Working models for the gravity field and
dynamical environment of Phobos
Shi, X.1,2, Willner, K.1, and Oberst, J.1,3
1. Department for Geodesy and Geoinformation Science, Technical University Berlin
2. Shanghai Astronomical Observatory, Chinese Academy of Sciences
3. German Aerospace Center, Institute of Planetary Research
2nd ISSI workshop of Phobos and Deimos
International Space Science Institute, Bern
March 28-April 1, 2011
Outline
➢
Introduction
➢
The gravity field of Phobos
➢
The dynamical environment of Phobos
➢
Landing site evaluation for Phobos-Grunt mission
➢
Conclusion and outlook
Introduction Gravity field Dynamical environment Landing site
Conclusion
1. Developing a model for Phobos’ gravity field is crucial for:
 investigating the origin and evolution of Phobos as well as its inner structure;

tracking and navigating any spacecraft near or landing on Phobos
2. Not enough knowledge has been acquired about the gravity field
of Phobos so far.
The GM value has been
estimated using spacecraft
radiometric data; most recent
value [Andert et al., 2010]:
➢
GM =0.7127±0.0021×10−3
The J2 value developed from
the close March 2010 flyby data
of Mars Express is still of great
uncertainty.
➢
(Andert et al., 2010)
Introduction Gravity field Dynamical environment Landing site
Conclusion
3. Phobos has a special dynamical surface environment:
➢
Highly irregular shape -> complex gravity field on the surface
➢
Relatively fast rotation -> considerable centrifugal effect
➢
Proximity to Mars -> strong tidal effect
4. Scope of this study
Developing working models for the gravity field of Phobos based on the
up-to-date shape model;
➢
Analyzing the dynamical environment of Phobos, in both senses of space
and time;
➢
Evaluating the dynamical environment of the proposed landing site area
for Phobos-Grunt mission.
➢
The Gravity Field of Phobos
Introduction Gravity field Dynamical environment Landing site
Conclusion
Three ways to model the gravity field of a small body
Harmonic Expansion
principle
Mass Concentration
('Mascon')
Polyhedron
Developing harmonic
coefficients as integrations
over the surface
Approximating the whole
body with a set of cubes
Approximating the surface
with a polyhedron
scenario
advantages
●
controls over truncating
degree and order
●
●
straightforward calculation
possibility of applying
varying density
●
disadvantages
●
converges only outside the
enclosing sphere
● accuracy depends on
truncating strategy
●
difference between the
gravity field of a cube and a
that of a sphere
● large computational effort
●
●
converges everywhere
outside the body
● exact mathematical
expression for polyhedron's
gravity field
large computational effort
complex convergence
situation on the surface
Introduction Gravity field Dynamical environment Landing site
Conclusion
In the case of Phobos
➢
Modeling parameters
shape model
Willner et al., 2010
density
1876 kg m-3 (Andert et al., 2010)
GM
0.7127 x 10-3 km3s-2 (Andert et al., 2010)
truncating degree of shape model for harmonic expansion
6
size of cube element
0.1 x 0.1 x 0.1 km
grid size of polyhedron
1.5 deg
➢
Volume estimation (km3)
Harmonic Expansion
5689.9
Mass Concentration
5689.3 (0.04km cube)
Polyhedron
5687.8
Introduction Gravity field Dynamical environment Landing site
Conclusion
Results from the three approaches and the comparison
Gravitational potential on a 14km sphere
Differences in pairs
The Dynamical Environment of Phobos
Introduction Gravity field Dynamical environment Landing site
Conclusion
Description of the dynamical environment
➢
dynamic height
Effective potential
H
dyn
W 0−W A
=
γ0
W A=W t +W c +W g
➢
“It is cumbersome to determine the position of
a particular potential surface for all latitudelongitude bins over Phobos as well as the
length of the curving plumb line for each
surface point. Consequently, the simplest
measure of height is the potential energy at
the surface.”
(Thomas, 1993)
surface acceleration
⃗r¨ ＝ r⃗¨t + r⃗¨c + r⃗¨g
r −⃗r
r⃗M
¨r⃗t =Δ R=GM ( ⃗M
− 3 )
3
∣r⃗M −⃗r ∣ r M
Tidal acceleration
r⃗¨c =−⃗
ω ×(ω
⃗ × ⃗r )
Centrifugal acceleration
Δv
r⃗¨g ≈G ρ ∑
( ⃗r − r⃗i )
3
r −⃗ri∣
i ∣⃗
Gravitational acceleration
(Davis et al., 1981)
Introduction Gravity field Dynamical environment Landing site
Dynamic Height (1)
Global map of dynamic height and its gradient
Conclusion
Introduction Gravity field Dynamical environment Landing site
Conclusion
Dynamic Height (2)
Contours of dynamic height on top of topography
(mid and low latitude)
(Background mosaic: Wählisch et al., 2010)
Introduction Gravity field Dynamical environment Landing site
Conclusion
Dynamic Height (3)
Dynamic height with Phobos at different distances to Mars
Introduction Gravity field Dynamical environment Landing site
Conclusion
Surface Acceleration
Magnitude of surface acceleration and its components
➢
Typical value of the
magnitude of surface
acceleration and its
components (cm sec-2)
➢
sub-Mars
point
north
pole
gravitational
0.58
0.62
tidal
0.13
0.043
centrifugal
0.065
0.0
total
0.40
0.67
global map of acceleration magnitude
Landing site area for Phobos-Grunt
Introduction Gravity field Dynamical environment Landing site
Conclusion
Preliminary evaluation of the proposed landing site area for Phobos-Grunt
Background: map in
sinusoidal projection (Willner
et al., 2011)
➢
Foreground: magnitude of
surface acceleration and its
tangential component
➢
(Willner et al., 2011)
Introduction Gravity field Dynamical environment Landing site
Conclusion
Conclusion
Working models for the gravity field of Phobos have been developed based
on the latest shape model; comparison of results from three approaches are
consistent;
➢
Analyses of the dynamical environment show consistent results with
previous works;
➢
The proposed landing site area for Phobos-Grunt mission has been
evaluated from both topographic and dynamical perspective, with preliminary
gravity and slope data reported to the Phobos-Grunt team.
➢
Outlook
Investigation of high resolution images of Phobos’ surface in combination
with surface gravity analysis could provide evidence for regolith mobility and
possible surface modifications;
➢
Ellipsoidal gravitational coefficients are more adaptive for trajectory study of
spacecrafts approaching or landing on Phobos (Garmier et al., 2001);
➢
Simultaneous radiometric tracking of Phobos-Grunt and YH-1 may provide
high precision data to improve Phobos' ephemeris, gravity field model, etc.
➢
Thanks for your attention!