1. 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 the European Community's Improving Human Potential Program under contract RTN2-2002-00217, MAGE

2. Observatoire Midi-Pyrénées Royal Observatory of Belgium S. Pireaux, JP. Barriot, P. Rosenblatt Collaborators:

3. 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 corrections on measurements

4. 2. THE CLASSICAL APPROACH: GINS Newton’s 2nd law of motion with - 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; - acceleration due to the ocean tide potential (single layer model); - acceleration due to change in satellite momentum owing to solar photon flux; - acceleration due to satellite colliding with residual gas molecules (hyp: free molecular flux); - acceleration due to gravitational interaction with Moon, Sun and planets; - acceleration due to gravitational relativistic effects; - acceleration induced by the redistribution of atmospheric masses (single layer model).

5. LAGEOS SEASAT 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) SEA SATellite Aims: -test oceanic sensors (to measure sea surface heights ) Design: Orbit: 800km Mission: June-October 1978 Examples: a high-, or respectively low-altitude satellite…

6. Cause LAGEOS 1SEASAT Earth monopole2.87.9 Earth oblateness1.0 10**-39.3 10 **-3 Low order geopotential harmonics (eg. l=2,m=2) 6.0 10**-65.4 10**-5 High order geopotential harmonics (eg.l=18,m=18) 6.9 10**-123.9 10**-7 Moon2.1 10**-61.3 10**-6 Sun9.6 10**-75.6 10**-7 Other planets (eg. Ve) 1.3 10**-107.3 10**-11 Indirect oblation (Moon-Earth)1.4 10**-11 General relativistic corrections9.5 10**-104.9 10**-9 Atmospheric drag3 10**-122 10**-7 Solar radiation pressure3.2 10**-99.2 10**-8 Earth albedo pressure3.4 10**-103.0 10**-8 Thermal emission1.9 10**-121.9 10**-9 Orders of magnitude [m/s²]… High satelliteLow satellite

7. a) Gravitational potential model for the Earth LAGEOS 1

8. with and b) Newtonian contributions from the Moon, Sun and Planets LAGEOS 1

9. c) Relativistic corrections on the forces LAGEOS 1

10. ,

11. ,

12. TAI J2000 (“inertial”) INTEGRATOR TAI J2000 (“inertial”) ORBIT PLANET EPHEMERIS DE403 For in and TDB GRAVITATIONAL POTENTIAL MODEL FOR EARTH GRIM4-S4 Earth rotation model ITRS (non inertial) d) diagram: GINS

13. 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

14. 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

15. 5. THE RELATIVISTIC APPROACH: (SC)RMI and first integral Need for symplectic integrator classical limit with W = GCRS generalized gravitational potential in metric The geodesic equation of motion for the appropriate metric, contains all needed gravitational relativistic effects. with = Christoffel symbol associated to GCRS metric = proper time

16. 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.

17. GCRS (“inertial”) INTEGRATOR TCG GCRS (“inertial”) ORBIT PLANET EPHEMERIS DE403 for in TDB GRAVITATIONAL POTENTIAL MODEL FOR EARTH GRIM4-S4 Earth rotation model METRIC MODEL IAU2000 GCRS metric ITRS (non inertial) c) diagram: RMI

18. d) Including non gravitational forces The generalized relativistic equation of motion includes non-gravitational forces measured by accelerometers classical limit with quadri-”force”

19. classical limit The principle of accelerometers: with evaluated at for the CM of satellite difference between the two equations at first order in : - test-mass, shielded from non-gravitational forces, at - satellite Center of Mass at

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