General-Relativistic Effects in Astrometry

S.A.Klioner, M.H.Soffel

Lohrmann Observatory, Dresden Technical University

2005 Michelson Summer Workshop, Pasadena, 26 July 2005

General-relativistic astrometry

• Newtonian astrometry

• Why relativistic astrometry?

• Coordinates, observables and the principles of relativistic modelling

• Metric tensor and reference systems

• BCRS, GCRS and local reference system of an observer

• Principal general-relativistic effects

• The standard relativistic model for positional observations

• Celestial reference frame

• Beyond the standard model

Modelling of positional observations in Newtonian physics

M. C. Escher

Cubic space division, 1952

Astronomical observation

physically preferred global inertialcoordinates

observables are directly related to the inertial coordinates

Modelling of positional observations in Newtonian physics

• Scheme:• aberration• parallax• proper motion

• All parameters of the model are defined in the preferred global coordinates:

( , ), ( , ), ,

Accuracy of astrometric observations

1 mas

1 µas10

100

10

100

1 as

10

100

1000

1 µas10

100

1 mas

10

100

1 as

10

100

1000

1400 1500 1700 1900 2000 21000 1600 1800

1400 1500 1700 1900 2000 21000 1600 1800

HipparchusUlugh Beg

Wilhelm IVTycho Brahe

Hevelius

FlamsteedBradley-Bessel

FK5

Hipparcos

Gaia

SIM

ICRF

GC

naked eye telescopes space

Accuracy-implied changes of astrometry: • underlying physics: general relativity vs. Newtonian physics• goals: astrophysical picture rather than a kinematical description

Why general relativity?

• Newtonian models cannot describe high-accuracy observations:

• many relativistic effects are many orders of magnitude larger than the observational accuracy

space astrometry missions would not work without relativistic modelling

• The simplest theory which successfully describes all available observational data:

APPLIED GENERAL RELATIVITYAPPLIED GENERAL RELATIVITY

Testing general relativity

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

Astronomical observation

physically preferred global inertialcoordinates

observables are directly related to the inertial coordinates

Astronomical observation

no physicallypreferred coordinates

observables have to be computed ascoordinate independent quantities

General relativity for space astrometry

Coordinate-dependentparameters

Relativistic reference system(s)

Equations ofsignal

propagation

Astronomicalreference

frames

Observational data

Relativisticequationsof motion

Definition ofobservables

Relativisticmodels

of observables

Metric tensor

x

2 2 2s x y

A

B

B

A

ds

2 22 2 2 2 2 2

1 1

i jij

i j

ds dx dy dr r d g dx dx

y

• Pythagorean theorem in 2-dimensional Euclidean space2R

• length of a curve in 2R

Metric tensor: special relativity

• special relativity, inertial coordinates

0( , ) ( , , , )ix x x ct x y z

• The constancy of the velocity of light in inertial coordinates

2 2 2 2ds c dt d x

2 2 2d c dtx

can be expressed as where2 0ds

00

0

1,

0,

diag(1,1,1).i

ij ij

g

g

g

Metric tensor and reference systems

• In relativistic astrometry the

• BCRS (Barycentric Celestial Reference System)

• GCRS (Geocentric Celestial Reference System)

• Local reference system of an observer

play an important role.

• All these reference systems are defined by

the form of the corresponding metric tensor.

BCRS

GCRS

Local RSof an observer

Barycentric Celestial Reference System

The BCRS:

• adopted by the International Astronomical Union (2000)• suitable to model high-accuracy astronomical observations

200 2 4

0 3

2

2 2( , ) ( , ) ,

4( , ) ,

2( , ) .

1

1

ii

ij ij

g w t w tc c

g w tc

g w tc

x x

x

x

, :iw w relativistic gravitational potentials

Barycentric Celestial Reference System

The BCRS is a particular reference system in the curved space-time of the Solar system

• One can use any

• but one should fix one

Geocentric Celestial Reference System

The GCRS is adopted by the International Astronomical Union (2000) to model physical processes in the vicinity of the Earth:

A: The gravitational field of external bodies is represented only in the form of a relativistic tidal potential.B: The internal gravitational field of the Earth coincides with the gravitational field of a corresponding isolated Earth.

Geocentric Celestial Reference System

The GCRS is adopted by the International Astronomical Union (2000) to model physical processes in the vicinity of the Earth:

A: The gravitational field of external bodies is represented only in the form of a relativistic tidal potential.B: The internal gravitational field of the Earth coincides with the gravitational field of a corresponding isolated Earth.

Geocentric Celestial Reference System

The GCRS is adopted by the International Astronomical Union (2000) to model physical processes in the vicinity of the Earth:

A: The gravitational field of external bodies is represented only in the form of a relativistic tidal potential.B: The internal gravitational field of the Earth coincides with the gravitational field of a corresponding isolated Earth.

200 2 4

0 3

2

2 21 ( , ) ( , ) ,

4( , ) ,

21 ( , ) .

aa

ab ab

G W T W Tc c

G W Tc

G W Tc

X X

X

X

, :aW W internal + inertial + tidal external potentials

Local reference system of an observer

The version of the GCRS for a massless observer:

A: The gravitational field of external bodies is represented only in the form of a relativistic tidal potential.

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

, :aW W internal + inertial + tidal external potentials

observer

Equations of translational motion

• The equations of translational motion (e.g. of a satellite) in the BCRS

200

0

24

3

2

2( , )

4( , )

2(

1 ,

,

1 ,

( , )

.)

2

i

ij ij

i

w tc

w tc

w

w tc

g

g

g

tc

x

x

x

x

• The equations coincide with the well-known Einstein-Infeld-Hoffmann (EIH) equations in the corresponding limit

23

1)

|(

|A

AB

BB A A B

tGMc

x x

Fx

xx

Equations of light propagation

• The equations of light propagation in the BCRS

200 4

0

2

3

2

21 ( , ) ,

,

1 .

2( , )

4( , )

2( , )

i

ij

i

ij

w tc

w tc

w t

t

gc

g wc

g

x

x

x

x

0 0 2( ) (

1)) ) ((t t c t

ct t xx x

• Relativistic corrections to the “Newtonian” straight line:

Observables I: proper time

Proper time of an observer can be related

to the BCRS coordinate time t=TCB using

• the BCRS metric tensor• the observer’s trajectory xi

o(t) in the BCRS

200

0

24

3

2

2( , )

4( , )

2(

1 ,

,

1 ,

( , )

.)

2

i

ij ij

i

w tc

w tc

w

w tc

g

g

g

tc

x

x

x

x

421

1 1pppN NA

t cA

d c

d

Observables II: proper direction

• To describe observed directions (angles) one should introduce spatial reference vectors moving with the observer explicitly into the formalism

• Observed angles between incident light rays and a spatial reference vector can be computed with the metric of the local reference system of the observer

The standard astrometric 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

t

observedrelated to the light raydefined in BCRS coordinates

Sequences of transformations

• Stars:

0 0 0 0

(1) (2) (3) (4) (5)

( ) ( ), , , , ,t ts n k l l

• Solar system objects:

(1) (2,3) (6)

orbitkns

(1) aberration(2) gravitational deflection(3) coupling to finite distance(4) parallax(5) proper motion, etc.(6) orbit determination

Aberration: s n

• Lorentz transformation with the scaled velocity of the observer:

2

1/ 22 2

2

1( 1) ,

(1 / )

1 / ,

21 ( , )o o

c v c

v c

w tc

nvn v

v x x

snv

• For an observer on the Earth or on a typical satellite:

• Newtonian aberration 20• relativistic aberration 4 mas• second-order relativistic aberration 1 as

• Requirement for the accuracy of the orbit: 1 as 1 mm/so xs

Gravitational light deflection: n k

• Several kinds of gravitational fields deflecting light

• monopole field• quadrupole field• gravitomagnetic field due to translational motion• gravitomagnetic field due to rotational motion• post-post-Newtonian corrections (ppN)

with Sun

without Sun

Gravitational light deflection: n k

body Monopole Quadrupole ppN

Sun 1.75106 180 11 53

(Mercury) 83 9

Venus 493 4.5

Earth 574 125

Moon 26 5

Mars 116 25

Jupiter 16270 90 240 152

Saturn 5780 17 95 46

Uranus 2080 71 8 4

Neptune 2533 51 10 3

max max max

• The principal effects due to the major bodies of the solar system in as• The maximal angular distance to the bodies where the effect is still >1 as

Gravitational light deflection: n k

• 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

Gravitational light deflection: n k

Jos de Bruijne, 2002

Parallax and proper motion: k l l0, 0, 0

• All formulas here are formally Euclidean:

0 0 0

( ) ( ) ( ), ,

| ( ) ( ) | | ( ) |

( ) ( ) ( ) ( )

o o s e s e

o o s e s e

s e s e s e e e

t t t

t t t

t t t t t

x X X

x X X

X X V

k l

• Expansion in powers of several small parameters:

1 AU | ( ) |,

| ( ) | | ( ) |

,

s e

s e s e

t

t t

0

V

X X

k l l l

Celestial Reference Frame

• All astrometrical parameters of sources obtained from astrometric observations are defined in BCRS coordinates:

• positions• proper motions• parallaxes• radial velocities• orbits of minor planets, etc.• orbits of binaries, etc.

• These parameters represent a realization (materialization) of the BCRS

• This materialization is „the goal of astrometry“ and is called

Celestial Reference Frame

Beyond the standard model• Gravitational light deflection caused by the gravitational fields generated outside the solar system

• microlensing on stars of the Galaxy, • gravitational waves from compact sources,• primordial (cosmological) gravitational waves, • binary companions, …

Microlensing noise could be a crucial problem for going well below 1 microarcsecond…

• Cosmological effects