S. Pireaux

1. MAGE Mid-term review (23/09/04):Scientific work in progress« Integrating the motion of satellites in a consistent relativistic framework » * Financial support provided through the European Community's Improving Human Potential Program under contract RTN2-2002-00217, MAGE S. Pireaux

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

3. Satellite motion current description: Newton’s law + relativistic corrections + other forces Z Z Relativistic corrections on measurements Y (X,Y,Z) = planetary crust frame Planetary potential model Y Planetary rotation model X 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 1. MOTIVATIONS: precise geophysics implies precise geodesy

4. - 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 the ocean tide potential (single layer model); - acceleration due to gravitational relativistic effects; - acceleration induced by the redistribution of atmospheric masses (single layer model). - acceleration due to Earth tide potential due to the Sun and Moon, corrected for Love number frequencies, ellipticity and polar tide; 2. THE CLASSICAL APPROACH: GINS Newton’s 2nd law of motion with

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

6. High satellite Low satellite Orders of magnitude [m/s²]…

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

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

9. c) Relativistic corrections on the forces LAGEOS 1

10. , LAGEOS 1

11. , LAGEOS 1

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

13. 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

14. (SC)RMI: Semi-Classical RMI (if non-gravitational forces are present) 4. GENERAL STRUCTURE OF THIS RELATIVISTIC STUDY … First developments for Earth satellites… en cours Part. 1: RELATIVISTIC TIME TRANSFORMATIONS Part. 2: METRIC PRESCRIPTIONS Part. 3: RMI: Relativistic Motion Integrator (if only gravitational forces) Then transpose this approach to others planets and missions: Mars, Mercury…

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

16. 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. 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) (…)

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

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

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

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