe GNC subsystem. Regarding the 3D boundary,it is possible to have stable AEPs above the polesof Phobos, but at a great distance (from 250km to1400km) and with high thrust.

4. Vertical-Displaced Circular Orbitsaround Phobos

The AEPs computed in the previous section area fixed solution in the rotating frame of the CR3BP.It is possible to further generalize the concept,which is to look at the dynamics in a general uni-formly rotating frame, and find the related AEPs.In the usual CR3BP, such solution results in a circu-lar orbit around a reference axis, which in our casewill be Phobos’ vertical axis: it is called a VerticalDisplaced Circular Orbit (VDCO). Therefore, if theangular velocity of the VDCO is opposite to the oneof the Phobos revolution, and the declination of theVDCO is opposite to the one of the Sun, a space-craft along this orbit experiences constant light orshadow conditions during a season, realizing a Sun-Synchronous (SS) seasonal orbit whose β-angle per-formance (expressing the mean time in light) is setby the choice of the initial phase along the VDCOwith respect to the Sun. In particular, for contin-uous shadowing (β = 0) the spacecraft tracks theposition of the anti-Sun moving clockwise around

Figure 12: SS-VDCOs around Phobos. AEPs in thex-z plane of the SS-rotating frame. On the top, linearizedstability region for AEPs in the Phobos 2BP (magnificationof the inner section). On the bottom, ∆v for one period forAEPs in the CR3BP.

Phobos: this is expressed in Fig.7. These displacedorbits are usually highly non-keplerian: here, we aregoing to consider the type III orbits of [28], whichhave fixed period, synchronous with the Sun aroundPhobos, and looking to incorporate low-thrust.

The approach is to start from the 2B dynam-ics around Phobos in a rotating frame R, definedby Eq.1 taking ω opposite to the Mars-Phobosmean motion and considering only the gravity ofthe moon. The model is axially symmetric aroundthe vertical axis, therefore it is defined either in thepositive x-z plane of the R frame, or in the polarcounterpart R-δ. The SS-VDCOs are AEPs of thisdynamics maintained by a constant acceleration inthe rotating frame’s components from Eq.8: in caseof complete light or shadow applications, the dec-

13

lination of the AEP must be equal to the seasonalanti-Sun declination, therefore the domain of inter-est is 0 ≤ δ ≤ θM , or 0 ≤ z ≤ xtanθM . However,shadowing is provided not only by strictly point-ing toward the anti-Sun, but remaining just insidethe eclipse wake: using a mean ellipsoidal model forPhobos’ shape, a SS equatorial circular orbit withradius 2.04 Phobos mean radii provides continuousshadow for all the Martian year.

The procedure is the same followed before for thehovering, computing the equi-thrust curves and therelated linear Lyapunov stability region. There isone natural equilibrium point that corresponds tothe keplerian equatorial circular orbit with a SS pe-riod achieved at the distance of R = 2.16 Phobosradii (mean altitude of 11.8km), and no other lo-cal minima of the thrust level. The three realms ofthe linear stability region in Fig.12 are similar tothe ones of the 3D stability region for AEPs of theCR3BP-CA, swapping the Mars-Phobos barycenterand Phobos with Phobos and the keplerian equilib-ria, and without the distortion of a second massivebody. For planar SS circular orbits this means thatorbits with radius in the interval [R, 3

√9/2R] are

linearly stable in the 2B dynamics, while stable SS-VDCOs could be obtained at higher distances withdeclination δ = 35.26 and high thrust.

We now look to the solution of VDCOs aroundPhobos in the framework of the CR3BP. The dy-namics will be described in the same rotating frameR of the VDCO centered in Phobos: the equationsof motion of this model are obtained from the pre-vious one adding the time-variant gravity of Mars.Since its location is not fixed in this frame, and con-sidering that now Phobos moves around Mars, anadditional time-variant apparent acceleration mustbe enabled in Eq.2, aA,T = aG1(0), equal to theMartian gravity at the origin of the R frame. Andfinally, the Phobos revolution affects also the iner-tial reference of the R frame: since the R framemust move clockwise with respect to the PhobosBCBF frame, and now the latter is rotating in theopposite way, Eq.2 must be used with ω = 0. In-deed, in a short-time analysis, the Sun is approxi-mately fixed with respect to an inertial frame cen-tered on Phobos, therefore realizing a SS-VDCOaround Phobos corresponds in maintaining an in-ertial fixed point dragged along the Mars-Phobosorbit.

The system is no longer axially symmetric, butthe required acceleration is periodic so we considerthe maximum level of the thrust profile and its cost

over one period, defined by the ∆v and presentedin Fig.12. Being the problem time-variant, no nat-ural equilibria are available. The max thrust levelfor a 100kg spacecraft at small-medium distancesfrom Phobos is affordable only with high thrusters.The minimum cost still happens in a region close to2 Phobos radii, up to the maximum Sun’s declina-tion at the solstices, but this is large (∆v= 50m/s)making the demand for the propulsion system veryhigh. Besides, a linearized Floquet stability anal-ysis of these SS-VDCOs showed us that they arehighly unstable.

Following the idea of exploiting the shadow ofPhobos to protect a spacecraft from directional so-lar radiation during the orbital station-keeping atthe Mars-Phobos orbital distance, in this section weanalyzed the most simple and straightforward or-bits around Phobos to track the anti-Sun motion,which are the SS-VDCOs. Due to the strong in-fluence of the Mars’ third body perturbation, theSS-VDCOs around Phobos require a huge amountof fuel consumption, which is infeasible over one pe-riod. A SS-VDCO around Mars is located out of itsSOI, while the cost of a SS-VDCO around Deimosat 5 mean radii (24.0km of mean altitude) has re-sulted to be of 16m/s per period, and it requireslighter thrusters.

5. Libration Point Orbits and their InvariantManifolds around Phobos

In the framework of the classical CR3BP [15],around each of the collinear LPs L1 and L2 thereexist a central manifold characterized by familiesof periodic orbits (POs) (the two branches ofplanar and vertical Lyapunov orbits, and the twobranches of Northern and Southern Halo orbits),and quasi-periodic orbits (QPOs) around them(known as Lissajous orbits). These Libration pointorbits (LPOs) are highly unstable and so theirnatural motion needs to be computed with highprecision to provide low cost tracking opportunities[15, 29, 30]. Moreover, these LPOs are separatricesof motion between transit and non-transit orbitsto enter or escape from the SOI of the secondmassive body: the boundary of these tubes is givenby the Invariant Manifolds (IMs) of the LPOsthat provide the energy-efficient trajectories tominimize the fuel consumption of spacecraft forinterplanetary transfer phases.

From our preliminary analysis of the dynamicsin proximity of Phobos in Section 2.2, we found

14

Figure 13: LPOs in the Mars-Phobos ER3BP-GH. Onthe top, the two iso-periodic families of POs around the os-cillating LPs. In the center, the families of 2-tori: A (red),B (green), C (magenta), D (cyan). On the bottom, exampleof 3-tori of different size and width around the LPs: threemedium-size QPOs of family AB (red and green with small-width, orange with high-width) and two high-width QPOsof family C (around the smallest and biggest orbits). Shapeharmonics series expansion for Phobos surface.

that the altitude of the LPs moves very close toPhobos’ irregular surface: in this situation, thedynamical approximation provided by the CR3BPfalls short, and in particular we found in Section2.3 that to describe the natural relative motioninside this moon’s SOI, its highly inhomogeneousgravity field, and its orbital eccentricity must betaken into account. Due to their highly unstablebehavior, measured by the Floquet stability in-dexes, the families of LPOs, computed with theclassical methodologies tailored for the CR3BP,are therefore not reliable for practical applications,because their reference signal tracking will requirea high station-keeping cost.

In [31]-[32] the dynamical substitutes of theLPOs were derived in a more realistic model thatconsiders these two major orbital perturbationsin proximity of Phobos. The modeling of thecomplete gravity field of convex bodies is providedby a spherical harmonics series expansion, knownas gravity harmonics (GHs). From a previouspaper [14] that collects the data obtained throughViking observations, we are provided with amodel of Phobos’ gravity field. The additionof the GHs in the dynamics is particulary suit-able for the Mars-Phobos system because theCR3BP frame and the Phobos BCBF frameare approximately fixed with respect to eachother (see Fig.2), which makes this system sounique to remain time-invariant. The equationsof motion of the Mars-Phobos ER3BP-GH arederived in the Hill’s frame of the Mars-Phobosorbit centered in Phobos, from Eq.1 consider-ing the angular velocity for an elliptical orbitω = [0; 0; 2

√G(m1 +m2)/a3(1 + e cos ν)2/(1− e2)3/2]

(where a and e are the semi-major axis and eccen-tricity of the Mars-Phobos orbit, and ν is the trueanomaly of Phobos), aA,T = aG1(0) like before forthe VDCOs, and defining the gravitational poten-tial of Phobos uG,2 as a truncated series expansionof GHs (J, λ)m,n through Legendre polynomialsPn,m (R is Phobos mean-volume radius):

uG = GmR

∞∑n=0

(Rr

)n+1 n∑m=0

Jn,m cosm (ψ − λn,m)Pmn (cosϑ)

(13)where the potential is defined as a function of thespherical coordinates r,ϑ,ψ of the Phobos BCBFframe, and so the gravity acceleration of Phobos inthe Hill’s frame is retrieved rotating back the com-ponents of the spherical gradient of the potential.

15

Figure 14: LPOs lighting conditions. Light function ofthe families of POs around L1 of the CR3BP-GH (parame-terized by the differential Jacobi integral with respect to L1,in logarithmic scale), averaged over 10 PO periods, at thedays of equinoxes (lower cluster) and solstices (upper clus-ter). Filled area spans values for different starting phases ofthe Sun (thick line is for mean values), where families colorsare coherent with Fig.13.

The procedure in [31]-[32] to compute the LPOsin an ER3BP-GH makes use of the numerical con-tinuation technique, which consists of the iterationof the classical differential corrector to computePOs and QPOs in a single dynamical system. First,we identify LPOs in the Mars-Phobos-spacecraftCR3BP using the classical tools of dynamical sys-tems theory, and then numerically continue a pa-rameter that incrementally increases the GHs ofPhobos, to find their dynamical substitutes in thefinal CR3BP-GH. The introduction of the GHs pro-duces families of POs and QPOs no longer symmet-ric and highly tilted and distorted from the classi-cal case. These new LPOs are then continued againusing the eccentricity as the perturbation parame-ter: despite the fact that the orbital eccentricity ofPhobos is not particularly high, the perturbationhas a significant effect on the LPOs, as an indirecteffect of the long-cited collapse of the SOI towardsthe moon. The effect of the eccentricity makes themotion to oscillate around these solutions with aconsiderable amplitude of 260m for the proximityof Phobos.

The resulting LPOs of this improved dynam-ical model considerably lower the station-keepingdemand exploiting the natural dynamics of the sys-tem. They are showcased in Fig.13: around eachcis/trans-side of Phobos, they are constituted bythe oscillating LPs and a 1-parameter family D ofiso-periodic POs vertically developed with the pe-riod of Phobos revolution around Mars, three 1-parameter families A, B, and CD (made by two

Figure 15: LPOs occulting conditions. Sky occultationfunction by the Phobos’ real bulk (modeled by the shapeharmonics series expansion) of the families of POs around L1

of the CR3BP-GH (parameterized by the differential Jacobiintegral with respect to L1, in logarithmic scale), averagedover 1 PO period. Families colors are coherent with Fig.13.Additional occultation by Mars’ bulk will be 3.4%.

branches C and D) of 2-tori QPOs, and two 2-parameter families AB and C of 3-tori QPOs, allof them very close to the surface of the moon andhighly unstable. Regarding the lighting conditionsand surface coverage, medium-small LPOs are actu-ally similar to close-range hovering points on the cisand trans-side of the moon, and the seasonal andannual light times are approximately the same ofthe ones computed in Section 2.5 (see Fig.8). LargeLPOs can cover also polar and lead/trail-regions,and Fig.14 shows that if used for short operationsthe light time can be tuned accordingly to the Sunphase along Phobos, with the families B and D al-lowing to increase/decrease the mean light functionup to the 15%.

In particular, due to their proximity to Pho-bos, the sky occultation produced by the natu-ral LPOs is relevant, and Fig.15 shows that largerLPOs around both L1−2 can provide passive radia-tion shielding over 20%. This outcome is obtainedusing a high order shape harmonics model [31]-[32]for Phobos’ bulk. Using Mars’ bulk to provide thesame shielding factor would require a low Martianorbit’s altitude under 850km, while LPOs aroundDeimos are too distant from the body to providerelevant natural shielding.

Since the orbits are close to Phobos, no homo-

16

Figure 16: IMs of the LPOs in the Mars-PhobosER3BP-GH. On the top, inside branch of the tube of un-stable IMs from the families of 3-tori LPOs. In the center,related performances of the trajectories that provide the minincidence at the touch-down from the family AB of L1 as afunction of the longitude and latitude of the landing site onPhobos surface modeled through shape harmonics series ex-pansion: landing velocity modulus, angle of incidence, down-ward vertical velocity. On the bottom, performances of thestable IMs of the same family that provide the min velocitytotal magnitude at the launch.

clinic nor heteroclinic connections4 are available tonaturally move around it, but the IMs of theseLPOs could be exploited as natural landing or take-off gateways to and from the surface of Phobos. Inparticular, as presented in Fig.16, the inside branchof the IMs has been computed and a related perfor-mance analysis has shown that high-efficient nat-ural tangential landing paths and low escape ve-locity injections (far less than the 2B ∆v value of11m/s) are available for a region of topographicalcollinear-faced sites on Phobos. These trajectorieshave the potential to be exploited for future sample-and-return missions to this moon, where free-fall isrequired to avoid contamination of the sample’s soilby the exhaust plume of the thrusters or rockets’nozzle.

5.1. Artificial LPOs and their IMs around Phobos

The natural LPOs computed in the ER3BP-GHaround Phobos are investigated in the framework ofthe addition of a constant acceleration representingSEP. The idea is the same used in Section 3 forhovering points: here we focus on the artificial or-bits around the displaced L1−2. Since we foundbefore, computing the dynamical substitutes of thenatural LPOs in the ER3BP-GH, that the effectsof the GHs and the eccentricity act in a differentway, with the first responsible for the change in po-sition, shape and orientation of the LPOs, and thesecond causing the motion to oscillate around them,to undertake an immediate analysis of the advan-tages and opportunities provided by the low-thrustpropulsion, only the periodic artificial LPOs are de-rived by numerical continuation from the families ofthe POs of the CR3BP-GH, since they provide thebackbone of both the QPOs in the same dynamicalmodel and the QPOs in the ER3BP-GH.

The numerical continuation is undertaken in-creasing the constant acceleration magnitude, withthe same differential approach for the differentialcorrector used for computing the POs from CR3BPto CR3BP-GH in [31]-[32]: the difference is thatthe dimension of the parameters of the problem isnow higher, since it depends also on the orienta-tion of the thrust vector; therefore, the analysis isundertaken six times to consider thrust directionsalong all coordinated axes ±x, ±y, and ±z. Fig.17

4They are natural connections of IMs of orbits aroundthe same or different equilibrium points in nonlinear systems[15].

17

Figure 17: Artificial LPOs and their IMs with con-stant acceleration in the Mars-Phobos CR3BP-GH.First two graphs show the example of one natural medium-size periodic LPO of the family A around L1 and the tra-jectories (propagation time of 2 LPO periods) of the innerbranch of its unstable IM, both modified by different levelsof constant acceleration magnitude (m/s2) along the direc-tion +x (crosses represent the current LP). On the bottom,example of a small periodic LPO of the family A aroundL1 with different levels of constant acceleration magnitude(m/s2), along all coordinate axes directions.

Figure 18: Periodic LPOs in the Mars-PhobosCR3BP-GH-CA. Summary of the POs (brown) obtainedby displacement of the four families of POs around eachLP of the C3BP-GH (blue) with constant acceleration(1mm/s2) along all coordinate axes directions (on the topdirections ±x, in the center directions ±y, on the bottomdirections ±z). Phobos shape harmonics model.

18

shows some examples of the effects that the ad-dition of a constant acceleration produces on thenatural LPOs, and Fig.18 showcases the resultingfamilies of POs in the Mars-Phobos CR3BP-GH-CA along all the thrusting coordinate directions. Inparticular, thrusting away from Phobos moves theLPOs closer to the moon, without great changesin the shape and orientation of larger orbits evenwith high thrust. On the contrary, thrusting to-wards Phobos moves the orbits further from themoon, and as the thrust level increases the effectof the GHs rapidly decreases and the LPOs tendto become similar to the families of the classicalCR3BP. The effect of the thrust along the othertwo coordinates axes is more complicated: the man-ifold moves accordingly to the thrust direction, withthe displacement of the artificial LP moving in ac-cordance to the equi-thrust surfaces computed be-fore in Fig.10; thrust in tangential direction main-tains the shape of families B and C, while the fam-ilies D are similar to vertical Lyapunov orbits, andthe families A are highly distorted; high thrust inthe vertical direction greatly modifies the manifold,since only two families of POs are now present, onesimilar to the Halo orbits (Southern for +z) andthe other distorted and perfectly lying on the y-zplane.

The addition of a constant acceleration aroundPhobos has revealed some interesting mission op-portunities. First, Fig.19 shows that the effect onthe period of the POs (the numerical continuationis undertaken with a fixed differential energy con-straint) is quite sensitive: in particular, since thedimension of the manifold of the LPOs is small dueto the proximity of Phobos, the range of periodsof the natural LPOs is limited, and the addition ofconstant low thrust allows cheap artificial LPOs tobe obtained with period in 2:1 resonance with theorbital period of Phobos around Mars. This meansthat they remain periodic also in the elliptical realscenario, which could be an advantage for designingthe insertion manoeuvers between the mission seg-ments. Second, the addition of this simple thrustprofile also affects the stability properties of thePOs: despite the LPOs remain unstable, the Flo-quet instability index could be massively loweredwith the thrust required to displace the LP far fromPhobos. This has a great impact on the frequencydemand for the GNC subsystem, reducing the dutycycle up to the 25% for artificial LPOs displaced atan altitude over 60km from Phobos along the Mars-Phobos radial. In particular, we see in Fig.19 that

Figure 19: Periodic LPOs in the Mars-PhobosCR3BP-GH-CA. On the top, example of one naturalmedium-size periodic LPO of the family A around L2 mod-ified by different levels of constant acceleration magnitudealong the direction −x: period and stability properties (sta-bility indexes of the two non-unit couples of eigenvalues ofthe monodromy matrix). Following graph shows the charac-teristic curves of the same properties for all the four familiesof POs around L1 with different constant acceleration mag-nitude (m/s2) along directions ±x.

19

Figure 20: IMs of the artificial LPOs with constant ac-celeration in the Mars-Phobos CR3BP-GH. First twofigures show the possible landing/take-off sites through theIM of the family A of POs around L1, third figure shows theregion of landing sites for all the families of artificial LPOsaround L1−2. Constant acceleration magnitude of 1mm/s2

along all coordinate axes directions (green line for CR3BP-GH, cyan for directions ±x, red for directions ±y, yellow fordirections ±z). Phobos shape harmonics.

trans-Phobos LPOs over the 70km altitude bound-ary of the Lyapunov stability region of the AEPs(the tip of the right leaf of Fig.11) become Floquetstable, while stable artificial LPOs along the y-axisare obtained displacing the equilateral LPs. Finally,displacing LPOs away from the natural SOI, in ad-dition to reducing instability, it has other importantadvantages: indeed, all the problems of the dynam-ical modeling of the relative motion in proximity ofthis moon are related to the collapse of the realmof attraction of Phobos, therefore the manifold ofLPOs is already too close in comparison to commoninterplanetary spacecraft operations (observations,science measures, descent start). Therefore, push-ing inward Phobos with a simple constant propul-sion profile enlarges the Phobos SOI, and there isa great advantage not only for mission operationsconstraints and light condition requirements, butin particular for the computational load of track-ing these orbits: the effect of Phobos’ gravity fieldquickly lowers with the distance, so the convergenceof the solution of the LPOs (and so its reliability)will be obtained with a far lower order of the trun-cated GHs model to be used in the numerical con-tinuation.

The IMs of the artificial periodic LPOs have beencomputed (Fig.20) and they have revealed that us-ing constant thrust, along an appropriate direction,allows to enlarge the region of landing and take-offsites to cover all the longitude range (mostly withthrust along ±y, but also inward Phobos), whilethe limit of the latitudes is also raised (thrust along±z) to become closer to the polar zones, that couldbe enclosed when considering also the families ofQPOs. Also, the displacement of the LPOs awayfrom Phobos and along the tangential axis enablesartificial heteroclinic connections between two man-ifolds that were not possible with the natural dy-namics: these trajectories could be exploited forfast orbital displacements around the two sides ofPhobos for close-range mission segments.

6. Quasi-Satellite Orbits around Phobos

The last class of orbits that can be used in a mis-sion to Phobos lies outside the SOI of the secondmassive body of a 3BP. The peculiar case of a smallplanetary satellite like Phobos is therefore very suit-able for the exploitation of these orbits because ofthe collapse of its SOI, which indirectly drags themanifold of these orbits closer to the body. They

20

constitute a family of QPOs which are called by dif-ferent names: Quasi-Satellite, Quasi-Synchronous,Distant Satellite, Distant Retrograde orbits. In par-ticular, planar QSOs could be recognized as thequasi-periodic solution around the Stromgren’s fclass of periodic orbits of Hill’s approximation of theplanar CR3BP, as indicated in the seminal papersof Henon [33, 34]. QSOs are more generally consid-ered as one of three kinds of co-orbital configura-tions in a CR3BP with 1:1 resonance together withTadpole and Horseshoe orbits: in [35] it is shownthat unstable QSOs evolve from and to Horseshoeorbits, which are linked together. Another usefulperspective is to relate the dynamics of QSOs to therelative motion between two close synchronous ke-plerian orbits around the primary body of the 3BP.This is the approach used in Formation Flying dy-namics, where the osculating orbit of the secondarybody, in our case Phobos, could be considered as achief spacecraft orbiting Mars, that the third bodyfollows in proximity. The solution of the kepleriansynchronous Formation Flying in the Hill’s rotatingframe of the chief is a relative retrograde ellipticalorbit, called epicycle [25]. This results in an arti-ficial satellite of the secondary body, but due onlyto the attraction of the primary. Such an ellipseis defined in the rotating Hill’s frame. The thirdbody never rotates around the chief in the inertialframe, centered on the chief, where the third bodyremains at one side. Only thanks to the spinningrotation of the secondary body, that for the case ofPhobos is synchronous with the Hill’s frame rota-tion, the third body rotates in the BCBF frame ofthe secondary in 1:1 resonance. In particular, forslightly eccentric keplerian orbits (as in the case ofPhobos) the epicycle resulting from a difference ineccentricity between third and secondary body is anellipse centered on the chief, with major axis alongthe tangential Hill’s axis and in 2:1 ratio with theminor axis; a difference in inclination or right as-cension of ascent node inclines the epicycle, and therelative motion is 3D. The QSO is the motion of theepicycle when the chief is not a spacecraft but a sec-ond massive body, thus the QSO is the solution ofthe 3B dynamics. Therefore a QSO in the 3B prob-lem is a QPO characterized by an oscillation of thewhole epicycle along the y-axis of the 3BP frame.In addition, 3D epicycles experience a secular pre-cession of their relative line-of-nodes: the relatedperiod grows as far as the size of the epicycle in-creases, which is the relative line-of-nodes becomesfixed in the Hill’s frame (1:1 resonance).

The analysis of the QSOs around Phobos in thispaper is conducted with the latter approach of along-range Martian Formation Flying, in a keple-rian perturbed (2B-P) model where the equationsof motion are the Gauss’ Planetary Equations [36],that use as state variable the equinoctial orbital el-ements (OEs) of the spacecraft around Mars, andthe ER3BP is retained using as forcing action the3B perturbation gravity of Phobos in the osculat-ing Hill’s frame centered on the moon. In terms ofkeplerian OEs, the QSO has short-period (rotationaround Phobos along the epicycle) and medium-period (tangential motion of the epicycle) oscilla-tions for semi-major axis, eccentricity and argu-ment of pericenter, while the inclination and rightascension experience a long-period oscillation (pre-cession of the epicycle): a 3D QSOs in the ER3BPis a torus with three phases. The 2B-P is suitableto be implemented with additional orbital pertur-bations: from our perturbation analysis in Section2.3, we recover that outside of the Phobos’ SOI, themajor effects are due to the eccentricity (alreadyembedded in the equations of motion) and the MarsJ2 GH. In particular, as a difference from the 3BP,in the 2B-P the orbital perturbations not due toPhobos are no longer differential. The differentialaction on the relative motion appears by computingin feedforward the osculating motion of Phobos andthe angular velocity of its Hill’s frame, with a ded-icated 2B-P under the effect of the perturbation.Recall that when defining an initial condition forthe QSO using mean OEs for the epicycle and Pho-bos and starting with true anomalies in accordancewith the 2B dynamics, it produces different QSOs.The same QSO is obtained with osculating OEs inaccordance to the perturbation, but their analyticalexpression is only available for particular cases, likeJ2 [37], but not for the 3B perturbation. As an ex-ample, starting with the spacecraft in perimars andinferior conjunction, and starting in Mars-Phobosquadrature and phase in accordance on the sameepicycle, it produces very different QSOs with thelatter having a far lower amplitude of the y-axis os-cillation.

The dynamical analysis of the QSOs requiresthe derivation of the secular derivatives of the 3Bperturbation in the 2B-P, in a similar way to thecase of J2. But the solution of the mean integralvalue of the 3B perturbation could not be under-taken analytically. This integral is solved numeri-cally in [35], and used for the stability analysis ofthe QSOs. The outcomes are two. First, any 2D

21

QSO large enough that the oscillation amplitudeof the epicycle does not make it fall towards thesecondary body (minimum distance of the epicy-cle) is always stable. Second, for any 3D QSOthe secular precession will rotate the epicycle withthe relative line-of-nodes towards the Hill’s x-axis(relative nodes in conjunction with the two centralbodies): such attitude becomes unstable as far asthe inclination of the QSO increases, therefore theQSO leaves the body neighborhood and become aHorseshoe orbit. This 3D stability condition of theQSOs requires the difference in inclination (in ra-dians) to be smaller than the difference in eccen-tricity. A linear Floquet stability analysis of QSOsin the CR3BP has highlighted the high stability ofthem [38]. In [39], a different approach has beenundertaken, which is a linearized stability analy-sis in the ER3BP around the epicycle, and resultsare applied to the Mars-Phobos case. The min-imum distance condition from Phobos is 29.4km(∆e = 0.00315), above which the planar QSO isstable. The 3D stability condition that bounds theadmissible difference in inclination to the differencein eccentricity below which the inclined QSO is sta-ble is ∆i/∆e < 96%. In addition, the period ofthe linearized precession motion is analytically de-rived. Similar numerical outcomes were obtainedfor the stability analysis of QSOs around Jupitermoon Europa [40]. In particular, we found thatthe minimum distance corresponds to a peculiarcondition presented by Henon in [34] on the sta-bility of the planar QPOs around the f family ofHill’s approximation of the planar CR3BP. As faras the dimension of the QPOs increases, the sta-bility is influenced by resonances with the familiesof multiple-revolution direct satellite orbits, wherethe last resonance 1:4 is encountered by an epicy-cle of minimum distance equal to 1.2 3

õ of the or-

bital semi-major axis: after that, the QPO remainsbounded even at infinite distance (within the Hill’sapproximation). This value for the Mars-Phobossystem leads to a distance of 28.7km, which is soclose to the value obtained above in the linearizedER3BP. In addition, [34] found a very different sen-sitivity of the stability of the QPO between the twovelocity components, with the radial one being farless critical. This means that the accuracy requiredby the GNC subsystem for the insertion manoeu-ver to the QSO is less critical in Mars-Phobos andPhobos-spacecraft common quadrature configura-tions, where the epicycle has only radial velocityon the orbital plane and the amplitude of the tan-

gential oscillation is the smallest using mean OEs.The stability of this class of orbits, combined

with the collapse of the SOI and the synchronousrotation, makes the QSOs attractive solutions to or-bit Phobos. A QSO will constitute the main orbitalmission segment around Phobos for the upcomingPhootprint mission [41], to observe the surface thor-oughly and identify the landing site where obtain-ing a sample of the soil to return back to Earth; itwill be used for a very long time (6-9 months), andrequired to remain stable during the solar conjunc-tions that would black out the communications tothe Earth for approximately 1 month. In this paperwe probe the nonlinear stability of the QSOs aroundPhobos. Regarding Phootprint, the stability hasbeen tested scanning the state-space in relative po-sition and velocity with a true-life simulator [38].Instead we conduct the analysis in the frameworkof the 2B-P (whose results, due to the fact thatwe are out of the Phobos SOI, are suitable to beinterfaced with a previous orbital segment aroundMars), and we use the linear stability region of [39](a trapezoid in the plane ∆e-∆i as shown in Fig.21)as a first guess in order to limit the boundary ofthe state-space in terms of osculating OEs whereconduct the nonlinear simulations. We used theSTK software, considering the default ephemeridesof Phobos, starting at 10 April 2013 08:35:30UTCG(perimars), and the additional perturbations of theMars GHs up to 15th degree/order, the PhobosGHs up to 4th degree/order, the Sun and SRP per-turbations. We further limit the region by settinga range of minimum altitudes of the epicycle fromPhobos between 20 and 60km, therefore already in-side the linear boundaries; regarding 3D QSOs thereference vertical size of Phobos is realized with just∆i = 0.001rad = 0.06. We simulate QSOs up toone year of propagation time: a QSO is consideredstable when it does not drift away by the end of thesimulation. This is very reliable and not too muchrestricting because the strong nonlinearity of the 3Bdynamics provokes behaviors drastically differentcrossing the stability region boundaries (as provedwith the AEPs). The resulting true-life stability re-gion boundary is presented in Fig.21: the minimumdistance requirement is significantly higher than thelinear one, and in the same range of ∆e the 3Dstability boundary is smaller than the linear one;for higher eccentricity such boundary asymptoti-cally reaches the linear one. This stability regionis related to positive ∆e-∆i, while all the other ini-tial conditions differences in osculating OEs at the

22

Figure 21: QSOs around Phobos. On the top-left, stability region of QSOs around Phobos tested by high-fidelity 1 yearsimulations, defined by initial conditions on osculating OEs around Mars for the spacecraft and Phobos: positive differences ineccentricity and inclination, starting at perimars epoch. On the top-right, example of a 3D stable QSO in the Phobos Hill’sframe for 1 year propagation. On the bottom-left, period of precession of the relative line-of-nodes that indicates the minimumtime for a complete Phobos surface coverage of the QSOs. On the bottom-right, example of the lighting conditions for a singlekeplerian epicycle at 50-110km distance range, remaining in the Mars and Phobos shadowing wakes for 34% of the time.

perimars are null. The starting ∆e of the epicy-cle does not define trivially the minimum distancefrom Phobos of the QSO for the smallest range ofeccentricities: the planar QSO at minimum stable∆e has a minimum altitude of 25km.

From a sample of trajectories simulated, the pe-riod of the secular precession for 3D QSOs was com-puted: this natural motion of the 3B dynamics isuseful for observation purposes because it allows toovercome the 1:1 resonance of the keplerian epicycleand provide a complete coverage of the surface ofPhobos. Fig.21 presents the related time requiredfor the QSOs of the stability region, and compares itwith the linearized solution from [39]: all the rangeof stable QSOs of interest provide a fast coverage

of the moon. Another important performance thatcould be used in the mission design to select theoptimal QSO inside the stable domain is the ∆vof the insertion manoeuver: considering a previousmission segment realized by a Martian Trailing or-bit (an AEP) with ∆ν = [0,−6] from Phobos(corresponding to a distance 0-1000km), we com-puted the ∆v budget to provide with the cheapeststrategy of impulsive manoeuvers the initial con-ditions of the QSO ∆e, ∆i and the phasing ∆ν.Since ∆e and ∆i are small their cost range is mod-erate (5-7.5m/s and 0-8m/s), and also the cost ofthe phasing (0-20m/s) could be lowered for distantAEPs linearly performing multiple laps. The priceto pay is instead the accuracy of the GNC subsys-

23

tem to insert precisely the spacecraft in this smallrange of initial conditions.

In this analysis we found a region of QSOs nat-urally stable with a high-fidelity perturbed model,potentially for a whole long-period mission scenario,with their distances from Phobos suitable for obser-vation, and the exploitation of the natural preces-sion motion provides a fast complete coverage of thesurface of the moon. Such fast precession, combinedwith the exploitation of these orbits for long peri-ods, allows a spacecraft to remain mostly in light.On the contrary, QSOs can be controlled to main-tain a 1:1 resonance with Phobos BCBF frame: thiswould provide constant lighting conditions, rangingfrom continuous light (during solstices) to continu-ous shadow (during equinoxes), controlling the β-angle thermal condition desired by fine tuning theinitial phase of the spacecraft along the epicyclewith respect to the VDCO of the Sun around Pho-bos. An example in Fig.21 shows that a planarepicycle in the middle of the stability region, dur-ing an equinoctial season could remain in shadowfrom the field of view of the Sun for the 34% of thetime.

7. Analysis of the Trade-offs and Applica-tions in the Space Mission Design aroundPhobos

The trailing/leading orbits around Mars analyzedin Section 3, starting from 25km distance from Pho-bos, are attractive configurations, because they arecheap and affordable by SEP even for heavy hu-man modules, they are stable to perturbations, andthey are mostly in full light. They are the best or-bits to start to approach Phobos SOI, however theirground-track on the moon is stationary and limited.Other distant configurations or close-range AEPsrequires either high thrust or high station-keepingcost for hovering over long-time: they can be usedonly for short and dedicated operations of small un-manned spacecraft.

Keplerian orbits around Phobos are infeasibledue to the collapse of the realm of attraction of themoon towards its surface. In Section 4 we investi-gated the artificial orbits around Phobos that wouldprovide a spacecraft continuous light or shadowconditions moving synchronous with the Sun onits seasonal surface of motion. Due to the pull ofMars these VDCOs require continuous propulsionand they are too expensive even for few revolutionsand also unstable.

Section 5 focuses on a class of interesting or-bits that exist about Phobos, which are the LPOs.They are computed in an improved system of therelative dynamics in proximity of Phobos, upgrad-ing the Mars-Phobos ER3BP system with the realgravity field of the moon, modeled with a grav-ity harmonics series expansion. These orbits arevery close to the moon surface, therefore they aresimilar to close-range points but with an extendedground-track and range of lighting conditions, andthe Phobos’ bulk occultation of the sky could pro-vide relevant passive shielding from the cosmic raysradiation. Despite their instability, the LPOs arenatural motion and so will require no propulsionand low station-keeping cost to provide observationon Phobos and communication bridges to managerobotic scouts on Mars and Phobos: however, theyrequire the high accuracy of an optical navigationsubsystem, and high-load on the guidance subsys-tem, whose reference signal must be computed withadvanced nonlinear techniques that need the acqui-sition of a high-fidelity gravity field of the moon. Inparticular, large quasi-periodic orbits enable cover-age also of the polar and lead/trail-regions, and wefound that in the Mars-Phobos elliptic system thereexist tall and inclined periodic orbits around eachside of the moon: a constellation, starting just fromone spacecraft on each side, would fly synchronousand so enable stationary communications betweenmost of the opposite sides of the moon (cis/trans,North/South, part of lead/trail) where different hu-man crews or rovers could be displaced, as well asrepeated access times to equatorial and middle lat-itude sites on Mars. Another useful application isto exploit their IMs as landing/taking-off gatewaysto and from the moon: in Section 5 we proved thatthere exist natural trajectories for a specific range oflongitude-latitude sites able to land tangentially, fa-cilitating a soft controlled touch-down, and departwith a very little escape velocity, less than 30% ofthe 2B ∆v value. The optimization of these per-formances to select the best trajectory at a givenlocation on Phobos will be paramount for sample-and-return missions (where also soil contaminationavoidance is necessary) as well as first manned ex-plorations of this moon. The addition of a simplepropulsive law, to obtain a constant acceleration,offers some advantages when using these LPOs forshort-phases. Section 5.1 shows that the surfacecoverage and landing/take-off targeting could beextended to the whole surface of Phobos, the in-stability of the orbits could be lowered, and the

24

Table 2: Summary of orbits around Phobos. First driver within a row is the altitude range (-). For IMs, ( / ) distinguishesthe branch direction. Propulsive thrust range is indicated considering a 100/10,000kg spacecraft mass.

Con

trol a

ctio

nG

uida

nce

erro

rN

avig

atio

n ac

cura

cy

0-4.

5km

x0m

/sre

lativ

e fix

ed p

ositi

on,

accu

rate

gra

vity

fiel

d m

odel

C3B

-GH

0.5/

50-0

/0N

(150

-0m

/s/T

)co

nsta

nt a

ccel

erat

ion

unst

able

: SK

3-

3m/s

/T e

very

7-7

min

10-1

0m0.

05-0

.05m

/s1-

1m0.

001-

0.00

1m/s

stat

iona

ry g

roun

d-tra

ckci

s/tra

ns-P

hobo

s reg

ion

sols

tices

: 50-

80%

equi

noxe

s: 4

0-70

% (c

is),

50-8

0% (t

rans

)de

dica

ted

prox

imity

ope

ratio

ns,

radi

atio

n sh

ield

(53-

13%

)

0-4.

5km

y0m

/sre

lativ

e fix

ed p

ositi

on,

accu

rate

gra

vity

fiel

d m

odel

C3B

-GH

0.5/

50-0

.3/3

0N (1

50-1

00m

/s/T

)co

nsta

nt a

ccel

erat

ion

unst

able

: SK

0.

6-0.

6m/s

/T e

very

14-

15m

in10

-10m

0.02

-0.0

2m/s

1-1m

0.00

1-0.

001m

/sst

atio

nary

gro

und-

track

lead

/trai

l-Pho

bos r

egio

nso

lstic

es: 5

0-80

%eq

uino

xes:

45-

75%

dedi

cate

d pr

oxim

ity o

pera

tions

, ra

diat

ion

shie

ld (5

0-16

%)

0-4.

5km

z0m

/sre

lativ

e fix

ed p

ositi

on,

accu

rate

gra

vity

fiel

d m

odel

C3B

-GH

1/10

0-0.

4/40

N (3

00-1

20m

/s/T

)co

nsta

nt a

ccel

erat

ion

unst

able

: SK

0.

9-0.

9m/s

/T e

very

11-

12m

in10

-10m

0.03

-0.0

2m/s

1-1m

0.00

1-0.

001m

/sst

atio

nary

gro

und-

track

Nor

th/S

outh

pol

e-Ph

obos

regi

onS/

W so

lstic

e: N

orth

/Sou

th li

ght,

Sout

h/N

orth

sh

adow

und

er 1

.4km

then

ligh

t, eq

uino

xes:

88%

dedi

cate

d pr

oxim

ity o

ps, c

ompl

ete

light

ing/

shad

owin

g, ra

diat

ion

shie

ld (5

8-20

%)

4.5-

70km

x0m

/sre

lativ

e fix

ed p

ositi

onC

3B0/

0-1.

5/15

0N (0

-400

m/s

/T)

cons

tant

acc

eler

atio

nun

stab

le: S

K

6-20

m/s

/T e

very

15-

9min

100-

100m

0.2-

0.5m

/s10

-10m

0.01

-0.0

1m/s

stat

iona

ry g

roun

d-tra

ckci

s/tra

ns-P

hobo

s reg

ion

sols

tices

: lig

hteq

uino

xes:

88%

shor

t clo

se o

bser

vatio

n

4.5-

60km

y0m

/sre

lativ

e fix

ed p

ositi

onC

3B0.

3/30

-0.0

4/4N

(100

-12m

/s/T

)co

nsta

nt a

ccel

erat

ion

unst

able

: SK

3-

3m/s

/T e

very

20-

21m

in10

0-10

0m0.

2-0.

2m/s

10-1

0m0.

01-0

.01m

/sst

atio

nary

gro

und-

track

lead

/trai

l-Pho

bos r

egio

nso

lstic

es: l

ight

equi

noxe

s: 8

8%lo

ng c

lose

obs

erva

tion

4.5-

250k

m z

0m/s

rela

tive

fixed

pos

ition

C3B

0.4/

40-1

.5/1

50N

(120

-400

m/s

/T)

cons

tant

acc

eler

atio

nun

stab

le: S

K

3-20

m/s

/T e

very

20-

9min

100-

100m

0.2-

0.5m

/s10

-10m

0.01

-0.0

1m/s

stat

iona

ry g

roun

d-tra

ckN

orth

/Sou

th p

ole-

Phob

os re

gion

sols

tices

: lig

hteq

uino

xes:

88%

shor

t clo

se o

bser

vatio

n

70-4

00km

x0m

/sre

lativ

e pe

riodi

c or

bit -

ci

s/tra

ns-M

ars o

rbit

E3B

- 2B

(Mar

s)1.

5/15

0-6/

600N

(420

-200

0m/s

/T)

feed

forw

ard

prof

ilest

able

1-10

km0.

5-2m

/s0.

1-1k

m0.

01-1

m/s

stat

iona

ry g

roun

d-tra

ckci

s/tra

ns-P

hobo

s reg

ion

sols

tices

: lig

hteq

uino

xes:

88%

shor

t sta

nd-b

y, m

easu

rem

ents

, ob

serv

atio

n, c

omm

-rel

ay

60-5

00km

0m

/sre

lativ

e pe

riodi

c or

bit -

le

ad/tr

ail-M

ars o

rbit

C3B

- 2B

(Mar

s)0.

04/4

-0.0

004/

0.04

N (1

2-0.

12m

/s/T

)fe

edfo

rwar

d pr

ofile

stab

le0.

1-10

km0.

05-2

m/s

0.01

-1km

0.01

-0.1

m/s

stat

iona

ry g

roun

d-tra

ckle

ad/tr

ail-P

hobo

s reg

ion

sols

tices

: lig

hteq

uino

xes:

88%

long

stan

d-by

, mea

sure

men

ts (a

cqui

sitio

n of

Pho

bos o

rbit

and

GM

), co

mm

-rel

ay

250-

1400

km z

0m/s

Nor

th/S

outh

-Mar

s orb

it2B

(Mar

s)1.

5/15

0-8/

800N

(420

-250

0m/s

/T)

feed

forw

ard

prof

ilest

able

10-5

0km

2-10

m/s

1-10

km0.

1-1m

/sst

atio

nary

gro

und-

track

Nor

th/S

outh

pol

e-Ph

obos

regi

onso

lstic

es: l

ight

equi

noxe

s: 8

8%sh

ort s

tand

-by,

mea

sure

men

ts,

obse

rvat

ion,

com

m-r

elay

0-10

km2-

7m/s

rela

tive

circ

ular

equ

ator

ial

orbi

t dai

ly S

un-s

ynch

2B(P

hobo

s)0.

5/50

-0.3

/30N

(100

-50m

/s/T

)fe

edfo

rwar

d pr

ofile

unst

able

: SK

30

-6m

/s/T

eve

ry 7

-15m

in10

0-10

0m0.

5-0.

2m/s

10-1

0m0.

01-0

.01m

/seq

uato

rial s

wat

hfix

ed, c

ontro

llabl

e (s

olst

ices

: 0-1

00%

, equ

inox

es: 0

-88%

)fix

ed li

ght/s

hado

w c

ondi

tions

(dire

ctio

nal

sola

r rad

iatio

n pr

otec

tion)

, com

m-r

elay

10-1

00km

7-30

m/s

rela

tive

circ

ular

orb

it at

hei

ght

0-0/

10-5

0km

dai

ly S

un-s

ynch

2B(P

hobo

s)0.

3/30

-1.5

/150

N (5

0-25

0m/s

/T)

feed

forw

ard

prof

ileun

stab

le: S

K

6-30

m/s

/T e

very

38-

21m

in1-

1km

1-1.

5m/s

100-

100m

0.1-

0.1m

/she

mis

pher

ic se

ctor

fixed

, con

trolla

ble

(sol

stic

es: 0

-100

%, e

quin

oxes

: 0-8

8%)

fixed

ligh

t/sha

dow

con

ditio

ns (d

irect

iona

l so

lar r

ad p

rote

ctio

n), h

emis

pher

e co

mm

-rel

ay

0-10

km0-

8m/s

rela

tive

perio

dic

(2:1

) or q

uasi

-per

iodi

cor

bit,

hig

h-co

mpu

tatio

nal t

echn

ique

s,ac

cura

te g

ravi

ty fi

eld

mod

elE3

B-G

Hno

t req

uire

dun

stab

le: S

K

0.15

-0.0

5m/s

/T e

very

20-

32m

in10

-10m

0.02

-0.0

1m/s

1-1m

0.00

1-0.

001m

/sst

atio

nary

gro

und-

track

on

wid

e Ph

obos

regi

onso

lstic

es: 1

00-6

0%eq

uino

xes:

80-

50%

(cis

), 88

-60%

(tra

ns)

shor

t clo

se o

bser

vatio

n, a

ppro

achi

ng m

anoe

uver

s, co

mm

-brid

ges,

radi

atio

n sh

ield

(23-

13%

)

0-80

km0-

20m

/s

rela

tive

perio

dic

(N:M

) or q

uasi

-per

iodi

cor

bit,

high

-com

puta

tiona

l tec

hniq

ues,

accu

rate

-ligh

ter g

ravi

ty fi

eld

mod

el

E3B

-GH

- E3

B0/

0-1.

5/15

0N (0

-400

m/s

/T)

cons

tant

acc

eler

atio

nun

stab

le: S

K

0.15

-0.2

m/s

/T e

very

19-

68m

in10

-100

m0.

02-0

.05m

/s1-

10m

0.00

1-0.

01m

/sst

atio

nary

gro

und-

track

on d

edic

ated

regi

on o

f who

le P

hobo

s sur

face

sols

tices

: 100

-60%

equi

noxe

s: 8

8-50

%

shor

t ded

icat

ed o

ps, r

adia

tion

shie

ld (3

5-3%

) -

shor

t sta

ndby

, mea

sure

men

ts (P

hobo

s gra

vity

fie

ld),

com

m-r

elay

0-10

km /

0-50

0km

0-15

m/s

/ 0-

60m

/s

rela

tive

traje

ctor

y fr

om L

PO /

Mar

s orb

it0.

5-2h

/ 10

-20h

E3B

-GH

not r

equi

red

SK0.

02m

/s e

very

20m

in /

3m/s

eve

ry 7

0min

10m

/ 1k

m

0.02

m/s

/ 0.

5m/s

1m /

100m

0.

001m

/s /

0.1m

/sla

ndin

g si

te c

is/tr

ans-

Phob

os re

gion

/ st

atio

nary

gro

und-

track

lead

/trai

l-Pho

bos r

egio

nco

ntro

llabl

e by

epo

ch 0

-100

% /

sols

tices

: lig

ht, e

quin

oxes

: 88%

natu

ral p

erfo

rman

ced

soft

land

ing

(fro

m 0

.5g)

/ a

ppro

ach

to P

hobo

s or i

nser

tion

man

ouev

eur t

o LP

O

0-10

km /

0-50

0km

0-15

m/s

/ 0-

60m

/s

rela

tive

traje

ctor

y fr

om ta

ke-o

ff si

te /

LPO

0.5-

2h /

10-2

0hE3

B-G

Hno

t req

uire

dSK

0.02

m/s

eve

ry 2

0 m

in /

3m/s

eve

ry 7

0min

10m

/ 1k

m

0.02

m/s

/ 0.

5m/s

1m /

100m

0.

001m

/s /

0.1m

/sta

ke-o

ff si

te c

is/tr

ans-

Phob

os re

gion

/ st

atio

nary

gro

und-

track

lead

/trai

l-Pho

bos r

egio

nco

ntro

llabl

e by

epo

ch 0

-100

% /

sols

tices

: lig

ht, e

quin

oxes

: 88%

natu

ral p

erfo

rman

ced

chea

p ta

ke-o

ff

(fro

m 3

m/s

) to

LPO

/ es

cape

from

Pho

bos

0-20

km /

0-50

0km

0-25

m/s

/ 0-

60m

/s

rela

tive

traje

ctor

y fr

om L

PO /

Mar

s orb

it0.

1-3h

/ 8-

20h

E3B

-GH

- E3

B0/

0-0.

1/10

N (0

-10m

/s /

0-1

00m

/s) c

onst

ant a

ccel

erat

ion

SK0.

02-0

.05m

/s e

very

20

min

/ 2-

3m/s

eve

ry 7

0min

10m

/ 1k

m

0.02

m/s

/ 0.

5m/s

1m

/ 10

0m

0.00

1m/s

/ 0.

1m/s

land

ing

site

on

dedi

cate

d re

gion

of w

hole

Ph

obos

surf

ace

(apa

rt ne

ar p

oles

) / st

atio

nary

gr

ound

-trac

k le

ad/tr

ail-P

hobo

s reg

ion

cont

rolla

ble

by e

poch

0-1

00%

/ so

lstic

es: l

ight

, equ

inox

es: 8

8%

perf

orm

ance

d so

ft la

ndin

g (f

rom

0.5

g)

/ app

roac

h to

Pho

bos o

r ins

ertio

n m

anou

eveu

r to

LPO

0-20

km /

0-50

0km

0-25

m/s

/ 0-

60m

/s

rela

tive

traje

ctor

y fr

om ta

ke-o

ff si

te /

LPO

0.1-

3h /

8-20

h

E3B

-GH

- E3

B0/

0-0.

1/10

N (0

-10m

/s /

0-10

0m/s

) con

stan

t acc

eler

atio

n

SK0.

02-0

.05m

/s e

very

20m

in /

2-3m

/s e

very

70m

in

10m

/ 1k

m

0.02

m/s

/ 0.

5m/s

1m /

100m

0.

001m

/s /

0.1m

/s

take

-off

site

on

dedi

cate

d re

gion

of w

hole

Ph

obos

surf

ace

(apa

rt ne

ar p

oles

) / st

atio

nary

gr

ound

-trac

k le

ad/tr

ail-P

hobo

s reg

ion

cont

rolla

ble

by e

poch

0-1

00%

/ so

lstic

es: l

ight

, equ

inox

es: 8

8%pe

rfor

man

ced

chea

p ta

ke-o

ff (f

rom

3m

/s)

to L

PO /

esca

pe fr

om P

hobo

s

20-5

00km

/ 20

-500

km0-

60m

/s /

0-60

m/s

rela

tive

traje

ctor

y fr

om L

PO, t

ake-

off

site

/ M

ars o

rbit,

LPO

, tak

e-of

f site

3-13

h / 5

-15h

E3B

-GH

/ E3

B0.

1/10

-1/1

00N

(10-

500m

/s /

50-6

00m

/s) c

onst

ant a

ccel

erat

ion

SK0.

1-0.

5m/s

eve

ry 2

0min

/ 1-

2m/s

eve

ry 7

0min

10m

/ 1k

m

0.02

m/s

/ 0.

5m/s

1m /

100m

0.

001m

/s /

0.1m

/sde

dica

ted

grou

nd-tr

ack

/ sta

tiona

ry

grou

nd-tr

ack

lead

-trai

l-Pho

bos r

egio

nco

ntro

llabl

e by

epo

ch 0

-100

% /

sols

tices

: lig

ht, e

quin

oxes

: 88%

trans

fer m

anoe

uver

aro

und

Phob

os /

appr

oach

/esc

ape

to/fr

om P

hobo

s or

inse

rtion

man

oeuv

er to

LPO

25-2

00km

8-50

m/s

rela

tive

quas

i-per

iodi

c or

bit,

high

-fid

elity

sim

ulat

ions

2B(M

ars)

-Pno

t req

uire

dna

tura

lly st

able

with

out S

K10

-100

m0.

01-0

.1m

/s10

-100

m0.

01-0

.1m

/sco

mpl

ete

in 5

-25d

ays

sols

tices

: lig

hteq

uino

xes:

88%

park

ing

orbi

ts (r

epai

rs, r

efue

lling

), m

easu

-re

men

ts (P

hobo

s gra

vity

fiel

d), c

ompl

ete

fast

map

ping

and

freq

uent

acc

ess c

omm

s

40-1

00km

10-2

5m/s

rela

tive

quas

i-per

iodi

c or

bit,

high

-fid

elity

sim

ulat

ions

2B(M

ars)

-Pif

requ

ired

by

-ang

leco

ntro

l sys

tem

natu

rally

stab

le w

ithou

t SK

10-1

00m

0.01

-0.1

m/s

10-1

00m

0.01

-0.1

m/s

stat

iona

ry sw

ath

adju

stab

le b

y de

dica

ted

cont

rol s

yste

mlo

ng st

and-

by, c

omm

-rel

ay, c

ontro

l of

light

ing

cond

ition

s (di

rect

iona

l so

lar r

adia

tion

prot

ectio

n)

Ligh

ting

Con

ditio

nsPo

tent

ial M

issi

onA

pplic

atio

nsM

aint

anan

ce S

trate

gyO

rbit

Alti

tude

Ran

geR

efer

ence

Sig

nal

Com

puta

tion

Dyn

amic

alM

odel

Orb

ital

Prop

ulsi

onSu

rfac

eC

over

age

AEP

VD

CO

L 1-2

-LPO

IM-L

PO

QSO

25

computation of the orbits themselves could be sim-plified to maintain them periodic also in the trueelliptic dynamics, and to lower the accuracy of themodel of the gravity field of the moon to be takeninto account. In particular, artificial heteroclinicconnections would now exist for fast orbital fly-bysaround two opposite sides of Phobos.

Finally, the QSOs are discussed in Section 6, andthey are the best solution for a precursor unmannedmission to Phobos. They are both natural orbitswith no need of propulsion, and self-stable up tovery long time with no need of station-keeping, andso they can be used as parking orbits with distancestarting from 25km from Phobos. In particular,closer 3D QSOs provide a fast complete coverage tomap the surface of Phobos and identify the landingsite, and they are mostly in light. 2D QSOs duringequinoctial seasons are suitable to be controlled toprovide nearly Sun-synchronous orbits around Pho-bos, enabling constant and adjustable lighting con-ditions: in particular, lighting condition schedulingcould be important for a first-generation mannedspacecraft while orbiting Mars.

8. Conclusions

In this paper we analyzed several kinds of orbitsaround the Martian moon Phobos, each one definedusing appropriate models of the relative dynamicswhere the reference signal is computed. The de-sign of a future space mission to Phobos requiresmultiple objectives and constraints to be satisfied:we collect the outcomes of our analysis in Table 2,where each orbit has a number of potential applica-tions and their performance can be assessed againstthe requirements of each mission segment.

A possible mission scenario for Phobos could beto start from establishing a trailing orbit aroundMars, with a phasing of 0.4-1.2, requiring a thrustto be maintained of less than 0.05-5N in accordanceto the class of spacecraft, where start to track thereal position of Phobos and acquire its gravity pa-rameter. Then perform a phasing manoeuver toreach a QSO at an intermediate distance from Pho-bos, where start to measure the real gravity fieldand map the surface of the moon. Finally, movealong the tube of the artificial invariant manifoldsdisplaced away from the moon with a constant ac-celeration along the x-axis, whose LPOs tracking isless affected in their numerical computation by theGHs perturbations, to directly reach with a probethe desired landing site on Phobos. In the case of

a sample-and-return mission, we then move in re-verse choosing the trajectory that provides the min-imum escape velocity, reaching a mothership thathas been left on a closer LPO or parked in the QSO,to remotely command the lighter probe. In partic-ular, the long-time stability of the QSOs aroundPhobos could be exploited as an orbital repositoryto send, in advance, unmanned propulsion modules,fuel stockpiles, and provisions, to remain parked ina secure low altitude Martian orbit without orbitalmaintenance costs and with short-period phasingmanoeuvers to dock the modules. To allow thefirst human expeditions to visit Mars and returnto the Earth, the spacecraft could make scheduledpit-stops at this orbital garage on Phobos’ orbit.

Acknowledgments

This work has been supported by the Euro-pean Commission through the Marie Curie fellow-ship PITN-GA-2011-289240 ”AstroNet-II”. We arethankful with Prof. J.J. Masdemont and Prof. G.Gomez (IEEC) for the assistance provided in thecomputation of LPOs in the CR3BP through high-order series expansions, and with Dr. J. Fontde-caba and Dr. V. Martinot (Thales Alenia Space)for providing the opportunity and the assistanceof undertaking the applied research of the QSOsaround Phobos.

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