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, 21st August 2006

Outline of the speachI. 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)]

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:

APressureRadiation

UA Egrad

UTidesEarth

grad

UTidesOcean

grad

ADrag cAtmospheri

A

E Bodies ngPerturbati

A icRelativist

A

Pressure cAtmospheri

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:

d

dXV

VVd

dV

VVGK ) , , ,T( ZYXX

with

3 ,2 ,1 ,0,,

K quadri-”force”

= Christoffel symbol wrt GCRS metric

= proper time

G

2cVVG

and first integral

IIa. Need for relativistic laser links:

2008

-201

2

GALILEO

Project: CNES, ESA, CE

Implied: GEMINI/ OCA

Goals: positioning, …

2014

-202

0

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

…

• 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 tij) 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…

gravitationalrelativistic effects

L (t)ij

of stations ? Coordinates Interdistance (L ) ij

• planets present

• 5 million km interdistance

5 x 10 km6

• at 20° behind

1 A

U

20°

geodesic motion

classical doppler,Sagnac effect…

60°

• rotation of

Photon travel time (tij) ?

station1

station 2 station 3 • double laser links

• relativistic modeling of orbitography/laser links required:

000 , , , , BBBBAABBph vxtxxntx

• Equation to be solved in terms of quantities at tA:

Photon orbit Receiving station orbit

(flight time, « direction ») = 1 + 2 (normalization) = 3 unknowns , ABABAB nttt

• Laser link:

A, tA = 0Emission:

tB = ?B,Reception:

photon? ABn

IIb. General method for relativistic laser-links

• Motion in background metric gh

in presence of gravitational sources (sce) :

1

2 1 22

42

dtcc

Ocr

GM

i

i

isce

k

i

kii

isce

dxcdtc

Ocr

VGM

1

4

53

22242

1

2 1 dzdydx

cO

cr

GM

i

i

isce

… with IAU2000 conventions 2 dxdxgds

... 1

1

1

1

/

432

2

cO

cO

cOdt

cdsd Proper- vs coordinate-time rates:

Proper vs coordinate time: ... t

Energy measured from spacecraft = ukgE

0// uudtdxv iii = spacecraft 4-velocity ddxu /

= photon 4-wave vector ddxk /

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)

1

AA

ABABBAAB tE

tttEtz

Order 1 :

terms in

Central body : presence, shape, orbital motion (during photon travel time)

Other bodies : presence, orbital motion

orbital motion:

2

2 ~

cr

GM

r

2 ~

2 sce

22volδtV

cr

GM

cr

GM

Order 2 : terms in

2

2 2 ~

cr

GM

Order 3/2 :

terms in

Central body: rotation, orbital motion

Other bodies: orbital motion

with = 1 for photons, for satellites

.

V

2 2 ~ sce2 dtc

dxccr

GM

dtc

dx

. c/V ~ sat

• Contributions from gravitational sources (sce) to h:

... (2)(3/2)(1)

hhhh

~ 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

bodies)(other (1)(Sun) )1(

hh

evaluated at tA

2/312/10

Otttt ABABABAB

c

Lt AB

AB0

0

000 ABAB xxL

order 0 : where (+ sign : photon travels from A to B)

c

vntt BAB

AB0

002/1

0

000

AB

ABAB

L

xxn

order 1/2 : where

2

020

00

.

.

AABA

ABAABA

xnr

nxnxP

00

00 .

. ln ,

rxn

rtcxnnPrrnt

AAB

AABABAB

2

0

0

2

20

01

2

1

c

vn

c

vtt BABB

AB

20

30

000

3

0

2

, 1 t

cr

xGMnt

c

GMn

B

BABAB

order 1 :

where

Classical

Classical kinematic terms

Kinematic terms Shapiro delay Velocity changeduring photon

flight time

• LISA Flight time solution:

• Numerical estimates of geometric time delays in s over a year

tAB order 0 : amplitude ~ 48 000 km/c

« flexing » of triangle

tAB = LAB/c0

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

tAB order 0 : « flexing » of triangle, amplitude ~ 48 000 km/c ;

tAB order 1/2 : amplitude ~ 960 km/c ;

Doppler

tAB = fct [ nAB , vB(tA)/c ]1/2

t23-t32… tAB is not symmetric (Sagnac+aberration term)1/2 1/2 1/2

• Numerical estimates of geometric time delays in s over a year

tAB order 0 : « flexing » of triangle, amplitude ~ 48 000 km/c ;

tAB order 1/2 : spacecraft Doppler, amplitude ~ 960 km/c ;

tAB order 1 : less than 30 m/c.

relativistic gravitational Einstein, Doppler, Shapiro effects

tAB = fct[ tAB , nAB , vB(tA)/c, GM/c², xA(tA), xB(tA) ]

1 0

810 7~

LISA configuration (spacecraft orbits: circular about CM

+velocity proportional to orbital radius)

=> (reduction factor ~ L/R)

)( 22/312/1

Ozzzz ABABABAB

? 10 2 6

? 10 2 10

?10 2 14

60)10(

~ 3

cos ~ 8 n

nAB

n

rLz

Naive estimate:

Order 1/2:c

vvnz AB 0002/1

.

Kinematic terms(Doppler)

• LISA Frequency shift solution:

free fall + LISA configuration (~ 60°) => compensation

4

30

2 .

2

1

r

LOvvnL

dt

d

c ABAB

1310 2~

L<<R=> compensation (reduction factor ~ L/R)

4

3

3

220

2 2

. 31

r

LO

r

L

r

xn

c

GM

BB

BAB

1210 6~

Einstein effectVelocity changeduring photon

flight time

Kinematic terms

00

230

0

002

00

2

0001 11 .

2

1 .

ABB

BABABAB

rrc

GM

r

xn

c

tGM

c

vv

c

vvnz

1010 2~ 1010 2~ 1510 2~ 1210 6~

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 simulationsTests of TDI data pre-processing, TDI-rangingsensitivity curvesrelevant 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

AemBrecAB ttt t

B

Brect

Brec

B

tA

Aemt

Ae

A

m

Proper time of

satellite A(physical

scale)

ASatellite A

regularly archives values of

AC

,,C

A

CA

and

321with

Satellite B regularly archives values of

BC

,,C

B

CB

and

321with

IIIa. Time scales

d/dt -1A

– t (s)ANumerical estimates

over a one year mission…

– t (s) linear trend removedA

IIIb. Illustration: LISA

Outline of the speachI. 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)]

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 correctionson

measurements

Geodesy: precise geophysics implies precise geodesy

LAGEOS 1 Laser GEOdymics Satellite 1Aims: - calculate station positions (1-3cm) - monitor tectonic-plate motion - measure Earth gravitational field - measure Earth rotationDesign: - spherical with laser reflectors - no onboard sensors/electronic - no attitude controlOrbit: 5858x5958km, i = 52.6°, around EarthMission: 1976, ~50 years (USA)

CHAllenging Minisatellite PayloadAims: - precise gravity and magnetic field, their space and time variationsDesign: - laser reflector, GPS receiver - drift meter - magnetometer, star sensor, accelerometersOrbit: 454 km initial, near polar, around EarthMission: ~5 years (Germany)

CHAMP

Geodesy examples: a high-, or respectively low-altitude satellite…

Cause LAGEOS 1 CHAMP

Earth monopole 2.8 8.6

Earth oblateness 1.0 10**-3 1.1 10 **-2

Low order geopotential harmonics (eg. l=2,m=2) 6.0 10**-6 6.4 10**-5

High order geopotential harmonics (eg.l=18,m=18) 6.9 10**-12 9.4 10**-7

Moon 2.1 10**-6 7.9 10**-7

Sun 9.6 10**-7 2.7 10**-7

Other planets (eg. Ve) 1.3 10**-10 9.8 10**-13

Indirect oblation (Moon-Earth) 1.4 10**-11 1.4 10**-11

General relativistic corrections (total) 9.5 10**-10 1.7 10**-8

Atmospheric drag 3 10**-12 3.5 10**-7

Solar radiation pressure 3.2 10**-9 3.2 10**-8

Earth albedo pressure 3.4 10**-10 3.3 10**-9

Thermal emission 1.9 10**-12 8.3 10**-9

High satellite Low satellite

Geodesy: orders of magnitude [m/s²]

a) Gravitational potential model for the Earth

LA

GE

OS

1

mSmCPGM

U lmlm

l

l

l

m

lm

l

EE

Esincos)(sin

XX

max

0 0

""body body 3rdcouplingMoon -J2E PB

n n

AAA with

and

b) Newtonian contributions from the Moon, Sun and Planets

26

0

m/s 10

0.34965593

02761036.1

58286072.0

XYZ

LA

GE

OS

1

33

body 3rd n

n

n

n

n

n X

X

XX

XXGMA

1

0

0

215 52

32

2

2

205

couplingMoon -J2

MoMo

Mo

Mo

E

Mo

Mo XXX

ZC

X

GMA

c) Relativistic correctionsAAAA

Precession Thirring-LensPrecession Sitter) (De GeodeticildSchwarzschR

L

AG

EO

S 1

28

0

m/s 10

0.210319-

524321.4

187604.0

XYZ

VXVXVX

GM

Xc

GMEEA 4

4 2

32

Schw

VA

GPGP

2 ,

211

0

m/s 10

0.928

141.2

245.0

XYZ

LA

GE

OS

1

XVX

GM

cE

GP

322

3

VA

LTPLTP

2 ,

212

0

m/s 10

40.10

83.34

13.0

XYZ

LA

GE

OS

1

23

2

3

X

XXSS

Xc

G E

ELTP

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

i

i

i

i

VdT

dXA

dT

Xd ,

2

2

PLANET EPHEMERIS

DE403

For in and

TDB

AE PB

AGP

EE vx ,

Earth ro

tation m

odel

GRAVITATIONAL POTENTIALMODEL FOR EARTH

GRIM4-S4

ITRS (non inertial)

TDBTTTAI

c) diagram: GINS

TAI

J2000 (“inertial”)

ORBIT

i

i

i VdT

dXX

,

with i=1,2,3 spatial indices

Ear

thro

tati

onm

odel

PLANET EPHEMERIS

DE403

for in

TDB

G

GCRS (“inertial”)

INTEGRATOR

d

dX

d

Xd ;

2

2

METRIC MODEL

GIAU2000

GCRS metric

GRAVITATIONAL POTENTIALMODEL FOR EARTH

GRIM4-S4

ITRS (non inertial)

TDBTCG

TCG

TCG

d) diagram: RMI

TCG

GCRS (“inertial”)

ORBIT

d

dXX

;

with =0,1,2,3 space-time indices

classical limit j

j

ji

i

iX

XX

W

dT

XdK

2

2

2

with evaluated at

for the CM of satellite

,G

K

X

X

d

Xd

d

dX

d

dX

d

dXX

Xd

Xd

d

dX

d

dX

cGK

2

12

2

2

difference between the two equations at first order in :

XX - test-mass, shielded from non-gravitational forces, at (geodesic eq.)

X- satellite Center of Mass at (generalized relativistic eq.)

Geodesy: principle of accelerometers…

[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

Relativistic time transformations

Geodesy: bibliography

[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.

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 (TA, Tsat, TC)

– Compute ground-satellite delay:

satcalibCsatAatmosphCsatAicrelativistCsatAAC

A TdTdTdTTT

T -22

2

2

-

• Date laser pulses:

– Departure from ground station: TA

– Arrival aboard: Tsat= TB

– Echo return on ground: TC

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

106.2 2113

x

Angular distance measurement = 2 10-9 rd

TIPO Telescope

TIPO (Télémétrie Interplanétaire Optique)

Scientific objectives of TIPO:

Method:

sce

sce orb.22

r

R 2

2 ~

2

T

δt

cr

GM

cr

GM vol

r

R sce orb.with ~ 1 for planets, << 1 for Sun

.

5 x 10 km6

Rorb. sce

r

Orbital motion of sces during photon flight time:

... (2)(3/2)(1)

HHHH

bodies)(other (1)(Earth) )1(

HH

Earth) sph.-(non (1)Earth) (sph. )1(

HH

Earth rotation:

orbital motion of sces : Sun

Moon

Jupiter

2

2

3 r

R4

c

GM

) ou 1( 4 sce3 r

Rr

VcGM

~ 10-15

~ 10-15

~ 10-18

~ 10-19

Sun Moon JupiterrR

rR

MmH

T ~

~ 10-15

~ 10-11

~ 10-13

~ 10-15

~ 10-12

~ 2H ~ 10-18

T2L2, rotation around the Earth:

~ 10-9

s vol photon:

0.1 s~ 10-10

)( .

2 22

Jcr

GM

,...)( .

2 32

Jcr

GM2

2 ~

cr

GM

Collaborations in LISA FRANCELISA 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