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Natural motion around the Martian moon Phobos


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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 theMars–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 closerange observation, communication, and passive radiation shielding for human spaceflight. Furthermore, their invariantmanifolds are demonstrated to provide high-performance natural landing and take-off pathways to and fromPhobos’ surface, and transfers fromand toMartian orbits. These orbits could be exploited in upcoming and future space missions targeting the exploration of this Martian moon.
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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
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
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
180of 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 0.30and 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-
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 ·108
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
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/||qq||, where Gis the gravitational constant, m
and qare 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 = [
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=1µin
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 WR3×3: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) =
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
c(x) = ˙
q+qTq+2u(q) = ˙
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 ,i,ω,ν,0of the Keplerian orbital elements of the body , and nREV,nROTare 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 N1(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
, and the Node of reference’s location N=π
. Finally,
the initial location of the body’s Prime Meridian from its Node of reference is given by N
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
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(2N2). 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:
PM2) = A3(π)(4)
where the location R1
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 uGdescribed by the series expansion of spherical harmonics is presented in Eq.(5)
(Kaula 1966), where Ris 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(qq). 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
eψ) defines the local TSE frame.
uG(q) = uG(r,ϑ,ψ) = Gm
n(ψ) = Cn,mcos mψ+Sn,msin mψ=Jn,mcos m(ψλn,m)
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 = [
∂ ϑ ;1
∂ ψ ], then rotated in the BCBF frame through the local attitude matrix of Eq.(6), where φis
the latitude.
T SE (q) = ABCBF
T SE (φ,ψ) = A1π
2A2(φ)A3(ψ) = A3(φ)A1π
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).
∂ ϑ
∂ ψ
∂ ϑ 1
∂ ϑ 2+1
∂ ϑ 1
∂ ψ
∂ ψ 1
∂ ϑ 1
∂ ψ 1
∂ ψ2+1
∂ ϑ
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) +
G2(q) = ABCBF
T SE (q)TT 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
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) =
(1e2)3P+2(1+ecosν)3esin ν
(1e2)3/2W 03x1
01x3 01x3 0
uG1(q,ν) + uG2(q)+(1µ)(1+ecos ν)2
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=(1e2)/(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, n2). The trajectory of a n-torus can be seen as dependent on nphases ϕrather than
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
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 60from both
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 L12is 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 (108), therefore the manifolds of the two
Natural Motion around the Martian Moon Phobos 13
LPs are almost symmetrical, and the location of L12is 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,
L12are 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´
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 ´
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´
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 ) = yyLP(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 L12, 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 107in 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 L12of 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
Table 2: CR3BP-GH collinear equilibria. Displacement of the LPs of the CR3BP-GH from their original value in the CR3BP.
EP xyz
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 (107), 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 L12are 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 106Lunits 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, 1.90as 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
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 101of
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 103-102, 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 6·104produce
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 3·104for
a, 9 ·104for µ, 3 ·102for e, 9 ·103for GHs’ coefficients, 2 ·104for x0, 9 ·102for ν0, 3 ·103for ω, 5 ·103for
ω, 6 ·101for θ, 8 ·104for 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 ·104, 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 ¨xω2
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 ·104for a(5km), 4 ·103for µ(103m3/s2for Phobos’ GM and 9 ·1010 m3/s2for Mars’ GM),
101for e(2 ·103), 4 ·102for GHs’ coefficients, 4·104for x0(6m for relative position and 2 ·103m/s for relative
velocity), 3 ·101for ν0(20), 102for ω(2 ·106rad/s), 2 ·102for ˙
ω(109rad/s2), 3 ·103for aP(8 ·106m/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 L12-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
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 L12can 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 L12, 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’
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 105m 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
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... UE to the significant growth in capability of space technology, challenging planetary resource missions to other planets and minor bodies are now possible. For example, Martian sample and return missions are being considered [1][2][3], including to its Moon's Phobos and Deimos, see for example [3][4]. One of the most challenging aspects of these sample and return missions involve the accurate landing of the spacecraft in such complex and hazardous environments [5][6]. ...
... A y B q g (22) where 4 , ...
... Moreover, the sensitivity to the relaxation factor s is also verified here. It is interesting that considering very small values for the relaxation factor (say 4 10  ) leads to significant changes in the trajectory and fuel consumption. Fig. 7(b) shows that the obstacle will be avoided after an incomplete envelope replacement. ...
In this paper, a minimum-fuel powered-descent optimal guidance algorithm that incorporates obstacle avoidance is presented. The approach is based on convex optimization which includes the obstacles using nonconvex functions. To convert these nonconvex obstacle constraints to convex ones, a simple linearization procedure is employed. It is proved that the optimal solution of the convex relaxation problem is also optimal for the original nonconvex one. The sensitivity of the multi-obstacle avoidance method to the relaxation factor and its effectiveness under different conditions are also investigated through simulations.
... Besides the polyhedron model, high-order spherical harmonic model is also an option. Zamaro and Biggs (2015) had used the spherical harmonic model to describe the Mars-Phobos system. ...
... Hénon (2003) had used the Keplerian orbits to study the orbits in the CR3BP. Zamaro and Biggs (2015) had calculated the periodic orbits in the Mars-Phobos system based on the periodic orbits in the CR3BP. ...
... The same kind of bifurcation is also introduced in the resonance orbits where 3D resonant orbits bifurcate from planar resonant orbits in the CR3BP (Vaquero and Howell 2014). The bifurcation is different in a asymmetrical system, such as the Mars-Phobos system (Zamaro and Biggs 2015) and the asteroid system (Jiang et al. 2015b). In another perspective, the bifurcation can also be considered as an intersection of two families instead of that one family bifurcates from the other. ...
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The motion of a massless particle in the gravity of a binary asteroid system, referred as the restricted full three-body problem (RF3BP), is fundamental, not only for the evolution of the binary system, but also for the design of relevant space missions. In this paper, equilibrium points and associated periodic orbit families in the gravity of a binary system are investigated, with the binary (66391) 1999 KW4 as an example. The polyhedron shape model is used to describe irregular shapes and corresponding gravity fields of the primary and secondary of (66391) 1999 KW4, which is more accurate than the ellipsoid shape model in previous studies and provides a high-fidelity representation of the gravitational environment. Both of the synchronous and non-synchronous states of the binary system are considered. For the synchronous binary system, the equilibrium points and their stability are determined, and periodic orbit families emanating from each equilibrium point are generated by using the shooting (multiple shooting) method and the homotopy method, where the homotopy function connects the circular restricted three-body problem and RF3BP. In the non-synchronous binary system, trajectories of equivalent equilibrium points are calculated, and the associated periodic orbits are obtained by using the homotopy method, where the homotopy function connects the synchronous and non-synchronous systems. Although only the binary (66391) 1999 KW4 is considered, our methods will also be well applicable to other binary systems with polyhedron shape data. Our results on equilibrium points and associated periodic orbits provide general insights into the dynamical environment and orbital behaviors in proximity of small binary asteroids and enable the trajectory design and mission operations in future binary system explorations.
... Based on the results of Hénon's analysis, retrograde relative orbits, also known in the literature as distant retrograde orbits or quasi-satellite orbits, have gained the attention of several space agencies and astrodynamics researchers looking for efficient trajectories to explore the solar system (e.g., [20][21][22]). Of particular relevancy to this work are the contributions by Lidov and Vashkov'yak [23], Gil and Schwartz [24], and Zamaro and Biggs [25], who first demonstrated how the eccentricity and irregular gravity field of Phobos can play a significant role in governing the dynamic evolution of satellites near the surface of the Martian moon. In spite of these efforts, work in the literature remains primarily focused on isolated periodic orbits whose period is resonant with the orbital period of Phobos around Mars [26]. ...
... The motion of the barycenter of Phobos is assumed to be Keplerian despite the asymmetric and irregular gravity field exerted on Mars. It is also known that the Martian satellite is in a tidal-locked configuration [25], completing one revolution around its maximum axis of inertia in the time span of an orbital period around Mars. Accordingly, the rotational spin rate of Phobos and the angular velocity of the reference frame may be approximated as ω P n Pẑ and Ω n P 1 − e 2 P −3∕2 γ 2ẑ , respectively, where γ 1 e P cos ν, ν is the true anomaly of Phobos with respect to Mars, n P is Phobos mean motion, and e P is its eccentricity. Relevant quantities of Phobos are tabulated in Table 1 based on the numerical values provided in [47]. ...
The Martian Moons eXploration mission, currently under development by JAXA, will be launched in 2024 with the goal of retrieving pristine samples from the surface of Phobos. Soon after arrival, the spacecraft will inject into retrograde relative trajectories known as quasi-satellite orbits and study the geophysical environment of the Martian moon for more than three years. This paper presents the orbit design and maintenance strategy of the Martian Moons eXploration mission in the framework of the elliptic Hill problem with ellipsoidal secondary. We first introduce a numerical continuation procedure on the eccentricity of Phobos to replace purely periodic solutions with families of quasi-periodic invariant tori. Two-dimensional torus maps can be then constructed and used to represent physical quantities of interest, as well as to generate reference trajectories at arbitrary epochs. Sensitivity and stability analyses are carried out to investigate the dynamical properties of retrograde relative trajectories in the elliptic case. Finally, a Linear Quadratic Regulator is implemented in order to assess the robustness of the computed trajectories under injection, navigation, and execution errors. Monte Carlo simulations demonstrate that the baseline quasi-satellite orbits of the Martian Moons eXploration mission can be maintained with as low as 6.265 m/s per month.
... Higher-order zonal coefficients (e.g., J 4 and J 6 ) typically result in negligible secular change in the pericenter distance (Murray & Dermott 1999) because their values are much smaller than J 2 ; e.g., J 2 = 1.96 × 10 −3 and J 4 = −1.54 × 10 −5 (Zamaro & Biggs 2015) for Mars. The J 6 term and higher-order terms are even smaller. ...
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Particles of various sizes can exist around Mars. The orbits of large particles are mainly governed by Martian gravity, while those of small particles could be significantly affected by non-gravitational forces. Many of the previous studies of particle dynamics around Mars have focused on relatively small particles (radius of $r_{\rm p} \lesssim 100 \, \mu m$) for $\lesssim 10^{4}$ years. In this paper, using direct numerical orbital integration and analytical approaches, we consider Martian gravity, Martian $J_{2}$, the solar radiation pressure (SRP) and the Poynting-Robertson (PR) force to study the giga-year dynamical evolution of particles orbiting near the Martian equatorial plane with radius ranging from micrometer to meter. We also newly study the effect of the planetary shadow upon the particle dynamics. Our results show that small particles ($r_{\rm p} \lesssim 10 \, \mu m$) initially at $\lesssim 8$ Martian radii (below the orbit of today's Deimos) are quickly removed by the SRP due to eccentricity increase, resulting in a collision with Mars at the pericenter distnace. The orbits of larger particles ($r_{\rm p} > 10 \, \mu m$) slowly decay due to the PR forces (timescale of $> 10^{4}$ years). The planetary shadow reduces the sunlit area in the orbit and thus the efficiency of the PR drag force is reduced. However, we show that, even including the planetary shadow, particles up to $\sim 10$ cm in radius, initially at $\lesssim 8$ Martian radii, eventually spiral onto the Martian surface within $\sim 10^{9}$ years. Smaller particles require less time to reach Mars, and vice versa. Our results would be important to better understand and constrain the nature of the remaining particle around Mars in a context of giant impact hypothesis for the formation of Phobos and Deimos.
... With reference to the former, the orbital dynamics around Phobos is particularly complex, since the small mass-ratio and length-scale of the Mars-Phobos system result in a sphere of influence of the Martian moon which is very close to its surface. Hence, the two-body problem is an inaccurate approximation of the spacecraft's dynamics in the vicinity of Phobos [12]. In the HLEPP an habitable station, the Deep Space Gateway (DSG), will fly a near rectilinear halo orbit (NRHO). ...
Full-text available
This paper discusses the derivation and simplification of equation sets for the relative motion dynamics characterization in the restricted three-body problem. As opposed to previous models proposed in the literature, the relative motion is studied in a frame local to the target spacecraft, the local-vertical local-horizon frame, and spacecraft state is expressed with respect to the primary about which they are orbiting. The exact description of the relative dynamics is derived, as well as simplified equation sets based on both the elliptic and the circular restricted three-body problems. The accuracy of the simplified sets is analyzed by means of extensive numerical simulations on a realistic rendezvous scenario with a target on lunar orbit.
... The distant retrograde orbits [11] and resonance orbits [12] around Phobos have been investigated. Zamaro and Biggs discussed the periodic orbits around Lagrange-points in Mars-Phobos systems and applied them to observation analysis [13]. The control hovering orbits in Phobos-fixed frame are analyzed and the fuel consumption of orbit keeping was evaluated [14]. ...
Conference Paper
View Video Presentation: Rendezvous and proximity operations are among the critical aspects in the future return to the Moon, and beyond. One of the most probable scenarios is the presence of a space station located in a quasi-periodic orbit about a Lagrangian point of the Earth-Moon system. Assuming a discontinuous presence of astronauts onboard, proximity operations will be heavily automated. This paper examines the relative dynamics between an active chaser and a target station in a three body setting. In particular, the dynamics and control issues are studied when some selected failures occur, which may require a new rendezvous attempt after one target orbit. The objective will be the determination and reallocation of a sequence of waypoints that guarantee passive safety of the rendezvous. The sequence of waypoints will be parametrized in terms of DV consumption and time of flight between.
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A dynamical model is developed in the body-fixed frame of Phobos, in which the high-precision gravity field and exact physical libration of Phobos, the gravity of Mars with J2, and the gravity perturbations of the Sun, Jupiter, and Earth are considered. The JPL development ephemeris are applied to calculate the positions of celestial bodies. Phobos is considered as a homogeneous polyhedron with 16 037 vertices to characterize its irregular shape and the corresponding gravity field. The physical libration of Phobos is incorporated into its rotational motion by using the results in ‘Report of the IAU WGCCRE’. With the proposed model, equivalent gravity and slope on Phobos surface are calculated and analysed. The liftoff velocity is also computed and presented. Besides, the orbital environment is also investigated. Instantaneous equilibrium points in the Mars–Phobos system are computed and demonstrated, and the acceleration of a particle in the vicinity of Phobos is analysed to find out the main influencing factor in different regions. Quasi-satellite orbits and libration point orbits, which were determined in the circular restricted three-body problem model, are simulated in different dynamical models. The results applying the newly developed high-fidelity dynamical model have shown significant differences with respect to existing models, suggesting that dynamical models with higher accuracy are needed for close-range orbital activities.
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This work includes analytical and numerical studies of spacecrafts orbiting two binary asteroid systems: 1999 KW4 and Didymos. The binary systems are modeled as full irregular bodies, such that the whole evolution of the results will show the impact of the irregular gravity field in the lifetime and dynamics of the spacecraft’s orbit. The equations of motion of the binary system and the spacecraft are derived from Lagrange Equations. The solar radiation pressure is consired in the dynamics of the spacecraft.Two distinct methods are used to search for stable orbits around the binary systems. One is called the grid search method, which defines the main body as a point mass to estimate the initial state of the spacecraft based on a circular Keplerian orbit. The second method is the search for periodic orbits based on zero-velocity surfaces.
Conference Paper
This paper presents a complete modelling, synthesis and analysis methodology of control compensators for descent and landing on small planetary bodies. These missions are scientifically very rewarding, but also extremely challenging due to the complex and poorly-known environment around those bodies, calling for the ability to manage competing robustness/performance requirements by design. Here, this is achieved using robust control tools and the recently-developed structured H-infinity framework. The proposed method is verified for three distinct landing trajectories on the Martian moon Phobos.
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Space missions which reach destinations such as the moon, asteroids, or the satellites of Jupiter are complex and challenging to design, requiring new and unusual kinds of orbits to meet their goals, orbits that cannot be found by classical approaches to the problem. In addition, libration point orbits are seeing greater use. This book guides the reader through the trajectory design of both libration point missions and a new class of low energy trajectories which have recently been discovered and make possible missions which classical trajectories (conic sections) could not. Low energy trajectories are achieved by making use of gravity as much as possible, using the natural dynamics arising from the presence of a third body (or more bodies). The term "low-energy" is used to refer to the low fuel and therefore low energy required to control the trajectory from a given starting condition to a targeted final condition. Low energy trajectory technology allows space agencies and aerospace companies to envision missions in the near future involving long duration observations and/or constellations of spacecraft using little fuel. A proper understanding of low energy trajectory technology begins with a study of the restricted three-body problem, a classic problem of astrodynamics, which we approach from a rigorous and geometric point of view. We develop systematic and computationally efficient methods for the design of both libration point orbits and low energy trajectories based on fundamental ideas of invariant manifold theory. Furthermore, we develop the computational techniques needed to design trajectories for a spacecraft in the field of N bodies by patching together solutions of the 3 body problem. These computational methods are key for the development of some NASA and ESA mission trajectories, such as low energy libration point orbit missions (e.g., the Genesis Discovery Mission and Terrestrial Planet Finder, to mention only two), low energy lunar missions and low energy tours of outer planet moon systems, such as the a mission to tour and explore in detail the icy moons of Jupiter. This book can serve as a valuable resource for graduate students and advanced undergraduates in aerospace engineering, as well as a manual for practitioners who work on libration point and deep space missions in industry and at government laboratories. The book contains a wealth of background material, but also brings the reader up to a portion of the research frontier. Furthermore, the book will also have appeal to students of applied mathematics, especially those with an interest in dynamical systems and real world applications.
Conference Paper
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Orbital dynamics with solar gravity perturbation in the vicinity of the irregular asteroids is studied in this paper. This study is conducted under the following assumptions: (1) the density of the asteroid is uniform; (2) the asteroid rotates around the principal axis of the largest moment of inertia; (3) the motion of a point mass is affected only by the gravitation of the asteroid and the sun. The polyhedral approach is used to model the shapes of the asteroid and analytical formulae of uniform-density polyhedron gravitational field are derived. The dynamical equation of a point mass under the effects of gravitations of the asteroid and the solar gravity perturbation is obtained. The motions in the gravity fields of 216 Kleopatra and 433 Eros, and the periodic orbits around the asteroid with perturbation of solar gravitation are analyzed. The periodic orbits in the vicinity of 216 Kleopatra with perturbation are calculated and a new family of periodic orbits of 216 Kleopatra is found. It is also found 12 families of periodic orbits around 433 Eros and their topology and stability are studied in some details. This paper shows that solar gravitation has little effect on the positions of asteroids equilibria in the body-fixed frame while that the non-spherical gravitation of the asteroid affects the motion of a mass point on the equilibria a lot. The solar gravitation is not strong enough to increase or decrease the families of periodic orbits, nor is it able to change the stabilities of periodic orbits. For unstable orbits, it will not lose its periodic characteristics as long as the original orbits can keep the characteristics in the vicinity of asteroids before solar gravitation is added and it is easier for orbits with larger Jacobi constant to keep its periodic characteristics, which provides the possibility of making use of unstable orbits to probe asteroids.
Invariant manifolds in the restricted three-body problem are a powerful tool for the design of spacecraft trajectories. This work presents an approach to compute families of quasi-periodic orbits and their linear stability via a stroboscopic map. The approach includes the generation of stable and unstable manifolds from hyperbolic quasi-periodic orbits. Including quasi-periodic orbits along with periodic orbits in the design space offers additional low-energy transfer options. Both the circular and elliptic restricted three-body problems are considered.
The existance of stable equilibrium regions in the CRTBP under constant thrust acceleration has been pointed out recently. This article analyses the case of the Mars-Phobos system, in which stable hovering can be obtained for a spacecraft coorbiting with the Moon at a distance of about 80 km from its center with less than 15 micro-g of constant acceleration. The system stability is investigated analytically and verified with a full numerical model accounting for all relevant perturbations.
A largely numerical study was made of families of three-dimensional, periodic, ‘halo’ orbits near the collinear libration points in the restricted three-body problem. Families extend from each of the libration points to the nearest primary. They appear to exist for all values of the mass ratio μ, from 0 to 1. More importantly, most of the families contain a range of stable orbits. Only near L1, the libration point between the two primaries, are there no stable orbits for certain values of μ. In that case the stable range decreases with increasing μ, until it disappears at μ=0.0573. Near the other libration points, stable orbits exist for all mass ratios investigated between 0 and 1. In addition, the orbits increase in size with increasing μ.