Transcript Fachspezifische Datenverarbeitung

Ephemerides in the relativistic framework:
_
time scales, spatial coordinates, astronomical
constants and units
Sergei A.Klioner
Lohrmann Observatory, Dresden Technical University
Paris Observatory, 6 April 2006
1
Contents
• Newtonian astrometry and Newtonian equations of motion
• Why relativity? 
• Coordinates, observables and the principles of relativistic modelling
• IAU 2000: BCRS and GCRS, metric tensors, transformations, frames
• Relativistic time scales and reasons for them:
TCB, TCG, proper times, TT, TDB, Teph
• Scaled-BCRS
• Astronomical units in Newtonian and relativistic frameworks
• Do we need astronomical units?
• Scaled-GCRS
• TCB/TCG-, TT- and TDB-compatible planetary masses
• TCB-based or TDB-based ephemeris: notes, rules and recipes
2
Accuracy of astrometric observations
naked eye
0
1400
Hipparchus
1000”
100”
10”
4.5
1500
1600
telescopes
1700
1800
1900
space
2000
2100
Ulugh Beg
1000”
100”
Wilhelm IV
Tycho Brahe
Flamsteed
Hevelius
Bradley-Bessel
orders of magnitude in 2000
years
1“
10”
1”
GC
100 mas
100 mas
FK5
10 mas
Hipparcos
further
4.5
orders
in 20 years
1 mas
100 µas
10 µas
1 µas
10 mas
1 mas
ICRF
Gaia
SIM
0
1400
1500
1600
1700
1800
1900
2000
100 µas
10 µas
1 µas
2100
1 as is the thickness of a sheet of paper seen from the other side of the Earth
3
Modelling of astronomical observations
in Newtonian physics
M. C. Escher
Cubic space division, 1952
4
Astronomical observation
physically preferred
global inertial
coordinates
observables are
directly related to
the inertial
coordinates
5
Modelling of positional observations
in Newtonian physics
• Scheme:
•
•
•
aberration
parallax
proper motion
• All parameters of the model are defined
in the preferred global coordinates:
( ,  ), (  ,  ),  ,
• Newtonian equations of motion:
6
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 or VLBI would not
work without relativistic modelling
• The simplest theory which successfully describes all
available observational data:
APPLIED RELATIVITY
7

Astronomical observation
physically preferred
global inertial
coordinates
observables are
directly related to
the inertial
coordinates
8
Astronomical observation
no physically
preferred coordinates
observables have
to be computed as
coordinate
independent
quantities
9
General relativity for astrometry
A relativistic reference
system
Relativistic
equations
of motion
Equations of
signal
propagation
Definition of
observables
Relativistic
models
of observables
Coordinate-dependent
parameters
Observational
data
Astronomical
reference
frames
10
General relativity for astrometry
Relativistic reference
systems
Relativistic
equations
of motion
Equations of
signal
propagation
Definition of
observables
Relativistic
models
of observables
Coordinate-dependent
parameters
Observational
data
Astronomical
reference
frames
11
The IAU 2000 framework in Manchester…
12
The IAU 2000 framework
• Three standard astronomical reference systems were defined
• BCRS (Barycentric Celestial Reference System)
BCRS
• GCRS (Geocentric Celestial Reference System)
• Local reference system of an observer
• All these reference systems are defined by
GCRS
the form of the corresponding metric tensors.
Technical details:
13
Brumberg, Kopeikin, 1988-1992
Damour, Soffel, Xu, 1991-1994
Klioner, Voinov, 1993
Klioner,Soffel, 2000
Soffel, Klioner,Petit et al., 2003
Local RS
of an observer
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
14
The Barycentric Celestial Reference System
• The BCRS is suitable to model processes in the whole solar system
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

g 0i  
1
2
 i (t, x)
3
i
3
w(t , x )  G  d x
 2 G 2  d x (t , x) | x  x | , w (t, x )  G  d x
,
| x  x  | 2c
t
| x  x |
3
 (t , x)
  T 00  T kk  / c 2 ,  i  T 0i / c, T  is the BCRS energy-momentum tensor
15
N-body problem in the BCRS
Equations of motion
in the PPN-BCRS:
Einstein-Infeld-Hoffman (EIH)
equations:
used in the JPL ephemeris
General relativity: software (usually ==1)
Lorentz, Droste, 1916
EIH, 1936
Damour, Soffel, Xu, 1992
PPN formalism:
Will, 1973; Haugan, 1979;
Klioner, Soffel, 2000
16
Geocentric Celestial Reference System
The GCRS is adopted by the International Astronomical Union (2000)
to model physical processes in the vicinity of the Earth…
Why not BCRS?
17
Geocentric Celestial Reference System
• Imagine a sphere (in inertial coordinates of special relativity),
which is then forced to move in a circular orbit around some point…
• What will be the form of the sphere for an observer at rest
relative to that point?
Lorentz contraction deforms the shape…
Direction of
the velocity
Additional effect due to acceleration (not a pure boost) and
gravitation (general relativity, not special one)
18
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.
19
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.
20
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

1
W (T , X )  WE (T , X )  Qa (T ) X a  WT (T , X ), W a (T , X )  WEa (T , X )   abcCb (T ) X c  WTa (T , X ).
2
21
internal + inertial + tidal external potentials
BCRS-GCRS transformation
• The coordinate transformations:
1
1
i i
T  t  2  A(t )  vE rE   4  B(t )  B i (t )rEi  B ij (t )rEi rEj  C  t , x    O  c 5  ,
c
c
1 1


X a  Ria (t )  rEi  2  vEi vEj rEj  D ij (t )rEj  D ijk (t )rEj rEk    O  c 4 
c 2


3
i
i
i
with C T , X   O  rE  , rE  x  xE (t )
where
xEi (t ) and vEi (t ) are the BCRS position and velocity of the Earth,
A(t ), B(t ), Bi (t ), Bij (t ), C(t, x), Dij (t ), Dijk (t ) are explicit functions,
and the orientation is CHOSEN to be kinematically non-rotating:
Ria (T )  ia
22
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)
23
Celestial Reference Frame
• All astrometric 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
24
Relativistic Time Scales: TCB and TCG
•
t = TCB
•T
Barycentric Coordinate Time = coordinate time of the BCRS
= TCG
Geocentric Coordinate Time = coordinate time of the GCRS
These are part of 4-dimensional coordinate systems so that
the TCB-TCG transformations are 4-dimensional:
T t
• Therefore:
1
1
i i
i
i
ij
i j
5
A
(
t
)

v
r

B
(
t
)

B
(
t
)
r

B
(
t
)
r
r

C
t
,
x

O
c








E
E
E
E
E
c2
c4
TCG  TCG(TCB, xi )
• Only if space-time position is fixed in the BCRS
TCG becomes a function of TCB.
25
( rEi  xi  xEi (t ) )
i
xi  xobs
(t )
Relativistic Time Scales: TCB and TCG
i
i
x

x
• Important special case
E (t ) gives the TCG-TCB relation
at the geocenter:
linear drift removed:
26
Relativistic Time Scales: proper time scales
•
proper time of each observer: what an ideal clock moving
with the observer measures…
• Proper time can be related to either TCB or TCG (or both) provided
that the trajectory of the observer is given:
i
xobs
(t ) and/or
a
X obs
(T )
The formulas are provided by the relativity theory:
d 
2
1

i
i
j
   g00  t, xobs (t )   g0i  t, xobs (t )  xobs
(t )  2 gij  t, xobs (t )  xobs
(t ) xobs
(t ) 
dt 
c
c

1/ 2
d 
2
1

a
a
b
  G00 T , Xobs (T )   G0a T , Xobs (T )  X obs
(T )  2 Gab T , Xobs (T )  X obs
(T ) X obs
(T ) / c 2 
dT 
c
c

1/ 2
27
Relativistic Time Scales: proper time scales
• Specially interesting case: an observer close to the Earth surface:
d
1 1 2

 1  2  X obs (T )  WE T , Xobs   "tidal terms"   O  c 4 
dT
c 2

1017
• But
1 2
X obs (T )  WE T , X obs   U G  const is the definition of the geoid!
2
• Therefore
d
1
 1  2 U G  " terms
dT
c
h
28
i
h(T ),  i (T )"   ...
is the height above the geoid
is the velocity relative to the rotating geoid
Relativistic Time Scales: TT
• Idea: let us define a time scale linearly related to T=TCG, but which
numerically coincides with the proper time of an observer on the geoid:
TT  (1  LG ) TCG
with
1
LG  2 U G
c
d
1

1

" terms
• Then
2 
d TT
c
h,  i "  ...
• To avoid errors and changes in TT implied by changes/improvements
in the geoid, the IAU (2000) has made LG to be a defined constant:
LG  6.969290134 10-10
• TAI is a practical realization of TT (up to a constant shift of 32.184 s)
• Older name TDT (introduced by IAU 1976): fully equivalent to TT
29
Relativistic Time Scales: TDB-1
• Idea: to scale TCB in such a way that the scaled TCB remains close to TT
• IAU 1976: TDB is a time scale for the use for dynamical modelling of the
Solar system motion which differs from TT only by periodic terms.
• This definition taken literally is flawed:
such a TDB cannot be a linear function of TCB!
But the relativistic dynamical model (EIH equations) used by e.g. JPL
is valid only with TCB and linear functions of TCB…
30
Relativistic Time Scales: Teph
• Since the original TDB definition has been recognized to be flawed
Myles Standish (1998) introduced one more time scale Teph differing
from TCB only by a constant offset and a constant rate:
Teph  K  TCB  Teph0
• The coefficients are different for different ephemerides.
• The user has NO information on those coefficients!
• The coefficients could only be restored by some numerical procedure
(Fukushima’s “Time ephemeris”)
•For JPL only the transformation from TT to Teph which matters…
• VSOP-based analytical formulas (Fairhead-Bretagnon) are used for this
31
transformations
Relativistic Time Scales: TDB-2
The IAU Working Group on Nomenclature in Fundamental Astronomy
suggested to re-define TDB to be a fixed linear function of TCB:
•
TDB to be defined through a conventional relationship with TCB:
TDB  TCB  LB   JDTCB  T0   86400  TDB0
• T0 = 2443144.5003725 exactly,
• JDTCB = T0 for the event 1977 Jan 1.0 TAI at the geocenter and
increases by 1.0 for each 86400s of TCB,
• LB = 1.550519768×10−8 exactly,
• TDB0 = −6.55 ×10−5 s exactly.
Using this “new TDB”, it is trivial to convert from TDB to TCB and back.
32
Gaia needs: TCB
• With all this involved situation with TDB/Teph the only unambiguous way
is to use TCB for all aspects of data processing:
• solar system ephemeris
• Gaia orbital data
• time parametrization of proper motions
• time parametrization of orbital solutions (asteroids and stars)
•…
• TCB was officially agreed to be the fundamental time scale for Gaia
33
Scaled BCRS: not only time is scaled
• If one uses scaled version TCB – Teph or TDB – one effectively uses
three scaling:
• time
t *  K  TCB  t0*
• spatial coordinates
x*  K  x
• masses (= GM) of each body
*  K  
(from now on “*” refer to quantities defined in the scaled BCRS;
these quantities are called TDB-compatible ones)
WHY THREE SCALINGS?
34
Scaled BCRS
• These three scalings
together leave
the dynamical equations
unchanged:
• for the motion of
the solar system bodies:
• for light propagation:
35
Scaled BCRS
• These three scalings lead to the following:
36
semi-major axes
a*  K  a
period
P*  K  P
mean motion
n *  K 1  n
the 3rd Kepler’s law
a 3n 2  
 a *3n*2   *
Quantities:
numerical values and units of measurements
• Arbitrary quantity
A
can be expressed by a numerical value
in some given units of measurements
AXX :
A   AXX AXX
• XX denote a name of unit or of a system of units, like SI
37
 AXX
Quantities:
numerical values and units of measurements
• Consider two quantities A and B, and a relation between them:
B  K  A, K  const
• No units are involved in this formula!
• The formula
A   AXX AXX should be used on both sides before
numerical values can be discussed.
• In particular,
is valid if and only if
38
 BXX  K   AXX
BXX  AXX
Scaled BCRS
• For the scaled BCRS this gives:
• Numerical values are scaled in the same way as quantities if and only if
the same units of measurements are used.
39
Astronomical units in the Newtonian framework
• SI units
• time
• length
• mass
tSI  second  s
xSI  meter  m
M SI  kilogram  kg
• Astronomical units
• time
• length
• mass
40
tA  day
xA  AU
M A  Solar mass  SM
Astronomical units in the Newtonian framework
• Astronomical units vs SI ones:
• time
• length
• mass
tA  d  tSI , d  86400
xA    xSI
M A    M SI
• AU is the unit of length with which the gravitational constant G takes
the value
GA  k 2  0.017202098952
• AU is the semi-major axis of the [hypothetic] orbit of a massless particle
which has exactly a period of
2 / k  365.256898326328
in the framework of unperturbed Keplerian motion around the Sun
41
days
Astronomical units in the Newtonian framework
• Values in SI and astronomical units:
• distance (e.g. semi-major axis)
• time (e.g. period)
• GM
42
 x  A   1  x SI
t  A  d 1  t SI
   A  d 2  3    SI
Astronomical units in the relativistic framework
Be ready for a mess!
43
Astronomical units in the relativistic framework
• Let us interpret all formulas above as TCB-compatible astronomical units
• Now let us define a different TDB-compatible astronomical units
*
*
t

day

d
 tSI
 A*
xA*  AU *   *  xSI
M A*  SM*   *  M SI
G    k
A*
*

* 2
• The only constraint on the constants:
k
*
* 3
* 2


M
  Sun  A*     d   K  1  L
B
   
k 2  M Sun      d 
A
44
* 2
Astronomical units in the relativistic framework
• Possibility I: Standish, 1995
d  d *  86400
k  k*
*
 M Sun
   M Sun   1
A*
A
 *  K 1/ 3 
• This leads to
 x   K
A*
*
2/3
  xA
t *   K  t  A
A
  *      A
A*
45
strange scaling…
Astronomical units in the relativistic framework
• Possibility II: Brumberg & Simon, 2004; Standish, 2005
d  d *  86400
k 
* 2
 M
*
Sun
  K  k  M Sun 
A*
A
*  
• This leads to
 x*   K   x  A
A*
t   K  t  A
A
*
    K     A
A*
*
46
2
Either k is different
or the mass of the Sun
is not one or both!
The same scaling as with SI:
 x*   K   x SI
SI
t *   K  t SI
SI
  *   K    SI
SI
How to extract planetary masses from the DEs
• From the DE405 header one gets:
*
11
TDB-compatible AU:   1.49597870691 10
• Using that (also can be found in the DE405 header!)
*
 Sun
  k 2  2.959122082855911025  10-4
A*
one gets the TDB-compatible GM of the Sun expressed in SI units
*
*
 Sun
   Sun
   *  86400-2 =1.32712440018 1020
SI
A*
3
• The TCB-compatible GM reads
(this value can be found in IERS Conventions 2003)
47
1
*
 Sun  
 Sun
 =1.32712442076  1020
SI
SI
1  LB
Do we need astronomical units?
• The reason to introduce astronomical units was that the angular
measurements were many order of magnitude more accurate than
distance measurements.
• Arguments against astronomical units
• The situation has changed crucially since that time!
• Solar mass is time-dependent just below current accuracy of
ephemerides
M Sun / M Sun
1013 yr 1
• Complicated situations with astronomical units in relativistic
framework
• Why not to define AU conventionally as fixed number of meters?
• Do you see any good reasons for astronomical units?
48
Scaled GCRS
• Again three scalings (“**” denote quantities defined in the scaled GCRS;
these TT-compatible quantities):
• time
T **  L  TT
• spatial coordinates
X **  L  X
• masses (=GM) of each body
 **  L  
• the scaling is fixed
L  1  LG  1  6.969290134  10-10
• Note that the masses are the same in non-scaled BCRS and GCRS…
• Example: GM of the Earth from SLR (Ries et al.,1992; Ries, 2005)
  **    398600441.5  0.4   106
• TT-compatible
 Earth  SI
  1   **    398600441.8  0.4   106
• TCG-compatible  
 Earth  SI 1  LG  Earth  SI
49
TCG/TCB-, TT- and TDB-compatible
planetary masses
• GM of the Earth from SLR:
  **    398600441.5  0.4   106
 Earth  SI
  1   **    398600441.8  0.4   106
• TCG/B-compatible  
 Earth  SI 1  LG  Earth  SI
• TT-compatible
• TDB-compatible
  *   1  LB   
   398600435.6  0.4   106
 Earth  SI
 Earth  SI
• GM of the Earth from DE:
50
Should the SLR mass be
used for ephemerides?
• DE403
  *    398600435.6  106
 Earth  SI
• DE405
  *    398600432.9   106
 Earth  SI
TCB-based or TDB-based ephemeris?
• Note 1: Teph defined by a fixed relation to TT may be a source of
inconsistency since newer ephemerides are not fully compatible
with the Teph –TT relation used for their development
(derived on the basis of VSOP87/DE200)
• Note 2: No good reasons to develop more accurate analytical formulas:
just like with the ephemeredes too many terms…
• Note 3: With fixed scaling constant K=1-LB (that is, with the re-defined
TDB) it is impossible to have different post-fit residuals when
using TDB or TCB.
The fits must be absolutely equivalent!
• Note 4: Once a TDB ephemeris is constructed, it is trivial to convert it to
TCB and vice verse: just use the three scalings given above!
51
Iterative procedure to construct ephemeris with
TCB or TDB in a fully consistent way
1. Use some apriori T(C/D)B–TT relation (based on some older
ephemeris) to convert the observational data from TT to T(C/D)B
2. (Re-) Construct the new ephemeris
3. Update the T(C/D)B–TT relation by numerical integration
using the new ephemeris
4. Convert the observational data from TT to T(C/B)D using
the updated T(C/D)B–TT relation
• This scheme works even if the change of the ephemeris is (very) large
• The iterations are expected to converge very rapidly (after just 1 iteration)
52