1. Relativistic time scales and relativistic time synchronization
Sergei A. Klioner
Lohrmann-Observatorium, Technische Universität Dresden
Problems of Modern Astrometry, Moscow, 24 October 2007

2. Relativistic astronomical time scales

3. General relativity for space astrometry
Relativistic reference
systems
Time scales
Equations of
signal
propagation
Relativistic
equations
of motion
Definition of
observables
Relativistic
models
of observables
Astronomical
reference
frames
Observational
data

4. The IAU 2000 framework
Three standard astronomical reference systems were defined
BCRS (Barycentric Celestial Reference System)
GCRS (Geocentric Celestial Reference System)
Local reference system of an observer
All these reference systems are defined by
the form of the corresponding metric tensors.
Technical details: Brumberg, Kopeikin,
Damour, Soffel, Xu,
Klioner, Voinov, 1993
Klioner,Soffel, 2000
Soffel, Klioner,Petit et al., 2003
BCRS
GCRS
Local RS
of an observer

5. Two kinds of time scales in relativity
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
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

6. Coordinate Time Scales: TCB and TCG
t = TCB Barycentric Coordinate Time = coordinate time of the BCRS
T = 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:
Therefore:
Only if space-time position is fixed in the BCRS
TCG becomes a function of TCB:

7. Coordinate Time Scales: TCB and TCG
Important special case gives the TCG-TCB relation
at the geocenter:
main feature: linear drift 1.4810-8
zero point is defined to be Jan 1, 1977
difference now: 14.7 seconds s
linear drift removed: s

8. with the observer measures…
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:
The formulas are provided by the relativity theory:
In BCRS:
In GCRS:

9. 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

10. 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

11. Proper time scales and TCG
Specially interesting case: an observer close to the Earth surface:
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:
can be neglected
in many cases
is the height above the geoid
is the velocity relative to the rotating geoid

12. 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:
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:
TAI is a practical realization of TT (up to a constant shift of s)
Older name TDT (introduced by IAU 1976): fully equivalent to TT

13. 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…

14. 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:
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

15. 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:
T0 = 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  ×10−8,
TDB0  −6.55 ×10−5 s.

16. Linear drifts between time scales
Pair
Drift per year
(seconds)
Difference at J2007
TT-TCG
TDB-TCB
geocenter

17. Scaled BCRS and GCRS

18. Scaled BCRS: not only time is scaled
If one uses scaled version TCB – Teph or TDB – one effectively uses three scaling:
time
spatial coordinates
masses (= GM) of each body
WHY THREE SCALINGS?

19. Scaled BCRS
These three scalings together leave the dynamical equations unchanged:
for the motion of the solar system bodies:
for light propagation:

20. Scaled GCRS
If one uses TT being a scaled version TCG one effectively uses three scaling:
time
spatial coordinates
masses of each body
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…

21. 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)
scaled GCRS (with TT)
Mass of the Earth
TT-compatible
TCB/G-compatible
TDB-compatible

22. 4-dimensional ephemerides

23. 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
position-dependent terms represent no problems
transformation at the geocenter:
each ephemeris defines its own transformation
analytical expressions of Fairhead & Bretagnon are used; these expressions are based on analytical ephemeris VSOP:
loss of accuracy is possible here!

24. 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
final 4D
ephemeris
changed? no
yes
update the TCB–TT relation
(by numerical integration using the new ephemeris)

25. 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.)

26. How to compute TT(TDB) from an ephemeris
Fundamental relativistic relation between TCG and TCB at the geocenter

27. How to compute TT(TDB) from an ephemeris
definitions of TT and TDB
1) TT(TCG) :
2) TDB(TCB) :

28. How to compute TT(TDB) from an ephemeris
two corrections
two differential equations

29. 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…

30. TT-TDB: DE405 vs. SOFA for full range of DE405
SOFA implements the
corrected Fairhead-Bretagnon
analytical series based
on VSOP-87
(about 1000 Poisson terms,
also non-periodic terms) ns ns

31. TT-TDB: DE405 vs. SOFA for ns ns

32. TT-TDB: DE405 vs. DE200 ns ns

33. TT-TDB: DE405 vs. DE403 ns ns

34. 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
relativistic
4-dim
ephemerides 

35. Clock synchronization

36. Clock synchronization: Newtonian physics
Newtonian physics: absolute time means absolute synchronization
two clocks are synchronized when they “beat” simultaneously
absolute time t *
space
non-simultaneous
simultaneous

37. 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 *
non-simultaneous *
time T
simultaneous
time t

38. Clock synchronization: special relativity
Einstein synchronization for two clocks at rest in some inertial reference system
Clock a
Clock b

39. Clock synchronization: general relativity
Coordinate synchronization and coordinate simultaneity (Allan, Ashby, 1986):
Two events and are called simultaneous if and only if
The relation of proper time of a clock and coordinate time
is a differential equation of 1st order:
This equation gives unique relation if the initial condition is given:
This can be postulated for one clock, but for different clock the values must be consistent with each other.

40. One-way synchronization
Clock a
Clock b
Observed:
Given:
Calculated:
Result: 

41. Two-way synchronization
Clock a
Clock b
Observed:
Given:
Calculated:
Result: 

42. Clock-transport synchronization
Observed:
Given:
Calculated:
Result: 

43. 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 = ns

44. Realizations of coordinate time scales
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

45. 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.

46. 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.)

47. IAU Commission 52 “Relativity in Fundamental Astronomy”
Created by the IAU in August 2006
President: S.Klioner
Vice-president: G.Petit

48. Backup slides

49. 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

50. 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

51. 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 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

52. 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

53. Transformations between TCB and TCG
Part one: TCG(TCB) at the geocenter

54. 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

55. 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

56. Transformations between TG and TCB
The same scheme as for the pair TCB/TCG

57. Transformations between TG and TCB
The same idea with two small corrections  
The initial conditions 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

58. 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…

59. 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
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

60. 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
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.

61. Clock calibration: how to go from OBT to TG
Gaia:
OBT recording event
site on the Earth:
OBT packet reception event
Signal propagation
OBT
Position, velocity
OBT calibration
UTC
Position, velocity
TG of the OBT recording event
Gaia orbit, hardware calibration
TCB of the OBT recording event

62. Note: only radial position is relevant!
Martin Hechler, February 2006

63. 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
A simple clock model in UTC will be fitted (linear drift with least squares)
Can we do better?
It depends on the accuracy of the clock and the synchronization…

64. 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
TG-TT as a function of TT
linear trend removed: ×10 –10
sec
500
1000
1500
2000
2500
3000
3500
0.05
0.1
0.15
0.2
500
1000
1500
2000
2500
3000
3500
-1.5 -1
-0.5
0.5 1
1.5
days

65. TT-TDB: DE200 vs. SOFA ns ns