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Relativistic modeling of the orbit of geodetic satellites
equipped with accelerometers
1
2
Observatoire
Midi-Pyrénées
3
Sophie PIREAUX , Jean-Pierre BARRIOT and Pascal ROSENBLATT,
Financial support provided through:
1
European Community's Improving Human Potential Program
under contract RTN2-2002-00217, MAGE;
2
CNES/INSU funding.
3
Belgian ESA/PRODEX 7 contract.
1,2 UMR 5562, Observatoire Midi-Pyrénées, Toulouse, France
Royal Observatory
of Belgium
[email protected] , [email protected] - tel. +33 (0)5 61 33 28 44 – Fax +33 (0)5 61 33 29 00
3
Royal Observatory of Belgium, Brussels, Belgium.
ABSTRACT
When non-gravitational forces are present, the relativistic equation of motion generalizes to include nongravitational forces encoded in a quadrivector (K ). Non-gravitational forces can be treated as perturbations, in the
sense that they do not modify the local structure of space-time (the metric).

Today, the motion of spacecrafts is still described according to the classical Newtonian equations plus the socalled "relativistic corrections", computed with the required precision using the Post-(Post-…) Newtonian
formalism.
We show here how to recover the classical equation for accelerometers from this relativistic formalism, hence
how to measure the force quadrivector.
The current approach, with the increase of tracking precision (Ka-Band Doppler, interplanetary lasers) and
clock stabilities (atomic fountains) is reaching its limits in terms of complexity, and can be furthermore error
prone. In the more appropriate framework of General Relativity, we study a method to numerically integrate the
native relativistic equations of motion for a weak gravitational field, taking into account not only gravitational
forces, but also non-gravitational ones (atmospheric drag, solar radiation pressure, albedo pressure, thermal
emission…).
When considering gravitational forces alone, the unperturbed satellite motion follows the geodesics of the
local space-time. For the appropriate metric at the required order, the latter equations contain all the
gravitational effects at the corresponding order. Indeed, computing the geodesic equations for the Geocentric
Coordinate Reference System (GCRS) metric (G ) (IAU conventions 2000) will take into account gravitational
multipole moment contributions from the central planetary gravitational potential, perturbations due to solar
system bodies, the Schwarzschild, geodesic and Lense-Thirring precessions.
The classical Newton plus relativistic corrections method briefly described below faces three major
problems.
First of all, it ignores that in General Relativity time and space are intimately related, as in the classical
approach, time and space are separate entities. Secondly, a (complete) review of all the corrections is needed
in case of a change in conventions (metric adopted), or if precision is gained in measurements. Thirdly, with
such a method, one correction can sometimes be counted twice (for example, the reference frequency
provided by the GPS satellites is already corrected for the main relativistic effect), if not forgotten.
For those reasons, a new approach was suggested. The prototype is called (Semi-Classical) Relativistic
Motion Integrator (whether non-gravitational forces are considered). In this relativistic approach, the
relativistic equations of motion are directly numerically integrated for a chosen metric.

THE CLASSICAL APPROACH:
*Géodésie
GINS*
Newton’s 2nd law of motion with [1]
par Intégrations Numériques Simultanées



A  gradU

E


A
Perturbating Bodies  E

Radiation Pressure
Earth Tides
where
A
Atmospheric Pressure
Relativistic
Ocean Tides


is a software developed by CNES



A  grad U  grad U  A  A

A
Atmospheric Drag

is due to satellite colliding with residual gas molecules (hyp: free molecular flux);
A
AD
is due to change in satellite momentum owing to solar photon flux; U is the

ET
RP
U
Earth tide potential due to the Sun and Moon, corrected for Love number frequencies, ellipticity and polar tide;
is the ocean tide potential (single layer model); and
A
OT
AP
is a gravitational acceleration induced by the redistribution of atmospheric masses (single layer model).
Figure 1-3: The following
Relativistic corrections on the forces A  A 
A
S

GM  4GM
V


c R  R
E
2
E
3
2





R  4 V  R V 


A
2
GP

GP


Geodetic(De Sitter) Precession
A
GINS ORGANIGRAM
Lens-Thirring Precession

A 
2
V
LTP
GP
3

v  grad U ( x )
2c
2

A
E
E
S


LTP
2

LTP
G

cR
3
ORBIT
V
 X ,Y , Z ;V ,V ,V 


3 S R R
  S 

R


E
E
x
Y
GRAVITATIONAL POTENTIAL
MODEL FOR EARTH
z
2
TAI
GRIM4-S4
ITRS (non inertial)
J2000 (“inertial”)
INTEGRATOR

A
PLANET EPHEMERIS
DE403
TAI
J2000 (“inertial”)
Figure 2
  0.187604 



4
.
524321


 - 0.210319 


10 m/s
8
  0.245 



2
.
141


  0.928 


2
XYZ0
Figure 3
10 m/s
11
  0.13 



34
.
83


  40.10 


2
XYZ0
10 m/s
12
XYZ0


RMI ORGANIGRAM
GRAVITATIONAL POTENTIAL
MODEL FOR EARTH
2

Method: GINS provides template orbits to validate the RMI orbits:
d X
W

dT
X
2
classical limit
i
2
2) (Schwarzschild + GRIM4-S4) metric => validate harmonic contributions
X
i

;U  
TCG
i  1, 2, 3
with
GRIM4-S4
ITRS (non inertial)
ORBIT
Need for
symplectic integrator
- simulations with 1) Schwarzschild metric => validate Schwarzschild correction
PB E
2

The motion of a test particle in free-fall (satellite subject to gravitational forces only) is
described by the geodesic equation of motion (1).
For the appropriate metric, it contains all needed gravitational effects (blue terms).
GP
E
TDB

dU




with X  (cT, X , Y , Z )
 d   U U
  0, 1, 2, 3
(1) 

dX
U  
and first integral G U U  c

d
Motion Integrator

for x , v  in A and A
TAI  TT  TDB
E
Figure 1

  TCG
METRIC MODEL
W = GCRS generalized
GCRS (“inertial”)
gravitational potential
in metric G
3) Kerr metric => validate Lens-Thirring correction
[5]

THE RELATIVISTIC APPROACH: RMI
*Relativistic
Schwarzschild
R

Earth
rotation
model
graphs were plotted with GINS
by selecting only certain
gravitational contributions.
Those examples show the
correction to tangential/ normal/
radial directions on the trajectory
of LAGEOS 1 satellite, due to
the selected effect, after one day.
Integration is carried out during
one day, from JD17000 to
JD17001. The corresponding
induced acceleration on
JD17000 is given below each
graph. We clearly see the orbital
periodicity of 6. 39
revolution/day in each figure.
Capital letters are used for
geocentric coordinates and
velocities; while lower cases are
used for barycentric quantities.
Name of planets/Sun are shown
by indices; no index is used in
case of the satellite.



IAU2000
GCRS metric
4) GCRS metric with(out) Sun, Moon, Planets => validate geodetic precession (other bodies contributions)
INTEGRATOR
RMI will go beyond GINS capabilities:
TCG  TDB

 dU

;U  

 d

TCG  
- will include 1) IAU 2000 standard GCRS metric [6,7]
PLANET EPHEMERIS
DE403
for G in 

2) IAU 2000 time transformation prescriptions [6,7]
G



GCRS (“inertial”)
3) IAU 2000/IERS 2003 new standards on Earth rotation [5,6]
TDB
4) Post Newtonian parameters in the metric and space-time transformations
- separate modules allow to easily update for metric, potential model (EGM96)… recommendations.
THE (SEMI CLASSICAL) RELATIVISTIC APPROACH: (SC)RMI
*(Semi


U
U
 

 dU



with K  quadri-”force” per volume and unit rest-mass

K
G




U
U




 d
c c 

(2) 

dX
U  

and first integral G U U  c
d
d X
W
classical limit(((

K
Measured by
Need for
dT
X


2
Classical) Relativistic Integrator
If non-gravitational forces are present, the satellite motion is described by the relativistic
equation of motion (2).
For the appropriate metric, it contains all needed gravitational effects (blue terms).
Non gravitational forces [8] are encoded in K measured by accelerometers.
When K  0, this equation reduces to the geodesic equation of motion (1).


THE PRINCIPLE OF ACCELEROMETERS:
2


2

2
i
2
accelerometers
i
symplectic integrator









1
dX
dX
d

X


dX
dX
dX
d

X





(3)  K G 


X
 2  



c d d 
d
X
d d
d d
2
Let the test-mass (m), shielded from non-gravitational forces, be located at X  X ;
and the satellite center of mass (CM) at X .
The test-mass motion is described by (1) while that of the satellite is described by (2).
Evaluating the difference between those two equations at first order in X gives (3).
The latter equation reduces to the geodesic deviation if K  0 .
It generalizes the classical equation for accelerometers.



2

REFERENCES:
[1] GRGS. Descriptif modèle de forces: logiciel GINS. Note technique du Groupe de Recherche en Géodesie Spatiale (GRGS), (2001).
[2] X. Moisson. Intégration du mouvement des planètes dans le cadre de la relativité générale (thèse). Observatoire de Paris (2000).
[3] A. W. Irwin and T. Fukushima. A numerical time ephemeris of the Earth. Astronomy and Astrophysics, 338, 642-652 (1999).
[4] SOFA homepage. The SOFA libraries. IAU Division 1: Fundamental Astronomy. ICRS Working Group Task 5: Computation Tools.
Standards of Fundamental Astronomy Review Board. ( 2003) http://www.iau-sofa.rl.ac.uk/product.html.
classical limit
with
G  ,   
evaluated at
X
2
d X
K 

dT
2

i
i
i
2

j
W
X
X X
K
for the CM of satellite
2
i
j
j
[5] D. D. McCarthy and G. Petit. IERS conventions (2003). IERS technical note 200?. (2003). http://maia.usno.navy.mil/conv2000.html.
[6] IAU 2000 resolutions. IAU Information Bulletin, 88 (2001). Erratum on resolution B1.3. Information Bulletin, 89 (2001).
[7] M. Soffel et al. The IAU 2000 resolutions for astrometry, celestial mechanics and metrology in the relativistic framework: explanatory supplement. astro-ph/0303376v1 ( 2003).
[8] A. Lichnerowicz. Théories relativistes de la gravitation et de l’électromagnétisme. Masson & Cie Editeurs 1955.