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Natural Motion around the Martian Moon Phobos

The Dynamical Substitutes of the Libration Point Orbits

in an Elliptic Three-Body Problem with Gravity Harmonics

M. Zamaro ·J.D. Biggs

Abstract The Martian moon Phobos is becoming an appealing destination for future scientiﬁc missions. The orbital

dynamics around this planetary satellite is particularly complex due to the unique combination of both small mass-ratio

and length-scale of the Mars-Phobos couple: the resulting sphere of inﬂuence of the moon is very close to its surface,

therefore both the classical two-body problem and circular restricted three-body problem (CR3BP) do not provide an

accurate approximation to describe the spacecraft’s dynamics in the vicinity of Phobos. The aim of this paper is to

extend the model of the CR3BP to consider the orbital eccentricity and the highly-inhomogeneous gravity ﬁeld of

Phobos, by incorporating the gravity harmonics series expansion into an elliptic R3BP, named ER3BP-GH. Following

this, the dynamical substitutes of the Libration Point Orbits (LPOs) are computed in this more realistic model of the

relative dynamics around Phobos, combining methodologies from dynamical systems theory and numerical continuation

techniques. Results obtained show that the structure of the periodic and quasi-periodic LPOs differs substantially from

the classical case without harmonics. Several potential applications of these natural orbits are presented to enable unique

low-cost operations in the proximity of Phobos, such as close-range observation, communication, and passive radiation

shielding for human spaceﬂight. Furthermore, their invariant manifolds are demonstrated to provide high-performance

natural landing and take-off pathways to and from Phobos’ surface, and transfers from and to Martian orbits. These

orbits could be exploited in upcoming and future space missions targeting the exploration of this Martian moon.

Keywords Phobos ·Libration Point Orbits ·Gravity Harmonics ·Invariant Manifolds ·Dynamical Systems Theory ·

Numerical Continuation ·Phobos surface

Mathematics Subject Classiﬁcation (2000) 70F15 ·70G60 ·70K42 ·70K43 ·37D05 ·34A12 ·33C55 ·70K40

PACS 45.10.-b ·95.10.Ce ·91.10.Sp ·95.55.Pe ·96.30.Hf ·96.25.Nc ·45.50.Pk ·05.45.-a

CR Subject Classiﬁcation J.2 ·G.1.7 ·G.1.5 ·G.1.2 ·G.1.10 ·I.0

1 Introduction

One of the most famous and historically investigated chaotic dynamical systems is the circular restricted problem of

three bodies (CR3BP), which models the motion of a point-mass subject to the gravitational ﬁeld generated by two

M. Zamaro ·J.D. Biggs

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

E-mail: mattia.zamaro@strath.ac.uk, james.biggs@strath.ac.uk

2 M. Zamaro, J.D. Biggs

spherical homogeneous bodies moving in circular orbits around their common center of mass. In the past few decades,

dynamical systems tools developed for this problem have provided an in-depth understanding of the spacecraft’s natural

dynamics (Koon et al. 2011). The orbits around the Libration points (LPOs) have been computed extensively, using

semi-analytical methods (such as Lindstedt-Poincar´

e series expansions (Koon et al. 2011; Masdemont 2004)) and nu-

merical techniques (such as numerical continuation (Koon et al. 2011; G´

omez and Mondelo 2001)), for the Sun-Earth

and Earth-Moon systems, where the approximation of the natural dynamics using the CR3BP is an accurate one (Far-

quhar 1968; Howell 1984)-(Conley 1969; McGehee 1969; Koon et al. 2000). The study of the invariant motions of these

systems has proved invaluable for real space missions, such as ISEE-3/ICE (1978), WIND (1994), SOHO (1995), ACE

(1997) and Genesis (2001), where they have been exploited as energy-efﬁcient trajectories to minimize the spacecraft’s

fuel consumption during station-keeping and orbital transfers.

However, future space missions are now targeting deep space celestial bodies, such as asteroids and planetary satel-

lites. In particular, the two moons of Mars, Phobos and Deimos, have been proposed as attractive destinations since the

early 80s (Singer 1981; O’Leary 1985). Scientiﬁc interest in these moons focuses on understanding their composition

and enigmatic origin, which would provide insight into the Solar System’s origin itself. The American Viking program

(1975) and Soviet Phobos-2 spacecraft (1988) were the ﬁrst missions to successfully provide images of the two moons.

Using this information, the modeling of their shape and gravity ﬁeld became an extensive area of research (Turner 1978;

Chao and Rubincam 1989) which continues today (Willner 2009). In addition, the Martian moons’ exploration has been

highlighted as a key milestone to the long term goal of a human mission to Mars (Sommerer 2014; Hopkins and Pratt

2011; Lee 2012). In particular Phobos, the inner-most and larger moon of the two, is considered to be an ideal candidate

for landing a spacecraft as a precursor to a landing mission on Mars (Brandenburg 2012; Troutman 2014; Price et al.

2014). Past missions designed to land on Phobos have all failed (Phobos-1 and Phobos-2, Phobos-Grunt). Currently,

ESA is scheduling a sample-and-return mission to Phobos (McCaughrean 2012; Koschny 2012) and NASA is targeting

this moon to test a new kind of rover for planetary exploration (Pandika 2012; Dorminey 2013).

This growing interest in undertaking a successful mission have fostered in recent years the need to probe suitable

orbits around Phobos. Early analysis focused on simple hovering station-keeping (Bombardelli 2012) and exploitation

of stable Quasi-Satellite orbits (Gil and Schwartz 2010; da Silva Pais Cabral 2011; Lidov and Vashkov’yak 1993). The

latter are orbits around Mars where the spacecraft ﬂies in close co-orbital conﬁguration with Phobos, and have been

applied in the design of past missions. In contrast, the LPOs have not received a lot of attention because the orbital

dynamics of the CR3BP, where the two primaries are Mars and Phobos, do not provide a suitable approximation of the

natural dynamics in the vicinity of the moon, where many strong perturbations are involved. In Wallace et al. (2012) a

multiple shooting method is used in a high-ﬁdelity perturbation model to correct the classical LPOs of the Mars-Phobos

CR3BP. The resulting solutions are only bounded for a speciﬁc time span and do not contain any information on the

natural frequencies of the perturbations, and are therefore not appropriate dynamical substitutes (G´

omez et al. 2002) nor

their performance analysis suitable to be optimized in the mission design loop.

The aim of this paper is to extend the model of the CR3BP to describe the accurate dynamics of a spacecraft in the

vicinity of Mars’ moon Phobos, and derive the dynamical substitutes of the LPOs in this new system. Phobos has a highly

inhomogeneous gravity ﬁeld and the eccentricity of its orbit around Mars has a signiﬁcant effect in designing orbits in

Natural Motion around the Martian Moon Phobos 3

Table 1: Physical and astrodynamical properties. All the values are rounded to 3 digits. Full values up to 16 digits are used in this paper,

and are retrieved from the following sources. Ephemerides: NASA JPL at 25th July 2012 00.00CT. Phobos axial tilt θat date: mean value at

epoch 1.08◦. Gravity models: Mars MGM1025, Phobos (Chao and Rubincam 1989).

Property Mars Phobos

m[kg] 6.42 ·1023 1.07 ·1016

Size [km] Rmean sphere Rmean ellipsoid

3.39 ·10313.1×11.1×9.30

Revolution T687d 7.65h

Rotation T24.6h 7.65h

θ[◦] 25.2 0.301

Gravity Field GHs GH s

J20.00196 0.105

J2,20.0000631 0.0147

J30.0000315 0.00775

J40.0000154 0.0229

Orbital Elements Sun-Ecliptic Mars-Equatorial

a[km] 2.28 ·1089.38 ·103

e0.0934 0.0156

i[◦] 1.85 1.07

its proximity. The inclusion of the eccentricity in the well-known elliptic R3BP (ER3BP) (Szebehely 1967) often uses a

rotating-pulsating frame to maintain a natural similarity with the CR3BP, which is particularly suitable for performing

numerical continuation with respect to the eccentricity parameter to obtain the corresponding dynamical substitutes of

Libration points and LPOs (Szebehely 1967; Broucke 1969). In the last two decades, the orbital dynamics around highly

irregular shaped bodies have been investigated using tetrahedrons to model their surface (Werner 1994), and applied in

the framework of a perturbed 2B dynamics to derive equilibrium points and orbits around them in proximity of asteroids

(Yu and Baoyin 2012b,a). These orbits are deﬁned in the asteroid’s body-ﬁxed frame and are not signiﬁcantly inﬂuenced

by the Sun’s third-body perturbation (Yu and Baoyin 2012b; Ni et al. 2014). The manifold of the LPOs is instead deﬁned

in the Sun-asteroid orbital frame and far from the asteroid, and is not signiﬁcantly inﬂuenced by the irregularities of its

gravity ﬁeld (Abouelmagd and Sharaf 2013). The two dynamical domains are therefore decoupled in the phase-space

of the asteroids’ orbital dynamics. On the contrary, the relative dynamics around a small planetary satellite like Phobos

cannot neglect any of these forces.

The outline of the paper is as follows. In Section 2 we derive a new model extending the classical CR3BP to an

ER3BP with an inhomogeneous gravity ﬁeld of the second massive body, described through a gravity harmonics series

expansion (ER3BP-GH). Subsequently in Section 3, we present the procedures of dynamical systems theory that use

the LPOs in the CR3BP as an initial guess in numerical continuation schemes (Keller 1977; Allgower and Georg 2003),

that allow us to compute their dynamical substitutes (G´

omez et al. 2002) in the new, more realistic, model. Section 4

presents the resulting periodic and quasi-periodic LPOs in the Mars-Phobos ER3BP-GH. In Section 5, the related ap-

plications and performances are presented, for example their lighting conditions, surface coverage and passive radiation

shielding. Furthermore, the invariant manifolds of the LPOs in this more realistic dynamical model are computed, and

due to the proximity of the moon, they are demonstrated as natural landing and take-off gateways to and from its surface.

In particular, we identify natural trajectories that provide the lowest angle of incidence and escape velocity to and from

their arrival and departure sites respectively. Section 6 concludes the paper and suggests potential applications for these

orbits and their invariant manifolds in future space missions to Phobos. In addition, we propose that these trajectories

4 M. Zamaro, J.D. Biggs

Fig. 1: Phobos. Shape harmonics model (Willner 2009), Hill’s frame.

Stickney is the large crater on the right side.

Fig. 2: Misalignment angle between the equatorial and orbital

planes of the Martian moons. Mean values in dotted lines.

can help to explain the formation of the craters and grooves of the moon, as well as the dust dynamics around it, and

the new model with gravity harmonics can provide further evidence to support the theory of accretion for the origin of

Phobos.

2 Modeling of the Relative Orbital Dynamics around Phobos

This section focuses on the derivation of an appropriate model to approximate the orbital dynamics of a spacecraft in the

proximity of Phobos, tailored to be used for the computation of the Libration Point Orbits (LPOs) in Section 3. Table

1 presents a summary of the physical and orbital parameters of Mars and Phobos that have been used in the analysis

undertaken in this paper. The standout characteristics of Phobos are its small size (even smaller than some asteroids) and

its irregular shape, as represented in Fig.1. Phobos moves in a relatively low altitude and almost circular and equatorial

orbit around Mars, and it rotates with synchronous period and almost zero-tilt with respect to its orbital motion.

Let’s now deﬁne two reference frames in Cartesian coordinates that will be used extensively in this paper.

–Phobos’ Hill’s frame has its origin in the moon’s barycenter and rotates with ﬁxed attitude with respect to its orbit

around Mars, with vertical z-axis perpendicular to the orbital plane, and radial x-axis pointing outwards from the

Mars-Phobos barycenter.

–Phobos’ body-centered body-ﬁxed frame (BCBF) has its origin in the moon’s barycenter and its attitude is ﬁxed with

respect to the moon’s geometry, with vertical z-axis aligned along the body’s spin axis, and x-axis pointing towards

the body’s Prime Meridian intersection on the equatorial plane.

The fact that not only the revolution of Phobos around Mars and the rotation around its spin axis are synchronous, but

that they are also respectively equatorial and zero-tilted (Duxbury and Callahan 1981), implies that Phobos’ attitude,

expressed by the body’s BCBF frame, is approximately ﬁxed in the orbital Hill’s frame. With this approximation, their

respective z-axes are coincident, while their x-axes differ only by the deﬁnition of the Prime Meridian. In particular,

Phobos’ Prime Meridian is formally identiﬁed by the location of the point constantly pointing towards Mars on the

body’s Equator (Archinal et al. 2010; Duxbury and Callahan 1981). Therefore the two frames differ by a rotation of

180◦of their x-yplane’s axes. Before the deﬁnition of the Prime Meridian, the actual misalignment between the two

Natural Motion around the Martian Moon Phobos 5

frames oscillates between a minimum of 0.30◦and a maximum of 1.90◦, and the dynamics of this libration motion is

much slower than the time-scale of a mission segment around Phobos (period of 2.26 terrestrial years), as presented in

Fig.2. In this paper, we approximate the two frame’s attitudes to be ﬁxed and with zero-tilt with respect to each other. It

is also interesting to recall that such nominal conﬁguration corresponds to the stable orientation provided by the central

body’s gravity-gradient torque: Phobos’ Hill’s and BCBF frames’ axes correspond to the body’s principal axes of inertia.

2.1 The Mars-Phobos CR3BP

The study of the Mars-Phobos-spacecraft system is conducted in the ﬁrst instance with the CR3BP model (Koon et al.

2011), which is to consider the gravitational effect of the spacecraft negligible with respect to the two massive bodies

(1: Mars; 2: Phobos) moving in a Keplerian circular orbit. The dynamics of the CR3BP is expressed in a third rotating

reference frame.

–The 3B frame corresponds to Phobos’ Hill’s frame except for the origin, which is the Mars-Phobos system’s barycen-

ter.

The CR3BP is expressed in non-dimensional units, setting the unit of mass to be the total mass of the system, and the

unit of length Lto be the semi-major axis of the orbit of the two primaries. In addition the time is set to correspond to

the mean anomaly of their orbit. The only parameter of the normalized time-invariant CR3BP is the mass ratio µ, the

normalized mass of the second body, which for the case of the Mars-Phobos couple is

µ=1.66059511088139 ·10−8

Note that the mass parameter of the system is very small, if compared to other cases of Sun-planet or planet-satellite

couples of primaries studied in the Solar System. In contrast the mass parameter of a Sun-asteroid R3BP is far lower

than our µ. In addition, the length unit of the system is very small too since the altitude of Phobos’ orbit is less than

twice the radius R1of Mars

R1/L=36%

This is an uncommon condition for all the general pairs of primaries of the Solar System.

The equations of motion of the CR3BP are stated in Eq.(1), where q= [x;y;z]is the position of the spacecraft in the

normalized 3B frame, and x= [q;˙

q]1is the resulting state of the vectorﬁeld fof the dynamical system reduced to ﬁrst

order. Such general notation will be maintained throughout this paper, where the position will refer to the frame of the

current model, which is identiﬁed by the vectorﬁeld’s subscript. We also use the common expressions In,0n,0nxm to

indicate respectively an identity, square and rectangular null matrix of dimensions nand m. In Eq.(1), u(q) = uG1(q) +

uG2(q)is the gravitational potential of the two massive bodies (only a function of q): in the CR3BP they are considered

spherical and homogeneous, therefore each term is uG⊕=Gm⊕/||q−q⊕||, where Gis the gravitational constant, m⊕

and q⊕are the mass and position of the body ⊕in the 3B frame, and are expressed in normalized units. The resulting

gravity acceleration is the Cartesian gradient ∇= [ ∂

∂x;∂

∂y;∂

∂z]of the potential. In particular, the gravity of both bodies is

time-invariant because in this rotating reference frame their positions are ﬁxed along the x-axis (x1=−µ,x2=1−µin

Lunits). Since the normalized angular velocity of the 3B frame is ω=ˆ

z= [0;0; 1], the notation is simpliﬁed by deﬁning

1The dot over a variable represents the time-derivative operator.

6 M. Zamaro, J.D. Biggs

the skew-symmetric matrix W∈R3×3:W·(·) = ˆ

z∧(·)and the related matrix P=W2, to yield the CR3BP in the system

of ﬁrst-order ordinary differential equations (ODEs).

˙

x=fCR3BP(x) =

03I3

−P−2W

x+

03x1

∇u(q)

(1)

The solution at time tis provided by the ﬂow map φf(t,t0,x0) = x(t), which depends from the initial condition x0at t0.

The CR3BP deﬁned above is a non-integrable time-invariant Hamiltonian system, whose integral of motion is usually

called the Jacobi integral c, which can be related to the mechanical energy per unit of mass εof the system. In relative

dynamics, the effective potential uef f is usually introduced to resemble the two classical components of the mechanical

energy.

c(x) = −˙

qT˙

q+qTq+2u(q) = −˙

qT˙

q+2ue f f (q) = −2ε(x)(2)

The Hill’s surfaces, or zero-velocity surfaces, are the iso-surfaces in the position domain of the Jacobi integral with

null velocity. For a given initial condition x0, the related Hill’s surface deﬁnes the boundary of the 3D position-space

where the natural motion is constrained. In particular, the Hill’s sphere of inﬂuence (SOI) is the region around each

body where the dynamics is dominated by its own gravity ﬁeld: placing a point at zero relative velocity inside the SOI

would result in an attraction towards the respective body. In the framework of the CR3BP, the second primary’s SOI

normalized radius is deﬁned as 3

pµ/3 (Koon et al. 2011). For Phobos, this radius is 0.17% of the distance from Mars,

which is an altitude over the moon’s mean sphere of only 5.5km. This means that the physical space that can be used

to orbit around Phobos is very close to the body at low energy, and considering the fact that Phobos is very irregular in

its shape, the maximum altitude of the Hill’s SOI boundary is of only 3.5km, as shown in Fig.3(left) using the moon’s

mean ellipsoid. Therefore it is impossible to naturally orbit around Phobos with a Keplerian motion.

As µand Ldecrease in the CR3BP, the SOI shrinks close to the surface of the secondary body. Thus, the two

peculiarities of the Mars-Phobos couple previously highlighted are responsible for the reduction of the exploitable

realm of attraction of the moon. This result signiﬁcantly affects the LPOs of the three-body dynamics, and will be

referred throughout this paper as the collapsing effect of the SOI. In particular an analysis of the physical and orbital

characteristics for all the major couples of primaries of the Solar System was undertaken by computing the related Hill’s

SOI radius. We considered two maximum thresholds for the altitude of the Hill’s SOI over the secondary’s mean radius:

10km for practical mission operation constraints, and 1 radius to deﬁne an outer boundary such that, within this altitude,

perturbations due to the secondary body signiﬁcantly inﬂuence the motion. Within these two absolute and relative

thresholds, in addition to Mars-Phobos, only ﬁve other couples of primaries fall inside, and therefore are characterized

by the same collapsing effect. They are all cases of planetary satellites: three moons of Saturn (Pan, Daphnis, Atlas),

one of Uranus (Cordelia), and one of Neptune (Naiad), while four inner moons of Jupiter have their Hill’s SOI inside

their body. Orbital dynamics in proximity of these bodies are therefore very complex: no Keplerian solutions (where the

central body is the secondary) nor classical LPOs of the CR3BP exist because the natural motion is highly inﬂuenced by

the perturbations of the secondary body, since these moons are all highly inhomogeneous. However, as the contraction

of the SOI indirectly drags the manifold of distant orbits (where the central body is the primary) closer to the secondary,

Natural Motion around the Martian Moon Phobos 7

Fig. 3: Left: Hill’s surface for L2energy. x-zprojection. Phobos mean sphere (dashed line) and ellipsoid (plain line). Axes normalized in

Lunits. Right: Differential perturbations analysis. Vertical dotted lines indicate Phobos major size and Hill’s SOI radius. Distance from

Phobos in mean radii.

the Quasi-Satellite orbits (Gil and Schwartz 2010) are also much closer than in classical systems studied with a CR3BP.

It is worth noting that despite the small µyielding a small normalized radius of the SOI, the collapsing effect is not

encountered in the case of a Sun-asteroid couple because Lis too large. For example, considering a distance of 1AU,

even with a hypothetical small bulk density of 1,000kg/m3, the relative altitude of the SOI of a body orbiting around

the Sun is more than 100 times its mean sphere’s radius.

2.2 Orbital Perturbations

An analysis is undertaken to quantify approximately the errors that occur in the Mars-Phobos system when it is ap-

proximated with a CR3BP. The ﬁrst orbital perturbation is due to the non-spherical gravity of both Mars and Phobos,

where the dominant term is known as J2and is related to the body’s oblateness, and has been evaluated including also its

apparent acceleration acting on the 3B frame. The second source of gravitational perturbation is the effect of additional

bodies, considered in their closest conjunction conﬁguration with Phobos, that in the framework of the 3BP is referred

as a fourth body perturbation. Furthermore, the radiation pressures of the Sun (SRP), Mars (MRP, enclosing also the

portion of the albedo of the SRP), and Phobos (PRP, with its albedo) are computed. In addition to these physical forces,

the modeling perturbations represented by the approximation of the dynamics as a circular R3BP are taken into account,

which is the effect of the eccentricity of the Mars-Phobos orbit. To derive a related physical acceleration value, we con-

sider the difference between the acceleration ﬁeld of the CR3BP and the one of the elliptic model of the Mars-Phobos

system, and take the maximum over the orbital phases of Phobos, with the reference of the same relative state with

respect to Phobos.

An important point to consider is that Phobos’ orbit around Mars is not Keplerian, but more closely follows a clas-

sical low-altitude J2perturbed orbit. Therefore, the analysis of every orbital perturbation which is not due to Phobos

(gravity and PRP) in its proximity must be conducted in the framework of the relative dynamics with respect to Phobos,

considering the resulting differential perturbation.

Since this analysis is undertaken to derive a basic reference for the orbital perturbations, it will be applied for the

8 M. Zamaro, J.D. Biggs

simple case of ﬁxed relative points. Also, as we are interested in the dynamics near Phobos, due to the small size of its

SOI, all the perturbations are nearly isotropic, and the only variable for this simple analysis is the radial distance from

Phobos along the Mars-Phobos direction (apart from the eccentricity, which has been evaluated along all three direc-

tions). Outcomes of the differential analysis are presented in Fig.3(right), where the magnitude of the perturbations aPis

shown as a ratio aP/a2with respect to the magnitude a2of Phobos’ Keplerian gravity term at the point, and they corre-

spond to Gil (2007). In conclusion, the CR3BP does not provide an accurate approximation to describe the Mars-Phobos

system’s dynamics: the gravity harmonics and the orbital eccentricity of Phobos are the main orbital perturbations in

proximity of the moon. Outside its Hill’s SOI boundary the eccentricity becomes the dominant term, with Mars J2the

second most inﬂuential. In this paper both the gravity harmonics and eccentricity are incorporated into the modeling of

the relative dynamics near Phobos, as will be described in Section 2.3 and 2.4.

2.3 The Mars-Phobos CR3BP-GH

The modeling of the external gravity ﬁeld of convex bodies is provided by a spherical harmonics series expansion,

known as gravity harmonics (GHs). Usually complex shaped asteroids require an appropriate discrete distribution of

tetrahedrons to model their surface and derive the related gravity ﬁeld, because both the shape and the ﬁeld cannot be

described by a single-valued function in the spherical coordinates of the body’s BCBF frame (Willner 2009). However, in

the case of Phobos the GHs have proved suitable to model its gravity ﬁeld. From a previous paper (Chao and Rubincam

1989) which collates the data obtained through Viking observations, we are provided with a model of Phobos’ gravity

ﬁeld: the related GHs are stated up to degree and order 4, with the most important terms summarized in Table 1. In

Werner (1994), the tetrahedron approach was applied to Phobos and results compared with a GHs series expansion that

is based on the same Viking’s data and is stated up to degree and order 6 (Martinec et al. 1989): the conclusion made

by the author was that the two techniques produce similar results. Therefore, the GHs technique was used to model the

gravity ﬁeld of Phobos in this paper. The advantage is that the GHs are analytical functions, with far less computational

cost than using a mesh of tetrahedrons, and the accuracy of the approximation could be naturally tuned by the terms of

the series to be considered, unlike the tetrahedron method. We also chose to use the lower order GHs model of Chao

and Rubincam (1989), and use the model of Martinec et al. (1989) as a more accurate benchmark reference: the error

between the two gravity ﬁelds at the SOI’s boundary is of the order of a few percent of the cumulated GHs perturbation

at this distance. The methodology developed in this paper will not change, and the outcomes will just depend on the

series’ coefﬁcients.

The key point is that Phobos’ BCBF frame, where the GHs are deﬁned, is aligned with the 3B frame, therefore the

equations of motion of the Mars-Phobos CR3BP with Phobos’ GHs (named CR3BP-GH) remain time-invariant in the

classical 3B frame. The actual misalignment was presented in Fig.2, and is evaluated in the following way. Consider the

four angles Ω⊕,i⊕,ω⊕,ν⊕,0of the Keplerian orbital elements of the body ⊕, and nREV⊕,nROT⊕are its revolution and

rotational spin rates. The orbital elements of Phobos are deﬁned in a frame centered in Mars, with the z-axis aligned

along Mars’ spin axis, and the x-axis pointing towards the Mars’ Node of reference ΩN1(Archinal et al. 2010). The

BCBF frame of a body ⊕is expressed by the right ascension αNP

⊕and declination δNP

⊕of its North Pole: these physical

Natural Motion around the Martian Moon Phobos 9

characteristics are deﬁned in the J2000 frame for every Solar System’s planet or satellite (Archinal et al. 2010). They

deﬁne the North Pole’s inclination iNP

⊕=π

2−δNP

⊕, and the Node of reference’s location ΩN⊕=π

2+αNP

⊕. Finally,

the initial location of the body’s Prime Meridian from its Node of reference is given by ΩN⊕

PM⊕,0. Let AF1

F2deﬁne the

attitude matrix that rotates a vector’s components from the frame F1 to the frame F2. The sought ABCBF

3Bis given by a

succession of elementary rotations, where the attitude matrix An(α)represents a rotation of αaround the axis n=1,2,3

of the current frame (x,y,z) of basis (ˆ

e1,ˆ

e2,ˆ

e3).

ABCBF

3B=A3(nREV2t+ν2,0+ω2)A1(i2)A3(Ω2)A1(iNP

1)·

·A3(ΩN1−ΩN2)A1(−iNP

2)A3−ΩN2

PM2,0−nROT2t(3)

This matrix is generally a function of the time t. The time-dependency is erased if and only if the rotation and revolution

are synchronous, and the cluster of the ﬁve central matrices of Eq.(3) is reduced to a single matrix A3(Ω2−ΩN2). This

central core of Eq.(3) represents the attitude matrix between the orbital and equatorial frames of the second body. The

ﬁrst condition nREV2=nROT2is true for Phobos, while the misalignment between Phobos’ orbital and equatorial planes

of the second condition is evaluated with the real ephemerides along time of Phobos (Duxbury and Callahan 1981): the

related Euler angle represents the tilt angle shown in Fig.2. Since this angle is relatively small (maximum 1.90◦) it is

approximated to be null, and the attitude matrix between the two frames becomes:

ABCBF

3B=A3(−ΩR1

2

PM2) = A3(−π)(4)

where the location ΩR1

2

PM2of Phobos’ Prime Meridian (x-axis of BCBF) with respect to the Mars-Phobos radial (x-axis of

3B) is constant and formally set to 180◦=πrad as explained at the beginning of Section 2. This approximation could

be estimated to produce a residual orbital perturbation on the new CR3BP-GH of two orders of magnitude lower than

the one provided by the cumulated GHs perturbation on the CR3BP. This, together with the inaccuracy of the GHs’

coefﬁcients, is a ﬁrst estimation of the savings that will be provided by tracking the orbits computed in a more accurate

description of the natural dynamics.

The gravitational potential uG⊕described by the series expansion of spherical harmonics is presented in Eq.(5)

(Kaula 1966), where R⊕is a reference radius, in our case the equivalent-volume radius of the body ⊕. It is still a

function of the position in the 3B frame, but the series expansion of GHs is expressed in term of its spherical coordinates

in the BCBF frame: qBCBF = (ABCBF

3B)T(q−q⊕). The transformation from Cartesian to spherical coordinates [r;ϑ;ψ] =

TT SE ([x;y;z]) is described by magnitude r, co-latitude ϑfrom the body’s North Pole, East-longitude ψwith respect to

the body’s Prime Meridian, and the related physical basis (ˆ

er,ˆ

eϑ,ˆ

eψ) deﬁnes the local TSE frame.

uG⊕(q) = uG⊕(r,ϑ,ψ) = Gm⊕

R⊕

∞

∑

n=0R⊕

rn+1n

∑

m=0

Cm

n(ψ)Pm

n(cosϑ)

Cm

n(ψ) = Cn,mcos mψ+Sn,msin mψ=Jn,mcos m(ψ−λn,m)

(5)

The double expansion is conducted with degree nand order m, using the Legendre associated polynomials Pm

n, functions

of the co-latitude, which are scaled with two n,m-dependent coefﬁcients linearly combined with harmonic functions of

the longitude: Cand S(or related magnitude Jand phase λ) are the Stokes coefﬁcients pair for each degree and order

10 M. Zamaro, J.D. Biggs

of the series expansion. The series are truncated at the maximum degree and order of the GHs coefﬁcients. The ﬁrst

term J0,0=1 provides the basic spherical term used in Section 2.1 in the CR3BP, therefore all the other terms constitute

the perturbation of the GHs. A similar series expansion is available for describing the real shape of Phobos (Willner

2009), which is expressed up to degree and order 17. The related shape harmonics will be used from now on to model

the surface of Phobos for practical applications, as represented in Fig.1.

From the deﬁnition of the potential, the gravitational acceleration of Phobos is obtained through the spherical gradi-

ent ∇T SE = [ ∂

∂r;1

r

∂

∂ ϑ ;1

rsinϑ

∂

∂ ψ ], then rotated in the BCBF frame through the local attitude matrix of Eq.(6), where φis

the latitude.

ABCBF

T SE (q) = ABCBF

T SE (φ,ψ) = A1−π

2A2(−φ)A3(ψ) = A3(−φ)A1−π

2A3(ψ)(6)

For the following applications that will be presented in Section 3, the Hessian matrix of the potential is required, and it is

obtained through the spherical Hessian HT SE , then rotated and counter-rotated through ABCBF

T SE . The Hessian in spherical

coordinates is computed from the Hessian deﬁnition for a general curvilinear coordinates basis, that makes use of the

metric matrix gand the Christoffels’ symbols Γof the coordinate transformation (Brannon 2004), in our case TTSE , and

is presented in Eq.(7).

HT SE =

∂2

∂r2

∂

∂r1

r

∂

∂ ϑ ∂

∂r1

rsinϑ

∂

∂ ψ

∂

∂r1

r

∂

∂ ϑ 1

r2

∂2

∂ ϑ 2+1

r

∂

∂r

1

r

∂

∂ ϑ 1

rsinϑ

∂

∂ ψ

∂

∂r1

rsinϑ

∂

∂ ψ 1

r

∂

∂ ϑ 1

rsinϑ

∂

∂ ψ 1

r2sin2ϑ

∂2

∂ ψ2+1

r

∂

∂r+cosϑ

r2sinϑ

∂

∂ ϑ

(7)

The gravitational potential of Phobos is extended in the GHs series expansion: J0,0=1, degree-1 terms are null for

barycentric frames, therefore the net gravitational perturbation of Phobos ap

G2, starting from J2,0=0.105, is added to the

CR3BP vectorﬁeld to derive the equations of motion of the CR3BP-GH in the classical 3B frame.

˙

x=fCR3BPGH(x) = fCR3BP (x) + fGH(x) = fCR3BP (x) +

03x1

ap

G2(q)

(8)

ap

G2(q) = ABCBF

3BABCBF

T SE (q)T∇T SE uG2(q)(9)

A related augmented Jacobi integral for the CR3BP-GH could be deﬁned as in Eq.(2), simply expanding the complete

gravity potential of Phobos with the GHs series.

Finally, it’s important to recall that Phobos’ orbit itself is highly non-Keplerian due to the strong Martian J2action,

but this perturbation (still time-invariant due to Phobos’ properties and the axial-symmetry of J2) is not considered in

our model, because in this paper we are interested in the relative dynamics near Phobos. Fig.3(right) shows that the

differential Mars-J2perturbation is negligible inside the moon’s SOI. In particular, the inclusion of the oblateness of the

massive bodies in the CR3BP has been investigated in Douskos and Markellos (2006), Arredondo et al. (2012), Abouel-

magd and Sharaf (2013): in addition to the direct gravitational effect, the inclusion of J2of the primary body affects also

the apparent acceleration, by increasing the angular velocity of the frame of reference by a factor p1+3

2J2(R1/L)2.

The differential Mars-J2perturbation reported in Fig.3(right) includes also this action. Apparent effects of additional

Natural Motion around the Martian Moon Phobos 11

Mars’ GHs are also negligible, as well as the ones of all Phobos’ GHs inside the SOI. For our focus on the LPOs, the

Mars-Phobos CR3BP-GH is therefore still deﬁned with the time-scale of the unperturbed mean motion of the Keplerian,

circular and equatorial orbit of Phobos around Mars.

2.4 The Mars-Phobos ER3BP-GH

Despite the fact that the orbital eccentricity e=0.0156 of Phobos is not particularly high, it can be seen in Fig.3(right)

that the differential perturbation has a signiﬁcant effect on the LPOs, as an indirect effect of the collapse of the SOI’s

boundary towards the moon. The addition of the eccentricity in the CR3BP has already been investigated in the past

(Szebehely 1967; Campagnola et al. 2008), and it has been extensively studied in an elliptic R3BP (ER3BP) in the

same barycentric and rotating 3B frame of the circular case. The frame of the ER3BP rotates not uniformly, but with a

time-varying ωand ˙

ω. Besides, the classical ER3BP uses a time-dependent length unit corresponding to the distance

between the two massive bodies (called a pulsating frame), and a non-uniform time variable corresponding to the true

anomaly ν.

In our case, since we are focused on studying close relative motion around Phobos, and also we need to include the

GHs of the moon, the classical pulsating frame of the ER3BP is avoided, because it produces different time-dependent

scalings for each GH at different degree, i.e. (1+ecosν)−n. To overcome this, the equations of motion of the Mars-

Phobos ER3BP with the moon’s GHs (named ER3BP-GH) are derived in Phobos’ Hill’s frame. This frame is still

rotating with the Mars-Phobos elliptic motion ω= [0;0; ˙

ν], but is centered at Phobos, and it uses the same ﬁxed length

unit L(semi-major axis) used for the previous models. Regarding the time variable, since the vectorﬁeld is used not only

for integration but also in continuation schemes in Section 3, the convenient choice is to consider a ﬁxed time unit too,

and use the mean anomaly. This is because the sensitivity of the ﬂow with respect to ν0could be retrieved appending ν

in the state vector x= [q;˙

q;ν]. Eq.(10) presents the equation of motion of the ER3BP-GH in the Phobos’ Hill’s frame.

˙

x=fER3BPGH(x) =

03I303x1

−(1+ecosν)4

(1−e2)3P+2(1+ecosν)3esin ν

(1−e2)3W−2(1+ecosν)2

(1−e2)3/2W 03x1

01x3 01x3 0

x+

+

03x1

∇uG1(q,ν) + ∇uG2(q)+(1−µ)(1+ecos ν)2

(1−e2)2h1

0

0i

(1+ecosν)2

(1−e2)3/2

(10)

It is worth noting that in this model the gravity potential of Phobos depends only on the relative position qin

the Hill’s frame, while all the other terms depend also on ν. In particular Mars’ pulsating position along the x-axis is

x1=−(1−e2)/(1+ecosν)Lunits. It is easy to see that the model of Eq.(10) reduces to the circular cases of Eq.(1),(8)

for e=0. Summarizing the eccentricity produces a time-dependent perturbation to the dynamics of the circular case,

and no integral of motion is available whatever reference frame is chosen to express the dynamics.

12 M. Zamaro, J.D. Biggs

Fig. 4: LPOs in the Mars-Phobos CR3BP. On the left, example of two planar (red) and vertical (green) Lyapunov orbits around each LP

and one Lissajous (black) orbit of high width around them. On the right, families of Northern (black) and Southern (blue) Halo orbits. Mean

ellipsoid for Phobos surface. From this point, axes in all the ﬁgures are normalized in Lunits unless otherwise indicated.

3 Computation of Periodic and Quasi-Periodic LPOs and their Dynamical Substitutes

The invariant solutions of the dynamical systems of Eq.(1),(8),(10) are trajectories that can be decomposed as the sum

of nperiodic functions, realizing an n-torus, such as equilibrium points (EPs, n=0), periodic orbits (POs, n=1), and

quasi-periodic orbits (nD-QPOs, n≥2). The trajectory of a n-torus can be seen as dependent on nphases ϕrather than

time.

In this section we use the procedures of dynamical systems theory that deﬁne such invariant solutions to compute the

LPOs of the Mars-Phobos-spacecraft system. First, we apply these techniques to the well-known case of the dynamics

approximated by a CR3BP, to compute the classical LPOs in this system. These solutions are then used as starting

guesses for the numerical continuation technique, that will be applied to ﬁnd the dynamical substitutes of the LPOs in

sequence, ﬁrst in the CR3BP-GH, and then in the Mars-Phobos ER3BP-GH. The resulting orbits are collected in Section

4.

3.1 LPOs of the Mars-Phobos CR3BP

In this section we introduce the structure of the natural invariant motions near Phobos and how they are computed in the

framework of the CR3BP of Eq.(1).

Equilibrium Points. The CR3BP is time-invariant therefore the stationary points of the effective potential of Eq.(2) are

the ﬁve equilibrium points in the rotating frame (Koon et al. 2011): the three collinear Libration points (LPs), aligned

with the two bodies (two close to the secondary, L1in inferior and L2in superior conjunction, as seen from the primary,

and L3in opposition), and the two equilateral L4and L5, equidistant from them and with elongation of 60◦from both

bodies.

The LPs are computed as roots of the vectorﬁeld of Eq.(1) using Newton’s method (Allgower and Georg 2003;

Koon et al. 2011). The collinear couple L1−2is very important in mission design, as their distance from the secondary

formally deﬁnes its SOI. Fig.3(left) shows the Hill’s surfaces for the L2energy. Due to the small mass parameter, the

energy difference between L1and L2is very small in non-dimensional units (10−8), therefore the manifolds of the two

Natural Motion around the Martian Moon Phobos 13

LPs are almost symmetrical, and the location of L1−2is close to the linearized approximation expressed by the Hill’s

SOI’s radius (L1is at 16.582km and L2is at 16.601km from Phobos’ barycenter). Regarding their stability properties,

L1−2are characterized by a vertical center manifold, and on the orbital plane by a center ×saddle behavior, which is

unstable. This means that the region of the phase-space around these LPs is characterized by two manifolds of periodic

orbits, their invariant manifolds, and transit (inside the invariant manifolds) and non-transit (outside them) orbits between

the realms of attraction of the two bodies (Koon et al. 2011).

Periodic Orbits. Since the CR3BP is time-invariant, the two center manifolds of periodic orbits around each LP are

organized in two 1-parameter families at increasing size and energy from them, called planar and vertical Lyapunov

orbits (Koon et al. 2011). As their energy increases, they encounter bifurcations: the ﬁrst one of the planar orbits gives

birth to the two symmetrical branches (with respect to the x-yplane) of Northern and Southern Halo orbits (Koon et al.

2011; G´

omez and Mondelo 2001); the second one of the planar orbits gives birth to the two symmetrical branches (with

respect to the x-axis) of axial orbits that represent a two-lane bridge to the ﬁrst bifurcation of the vertical orbits (G´

omez

and Mondelo 2001); further bifurcations provide multiplication of the orbits’ period (G´

omez and Mondelo 2001).

Analytical and semi-analytical methodologies, such as the Lindstedt-Poincar´

e techniques (Koon et al. 2011; Mas-

demont 2004; Hou and Liu 2011), allow high-order approximations of the nonlinear dynamics to compute periodic and

quasi-periodic solutions of a dynamical system. They can provide very accurate initial conditions to propagate in the

full nonlinear dynamics, but if the POs are highly unstable even very high-orders do not have a very extended region of

convergence (Koon et al. 2011; Masdemont 2004; G ´

omez and Mondelo 2001). In these cases, numerical methodologies

must be applied to provide reliable solutions for practical applications, where low-order analytical solutions are used

as starting guesses (G´

omez and Mondelo 2001). Such techniques search for an initial condition of a PO applying its

deﬁnition as an equilibrium ﬁxed point of a stroboscopic map (for time-variant systems) or a Poincar´

e map (for time-

invariant systems) (Koon et al. 2011) in a classical Newton’s method. Thus, the numerical methodology requires the

integration of the equations of motion and the associated variational equations at each step of Newton’s method. This

procedure is commonly known as a differential corrector (Stoer and Bulirsch 1983), or shooting method, and referred to

as single shooting for the case when only one trajectory is propagated at each iteration, and multiple shooting when N

independent trajectories are propagated at each iteration of the differential corrector.

The families of Lyapunov orbits of the Mars-Phobos CR3BP are computed in this paper using the Lindstedt-Poincar ´

e

method. The method approximates the nonlinear terms of the gravity potential as a series expansion of Legendre poly-

nomials, and computes the coefﬁcients of the series expansion of only the periodic normal modes of these approximated

dynamics (Koon et al. 2011; Masdemont 2004). The two families of Halo orbits are computed through the numerical

differential corrector (Koon et al. 2011; G´

omez and Mondelo 2001; Howell 1984). The Poincar´

e map is taken at the sur-

face of section g(x) = y, and due to the symmetries of the CR3BP, the objective function is to erase the ﬁnal velocities ˙x

and ˙zcorrecting the initial state xand ˙y. The initial guess for the differential corrector is given by Richardson’s solution

(Richardson 1980), which is a Lindstedt-Poincar´

e expansion at order 3. These four families of POs around each LP have

been computed up to the intersection with Phobos, and they are all unstable as described by Floquet theory (Koon et al.

2011; Howell 1984). Further bifurcation branches, in particular the one of the axial orbits, are not considered due to the

14 M. Zamaro, J.D. Biggs

collapse of the SOI of this system, as they are placed after the intersection with Phobos. Fig.4 shows some example of

POs that we computed using these methods in the Mars-Phobos CR3BP.

Quasi-Periodic Orbits. The double center manifold of the CR3BP is then characterized by families of 2-tori, which are

quasi-periodic orbits with two phases, at increasing size around each PO with central part. Due to the pattern of the

projections on the coordinate planes, they are referred as Lissajous orbits (Koon et al. 2011). Each PO constitutes the

backbone of a family of iso-energetic QPOs: on the map where the initial condition of the PO represents an invariant

point, each iso-energetic 2D-QPO represents an invariant closed curve.

In this paper, the manifold of the 2-parameter family of Lissajous orbits around the Lyapunov orbits is computed

directly with the Lindstedt-Poincar´

e expansion of 2-tori (Koon et al. 2011; Masdemont 2004). These orbits are all

unstable as described by Floquet theory (Koon et al. 2011). Fig.4 shows examples of QPOs computed in the Mars-

Phobos CR3BP.

Invariant Manifolds. Finally, the presence of the saddle manifold implies that the LPOs are separatrices of motion

between transit and non-transit orbits to enter or escape from the SOI of the second massive body. The boundary of these

tubes is given by the Invariant Manifolds (IMs) of the LPOs that provide the energy efﬁcient trajectories to minimize the

fuel consumption of spacecraft for interplanetary transfer phases (Koon et al. 2011; Conley 1969). The local direction

of the IMs is represented by the eigenvectors of the set of hyperbolic eigenvalues of the linearized dynamics of the

LPO. There are two kinds of IMs: the linear span of the stable eigenvectors along the orbit and the one of the unstable

set, and each of them has two branches, deﬁned by the sign of the eigenvector. For each saddle manifold, to derive the

complete tube of each of the four branches, one should integrate the initial conditions provided by the IM along the orbit

and derive the set of trajectories. This procedure is known as the globalization of the manifold and it has been widely

applied in the CR3BP (Koon et al. 2011; G´

omez et al. 1991).

In this paper, due to the proximity of the LPOs to Phobos, all the inside branch of their IMs intersects Phobos,

therefore no homoclinic nor heteroclinic connections of the two LPs manifolds exist. In Section 5.4 we compute the

inside branch of the IMs of the LPOs in the ER3BP-GH and derive the performances of these natural trajectories to

arrive (along the unstable IMs) and depart (along the stable IMs) from the moon’s 3D surface.

3.2 Dynamical Substitutes of the LPOs

This paper focuses on the computation of the natural orbits in close proximity of Phobos. Due to the collapsing effect of

its SOI, these are the families of periodic and quasi-periodic LPOs computed in Section 3.1. However, in Section 2.2 we

found that the CR3BP does not provide a suitable approximation of the relative dynamics, which in this region should

take into account two important orbital perturbations. This led us in Section 2.3 and 2.4 to deﬁne two additional dynam-

ical models, the CR3BP-GH that encloses the modeling of the gravity ﬁeld of the moon with GHs, and the ER3BP-GH

that considers Phobos moving in an elliptical orbit around Mars. The procedure that we present in this section is to use

the LPOs of the CR3BP as starting solutions of a numerical continuation along the three different dynamics, computing

Natural Motion around the Martian Moon Phobos 15

in sequence their dynamical substitutes in the new models. A dynamical substitute, as the name suggests, is the equiv-

alent of the classical solution of the original dynamics in a perturbed dynamical system, and was introduced in G´

omez

et al. (2002). The dynamical substitute is an orbit that not only is a solution of the new dynamics, but also maintains in

its phase-space the natural and forcing frequencies of the perturbation.

The numerical continuation is a well-known numerical technique used to compute the curve of solutions of a para-

metric nonlinear equation (Keller 1977; Allgower and Georg 2003). This method starts from a known solution of the

equation, which is at a value of the parameter that makes it simpler to solve analytically or numerically. The numerical

continuation consists in incrementing the parameter slowly, solving Newton’s methods by iteratively using, as an initial

guess, the solution found at the previous step, until the solution for the desired value of the parameter is obtained. In this

paper the numerical continuation is applied to ﬁnd EPs, POs, and QPOs solutions of differential equations, therefore

the continuation parameter weights the level of perturbation between different dynamics, and at each step the related

differential correction scheme to compute the invariant solution is performed. Since the nonlinear effects of the pertur-

bation could be signiﬁcant, to overcome possible turning points in the continuation curve the numerical continuation

is implemented with the pseudo-arclength algorithm (Keller 1977; Allgower and Georg 2003). Also, to enhance the

convergence of the differential corrector and speed up the continuation, we use a predictor for the initial guesses of the

differential corrector along the continuation curve, and an adaptive step strategy based on the control of the number of

Newton’s method iterations (Allgower and Georg 2003; Masdemont and Mondelo 2004).

3.2.1 Dynamical Substitutes of the LPOs in the Mars-Phobos CR3BP-GH

In this section we present the approach used to compute the dynamical substitutes of the LPOs from the model of

Eq.(1) to the system described by Eq.(8). We differentiate the procedure undertaken by the phase-space dimension of

the invariant solutions, which are EPs, POs, and 2D-QPOs. As the perturbation of Phobos’ GHs is time-invariant in the

3B frame, the dynamical substitutes of the invariant solutions maintain the dimension of their original phase-space in

the CR3BP.

Equilibrium Points. The EPs of the CR3BP-GH are calculated similarly to the CR3BP, with the potential function of

Eq.(2) now considering the complete gravity ﬁeld of Phobos through Eq.(5), without need of numerical continuation.

Also, it must be reported that the truncated GHs model mathematically provides additional EPs to the system, but they

are not useful as will be discussed in Section 4.1.

Periodic Orbits. To derive the substitutes of the periodic LPOs in the Mars-Phobos CR3BP-GH, the numerical contin-

uation is used starting from the initial conditions of the families of POs computed in the basic CR3BP. The differential

corrector scheme’s step looks for invariant points of a Poincar´

e map. The continuation parameter is, intuitively, a scaling

factor σGH , from 0 to 1, to weight the GHs acceleration in the intermediate vectorﬁeld h.

˙

x=h(x,σGH ) = fCR3BP (x) + σGH fGH (x)→φh(t,t0,x0,σGH)(11)

16 M. Zamaro, J.D. Biggs

Fig. 5: POs in the Mars-Phobos CR3BP-GH. Numerical continuation with respect to the GHs of a small-energy planar and vertical Lyapunov

orbit around L1of the CR3BP (blue) to their dynamical substitute in the CR3BP-GH (black) belonging to the families A and B respectively,

together with the current LP.

The problem is conservative and an appropriate Jacobi integral of the augmented vectorﬁeld is available as was addressed

in Section 2.3. Therefore, POs are continued in a differential way with respect to the state of the current LP, using a

variant Poincar´

e map g(x,σGH ) = y−yLP(σGH ), and with a ﬁxed-energy constraint with respect to the energy of the

LP. As the two dynamics have different Jacobi integrals, the dynamical substitute is naturally identiﬁed as the orbit with

the same energy gap ∆c(x) = c(x)−c(xLP)from the energy level of the LP in the related dynamics. Therefore, the

numerical continuation of the POs embeds also the computation of the current LP: this choice of a differential numerical

continuation has been proved to be 33% faster than continuing the absolute state, since it requires less differential

corrector iterations for a given step size.

The numerical continuation is performed for a sampling of POs for each of the four families of the CR3BP, for both

LPs. Fig.5 shows what ”physically” happens: as σGH is increased from 0 to 1, the small planar and vertical Lyapunov

orbits follow the corresponding LP and are highly distorted, mostly in the second half of the continuation. In particular,

the vertical Lyapunov orbit deploys itself in a 2:1 elliptical orbit too, and counterintuitively, substitutes of planar orbits

are mostly vertical and substitutes of vertical ones are mostly planar at σGH =1. These results demonstrate the effect of

the strong 3D-coupled dynamics. In this sense, the Hessian of Eq.(7) applied in Eq.(9) maintains the decoupling of the

x-yand zdynamics of Eq.(8), when linearized around L1−2, only for even zonal harmonics or sectorial harmonics. It is

worth noting that with this continuation scheme, turning points of the continuation curve are present: planar Lyapunov

and Northern and Southern Halo orbits of the CR3BP are linked together under the framework of the augmented state

space [x;σGH ]of the vectorﬁeld h.

After obtaining this sampling of dynamical substitutes, the new families of POs in the CR3BP-GH can be visually

identiﬁed, and will be discussed in Section 4. To reﬁne the families of POs identiﬁed with this approach, we then take

one of the POs just computed for each of them and perform a numerical continuation with respect to the differential

Jacobi integral ∆cused as a ﬁxed constraint previously, in the ﬁnal CR3BP-GH for σGH =1.

Natural Motion around the Martian Moon Phobos 17

Fig. 6: Left: QPOs in the Mars-Phobos ER3BP-GH. Numerical continuation of the invariant curve on the stroboscopic map with respect

to the eccentricity, from a medium-energy PO of the A family around L1of the CR3BP-GH (blue point) to the dynamical substitute in the

ER3BP-GH (black). Right: QPOs in the Mars-Phobos ER3BP-GH. Numerical continuation of the invariant curve on the stroboscopic map

with respect to the stroboscopic period, from a low-size QPO (blue) to the QPO intersecting Phobos’ surface (black) of the A family around

L1in the ER3BP-GH.

Quasi-Periodic Orbits. To compute the families of 2-tori around each PO of the families with central part of the CR3BP-

GH, we used the numerical approach of dynamical systems theory that derives the invariant curve of these QPOs on an

iso-energetic Poincar´

e map. This invariant object provides a closed curve of initial condition for the surface of motion,

and it is described by the truncated Fourier series CF(c,θ)function of the curvilinear coordinate θ, and the Fourier

coefﬁcients c(2NC+1 for each coordinate variable). Different differential corrector schemes based on Fourier analysis

have been used, see G´

omez and Mondelo (2001), Kolemen et al. (2012), Olikara and Scheeres (2012). In this paper

the approach of Kolemen et al. (2012) is adopted, where the curvilinear coordinate that parameterizes the invariant

curve is the polar anomaly between two state components. Since the invariant curve is continuous, it is discretized in

NSsample points. The differential corrector consists of a multiple shooting scheme that forces the images of the sample

points through the Poincar´

e map to lie on the same Fourier series CF, by ﬁne tuning the Fourier coefﬁcients in a least

mean square (LMS) ﬁtting algorithm. The gap between the curvilinear coordinates of image and initial condition along

the invariant curve constitutes its rotation number proﬁle; for the nonlinear CR3BP-GH, this is not ﬂat and unknown

a priori. As the CR3BP-GH is time-invariant, the family of iso-energetic invariant curves around a PO backbone is

obtained embedding the LMS algorithm in a numerical continuation with respect to a sizing parameter of the curve.

Once one invariant curve is identiﬁed, it’s possible to simulate the QPO by numerically integrating through iterative

mappings, interpolating the intermediate initial condition along the invariant curve; this is necessary also to derive its

rotation number proﬁle. These surfaces of motion represent the dynamical substitutes of the Lissajous orbits of the

CR3BP around Lyapunov and Halo orbits.

To perform the stability analysis for QPOs and the computation of the related IMs, we modiﬁed the method of

Kolemen et al. (2012) in order to provide a 1-1 mapping that determines the invariant curve in a way that resembles the

invariance condition of POs. This lies in the fact that we approximate the invariant curve with a truncated Fourier series

expansion. The idea is that the set of mappings of the initial conditions along the starting curve CF(c0,θ)constitutes

a second curve that could also be deﬁned by a related set of ﬁnal Fourier coefﬁcients cf. This is a map from a set

of Fourier coefﬁcients to another (at the same truncated order), and if the curve is invariant under the usual map, the

18 M. Zamaro, J.D. Biggs

Fig. 7: QPOs in the Mars-Phobos ER3BP-GH. On the left, invariant curves on the stroboscopic map of the D family around L1, from a

low-size QPO (blue) to the QPO intersecting Phobos’ surface (black). The invariant curve highlighted in red corresponds to a QPO with

the resonant stroboscopic period π. On the right, the related proﬁle of the curve’s parametrization variable (corresponding to Phobos’ true

anomaly) along iterated mappings shows that the proﬁle at the second-return frequency (red) is ﬂat: as the magniﬁcation highlights, the total

change of the Phobos’ true anomaly is just 10−7◦in 100 stroboscopic periods, therefore this QPO constitutes a closed family of POs in the

ER3BP-GH, since every point along the curve is invariant.

initial Fourier coefﬁcients c0are a ﬁxed point of the new map. In this case, moving from sampled points (the set of ﬁnal

conditions) to coefﬁcients, the discrete Fourier transform is used to deﬁne the new map, which is solved using LMS

because the images’ sampling is not uniform. The complexity has been transferred to the map itself, because the same

LMS algorithm is now embedded inside it and no longer required to solve the linear system of the Newton’s method

iterations, which is now characterized by a square Jacobian matrix. Indeed the map of the new method requires the

computation of the full inverse of a large-scale left pseudo-inverse matrix with dimension (2NC+1), requiring far more

computational time than the previous one. However it provides, as a byproduct, a monodromy matrix for the stability

analysis of the QPO. The related eigenvalues for an invariant curve are organized in circles on the complex plane (Jorba

2001; Adomaitis 2007), and the Floquet linear stability is therefore given by their magnitude. Their eigenvectors are

deﬁned in the space of the Fourier coefﬁcients, therefore they provide a curve of initial conditions for the globalization

of the IMs.

3.2.2 Dynamical Substitutes of the LPOs in the Mars-Phobos ER3BP-GH

In this section we present the approach used to compute the dynamical substitutes of the LPOs from the model of

Eq.(8) to the one of Eq.(10). The ER3BP-GH is a time-variant dynamical system, sum of its conservative vectorﬁeld

(the CR3BP-GH in the Hill’s frame) plus a forcing term (the elliptic perturbation). The general solution of such ODEs

is constituted by its homogeneous term (sum of the normal modes of the conservative part) plus a particular solution

(dependent from the kind of forcing term). The elliptic perturbation on the CR3BP-GH, as we see in Eq.(10), is function

of ν, and it is a periodic signal at the orbital frequency of the primaries: the particular solution is a signal at the same

frequency. This means that the solutions of the elliptic dynamics in the phase-space have one phase constrained to

follow the dynamics of ν(t), which is appended in the vectorﬁeld and so explicitly retrievable from the integration of

Natural Motion around the Martian Moon Phobos 19

the equations of motion. The vectorﬁeld of the ER3BP-GH is a natural homotopy hwith respect to the eccentricity

˙

x=fER3BPGH (x) = h(x,e)→φh(t,t0,x0,e)(12)

and the numerical continuation with respect to the eccentricity of the invariant solutions computed in the CR3BP-GH

increments by one their phase-space and produces dynamical substitutes in the ER3BP-GH that oscillate around the

original ones with period 2π.

Periodic Orbits. Since the dynamics of the ER3BP-GH are time-variant, and the eigenvalues of the LPs of the CR3BP-

GH are different from the orbital frequency of Phobos, no physical EPs exist in this non-pulsating system. The LPs now

become POs, as they oscillate along the LPs of the circular case.

Apart from the dynamical substitutes of the ancestor EPs, isolated POs in the ER3BP-GH develop when the original

PO in the circular case is resonant with ν. In this case, the characteristic curves of the period of the POs for the families

of the CR3BP-GH show us in Section 4.2 that one family around each LP has one PO with period π, which is the

usual overresonance 2:1 (resonances with high denominators are not considered). The dynamical substitutes of these

two synchronous POs in the ER3BP-GH are still POs, with period 2π: this gives rise to two iso-periodic families of

POs in the elliptic problem, since the circular problem is a singular case for the deﬁnition of ν0. The computation

of these POs could be undertaken with the differential corrector used previously in Section 3.2.1 for POs, but using

a stroboscopic map with propagation time of 2π, and embedded in a numerical continuation where the parameter is

the eccentricity. This approach was ﬁrst undertaken in Broucke (1969), to compute dynamical substitutes of POs from

CR3BP to ER3BP. This continuation produces different dynamical substitutes starting from different ν0on the same

stroboscopic map. Single shooting was used in Broucke (1969) to compute only symmetrical dynamical substitutes,

starting the continuation only at pericenter or apocenter conditions. For our asymmetric ER3BP-GH, the curve of initial

conditions on the stroboscopic map is required. In particular, due to the overresonance, one half of this curve could be

avoided, since the initial conditions correspond to the ﬂows starting from the other half after a time of π.

2D Quasi-Periodic Orbits. Tori with two phases are the dynamical substitutes of the POs of the CR3BP-GH in the

general non-resonant case. They are derived with the differential corrector that computes their invariant curve, used

previously in Section 3.2.1 to compute families of 2-tori around a PO in the CR3BP-GH, but with some important

differences. First, the invariant curves are now deﬁned on a stroboscopic map, with the period ﬁxed to the one of the

original PO. Second, the invariant curve parameter is ﬁxed to be νon the stroboscopic map, so the related return value

after one revolution is directly retrievable by integration of the equations of motion. Third, the only constraint to add

regards the longitudinal phase indeterminacy of the torus (G´

omez and Mondelo 2001). The initial condition of the

PO computed in the CR3BP-GH is translated to Phobos’ Hill’s frame of its circular orbit around Mars, and then the

differential corrector is embedded in a numerical continuation with respect to the eccentricity to ﬁnd the invariant curve

in the ER3BP-GH. An example is shown in Fig.6(left). This procedure has already been used in Olikara and Scheeres

(2012) to ﬁnd dynamical substitutes of periodic LPOs from CR3BP to ER3BP. Once a QPO is obtained, the reﬁnement

for the related family can be achieved by changing the continuation parameter to be the stroboscopic time, in the ﬁnal

20 M. Zamaro, J.D. Biggs

Fig. 8: 3-Tori in the Mars-Phobos ER3BP-GH. Example of a small-width and medium-size 3D-QPO of the A family (on the left) and of

a large-width and low-size 3D-QPO of the A family (on the right), both around L1of the ER3BP-GH. The 3D-QPOs in black, obtained by

multiple shooting of the 2D-QPOs of the CR3BP-GH in blue, oscillate around these backbones with an amplitude equal to the motion of the

LP in the ER3BP-GH, also plotted in the graph. Integration up to the ﬁrst transversal revolution of the backbone 2D-QPO in the CR3BP-GH.

Fig. 9: Hill’s surfaces in CR3BP-J2at different energies. x-yand x-zprojections, with magniﬁcation around Northern additional EP.

ER3BP-GH with the eccentricity of Phobos. An example of this second scheme is shown in Fig.6(right).

This algorithm for QPOs naturally embeds the resonant cases N:M: when the period of the ancestor PO is in an

integer ratio M:Nwith the orbital period, the invariant curve of its dynamical substitute on a multiple-return stroboscopic

map (propagation time of Ntimes the orbital period) has a ﬂat rotation number proﬁle, whose value is zero. Therefore

this method for QPOs is used to continue the two synchronous POs in 2:1 resonance of the CR3BP-GH, and it is

clearly superior to the dedicated one of Broucke (1969) for POs, because it provides not only one initial condition in the

ER3BP-GH, but directly a curve of invariant points that describes the iso-periodic family of POs, as shown in Fig.7.

3D Quasi-Periodic Orbits. Tori with three phases are the dynamical substitutes of the QPOs of the CR3BP-GH, and

each of them is identiﬁed by an invariant 2-torus. The differential corrector method for computing QPOs is naturally

extended to a phase-space with higher dimensions using a multi-dimensional Fourier series to describe the invariant

object. However, for an ER3BP this is high computationally demanding. In this paper, to conclude the analysis of the

manifold of LPOs in the ER3BP-GH, we used the differential corrector scheme, embedded in a numerical continuation

with respect to the eccentricity, of the classical multiple shooting (Masdemont and Mondelo 2004; Stoer and Bulirsch

1983) that satisﬁes the continuity between the subintervals where the QPO trajectory is sampled for a ﬁnite propagation

time. The resulting linear system of the differential corrector is in this case underdetermined, and the solution at each

Natural Motion around the Martian Moon Phobos 21

Fig. 10: Hill’s surface in CR3BP-GH for L2energy. Projections on coordinates planes. Phobos real shape, mean sphere (dashed line) and

ellipsoid (plain line).

step is taken as the one with the minimum norm correction. This does not guarantee to ﬁnd a bounded orbit for high

perturbations, since the solution is not a dynamical substitute. However as Fig.8 shows, the orbits obtained oscillate

around the ancestor 2D-QPOs with an amplitude similar to that of the LPs even for a high time span set in the differential

corrector. Therefore we conclude that such trajectories describe appropriate 3-tori in the deﬁned time span, which in

terms of number of longitudinal revolutions is longer than that of typical applications of LPOs in mission segments. The

IMs are obtained from the hyperbolic eigenspace of the state-transition matrix within each subinterval.

4 LPOs around Phobos and Analysis of the Results

In this section the LPOs obtained with the methodologies described in Section 3 are presented and analyzed. The related

performances for potential station-keeping applications are collected in Section 5.

4.1 Dynamical Substitutes of the Equilibrium Points and Hill’s Surfaces

The collinear EPs L1−2of the classical CR3BP represent the ﬁrst two intersections of the Hill’s surfaces at increasing

energy: L1between the realms of admissible motion around Mars and Phobos, and L2between the exterior realm around

both bodies and the realm around Phobos. This holds for the Mars-Phobos CR3BP-GH. However, the additional terms

of the GHs in the augmented potential deﬁnition of Eq.(2) have an energizing effect on the Phobos realm, so the two

intersections occur farther away from Phobos. Another effect of the GHs is to create additional lobes on the Hill’s

surface. They are evident at the ”birth” of the realm (which is at low energy), and as the energy increases the Hill’s

surfaces grow in dimension and their lobes collapse with each other. This continues until there are no more lobes nor

residual internal holes and the realm becomes ”adult”, which is at an energy such that the Hill’s surfaces tend to the

ones of the classical CR3BP. The points where the surfaces attach to each other are additional EPs of the model. An

example is shown in Fig.9 for the case of only adding J2to the CR3BP, where two additional and symmetrical EPs are

born in the x-zplane, close to the z-axis of Phobos’ Hill’s frame and slightly pointing towards Mars. The result for a

generic CR3BP with the addition of J2for either one or both primaries was ﬁrst indicated in Douskos and Markellos

22 M. Zamaro, J.D. Biggs

Fig. 11: Convergence analysis of the CR3BP-GH. L1and L2displacement from the original value in the CR3BP. Error respect to the solution

with all the harmonics is normalized by the altitude over Phobos’ surface. Dotted lines refer to increasing zonal harmonics degree starting from

J2.

Table 2: CR3BP-GH collinear equilibria. Displacement of the LPs of the CR3BP-GH from their original value in the CR3BP.

EP ∆x∆y∆z

L1-743m 346m -195m

L2643m 345m -101m

(2006). These points are usually inside the central body’s ﬁgure. This is also the case in our system with GHs up to

fourth degree. The 4 additional equilibria are not valid because interior, with the outermost located 2km below Phobos’

surface. The ”adult” realm, once the last additional EPs is born, will then intersect the Hill’s surfaces of the Martian and

exterior realms to provide the new LPs of the CR3BP-GH. Table 2 provides the location of the dynamical substitutes

of these LPs, and Fig.10 shows the pattern of the Hill’s surface at the energy of L2. Due to the low mass parameter, as

it was in the CR3BP, the energy difference between L1and L2is small in non-dimensional units (10−7), therefore the

related Hill’s surfaces at the energy of L1and L2, apart from the small region around each EP, are practically unchanged:

they identify the boundary of the Hill’s SOI. The asymmetric displacement of L1and L2from their original values in the

basic CR3BP model is signiﬁcant, and accounts for 20% of their altitude over Phobos. This shows how inaccurate the

CR3BP is for practical applications.

Note in Fig.10 that the Hill’s surfaces at the energy of the LPs around Phobos are very close to its surface. This

was already evident in a classical CR3BP, just considering a mean ellipsoid as reference from the Phobos’ shape as

shown in Fig.3(left). However, the inclusion of the GHs improves the correspondence between the two boundaries in

particular in the vertical dimension, where no other way of escape at low energy is available because the gateway of

the invariant manifolds of the LPs is close to the orbital plane. The fact that the body’s shape ﬁlls entirely its SOI is

considered one of the proofs of the theory of accretion proposed by Craddock (2011) (see also Rosenblatt and Charnoz

(2012)) to explain the origin of Phobos, in contrast to the asteroid capture (Rosenblatt 2011; Lee 2012; Koschny 2012).

The theory of accretion speculates that a previous small body impacted Mars; the ejecta from the Red Planet created a

ring of dust around Mars, that eventually built up to create Phobos. The results obtained with our enhanced CR3BP-GH

suggest that the transportation of mass at low energy could be responsible for the current shape of Phobos, in particular

Natural Motion around the Martian Moon Phobos 23

for the vertical dimension. An additional consideration on the range of dimensions of these transferred materials will be

presented in Section 5.5.

The properties of L1−2are an important summarizing ﬁgure to conduct a convergence analysis on the CR3BP-GH

as a function of the number of GHs. This is done incrementally increasing the terms of the series expansion included in

the dynamics. Using the benchmark model of Martinec et al. (1989) as a reference, we see in Fig.11 that convergence

under 2% is achieved at degree and order 4 for the static location of the LPs. Considering the dynamical properties,

an even better convergence is achieved for the eigenvalues and the saddle’s eigenvectors, which represent the direction

of the IMs. For the eigenvectors of the two centers the outcome depends on the components. Identifying each center

manifold as the state-space’s section around the LP where each family of POs develops, the same ﬁgure of convergence

is achieved for the in-section components, while the out-of-section components are still oscillating at high amplitude, in

particular over 10% for L2. Therefore the outcomes are mostly accurate for the IMs and the longitudinal behavior of an

invariant motion, while a more reﬁned model is needed for the quasi-periodic motion around them.

Finally, the dynamical substitutes of the LPs in the ER3BP-GH are POs, similar to 1D oscillations around each LP of

the CR3BP-GH, with an amplitude of 260m, which is signiﬁcant at the length scale of Phobos for practical applications,

and a direction tilted from the x-axis by effect of the GHs: they can be seen in Fig.8. It is well-known that in the ER3BP

the EPs and the related Hill’s surfaces move with the pulsating reference frame, therefore the concept of a pulsating

SOI for Phobos can be extended also to the ER3BP-GH. This could be simulated considering the two extreme phases of

Phobos at perimars and apomars, changing the length unit to rescale the non-dimensional CR3BP-GH. The difference

from the classical ER3BP is that the outcomes of the non-dimensional circular problem are not maintained at different

phases of Phobos, and should therefore be recomputed because the pulsating length unit provides a different scaling of

the GHs (as pointed out in Section 2.4). However at the L2energy, the Hill’s surfaces obtained at the extreme phases

are similar to the ones of the CR3BP-GH, therefore the eccentricity effect does not seem to be a relevant factor for the

accretion theory’s validation.

4.2 Dynamical Substitutes of the Periodic Orbits

The periodic LPOs of the Mars-Phobos CR3BP are the 1-parameter families of planar and vertical Lyapunov orbits and

Northern and Southern Halo orbits. They have been computed with classical semi-analytical and numerical techniques,

and an example is shown in Fig.4.

The dynamical substitutes of these POs in the CR3BP-GH are obtained with the continuation schemes discussed in

Section 3.2.1, and they are showcased in Fig.12. They are constituted by two 1-parameter families A and B originating

from each LP, and by a detached 1-parameter family CD, made by two branches C and D connected by a common orbit

at the lowest energy. The low-energy orbits of the families A and B are highly distorted from their ancestor Lyapunov

orbits in the CR3BP as we see in Fig.5 and Fig.13, because the highly inhomogeneous gravity ﬁeld of Phobos produces

a 3D coupled linearized state-matrix at the LP. Besides, the force ﬁeld at the LP is no longer symmetric and highly tilted

from the one of the classical CR3BP, and the structure of the saddle ×center ×center manifolds is no longer aligned

with the coordinates axes and planes. The orientation of the centers’ manifolds is given by the couples of ”relative incli-

24 M. Zamaro, J.D. Biggs

Fig. 12: Periodic LPOs in the Mars-Phobos CR3BP-GH. Graphical visualization of the new A (red), B (green), C (magenta), and D (cyan)

families of POs around each LP in the CR3BP-GH. Shape harmonics series expansion for Phobos’ surface.

Fig. 13: Periodic LPOs in the Mars-Phobos CR3BP-GH. On the left, behavior of the A and B families around L1at low energy. On the

right, behavior of the C and D families around L1showing their actual connection through the ancestor orbit (in blue) as two branches of a

single CD family.

nation / ascending node’s right ascension” of each family’s small-energy PO. Around L1, family A has 46◦/278◦, family

B has 82◦/98◦, and around L2, family A has 26◦/43◦, family B has 71◦/223◦. Interestingly, the two POs around each LP

have opposite nodes (see also Fig.13). The effect of the complex gravity ﬁeld of Phobos is mostly local and is therefore

averaged for larger orbits, that resemble the shape of the LPOs of the classical CR3BP. The shape of larger orbits of

the family A is similar to the Southern Halo, B to the planar Lyapunov, C to the Northern Halo, and D to the vertical

Lyapunov. However, the tilt is still maintained, and this represents the most signiﬁcant effect of the addition of the GHs

of the second massive body. This information can be evaluated from the eigenspace at the LP. Regarding the properties

of these new POs, the period is close to the natural one of the LP as shown in Fig.14, and the Floquet stability analysis

presented in Fig.15 shows that all the orbits are unstable. The families A and C, like the Halo orbits, have their maximum

stability index sλ(Howell 1984) (a measure proportional to the instability) that rapidly decreases with increasing size of

the orbit; the maximum sλfor the family B is less sensitive to the size; the branch D is instead characterized by a saddle

×saddle ×center manifold, therefore no QPOs exist around these POs.

Natural Motion around the Martian Moon Phobos 25

Fig. 14: LPOs characteristic curves. On the left, characteristic curves of the period of the families of POs around L1in the CR3BP-GH,

parameterized by the non-dimensional differential Jacobi integral with respect to the LP. Magniﬁcation shows that the family A’s characteristic

curve is not monotonic. On the right, characteristic curves of the period of the ﬁrst transversal revolution around the invariant curve on the

second-return stroboscopic map of the families of 2-tori around L1in the ER3BP-GH, parameterized by the non-dimensional stroboscopic

period. The asymptote for the D family at the synchronous period T=πcorresponds to the 2:1 resonant QPO that is actually a family of POs.

Fig. 15: LPOs stability properties. Characteristic curves of the stability indexes of the two non-unit couples of eigenvalues of the monodromy

matrix of the families of POs in the CR3BP-GH, parameterized by the non-dimensional differential Jacobi integral with respect to the LP.

Families’ colors are coherent with Fig.12.

As explained in Section 3.2.1, the families of periodic LPOs in the CR3BP-GH have been obtained with two con-

tinuation schemes. The ﬁrst uses as continuation parameter the GHs’ magnitude σGH to derive the dynamical substitute

of a PO from the original CR3BP, and the second uses as continuation parameter the energy ∆cto compute the family

of POs in the CR3BP-GH. With this second continuation scheme, we found that the families C and D are actually two

branches of the same global family CD, since they are connected through an ancestor orbit at the lowest energy as we

see in Fig.13. In particular the reverse numerical continuation of this ancestor orbit with the ﬁrst scheme, from σGH =1

to the original CR3BP for σGH =0, produces as continuation curve a closed loop, which does not provide a dynamical

substitute in the original CR3BP at the same energy ∆c. We underline that a similar procedure to compute the POs in

the CR3BP-GH could be undertaken by-passing the ﬁrst scheme by computing analytically the LPOs in the linearized

dynamics of Eq.(8), which represent the dynamical substitutes of the small Lyapunov orbits showed in Fig.5. After, the

second scheme is performed from these solutions. This allows to compute the families A and B, but not CD, which

would require a bifurcation analysis. However, the proﬁle of the stability indexes of the three families in Fig.15 does

26 M. Zamaro, J.D. Biggs

Fig. 16: Convergence analysis of the LPOs in the Mars-Phobos CR3BP-GH. The ﬁgure on the left shows the different dynamical substitutes

of a medium-energy planar Lyapunov orbit around L1of the CR3BP (blue), obtained incrementally increasing the degree and order of the

harmonics included in the partial CR3BP-GH (red for degree 2, purple for degree 3, green for degree 4, with color darkened by increasing

order, and ﬁnal orbit at fourth degree and order in black). Crosses represent the current EP. On the right, related time such that the ﬂow of

these solutions in the full CR3BP-GH remains within a position error of 10m from their reference signal. Dotted lines refer to increasing zonal

harmonics degree starting from J2.

not suggest that the family CD bifurcates from neither A nor B, even with a bridge of POs at high-energy intersecting

Phobos’ ﬁgure. The strategy of two continuation schemes in cascade allowed us instead to compute easily the family

CD, where the ﬁrst scheme ﬁnds a PO ”seed” to feed the second.

Finally, we investigated the convergence of the numerical continuation procedure, as a function of the degree and

order of the Mars-Phobos CR3BP-GH, as was done in Section 4.1 for the EPs. One sample medium-energy orbit of the

A family has been chosen to evaluate the cost of the different truncated models in Fig.16. Compared to the case of the

EPs, the outcome is that a worse convergence of 5% is achieved for both static and dynamical properties, in particular

regarding the center manifold longitudinal to the PO. To have an idea of the price to pay, we simulated the dynamical

substitutes found at each different degree and order in the complete CR3BP-GH up to degree and order 4. Considering

the data at degree and order 3, station-keeping at a frequency of more than ten times per LPO’s period is required to

track the reference signal within 10m.

The dynamical substitutes of the POs in the ER3BP-GH are still organized in the same 1-parameter families A, B,

and CD, but now they are constituted by invariant 2-tori with one phase set to be the true anomaly of Phobos around

Mars, and are showcased in Fig.17. The width of their invariant curves, obtained with the continuation schemes dis-

cussed in Section 3.2.2, is comparable to the LP oscillation’s amplitude. Simulating these QPOs by multiple-return

stroboscopic maps, the rotation number proﬁle of the transversal phase around the invariant curve is obtained. This

allows us to determine how many longitudinal revolutions are needed to achieve a full lap of the invariant curve, and

is presented in Fig.14. This number increases when the period of the ancestor PO tends to be in an integer ratio with

the orbital period. In particular, in Fig.14 we see that large 2D-QPOs of the families B and D in the ER3BP-GH are

close to the resonance 2:1, therefore after two revolutions the spacecraft will ﬂy back near the initial condition. For the

family B around each LP, the POs with period πslightly intersect Phobos, therefore are not considered in this paper.

For the synchronous case of the family D, the invariant curve is resonant as presented in Fig.7, and Fig.17 shows the

Natural Motion around the Martian Moon Phobos 27

Fig. 17: LPOs in the Mars-Phobos ER3BP-GH. On the left, the two iso-periodic families of POs around the oscillating LPs. On the right,

the families of 2-tori: A (red), B (green), C (magenta), D (cyan).

large, vertically-elongated, inclined POs, similar to eight-shaped orbits, that exist in the ER3BP-GH around both sides

of Phobos.

4.2.1 Sensitivity Analysis and Tracking Performance of the LPOs in the ER3BP-GH

The POs of the family D in the ER3BP-GH are chosen to conduct a ﬁrst-order sensitivity analysis to assess the ef-

fect of the uncertainties of the modeling parameters, navigation sensors, and the physical perturbations in tracking the

reference signal of the orbit computed. The initial condition of this reference trajectory is propagated in the perturbed

dynamics, and the error of the resulting ﬂow from the reference signal is computed. Two metrics are used to assess the

tracking performance: the position error from the guidance law after a prescribed propagation time, and the time that

the ﬂow freely tracks the reference signal within a prescribed position error. Thus, two thresholds are introduced. The

prescribed propagation time is set to be one tenth of the Mars-Phobos (and the PO itself) orbital period, as this is close

to the reference value of 44min required for a transponder’s signal’s round-trip at the farthest Earth-Mars distance. The

prescribed position error is set to be 10m, as this is close to the value of 10−6Lunits commonly used for the Sun-Earth

and Earth-Moon systems (Koon et al. 2011). Fig.18 presents the analysis for several parameters, normalized in abscissa

by the sensitivity number S. For semi-major axis a, mass parameter µ, eccentricity e, GHs’ coefﬁcients, and initial con-

dition x0of relative position and velocity, Sweights these uncertainties with respect to the related reference value of the

ER3BP-GH. For the other actions, 1rad is used for the initial true anomaly ν0, the mean motion (equal to 1/time unit)

for the angular velocity ωof the BCBF frame, 1/(time unit)2for its angular acceleration ˙

ω, 1.90◦as the maximum tilt θ

between BCBF and Hill’s frames, the Phobos gravity at the SOI’s boundary for the magnitude of a harmonic perturbing

acceleration aPat the orbital frequency. Furthermore, the GHs model of Martinec et al. (1989) is used as a benchmark

representing a more realistic dynamical model of the spacecraft around Phobos, to relate the sensitivity of the other per-

turbations. This test indicates that station-keeping is required at least every 17min for the POs around L1. In this sense,

not including the GHs would have required maintenance actions at least every 4min, while not including the eccentricity

would have required station-keeping at least every 7min. The savings on the duty cycle are 76% and 58% respectively,

28 M. Zamaro, J.D. Biggs

Fig. 18: Sensitivity and performance analysis of the LPOs in the Mars-Phobos ER3BP-GH. The ﬁrst ﬁgure shows the propagation time

where the spacecraft remains within the error of 10m from the reference signal of one PO of the D family around L1in the ER3BP-GH, when

propagated in perturbed dynamics where the modeling parameters, navigation inaccuracies, and physical forces in the legend are weighted

by the sensitivity factor S. The blue cross represents the benchmark dynamics. The horizontal dotted line is the PO period. The second ﬁgure

shows the related equivalent perturbing acceleration (normalized by the Phobos gravity at L1) that produces the same ﬁnal response as a

constant input in a forced undamped unstable 1D oscillator. The second ordinate axis represents the associated ∆v consumption over one

period.

which highlight the importance of including these forces in the dynamical model to design orbits around Phobos.

The second conclusion arises from the comparison of the sensitivity of the perturbations. In Section 2.3 we es-

timated that in our ER3BP-GH, the residual perturbations of both the benchmark dynamics and the frame’s libration

motion account for a few percent of the cumulated GHs perturbation at the SOI, which itself is of the order of 10−1of

the Keplerian gravity (see Fig.3(right)). This ﬁrst estimation is now conﬁrmed for the LPOs computed in the ER3BP-

GH, as the results of the benchmark and the maximum tilt in the ﬁrst graph of Fig.18 fall within the ones of an additional

relative perturbing acceleration aPbetween 10−3-10−2, and an inaccuracy of µof the same range. Inaccuracies of the

semi-major axis and the relative non-dimensional state are similar, and very sensitive (inaccuracies over 6·10−4produce

initial errors larger than the 10m threshold), while inaccuracies of the true anomaly and rotational rates of the frame are

allowed to be higher. Summarizing, the ﬁrst graph of Fig.18 provides the maximum inaccuracies: if the parameters

are acquired within these inaccuracies, station-keeping at the frequency of the time threshold chosen will maintain the

spacecraft within the deadband deﬁned by the position threshold required. Relative inaccuracies larger than 3·10−4for

a, 9 ·10−4for µ, 3 ·10−2for e, 9 ·10−3for GHs’ coefﬁcients, 2 ·10−4for x0, 9 ·10−2for ν0, 3 ·10−3for ω, 5 ·10−3for

˙

ω, 6 ·10−1for θ, 8 ·10−4for aP, require the control load every 44min to be increased, or the guidance performance of

10m to be relaxed. In particular, we see that the value for ωis larger than the perturbation on the mean motion of the

Hill’s frame introduced by Mars J2, which is 2 ·10−4, conﬁrming that this term could be disregarded from our model as

explained in Section 2.3.

It is now possible to plot these actions deriving an associated ﬁgure of perturbing acceleration, in order to univer-

sally compare the sensitivity effects with the preliminary orbital perturbations analysis in Fig.3(right). This is done with

an equivalent model that considers only the unstable normal mode, which is responsible for the drift of the perturbed

trajectory from the reference signal. In the 1D linear ODE ¨x−ω2

0x=aP, the state xrepresents the magnitude of the

position error from the reference signal, and the natural pulsation ω0is taken as the maximum unstable Floquet ex-

ponent of the PO (2.26 of the mean motion). The system is forced by the constant acceleration aP, and the analytical

solution is available. Thus, the equivalent constant acceleration that produces the same ﬁnal response of the numerical

Natural Motion around the Martian Moon Phobos 29

Fig. 19: Quasi-Periodic LPOs in the Mars-Phobos CR3BP-GH. On the left, graphical visualization of the family of QPOs that connects the

two medium iso-energetic POs of the A and B families around L1of the CR3BP-GH. Related two backbones POs displayed in black. On the

right, family of QPOs around a medium-energy PO of the C family around L1of the CR3BP-GH.

simulation is computed for every perturbing action, and is presented in the second graph of Fig.18, in terms of the same

ratio aP/a2used in Fig.3(right). The two cases (response at prescribed propagation time and at prescribed position er-

ror) have produced a similar output, so the graph refers to the ﬁrst one. For a linear approximation comparison, related

∆v consumptions over one period are indicated in parallel. The LPO computed in our model would require 0.5m/s

per period in the more accurate GHs and maximum-misaligned model, that constitute the main residual perturbations

assuming the remaining model parameters and navigation variables are acquired within the following relative and ab-

solute inaccuracies: 5 ·10−4for a(5km), 4 ·10−3for µ(103m3/s2for Phobos’ GM and 9 ·1010 m3/s2for Mars’ GM),

10−1for e(2 ·10−3), 4 ·10−2for GHs’ coefﬁcients, 4·10−4for x0(6m for relative position and 2 ·10−3m/s for relative

velocity), 3 ·10−1for ν0(20◦), 10−2for ω(2 ·10−6rad/s), 2 ·10−2for ˙

ω(10−9rad/s2), 3 ·10−3for aP(8 ·10−6m/s2).

This represents a considerable saving from the 11m/s and 3.2m/s per period if, respectively, the GHs and eccentricity

are not included in the dynamical model.

4.3 Dynamical Substitutes of the Quasi-Periodic Orbits

The quasi-periodic LPOs of the Mars-Phobos CR3BP are the 2-parameters families of Lissajous orbits around every

Lyapunov and Halo orbit that do not intersect Phobos. Examples of QPOs computed with the Lindstedt-Poincar´

e algo-

rithm are shown in Fig.4.

The 2D-QPOs in the CR3BP-GH have been derived with the numerical technique that computes their invariant

curve around the backbone PO as discussed in Section 3.2.1. This differential corrector scheme allows to directly com-

pute all the family of iso-energetic QPOs in the ﬁnal dynamics, so no continuation between different models is required.

Similarly to the CR3BP for the two Lyapunov families, the 2-tori of the families A and B are connected to each other

because they are part of the same 4D center manifold originating from the EP, as shown in Fig.19. Simulation of the

QPOs by iterative mappings allows to display the surfaces of toroidal motion presented in Fig.19-20, as well as retrieve

the rotation number property. Cases of resonances with high denominator are present.

30 M. Zamaro, J.D. Biggs

Fig. 20: LPOs in the Mars-Phobos CR3BP-GH and ER3BP-GH. Graphical visualization of 2-tori in the CR3BP-GH (on the left) and

related 3-tori’s orbits in the ER3BP-GH (on the right), of different size and width around the LPs. Summarizing examples: three medium-size

QPOs of family AB (red and green with small-width, orange with high-width) and two high-width QPOs of family C (of small and large-size).

Finally, a sampling of these QPOs has been continued to consider the additional forced phase introduced by the

elliptic dynamics, as discussed in Section 3.2.2. As shown in Fig.8 and Fig.20, the motion in the ER3BP-GH oscillates

around the solution found in the circular case with an amplitude similar to the one of the LPs.

The families of QPOs in the CR3BP-GH and ER3BP-GH are all unstable, and all the inside branch of their IMs

intersects Phobos’ surface, which is modeled by the high-order shape harmonics model. The exploitation of the IMs in

the ER3BP-GH is undertaken in Section 5.4.

5 Applications of the LPOs and their Invariant Manifolds

In this section we present the performance of the LPOs computed in Section 4, as well as their IMs, with respect to a

number of potential applications that can be required in the design of a space mission to Phobos.

5.1 Lighting Conditions

The kinematics of the Sun are included in the rotating frame of the R3BP to evaluate the instantaneous ﬁeld of view

of the Sun for a point around Phobos. As the tilt between the equatorial planes of Mars and Phobos is approximated to

zero, the orbital plane of the R3BP is inclined with respect to the Solar ecliptic plane by Mars’ rotational tilt θM=25.2◦.

The Sun position vector in the rotating frame rotates clockwise with an angular velocity equal to the difference between

Phobos and Mars revolution rates (dominated by the ﬁrst), with a ﬁxed declination in the range [−θM,θM]according

to the seasonal phase of Mars. Seasons of Phobos correspond with Mars’ ones, so are referred without any distinction.

Due to Mars’ high eccentricity, seasons are unequal, with the Northern Hemisphere of Mars and Phobos experiencing a

summer longer than that occurring in the South.

The analysis of the lighting conditions in this system is undertaken using eclipse modeling, which is to derive the

zones of light and shadow produced by Mars and Phobos when illuminated by the Sun. This is described by a scalar

light function ﬁeld L, ranging from 0 to 1 to express the ratio of incident light with respect to the complete light case.

For both Sun-Mars and Sun-Phobos couples the analysis could be undertaken with a cylindric model for the positions

Natural Motion around the Martian Moon Phobos 31

Fig. 21: LPOs lighting conditions. Light function of the families of POs of the CR3BP-GH (parameterized by the non-dimensional differential

Jacobi integral with respect to the LP), averaged over 10 PO periods, at the days of equinoxes (lower cluster) and solstices (upper cluster).

Filled area spans values for different starting phases of the Sun (thick line is for mean values), where families’ colors are coherent with Fig.12.

inside and close to the SOI’s boundary of the moon. The shadowing wake of Phobos is limited by the size of the body,

while Mars’ wake covers all Phobos’ neighborhood during an equinoctial season for 12% of the orbital period, but its

effect becomes null at the solstices. Also, as the revolution period of the Sun is almost double that of the LPOs, the

average condition for points around L1and L2is to remain mostly in light, with the L1-side experiencing more shadow

time with the exception of the two solstice’s seasons.

The analysis is undertaken for the POs of the CR3BP-GH, since they provide the backbone of both the QPOs in the

same dynamical model and the QPOs in the ER3BP-GH, and the outcomes are presented in Fig.21. Due to the additional

phase of the Sun, we deﬁne the light function of the LPO averaging over 10 revolutions of the trajectory, and results will

depend on the initial condition of the Sun around Phobos.

As presented in Fig.21, medium-small LPOs within the families are similar to close-range hovering points on the

two L1−2-sides of the moon and the dependence of the Sun phase is mild (less than 5%). Such LPOs of the families A

and B are more exposed to light during the solstices than the equinoxes. At the equinoxes, all large orbits of the families

A, B, and C experience more lighting time on the L1-side, while the ones of A and C on the L2-side experience more

shadowing time than their medium-size orbits. Instead both families D experience more lighting time. At the solstices,

more shadowing/lighting is obtained by the family A/C in summer, and by C/A in winter, while the families B and D are

not altered by the type of solstice. The light time of large LPOs, if used for short operations, can be tuned accordingly

to the Sun phase along Phobos, with the families B and D allowing to increase/decrease the mean light function up to

15%.

5.2 Surface Coverage

One of the most important requirements for an observation mission is the region of the surface that the instruments can

point while the spacecraft moves along the chosen orbit. Fig.22 collects the regions of full geometrical coverage for the

larger LPOs of every family of POs in the CR3BP-GH around both LPs. Thus, these outcomes consider no technical

constraints and data of the instruments, such as the minimum elevation angle on the ground and the maximum angles

of view that deﬁne the ﬁeld of view of the instrument around its beam’s boresight. The results are obtained using the

high-order Phobos’ shape harmonics.

32 M. Zamaro, J.D. Biggs

Fig. 22: LPOs surface coverage. On the left, colored areas ﬁll the regions of full geometrical coverage on Phobos’ surface of the largest orbits

of the families of POs around L1and L2of the CR3BP-GH. Families’ colors are coherent with Fig.12. On the right, surface coverage of the

synchronous POs of the family D around L1and L2of the ER3BP-GH.

The family B provides the least surface coverage of all the families, allowing to point only at the regions on the

cis (L1) and trans (L2) side of the moon, and covering respectively 28% and 31% of the whole surface during a full

orbit. The large POs of the family D, which are vertically-elongated and inclined, allow to cover both Poles and also

a fair region on the leading and trailing-side of the moon, resulting in a 45%/43% coverage ratio. Finally, the families

A (47%/44%) and C (39%/39%) cover the South and North polar sides respectively, and most importantly the leading

and trailing-sides. Fig.22 shows that the union of the areas provided by all the families covers the whole surface. In

particular, a combination of a large D orbit and an A or C orbit on the other side, as well as one A and one C on each

side, could ﬁll most of the map. Finally, Fig.22 shows the surface coverage of the D families of POs of the ER3BP-GH.

This case is very interesting because the orbits are synchronous, and the combined surface coverage is 73% missing only

two sectors on the leading and trailing-sides. A constellation, using a minimum of one spacecraft on each side, would

enable a stationary communications bridge between most of the opposite sides of the moon (cis/trans, North/South, part

of lead/trail) where different human crews or rovers could be displaced, as well as repeated access times to equatorial

and middle latitude sites on Mars.

5.3 Radiation Shielding

In this section we consider the possibility that a manned spacecraft during a Mars orbital mission segment could exploit

Phobos as a natural and passive shield against the isotropic cosmic rays, whose hazardous radiation is currently consid-

ered the showstopper in designing a safe manned mission in deep space (Genta et al. 2014). The idea is that the ﬂuence

of the incoming isotropic particles (Galactic Cosmic Rays and part of Solar Energetic Particle Events) on a spacecraft

is lowered proportionally to the ﬁlling fraction in the sky of the apparent size of the body’s bulk, as seen by the space-

craft’s location. In this paper we refer to this action as sky occultation. The computation is similar to the full geometrical

coverage undertaken in Section 5.2, just considering the solid angle covered by Phobos (representing its apparent shape)

instead of the footprint’s area on its surface. Thus, the occultation function ﬁeld Ois the ﬁlling ratio of this area with

respect to the background sky of 4πsr.

As in Section 5.1, the analysis in Phobos neighborhood considers also the effect of Mars, whose bulk provides a

constant occultation of 3.4% in the rotating frame of the R3BP. This effect is weighted by the Mars-Phobos light func-

Natural Motion around the Martian Moon Phobos 33

Fig. 23: LPOs occulting conditions. Sky occultation function by the Phobos’ bulk of the families of POs around L1and L2of the CR3BP-GH

(parameterized by the non-dimensional differential Jacobi integral with respect to the LP), averaged over 1 PO period. Families’ colors are

coherent with Fig.12. Additional occultation by Mars’ bulk will be 3.4% on the L1-side.

tion, where Mars is the radiating body and Phobos is the central body. Unlike the Sun-Mars and Sun-Phobos couples,

this light function must be computed with the accurate dual-cone model, since due to the proximity of Mars, the shadow

cone’s vertex of Phobos is located only at 2.77 Phobos mean radii in the anti-Mars direction. Therefore, the combined

occultation is a few percent greater for LPOs on the L1-side of the moon, than on the L2-side. The major effect is in-

deed given by Phobos’ bulk, and due to their proximity to Phobos, the sky occultation produced by the natural LPOs is

relevant. Fig.23 shows that larger LPOs around both L1−2can provide passive radiation shielding from over 20% of the

incoming isotropic cosmic rays. This result is obtained using the high-order shape harmonics model for Phobos’ bulk.

5.4 Natural Landing and Take-Off

Since the LPOs are close to Phobos, no homoclinic nor heteroclinic connections are available to naturally move around

it, because all the inside branch of their IMs intersect Phobos. This is a unique feature of the Mars-Phobos CR3BP,

because the natural trajectories, at low energy, to transfer between the SOIs of the two primaries in the Sun-Earth and

Earth-Moon systems ﬂy far from the bodies. In Baoyin and McInnes (2006), a selection of the smallest planar Lyapunov

orbits whose IMs naturally intersect the smallest primary of the Sun-Earth and Earth-Moon systems, and provide whole-

coverage along the body’s orbital equator, has been undertaken. These orbits are very large and so require high energy

relative to the one of L1−2, and the analysis undertaken is 2D. The same approach could be extended considering

impulsive propulsion to modify the IMs at the starting point at the LP (Baoyin and McInnes 2006) or along a smaller

orbit, as well as including low-thrust continuous propulsion to modify directly the LPO. Instead in our unique case,

a spacecraft could land or take-off from Phobos using the gateway of small-energy LPOs and with no need of thrust.

In this section we ﬁnd the landing and take-off sites on the 3D surface of Phobos and derive the performances at the

touch-down and launch that are naturally provided by these trajectories.

The inside branch of the unstable and stable IM of the families of POs and QPOs of the ER3BP-GH has been

computed with the techniques tailored for each invariant object introduced in Section 3, and propagated in the related

dynamics up to the intersection with Phobos’ surface modeled with the high-order shape harmonics model. Examples

34 M. Zamaro, J.D. Biggs

Fig. 24: IMs of the LPOs in the Mars-Phobos ER3BP-GH. On the left, inside branch of the tube of unstable IMs from the families of 3-tori

LPOs. On the right, stable IMs from the families D of 2-tori LPOs.

of these tubes are shown in Fig.24, while the region of possible landing and take-off topographical sites is provided in

Fig.25. This area is restricted to the collinear-faced sites on Phobos, due to the collapsing effect of the SOI. In particular,

large tori within the families AB and C can extend the region in both longitude and latitude, while the POs and the

QPOs of the D branch are more constrained. Also the region of the big crater Stickney is accessible with both stable and

unstable IMs of large QPOs of the families AB and C.

We propose using the IMs as natural de-orbiting trajectories that evolve from the LPOs, since they are the asymptotic

separatrices of their center manifold. Since the perturbation of the GHs grows exponentially as the proximity to Phobos

increases, the globalization of the manifolds, using a low-order series expansion, produces an inaccurate approximation

in the ﬁnal stages of the landing (and ﬁrst stages of the take-off). This is a common problem for any lander, where the

ﬁnal manoeuvers are managed by the Guidance, Navigation and Control (GNC) subsystem to satisfy the performances

required. Results in this section should therefore be considered as an estimation of these performances at the touch-down

and launch, more reﬁned than the CR3BP, to lower the load on the GNC subsystem in the real scenario. Focusing on the

landing, apart from site and time of ﬂight, the velocity performances at the touch-down considered are: total magnitude,

angle of incidence and vertical component. The desired trajectory would minimize the longitudinal control as well as

yield a soft landing. In this paper the IM ﬁnal condition is considered as an input to the ﬁnal descent undertaken by

the GNC subsystem, therefore the aim considered is to minimize the angle of incidence for the landing. The GNC will

then be required to brake the spacecraft horizontally to verticalize the ﬁnal stages of the descent, starting from a small

vertical velocity and so facilitating the shock absorbtion at touch-down. An indicative ﬁgure for the shock force could

be derived from the law of conservation of momentum, considering an impulse of 0.1s (inversely proportional to the

rigidity of the absorbtion structure): numerically the shock force in terrestrial gis approximately equal to the vertical

velocity in m/s. Instead for the take-off the ﬁrst driver is to minimize the escape velocity (total magnitude).

For an m-parameter family of invariant n-tori, the number of trajectories of the tube of IMs is organized in a (n+m)-

dimensional matrix parameterized by the n phases and the m parameters. To exploit the outcomes in the mission design,

the performances are reparameterized by the topographical map of the landing/take-off sites, to represent the properties

of the optimal trajectory available for each couple of longitude and latitude. Fig.26 presents the velocity performances

for the natural landing and take-off opportunities at Phobos. The analysis of these performances shows that highly

Natural Motion around the Martian Moon Phobos 35

Fig. 25: IMs of the LPOs in the Mars-Phobos ER3BP-GH. On the left, outer boundaries of the region of the possible landing sites through

the IM of all the families of tori around both LPs. On the right, outer boundaries of the region of the possible take-off sites through the IM of

all the families of tori around both LPs. Families’ colors are coherent with Fig.12, with red associated to the global family AB.

efﬁcient natural trajectories are able to land tangentially, facilitating a soft controlled touch-down, and depart with a

very small escape velocity, less than 30% of the 2B ∆v value, for an extended region of topographical collinear-faced

sites on Phobos. These trajectories have the potential to be exploited for future sample-and-return missions to this moon,

where free-fall is required to avoid contamination of the sample’s soil by the exhaust plume of the thrusters or rockets’

nozzle.

5.5 Extension of the LPOs and IMs to Model Natural Impacts on the Surface and the Dust Dynamics around Phobos

The LPOs and IMs presented in this paper have been applied to model the motion of a spacecraft about Phobos. Ac-

cording to our analysis of the orbital perturbations in Section 2.2, SRP is not signiﬁcant if the spacecraft does not use

solar sails. SRP becomes important also in the well-known photo-gravitational 3BP for the orbital dynamics of dust

grains around the secondary body, because the perturbation aPis inversely proportional to the grain radius. This is very

important for an asteroid, where the SRP could be higher than the Sun 3B perturbation, and affect even a spacecraft.

The ratio aP/awith the full Solar gravity magnitude ais known as the lightness number λ. However, an analysis on

the order of magnitude of the physical characteristics leads to an interesting outcome for the Mars-Phobos system. The

objective here is to study the transition from a classical CR3BP to a photo-gravitational CR3BP. This means to identify

the conditions that make the SRP a signiﬁcant force in the modeling of the dynamics around a body of the Solar System.

The transition between the two models is assessed referring λto the second massive body gravity, which is aP/a2. The

dust grain is modeled as a sphere, with common physical values (specular and diffusive reﬂectance coefﬁcients both set

to 5%, and density to 3,000kg/m3). Thus λ, for a given couple of primaries in the Solar System, is now a function of

the grain radius and the distance from the secondary. Assuming a threshold λ=1% to identify when the SRP becomes

relevant enough to be included in the modeling of the dynamics, the transition grain radius is computed as a function

of the distance from the secondary. This radius does not change order of magnitude inside the SOI of Phobos, due to

its small altitude over the body. Its value is quite low: 20µm at the Hill’s radius, and 9µm on the mean surface. As a

difference, if Phobos had been an asteroid orbiting the Sun at the Sun-Mars distance, its gravity ﬁeld at the LPs would

have been 10,000 times smaller, so the equivalent dust grain radius that requires the transition to a photo-gravitational

problem would rise to 20cm.

36 M. Zamaro, J.D. Biggs

Fig. 26: IMs of the LPOs in the Mars-Phobos ER3BP-GH. On the top, performances of the trajectories that provide the min incidence at

the touch-down from all the families of tori around both LPs, as a function of the longitude and latitude of the landing site on Phobos surface

modeled through shape harmonics: velocity magnitude, angle of incidence, downward vertical velocity. On the bottom, performances of the

trajectories that provide the min velocity magnitude at the launch from all the families of tori around both LPs.

Therefore, the results obtained in this paper focused on the natural motion around Phobos in the ER3BP-GH, that

do not consider the SRP, in addition to common spacecraft, also provide a good approximation for the natural motion of

dust grains down to 10−5m scale. This is a broad range considering the common distribution of minor bodies’ regolith

(m −0.01nm). So for Phobos, only very small dust grains require the photo-gravitational model to be used. This means

that the IMs of the ER3BP-GH could be the trajectories followed by a large dimensional range of meteoroids that have

impacted Phobos, as well as boulders of regolith that lifted off from its surface, to constitute the orography of its craters

distribution. In this sense, Fig.25 highlights that the medium-small craters are densely accumulated on the two collinear

sides where the IMs of the LPOs impact or leave the moon, and their boundaries link neatly together all the larger craters,

in particular Stickney and the craters on the Northern sub-Mars sector. With the help of the map in Fig.1 of Ramsley

and Head III (2013b), it is immediately noticeable that most of the characteristic grooves of Phobos run parallel to the

boundary of the regions of landing and take-off sites computed. In particular, recall that only the IMs of the practical

LPOs that do not intersect Phobos were considered in this paper, but larger orbits (which refer to higher energy trans-

ferred material) provide additional impact trajectories themselves. Their footprint will correspond to the pattern of the

remaining grooves, since they both develop farther concentrically on the L1and L2-sides of the moon. Indeed, extending

the above calculation for all the major couples of primaries of the Solar System, the two Martian moons have the highest

Natural Motion around the Martian Moon Phobos 37

values of transition grain radius on the mean surface among the bodies considered. This means that Phobos and Deimos

could experience rings of dust clouds of micrometer scale around them (Krikov et al. 1996) and some authors theorize

that dust dynamics on the surface is responsible for the origin of the grooves of Phobos (Ramsley and Head III 2013a,b;

Dombard et al. 2010). We suggest that the footprint of IMs and large LPOs could have originated in the past the craters

and the grooves of Phobos, and that current dust clouds of grain dimension over 20µm could correspond to the LPOs

presented in this paper.

6 Conclusions

The analysis of the natural dynamics inside Phobos’ realm of attraction undertaken in this paper has revealed some

potential mission opportunities around the Libration points L1and L2. First the classical families of periodic and quasi-

periodic Libration Point orbits (LPOs) in the basic CR3BP have been computed for the Mars-Phobos system. The

LPOs are well-known exploitable solutions for orbiting about the smaller primary and they provide energy-efﬁcient

trajectories for interplanetary transfers of spacecraft through their Invariant Manifolds; however they are highly unstable,

and station-keeping is required. For the Mars-Phobos system, the altitude of the Libration points moves very close to

Phobos’ irregular surface. In this situation, the basic dynamical approximation provided by the CR3BP falls short, and

in particular two differential orbital perturbations become relevant in describing the natural relative motion around this

moon. They are shown to be its highly inhomogeneous gravity ﬁeld and its orbital eccentricity.

The dynamical substitutes of the LPOs have been computed in an improved system of the relative dynamics in

proximity of Phobos, upgrading the Mars-Phobos ER3BP with the real gravity ﬁeld of the moon, modeled with a

gravity harmonics series expansion. The solutions are derived through numerical continuation techniques that maintain

the phase-space of the perturbations. In particular, the effect of the gravity harmonics produces families of periodic and

quasi-periodic orbits highly distorted and tilted from the classical case, whereas the effect of the eccentricity causes the

orbits to oscillate around these solutions with a signiﬁcant amplitude for the proximity of Phobos.

Around each collinear side of Phobos, the new LPOs in the ER3BP-GH are constituted by the oscillating Libration

points and a 1-parameter family of iso-periodic periodic orbits vertically-elongated with the period of Phobos revolution

around Mars, three 1-parameter families of 2-tori quasi-periodic orbits, and two 2-parameter families of 3-tori quasi-

periodic orbits. These orbits are very close to the moon’s surface, therefore they are similar to close-range points but with

an extended ground-track and range of lighting conditions, and the Phobos’ bulk occultation of the sky could provide

relevant passive shielding from the cosmic rays radiation. Despite their high instability, the LPOs are natural motions

and so will require no propulsion and low station-keeping cost to provide: close-range observation of Phobos, mission

segments to directly interface with landing and take-off phases, proximity parking orbits for mothership modules to

remotely command lighter probes deployed on the surface, and communication bridges to manage robotic scouts on

Mars and Phobos. However, they require the high accuracy of an optical navigation subsystem, and high load on the

guidance subsystem, whose reference signal must be computed with advanced analytical and numerical techniques of

dynamical systems theory that need the acquisition of a high-ﬁdelity gravity ﬁeld of the moon. In this sense, in this paper

we assessed the accuracies required for the modeling parameters of the ER3BP-GH against the GNC requirements,

38 M. Zamaro, J.D. Biggs

to track a spacecraft along the reference signal of the new orbits computed in our model. A preliminary estimation

proved that such LPOs considerably lower the control load and enhance the fuel savings between one and two orders of

magnitude, enabling feasible station-keeping by exploiting the natural dynamics around Phobos.

Another useful application is to exploit the Invariant Manifolds of these new LPOs as landing or take-off gateways

to and from the surface of Phobos. In this paper it was proved that there exist natural trajectories for a speciﬁc range of

longitude-latitude sites able to facilitate a soft controlled landing, and depart with very low escape velocity injections,

less than the 2B ∆v value. The optimization of these performances to select the best trajectory at a given location on

Phobos will be paramount for sample-and-return missions (where soil contamination avoidance is necessary) as well as

ﬁrst manned explorations of this moon.

In addition, the new natural dynamics upgraded with the inhomogeneous gravity ﬁeld of the moon could provide

further evidence to the unique formation of Phobos by Mars’ ejecta accretion, and the natural LPOs and Invariant

Manifolds could also be related to the craters and grooves distribution’s formation of the moon, as well as model the

dynamics of the theorized clouds of dust existing on Phobos.

Acknowledgements This work has been supported by the European Commission through the Marie Curie fellowship PITN-GA-2011-289240

”AstroNet-II”. We are thankful to Prof. J.J. Masdemont and Prof. G. G´

omez (IEEC) for the assistance provided in the computation of LPOs in

the CR3BP through high-order series expansions in the early stage of this research.

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