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SCRMI: a Semi-Classical Relativistic Motion Integrator,
to model the orbits of space probes around the Earth
and other planets.
Observatoire
Midi - Pyrénées
3
Sophie PIREAUX , Jean-Pierre BARRIOT and Pascal ROSENBLATT,
1
Financial support provided through:
European Community's Improving Human Potential Program
under contract RTN2-2002-00217, MAGE;
2
CNES/INSU funding.
1,2
1
2
Royal Observatory
of Belgium
UMR 5562, Observatoire Midi-Pyrénées, Toulouse, France
[email protected] , [email protected] - tel. +33 (0)5 61 33 28 44 – Fax +33 (0)5 61 33 29 00
3
Belgian ESA/PRODEX 7 contract.
3
Royal Observatory of Belgium, Brussels, Belgium.
ABSTRACT
Today, the motion of spacecrafts is still described according to the classical Newtonian equations plus the
so-called « relativistic corrections», computed with the required precision using the Post-(Post-) Newtonian
formalism.
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 is furthermore error prone.
Those problems are especially relevant when modeling tiny geophysical effects through orbitography, like for
example the waxing-waning mechanism of the Martian polar caps, with temporal frequencies in the same band
as relativistic effects.
In the more appropriate frame 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 forces (atmospheric drag, solar radiation pressure, albedo pressure, thermal
emission…). The latter are treated as perturbations, in the sense that we assume that both the local structure of
space-time (G ) is not modified by these forces, and that the unperturbed satellite motion follows the geodesics
of the local space-time. We also use a symplectic integrator to compute the unperturbed geodesic motion, in
order to insure the constancy of the norm of the proper velocity quadrivector (U ).
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.


Newton’s 2nd law of motion with [1]
THE CLASSICAL APPROACH: GINS*
*Géodésie

par Intégrations Numériques Simultanées


A  gradU

E


A
Perturbating Bodies  E
A

Radiation Pressure
Earth Tides
where
A
Atmospheric Pressure
Relativistic
Ocean Tides


is a software developed by CNES



A  grad U  grad U  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
OT
A
AP
is a gravitational acceleration induced by the redistribution of atmospheric masses (single layer model).
Figure 1-10: The following graphs were plotted 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, as well as the additional periodicities due to J2 (Fig.2) and
higher orders in the gravitational potential (Fig.3). 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.
Relativistic corrections on the forces A  A 
Gravitational potential model for the Earth
U
E
GM

R

lmax
E
l 0
l
m 0
a 
  P (sin  )C cos m  S sin m 
R
lm
lm
lm

with
A
a semi-major axis of the Earth, C , S , the normalized
lm
E
lm

S
harmonic coefficients, given in GRIM4-S4 model
GM  4GM
V


c R  R
E
2
E
2
3





R  4 V  R V 


GP

Figure 2
Figure 3
Newtonian contributions from the Moon, Sun and Planets




A A   A
PB  E
J2 - Moon coupling
body "n "
with
3rd body n
A
J2 - Moon coupling
3 GM

2 R
Mo
Mo
5

 Z
5 C a  5
R

2
20
2
Mo
2
E
Mo
 0 
 

 1 R  2 R  0 

1  
 
Mo
Mo

and
A
3rd body n
 RR
R 

 GM

 RR
R 


n
3
v  grad U ( x )
2c
2
E
E
S
10 m/s
8
  0.245 


  2.141 
  0.928 


2
XYZ0
Geodetic(De Sitter) Precession
A
Lens-Thirring Precession


LTP

G
cR
2

LTP
3


3 S R R
  S 

R


E
E
2
Figure 6
10 m/s
11
  0.13 


  34.83 
  40.10 


2
XYZ0
10 m/s
12
2
XYZ0
n
3
n
LTP
GP


A
A  2  V
Figure 5
Figure 4
  0.187604 


  4.524321 
 - 0.210319 





A  2  V
GP
Figure 1
Schwarzschild
R
l
E



3
n
Earth tides
n
C ,  S
with l=2,3 in the
Earth gravitational
potential, due to Sun and
Moon.
lm
(…)
lm
ORBIT
 X ,Y , Z ;V ,V ,V 
x
Y
GRAVITATIONAL POTENTIAL
MODEL FOR EARTH
z
TAI
GRIM4-S4
ITRS (non inertial)
J2000 (“inertial”)
INTEGRATOR

10 m/s
6
  0.5313475 



5
.
5674979


  1.3066652 


2
XYZ0
Figure 9
10 m/s
7
  0.314 



0
.
560


  0.257 


2
XYZ0
A
 10 m/s
11


J2000 (“inertial”)
PB E



GRAVITATIONAL POTENTIAL
MODEL FOR EARTH


Need for
symplectic integrator
X

;U  
TCG
measured
by
accelerometers
GRIM4-S4
ITRS (non inertial)
ORBIT
  TCG
METRIC MODEL
GCRS (“inertial”)
4) GCRS metric with(out) Sun, Moon, Planets => validate geodetic precession (other bodies contributions)
RMI will go beyond GINS capabilities:
INTEGRATOR
- will include 1) IAU 2000 standard GCRS metric [6,7]
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.
GP
E
TDB
 dU






with K  quadri-”force”



K
G

U
U



U
U


 d
X  (T, X , Y , Z )


  0, 1, 2, 3
dX
U  
+ first integral G U U  1

d
3) Kerr metric => validate Lens-Thirring correction
- separate modules allow to easily update for metric, potential model (EGM96)… prescriptions.


2) (Schwarzschild + GRIM4-S4) metric => validate harmonic contributions
4) Post Newtonian parameters in the metric and space-time transformations

for x , v  in A and A
E
- simulations with 1) Schwarzschild metric => validate Schwarzschild correction
3) IAU 2000/IERS 2003 new standards on Earth rotation [5,6]
TAI  TT  TDB
XYZ0
Method: GINS provides template orbits to validate the RMI orbits:
2) IAU 2000 time transformation prescriptions [6,7]
DE403
TAI
2
Figure 10
THE (SEMI CLASSICAL) RELATIVISTIC APPROACH: (SC)RMI
For the appropriate metric, the relativistic equation of motion 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.
PLANET EPHEMERIS
[5]
  0.58286072 



1
.
02761036


  0.34965593 


Figure 8
Earth
rotation
model
Figure 7
TCG  

 dU

;U  

 d


IAU2000
GCRS metric
G
TCG  TDB
PLANET EPHEMERIS
DE403
for G in 

GCRS (“inertial”)
TDB
[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.

