Relativity and Space Geodesy S. Pireaux UMR 6162 ARTEMIS, Obs. de la Côte d’Azur, Av. de Copernic, Grasse, France IAU Commission 31: TIME AND ASTRONOMY, IAU General Assembly, Prague, 21 st August 2006

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, (2005)]

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 ‘’

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

IIa. Need for relativistic laser links: GALILEO Project: CNES, ESA, CE Implied: GEMINI/ OCA Goals: positioning, … 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. … T2L Implied: GEMINI, ARTEMIS, through SIR ILIADE of OCA Goal: metrology, planetodesy, … TIPO …

LISA = space GW detector complementary to ground detectors LISA (Laser Interferometer Space Antenna) good precision required on arm length:  L/L ~ 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

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:

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

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

 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)

 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  :

~ Sun rotation: Orbital motion of sces: Sun Jupiter Venus ~ ~ ~ (<<) ~ Presence: Orbital motion: ~ ~ Presence: Orbital motion: ~ ~ m s ~ ~ 50 Photon flight: km Orders of magnitude : IIc. Illustration: LISA, rotation around the Sun

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:

Numerical estimates of geometric time delays in s over a year  t AB order 0 : amplitude ~ 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

Numerical estimates of geometric time delays in s over a year  t AB order 0 : « flexing » of triangle, amplitude ~ 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

Numerical estimates of geometric time delays in s over a year  t AB order 0 : « flexing » of triangle, amplitude ~ 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

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:

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

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

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

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

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, (2005)]

Other transparencies

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

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…

Cause LAGEOS 1CHAMP Earth monopole Earth oblateness1.0 10** **-2 Low order geopotential harmonics (eg. l=2,m=2) ** **-5 High order geopotential harmonics (eg.l=18,m=18) ** **-7 Moon2.1 10** **-7 Sun9.6 10** **-7 Other planets (eg. Ve) ** **-13 Indirect oblation (Moon-Earth)1.4 10**-11 General relativistic corrections (total)9.5 10** **-8 Atmospheric drag3 10** **-7 Solar radiation pressure3.2 10** **-8 Earth albedo pressure3.4 10** **-9 Thermal emission1.9 10** **-9 High satelliteLow satellite Geodesy: orders of magnitude [m/s²]

a) Gravitational potential model for the Earth LAGEOS 1

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

c) Relativistic corrections LAGEOS 1

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 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…

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.

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

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

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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[Damour et al 1991] Physical Review D, 43, 10, [Damour et al 1992] Physical Review D, 45, 4, [Damour et al 1993] Physical Review D, 47, 8, [Damour et al 1994] Physical Review D, 49, 2, [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. [IERS 2003] IERS website. [Klioner 1996] International Astronomical Union, 172, 39K, [Klioner et al 1993] Physical Review D, 48, 4, [Klioner et al 2003] astro-ph/ v1 [Soffel et al 2003] prepared for the Astronomical Journal, asro-ph/ v1 [GRGS 2001] Descriptif modèle de forces: logiciel GINS [Moisson 2000] (thèse). Observatoire de Paris [McCarthy Petit 2003] IERS conventions Metric prescriptions RMI

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

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:

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

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

Earth rotation: orbital motion of sces : Sun Moon Jupiter ~ ~ ~ Sun Moon Jupiter ~ ~ ~ ~ ~ ~ T2L2, rotation around the Earth: ~ s vol photon: 0.1 s ~

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