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