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Relativity and Space Geodesy S. Pireaux UMR 6162 ARTEMIS , Obs. de la Côte d’Azur, Av. de Copernic, 06130 Grasse, Fr

Relativity and Space Geodesy S. Pireaux UMR 6162 ARTEMIS , Obs. de la Côte d’Azur, Av. de Copernic, 06130 Grasse, France sophie.pireaux@obs-azur.fr. IAU Commission 31: TIME AND ASTRONOMY, IAU General Assembly, Prague, 21 st August 2006. Outline of the speach.

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Relativity and Space Geodesy S. Pireaux UMR 6162 ARTEMIS , Obs. de la Côte d’Azur, Av. de Copernic, 06130 Grasse, Fr

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  1. Relativity and Space Geodesy S. Pireaux UMR 6162 ARTEMIS, Obs. de la Côte d’Azur, Av. de Copernic, 06130 Grasse, Francesophie.pireaux@obs-azur.fr IAU Commission 31: TIME AND ASTRONOMY, IAU General Assembly, Prague, 21st August 2006

  2. Outline of the speach [ Pireaux, Barriot, Rosenblatt, Acta A 2005] [ Pireaux et Barriot, Cel. Meca en prépa] II. Native relativistic approach wrt photon trajectory: laser-links (time transfer, frequency shift) a. Needed in: LISA, Tippo, T2L2, Galileo … b. General method for relativistic laser-links c. Illustration: LISA [B. Chauvineau, T. Régimbau, J.-Y. Vinet, S. Pireaux, Phys. Rev. D 72, 122003 (2005)] III. Caution with relativistic time-scales a. Relativistic time-scales b. Illustration: LISA I. Native relativistic approach wrt spacecraft trajectory : orbitography a. Needed in: precise planetary gravitational field modeling, orbitography b. Illustration: classical vsRMI prototype –Relativistic Motion Integrator- method

  3. I. Native relativistic approach wrt spacecraft trajectory : orbitography precise orbitography good model of perturbations relativistic gravitation: • Schwarzschild precession • geodesic ‘’ • Lense-Thirring‘’ • Include IAU 2000 standards regarding General Relativity: • GCRS metric- time transformation- Earth rotation- … Ia. Needed in: - precise planetary gravitational field modeling - orbitography CHAMP GRACE • A good planetary gravitational field model? GOCE STELLA or LAGEOS

  4. Ib. Illustration: classical method: numericaly integrate Newton’s second law of motion: RMI (Relativistic Motion Integrator) prototype method: numericaly integrate relativistic equation of motion (for a given metric): quadri-”force” with = proper time = Christoffel symbol wrt GCRS metric and first integral Simplectic integrator

  5. II. Native relativistic approach wrt photon trajectory: laser-links … 2008 2014-2020 2008-2012 T2L2 TIPO Project:CNES, ESA, NASA Implied:LISAFrance Goals:Time Delay Interferom. Project: CNES, ESA, CE Implied: GEMINI/ OCA Goals: positioning, … LISA GALILEO Project: CNES Implied: GEMINI/OCA Goals: metrology, geodesy, clocks synchro. … Implied:GEMINI, ARTEMIS, through SIR ILIADE of OCA Goal: metrology, planetodesy, … IIa. Need for relativistic laser links:

  6. LISA (Laser Interferometer Space Antenna) • GW detection through measurement of phase shift due to DL • good precision required on arm length: DL/L ~ 10-23 • laser frequency noise and optical bench noise >>> GW signal TDI pre-processing of data required TDI observables = time-delayed (wrt photon flight time tij) combination of data fluxes from = laser links, in close loops, in order to cancel bench and frequency noise • LISA = space GW detector complementary to ground detectors

  7. 6 . 5 x 10 km • equilateral L (t) 1 AU 20° 60° ij • 5 million km interdistance • at 20° behind • rotation of • rotation around gravitational relativistic effects • planets present . Photon travel time (tij) ? • double laser links geodesic motion classical doppler, Sagnac effect… station 3 station 2 . • planets and present station1 light deflection… • relativistic modeling of orbitography/laser links required: Coordinates Interdistance (L ) of stations ? ij • 3 (drag-free) stations3 test masses

  8. Emission: tA = 0 A, photon Reception: B, tB = ? • Equation to be solved in terms of quantities at tA: Photon orbit Receiving station orbit (flight time, « direction ») = 1 + 2 (normalization) = 3 unknowns IIb. General method for relativistic laser-links • Laser link:

  9. Proper- vs coordinate-time rates: • Proper vs coordinate time: • Motion in background metric gab = hab + hab in presence of gravitational sources (sce) : … with IAU2000 conventions

  10. Frequency shift = = relative difference between (if transfer from A to B) • frequency of photon, emitted by A, measured when received at B • proper frequency of photon when emitted by A (= proper frequency of identical oscillators aboard A and B) • Energy measured from spacecraft = where = spacecraft 4-velocity = photon 4-wave vector

  11. Order 1 : • terms in • Central body : presence, shape, orbital motion (during photon travel time) •  Other bodies : presence, orbital motion • orbital motion: • Order 3/2 : • terms in • Central body: rotation, orbital motion •  Other bodies: orbital motion • with = 1 for photons, for satellites • Order 2 : terms in • Contributions from gravitational sources (sce) to hab :

  12. IIc. Illustration: LISA, rotation around the Sun ~ 10-16 Sun rotation: Orbital motion of sces: Sun Jupiter Venus (<<) ~ 10-13 ~ 10-15 ~ 2 . 10-16 ~ 10-17 Presence: Orbital motion: ~ 2 . 10-12 ~ 10-18 Presence: Orbital motion: ~ 10-8 ~ 50 m s Photon flight: 5 . 10+6 km ~ 2 . 10-16 ~ 2 . 10-7 • Orders of magnitude :

  13. order 0 : where (+ sign : photon travels from A to B) evaluated at tA Classical • order 1/2 : where Classical kinematic terms • order 1 : • where Shapiro delay Kinematic terms Velocity change during photon flight time • LISA Flight time solution:

  14. 0 tAB = LAB/c 4 month period (rotation D around its center of mass) 6 month period 1 au périhélie 1 à l’aphélie 1 year period (rotation around the Sun) • Numerical estimates of geometric time delays in s over a year • tABorder 0 : amplitude ~ 48 000 km/c • « flexing » of triangle

  15. Numerical estimates of geometric time delays in s over a year • tABorder 0 :« flexing » of triangle, amplitude ~ 48 000 km/c ; • tABorder 1/2 : amplitude ~ 960 km/c ; • Doppler 1/2 tAB = fct [ nAB , vB(tA)/c ] 1/2 1/2 1/2 t23-t32… tAB is not symmetric (Sagnac+aberration term)

  16. Numerical estimates of geometric time delays in s over a year • tABorder 0 :« flexing » of triangle, amplitude ~ 48 000 km/c ; • tABorder 1/2 :spacecraft Doppler, amplitude ~ 960 km/c ; • tABorder 1 : less than 30 m/c. 0 1 tAB = fct[ tAB , nAB , vB(tA)/c, GM/c², xA(tA), xB(tA) ] relativistic gravitational Einstein, Doppler, Shapiro effects

  17. Order 1/2: Kinematic terms (Doppler) LISA configuration (spacecraft orbits: circular about CM +velocity proportional to orbital radius) => (reduction factor ~ L/R) • LISA Frequency shift solution: • Naive estimate:

  18. Einstein effect Velocity change during photon flight time Kinematic terms L<<R=> compensation (reduction factor ~ L/R) free fall + LISA configuration (~ 60°) => compensation • Order 1:

  19. LISACODE • collaboration of ARTEMIS (Côte d’Azur) – APC (Paris),in LISA FRANCE • aims at • includes without planets mission simulations Tests of TDI data pre-processing, TDI-ranging sensitivity curves relevant order of magnitude estimates … relativistic laser links (time transfer + freq. shift) classical orbito. coordinate time only • Laser link :Sun alone sufficient, but relativistic description of its field necessary • Ephemeris of stations :presence of planets necessary, to provide initial conditions for photon flight times • Time scales: careful with archives and coherence

  20. III. Caution with relativistic time-scales Satellite A regularly archives values of B A Satellite B regularly archives values of t t t Barycentric coordinate time (artificial scale) Proper time ofsatellite B (physical scale) Proper time ofsatellite A (physical scale) IIIa. Time scales

  21. A A t – t (s) dt/dt -1 A t – t (s) linear trend removed IIIb. Illustration: LISA • Numerical estimates • over a one year mission…

  22. Outline of the speach [ Pireaux, Barriot, Rosenblatt, Acta A 2005] [ Pireaux et Barriot, Cel. Meca en prépa] II. Native relativistic approach wrt photon trajectory: laser-links (time transfer, frequency shift) a. Needed in: LISA, Tippo, T2L2, Galileo … b. General method for relativistic laser-links c. Illustration: LISA [B. Chauvineau, T. Régimbau, J.-Y. Vinet, S. Pireaux, Phys. Rev. D 72, 122003 (2005)] III. Caution with relativistic time-scales a. Relativistic time-scales b. Illustration: LISA I. Native relativistic approach wrt spacecraft trajectory : orbitography a. Needed in: precise planetary gravitational field modeling, orbitography b. Illustration: classical vsRMI prototype –Relativistic Motion Integrator- method

  23. Other transparencies

  24. 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 Geodesy: precise geophysics implies precise geodesy

  25. LAGEOS 1 CHAMP Geodesy examples: a high-, or respectively low-altitude satellite… Laser GEOdymics Satellite 1 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°, around Earth Mission: 1976, ~50 years (USA) CHAllenging Minisatellite Payload Aims: - precise gravity and magnetic field, their space and time variations Design: - laser reflector, GPS receiver - drift meter - magnetometer, star sensor, accelerometers Orbit: 454km initial, near polar, around Earth Mission: ~5 years (Germany)

  26. High satellite Low satellite Geodesy: orders of magnitude [m/s²]

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

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

  29. c) Relativistic corrections LAGEOS 1

  30. , LAGEOS 1

  31. , LAGEOS 1

  32. Geodesy: a modern view… • Classical approach:“Newton” + relativistic corrections for precise satellite dynamics and time measurements. • Advantages: - Well-proven method. - Might be sufficient for current applications. • Drawbacks: - To be adapted to the adopted space-time transformations and to the level of precision of data • 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

  33. 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-newtonian parameters in metric and time transformations - 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) (…)

  34. ORBIT TAI J2000 (“inertial”) GRAVITATIONAL POTENTIAL MODEL FOR EARTH GRIM4-S4 ITRS (non inertial) INTEGRATOR TAI J2000 (“inertial”) PLANET EPHEMERIS DE403 For in and TDB c) diagram: GINS Earth rotation model with i=1,2,3 spatial indices

  35. GRAVITATIONAL POTENTIAL MODEL FOR EARTH GRIM4-S4 ORBIT ITRS (non inertial) TCG GCRS (“inertial”) METRIC MODEL IAU2000 GCRS metric INTEGRATOR GCRS (“inertial”) PLANET EPHEMERIS DE403 for in TDB d) diagram: RMI Earth rotation model with a=0,1,2,3 space-time indices

  36. satellite Center of Mass at (generalized relativistic eq.) - test-mass, shielded from non-gravitational forces, at (geodesic eq.) difference between the two equations at first order in : with evaluated at for the CM of satellite classical limit Geodesy: principle of accelerometers…

  37. Geodesy: 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

  38. 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.

  39. T2L2 (optical telemetry with 2 laser links) Principle of ground-space time transfer: • Date laser pulses: • Departure from ground station: TA • Arrival aboard: Tsat= TB • Echo return on ground: TC Clock Retro-reflectors • Follow evolution of time aboard wrt ground time: • Rebuild triplets (TA, Tsat, TC) • Compute ground-satellite delay: Detection Clock Laser telemetry station

  40. Principle of ground-ground time transfer: Common view On-board oscillator noise sx(0.1 s) Non-Common view On-board oscillator noise sx(t3)

  41. TIPO (Télémétrie Interplanétaire Optique) Radial distance measurement : centimetric over 1 day Angular distance measurement dq = 2 10-9 rd Method: Scientific objectives of TIPO: • Mesure PPN parameter g (Shapiro effect) • Planet Telemetry • Asteroid masses • Pioneer effect • … TIPO Telescope

  42. 6 5 x 10 km with ~ 1 for planets, << 1 for Sun . r Rorb. sce Orbital motion of sces during photon flight time:

  43. ~ 10-18 Earth rotation: orbital motion of sces : Sun Moon Jupiter ~ 10-15 ~ 10-15 ~ 10-18 ~ 10-19 ~ 10-11 Sun Moon Jupiter ~ 10-13 ~ 10-15 vol photon: 0.1 s s ~ 10-10 T2L2, rotation around the Earth: ~ 10-12 ~ 10-9 ~ 10-15

  44. UMR ARTEMIS, OCA: - B. Chauvineau: gravitation relativiste - S. Pireaux: gravitation relativiste, théories alternatives - T. Régimbau: modélisation d'ondes gravitationnelles - fond stochastique- - J-Y. Vinet: Time-Delay Interferometry Collaborations in LISA FRANCE LISA France: - APC, Paris 7 - ARTEMIS, OCA - CNES - IAP Paris - LAPP Annecy - LUTH Observatoire de Paris-Meudon - ONERA - Service d'Astrophysique CEA

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