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* Financial support provided through the
European Community's Improving Human Potential Program
under contract RTN2-2002-00217, MAGE
MAGE Mid-term review (23/09/04):
Scientific work in progress
« Integrating the motion of satellites
in a consistent relativistic
framework »
S. Pireaux
Collaborators:
• S.
Pireaux, JP. Barriot,
Observatoire
Midi-Pyrénées
• P. Rosenblatt
Royal Observatory of Belgium
1. MOTIVATIONS:
precise geophysics implies precise geodesy
Satellite motion current description: Newton’s law
+
relativistic corrections
+
other forces
Relativistic corrections
Z
Z
on
measurements
Y
Y
X
(X,Y,Z) = planetary crust frame
Planetary potential model
Planetary rotation model
X
(X,Y,Z) = quasi inertial frame
Satellite motion
Errors in relativistic corrections, time or space transformations…
Mis-modeling in the planetary potential or the planetary rotation model
better use relativistic formalism directly
2. THE CLASSICAL APPROACH: GINS
Newton’s 2nd law of motion with
- acceleration due to the Earth gravitational potential;
- acceleration due to gravitational interaction with Moon, Sun and planets;
- acceleration due to satellite colliding with residual gas molecules (hyp: free molecular flux);
- acceleration due to change in satellite momentum owing to solar photon flux;
- acceleration due to Earth tide potential due to the Sun and Moon,
corrected for Love number frequencies, ellipticity and polar tide;
- acceleration due to the ocean tide potential (single layer model);
- acceleration due to gravitational relativistic effects;
- acceleration induced by the redistribution of atmospheric masses (single layer model).

A  gradU
E


 grad



A

Pertu rb atin g Bo d ies E
A
Atmo sp h eric Drag
Earth Tides
 grad

A
RadiationPressure



U

U
Ocean Tides

A
Relativistic

A
Atmo sp h eric Pressu re
Examples: a high-, or respectively low-altitude satellite…
LAGEOS
Laser GEOdymics Satellite
Aims: - calculate station positions (1-3cm)
- monitor tectonic-plate motion
- measure Earth gravitational field
- measure Earth rotation
Design: - spherical with laser reflectors
- no onboard sensors/electronic
- no attitude control
Orbit: 5858x5958km, i = 52.6°
Mission: 1976, ~50 years (USA)
SEASAT
SEA SATellite
Aims: -test oceanic sensors
(to measure sea surface heights )
Design:
Orbit: 800km
Mission: June-October 1978
Orders of magnitude [m/s²]…
Cause
LAGEOS 1
SEASAT
Earth monopole
2.8
7.9
Earth oblateness
1.0 10**-3
9.3 10 **-3
Low order geopotential harmonics (eg. l=2,m=2)
6.0 10**-6
5.4 10**-5
High order geopotential harmonics (eg.l=18,m=18) 6.9 10**-12
3.9 10**-7
Moon
2.1 10**-6
1.3 10**-6
Sun
9.6 10**-7
5.6 10**-7
Other planets (eg. Ve)
1.3 10**-10
7.3 10**-11
Indirect oblation (Moon-Earth)
1.4 10**-11
1.4 10**-11
General relativistic corrections
9.5 10**-10
4.9 10**-9
Atmospheric drag
3
2
Solar radiation pressure
3.2 10**-9
9.2 10**-8
Earth albedo pressure
3.4 10**-10
3.0 10**-8
Thermal emission
1.9 10**-12
1.9 10**-9
10**-12
10**-7
High satellite Low satellite
a) Gravitational potential model for the Earth
E
GM

X
LAGEOS 1
U 

lmax
E
l 0
l
m 0

 
X

E

 P (sin  )C cos m  S sin m 


l
lm
lm
lm
b) Newtonian contributions from the Moon, Sun and Planets



A A   A
PB  E
J2 - Moon coupling
body "n "
3rd body n

with
A
J2 - Moon coupling

3 GM

2 X
Mo
5
Mo

Z
5 C   5 
 X


and
2
20
Mo
2
E
A
3rd body n
LAGEOS 1
2
Mo

 1 X


Mo
 X X
 GM 
 X X

Mo
n
n
X 

X 
n
3
n
  0.58286072 


  1.02761036 
  0.34965593 


 0 
 
 2 X  0 
1  
 
10 m/s
6
XYZ0
3
n
2

c) Relativistic corrections on the forces
A A 
R

A
LAGEOS 1
Schw
GM


c X
2
E
3
 4GM


 V


 X
E
2


Schwarzschild
A


Geodetic(De Sitter) Precession
A
Lens-Thirring Precession

 
X  4 V  X V 





  0.187604 


  4.524321 
 - 0.210319 


10 m/s
8
XYZ0
2

A
LAGEOS 1
GP
2

GP
V
,


3 GM
 
 VX
2c X
E
GP
2
3
  0.245 


  2.141 
  0.928 



10 m/s
11
XYZ0
2

A 
2
LAGEOS 1
LTP
LTP
V
,

LTP


 

G
3S X X 
    S 



c X 
X

2
3
  0.13 


  34.83 
  40.10 


E
E
2
10 m/s
12
XYZ0
2
d) diagram: GINS
ORBIT
 X ,Y , Z ;V ,V ,V 
x
Y
z
GRAVITATIONAL POTENTIAL
MODEL FOR EARTH
TAI
GRIM4-S4
ITRS (non inertial)
J2000 (“inertial”)
INTEGRATOR

A
PLANET EPHEMERIS
TAI
DE403
J2000 (“inertial”)
TAI  TT  TDB
For x , v  in
E
E
A andA
GP
TDB

PB E
3. THE IDEA…
 Classical approach: “Newton” + relativistic corrections for
precise satellite dynamics and time measurements.
 Advantages: - Well-proven method.
- Might be sufficient for current application.
 Drawbacks: - To be adapted to the level of precision of data
and to the adopted space-time transformations
 Alternative and pioneering effort: develop a satellite motion
integrator in a pure relativistic framework.
 Advantages: - To easily take into account all relativistic effects
with “metric” adapted to the precision of measurements
and adopted conventions.
- Same geodesic equation for photons (light signals)
massive particles (satellites without non-grav forces)
- Relativistically consistent approach
4. GENERAL STRUCTURE OF
THIS RELATIVISTIC STUDY …
First developments for Earth satellites…

Part. 1: RELATIVISTIC TIME TRANSFORMATIONS

Part. 2: METRIC PRESCRIPTIONS
en cours Part. 3:
RMI: Relativistic Motion Integrator (if only gravitational forces)
(SC)RMI: Semi-Classical RMI (if non-gravitational forces are present)
Then transpose this approach to others planets and missions:
Mars, Mercury…
5. THE RELATIVISTIC APPROACH: (SC)RMI
The geodesic equation of motion for the appropriate metric,
contains all needed gravitational relativistic effects.
 dV






V
V

 d


with
V   dX

d X  (cT, X , Y , Z )
Need for
symplectic integrator

and first integral

GV V   c
  0, 1, 2, 3
 = proper time
 = Christoffel symbol associated to GCRS metric G


d X
W

dT
X
2
classical limit

i
2
i
with
i  1, 2, 3
W = GCRS generalized
gravitational potential
in metric G
2
a) Method: GINS provides template orbits to validate the RMI orbits
- simulations with 1) Schwarzschild metric => validate Schwarzschild correction
2) (Schwarzschild + GRIM4-S4) metric => validate harmonic contributions
3) Kerr metric => validate Lens-Thirring correction
4) GCRS metric with(out) Sun, Moon, Planets => validate geodetic precession
(other bodies contributions)
(…)
b) RMI goes beyond GINS capabilities:
- (will) includes
1) IAU 2000 standard GCRS metric
2) IAU 2000 time transformation prescriptions
3) IAU 2000/IERS 2003 new standards on Earth rotation
4) (post)-post-Newtonian parameters ( ,  ,  ...) in metric and space-time transfo
- separate modules allow easy update for metric, Earth potential model (EGM96)… prescriptions
- contains all relativistic effects, different couplings at corresponding metric order.
c) diagram: RMI
GRAVITATIONAL POTENTIAL
MODEL FOR EARTH
GRIM4-S4
ORBIT
;V

ITRS (non inertial)

TCG
GCRS (“inertial”)
  TCG
Earth
rotation
model
X

METRIC MODEL
IAU2000
GCRS metric G

INTEGRATOR
TCG  

 dV

;V  

 d


GCRS (“inertial”)
TCG  TDB
PLANET EPHEMERIS
DE403
for G in 

TDB


d) Including non gravitational forces
The generalized relativistic equation of motion
includes non-gravitational forces
  V V 
 dV



 K  G 



V
V


 d
c c 



dX
with
V  

K  quadri-”force”
d



measured
by
accelerometers
d X
W

K
dT
X
2
classical limit
i
2
i
i

The principle of accelerometers:

- satellite Center of Mass at X
- test-mass, shielded from non-gravitational forces, at X  X


difference between the two equations at first order in X :





 
dX 
dX  dX 
  1 dX dX  d X
 dX

 K G 


X
 2  


c d d 
d
X
d d
d d

2
2
2
G  ,   
with
K
d X
K 

dT
2
classical limit
i
i
2

j
W
X
X X
2
i
j
j
evaluated at
X
for the CM of satellite
BIBLIOGRAPHY
Relativistic time transformations
[Bize et al 1999] Europhysics Letters C, 45, 558
[Chovitz 1988]
Bulletin Géodésique, 62,359
[Fairhaid_Bretagnon 1990] Astronomy and Astrophysics, 229, 240-247
[Hirayama et al 1988]
****
[IAU 1992]
IAU 1991 resolutions. IAU Information Bulletin 67
[IAU 2001a]
IAU 2000 resolutions. IAU Information Bulletin 88
[IAU 2001b]
Erratum on resolution B1.3. Information Bulletin 89
[IAU 2003]
IAU Division 1, ICRS Working Group Task 5: SOFA libraries.
http://www.iau-sofa.rl.ac.uk/product.html
[IERS 2003]
IERS website.
http://www.iers.org/map
[Irwin-Fukushima 1999]
Astronomy and Astrophysics, 348, 642-652
[Lemonde et al 2001]
Ed. A.N.Luiten, Berlin (Springer)
[Moyer 1981a]
Celestial Mechanics, 23, 33-56
[Moyer 1981b]
Celestial Mechanics, 23, 57-68
[Moyer 2000]
Monograph 2: Deep Space Communication and Navigation series
[Soffel et al 2003] prepared for the Astronomical Journal, asro-ph/0303376v1
[Standish 1998]
Astronomy and Astrophysics, 336, 381-384
[Weyers et al 2001]
Metrologia A, 38, 4, 343
Metric prescriptions
[Damour et al 1991] Physical Review D, 43, 10, 3273-3307
[Damour et al 1992] Physical Review D, 45, 4, 1017-1044
[Damour et al 1993] Physical Review D, 47, 8, 3124-3135
[Damour et al 1994] Physical Review D, 49, 2, 618-635
[IAU 1992]
IAU 1991 resolutions. IAU Information Bulletin 67
[IAU 2001a]
IAU 2000 resolutions. IAU Information Bulletin 88
[IAU 2001b]
Erratum on resolution B1.3. Information Bulletin 89
[IAU 2003]
IAU Division 1, ICRS Working Group Task 5: SOFA libraries.
http://www.iau-sofa.rl.ac.uk/product.html
[IERS 2003]
IERS website.
http://www.iers.org/map
[Klioner 1996] International Astronomical Union, 172, 39K, 309-320
[Klioner et al 1993] Physical Review D, 48, 4, 1451-1461
[Klioner et al 2003] astro-ph/0303377 v1
[Soffel et al 2003] prepared for the Astronomical Journal, asro-ph/0303376v1
RMI
[GRGS 2001]
Descriptif modèle de forces: logiciel GINS
[Moisson 2000]
(thèse). Observatoire de Paris
[McCarthy Petit 2003]
IERS conventions 2003
http://maia.usno.navy.mil/conv2000.html.