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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
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
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