Transcript Fachspezifische Datenverarbeitung

Relativistic time scales and
relativistic time synchronization
Sergei A. Klioner
Lohrmann-Observatorium, Technische Universität Dresden
Problems of Modern Astrometry, Moscow, 24 October 2007
1
TCB
Teph
TDT
TAI
GPS time
Relativistic astronomical time scales
UTC
TT

2
TCG
TDB
TT (BIPM)
General relativity for space astrometry
Relativistic reference
systems
Relativistic
equations
of motion
Equations of
signal
propagation
Time scales
Definition of
observables
Relativistic
models
of observables
Observational
data
Astronomical
reference
frames
3
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:
4
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
Two kinds of time scales in relativity
1. Proper time of an observer:
reading of an ideal clock located and moving together with the observer
- defined and meaningful only for the specified observer
2. Coordinate time scale:
one of the 4 coordinates of some 4-dimensional relativistic reference
system
- defined for any events in the region of space-time where the reference
system is defined
5
Coordinate 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:
( rEi  xi  xEi (t ) )
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:
i
xi  xobs
(t )
i
TCG  TCG(TCB, xobs
(TCB))  TCG(TCB)
6
Coordinate Time Scales: TCB and TCG
• Important special case
at the geocenter:
xi  xEi (t ) gives the TCG-TCB relation
15
10
5
main feature: linear drift 1.4810-8
zero point is defined to be Jan 1, 1977
difference now: 14.7 seconds
s
0
-5
-10
-15
1950
1960
1970
1980
1990
2000
2010
0.0015
0.001
0.0005
linear drift removed:
s
0
-0.0005
-0.001
-0.0015
1970 1972 1974 1976 1978 1980 1982 1984
7
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:
In BCRS:
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

In GCRS:
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 ) 
dT 
c
c

1/ 2
8
Proper Time Scales
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
9
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  
Local Positional Invariance
One aspect of the LPI can be tested
by measuring the gravitational red
shift of clocks
degree of the violation of
the gravitational red shift

U
 1    2

c
10
Proper time scales and TCG
• 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
• Idea: let us define a time scale linearly related to T=TCG, but which
is numerically close to the proper time of an observer on the geoid:
TT  (1  LG ) TCG, LG  6.969290134 10-10
d
1
 1  2 " terms
d TT
c
h
11
i
h,  i " " tidal terms" ...   ...
is the height above the geoid
is the velocity relative to the rotating geoid
can be neglected
in many cases
Coordinate Time Scales: TT
• Idea: let us define a time scale linearly related to T=TCG, but which
is numerically close to the proper time of an observer on the geoid:
TT  (1  LG ) TCG, LG  6.969290134 10-10
d
1
 1  2 " terms
d TT
c
h
i
h,  i " " tidal terms" ...   ...
can be neglected
in many cases
is the height above the geoid
is the velocity relative to the rotating geoid
• 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)
12
• Older name TDT (introduced by IAU 1976): fully equivalent to TT
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 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…
13
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  R  TCB  Teph0
• The coefficients are different for different ephemerides.
• The user has NO information on those coefficients from the ephemeris.
• The coefficients could only be restored by some additional numerical
procedure (Fukushima’s “Time ephemeris”)
• Teph is de facto defined by a fixed relation to TT:
by the Fairhead-Bretagnon formula based on VSOP-87
14
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,
• TDB0  −6.55 ×10−5 s.
15
Linear drifts between time scales
16
Pair
Drift per year
(seconds)
Difference at J2007
(seconds)
TT-TCG
0.021993
0.65979
TDB-TCB
0.489307
14.67921
TCB-TCG @ geocenter 0.467313
14.01939
Scaled BCRS and GCRS
17
Scaled BCRS: not only time is scaled
• If one uses scaled version TCB – Teph or TDB – one effectively uses
three scaling:
• time
t *  F  TCB  t0*
• spatial coordinates
x*  F  x
• masses (= GM) of each body
*  F  
F  1  LB
WHY THREE SCALINGS?
18
Scaled BCRS
• These three scalings
together leave
the dynamical equations
unchanged:
• for the motion of
the solar system bodies:
• for light propagation:
19
Scaled GCRS
• If one uses TT being a scaled version TCG one effectively uses
three scaling:
• time
T **  TT  L  TCG
• spatial coordinates
X **  L  X
• masses of each body
 **  L  
L  1  LG
• International Terrestrial Reference Frame (ITRF) uses such scaled GCRS
coordinates and quantities
• Note that the masses are the same in non-scaled BCRS and GCRS…
20
Scaled masses
The masses  are the same in non-scaled BCRS and GCRS,
but not the same with the scaled versions
scaled BCRS (with TDB)
 *  F   , F  1  LB
scaled GCRS (with TT)
 **  L   , L  1  LG
Mass of the Earth
21
TT-compatible
 **   398600441.5  0.4   106
TCB/G-compatible
  1  LG   **   398600441.8  0.4   106
TDB-compatible
 *  1  LB     398600435.6  0.4   106

1




4-dimensional ephemerides
22
Time scales important for ephemerides
• Equations of motion are parametrized in TCB or TDB
• Observations are tagged with TT (or UTC or TAI…)
• Time tags of observations must be recalculated into TCB or TDB
T t
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
• position-dependent terms represent no problems
• transformation at the geocenter:
each ephemeris defines its own transformation
23
analytical expressions of Fairhead & Bretagnon are used;
these expressions are based on analytical ephemeris VSOP:
loss of accuracy is possible here!
Iterative procedure to construct ephemeris with
TCB or TDB in a fully consistent way
a priori TCB–TT relation (from an old ephemeris)
convert the observational data from TT to TCB
construct the new ephemeris
changed?
no
yes
update the TCB–TT relation
(by numerical integration using the new ephemeris)
24
final 4D
ephemeris
Notes on the iterative procedure
• 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)
• The time ephemeris (TT-TDB relation) becomes a natural part of
any new ephemeris of the Solar system:
Self-consistent 4-dimensional ephemerides should be
produced in the future
Consequence of not doing it:
e.g. TEMPO2 does it internally, but the user does not have
the full dynamical dynamical model of the ephemeris (asteroids etc.)
25
How to compute TT(TDB) from an ephemeris
-
26
Fundamental relativistic relation between TCG and TCB at the geocenter
How to compute TT(TDB) from an ephemeris
-
definitions of TT and TDB
1) TT(TCG) :
2) TDB(TCB) :
27
How to compute TT(TDB) from an ephemeris
TT  TDB  TDB(TDB)
-
two corrections
-
two differential equations
TDB  TT  TT (TT )
d
1
1


TDB   LB  2  (TDB)  1  LB  LG   LG  4  (TDB)
dTDB
c
c


d
1
TT  2  (TT  TT ) 1  LB  LG 
dTT
c
1
 4   (TT  TT )   2 (TT  TT )    LB  LG 1  LG 
c
28
Representation with Chebyshev polynomials
-
Any of those small functions can be represented
by a set of Chebyshev polynomials
-
The conversion of a tabulated y(x) into an is a well-known task…
29
TT-TDB: DE405 vs. SOFA for full range of DE405
75
-
SOFA implements the
50
corrected Fairhead-Bretagnon
25
analytical series based
ns 0
on VSOP-87
-25
(about 1000 Poisson terms,
also non-periodic terms)
-50
1600
1700
1800
1900
2000
2100
2200
1700
1800
1900
2000
2100
2200
15
10
5
0.868 ns- 8.28 1018 t 
0
-5
-10
ns
-15
1600
30
TT-TDB: DE405 vs. SOFA for 1960-2020
10
5
ns
0
-5
-10
1960
1970
1980
1990
2000
2010
2020
2010
2020
8
6
0.868 ns- 8.28 1018 t 
4
2
ns
0
1960
31
1970
1980
1990
2000
TT-TDB: DE405 vs. DE200
125
100
75
50
ns
25
0
-25
-50
1700
1800
1900
2000
2100
1900
2000
2100
4
2
1.171 ns- 1.087 1017 t 
0
-2
ns
-4
1700
32
1800
TT-TDB: DE405 vs. DE403
0.4
0.3
0.2
ns
0.1
0
-0.1
-0.2
1600
1700
1800
1900
2000
2100
2200
2100
2200
0.075
0.05
0.025
20
0.005 ns- 3 10 t 
0
-0.025
-0.05
-0.075
ns
-0.1
1600
33
1700
1800
1900
2000
4-dimensional ephemerides
-
IMCCE (Fienga, 2007) and JPL (Folkner, 2007) have agreed to include
time transformation (TT-TDB) into the future releases of the ephemerides
-
The Paris Group have implemented already the algorithms
as discussed above
conventional space ephemeris
+
time ephemeris
34

relativistic
4-dim
ephemerides
Clock synchronization
35
Clock synchronization: Newtonian physics
-
Newtonian physics: absolute time means absolute synchronization
two clocks are synchronized when they “beat” simultaneously
space
non-simultaneous
event 2
event 2
*
*
* event 1
event 1
simultaneous
*
t1  t2
36
t1  t2
absolute time t
Clock synchronization: special relativity
-
Special relativity: time is relative and synchronization is also relative
two events (e.g. two clocks showing 00:00:00 exactly) can be simultaneous
in one inertial reference system and non-simultaneous in another one
space
event 2
*
event 1
*
non-simultaneous
time T
T1  T2
t1  t2
37
simultaneous
time t
Clock synchronization: special relativity
Einstein synchronization
for two clocks at rest in some inertial reference system
t0
t1
t2
Clock a
Clock b
1
t1   t2  t0 
2
38
Clock synchronization: general relativity
-
Coordinate synchronization and coordinate simultaneity
(Allan, Ashby, 1986):
i
Two events (t1, x1 ) and (t2 , x2i ) are called simultaneous if and only if
-
t1  t2
The relation of proper time of a clock and coordinate time
is a differential equation of 1st order:
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
-
This equation gives unique relation if the initial condition is given:
 (t0 )   0
39
This can be postulated for one clock, but for different clock the values
must be consistent with each other.
One-way synchronization
Clock a
Clock b
Observed:
 a 0 ,  b1
Given:
t0 , xai (t ), xbi (t )
Calculated:
Result:
40
t1

 b (t1 )   b1
1
t1  t0  R   gr ,
c
R  xai (t0 )  xbi (t1 )
Two-way synchronization
Clock a
Clock b
Observed:
 a 0 ,  b1,  a 2
Given:
t0 , xai (t ), xbi (t )
Calculated:
Result:
41
t1

 b (t1 )   b1
 a 2   a 0  t2  t0 ,
1
t1  t0   t2  t0      gr ,
2
1
   R2  R1 
2c
Clock-transport synchronization
Observed:
 a 0 ,  c0 ,  c1,  b1
Given:
t0 , xai (t ), xbi (t ), xci (t )
Calculated: t1
Result:
42

 b (t1 )   b1
 c1   c0  t1  t0
Clock-transport synchronization: experiments
Hafele & Keating
(1972):
comparison with
ground-based clocks
eastward flight:
tg + tv =
+144 ns -184 ns = - 40 ns
westward flight:
+179 ns + 96 ns = + 275 ns
43
Realizations of coordinate time scales
•
44
General principle:
•
a physical process is observed (no matter if periodic or not)
•
a relativistic model of that process is used to predict observations
as a function of coordinate time
•
events of observing some particular state of the process
realize particular values of coordinate time
Realizations of coordinate time scales
Example 1:
Realizations of TT (or TCG) using atomic clocks:
- clocks themselves realize proper times along their trajectories
- moments of TT are computed from proper time of each clock
- clocks are synchronized with respect to TT
- different clocks are combined (averaging, etc.)
result: TAI, TT(BIPM), TT(USNO), TT(OBSPM), TT(GPS), etc.
45
Realizations of coordinate time scales
Example 2:
Realization of TDB (or TCB) using pulsar timing
- pulsars themselves realize proper times along their trajectories
- moments of TCB are computed from the times of arrivals
of the pulses to the observing site
- different pulsars are combined (averaging, etc.)
46
IAU Commission 52
“Relativity in Fundamental Astronomy”
•
•
•
Created by the IAU in August 2006
President: S.Klioner
Vice-president: G.Petit
http://astro.geo.tu-dresden.de/RIFA
47
Backup slides
48
Time transformations in relativity
•
Time transformations are defined only for space-time events:
-
An event is something that happened at some moment of time
somewhere in space
-
Time transformation in relativity is not defined
if the place of the event is not specified!
E.g. One cannot transform TT into TCB if the place is unknown
49
Time transformations in relativity
-
Apparent “exceptions”
1) TT can be always transformed into TCG and back:
2) TDB can be always transformed into TCB and back:
3) proper time of an observer can be always transformed
into TCB and back: the place is specified implicitly,
since proper time is defined at the location of the observer
50
Time scales in data processing
1. TCB is the coordinate time of BCRS.
- TCB is intended to be the time argument of final Gaia catalogue, etc.
- TCB is defined for any event in the solar system and far beyond it.
2. TT is a linear function of the coordinate time TCG of GCRS.
- TT will be used to tag the events at the Earth’s bound observing sites
(for example, for OBT-UTC correlation)
- The mean rate of TT is close to the mean rate of an observer on the
geoid.
- UTC=TT+32.134 s + leap seconds
(3.) TDB is a specific linear function of TCB
- the linear drift between TDB and TT is made as small as possible
- obsolete time scale used in some ephemerides
- non-zero probability to have it for Gaia ephemeris from ESOC
51
Time scales in data processing
4. Proper times of each observing station
- is automatically recomputed to UTC and, therefore, TT
5. TG is the proper time of the observer
- TG is an ideal form of OBT (an ideal clock on Gaia would show TG)
- TG is an intermediate step in converting OBT into TCB
6. OBT is a realization of TG with all technical errors…
- OBT will be used to tag the observations
52
Transformations between TCB and TCG
-
53
Part one: TCG(TCB) at the geocenter
Transformations between TCB and TCG
-
Part one: TCG(TCB) at the geocenter
practical calculations:
define two small corrections
obeying two differential equations
and solve these two with the conventional initial conditions given by IAU, 1991
54
Transformations between TCB and TCG
-
Part one: position dependent terms
-
For a fixed site on the Earth:
a quasi-periodic signal (period of 1 day) with an amplitude of 2.2 s
55
Transformations between TG and TCB
-
56
The same scheme as for the pair TCB/TCG
Transformations between TG and TCB
-
The same idea with two small corrections

-
The initial conditions
-
57

for some fixed
for simulations any
for real Gaia a moment of time for which Gaia ephemeris is already defined!
a parameter in the Gaia parameter database
Representation with Chebyshev polynomials
-
Any of those small functions can be represented
by a set of Chebyshev polynomials
-
The conversion of tabulated y(x) into an is a well-known task…
58
Clock calibration: how to go from OBT to TG
-
Observational data available:
-
OBT is generated onboard and stored into some special data packets
-
After a short (partially known) hardware delay the packet is sent to the
Earth
-
After the propagation delay it reaches the antenna on the Earth
-
-
59
BCRS distance between Gaia and the antenna
Solar plasma delay
Ionosphere and troposphere delay
Relativistic propagation delay (Shapiro effect)
After a short (partially known) hardware delay it recorded by the
hardware of the observing station with a tag of UTC of reception
Clock calibration: how to go from OBT to TG
-
Relativistic modelling:
-
UTC is recomputed into TT
-
TT of the reception is transformed into TCB of the reception
-
TCB of the emission (recording of the OBT) is computed from
-
60
BCRS distance between Gaia and the antenna
Solar plasma delay
Ionosphere and troposphere delay
Relativistic propagation delay (Shapiro effect)
Hardware delays on the Earth and in the satellite
-
TCB of the OBT recording is transformed into TG
-
TG and OBT are compared and some parameters of the model of the clock
are fitted to get the calibration of model of the OBT.
Clock calibration: how to go from OBT to TG
Gaia:
site on the Earth:
OBT recording event
OBT packet reception event
Signal propagation
OBT calibration
OBT
Position, velocity
UTC
Position, velocity
TG of the OBT recording event
Gaia orbit, hardware calibration
TCB of the OBT recording event
61
Note: only radial position is relevant!
- Martin Hechler, February 2006
62
OBT-UTC correlation
-
Similar thing called OBT-UTC calibration will be done by ESOC
-
OBT will be converted into UTC
-
Relativistic models are not clear
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A simple clock model in UTC will be fitted (linear drift with least squares)
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Can we do better?
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It depends on the accuracy of the clock and the synchronization…
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Relation between TG and TT
• The mean rate of the proper time on the Gaia orbit is different
from Terrestrial Time by about 6.9 ×10 –10
• Periodic terms of order 1 – 2 s
linear trend removed: 6.926 ×10 –10
TG-TT as a function of TT
0.2
sec
1.5
1
0.15
0.5
0.1
500
-0.5
0.05
-1
days -1.5
500
64
1000
1500
2000
2500
3000
3500
1000
1500
2000
2500
3000
3500
TT-TDB: DE200 vs. SOFA
10
0
-10
-20
ns
-30
-40
1700
1800
1900
2000
2100
1700
1800
1900
2000
2100
10
5
2.773 ns  2.60 1018 t 
0
-5
-10
ns
65