MAGE Mid-term review (23/09/04):

Scientific work in progress

« Integrating the motion of satellites in a consistent relativistic

framework »

S. Pireaux

* Financial support provided through theEuropean Community's Improving Human Potential Program

under contract RTN2-2002-00217, MAGE

Observatoire Midi-Pyrénées

Royal Observatory of Belgium

• S. Pireaux, JP. Barriot,

• P. Rosenblatt

Collaborators:

1. MOTIVATIONS: precise geophysics implies precise geodesy

Y

Z

X Planetary rotation model

(X,Y,Z) = planetary crust frame Planetary potential model

better use relativistic formalism directly

Errors in relativistic corrections, time or space transformations…

Mis-modeling in the planetary potential or the planetary rotation model

Satellite motion current description: Newton’s law + relativistic corrections + other forces

X

Y

Z

Satellite motion(X,Y,Z) = quasi inertial frame

Relativistic correctionson

measurements

2. THE CLASSICAL APPROACH: GINS

Newton’s 2nd law of motion with

UA Egrad

- acceleration due to the Earth gravitational potential;

- acceleration due to Earth tide potential due to the Sun and Moon, corrected for Love number frequencies, ellipticity and polar tide;

UTidesEarth

grad

- acceleration due to the ocean tide potential (single layer model);

UTidesOcean

grad

- acceleration due to change in satellite momentum owing to solar photon flux;

APressureRadiation

- acceleration due to satellite colliding with residual gas molecules (hyp: free molecular flux);

ADrag cAtmospheri

- acceleration due to gravitational interaction with Moon, Sun and planets;

A

E Bodies ngPerturbati

- acceleration due to gravitational relativistic effects;

A icRelativist

- acceleration induced by the redistribution of atmospheric masses (single layer model).

APressure cAtmospheri

LAGEOS

SEASAT

Laser GEOdymics SatelliteAims: - calculate station positions (1-3cm) - monitor tectonic-plate motion - measure Earth gravitational field - measure Earth rotationDesign: - spherical with laser reflectors - no onboard sensors/electronic - no attitude controlOrbit: 5858x5958km, i = 52.6°Mission: 1976, ~50 years (USA)

SEA SATelliteAims: -test oceanic sensors (to measure sea surface heights )Design:Orbit: 800km Mission: June-October 1978

Examples: a high-, or respectively low-altitude satellite…

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 10**-12 2 10**-7

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

Orders of magnitude [m/s²]…

High satellite Low satellite

a) Gravitational potential model for the Earth

LA

GE

OS

1

mSmCPGM

U lmlm

l

l

l

m

lm

l

EE

Esincos)(sin

XX

max

0 0

""body body 3rdcouplingMoon -J2E PB

n n

AAA with

and

b) Newtonian contributions from the Moon, Sun and Planets

26

0

m/s 10

0.34965593

02761036.1

58286072.0

XYZ

LA

GE

OS

1

33

body 3rd n

n

n

n

n

n X

X

XX

XXGMA

1

0

0

215 52

32

2

2

205

couplingMoon -J2

MoMo

Mo

Mo

E

Mo

Mo XXX

ZC

X

GMA

c) Relativistic corrections on the forcesAAAA

Precession Thirring-LensPrecession Sitter) (De GeodeticildSchwarzschR

LA

GE

OS

1

28

0

m/s 10

0.210319-

524321.4

187604.0

XYZ

VXVXVX

GM

Xc

GMEEA 4

4 2

32

Schw

VA

GPGP

2 ,

211

0

m/s 10

0.928

141.2

245.0

XYZ

LA

GE

OS

1

XVX

GM

cE

GP

322

3

VA

LTPLTP

2 ,

212

0

m/s 10

40.10

83.34

13.0

XYZ

LA

GE

OS

1

23

2

3

X

XXSS

Xc

G E

ELTP

TAI

J2000 (“inertial”)

INTEGRATOR

A

TAI

J2000 (“inertial”)

zYx

VVVZYX ,,;,,

ORBIT

PLANET EPHEMERIS

DE403

For in and

TDB

AE PB

AGP

EE vx ,

GRAVITATIONAL POTENTIALMODEL FOR EARTH

GRIM4-S4

Earth ro

tation m

odel

ITRS (non inertial)

TDBTTTAI

d) diagram: GINS

3. THE IDEA…

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

Advantages: - Well-proven method. - Might be sufficient for current application.

Classical approach: “Newton” + relativistic corrections for precise satellite dynamics and time measurements.

Alternative and pioneering effort: develop a satellite motion integrator in a pure relativistic framework.

Drawbacks: - To be adapted to the level of precision of data and to the adopted space-time transformations

Part. 3: RMI: Relativistic Motion Integrator (if only gravitational forces)

Part. 1: RELATIVISTIC TIME TRANSFORMATIONS

Part. 2: METRIC PRESCRIPTIONS

4. GENERAL STRUCTURE OF THIS RELATIVISTIC STUDY …

First developments for Earth satellites…

Then transpose this approach to others planets and missions: Mars, Mercury…

(SC)RMI: Semi-Classical RMI (if non-gravitational forces are present)

en cours

5. THE RELATIVISTIC APPROACH: (SC)RMI

2cVVG

and first integral

Need for symplectic integrator

classical limiti

i

X

W

dT

Xd

2

2 with W = GCRS generalized gravitational potential in metric

3 ,2 ,1 i

G

The geodesic equation of motion for the appropriate metric,

contains all needed gravitational relativistic effects.

with

d

dXV

VVd

dV

) , , ,T( ZYXcX

3 ,2 ,1 ,0

= Christoffel symbol associated to GCRS metric

= proper time

G

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.

... , ,

GCRS (“inertial”)

INTEGRATOR

TCG

GCRS (“inertial”)

ORBIT

VX ;

V

d

dV;

PLANET EPHEMERIS

DE403

for in

TDB

G

GRAVITATIONAL POTENTIALMODEL FOR EARTH

GRIM4-S4

Ear

thro

tati

onm

odel

METRIC MODEL

GIAU2000

GCRS metric

ITRS (non inertial)

TDBTCG

TCG

TCG

c) diagram: RMI

d) Including non gravitational forces

The generalized relativistic equation of motion includes non-gravitational forces

measuredby

accelerometers

classical limitii

i

KX

W

dT

Xd

2

2

with

d

dXV

VVd

dV

K quadri-”force”

c

V

c

VGK

classical limit j

j

ji

i

iX

XX

W

dT

XdK

2

2

2

The principle of accelerometers:

X

d

Xd

d

dX

d

dX

d

dXX

Xd

Xd

d

dX

d

dX

cGK

2

12

2

2

with evaluated at

for the CM of satellite

,G

K

X

difference between the two equations at first order in :

XX - test-mass, shielded from non-gravitational forces, at

X- satellite Center of Mass at

BIBLIOGRAPHY

[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

Relativistic time transformations

[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

[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.

Metric prescriptions

RMI