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 inertial
coordinates
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
naked eye
0
1400
1500
Hipparchus
1000
telescopes
1600
1700
1800
1900
space
2000
2100
Ulugh Beg
1000
Hevelius
100
Wilhelm IV
Tycho Brahe
10
100
Flamsteed
10
Bradley-Bessel
1 as
1 as
GC
100
100
FK5
10
10
Hipparcos
1 mas
ICRF
100
1 mas
100
Gaia
10
1 µas
10
SIM 1 µas
0
1400
1500
1600
1700
1800
1900
2000
2100
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 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)
G / G  5  1014 yr -1
• Pulsar timing: indirect evidence for gravitational radiation
Astronomical observation
physically preferred
global inertial
coordinates
observables are
directly related to
the inertial
coordinates
Astronomical observation
no physically
preferred coordinates
observables have
to be computed as
coordinate
independent
quantities
General relativity for space astrometry
Relativistic reference
system(s)
Relativistic
equations
of motion
Equations of
signal
propagation
Definition of
observables
Relativistic
models
of observables
Coordinate-dependent
parameters
Observational
data
Astronomical
reference
frames
Metric tensor
• Pythagorean theorem in 2-dimensional Euclidean space
s
y
2
  x   y
2
R2
2
x
• length of a curve in
R2
B
A
  ds
B
A
2
2
ds  dx  dy  dr  r d   gij dx dx
2
2
2
2
2
2
i
i 1 j 1
j
Metric tensor: special relativity
• special relativity, inertial coordinates
x   ( x 0 , xi )  (ct, x, y, z )
• The constancy of the velocity of light in inertial coordinates
dx 2  c 2dt 2
can be expressed as
ds 2  0 where ds 2  c 2dt 2  dx 2
g 00  1,
g 0i  0,
gij   ij  diag(1, 1, 1).
Metric tensor and reference systems
• In relativistic astrometry the
BCRS
• BCRS (Barycentric Celestial Reference System)
• GCRS (Geocentric Celestial Reference System)
• Local reference system of an observer
GCRS
play an important role.
• All these reference systems are defined by
the form of the corresponding metric tensor.
Local RS
of an observer
Barycentric Celestial Reference System
The BCRS:
• adopted by the International Astronomical Union (2000)
• suitable to model high-accuracy astronomical observations
2
2 2
g 00  1  2 w(t , x )  4 w (t , x ) ,
c
c
4 i
g 0i   3 w (t , x ) ,
c
2


gij   ij 1  2 w(t , x )  .
 c

w , wi :
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.
2
2 2
G00  1  2 W (T , X )  4 W (T , X ) ,
c
c
4 a
G0 a   3 W (T , X ) ,
c
2


Gab   ab  1  2 W (T , X )  .
 c

W , W a : 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.
observer
W , W a : internal + inertial + tidal external potentials
• Modelling of any local phenomena:
observation,
attitude,
local physics (if necessary)
Equations of translational motion
• The equations of translational motion
(e.g. of a satellite) in the BCRS
g00  1 
2
2 2
w
(
t
,
x
)

w (t , x ) ,
2
4
c
c
4 i
w (t , x ) ,
3
c
2


gij   ij  1  2 w(t , x )  .
 c

g0i  
• The equations coincide with the well-known Einstein-Infeld-Hoffmann (EIH)
equations in the corresponding limit
x A  xB
1
x A    GM B
 2 F (t )
3
| x A  xB | c
B A
Equations of light propagation
• The equations of light propagation
in the BCRS
• Relativistic corrections to
the “Newtonian” straight line:
x (t )  x0 (t )  c (t  t0 ) 
1
 x (t )
2
c
g00  1 
2
2 2
w
(
t
,
x
)

w (t , x ) ,
2
4
c
c
4 i
w (t , x ) ,
3
c
2


gij   ij  1  2 w(t , x )  .
 c

g0i  
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 xio(t) in the BCRS
d
1
1
 1  2 ApN  4 AppN
dt
c
c
g00  1 
2
2 2
w
(
t
,
x
)

w (t , x ) ,
2
4
c
c
4 i
w (t , x ) ,
3
c
2


gij   ij  1  2 w(t , x )  .
 c

g0i  
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 t  
•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
observed
related to the light ray
defined in BCRS coordinates
Sequences of transformations
• Stars:
(1)
(2)
s  n  
(3)
 k
(4)
 l (t ),  (t )  l0 ,  0 , 0 ,  0 ,
• Solar system objects:
(1)
s  n
(2,3)
 k
(6)
 orbit
(1) aberration
(2) gravitational deflection
(3) coupling to finite distance
(4) parallax
(5) proper motion, etc.
(6) orbit determination
(5)
Aberration: s  n
• Lorentz transformation with the scaled velocity of the observer:

v n 
1

s    n    (  1) 2  v 
,
v    (1  v  n / c)
c

  1  v / c
2

2 1/ 2
,
2


v  xo 1  2 w(t , xo ) 
 c

• For an observer on the Earth or on a typical satellite:
• Newtonian aberration
• relativistic aberration
• second-order relativistic aberration
• Requirement for the accuracy of the orbit:
20
 4 mas
 1 as
 s  1as   xo  1 mm/s
Gravitational light deflection: n    k
with Sun
without Sun
• 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)
Gravitational light deflection: n    k
• 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
body
Monopole
 max
Quadrupole
 max
1.75106
180 
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
Sun
(Mercury)
ppN
11
 max
53
Gravitational light deflection: n    k
• A body of mean density  produces a light deflection not less than 
if its radius:
  
R
3 
1
g/cm


Ganymede
Titan
Io
Callisto
Triton
Europe
35
32
30
28
20
19
1/ 2
1/ 2
  


1
μas


 650 km
Pluto
Charon
Titania
Oberon
Iapetus
Rea
Dione
Ariel
Umbriel
Ceres
7
4
3
3
2
2
1
1
1
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:
xo (to )  X s (te )
X s (te )
k
, l
,
| xo (to )  X s (te ) |
| X s (te ) |
X s (te )  X s (te 0 )  Vs (te 0 ) (te  te 0 ) 
• Expansion in powers of several small parameters:
1 AU
| Vs (te ) |

, 
| X s ( te ) |
| X s ( te ) |
k  l 
, l  l0 
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