ISTS-2017-d-009／ISSFD-2017-009
Quasi-Satellite Orbits around Phobos for the Sample Return Mission
By Javier Martín and Simone Centuori1)
1)

Deimos Space SLU, Spain(Received April 6th, 2017)
This paper illustrates the full process of generation of quasi-satellite orbits (QSOs) through a parametric optimizationof the initial conditions and control burns, and the selection of the best orbits for observation of the surface of the moons ofMars. QSOs are particular, short-term stable solutions to the three-body problem that can be exploited to remain in close
proximity (< 100 km) to the Martian moons for periods in the order of weeks with very low control requirements. In thecontext of the joint ESA-Roscomos Phobos Sample Return (PhSR) mission, they are used to characterize the surface of theMartian moons Phobos and Deimos, where the dynamical environment is highly perturbed and dominated by the planet tothe point that the Hill sphere of Phobos is below its surface at some points. The objective for the trajectory optimization is
obtaining a high-resolution image map of both moons with different instruments and, in the case of Phobos, establishing thebest landing site for the sampling procedure.
Key Words: Phobos, Quasi-Satellite Orbits, Optimization

Nomenclature
LOF
v
r
T
w
ω

S/C
Subscripts
0fMmmoons and maintains stability in the order of days/weeks,
which results in a small number of control burns required
during the full observation campaign.

: local orbital frame: velocity: position: orbital period: QSO amplitude, projection of the relative
position on the target’s local orbital plane: angular frequency = 2𝜋 𝑇 −1
: spacecraft

2. Dynamical environment near the moons of Mars
2.1. Physical characteristics
Phobos and Deimos are small bodies, with dimensions in
the order of 10 km and markedly non-spherical, see Table 1.
They orbit extremely close to Mars, with Phobos even inside
the areosynchronous radius: it orbits faster than Mars rotates,
so it rises on the west and sets on the east. It is also getting
closer to the planet, and will likely end up being torn apart by
tidal forces in time scales of tens of millions of years.

: initial: final: Mars: moon (Phobos/Deimos)

Table 1: Physical and orbital characteristics of Phobos and Deimos.
Items
Ellipsoid axes (km)GM (km3/s2)
Equivalent-vol. radius (km)Semi-major axis (km)Eccentricity (-)
Inc. w.r.t. Mars equator (deg)
Source (physical data)
Source (ephemerides)

1. Introduction
Phobos Sample Return (PhSR) is a joint ESA-Roscomos
mission that aims to characterize Phobos and to collect asample of its surface to bring back to Earth1). The mainscience goal of the mission is to understand the formation of
the Martian moons Phobos and Deimos and to put constraints
on the evolution of the solar system. To do this, samples from
Phobos (the moon with the older expected surface) have to be
returned to Earth and thus potential landing sites need to bestudied beforehand to establish sampling usefulness.
The study of the Phobos surface takes place in a complex
dynamical environment that is extremely dominated by Mars:due to the small size of Phobos and its proximity to the planet,even a semblance of Keplerian orbits around the moon is
impossible2). Given that the mission needs to return a sample
to Earth, the usual mass constraints in space missions aremuch more stringent due to the need to carry the sampling andreturn vehicle. Thus, quasi-satellite orbits (QSOs) are used toprovide a trajectory that remains in close proximity to the

PhobosDeimos
13.0x11.4x9.2
7.8x6.0x5.1
7.07·10−4
9.62·10−5
11.35
6.25
9378
23458
0.01500.00041.07
0.94
Refs.3) and 4)
Refs. 3) and 5)
JPL MAR097

Both moons are nearly tidally locked to Mars, althoughlibration motions and small deviations from a perfect longmoon axis-planet alignment exist.
2.2. Accelerations near Phobos
The preponderance of Mars in the system is so large thatthe centre of mass of the system is only a few metres off thatof Mars itself – compared to about 4800km for Earth-Moonsystem. The gravitational parameters of Phobos and Deimos
are 8-9 orders of magnitude smaller than that of Mars, which
together with the close proximity to the planet makes it
impossible to actually orbit them in a Keplerian sense.

1

As an example, Fig. 1 shows the accelerations felt by a test
particle on the Mars-Phobos axis, close to the latter. Only atradial distances below 20 km (corresponding to altitudes well
below 10 km) does the Phobos gravitational acceleration
surpass the Martian tidal term – that is, the Phobos-relative
acceleration due to Mars’ gravity.

since it was first developed to analyse the rendezvous of a S/Cwith a target in a circular orbit, like a space station in Earthorbit. Its applicability to the problem of orbits around othersmall bodies such as comets or asteroids in orbit around the
Sun is usually limited, since the gravity of the small bodyneeds to be taken into account from much further away due tothe smaller influence of the Sun gravity tide term. However,in the case of Mars, the extreme proximity of the primary
makes the equations applicable as a first approximation fororbits that are not too close to the moon.
3.2. Trajectory characteristics
A quick analysis of the equations shows that, in theunperturbed case with ∆γ = 0, the out-of-plane motion (the Y
direction) is decoupled from the in-plane motion (X-Z). In
particular, unperturbed motion in the Y direction is simply an
oscillation about the X-Z plane with the same period as thetarget orbit. This is a positive discovery, since it means thatany trajectory that describes a “pseudo-orbit” in the X-Z plane
can be modified to have an arbitrary target-relative inclination,
thus providing better global observability of the surface.
Focusing on the motion in the X-Z plane, it is readily
apparent that there is a family of closed orbits which describe
elliptical pseudo-orbits around the target with the same periodof the target’s orbit around the primary. However, unlike
Keplerian orbits, the target is in the centre of the ellipse andnot in one of the focal points. The shape of the ellipse is fixed,with the major axis, aligned to the X direction, twice the sizeof the minor axis. There are no other families of closed orbits
in the unperturbed problem.
Including the third dimension once again, three parameters
define a closed trajectory: the size of the ellipse, the“inclination” and the phasing between the in-plane andout-of-plane motions, that is, the position of the nodal points
on the local orbital plane of the target. The effect of varying
these parameters can be seen in Fig. 2, which shows
trajectories with a fixed size but varying inclinations, and Fig.
3, which does the same with a fixed inclination but varying
phasing parameters. From the figures, it is apparent that theorbits could provide a rather complete coverage for target
surface observations.

Fig. 1. Gravitational accelerations of Mars and Phobos on a particle on theMars-Phobos axis. Note that the “Mars TB tide” is the effect of Mars’
gravity considered in a Phobos-centred, non-rotating frame.

It seems clear that, for altitudes well over the mentioned~10 km, it makes more sense to approach the problem fromthe point of view of co-orbital motion that remains close to thetarget moon, instead of trying to model a (heavily) perturbed“orbit” around it. The following sections will analyse theproblem of relative motion between two co-orbiting objects,
starting from the simplified case of rendezvous dynamics andarriving at quasi-satellite orbits.
3. Clohessy-Wiltshire motion
3.1. Dynamics
Considering only two bodies in orbit around a primary(“target” and “chaser” in the literature), the relative motion of
the chaser w.r.t. the target can be obtained by writing the
dynamics of the chaser in a frame centred at the target. If thetarget follows a circular orbit, a common choice is a frame
where X points towards the target velocity and Z towards theplanet, while Y is collinear with the orbital momentum andchosen to follow the right-hand rule. Note that, if the target’s
orbit is not circular, either the X/V or the Z/R alignment needs
to be broken.
The driving force in such a system is likely to be the“gravitational tide” term: the difference of gravitational
acceleration between the chaser and the target. In thisformulation, the tide caused by a point-mass primary is split
off, leaving a differential acceleration ∆γ on the right handside. This term represents all other influences (e.g. thenon-spherical gravity influences, solar radiation pressure,atmospheric drag, thrust commands, etc.). If the distance
between the two bodies is small compared to the size of theorbit, the equations can be linearized as follows 6):
𝑥̈ − 2𝜔𝑧̇ = Δ𝛾𝑥𝑦̈ + 𝜔2 𝑦 = Δ𝛾𝑦(1)𝑧̈ + 2𝜔𝑥̇ − 3𝜔2 𝑧 = Δ𝛾𝑧This is commonly described as “rendezvous dynamics”,

Fig. 2. Closed trajectories in unperturbed Clohessy-Wiltshire motion
around a tidally-locked ellipsoidal target, generated with a fixed node on
the local orbital plane of the target and varying inclinations 0-45 deg.

2

not linearized – in order to generate trajectories at a wide
range of distances to the target, the full system is used.
Additional perturbations are also introduced, although thisis not a fundamental difference, since they could have also
been represented in C-W dynamics through the ∆γ term as
described above. In the particular case of the PhSR mission,
all perturbations were gravitational in nature: from the target
moon (either as a point mass or with expansions up to degreeand order 3), from the Sun as a third body and from thenon-spherical gravity field of Mars.
The expected results vary depending on the distance to thetarget moon: for close orbits below the limit mentioned in thesection above, it is expected that the trajectories become morecircular due to the influence of the gravity of the target. On
the other hand, for large distances it is expected that
trajectories will look significantly like the C-W solutions.
4.2. Design and optimization
The computation of the QSO trajectory is no longer
possible through direct symbolic integration. Instead, theobtained orbits are the result of an optimization, in which theinitial position is selected with parameters similar to the C-W
case and the initial velocity is the optimization variable.
The objective for the optimization is, in abstract terms, thesurvival and stability of the QSO for a certain amount of time.
Several functions to represent this objective were tested,falling in two categories:
 Those that minimize the variation of some function of
the moon-relative position, introducing a pressure to
make the orbit more circular.
 Those that minimize the variation of that distance at
the crossings of a surface of section, thus leaving the
distance at other points unconstrained.
In both cases, the function of the position could be either
the 3D distance between the S/C and the target moon (the“radius”), or its projection on the local orbital plane of thetarget (the “amplitude” or w). Note that, unlike in thelinearized case, there is a coupling between the in-plane andout-of-plane motions.
The initial guess for the optimization variable is thesolution to the unperturbed C-W equations. However, for longintegration periods >10 Tm, it was observed that behaviour ofthe orbit was sometimes too unstable for a direct applicationof the optimization algorithm. In particular, the use of a C-W
initial guess did not provide a good enough starting point
because the perturbations tended to move the S/C away fromthe vicinity of the target moon in a few periods, and in someoccasions to crash into the moon.
In order to circumvent these problems, a grid-scan step in
the initial velocity was introduced before the optimization
itself. The result of this strategy can be seen in Fig. 4 for avery close pseudo-orbit, started with 50 deg of target-relative
inclination and an initial position of 𝑧0 = 𝑅𝑚 + 20km where
Rm is the largest radius of Phobos. Even in a case that is very
perturbed by the low altitude and high initial inclination, thegrid-scanning algorithm succeeded in finding an initialcondition for the optimizer to refine into a trajectory thatremained close to Phobos (between 28 and 150 km of central
distance) for 30 days without any controls.

Fig. 3. Closed trajectories in unperturbed Clohessy-Wiltshire motion
around a tidally-locked ellipsoidal target, generated with a fixed inclinationof 30 deg and varying the in-plane/out-of-plane phase.

It is important to remark, however, that the orientation of
the elliptical trajectory in the X-Z plane is fixed, with thelargest axis oriented perpendicular to the target-primary line.
This is a significant disadvantage of this family of trajectories,
since a tidally-locked body like the moons of Mars would
usually have its largest axis substantially aligned with thetarget-primary line.
Another important concern is that the formulation of theunperturbed C-W motion is disregarding important terms in
the acceleration, mainly the gravity of the target itself and thenon-sphericity of the primary (especially for Phobos, which isextremely close to Mars). Thus, unperturbed solutions canonly be used as an initial guess for the trajectory design andoptimization procedure. However, most of these terms could
be taken into account if necessary through the ∆γ term,
although the applicability of the linearization of the dynamics
might be called into question.
Furthermore, as shown in Fig. 1, the influence of the moons
falls off rapidly and the effect of Phobos’ gravity is overtakenby the Martian tidal term for altitudes over 20 km. Thus, real
trajectories are expected to resemble unperturbed C-W closed
orbits outside that area of high influence of the moon.
4. Quasi-satellite orbits
4.1. Dynamics
The assumptions used in the generation of quasi-satellite
orbits (QSOs) are significantly relaxed from those in C-W
dynamics. In particular, the target is not assumed to be in astrictly circular orbit, although the eccentricity is still assumed
to be small (as it is in the case of Phobos and Deimos). Thus,
the C-W local orbital frame is not well defined, and it isreplaced by a VVLH frame – the X axis alignment with target
velocity is kept, at the expense of the exact alignment of Zwith the target-primary line.
The main difference from a mathematical point of view,
however, is that the equations used to propagate the QSOs are3

Fig. 4. Results of optimization for a QSO around Phobos, with an initialaltitude of 20 km over the largest Phobos radius and an inclination of
50 deg. Red: failure to converge using the initial guess from the C-W
solution. Blue: converged trajectory which remains in close proximity to
Phobos for at least 100 Tm.

Fig. 6. Instantaneous distance to Phobos in the previous QSO. The
envelope is stable, with a superimposed oscillation due to the precession of
the node in the local orbital plane of Phobos.

4.3. Trajectory characteristics
The converged trajectories obtained by the above procedure
have a shape and properties that matches the behaviour
expected by the addition of the mentioned perturbations to theC-W problem. In particular:
 For “distant” orbits, the shape of the orbit is very
similar to the unperturbed C-W solution. The main effectof the gravity of the target is a secular variation in thephase between the out-of-plane and the in-plane motions.
This is shown in Fig. 5, where a single trajectory close to
Phobos sweeps an elliptical cylinder in 20 days – incontrast, the phase is fixed parameter of the orbit in C-W
dynamics, as seen in Fig. 2. The projection of the
trajectory on the target LOF is very close to a 2:1 ellipse
and remarkably stable (see Fig. 6), but the nodal point of
the S/C on the local orbital plane of the target precesses
as shown on Fig. 7.
 High-inclination orbits are severely destabilized by
the same effect. In particular, problems first appear at
inclinations over 20-30 deg (depending on the amplitudeof the orbit) and worsen progressively.
The resulting trajectories require more intermediate
controls to survive for the same time as a similar but lessinclined trajectory. The evolution of the distance to thetarget moon is more unstable even when the trajectory
converges; see e.g. the “good” case Fig. 4. As a result, theusage of QSOs with inclinations larger than 50-60 deg
was not pursued in the project.

Fig. 7. Argument of the nodal points of the S/C on the Phobos local
orbital plane for the previous QSO. Angles are measured from +Vm
towards the Mars-pointing direction.

 Finally, orbits very close to the target are even more
distorted by its gravity. The projection of the trajectoryon the local orbital plane is markedly less elongated, withan axes ratio as low as 1.75:1 for otherwise stable orbits,
or even lower values of 1.35:1 for more controlled
trajectories. Both cases are represented in in Fig. 8, whichcontains a 30-day uncontrolled QSO (in blue) and a
20-day trajectory with controls every 5 days (in red).
The change is also very noticeable as a reduction of thepseudo-period (time between two crossings of the YZ
plane of the LOF), as can be seen in Fig. 9. In the limit
case, if the target was a point mass, the orbit would get
more and more circular for lower altitudes, as the gravityof the target started dominating its vicinity.

Fig. 8. QSOs in extremely close proximity to Phobos. Blue: 30-day
uncontrolled QSO with w0 = 33 km on the Mars-Phobos line and an
inclination of 30 deg. Red: 20-day QSO with controls at 5-day intervals,
w0 = 20 km and an inclination of 15 deg.

Fig. 5. QSO around Phobos, for a 20-day trajectory case with w0 = 53 km
on the Mars-Phobos line and an inclination of 20 deg. Four intermediate
controls were introduced at fixed intervals of 5 days, all below 1.5 cm/s.
The first revolution is highlighted.

4These scientific requirements point to a trajectory for the
global characterization phase with a substantial uncontrolled
part, allowing the determination of the gravitational androtational parameters. On the other hand, the surface
observation requirements would be better achieved by using
trajectories with less or even no variation in the distance to thetarget moon, in order to have a constant resolution and requireless post-processing on the ground side. However, as stated inprevious sections, the shape of a QSO is an ellipse with thetarget at the centre, and whose major axis is aligned to thetarget velocity, thus nearly perpendicular to the target-primaryline for a quasi-circular orbit like that of Phobos or Deimos.
Thus, the summary of trade-offs for the choice of QSOs as
for the global characterization phases in PhSR was:
 In favour: QSOs can be generated as long arcs withno controls, leading to a small GNC ∆V cost. This isparticularly important given the tight mass constraints.
 In favour: the existence of long periods without
controls makes it easier to determine the gravitational androtational parameters of the moons because it means lessthruster noise in the measurements.
 Against: the shape and orientation of the QSO is ill
suited for surface imaging, since the obtained data will
necessarily have variable resolution. This implies not justthat a smaller QSO size will have to be chosen to achieve
the resolution requirements, but also that the dataobtained will need more post-processing.
All in all, the choice was made in favour of the QSO
because of the very tight mass requirements in the mission.
5.2. QSO design and choice
The design and optimization procedure was implemented
using a custom toolbox built on top of the open-source Orekit
framework. The orbit propagation used a Dormand-Prince
integration scheme of order 8(5,3) in local extrapolation mode,
although in specific cases this was modified to use a classical
Runge-Kutta scheme of order 4 for performance. Due tolimitations in Orekit, the dynamics equations were definedand integrated in a non-rotating frame centred at the target.
In order to determine the best QSO for each scenario of themission, a two-dimensional map in the initial amplitude andinclination was generated, computing one trajectory for eachpoint in the grid and for each scenario, so that the stability andperformance of each QSO could be evaluated. The parameters
for the generation are described in Table 3, where the initialsize parameter is the amplitude w0 minus the largest sphericalradius of the respective moon Rm.

Fig. 9. Instantaneous distance for the previous QSOs, limited to the firstten Phobos periods (around 3.2 days). Notice that the QSO pseudo-periods
are shorter than Tm: 83% and 52%, respectively.

5. Application to the Phobos Sample Return project
5.1. Project requirements
In the context of the Phobos Sample Return (PhSR) mission,
the scientific requirements7) describe the study of the shape,
structure and composition of the moons of Mars. In particular,
this “global characterization phase” requires the measurementof the gravity field and the generation of detailed surface andelevation maps of Phobos and Deimos (although visiting thelatter is optional in some mission scenarios) using a variety of
instruments in the visible and infrared bands of the spectrum.
In the case of Phobos, this phase allows the determination of
up to five tentative landing sites that can be further studied ina subsequent local characterization phase.
The specific requirements for the global characterization
phase are different for each instrument, and are detailed inTable 2, although the requirements for the IR instruments wereinvalidated during the project in favour of a redesign withperformances similar to the NAC. The resulting surface maps
need to cover at least 50% of Phobos and 10% of Deimos,
while ensuring that possible landing areas (defined as thelatitude band within 20 deg of latitude of the sub-solar point at
landing) are fully covered. Other limitations also apply, e.g.
the narrow-angle camera can only take images when the solar
elevation is between 30 and 60 deg.
Table 2: Phobos surface observation requirements for each instrument.
Resolution Req.
Pixel size
Image size
Field of view
CCD size
Focal length
Max altitude

[m]
[µm]
[px]
[deg]
[mm]
[mm]
[km]

NAC

VisNIR

MidIR

3
10
2048
1.8
20
652
97.78

30
30
320
4.6
10
120
59.75

30
25
315
9
8
50
30.02

Table 3: Parameters for the generation of the QSO maps.

Furthermore, indirect observational requirements specify
that the coefficients of the gravitational field of Phobos up todegree and order 2 must be measured with a precision of 2%,
and its libration around the tidally-locked alignment must be
determined within 1%. For Deimos, only the point-massgravitational parameter must be determined, to 1% precision.

Parameter

PhobosDeimos

Initial size parameter [km]Size parameter grid step [km]Initial inclination [deg]Inclination grid step [deg]Trajectory duration [days]
Time between controls [days]

20‒50
2
40‒50
5
60
30

22‒28
2
40‒50
5
30
‒

Each trajectory begins at a scenario-specific date on the YZ
plane, that is, the initial amplitude is given at the short axis of5the ellipse and the initial nodes are at 0 and 180 deg. The
initial guess for the velocity was obtained from a grid-scanalgorithm around the C-W solution, as stated earlier. Fromthat result, the optimization was launched with a Nelder-Meadsimplex algorithm9), and the final results were evaluated interms of the minimum/maximum distance to the target. In alater stage, the performance properties of the solutions interms of other functions (finer coverage, Earth communication,etc.) were evaluated by other consortium partners, leading to a
final trade-off in which a QSO was chosen for each scenario.
5.3. Results: Phobos surface coverage
The stability of each QSO in the map can be observed inFig. 10 and Fig. 11 in terms of the minimum and maximum
distance to the target respectively, before being filtered byobservation and communications performance as describedabove. Note that blank areas represent trajectories that failed
to converge to a stable QSO.

elevation can be seen in Fig. 12 and Fig. 13, respectively,
where the dashed lines show the observability limits of 98 km
altitude (a radial distance of 107 km, since in the worst-casethe altitude is measured over the smallest radius of Phobos)
and the 30‒60 deg band of solar elevation.

Fig. 12. Instantaneous target range in first 30 days of the Phobos QSO.

Fig. 13. Instantaneous Sun elevation in first 30 days of the Phobos QSO.

Additionally, the ground track of the S/C was analysed
together with the observability limitations in order to verify
that the coverage of was adequately distributed in the bands ofinterest, without leaving zones with a particularly sparse cover.
Results are shown in Fig. 14.

Fig. 10. Minimum distance to Phobos for each QSO in the map for the
ESA-standalone scenario of the PhSR mission.

Fig. 14. Ground track of the orbit selected for the global characterizationof Phobos, limited to points in which the NAC instrument can operate.
Fig. 11. Maximum distance to Phobos for each QSO in the map for the
ESA-standalone scenario of the PhSR mission.

5.4. Results: Deimos surface coverage
The obtained Deimos QSO map is very similar to that of
Phobos, but with an important difference: the much larger
distance to Mars. The main effect of this change is a slower
motion, since the period of Deimos is nearly four times that of
Phobos. Since the time allotted for the observation of Deimos
is also halved (30 days vs. 60), the total number of revolutions

The trajectory that was finally chosen was the same for the
ESA-Standalone and the ESA-Roscosmos joint mission
scenarios. The trajectory size parameter (𝑤0 − 𝑅𝑚 ) is 38 km
and the initial inclination is 45 deg, with a single control burnof just 1.2 cm/s. The resulting evolution of the range and solar

6around Deimos is approximately 1/8 than around Phobos,
although this is compensated by the fact that only 10% of thesurface needs to be covered per the requirements7).
On the other hand, orbits near Deimos are inherently morestable due to the weaker perturbing gravity fields, both fromthe much further primary and the smaller moon. Only one
case failed to converge to a valid 30 day ballistic arc.
The orbit finally chosen for the ESA-Roscosmos joint
scenario has a size parameter of 38 km and an inclination of
45 deg. As a consequence of the slower motion, the imaging
coverage is sparser, as shown in Fig. 15.

Fig. 17. Plot of the Deimos QSO in the body-fixed frame of the moon.

Acknowledgments
The authors wish to acknowledge the collaboration of otherpartners in the consortium, including key personnel fromThales Alenia Space, in the coverage and trade-off analysis of
the obtained QSO trajectories.
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Fig. 15. Ground track of the orbit selected for the observation of Deimos,
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Quasi-satellite orbits can be thought of a particular solution
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around Phobos (Fig. 16) and Deimos (Fig. 17). QSOs are able
to provide coverage for all longitudes of a tidally-locked body,although the observation distance is necessarily variable.

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Fig. 16. Plot of the Phobos QSO in the body-fixed frame of the moon.

7