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Testing Relativity with Space Astrometry Missions

Sergei A.Klioner

Lohrmann-Observatorium, Technische Universität Dresden

SKA/LISA/Gaia workshop, Birmingham, 31 March 2006

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Accuracy of astrometric observations

1 mas

1 µas10 µas

100 µas

10 mas

100 mas

1“

10”

100”

1000”

1 µas10 µas

100 µas

1 mas

10 mas

100 mas

1”

10”

100”

1000”

1400 1500 1700 1900 2000 21000 1600 1800

Ulugh Beg

Wilhelm IVTycho Brahe

HeveliusFlamsteed

Bradley-Bessel

FK5

Hipparcos

Gaia

SIM

ICRF

GC

naked eye telescopes space

1400 1500 1700 1900 2000 21000 1600 1800

Hipparchus

4.5 orders of magnitude in 2000 years

further 4.5 orders in 20 years

1 as is the thickness of a sheet of paper seen from the other side of the Earth

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Relativity as a driving force for Gaia

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The IAU 2000 framework

• Three standard astronomical reference systems were defined

• BCRS (Barycentric Celestial Reference System)

• GCRS (Geocentric Celestial Reference System)

• Local reference system of an observer

• All these reference systems are defined by

the form of the corresponding metric tensors.

Technical details: Brumberg, Kopeikin, 1988-1992 Damour, Soffel, Xu, 1991-1994 Klioner, Voinov, 1993

Soffel, Klioner, Petit et al., 2003

BCRS

GCRS

Local RSof an observer

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Relativistic Astronomical Reference Systems

particular reference systems in the curved space-time of the Solar system

• One can use any

• but one should fix one

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Barycentric Celestial Reference System

• The BCRS is suitable to model processes in the whole solar system

200 2 4

0 3

2

2 21 ( , ) ( , ) ,

4( , ) ,

21 ( , ) .

ii

ij ij

g w t w tc c

g w tc

g w tc

x x

x

x

23 3 3

2 2

00 2 0

( , ) 1 ( , )( , ) ( , ) | | , ( , ) ,

| | 2 | |

/ , / , is the BCRS energy-momentum tensor

ii

kk i i

t tw t G d x G d x t w t G d x

c t

T T c T c T

x x

x x x x xx x x x

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Local Reference System of an Observer

The version of the GCRS for a massless observer:

The gravitational field of external bodies is represented only in the form of relativistic tidal potentials.

( , ) ( ) ( , ),

1( , ) ( ) ( , ).

2

aa T

a c aabc b T

W T Q T X W T

W T C T X W T

X X

X X

• the BCRS-induced tetrad is the local coordinate basis at the origin of that reference system…

• Modelling of any local phenomena: observation, attitude, local physics (if necessary)

2X

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General structure of the model

• s the observed direction • n tangential to the light ray

at the moment of observation• tangential to the light ray

at • k the coordinate direction

from the source to the observer• l the coordinate direction

from the barycentre to the source

• the parallax of the source in the BCRS

The model must be optimal:

t

observedrelated to the light raydefined in the BCRS coordinates

Klioner, Astron J, 2003; PhysRevD, 2004:

91 10 objects 30 years!s

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Current accuracies of relativistic tests

Several general-relativistic effects are confirmed with the following precisions:

• VLBI ± 0.0003

• HIPPARCOS ± 0.003

• Viking radar ranging ± 0.002

• Cassini radar ranging ± 0.000023

• Planetary radar ranging ± 0.0001

• Lunar laser ranging I ± 0.0005

• Lunar laser ranging II ± 0.007

Other tests:

• Ranging (Moon and planets)

• Pulsar timing: indirect evidence for gravitational radiation

14 -1/ 5 10 yrG G

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Why to test further?

Just an example…

• Damour, Nordtvedt, 1993-2003:

Scalar field (-1) can vary on cosmological time scales so that it asymptotically vanishes with time.

• Damour, Polyakov, Piazza, Veneziano, 1994-2003:

The same conclusion in the framework string theory and inflatory cosmology.

• Small deviations from general relativity are predicted for the present epoch:

5 81 4 10 5 10

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Gaia’s goals for testing relativity

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6

4

7

10

10

10

a lot more...

J

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Fundamental physics with Gaia

Global tests Local tests

Local Positional Invariance

Local Lorentz Invariance

Light deflection

One single

Four different ‘s

Differential solutions

Asteroids

Pattern matching

Perihelion precession

Non-Schwarzschild effects

SEP with the Trojans

Stability checks for

Alternative angular dependence

Non-radial deflection

Higher-order deflection

Improved ephemeris

SS acceleration

Primordial GW

Unknown deflector in the SS

Monopole

Quadrupole

Gravimagnetic

Consistency checks

J_2 of the Sun

/G G

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Necessary condition: consistency of the whole data processing chain

• Any kind of inconsistency is very dangerous for the quality and reliability of the estimates

• The whole data processing and all the auxiliary information should be assured to be compatible with the PPN formalism (or at least GR)

• planetary ephemeris: coordinates, scaling, constants• Gaia orbit: coordinates, scaling, constants• astronomical constants • ???

• Monitoring of the consistency during the whole project

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Example: consistency of the Gaia orbit

L2 X

Y

Z

Sun E

Z

Y

• Gaia have very tough requirements for the accuracy of its orbit:

1-2 mm/s in velocity

(this allows to compute aberration with an accuracy of 1 as)

• Example of the non-Schwarzschild relativistic effects for a Lissajous orbit the Lagrange point L2 over 200 days (km)

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Global vs. local tests

• It is natural to divide all tests into two groups:

• global tests

• are related to the global solution• should use the whole Gaia data or at least as much as possible

• local tests

• special additional solutions (e.g. differential or orbital ones)• relatively small amount of data

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Global test: gravitational red shift

• Depending on the final design and clock synchronization mode it could be possible to test the gravitational red shift of the on-board clock (Local Positional Invariance)

• Currently, the best accuracy for the red shift comes from the GP-A: 10 –4

(Vessot, 1979)

• Several dedicated and semi-dedicated missions were cancelled

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Global test: gravitational red shift

• The mean rate of the proper time on a Lissajous orbit is different from Terrestrial Time only

by 4 ×10 –12

• Cancellation: lower potential and larger velocity than on the Earth

• The gravity term is still 6 ×10 –10

• We could be sensitive to for the secular drift

• Still unclear if technically feasible…

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Global test: local Lorentz invariance

• Mansouri & Sexl (1977) suggested a test framework against which one can test special relativity

Robertson (1949) discussed similar ideas

• Lorentz transformations with additional numerical parameters

• Many experiments can be interpreted in terms of constrains on those parameters: e.g. Michelson-Morley and similar

• The idea is to use Gaia data to check if the special-relativistic formula for aberration is correct

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1/ 22 2

2

1( 1) ,

(1 / )

1 / ,

11 (1 ) ( , )o o

c v c

v c

x w tc

vv

v

v x

nns

nstandard Lorentztransformations}

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Global test: PPN from light deflection

• Several kinds of gravitational fields deflecting light at the 1 muas level

• monopole field• quadrupole field• gravitomagnetic field due to translational motion• gravitomagnetic field due to rotational motion

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Monopole gravitational light deflection

body (as) >1as

Sun 1.75 180

Mercury 83 9

Venus 493 4.5

Earth 574 125

Moon 26 5

Mars 116 25

Jupiter 16270 90

Saturn 5780 17

Uranus 2080 71

Neptune 2533 51

• Monopole light deflection: distribution over the sky on 25.01.2006 at 16:45 equatorial coordinates

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Monopole gravitational light deflection

body (as) >1as

Sun 1.75 180

Mercury 83 9

Venus 493 4.5

Earth 574 125

Moon 26 5

Mars 116 25

Jupiter 16270 90

Saturn 5780 17

Uranus 2080 71

Neptune 2533 51

• Monopole light deflection: distribution over the sky on 25.01.2006 at 16:45 equatorial coordinates

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Gravitational light deflection

• A body of mean density produces a light deflection not less than if its radius:

1/ 2 1/ 2

3650 km

1 g/cm 1μasR

Ganymede 35Titan 32Io 30Callisto 28Triton 20Europe 19

Pluto 7Charon 4Titania 3Oberon 3Iapetus 2Rea 2Dione 1Ariel 1Umbriel 1Ceres 1

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Global test: PPN from light deflection

• Most precise test possible with Gaia

Preliminary analysis: ESA, 2000; Mignard, 2001; Vecchiato et al., 2003:

610

• Advantages of the Gaia experiment

• optical, • deflection (not Shapiro),• wide range of angular distances,• full-scale simulations of the experiments

• Problems with some of the „current best estimates“ of

1. special fits of the post-fit residuals of a standard solution (missed correlations lead to wrong estimates of the uncertainty);

2. no special simulations with simulated data to check what kind of effects we are really sensitive to

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Global test: PPN from light deflection

• Specific Gaia-related problems in the test:

• Correlations:

• parallax zero point (90%)• special kinds of systematic errors in the velocity of the satellite• …

• Special care should be taken with the stability of the estimate:

• barely undetected binaries,• source structure and stability,• …

• A series of global deflection tests!

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Global test: PPN from light deflection

I. Main experiment: one single for all deflecting bodies.

• highest accuracy expected• other bodies (Jupiter) de-correlate and parallaxes

II. Individual for each deflecting body (at least: Sun, Jupiter, Earth)

Jupiter

Earth

Saturn

…important since this can be interpreted in terms of Equivalence Principle

3 410 10

210

310

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Global test: PPN from light deflection

III. Stability check: dependence of on various parameters

• data divided into several time spans• linear drift in (equivalent to linear drifts in M and/or G)• dependence on the brightness• dependence on the angular distance to the Sun• …

IV. Alternative angular dependence: higher-order PPN/PPL terms

V. Alternative angular dependence: a

(-1 in General Relativity)

VI. Alternative non-radial deflection patterns: vector spherical harmonics

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Global test: pattern matching in positions/proper motions

I. Secular change of the secular aberration due to acceleration of the Solar system relative to the Galaxy.

II. Deflection on very low frequency gravitational waves:

- constrain the flux at 10-7 to 10-8 Hz- detailed sensitivity study: to be done

similar study done for VLBI: Pyne et al. 1996, 1997

III. Deflection pattern due to hypothetical unknown massive body within the Solar system

- case with almost no proper motion: Gaudi & Bloom 2005

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Global test: acceleration of the solar system

• Acceleration of the Solar system relative to remote sources leads to a time dependency of secular aberration: 5 as/yr

• constraint for the galactic model• important for the binary pulsar test of relativity (at 1% level)

O. Sovers, 1988: first attempts to use geodetic VLBI data

4 5, 9 5, 4 5 / x y za a a as yr

4.2 1.5, 2.6 1.6, 6.1 2.3 / x y za a a as yr

0.2, 3.7, 2.1 / x y za a a as yr Circular orbit about the galactic centre gives:

O. Titov, S.Klioner, 2003-…: > 3.2 106 observations, OCCAM

M.Eubanks, S.Klioner, …, 1992-1997: 1.5 106 observations,CALC/SOLVEVery hard business: the VLBI estimates are not reliable(dependent on the used data subset: source stability, network, etc)

Gaia will have better chances, but it will be a challenge.

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Local test: differential deflection due to Jupiter and Saturn

The accuracy of ephemerides is not sufficient (by a factor of 100!) to predict

deflection with an accuracy of 1 as: exclude from the global solution.

Differential solution could allow one to

I. measure the light-deflection parameters γ for each of these planets(NOTE: this is independent of global solution)

II. quadrupole light defection (Crosta, Mignard, 2004,…)

III. measure the light deflection due to the gravimagnetic field induced by translational motion of the planets

3

2

110

310

1 410

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Local test: relativistic effects in asteroids

( cty) ( cty)e ( )a AU e ( )i Object

Mercury 42.98 8.84 0.39 0.21 7.00

Venus 8.62 0.06 0.72 0.01 3.39

Earth 3.84 0.06 1.00 0.02 0.00

Mars 1.35 0.12 1.52 0.09 1.85

I. Schwarzschild effects due to the Sun: perihelion precession

Historically the first test of general relativity

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Perihelion precession (the first 20001 asteroids)

( cty) ( cty)e ( )a AU e ( )i Object number

Mercury 42.98 8.84 0.39 0.21 7.00

Phaethon 3200 10.13 9.01 1.27 0.89 22.17

Icarus 1566 10.06 8.31 1.08 0.83 22.85

Talos 5786 9.98 8.25 1.08 0.83 23.24

Hathor 2340 7.36 3.31 0.84 0.45 5.85

Ra-Shalom 2100 7.51 3.28 0.83 0.44 15.75

Cruithne 3753 5.25 2.70 1.00 0.51 19.81

Khufu 3362 5.05 2.37 0.99 0.47 9.92

1992 FE 5604 5.55 2.25 0.93 0.41 4.80

Castalia 4769 4.30 2.08 1.06 0.48 8.89

Epona 3838 2.72 1.91 1.50 0.70 29.25

Cerberus 1865 4.05 1.89 1.08 0.47 16.09

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Perihelion precession (12.09.05: 253113)( cty) ( cty)e ( )a AU e ( )i Object number

Mercury 42.98 8.84 0.39 0.21 7.00

2004 XY60 32.14 25.63 0.64 0.80 23.79

2000 BD19 26.83 24.02 0.88 0.90 25.68

1995 CR 19.95 17.33 0.91 0.87 4.03

1999 KW4 66391 22.06 15.19 0.64 0.69 38.89

2004 UL 15.06 13.96 1.27 0.93 23.66

2001 TD45 17.12 13.30 0.80 0.78 25.42

1999 MN 18.48 12.30 0.67 0.67 2.02

2000 NL10 14.45 11.80 0.91 0.82 32.51

1998 SO 16.39 11.45 0.73 0.70 30.35

1999 FK21 85953 16.19 11.38 0.74 0.70 12.60

2004 QX2 11.05 9.97 1.29 0.90 19.08

2002 AJ129 10.70 9.79 1.37 0.91 15.55

2000WO107 12.39 9.67 0.91 0.78 7.78

2005 EP1 12.50 9.60 0.89 0.77 16.19

Phaethon 3200 10.13 9.01 1.27 0.88 22.17

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I. Schwarzschild effects due to the Sun: perihelion precession Mignard, 2001; Hestroffer, Berthier, 2005:

Preliminary results with limited number of sources and with perihelion advance only:

2

4

7

10

10J

Local test: relativistic effects in asteroids

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II. Non-Schwarzschild effects

• Orbital consequences of the EIH equations for asteroids are still poorly known.

• Especially interesting for resonant asteroids for which the relativistic effects of e.g. Jupiter can be enhanced

Local test: relativistic effects in asteroids

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Maximal „post-Sun“ perturbations in meters

2 | |N Sun pNx x

1 2 3 4 5

0.5

1

5

10

50

0 0.2 0.4 0.6 0.8

0.01

0.1

1

10

100a

e

2 4 6 8

20

40

60

80

e

20000 Integrations over 200 days

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III. Special test: SEP with Trojan and other resonant asteroids

• The effect is historically the first example of observable effect due to a violation of the Strong Equivalence Principle: (Nordtvedt, 1968)

shift of L4 and L5 by 1” for =1

• The effect is hidden in the PPN-EIH equations of motion

• Orellana, Vucetich, 1988-1993: =-0.54±0.48

12 Trojans, 100-200 observations for each, accuracy 1”:

• One can hope to do much better with Gaia

• Rigorous theoretical analysis still has to be done…

Local test: relativistic effects in asteroids

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Global/local tests: improve ephemeris and redo

A short-arc (5 years) ephemeris with highest possible accuracy is necessary

Observations relevant for the solar system ephemeris:

• direct observations of the giant planets• indirect: from differential light deflection• indirect: from natural satellites• masses of hundreds of asteroids

(marginally important for the giant planets)

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Gaia provides the ultimate test for the existing of black holes?

• Fuchs, Bastian, 2004: Weighing stellar-mass black holes in binaries

•Astrometric wobble of the companions (just from binary motion)

V(mag) (as)

Cyg X-1 9 28

V1003 ScoGROJ1655-40

17 16

V616 MonA0620-00

18 16

V404 CygGS2023+338

19 50

V381 NorXTEJ1550-564

20 18

• Already known objects:

• Unknown objects, e.g. binaries with “failed supernovae” (Gould, Salim, 2002)

• Gaia advantage: we record all what we see!

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Search for the optimal strategy for Gaia

• The mission would survive without fundamental physics tests:

the tests cannot be “too heavy” so that they “disturb” the main goals…

• But the tests are more than welcome and they are “for free”: