IAC-14-C1.2.1 IDENTIFICATION OF NEW ORBITS TO ENABLE ...strathprints.strath.ac.uk/49981/3/Zamaro_M_Biggs... · Mattia Zamaro Advanced Space Concepts Laboratory, Glasgow, United Kingdom,

65th International Astronautical Congress, Toronto, Canada. Copyright c©2014 by M. Zamaro and J.D. Biggs.Published by the International Astronautical Federation, with permission and released to the IAF to publish in all forms.

IAC-14-C1.2.1

IDENTIFICATION OF NEW ORBITS TO ENABLE FUTURE MISSION OPPORTUNITIESFOR THE HUMAN EXPLORATION OF THE MARTIAN MOON PHOBOS

Mattia ZamaroAdvanced Space Concepts Laboratory, Glasgow, United Kingdom, [email protected]

James D. BiggsAdvanced Space Concepts Laboratory, Glasgow, United Kingdom, [email protected]

One of the paramount stepping stones towards NASA’s long-term goal of undertaking human missions to Mars is theexploration of the Martian moons. Since a precursor mission to Phobos would be easier than landing on Mars itself,NASA is targeting this moon for future exploration, and ESA has also announced Phootprint as a candidate Phobossample-and-return mission.Orbital dynamics around small planetary satellites is particularly complex because many strong perturbations are in-volved, and the classical circular restricted three-body problem (R3BP) does not provide an accurate approximation todescribe the system’s dynamics. Phobos is a special case, since the combination of a small mass-ratio and length-scalemeans that the sphere-of-influence of the moon moves very close to its surface. Thus, an accurate nonlinear model ofa spacecraft’s motion in the vicinity of this moon must consider the additional perturbations due to the orbital eccen-tricity and the complete gravity field of Phobos, which is far from a spherical-shaped body, and it is incorporated intoan 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 of potential mission ap-plications in the Mars-Phobos system are proposed: these results could be exploited in upcoming unmanned missionstargeting the exploration of this Martian moon. These applications include: low-thrust hovering and orbits aroundPhobos for close-range observations; the dynamical substitutes of periodic and quasi-periodic Libration Point Orbitsin the ER3BP-GH to enable unique low-cost operations for space missions in the proximity of Phobos; their manifoldstructure for high-performance landing/take-off maneuvers to and from Phobos’ surface and for transfers from andto Martian orbits; Quasi-Satellite Orbits for long-period station-keeping and maintenance. In particular, these orbitscould exploit Phobos’ occulting bulk and shadowing wake as a passive radiation shield during future manned flights toMars to reduce human exposure to radiation, and the latter orbits can be used as an orbital garage, requiring no orbitalmaintenance, where a spacecraft could make planned pit-stops during a round-trip mission to Mars.

I INTRODUCTION

Since the discovery of Phobos and Deimos in 1877, thetwo natural satellites of Mars have become increasinglyinteresting astronomical objects to investigate. Phobos iscloser to Mars than Deimos and almost double its size, butdespite this, they are very similar, since they share com-mon physical, orbital and geometrical features. Their ori-gin is still largely unknown [1, 2], and is currently debatedto have been either an asteroid capture by Mars, or coales-cence from proto-Mars or Solar System material, or evenaccretion of material from Mars ejected from its surfaceafter an impact with a previous small body. This puzzle issupported by the mysterious composition of these moonsinferred from infrared spectral analysis: due to their rela-tive low density and high porosity, they could hide a con-siderable amount of iced water [2], which is an attractivein-situ resource that could be exploited by human mis-sions. In addition, it is speculated that the Martian moons’rocks could provide evidence of alien life [3]. Phobos hassome unique characteristics that make it also astrodynami-cally interesting. The low altitude of its orbit around Mars

has produced speculation on its evolution: due to its tidalinteraction with Mars, its altitude is currently decreasing,which means Phobos will eventually crash into Mars orbreak up into a planetary ring [4].

Due to its proximity to Mars, Phobos is currently ofgreat interest for future missions to the Red Planet.During its Ministerial Council Meeting of November2012, ESA confirmed post-2018 mission concepts: theMars Robotic Exploration Programme would include amission (Phootprint) to return back to Earth a samplefrom Phobos [5, 6]. Another sample-and-return missionto this moon is currently proposed by NASA InnovativeAdvanced Concepts team, that will use two CubeSats pro-pelled by a solar sail and joined by a tether mechanism[7]. In addition, NASA has identified a mission to Pho-bos as a key milestone to be achieved before bringinghumans to Mars [8, 9, 10, 11], since the absence of atmo-sphere on Phobos and Deimos makes landing and take-offeasier for a manned spacecraft than on Mars. Therefore,the Martian moons could be exploited as outposts for as-tronauts: Phobos’ proximity and fast orbital period can

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provide a relay for robotic exploration on Mars, and pro-tection from space radiation hazards for manned space-craft orbiting Mars (Phobos’ bulk and shadow shieldingthe spacecraft). At the beginning of 2013, with the de-velopment of a new rover platform for the exploration ofminor bodies, consisting of robotic hedgehogs, it has beenreported that NASA is taking into consideration a mission(Surveyor) to Phobos as a test-bed for this new technol-ogy [12].

The purpose of this paper is to present a breakdown ofdifferent kinds of orbits that could be exploited in futurespace missions to Phobos. Section II introduces the readerto the physical environment connected to the orbital dy-namics and constraints of a spacecraft in the vicinity ofPhobos. The following sections III-VI showcase each ofthe different kinds of orbits around this moon, such as:hovering points using Solar Electric Propulsion (SEP);Vertical Displaced Circular Orbits with low-thrust; natu-ral Libration Point Orbits and their Invariant Manifold tra-jectories, and their artificial equivalent with constant low-thrust; Quasi-Satellite Orbits around Phobos. Section VIIprovides a summary of the different solutions focusing ontheir applications in space missions to Phobos, and it con-cludes the paper suggesting their potential usefulness in areal-world mission scenario.

II PRELIMINARY ANALYSIS FORA SPACE MISSION AROUND PHOBOS

In this section we introduce the basic design aspectsof the dynamics and physics of a spacecraft in orbit ofPhobos.

II.I Physical and Astrodynamical CharacteristicsThe immediately noticeable characteristics of Phobos

are its small size (even smaller than some asteroids) andits irregular shape: in particular the surface is markedby a dense texture of grooves and by several big craters,one of them, named Stickney, is by far the largest and islocated on the face of the moon pointing towards Mars.Phobos has an almost circular and equatorial orbitaround Mars, and it rotates with synchronous period andalmost zero-tilt with respect to its orbital motion. Thelow altitude of its orbit is lower than that required for aMars-synchronous rotation: Phobos rises from the Westmore than three times a day as seen from Mars’ surfacenear its Equator, whereas near the poles Phobos is neverseen, since it is always under the horizon. Table 1 presentsa summary of the physical and orbital parameters of Marsand Phobos that have been used in the analysis of the or-bits undertaken in this paper.

II.II Relative DynamicsThe general equations of motion (EoM) of the relative

Table 1: Physical and astrodynamical properties.Ephemerides source: NASA JPL at 25th July 2012 00.00CT(ICRF/J2000.0). Phobos axial tilt θ at date: mean value at epoch1.08. Gravity models: Mars MGM1025, Phobos [13].

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

orbital dynamics that will be used to describe the differentkinds of orbits presented in this paper are stated 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 theapparent acceleration of the general relative frame of ref-erence. aA is presented in Eq.2 as a function of theframe’s translational acceleration aA,T and angular ve-locity ω with respect to an inertial reference; aG is thesum of the gravity accelerations of the celestial bodies ofinterest, each defined as the gradient of the gravitationalpotential uG,⊕ = Gm⊕/||q− q⊕||, whereG is the gravi-tational constant,m⊕ and q⊕ are the mass and position ofthe body ⊕; aP indicates the thrusting acceleration of thepropulsion system of the spacecraft required for artificialorbits, while for natural orbits aP = 0. These three termsconstitute the model of the dynamics where the referencesignal of the orbit over time q(t) is solved, to be used bythe guidance system in the mission. This motion will beperturbed in the real world by the disturbance aD, con-sisting of the forces not considered in the model, and bythe perturbations on the initial condition q0, due to theinaccuracies of the navigation system; to track the guid-ance law, such perturbations need to be counteracted bythe station-keeping action aC of the orbital control sys-tem of the spacecraft, either planned in feedforward orperformed in feedback.

The study of the dynamics of a spacecraft about Pho-bos is conducted in the first instance with the model of theclassical circular restricted three-body problem (CR3BP)[14], where the two massive bodies are Mars (1) and Pho-bos (2), and the frame of reference is centered in the Mars-Phobos barycenter and aligned with the rotating Hill’s

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

Figure 2: Misalignment angle betweenthe equatorial and orbital planes of theMartian moons. Mean values dotted.

Figure 3: Differential perturbationsanalysis. Vertical dotted lines indicatePhobos major size and Hill’s SOI radius.

frame of the Mars-Phobos orbit, which is considered ke-plerian, circular, and equatorial. The EoM of the CR3BPare derived from Eq.1 using a constant vertical ω corre-sponding to the Mars-Phobos mean motion, and consid-ering the gravity aG1 and aG2 of the two massive bodiesplaced in fixed position in this frame’s choice. Using non-dimensional units, the only parameter of the CR3BP is themass factor µ, the normalized mass of the secondary bodywith respect to the total mass of the primaries, while thesemi-major axis of their orbit provides the length unit l.For the case of Mars and Phobos, two peculiarities are ev-ident from the physical parameters of Table 1: the massparameter of the system is very small (µ = 1.66·10−8),if compared to other cases studied so far in the Solar Sys-tem, and the length unit of the system is very small toosince the altitude of Phobos’ orbit is less than twice theradius R1 of Mars (R1/l = 36%), an unusual conditionfor a pair of primaries in our Solar System.

The Hill’s sphere of influence (SOI) is the regionaround each body where the dynamics is dominated byits own gravity field, and its radius for Phobos is 0.17%of the distance from Mars, and considering the fact thatPhobos is very irregular in shape the related maximumaltitude is only 3.5km, therefore it’s impossible to nat-urally orbit around Phobos with a Keplerian motion,as shown in Fig.1. In particular the microgravity envi-ronment of Phobos is characterized by a keplerian escapevelocity at its mean surface of only 11m/s, which meansa human being (or a rover) could auto-inject itself out ofthe body with a very small force.

The collapse of the realm of attraction of the secondarybody of the CR3BP towards its surface is a result of thetwo peculiarities of the Mars-Phobos system, which hassome additional physical and orbital features: not onlythe revolution of Phobos around Mars and the rotationaround its spin axis are synchronous, but they are also re-spectively equatorial and zero-tilted [15], therefore Pho-bos’ attitude, expressed by the body-centered body-fixed

reference frame BCBF, is approximately fixed in the ro-tating frame of the CR3BP, and so they differ only bythe definition of the Prime Meridian (PM) [16, 15]. Be-fore this choice, the actual misalignment between the twoframes oscillates between a minimum of 0.30 and a max-imum of 1.90, and the dynamics of this libration motionis much slower than the time-scale of a mission segmentaround Phobos (period of 2.26 terrestrial years), as pre-sented in Fig.2.

II.III Orbital PerturbationsAn analysis is undertaken to quantify approximately the

errors that occur in the Mars-Phobos system when it isapproximated with a CR3BP. The major physical orbitalperturbations are distinguished by gravitational and non-gravitational forces. In the first class, we have the net termthat when added to the basic newtonian point-mass forceprovides the true gravity pull of a general non-sphericaland non-uniform body, which is usually modeled with aspherical harmonics series expansion known as gravityharmonics (GHs): for a first analysis, we consider thedominant term for both Mars and Phobos gravity fields,which is known as J2 and is related to the oblateness ofthe body; the second type of gravitational perturbation isthe basic gravitational term of additional bodies, that inthe framework of the 3BP is referred as a forth body per-turbation, which is the sum of the body gravity and itsapparent force on the 3BP frame: for a basic analysis, weconsider the perturbing body in the closest conjunctionconfiguration with Phobos. In the second class, we havethe pressure disturbances of the atmospheric drag and theelectromagnetic radiation: Phobos does not have an at-mosphere, and the atmosphere of Mars is negligible atPhobos’ altitude, therefore we consider the radiation pres-sures of the Sun (SRP), Mars (MRP, enclosing also theportion of the albedo of the SRP), and Phobos (PRP, withits albedo). These perturbations require knowledge of ad-ditional technical parameters of the structure and the sub-

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systems of the spacecraft, so for a first analysis we con-sider common mean values for them. In addition to thesephysical actions, we must consider the modeling pertur-bations represented by the approximation of the dynamicsas a circular R3BP, which is the effect of the eccentricityof the Mars-Phobos orbit. To derive a related physical ac-celeration value, we consider the difference between theacceleration field of the CR3BP and the one of the ellipticmodel of the Mars-Phobos R3B system, where ω and theprimaries’ positions are variable, and we take the maxi-mum over the orbital phases of Phobos, with the referenceof the same relative state with respect to Phobos.

An important point to consider is that Phobos’ orbitaround Mars is not keplerian, but it seems to follow a clas-sical low altitude J2 perturbed orbit. Therefore, the anal-ysis of every orbital perturbation which is not due to Pho-bos (gravity and PRP) in its proximity must be conductedin the framework of the relative dynamics with respect toPhobos, considering the resulting differential perturba-tion.

Since this analysis is undertaken to derive a basic refer-ence for the orbital perturbations, it will be applied for thesimple case of fixed relative points. Also, as we are inter-ested in the dynamics near Phobos, due to the small size ofits SOI, all the perturbations are nearly isotropic, and theonly variable for this simple analysis is the radial distancefrom Phobos along the Mars-Phobos direction (apart fromthe eccentricity, which has been evaluated along all threedirections). Outcomes of the differential analysis are pre-sented in Fig.3, where the perturbations are shown as a ra-tio aP/a2 with respect to Phobos’ keplerian gravity termin the point, and they correspond to [17]. In conclusion,the CR3BP does not provide an accurate approximation todescribe the Mars-Phobos system’s dynamics: the grav-ity harmonics and the orbital eccentricity of Phobosare the main orbital perturbations in proximity of themoon, and outside its Hill’s SOI boundary the eccen-tricity becomes the dominant term, with Mars J2 be-ing the second largest.

II.IV Radiation EnvironmentThe microgravity that characterizes the space environ-

ment has important implications on the spacecraft struc-ture and subsystems design, as well as for human crewphysiological and biological effects. In particular, the ion-izing part of the space radiation in the Solar System,which is not shielded by the atmosphere and magneticfield as it happens here on the Earth surface, is currentlyconsidered the most challenging engineering aspect in de-signing a safe manned mission in deep space [18].

The Sun’s activity is variable and constituted by grad-ual radiative and particle production and by impulsiveparticle events, the latter emission is collectively calledSolar Energetic Particle Events (SEPEs): they are high-

energy charged particles and they constitute the hazardousionizing part for the organic tissues. The low-energycharged gradual particles (mainly protons and electrons)constitute a plasma flow, called the solar wind. Its in-teraction with the magnetic field of a planet producesthe so-called Magnetosphere: this is a region where thecharged particles remain trapped by the magnetic fieldlines, known as radiation belts, but despite they are haz-ardous when crossed, they provide the natural shield thatprotects life on the Earth’s surface (in combination withthe ozone layer counteracting the UV rays), satellites inLEO, and in particular the crew of the ISS, from outerspace radiation.

In addition to the Sun, there is a second very impor-tant radiation source in the Solar System known as Galac-tic Cosmic Rays (GCRs). These are gradual high-energycharged particles that originate from interstellar space.

Focusing now on the Mars-Phobos system, the Marsmagnetic field is very weak, so no trapped particles (andrelated shielding) constitute the radiation environment fora mission following the orbit of Phobos, which is similarto a deep space environment at the Sun-Mars distance,constituted by two main sources: the protons from SEPEsand protons and alpha particles from GCRs. For applica-tions to future manned mission to Phobos’ orbital environ-ment, we conducted an estimation analysis with the open-source SPENVIS program [19] and its dedicated modelfor Mars MEREM [20]. To derive an approximated fig-ure of the gross effect of the radiation environment (with-out any shielding effect of the spacecraft structure) to hu-man factors, we consider the dosimetry quantity called theEffective Dose (Ef.D., whose IS unit is the Sievert, Sv),which represents the amount of energy that the radiationdeposits in 1kg of the material’s mass, averaged throughboth the incoming radiation and the reference tissue com-positions.

The result obtained for the gross radiation hazards fora mission in Phobos’ orbit from 2010 to 2030 is Ef.D.=1.9Sv/y, 1.1Sv/y from SEPEs protons and 0.8Sv/yfrom GCRs protons and alpha particles. This should nowbe compared with the estimated allowable dose amountfor astronauts, which is based on the recommendations ofthe National Council for Radiological Protection (NCRP)and is currently used by both NASA and ESA [21]. Con-sidering the case of a 35-year old astronaut, the figure de-rived from our analysis, for a Martian orbital segment ofone year without any structural shielding, falls inside therange 1.75-2.5Sv that indicates the maximum amount ofradiation dose that such human crew could be allowed toabsorb throughout the entire mission. Thus, the develop-ment of a strong shielding strategy for crewed missions isrequired. An interesting idea that has recently gained at-tention, is that a manned spacecraft during a Mars orbitalmission segment could exploit Phobos as a passive radia-

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Figure 4: Sun-Mars eclipse at Phobos.Eclipse times are defined for a singleeclipse during one Phobos’ rotation.

Figure 5: Sun-Phobos eclipse. Lightfunction of the dual-cone model. CR3BPframe, Sun in superior conjunction.

Figure 6: Sun-Phobos eclipse. Meanlight function versus distance from Pho-bos on the anti-Sun surface of motion.

tion shield: staying in its shadowing wake would theoret-ically counteract the gross Ef.D. of the directional part ofthe SEPEs, while simply remaining close to the moon willblock any incoming isotropic particles (remaining SEPEsand GCRs) as much as its bulk covers the sky. Followingthis idea, in this paper we investigate possible orbits to beused also for shielding purposes about Phobos. However,some additional albedo effects (of the GCRs neutrons)could be relevant, and also covering the field of view ofthe Sun has not been proved to be relevant, as the scatter-ing effect of the particles along the Heliosphere’s lines offield is still a current topic of research [22].

II.V Lighting ConditionsIn this subsection we analyze and quantify the lighting

conditions around Phobos, in particular we describe theshadowing opportunities that could exploit Phobos as anatural shield against the directional solar radiation.

Since we are interested in Phobos’ neighborhood, theanalysis is conducted in the CR3BP reference frame butcentered in Phobos, aligned with its BCBF frame. Thekinematics must now consider also the motion of the Sunin this frame, using a restricted four-body model to eval-uate the field of view of the Sun over time for pointsaround Phobos. In particular, the Mars heliocentric or-bit is the second most elliptic among the Solar Systemplanets, and Phobos’ orbit is equatorial, with the resultingorbital plane inclined with respect to its ecliptic planeby Mars’ rotational tilt θM = 25.19. The resulting Sunadimensionalized position vector in the CR3BP frame ro-tates clockwise with an angular velocity equal to the dif-ference between Phobos and Mars revolution rates (dom-inated by the first), with a fixed declination in the range[−θM , θM ] according to the seasonal phase of Mars (sea-sons of Phobos correspond chronologically with Mars’ones, so we refer to them without any distinction).

The analysis of the shadowing effects in this systemis undertaken using eclipse modeling, which is to derive

the zones of light and shadow produced by a shadowingcentral body ⊕ when illuminated by a radiating body ,described by a scalar light function field L,⊕, rangingfrom 0 to 1 to express the ratio of incident light with re-spect to the complete light case (the shadow function Sis the 1-complement of L). The most accurate dual-conemodel in Fig.5 is able to discriminate positions of com-plete light (L = 1), complete shadow (L = 0) inside aconic wake, and penumbra (L ∈ (0, 1)). When the posi-tions of interest are very close to ⊕ and when there is agreat difference between the two bodies dimensions, theanalysis could be simplified to a cylindric model, wherethe shadowing wake is cylindrical and the transition zonesof penumbra vanish, so positions are only in completelight or shadow.

The approach we used is to analyze the shadowing ef-fect of each couple of bodies. The first case is the Sun-Mars couple. Here we are interested in the value of L atthe Phobos location. Using the dual-cone model the Pho-bos SOI is a near and small domain in the Mars shadowingwake, therefore the analysis of this couple is undertakenwith the cylindric model. The Sun-Mars L1 over time atPhobos has a small-period variation due to the fast revolu-tion of Phobos, and a long-period variation due to Mars’revolution; the latter motion inclines the Mars shadowcone with respect to the 3BP orbital plane, such that dur-ing winter and summer Phobos is constantly in light,without Martian eclipses: due to Mars high eccentricity,seasons are unequal, with the Northern Hemisphere ofMars and Phobos experiencing a summer longer than thatoccurring in the South. Fig.4 summarizes the outcomes:the maximum eclipse time at the equinoxes is 54min, cor-responding to 12% of Phobos daytime; summer’s totallight period is about 164 days (3 Martian months), andin the winter this is about 110 days (2 months).

The second case is the shadowing effect provided bythe Sun-Phobos couple. For this preliminary analysis,the mean spherical shape of Phobos is considered. Like

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Figure 7: Sun-Mars-Phobos eclipse. Anti-Sun daily orbit inspherical coordinates around Phobos over the Martian year.

for the Sun-Mars L, the cylindric approximation is suit-able in proximity of Phobos, but now the Sun-Phobos L2

is a time-variant 3D field: since this is axially symmetricalong their rotating conjunction line, it is defined on thehalf-plane whose reference axis is such line. The meanintegral value L2 along one Phobos’ revolution period, fora given distance to Phobos, is minimum on the surface ofmotion where the conjunction line between the anti-Sunand Phobos revolves, shortly becoming 1 in points out ofthe surface: such minimum value rapidly increases withthe distance from Phobos. This is shown in Fig.6: fromL = 50% at the body’s surface, L = 78% at the SOI’sboundary, L = 83% at 2 Phobos radii.

The last shadowing case is the Mars-Phobos couple.Since the radiation of Mars (without the albedo) is insidethe IR spectra, such eclipse analysis is neglected becauseit brings little variation to the radiation hazards thatwe aim to reduce (MRP flux at Phobos is two orders ofmagnitude lower than SRP one).

The conclusion of the analysis of the shadowing effectsis now obtained combining the previous single couplesinto the system of three massive bodies of the R4BP,focusing the analysis in Phobos’ neighborhood. TheSun-Mars L1 is simply a scalar value along the Martianyear, while the Sun-Phobos L2 field must now considerthe real dynamics of the Sun, which is moving in theCR3BP frame of reference (actually we are going to con-sider the direction of the anti-Sun ′ because it’s moreimmediate to relate it with the position of the shadowingwake of Phobos). Fig.7 shows a complete orbit of the anti-Sun for every season: the cylindrical shadowing wakeof Phobos revolves around its spin axis with the periodof Phobos, and varies its inclination with the seasonθ′ ∈ [−θM , θM ] (when the anti-Sun is in the NorthernHemisphere, we are in winter). Besides, Phobos realmis in complete shadow when a Martian eclipse occurs,which is when the anti-Sun is close to the positive direc-tion of the 3BP x-axis frame, enduring from a maximumat the equinoxes to zero during summer and winter.

The approach taken to compute the light function of theR4B model in proximity of Phobos follows the following

Figure 8: Lighting conditions around Phobos. On the left,light field around Phobos in the radial-vertical plane of theCR3BP frame, averaged over a year, a spring equinox month, asummer solstice month. On the right, corresponding plots func-tion of the declination, evaluated for some radial distances, showalso right ascension dependency, where upper/lower border ofthe filled area is for points in superior/inferior conjunction posi-tions, and black line is for points in quadrature.

procedure. First we compute the Sun-Phobos mean inte-gral field L2 along one Phobos’ revolution for differentseasons. Due to the axial-symmetry of L, the desired in-tegration of a 3D field along time is simplified to an un-coupled 1D integral in polar coordinates, and it is alsosymmetrical along the spin axis. For the particular caseof a cylindrical light function, the analytical solution isderived in Eq.4, where r and ϕ are the radial distance anddeclination over the equatorial plane of the point consid-ered, ad R is Phobos’ radius.

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 of theMartian year is then straightforward.

But before doing this, the second step is to extend theSun-Phobos daily L2 to take into consideration the cor-rection due to the Sun-Mars L1 at the current day pre-viously derived, to obtain the aimed Sun-Mars-Phobosmean 3D light field L12 that models a coupled 3B eclipse.This is far from an easy operation, but since both 2B

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Figure 9: Sky occultation around Phobos. Occultation fieldaround Phobos in the radial-vertical plane of the CR3BP frame.

light fields are cylindrical, and their shadowing wakeshave very different dimensions, the instantaneous com-bined L12 around Phobos is the logical conjunction ∧ ofthe two single instantaneous light functions, which resultsin their product: L12 = L1 ∧ L2 = L1L2. This allowsus to compute the sought mean integral field L12 overone Phobos’ revolution with only the information of thetwo single daily values L1 and L2, with the deal of intro-ducing a further variable ψ, which is the right ascensionwith respect to Phobos of the point analyzed. This is de-scribed in Eq.5. L2 is axially symmetric along the spinaxis, being the same for points with same ϕ: they havethe same profile L2(t), but shifted along time for differentψ. Therefore the integral of the product L1L2 producesa different L12, expressed by the correction term ∆t/Twhich is the percentage of time in one Phobos’ revolu-tion period T such that L1 is in shadow and L2 is in light.This correction will be maximum for points facing Marsand minimum for those on the other side as presented inEq.6, while the values in quadrature become closer to themaximum 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 distinguishseasonal and yearly averaging. The seasonal tilt inclinesthe shadow wake, so complete light and one cone ofcomplete shadow appear in the Phobos polar regions:the cone’s maximum altitude, using a mean ellipsoidalmodel for Phobos shape, is of 1.4km. Instead duringspring and fall, no complete shadow zones are present,and the minimal daily L at the day of equinoxes is 38%along the sub-Mars meridian onto the surface of the moon(this without considering its orography and morphology).Considering now the annual L, yearly averaging drasti-cally increases the light conditions. Due to Mars’ eccen-

Figure 10: AEPs of the Mars-Phobos system. Iso-surfacesslices of propulsive acceleration magnitude (logarithmic scale).

tricity, Southern regions experience more shadow timethan the upper counterpart; due to the Martian eclipses,points in-between Mars and Phobos and close to themoon experience more shadow time per annum.

In conclusion, this analysis provides the lighting con-ditions for a spacecraft orbiting Phobos. Focusing on theshadowing opportunities, a fixed observation point in the3BP frame could provide relevant reduction of the FOV ofthe Sun for long-period station-keeping only if it is insidethe SOI and onto the equatorial plane, and pointing Mars;instead, a shorter period could provide continuous shad-owing opportunities for points over the poles during thesolstice seasons, in particular inside the Southern polarcone during summer. Instead, for middle seasons the min-imum of the light field moves towards lower latitudes, andrelevant reduction of the Sun FOV is obtained only veryclose to the surface. Therefore during equinoctial or longobservation periods, the lighting conditions around Pho-bos are close to experience continuous light, up to 88%due to the unavoidable Martian eclipses. Suitable shad-owing exploitation could be obtained only using orbitsthat track the daily anti-Sun path, such a vertical displacedcircular orbit around the spin axis. We acknowledge thata complete shadowing is not possible with current tech-nologies, because a spacecraft requires sunlight for theelectrical power generation from the solar arrays, withthe solar flux decreasing with the distance from the Sun.

II.VI Sky OccultationIn this subsection we consider the possible exploitation

of Phobos as a natural shield against the isotropic cos-mic rays (SEPEs and GCRs): the idea is that the incom-ing radiation on a spacecraft is lowered proportionallyto the filling fraction in the sky of the apparent size ofthe body’s bulk, as seen by the spacecraft’s location. Inastronomy, when a body is totally or partially hidden bythe bulk of another one that passes between it and the ob-server, we speak about occultations or transits. Since inthis case the hidden body is the total background sky, inthis paper we refer to this action as sky occultation.

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Figure 11: AEPs of the Mars-Phobos system. On the left, planar stability region (in green) around Phobos. Following figuresshow the inner boundary of the 3D stability region around Phobos (in the last picture, only one half of the lower hemisphere isrepresented to visualize Phobos mean sphere).

A point-like observer sees an object with an appar-ent shape that corresponds to the area that it covers on asphere centered on the observer, and radius equal to theirdistance. In 3D geometry, the area subtends the 2D solidangle Ω on the unit sphere (whose IS unit is the stera-dian, sr), and the total spherical surface has Ω = 4πsr.For a general body placed at distance d from the observer,Ω =

∫∫ 2π,π

0,0M(ϑ, φ) sinϑdϑdφ. M is the mask func-

tion of the body, which is a binary function of the polarand azimuthal spherical coordinates ϑ and φ centered onthe observer, whose value is 1 or 0 if the related directionfrom the observer intersects or not the body: it representsthe apparent shape of the body on the unit sphere. Theratio with 4πsr represents the filling fraction of the bodywith respect to the background.

The occulting bodies in our case are Mars and Phobos,while the Sun is neglected because it is very small as seenfrom Phobos. The approach is similar to the one under-taken for the lighting conditions, defining an occultationfunction field O⊕ which represents the bulk/sky fillingfraction of the occulting body ⊕. This analysis is easierbecause in the CR3BP frame Mars and Phobos are fixed,and their O does not depend on time. For a first anal-ysis, we consider the mean spherical shapes for the twobodies, of radius R. In this case, the occulting body fillsa spherical cap on the unit sphere, with apparent angu-lar radius α = arcsin(R/d), and O = 1−cosα

2 , which isspherically-symmetric. First we consider the occultationof Mars, at the Phobos location, since the region of inter-est is small: the result is O1 = 3.4%. Second we considerthe occultation of Phobos, which depends only from theradial distance from the body. This function starts fromO2 = 50% on the surface (astronauts staying inside of adeep crater would we shielded also laterally by the moun-tain ridge), and then decreases rapidly: O2 = 13% at theSOI’s boundary, O2 = 7% at 2 Phobos radii. The conclu-sion of the analysis of the sky occultation is obtained com-bining the previous single effects. This requires to deter-mine if the apparent shapes of the two bodies’ bulks inter-

sect, and how much they overlap: such axially-symmetric3D field corresponds to the light function of the Mars-Phobos coupleL1,2 that we avoided to compute in the pre-vious lighting conditions analysis, but we do need here. Inparticular, this light function must be computed with theaccurate dual-cone model, since due to the proximity ofMars, the shadow cone vertex of Phobos is located only at2.77 Phobos radii in the anti-Mars direction, therefore itsinclination inside the SOI is not negligible. The resulting2B combined occultation function is O12 = O2 +L1,2O1

and it is shown in Fig.9.This analysis highlighted that mild but relevant reduc-

tion of the isotropic SEPEs and GCRs by using the bulk ofPhobos to occult part of the celestial sphere is obtainedinside the SOI of the moon. Besides, points on Mars’ sideand over the poles experience an additional but small re-duction due to the occultation of Mars. Orbits that re-main inside the Phobos’ SOI are therefore suitable toenhance the radiation protection of the spacecraft byexploiting Phobos’ bulk as a passive radiation shield.

III HOVERING POINTS AROUND PHOBOSA simple trajectory for a mission around Phobos is pro-

vided by maintaining a fixed position with respect toits BCBF frame: due to the small µ, this is similar to aMartian keplerian orbit close to Phobos, analogous to theTrailing/Leading configurations used in Formation Fly-ing [23]. The analysis of these trajectories is undertakenadding a constant propulsive acceleration aP to the EoMof the CR3BP, which will be referred CR3BP-CA. Theaim of the hovering in a point is therefore to counteractthe natural acceleration of the CR3BP, and thus leading toan Artificial Equilibrium Point (AEP).

aP = aA − aG (7)

Recall that SEP Hall/ion thrusters operate roughly inthe medium range of 0.01mN -0.1N , new generationFEEP and colloid thrusters provide low-thrust down to

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1µN scale, and electrothermal propulsion (resistojets andarcjets) supplies the higher ranges up to 1N , but the levelsof propulsive acceleration must be scaled accordingly tothe mass of the spacecraft (100kg for a medium-size inter-planetary satellite, 100 times larger for a big manned mod-ule). Fig.10 presents the iso-surfaces of the thrusting ac-celeration level required to hover around Phobos: as faras the propulsion grows, AEPs could be further displacedfrom the natural equilibrium points of the CR3BP,which are the three collinear libration points (LPs) alignedwith the two bodies (two close to the secondary, L1 ininferior and L2 in superior conjunction, and L3 in oppo-sition), and the two equilateral LPs L4 and L5 equidis-tant from them in the two quadrature configurations; inparticular, the cis/trans couple of LPs L1−2 is located onthe boundary of the Hill’s SOI of the secondary, at an al-titude of 3.5km from Phobos. Despite this proximity,the thrust level required to establish an AEP displacing acollinear LP is very demanding. Instead, displacing theequilateral LPs L4−5 is very cheap and effective, ar-riving close to Phobos along the y-axis still with smallvalues of thrust; on the other end, establishing AEPs overpolar regions requires high thrust levels.

The next step in the hovering analysis is to look for thestability of the AEPs analyzing the linearized CR3BP-CA: since the propulsive acceleration is constant, the lin-earized system coincides with the one of the CR3BP. The3D stability requires the computation of the eigenvaluesλ of the 6D linearized state-matrix, that contains the 3DHessian matrix Hu of the gravity potential evaluated atthe current AEP. Due to the sparse structure within thestate-matrix, it is possible to derive a condensed analyti-cal expression of the three couples of opposite eigenvaluesin terms of the three scalar invariants of the hessian ma-trix (which is symmetric) I1, I2, and I3. To do so, weintroduce some coefficients: A, B and D are strictly realscalars by definition, instead C is complex. The expres-sion of the three couples of eigenvalues is expressed in aconvenient symmetrical form in the complex field in Eq.9.

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

(8)

λ = ±√A+ B

C e(π−θ)i + Ceθi, θ = 0,±2π/3 (9)

The linear Lyapunov marginal stability for the AEP re-quires all the eigenvalues to be purely imaginary, whichis their squares to be real and negative. To solve thesetwo constraints, we use the magnitude-phase notation forC = |C|eθC : the first constraint is satisfied by one sim-ple relationship betweenB and C, which is the necessarycondition for real eigenvalues. Using this, Eq.9 could be

Figure 12: SS-VDCOs around Phobos. AEPs in the x-z planeof the SS-rotating frame. On the left, linearized stability regionfor AEPs in the Phobos 2BP (magnification of the inner section).On the right, ∆v for one period for AEPs in the CR3BP.

rewritten in a compact way where the six solutions derivefrom the same definitions of C as a cubic root of Eq.8.The second stability constraint consists of one simpleinequality, which considers the algebraic root of C withmaximum real part (there is at least one that is strictlypositive-definite).

λ = ±√A+ 2 |C| cos (θC + θ), θ = 0,±2π/3 (10)

B = −|C|2A+ 2maxRe C < 0

(11)

This approach extends the result obtained for only theplanar case of the CR3BP-CA of [24]. For the Mars-Phobos system, the 3D stability region is made of threerealms: one central ring, and two symmetric half hyper-bolic coronas placed at very high out-of-plane altitudes.More interesting as presented in Fig.11, the inner surfaceof the ring is distorted in proximity to the second mas-sive body leaving outside the body’s SOI. The planar sta-bility region in the orbital plane is a thin corona extendingalong the Mars-Phobos orbital distance, that comprisesthe equilateral LPs and cuts off the three collinear LPs,and in proximity of the secondary body, the inner sta-bility region boundary is distorted to represent a three-leaves clover: this corresponds to the outcome in [25].Recall that this is the linearized stability region, which forthe case of marginal stability encountered by the AEPsdoes not assure stability in the original CR3BP-CA. How-ever, it has been proved by analysis of higher order termsthat the AEPs of the linearized stability region are alsostable in the full nonlinear dynamics apart from singularcases that lie on co-1D domains of the stability region,where resonance effects are present [24].

If we compare the stability region with the equi-thrust curves in the orbital plane, it can be seen that theplanar stability boundary is not too far from Pho-bos, starting from 25km (in the petal-head connectionborder, requiring 1.9mm/s2), 71km (on the tip of thetop and down leaves, 0.4mm/s2) or 81km (along thex-axis, 12mm/s2) and arriving to 400km (along the x-

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Figure 13: LPOs in the Mars-Phobos ER3BP-GH. On the left, the two iso-periodic families of POs around the oscillating LPs.In the center, the families of 2-tori: A (red), B (green), C (magenta), D (cyan). On the right, example of 3-tori of different size andwidth around the LPs: three medium-size QPOs of family AB (red and green with small-width, orange with high-width) and twohigh-width QPOs of family C (around the smallest and biggest orbits). Shape harmonics series expansion for Phobos surface.

axis, where the outer boundary lies). Therefore six attrac-tive positions for medium distance observation of Phobosare identified: four minimum-distance AEPs, and twominimum-control AEPs, all obtained by the displace-ment of the equilateral LPs and affordable by current lightelectric thrusters. Further trailing/leading orbits aroundMars provide attractive cheap, stable, in-light fixed po-sitions with respect to Phobos at long distances from themoon. All the AEPs available with a low-thrust level arenot stable near the collinear LPs, and AEPs closer to Pho-bos, used to maximize the shadowing time and the skyoccultation ratio, or perform short dedicated operations,are feasible only with heavy or multiple thrusters, andmust take into consideration in the model also the com-plete inhomogeneous gravity of Phobos: this increases theprecision, frequency and computational load of the GNCsubsystem. Regarding the 3D boundary, it is possible tohave stable AEPs above the poles of Phobos, but at a greatdistance (from 250km to 1400km) and with high thrust.

IV VERTICAL DISPLACED CIRCULAR ORBITSAROUND PHOBOS

The AEPs computed in the previous section are a fixedsolution in the rotating frame of the CR3BP. It’s possibleto further generalize the concept, which is to look at thedynamics in a general uniformly rotating frame, and findthe related AEPs. In the usual CR3BP, such solution re-sults in a circular orbit around a reference axis, which inour case will be Phobos’ vertical axis: it’s called a Verti-cal Displaced Circular Orbit (VDCO). Therefore, if theangular velocity of the VDCO is opposite to the one ofthe Phobos revolution, and the declination of the VDCOis opposite to the one of the Sun, a spacecraft along thisorbit experiences constant light or shadow conditionsduring a season, realizing a Sun-Synchronous (SS) sea-sonal orbit whose β-angle performance (expressing themean time in light) is set by the choice of the initial phasealong the VDCO with respect to the Sun. In particular, for

continuous shadowing (β = 0) the spacecraft tracks theposition of the anti-Sun moving clockwise around Pho-bos: this is expressed in Fig.7. These displaced orbits areusually highly non-keplerian: here, we are going to con-sider the type III orbits of [26], which have fixed period,synchronous with the Sun around Phobos, and looking toincorporate low-thrust.

The approach is to start from the 2B dynamics aroundPhobos in a rotating frame R, defined by Eq.1 taking ωopposite to the Mars-Phobos mean motion and consid-ering only the gravity of the moon. The model is axi-ally symmetric around the vertical axis, therefore it is de-fined either in the positive x-z plane of the R frame, orin the polar counterpart R-δ. The SS-VDCOs are AEPsof this dynamics maintained by a constant acceleration inthe rotating frame’s components from Eq.7: in case ofcomplete light or shadow applications, the declination ofthe AEP must be equal to the seasonal anti-Sun declina-tion, therefore the domain of interest is 0 ≤ δ ≤ θM , or0 ≤ z ≤ xtanθM . However, shadowing is provided notonly by strictly pointing toward the anti-Sun, but remain-ing just inside the eclipse wake: using a mean ellipsoidalmodel for Phobos’ shape, a SS equatorial circular orbitwith radius 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 the re-lated linear Lyapunov stability region. There is one nat-ural equilibrium point that corresponds to the keple-rian equatorial circular orbit with a SS period achievedat the distance of R = 2.16 Phobos radii (mean altitudeof 11.8km), and no other local minima of the thrust level.The three realms of the linear stability region in Fig.12are similar to the ones of the 3D stability region for AEPsof the CR3BP-CA, swapping the Mars-Phobos barycen-ter and Phobos with Phobos and the keplerian equilibria,and without the distortion of a second massive body. Forplanar SS circular orbits this means that orbits with ra-

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Figure 14: LPOs lighting conditions. Light function of thefamilies of POs around L1 of the CR3BP-GH (parameterized bythe differential Jacobi integral with respect to L1, in logarithmicscale), averaged over 10 PO periods, at the days of equinoxes(lower cluster) and solstices (upper cluster). Filled area spansvalues for different starting phases of the Sun (thick line is formean values), where families colors are coherent with Fig.13.

dius in the interval [R, 3√

9/2R] are linearly stable in the2B dynamics, while stable SS-VDCOs could be obtainedat higher distances with declination δ = 35.26 and highthrust.

We now look to the solution of VDCOs around Phobosin the framework of the CR3BP. The dynamics will bedescribed in the same rotating frame R of the VDCO cen-tered in Phobos: the EoM of this model are obtained fromthe previous one adding the time-variant gravity of Mars.Since its location is not fixed in this frame, and consider-ing that now Phobos moves around Mars, an additionaltime-variant apparent acceleration must be enabled inEq.2, aA,T = aG1(0), equal to the Martian gravity at theorigin of the R frame. And finally, the Phobos revolutionaffects also the inertial reference of the R frame: since theR frame must move clockwise with respect to the PhobosBCBF frame, and now the latter is rotating in the oppositeway, Eq.2 must be used with ω = 0. Indeed, in a short-time analysis, the Sun is approximately fixed with respectto an inertial frame centered on Phobos, therefore realiz-ing a SS-VDCO around Phobos corresponds in maintain-ing an inertial fixed point dragged along the Mars-Phobosorbit.

The system is no longer axially symmetric, but the re-quired acceleration is periodic so we consider the maxi-mum level of the thrust profile and its cost over one pe-riod, defined by the ∆v and presented in Fig.12. Beingthe problem time-variant, no natural equilibria are avail-able. The max thrust level for a 100kg spacecraft at small-medium distances from Phobos is affordable only withhigh thrusters. The minimum cost still happens in a regionclose to 2 Phobos radii, up to the maximum Sun’s declina-tion at the solstices, but this is large (∆v= 50m/s) mak-ing the demand for the propulsion system very high.Besides, a linearized Floquet stability analysis of theseSS-VDCOs showed us that they are highly unstable.

Figure 15: LPOs occulting conditions. Sky occultation func-tion by the Phobos’ real bulk (modeled by the shape harmon-ics series expansion) of the families of POs around L1 of theCR3BP-GH (parameterized by the differential Jacobi integralwith respect to L1, in logarithmic scale), averaged over 1 POperiod. Families colors are coherent with Fig.13. Additionaloccultation by Mars’ bulk will be 3.4%.

Following the idea of exploiting the shadow of Pho-bos to protect a spacecraft from directional solar radiationduring the orbital station-keeping at the Mars-Phobos or-bital distance, in this section we analyzed the most sim-ple and straightforward orbits around Phobos to trackthe anti-Sun motion, which are the SS-VDCOs. Due tothe strong influence of the Mars’ third body perturbation,the SS-VDCOs around Phobos require a huge amount offuel consumption, which is infeasible over one period. ASS-VDCO around Mars is located out of its SOI, whilethe cost of a SS-VDCOs around Deimos at 5 mean radii(24.0km of mean altitude) has resulted to be of 16m/sper period, and it requires lighter thrusters.

V LIBRATION POINT ORBITS AND THEIRINVARIANT MANIFOLDS AROUND PHOBOS

In the framework of the classical CR3BP [14], aroundeach of the collinear LPs L1 and L2 there exist acentral manifold characterized by families of periodicorbits (POs) (the two branches of planar and verticalLyapunov orbits, and the two branches of Northernand Southern Halo orbits), and quasi-periodic orbits(QPOs) around them (know as Lissajous orbits). TheseLibration point orbits (LPOs) are highly unstable andso their natural motion needs to be computed with highprecision to provide low cost tracking opportunities[14, 27, 28]. Moreover, these LPOs are separatrices ofmotion between transit and non-transit orbits to enteror escape from the SOI of the second massive body:the boundary of these tubes is given by the InvariantManifolds (IMs) of the LPOs that provide the energy-efficient trajectories to minimize the fuel consumption ofspacecraft for interplanetary transfer phases.

From our preliminary analysis of the dynamics inproximity of Phobos in Section II.II, we found that the

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Figure 16: IMs of the LPOs in the Mars-Phobos ER3BP-GH. On the left, inside branch of the tube of unstable IMs from thefamilies of 3-tori LPOs. In the center, related performances of the trajectories that provide the min incidence at the touch-downfrom the family AB of L1 as a function of the longitude and latitude of the landing site on Phobos surface modeled through shapeharmonics series expansion: landing velocity modulus, angle of incidence, downward vertical velocity. On the right, performancesof the stable IMs of the same family that provide the min velocity total magnitude at the launch.

altitude of the LPs moves very close to Phobos’ irregularsurface: in this situation, the dynamical approximationprovided by the CR3BP falls short, and in particular wefound in Section II.III that to describe the natural relativemotion inside this moon’s SOI, its highly inhomoge-neous gravity field, and its orbital eccentricity must betaken into account. Due to their highly unstable behavior,measured by the Floquet stability indexes, the familiesof LPOs, computed with the classical methodologiestailored for the CR3BP, are therefore not reliable forpractical applications, because their reference signaltracking will require a high station-keeping cost.

In [29] the dynamical substitutes of the LPOs werederived in a more realistic model that considers thesetwo major orbital perturbations in proximity of Phobos.The modeling of the complete gravity field of convexbodies is provided by a spherical harmonics seriesexpansion, known as gravity harmonics (GHs). Froma previous paper [13] that collects the data obtainedthrough Viking observations, we are provided with amodel of Phobos’ gravity field. The addition of theGHs in the dynamics is particulary suitable for theMars-Phobos system because the CR3BP frame andthe Phobos BCBF frame are approximately fixed withrespect to each other (see Fig.2), which makes this systemso unique to remain time-invariant. The EoM of theMars-Phobos ER3BP-GH are derived in the Hill’s frameof the Mars-Phobos orbit centered in Phobos, from Eq.1considering 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 eccentricityof the Mars-Phobos orbit, and ν is the true anomaly ofPhobos), aA,T = aG1(0) like before for the VDCOs,and defining the gravitational potential of Phobos uG,2 asa truncated series expansion of GHs (J, λ)m,n throughLegendre polynomials Pn,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)

(12)where the potential is defined as a function of the spher-ical coordinates r,ϑ,ψ of the Phobos BCBF frame, andso the gravity acceleration of Phobos in the Hill’s frameis retrieved rotating back the components of the sphericalgradient of the potential.

The procedure in [29] to compute the LPOs in anER3BP-GH makes use of the numerical continuation(NC) technique, which consists of the iteration of theclassical differential corrector (DC) to compute POs andQPOs in a single dynamical system. First, we iden-tify LPOs in the Mars-Phobos-spacecraft CR3BP usingthe classical tools of dynamical systems theory, and thennumerically continue a parameter that incrementally in-creases the GHs of Phobos, to find their dynamical sub-stitutes in the final CR3BP-GH. The introduction of theGHs produces families of POs and QPOs no longer sym-metric and highly tilted and distorted from the classicalcase. These new LPOs are then continued again using theeccentricity as the perturbation parameter: despite the factthat the orbital eccentricity of Phobos is not particularlyhigh, the perturbation has a significant effect on the LPOs,as an indirect effect of the long-cited collapse of the SOItowards the moon. The effect of the eccentricity makes themotion to oscillate around these solutions with a consid-erable amplitude of 260m for the proximity of Phobos.

The resulting LPOs of this improved dynamicalmodel considerably lower the station-keeping demandexploiting the natural dynamics of the system. Theyare showcased in Fig.13: around each cis/trans-side ofPhobos, they are constituted by a 1-parameter family D ofiso-periodic POs vertically developed with the period ofPhobos revolution around Mars, three 1-parameter fami-

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Figure 17: Artificial LPOs and their IMs with constant acceleration in the Mars-Phobos CR3BP-GH. First two graphs showthe example of one natural medium-size periodic LPO of the family A around L1 and the trajectories (propagation time of 2 LPOperiods) of the inner branch of its unstable IM, both modified by different levels of constant acceleration magnitude (m/s2) alongthe direction +x (crosses represent the current LP). On the right, example of a small periodic LPO of the family A around L1 withdifferent levels of constant acceleration magnitude (m/s2), along all coordinate axes directions.

lies A, B, and CD (made by two branches C and D) of2-tori QPOs, and two 2-parameter families AB and C of3-tori QPOs, all of them very close to the surface of themoon and highly unstable. Regarding the lighting condi-tions and surface coverage, medium-small LPOs are actu-ally similar to close-range hovering points on the cis andtrans-side of the moon, and the seasonal and annual lighttimes are approximately the same of the ones computed inSection II.V (see Fig.8). Large LPOs can cover also polarand lead/trail-regions, and Fig.14 shows that if used forshort operations the light time can be tuned accordinglyto the Sun phase along Phobos, with the families B and Dallowing to increase/decrease the mean light function upto the 15%.

In particular, due to their proximity to Phobos, the skyoccultation produced by the natural LPOs is relevant, andFig.15 shows that larger LPOs around both L1−2 canprovide passive radiation shielding over 20%. Thisoutcome is obtained using a high order shape harmonicsmodel [29] for Phobos’ bulk. Using Mars’ bulk to providethe same shielding factor would require a low Martian or-bit’s altitude under 850km, while LPOs around Deimosare too distant from the body to provide relevant naturalshielding.

Since the orbits are close to Phobos, no homoclinicnor heteroclinic connections are available to naturallymove around it, but the IMs of these LPOs could be ex-ploited as natural landing or take-off gateways to andfrom the surface of Phobos. In particular, as presentedin Fig.16, the inside branch of the IMs has been computedand a related performance analysis has shown that high-efficient natural tangential landing paths and low es-cape velocity injections (far less than the 2B ∆v value)are available for a region of topographical collinear-faced sites on Phobos. These trajectories have the poten-tial to be exploited for future sample-and-return missionsto this moon, where free-fall is required to avoid contam-ination of the sample’s soil by the exhaust plume of thethrusters or rockets’ nozzle.

V.I Artificial LPOs and their IMs around PhobosThe natural LPOs computed in the ER3BP-GH around

Phobos are investigated in the framework of the additionof a constant acceleration representing SEP. The idea isthe same used in Section III for hovering points: here wefocus on the artificial orbits around the displaced L1−2.Since we found before, computing the dynamical substi-tutes of the natural LPOs in the ER3BP-GH, that the ef-fects of the GHs and the eccentricity act in a different way,with the first responsible for the change in position, shapeand orientation of the LPOs, and the second causing themotion to oscillate around them, to undertake an immedi-ate analysis of the advantages and opportunities providedby the low-thrust propulsion, only the periodic artificialLPOs are derived by NC from the families of the POs ofthe CR3BP-GH, since they provide the backbone of boththe QPOs in the same dynamical model and the QPOs inthe ER3BP-GH.

The NC is undertaken increasing the constant accel-eration magnitude, with the same differential approachfor the DC used for computing the POs from CR3BP toCR3BP-GH in [29]: the difference is that the dimensionof the parameters of the problem is now higher, since itdepends also on the orientation of the thrust vector; there-fore, the analysis is undertaken six times to consider thrustdirections along all coordinated axes ±x, ±y, and ±z.Fig.17 shows some examples of the effects that the ad-dition of a constant acceleration produces on the naturalLPOs, and Fig.18 showcases the resulting families of POsin the Mars-Phobos CR3BP-GH-CA along all the thrust-ing coordinate directions. In particular, thrusting awayfrom Phobos moves the LPOs closer to the moon, withoutgreat changes in the shape and orientation of larger orbitseven with high thrust. On the contrary, thrusting towardsPhobos moves the orbits further from the moon, and asthe thrust level increases the effect of the GHs rapidly de-creases and the LPOs tend to become similar to the fami-lies of the classical CR3BP. The effect of the thrust alongthe other two coordinates axes is more complicated: the

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Figure 18: Periodic LPOs in the Mars-Phobos CR3BP-GH-CA. Summary of the POs (brown) obtained by displacement ofthe four families of POs around each LP of the C3BP-GH (blue) with constant acceleration (1mm/s2) along all coordinate axesdirections (on the left directions ±x, in the center directions ±y, on the right directions ±z). Phobos shape harmonics model.

manifold moves accordingly to the thrust direction, withthe displacement of the artificial LP moving in accordanceto the equi-thrust surfaces computed before in Fig.10;thrust in tangential direction maintains the shape of fam-ilies B and C, while the families D are similar to verticalLyapunov orbits, and the families A are highly distorted;high thrust in the vertical direction greatly modifies themanifold, since only two families of POs are now present,one similar to the Halo orbits (Southern for +z) and theother distorted and perfectly lying on the y-z plane.

The addition of a constant acceleration around Pho-bos has revealed some interesting mission opportunities.First, Fig.19 shows that the effect on the period of thePOs (the NC is undertaken with a fixed differential en-ergy constraint) is quite sensitive: in particular, since thedimension of the manifold of the LPOs is small due to theproximity of Phobos, the range of periods of the naturalLPOs is limited, and the addition of constant low thrustallows cheap artificial LPOs to be obtained with periodequal to the 2:1 orbital overresonance of Phobos aroundMars. This means that they remain periodic also in theelliptical real scenario, which could be an advantage fordesigning the insertion manoeuvers between the missionsegments. Second, the addition of this simple thrust pro-file also affects the stability properties of the POs: de-spite the LPOs remain unstable, the Floquet instability in-dex could be massively lowered with the thrust requiredto displace the LP far from Phobos. This has a great im-pact on the frequency demand for the GNC subsystem,reducing the duty cycle up to the 25% for artificial LPOsdisplaced at an altitude over 60km from Phobos along theMars-Phobos radial. In particular, we see in Fig.19 thattrans-Phobos LPOs over the 70km altitude boundary ofthe Lyapunov stability region of the AEPs (the tip of theright leaf of Fig.11) become Floquet stable, while stableartificial LPOs along the y-axis are obtained displacingthe equilateral LPs. Finally, displacing LPOs away fromthe natural SOI, in addition to reducing instability, it has

other important advantages: indeed, all the problems ofthe dynamical modeling of the relative motion in proxim-ity of this moon are related to the collapse of the realmof attraction of Phobos, therefore the manifold of LPOsis already too close in comparison to common interplan-etary spacecraft operations. Therefore, pushing inwardPhobos with a simple constant propulsion profile enlargesthe Phobos SOI, and there is a great advantage not onlyfor mission operations constraints and light condition re-quirements, but in particular for the computational loadof tracking these orbits: the effect of Phobos’ gravity fieldquickly lowers with the distance, so the convergence ofthe solution of the LPOs (and so its reliability) will beobtained with a far lower order of the truncated GHsmodel to be used in the NC.

The IMs of the artificial periodic LPOs have beencomputed (Fig.20) and they have revealed that using con-stant thrust, along an appropriate direction, allows to en-large the region of landing and take-off sites to coverall the longitude range (mostly with thrust along ±y,but also inward Phobos), while the limit of the latitudes isalso raised (thrust along±z) to become closer to the polarzones, that could be enclosed when considering also thefamilies of QPOs. Also, the displacement of the LPOsaway from Phobos and along the tangential axis enablesartificial heteroclinic connections between two manifoldsthat were not possible with the natural dynamics: thesetrajectories could be exploited for fast orbital displace-ments around the two sides of Phobos for close-range mis-sion segments.

VI QUASI-SATELLITE ORBITS AROUND PHOBOSThe last class of orbits that can be used in a mission to

Phobos lies outside the SOI of the second massive bodyof a 3BP. The peculiar case of a small planetary satellitelike Phobos is therefore very suitable for the exploitationof these orbits because of the collapse of its SOI, whichindirectly drags the manifold of these orbits closer to the

IAC-14-C1.2.1 Page 14 of 20


Figure 19: Periodic LPOs in the Mars-Phobos CR3BP-GH-CA. On the left, example of one natural medium-size periodic LPOof the family A around L2 modified by different levels of constant acceleration magnitude along the direction −x: period andstability properties (stability indexes of the two non-unit couples of eigenvalues of the monodromy matrix). Following graph showsthe characteristic curves of the same properties for all the four families of POs around L1 with different constant accelerationmagnitude (m/s2) along directions ±x.

body. They constitute a family of QPOs which are calledby different names: Quasi-Satellite, Quasi-Synchronous,Distant Satellite, Distant Retrograde orbits. In particular,planar QSOs could be recognized as the quasi-periodicsolution around the Stromgren’s f class of periodic orbitsof Hill’s approximation of the planar CR3BP, as indicatedin the seminal papers of Henon [30, 31]. QSOs are moregenerally considered as one of three kinds of co-orbitalconfigurations in a CR3BP with 1:1 resonance togetherwith Tadpole and Horseshoe orbits: in [32] it is shownthat unstable QSOs evolve from and to Horseshoe orbits,which are linked together. Another useful perspectiveis to relate the dynamics of QSOs to the relative motionbetween two close synchronous keplerian orbits aroundthe primary body of the 3BP. This is the approach usedin Formation Flying (FF) dynamics, where the osculatingorbit of the secondary body, in our case Phobos, couldbe considered as a chief spacecraft orbiting Mars, thatthe third body follows in proximity. The solution of thekeplerian synchronous FF in the Hill’s rotating frame ofthe chief is a relative retrograde elliptical orbit, calledepicycle [23]. This results in an artificial satellite of thesecondary body, but due only to the attraction of theprimary. Such an ellipse is defined in the rotating Hill’sframe. The third body never rotates around the chief inthe inertial frame, centered on the chief, where the thirdbody remains at one side. Only thanks to the spinningrotation of the secondary body, that for the case ofPhobos is synchronous with the Hill’s frame rotation, thethird body rotates in the BCBF frame of the secondaryin 1:1 resonance. In particular, for slightly eccentrickeplerian orbits (as in the case of Phobos) the epicycleresulting from a difference in eccentricity between thirdand secondary body is an ellipse centered on the chief,with major axis along the tangential Hill’s axis and in2:1 ratio with the minor axis; a difference in inclinationor right ascension of ascent node inclines the epicycle,and the relative motion is 3D. The QSO is the motion

of the epicycle when the chief is not a spacecraft but asecond massive body, thus the QSO is the solution ofthe 3B dynamics. Therefore a QSO in the 3B problemis a QPO characterized by an oscillation of the wholeepicycle along the y-axis of the 3BP frame. In addition,3D epicycles experience a secular precession of theirrelative line-of-nodes: the related period grows as far asthe size of the epicycle increases, which is the relativeline-of-nodes becomes fixed in the Hill’s frame (1:1resonance).

The analysis of the QSOs around Phobos in this paperis conducted with the latter approach of a long-rangeMartian FF, in a keplerian perturbed (2B-P) modelwhere the EoM are the Gauss’ Planetary Equations [34],that use as state variable the equinoctial orbital elements(OEs) of the spacecraft around Mars, and the ER3BPis retained using as forcing action the 3B perturbationgravity of Phobos in the osculating Hill’s frame centeredon the moon. In terms of keplerian OEs, the QSOhas short-period (rotation around Phobos along theepicycle) and medium-period (tangential motion of theepicycle) oscillations for semi-major axis, eccentricityand argument of pericenter, while the inclination andright ascension experience a long-period oscillation(precession of the epicycle): a 3D QSOs in the ER3BP isa torus with three phases. The 2B-P is suitable to be im-plemented with additional orbital perturbations: fromour perturbation analysis in Section II.III, we recover thatoutside of the Phobos’ SOI, the major effects are due tothe eccentricity (already embedded in the EoM) and theMars J2 GH. In particular, as a difference from the 3BP,in the 2B-P the orbital perturbations not due to Phobosare no longer differential. The differential action on therelative motion appears by computing in feedforward theosculating motion of Phobos and the angular velocityof its Hill’s frame, with a dedicated 2B-P under theeffect of the perturbation. Recall that when defining aninitial condition for the QSO using mean OEs for the

IAC-14-C1.2.1 Page 15 of 20


Figure 20: IMs of the artificial LPOs with constant acceleration in the Mars-Phobos CR3BP-GH. First two figures show thepossible landing/take-off sites through the IM of the family A of POs around L1, third figure shows the region of landing sites forall the families of artificial LPOs around 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 for directions ±z). Phobos shape harmonics.

epicycle and Phobos and starting with true anomalies inaccordance with the 2B dynamics, it produces differentQSOs. The same QSO is obtained with osculating OEsin accordance to the perturbation, but their analyticalexpression is only available for particular cases, like J2

[35], but not for the 3B perturbation. As an example,starting with the spacecraft in perimars and inferiorconjunction, and starting in Mars-Phobos quadrature andphase in accordance on the same epicycle, it producesvery different QSOs with the latter having a far loweramplitude of the y-axis oscillation.

The dynamical analysis of the QSOs requires thederivation of the secular derivatives of the 3B pertur-bation in the 2B-P, in a similar way to the case of J2.But the solution of the mean integral value of the 3Bperturbation could not be undertaken analytically.This integral is solved numerically in [32], and usedfor the stability analysis of the QSOs. The outcomesare two. First, any 2D QSO large enough that theoscillation amplitude of the epicycle does not make itfall towards the secondary body (minimum distance ofthe epicycle) is always stable. Second, for any 3D QSOthe secular precession will rotate the epicycle with therelative line-of-nodes towards the Hill’s x-axis (relativenodes in conjunction with the two central bodies): suchattitude becomes unstable as far as the inclination ofthe QSO increases, therefore the QSO leaves the bodyneighborhood and become a Horseshoe orbit. This 3Dstability condition of the QSOs requires the difference ininclination (in radians) to be smaller than the differencein eccentricity. A linear Floquet stability analysis ofQSOs in the CR3BP has highlighted the high stabilityof them [33]. In [36], a different approach has beenundertaken, which is a linearized stability analysis in theER3BP around the epicycle, and results are applied to theMars-Phobos case. The minimum distance conditionfrom Phobos is 29.4km (∆e = 0.00315), above whichthe planar QSO is stable. The 3D stability condition

that bounds the admissible difference in inclination tothe difference in eccentricity below which the inclinedQSO is stable is ∆i/∆e < 96%. In addition, theperiod of the linearized precession motion is analyticallyderived. Similar numerical outcomes were obtainedfor the stability analysis of QSOs around Jupiter moonEuropa [37]. In particular, we found that the minimumdistance corresponds to a peculiar condition presentedby Henon in [31] on the stability of the planar QPOsaround the f family of Hill’s approximation of the planarCR3BP. As far as the dimension of the QPOs increases,the stability is influenced by resonances with the familiesof multiple-revolution direct satellite orbits, where thelast resonance 1:4 is encountered by an epicycle of mini-mum distance equal to 1.2 3

√µ of the orbital semi-major

axis: after that, the QPO remains bounded even at infinitedistance (within the Hill’s approximation). This value forthe Mars-Phobos system leads to a distance of 28.7km,which is so close to the value obtained above in thelinearized ER3BP. In addition, [31] found a very differ-ent sensitivity of the stability of the QPO between thetwo velocity components, with the radial one being farless critical. This means that the accuracy required by theGNC subsystem for the insertion manoeuver to the QSOis less critical in Mars-Phobos and Phobos-spacecraftcommon quadrature configurations, where the epicyclehas only radial velocity on the orbital plane and theamplitude of the tangential oscillation is the smallestusing mean OEs.

The stability of this class of orbits, combined withthe collapse of the SOI and the synchronous rotation,makes the QSOs attractive solutions to orbit Phobos. AQSO will constitute the main orbital mission segmentaround Phobos for the upcoming Phootprint mission[38], to observe the surface thoroughly and identify thelanding site where obtaining a sample of the soil to returnback to Earth; it will be used for a very long time (6-9months), and required to remain stable during the solar

IAC-14-C1.2.1 Page 16 of 20


Figure 21: QSOs around Phobos. On the top-left, stability re-gion of QSOs around Phobos tested by high-fidelity 1 year sim-ulations, defined by initial conditions on osculating OEs aroundMars for the spacecraft and Phobos: positive differences in ec-centricity and inclination, starting at perimars epoch. On the top-right, example of a 3D stable QSO in the Phobos Hill’s frame for1 year propagation. On the bottom-left, period of precession ofthe relative line-of-nodes that indicates the minimum time for acomplete Phobos surface coverage of the QSOs. On the bottom-right, example of the lighting conditions for a single keplerianepicycle at 50-110km distance range, remaining in the Mars andPhobos shadowing wakes for 34% of the time.

conjunctions that would black out the communicationsto the Earth for approximately 1 month. In this paperwe probe the nonlinear stability of the QSOs aroundPhobos. Regarding Phootprint, the stability has beentested scanning the state-space in relative position andvelocity with a true-life simulator [33]. Instead weconduct the analysis in the framework of the 2B-P (whoseresults, due to the fact that we are out of the PhobosSOI, are suitable to be interfaced with a previous orbitalsegment around Mars), and we use the linear stabilityregion of [36] (a trapezoid in the plane ∆e-∆i as shownin Fig.21) as a first guess in order to limit the boundaryof the state-space in terms of osculating OEs whereconduct the nonlinear simulations. We used the STKsoftware, considering the default ephemerides of Phobos,starting at 10 April 2013 08:35:30UTCG (perimars), andthe additional perturbations of the Mars GHs up to 15thdegree/order, the Phobos GHs up to 4th degree/order, theSun and SRP perturbations. We further limit the regionby setting a range of minimum altitudes of the epicyclefrom Phobos between 20 and 60km, therefore alreadyinside the linear boundaries; regarding 3D QSOs thereference vertical size of Phobos is realized with just∆i = 0.001rad = 0.06. We simulate QSOs up to oneyear of propagation time: a QSO is considered stablewhen it does not drift away by the end of the simula-

tion. This is very reliable and not too much restrictingbecause the strong nonlinearity of the 3B dynamicsprovokes behaviors drastically different crossing thestability region boundaries (as proved with the AEPs).The resulting true-life stability region boundary ispresented in Fig.21: the minimum distance requirementis significantly higher than the linear one, and in thesame range of ∆e the 3D stability boundary is smallerthan the linear one; for higher eccentricity such boundaryasymptotically reaches the linear one. This stabilityregion is related to positive ∆e-∆i, while all the otherinitial conditions differences in osculating OEs at theperimars are null. The starting ∆e of the epicycle doesnot define trivially the minimum distance from Phobosof the QSO for the smallest range of eccentricities: theplanar QSO at minimum stable ∆e has a minimumaltitude of 25km.

From a sample of trajectories simulated, the periodof the secular precession for 3D QSOs was computed:this natural motion of the 3B dynamics is useful forobservation purposes because it allows to overcome the1:1 resonance of the keplerian epicycle and provide acomplete coverage of the surface of Phobos. Fig.21presents the related time required for the QSOs of thestability region, and compare it with the linearizedsolution from [36]: all the range of stable QSOs ofinterest provide a fast coverage of the moon. Anotherimportant performance that could be used in the missiondesign to select the optimal QSO inside the stable domainis the ∆v of the insertion manoeuver: considering aprevious mission segment realized by a Martian Trailingorbit (an AEP) with ∆ν = [0,−6] from Phobos(corresponding to a distance 0-1000km), we computedthe ∆v budget to provide with the cheapest strategy ofimpulsive manoeuvers the initial conditions of the QSO∆e, ∆i and the phasing ∆ν. Since ∆e and ∆i are smalltheir cost range is moderate (5-7.5m/s and 0-8m/s), andalso the cost of the phasing (0-10m/s) could be loweredfor distant AEPs linearly performing multiple laps. Theprice to pay is instead the accuracy of the GNC subsystemto insert precisely the spacecraft in this small range ofinitial conditions.

In this analysis we found a region of QSOs naturallystable with a high-fidelity perturbed model, potentiallyfor a whole long-period mission scenario, with theirdistances from Phobos suitable for observation, and theexploitation of the natural precession motion provides afast complete coverage of the surface of the moon. Suchfast precession, combined with the exploitation of theseorbits for long periods, allows a spacecraft to remainmostly in light. On the contrary, QSOs can be controlledto maintain a 1:1 resonance with Phobos BCBF frame:this would provide constant lighting conditions, rangingfrom continuous light (during solstices) to continuous

IAC-14-C1.2.1 Page 17 of 20


shadow (during equinoxes), controlling the β-anglethermal condition desired by fine tuning the initial phaseof the spacecraft along the epicycle with respect to theVDCO of the Sun around Phobos. An example in Fig.21shows that a planar epicycle in the middle of the stabilityregion, during an equinoctial season could remain inshadow from the field of view of the Sun for the 34% ofthe time.

VII CONCLUSIONSIn this paper we analyzed several kinds of orbits

around the Martian moon Phobos, each one defined us-ing appropriate models of the relative dynamics where thereference signal is computed. The design of a future spacemission to Phobos requires multiple objectives and con-straints to be satisfied: we collect the outcomes of ouranalysis in Table 2, where each orbit has a number of po-tential applications and their performance can be assessedagainst the requirements of each mission segment.

Trailing/Leading orbits around Mars, starting from25km distance from Phobos, are attractive configurations,because they are cheap and affordable by SEP even forheavy human modules, they are stable to perturbations,and they are mostly in full light. They are the best orbitsto start to approach Phobos SOI, however their ground-track on the moon is stationary and limited. Other distantconfigurations or close-range AEPs requires either highthrust or high station-keeping cost for hovering over long-time: they can be used only for short and dedicated oper-ations of small unmanned spacecraft.

Keplerian orbits around Phobos are infeasible due tothe collapse of the realm of attraction of the moon towardsits surface. In this paper we investigated the artificial or-bits around Phobos that would provide a spacecraft con-tinuous light or shadow conditions moving synchronouswith the Sun on its seasonal surface of motion. Due to thepull of Mars these VDCOs require continuous propulsionand they are too expensive even for few revolutions andalso unstable.

We then analyzed the LPOs computed in an improvedsystem of the relative dynamics in proximity of Phobos,upgrading the Mars-Phobos ER3BP system with thereal gravity field of the moon, modeled with a gravityharmonics series expansion. These orbits are very closeto the moon surface, therefore they are similar to close-range points but with an extended ground-track and rangeof lighting conditions, and the Phobos’ bulk occultationof the sky could provide relevant passive shielding fromthe cosmic rays radiation. Despite their instability, theLPOs are natural motion and so will require no propul-sion and low station-keeping cost to provide observationon Phobos and communication bridges to manage roboticscouts on Mars and Phobos: however, they require the

high accuracy of an optical navigation subsystem, andhigh-load on the guidance subsystem, whose referencesignal must be computed with advanced nonlinear tech-niques that need the acquisition of a high-fidelity grav-ity field of the moon. In particular, large quasi-periodicorbits enable coverage also of the polar and lead/trail-regions, and we found that in the Mars-Phobos ellipticsystem there exist tall and inclined periodic orbits aroundeach side of the moon: a costellation, starting just fromone spacecraft on each side, would fly synchronous andso enable stationary communications between most of theopposite sides of the moon (cis/trans, North/South, partof lead/trail) where different human crews or rovers couldbe displaced, as well as repeated access times to equa-torial and middle latitude sites on Mars. Another usefulapplication is to exploit their IMs as landing/taking-offgateways to and from the moon: in this paper we provedthat there exist natural trajectories for a specific range oflongitude-latitude sites able to land tangentially, facilitat-ing a soft controlled touch-down, and depart with a verylittle escape velocity, less than 30% of the 2B ∆v value.The optimization of these performances to select the besttrajectory at a given location on Phobos will be paramountfor sample-and-return missions (where also soil contami-nation avoidance is necessary) as well as first manned ex-plorations of this moon. The addition of a simple propul-sive law, to obtain a constant acceleration, offers someadvantages when using these LPOs for short-phases: thesurface coverage and landing/take-off targeting could beextended to the whole surface of Phobos, the instabilityof the orbits could be lowered, and the computation of theorbits themselves could be simplified to maintain them pe-riodic also in the true elliptic dynamics, and to lower theaccuracy of the model of the gravity field of the moon tobe taken into account. In particular, artificial heteroclinicconnections would now exist for fast orbital fly-bys aroundtwo opposite sides of Phobos.

Finally, the QSOs are the best solution for a precur-sor unmanned mission to Phobos. They are both natu-ral orbits with no need of propulsion, and self-stable upto very long time with no need of station-keeping, and sothey can be used as parking orbits with distance startingfrom 25km from Phobos. In particular, closer 3D QSOsprovide a fast complete coverage to map the surface ofPhobos and identify the landing site, and they are mostlyin light. 2D QSOs during equinoctial seasons are suitableto be controlled to provide nearly Sun-synchronous orbitsaround Phobos, enabling constant and adjustable lightingconditions: in particular, lighting condition schedulingcould be important for a first-generation manned space-craft while orbiting Mars.

In conclusion, a possible mission scenario for Phoboscould be to start from establishing a trailing orbit aroundMars, with a phasing of 0.4-1.2, requiring a thrust to

IAC-14-C1.2.1 Page 18 of 20


be maintained of less than 0.05-5N in accordance to theclass of spacecraft, where start to track the real positionof Phobos and acquire its gravity parameter. Then per-form a phasing manoeuver to reach a QSO at an inter-mediate distance from Phobos, where start to measure thereal gravity field and map the surface of the moon. Finally,move along the tube of the artificial invariant manifoldsdisplaced away from the moon with a constant accelera-tion along the x-axis, whose LPOs tracking is less affectedin their numerical computation by the GHs perturbations,to directly reach with a probe the desired landing site onPhobos. In the case of a sample-and-return mission, wethen move in reverse choosing the trajectory that providesthe minimum escape velocity, reaching a mothership thathas been left on a closer LPO or parked in the QSO, toremotely command the lighter probe. In particular, thelong-time stability of the QSOs around Phobos couldbe exploited as an orbital repository to send, in advance,unmanned propulsion modules, fuel stockpiles, and pro-visions, to remain parked in a secure low altitude Martianorbit without orbital maintenance costs and with short-period phasing manoeuvers to dock the modules. To al-low the first human expeditions to visit Mars and return tothe Earth, the spacecraft could make scheduled pit-stopsat this orbital garage on Phobos’ orbit.

ACKNOWLEDGMENTSThis work has been supported by the European Commis-sion through the Marie Curie fellowship PITN-GA-2011-289240 ”AstroNet-II”. We are thankful with Prof. J.J.Masdemont and Prof. G. Gomez (IEEC) for the assis-tance provided in the computation of LPOs in the CR3BPthrough high-order series expansions, and with Dr. J.Fontdecaba and Dr. V. Martinot (Thales Alenia Space)for providing the opportunity and the assistance of under-taking the applied research of the QSOs around Phobos.

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

Orb

itA

ltitu

deR

efer

ence

Sign

alD

ynam

ical

Orb

ital

Mai

nten

ance

Stra

tegy

Surf

ace

Lig

htin

gPo

tent

ialM

issi

onR

ange

Com

puta

tion

Mod

elPr

opul

sion

Con

trol

actio

nG

uida

nce

erro

rN

avig

atio

nac

cura

cyC

over

age

Con

ditio

nsA

pplic

atio

nsA

EP

0-4.

5km

xre

lativ

efix

edpo

sitio

n,C

3B-G

H0.

5/50

-0/0

N(1

50-0

m/s

/T)

unst

able

:SK

10-1

0m1-

1mst

atio

nary

grou

nd-t

rack

sols

tices

:50-

80%

dedi

cate

dpr

oxim

ityop

erat

ions

,0m

/sac

cura

tegr

avity

field

mod

elco

nsta

ntac

cele

ratio

n3-

3m/s

/Tev

ery

7-7m

in0.

05-0

.05m

/s0.

001-

0.00

1m/s

cis/

tran

s-Ph

obos

regi

oneq

uino

xes:

40-7

0%(c

is),

50-8

0%(t

rans

)ra

diat

ion

shie

ld(5

3-13

%)

0-4.

5km

yre

lativ

efix

edpo

sitio

n,C

3B-G

H0.

5/50

-0.3

/30N

(150

-100

m/s

/T)

unst

able

:SK

10-1

0m1-

1mst

atio

nary

grou

nd-t

rack

sols

tices

:50-

80%

dedi

cate

dpr

oxim

ityop

erat

ions

,0m

/sac

cura

tegr

avity

field

mod

elco

nsta

ntac

cele

ratio

n0.

6-0.

6m/s

/Tev

ery

14-1

5min

0.02

-0.0

2m/s

0.00

1-0.

001m

/sle

ad/tr

ail-

Phob

osre

gion

equi

noxe

s:45

-75%

radi

atio

nsh

ield

(50-

16%

)0-

4.5k

mz

rela

tive

fixed

posi

tion,

C3B

-GH

1/10

0-0.

4/40

N(3

00-1

20m

/s/T

)un

stab

le:S

K10

-10m

1-1m

stat

iona

rygr

ound

-tra

ckS/

Wso

lstic

e:N

orth

/Sou

thlig

ht,S

outh

/Nor

thde

dica

ted

prox

imity

ops,

com

plet

e0m

/sac

cura

tegr

avity

field

mod

elco

nsta

ntac

cele

ratio

n0.

9-0.

9m/s

/Tev

ery

11-1

2min

0.03

-0.0

2m/s

0.00

1-0.

001m

/sN

orth

/Sou

thpo

le-P

hobo

sre

gion

shad

owun

der1

.4km

then

light

,equ

inox

es:8

8%lig

htin

g/sh

adow

ing,

radi

atio

nsh

ield

(58-

20%

)4.

5-70

kmx

rela

tive

fixed

posi

tion

C3B

0/0-

1.5/

150N

(0-4

00m

/s/T

)un

stab

le:S

K15

-9m

in10

0-10

0m0.

01-0

.01m

/sst

atio

nary

grou

nd-t

rack

sols

tices

:lig

htsh

ortc

lose

obse

rvat

ion

0m/s

cons

tant

acce

lera

tion

6-20

m/s

/Tev

ery

15-9

min

0.2-

0.5m

/s0.

01-0

.01m

/sci

s/tr

ans-

Phob

osre

gion

equi

noxe

s:88

%4.

5-60

kmy

rela

tive

fixed

posi

tion

C3B

0.3/

30-0

.04/

4N(1

00-1

2m/s

/T)

unst

able

:SK

100-

100m

10-1

0mst

atio

nary

grou

nd-t

rack

sols

tices

:lig

htlo

ngcl

ose

obse

rvat

ion

0m/s

cons

tant

acce

lera

tion

3-3m

/s/T

ever

y20

-21m

in0.

2-0.

2m/s

0.01

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red

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stat

iona

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Phob

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/app

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Phob

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rtio

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/10-

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ever

y70

min

man

ouev

eurt

oL

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00km

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tive

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ryE

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Hno

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/1km

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take

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iona

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ound

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ckle

ad/tr

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Phob

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ht,e

quin

oxes

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(fro

m3m

/s)t

oL

PO/e

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efr

omPh

obos

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00km

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ecto

ryE

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H-

0/0-

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10N

(0-1

0m/s

/SK

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/1km

1m/1

00m

land

ing

site

onde

dica

ted

regi

onof

who

leco

ntro

llabl

eby

epoc

h0-

100%

/pe

rfor

man

ced

soft

land

ing

(fro

m0.

5g)

0-25

m/s

/0-6

0m/s

from

LPO

/Mar

sor

bits

E3B

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)con

stan

tacc

eler

atio

n0.

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.05m

/sev

ery

20m

in0.

02m

/s/0

.5m

/s0.

001m

/s/0

.1m

/sPh

obos

surf

ace

(apa

rtne

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les)

/sta

tiona

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lstic

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ight

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inox

es:8

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obos

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00km

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10N

(0-1

0m/s

/SK

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1m/1

00m

take

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site

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ted

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onof

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rfor

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ced

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pta

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rom

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/s/0

-60m

/sfr

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ffsi

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anta

ccel

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ion

0.02

-0.0

5m/s

ever

y20

min

/0.

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/s/0

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/s0.

001m

/s/0

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/sPh

obos

surf

ace

(apa

rtne

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les)

/sta

tiona

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lstic

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ight

,equ

inox

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ape

from

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1-3h

/8-2

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3m/s

ever

y70

min

grou

nd-t

rack

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/trai

l-Ph

obos

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-500

km/2

0-50

0km

rela

tive

traj

ecto

ryfr

omL

PO,t

ake-

off

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-GH

/0.

1/10

-1/1

00N

(10-

500m

/s/

SK10

m/1

km1m

/100

mde

dica

ted

grou

nd-t

rack

/sta

tiona

ryco

ntro

llabl

eby

epoc

h0-

100%

/tr

ansf

erm

anoe

uver

arou

ndPh

obos

/0-

60m

/s/0

-60m

/ssi

te/M

ars

orbi

t,L

PO,t

ake-

offs

iteE

3B50

-600

m/s

)con

stan

tacc

eler

atio

n0.

1-0.

5m/s

ever

y20

min

/0.

02m

/s/0

.5m

/s0.

001m

/s/0

.1m

/sgr

ound

-tra

ckle

ad/tr

ail-

Phob

osre

gion

sols

tices

:lig

ht,e

quin

oxes

:88%

appr

oach

/esc

ape

to/f

rom

Phob

os3-

13h

/5-1

5h1-

2m/s

ever

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min

orin

sert

ion

man

oeuv

erto

LPO

QSO

25-2

00km

rela

tive

quas

i-pe

riod

icor

bit,

2B(M

ars)

-Pno

treq

uire

dna

tura

llyst

able

with

outS

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m10

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mco

mpl

ete

in5-

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ysso

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es:l

ight

park

ing

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t(re

pair

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fuel

ling)

,mea

su-

8-50

m/s

high

-fide

lity

sim

ulat

ions

0.01

-0.1

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0.01

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m/s

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%re

men

ts(P

hobo

sgr

avity

field

),co

mpl

ete

fast

map

ping

and

freq

uent

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mm

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kmre

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asi-

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odic

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(Mar

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m10

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ons

(dir

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nal

sola

rrad

iatio

npr

otec

tion)

IAC-14-C1.2.1 Page 19 of 20


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IAC-14-C1.2.1 Page 20 of 20

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