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Basis for a native relativistic software integrating
Observatoire
Midi-Pyrennees
the motion of satellites
Sophie PIREAUX
1
1
UMR 5562, Observatoire Midi-Pyrénnées, Toulouse, France
Financial support provided through the
European Community's Improving Human Potential Program
under contract RTN2-2002-00217, MAGE
[email protected] - tel. +33 (0)5 61 33 28 44 – Fax +33 (0)5 61 33 29 00
INTRODUCTION
And finally, a correction on the clock frequency, owing to the presence of a gravitational field, is considered, leading
to a relativistic Doppler effect. This is a natural consequence of the distinction between proper time and coordinate
time in relativistic theories.
We introduce here a relativistically consistent software called RMI (Relativistic Motion Integrator). We
compare it with the GINS software [2] as an example of the classical approach consisting in the Newtonian
formalism plus relativistic corrections.
GINS is a software routinely used to evaluate the gravity potential (GRIM5, EIGENS 1-2) from the
determination of orbit perturbations, or for precise orbit determinations around the Earth (CHAMP,
GRACE...) and around Mars (MGS, modeling of future missions). It is based upon the usual formalism used
in spatial geodesy, i.e. it relies upon the classical Newtonian description of motion, to which relativistic
corrections are added. The number/type of corrections needed depends upon the precision in the
measurements (Clock precision/stability).
The relativistic corrections on the forces already taken into account in GINS are:
the Schwarzschild effect, function of the position and speed of the satellite; the Lens-Thirring effect, due to
the rotation of the attracting body; the geodetic precession, function of the chosen coordinate system and due
to the non inertial motion of the gravitational source in the solar system.
Corrections are also applied on the measurements:
a datation correction, which stems from the transformation between the Universal Time Coordinate and the
International Atomic Time, or the time referred to in the ephemeris; a relativistic time delay correction, due to
the curvature of space-time through which light travels.



Newton’s 2nd law of motion with [1]
THE CLASSICAL APPROACH: GINS
The classical Newton plus relativistic corrections method briefly described here 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. In this relativistic approach, the geodesic equations of
motion are directly numerically integrated for a chosen metric.
A  gradU

E


A
Perturbating Bodies  E
A

Radiation Pressure
Earth Tides
A
Atmospheric Pressure
Relativistic
Ocean Tides


where



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;

is the ocean tide potential (single layer model); and
A
RP
ET
U
Earth tide potential due to the Sun and Moon, corrected for Love number frequencies, ellipticity and polar tide;
is the
U
OT
AP
is a gravitational acceleration induced by the redistribution of atmospheric masses (single layer model).
Figure 1-9: 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 JD17001 to JD17002. 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 
The 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
3
2





R  4 V  R V 


GP


Figure 2
Figure 4
Figure 3
  0.187604 


  4.524321 
 - 0.210319 


The 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
XYZ0
n
3
n
S
10 m/s
11
Geodetic(De Sitter) Precession
A
Lens-Thirring Precession


LTP


LTP
GM
cR
2
E
3


3 S R R
  S 

R


E
E
2
Figure 6
  0.13 


  34.83 
  40.10 


2
XYZ0
10 m/s
12
2
XYZ0
ORBIT
3
n
  0.245 


  2.141 
  0.928 


2
 RR
R 

 GM

 RR
R 


n
2
E
Figure 5
10 m/s
8
LTP
3
v  grad U ( x )
2c
E

A
A  2  V
GP



A  2  V
GP
Figure 1
Schwarzschild
R
l
E



n
 X ,Y , Z ;V ,V ,V 
x
Y
GRAVITATIONAL POTENTIAL
MODEL FOR EARTH
z
TAI
(…)
GRIM4-S4
ITRS (non inertial)
J2000 (“inertial”)
INTEGRATOR

A
PLANET EPHEMERIS
DE403
TAI
10 m/s
6
2
XYZ0
  0.5313475 


  5.5674979 
  1.3066652 


Figure 9
10 m/s
7
2
XYZ0
  0.314 


  0.560 
  0.257 


E
J2000 (“inertial”)
 10 m/s
11
PB E
E
TDB
2
XYZ0
For the appropriate metric, the geodesic equation of motion,
contains all needed relativistic effects.(For a massive
particle, the affine parameter    the proper time)
Method: GINS provides template orbits to validate the RMI orbits:
THE RELATIVISTIC APPROACH: RMI
GRAVITATIONAL POTENTIAL
MODEL FOR EARTH
 dU






U
U

 d


dX

U 

d

GRIM4-S4
ITRS (non inertial)
ORBIT
X

;U  
- simulations with 1) Schwarzschild metric => validate Schwarzschild correction
TCG
2) (Schwarzschild + GRIM4-S4) metric => validate harmonic contributions
GCRS (“inertial”)
4) GCRS metric with(out) Sun, Moon, Planets => validate geodetic precession (other bodies contributions) (…)
RMI goes beyond GINS capabilities:
INTEGRATOR
- includes 1) IAU 2000 standard GCRS metric [6,7]
2) IAU 2000 time transformation prescriptions [6,7]
3) IAU 2000/IERS 2003 new standards on Earth rotation [5,6]
- separate modules allow to easily update for metric, potential model (EGM96)… prescriptions.
  TCG
METRIC MODEL
3) Kerr metric => validate Lens-Thirring correction
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).
GP
for x , v  in A and A
[5]
  0.58286072 


  1.02761036 
  0.34965593 


Figure 8
TAI  TT  TDB

Earth
rotation
model
Figure 7

TCG  

 dU

;U  

 d


IAU2000
GCRS metric
G
TCG  TDB
PLANET EPHEMERIS
DE403
for G in 



GCRS (“inertial”)
TDB
[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.
[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).