1. 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

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

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

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

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

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

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

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

9. Motion in background metric g     h  in presence of gravitational sources (sce) : … with IAU2000 conventions  Proper- vs coordinate-time rates:  Proper vs coordinate time:

10.  Energy measured from spacecraft = = spacecraft 4-velocity = photon 4-wave vector where  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)

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

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

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

14. Numerical estimates of geometric time delays in s over a year  t AB order 0 : amplitude ~ 48 000 km/c « flexing » of triangle t AB = L AB /c 0 1 year period (rotation around the Sun) 4 month period (rotation  around its center of mass) 1 au périhélie 1 à l’aphélie 6 month period

15. Numerical estimates of geometric time delays in s over a year  t AB order 0 : « flexing » of triangle, amplitude ~ 48 000 km/c ;  t AB order 1/2 : amplitude ~ 960 km/c ; Doppler t AB = fct [ n AB, v B (t A )/c ] 1/2 t 23 -t 32 … t AB is not symmetric (Sagnac+aberration term) 1/2

16. Numerical estimates of geometric time delays in s over a year  t AB order 0 : « flexing » of triangle, amplitude ~ 48 000 km/c ;  t AB order 1/2 : spacecraft Doppler, amplitude ~ 960 km/c ;  t AB order 1 : less than 30 m/c. relativistic gravitational Einstein, Doppler, Shapiro effects t AB = fct[ t AB, n AB, v B (t A )/c, GM/c², x A (t A ), x B (t A ) ] 1 0

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

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

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

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

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

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

23. Other transparencies

24. 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 Geodesy: precise geophysics implies precise geodesy

25. LAGEOS 1 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: 454 km initial, near polar, around Earth Mission: ~5 years (Germany) CHAMP Geodesy examples: a high-, or respectively low-altitude satellite…

26. Cause LAGEOS 1CHAMP Earth monopole2.88.6 Earth oblateness1.0 10**-31.1 10 **-2 Low order geopotential harmonics (eg. l=2,m=2) 6.0 10**-66.4 10**-5 High order geopotential harmonics (eg.l=18,m=18) 6.9 10**-129.4 10**-7 Moon2.1 10**-67.9 10**-7 Sun9.6 10**-72.7 10**-7 Other planets (eg. Ve) 1.3 10**-109.8 10**-13 Indirect oblation (Moon-Earth)1.4 10**-11 General relativistic corrections (total)9.5 10**-101.7 10**-8 Atmospheric drag3 10**-123.5 10**-7 Solar radiation pressure3.2 10**-93.2 10**-8 Earth albedo pressure3.4 10**-103.3 10**-9 Thermal emission1.9 10**-128.3 10**-9 High satelliteLow satellite Geodesy: orders of magnitude [m/s²]

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

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

29. c) Relativistic corrections LAGEOS 1

30. ,

31. ,

32.  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 applications.  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 adopted space-time transformations and to the level of precision of data Geodesy: a modern view…

33. 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-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.

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

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

36. classical limit 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 (geodesic eq.) - satellite Center of Mass at (generalized relativistic eq.) Geodesy: principle of accelerometers…

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38. [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.http://maia.usno.navy.mil/conv2000.html Metric prescriptions RMI

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

40. Common view On-board oscillator noise  x (0.1 s) Non-Common view On-board oscillator noise  x (  3 ) Principle of ground-ground time transfer:

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

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

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

44. 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 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