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

ephT

TCB

TCGTTTDB

TAItimeGPS

UTC

(BIPM)TT

TDT

3

General relativity for space astrometry

Time scalesRelativistic reference

systems

Equations ofsignal

propagation

Astronomicalreference

frames

Observational data

Relativisticequationsof motion

Definition ofobservables

Relativisticmodels

of observables

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, 1988-1992 Damour, Soffel, Xu, 1991-1994 Klioner, Voinov, 1993 Klioner,Soffel, 2000 Soffel, Klioner,Petit et al., 2003

BCRS

GCRS

Local RSof an observer

5

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

3. 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:

52 4

1 1( ) ( ) ( ) ( ) ,i i i i ij i j

E E E E ET t A t v r B t B t r B t r r C t O cc c

x

( , )iTCG TCG TCB x

( )i iobsx x t

( ( ) ) i i iE Er x x t

( , ( )) ( )iobsTCG TCG TCB x TCB TCG TCB

7

Coordinate Time Scales: TCB and TCG• Important special case gives the TCG-TCB relation at the geocenter:

( )i iEx x t

linear drift removed:

1950 1960 1970 1980 1990 2000 2010

-15

-10

-5

0

5

10

15

s

s

1970 1972 1974 1976 1978 1980 1982 1984

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0.0015

main feature: linear drift 1.4810-8

zero point is defined to be Jan 1, 1977difference now: 14.7 seconds

8

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:

and/or( ) ( )i aobs obsx t X T

1/ 2

00 0 2

2 1, ( ) , ( ) ( ) , ( ) ( ) ( )i i j

obs i obs obs ij obs obs obs

dg t t g t t x t g t t x t x t

dt c c

x x x

1/ 2

00 0 2

2 1, ( ) , ( ) ( ) , ( ) ( ) ( )a a b

obs a obs obs ab obs obs obs

dG T T G T T X T G T T X T X T

dT c c

X X X

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 xi

o(t) in the BCRS

00

0 3

22

4

2

1 ,

,

.

2( , )

4( , )

2( ,

2( , )

1 )

ii

ij ij

w tc

w tc

g

g

w tc

w t

g

cx

x

x

x

421

1 1pppN NA

t cA

d c

d

10

Local Positional Invariance

One aspect of the LPI can be tested by measuring the gravitational red shift of clocks

degree of the violation ofthe gravitational red shift

21

U

c

11

Proper time scales and TCG

• Specially interesting case: an observer close to the Earth surface:

2 42

1 11 ( ) , "tidal terms"

2

Xobs E obs

dX T W T O c

dT c

1710

h is the height above the geoid

iis the velocity relative to the rotating geoid

• 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:

-10(1 ) , 6.969290134 10G GTT L TCG L

2

11 " terms , " " tidal terms" ... ...id

hd TT c

can be neglectedin many cases

12

Coordinate Time Scales: TT

h is the height above the geoid

iis the velocity relative to the rotating geoid

• 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:

-10(1 ) , 6.969290134 10G GTT L TCG L

2

11 " terms , " " tidal terms" ... ...id

hd TT c

can be neglectedin many cases

• 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:

-106.969290134 10 GL

• 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

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:

0eph ephT R TCB T

• 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-2The 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 = 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.

0 086400 B TCBTDB TCB L JD T TDB

16

Linear drifts between time scales

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

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?

* *0

*

*

t F TCB t

F

F

x x

1 BF L

19

• These three scalings together leave the dynamical equations unchanged:

• for the motion of the solar system bodies:

• for light propagation:

Scaled BCRS

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…

**

**

**

T TT L TCG

X L X

L

1 GL L

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

*

**

, 1

, 1

B

G

F F L

L L L

** 6

1 ** 6

* 6

398600441.5 0.4 10

1 398600441.8 0.4 10

1 398600435.6 0.4 10

G

B

L

L

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!

52 4

1 1( ) ( ) ( ) ( ) ,i i i i ij i j

E E E E ET t A t v r B t B t r B t r r C t O cc c

x

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

update the TCB–TT relation (by numerical integration using the new ephemeris)

changed?

yes

final 4Dephemeris

no

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( )

( )

TT TDB TDB TDB

TDB TT TT TT

2 4

1 1( ) 1 ( )B B G G

dTDB L TDB L L L TDB

dTDB c c

- two differential equations

2

24

1( ) 1

1( ) ( ) 1

B G

B G G

dTT TT TT L L

dTT c

TT TT TT TT L L Lc

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

180.868 ns- 8.28 10 t

1600 1700 1800 1900 2000 2100 2200

-50

-25

0

25

50

75

1600 1700 1800 1900 2000 2100 2200

-15

-10

-5

0

5

10

15

ns

ns

- SOFA implements thecorrected Fairhead-Bretagnon analytical series basedon VSOP-87(about 1000 Poisson terms, also non-periodic terms)

31

TT-TDB: DE405 vs. SOFA for 1960-2020

180.868 ns- 8.28 10 t

1960 1970 1980 1990 2000 2010 2020-10

-5

0

5

10

1960 1970 1980 1990 2000 2010 2020

0

2

4

6

8

ns

ns

32

TT-TDB: DE405 vs. DE200

171.171 ns- 1.087 10 t

1700 1800 1900 2000 2100

-50

-25

0

25

50

75

100

125

1700 1800 1900 2000 2100

-4

-2

0

2

4

ns

ns

33

TT-TDB: DE405 vs. DE403

200.005 ns- 3 10 t

1600 1700 1800 1900 2000 2100 2200

-0.2

-0.1

0

0.1

0.2

0.3

0.4

1600 1700 1800 1900 2000 2100 2200

-0.1

-0.075

-0.05

-0.025

0

0.025

0.05

0.075

ns

ns

34

4-dimensional ephemerides

- IMCCE (Fienga, 2007) and JPL (Folkner, 2007) have agreed to includetime 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 synchronizationtwo clocks are synchronized when they “beat” simultaneously

absolute time t

*

* *

*

space

1 2t t1 2t t

event 1

event 2

event 1

event 2

non-simultaneous

simultaneous

37

Clock synchronization: special relativity

- Special relativity: time is relative and synchronization is also relativetwo events (e.g. two clocks showing 00:00:00 exactly) can be simultaneousin one inertial reference system and non-simultaneous in another one

time t

*

*

space

1 2t t

event 1

event 2

time T

1 2T T

non-simultaneous

simultaneous

38

Clock synchronization: special relativity

Einstein synchronization for two clocks at rest in some inertial reference system

Clock aClock b

0t

1t

2t

1 2 0

1

2 t t t

39

Clock synchronization: general relativity

- Coordinate synchronization and coordinate simultaneity (Allan, Ashby, 1986):

Two events and are called simultaneous if and only if

1/ 2

00 0 2

2 1, ( ) , ( ) ( ) , ( ) ( ) ( )i i j

obs i obs obs ij obs obs obs

dg t t g t t x t g t t x t x t

dt c c

x x x

1 1( , )it x 2 2( , )it x 1 2t t

- The relation of proper time of a clock and coordinate timeis a differential equation of 1st order:

- This equation gives unique relation if the initial condition is given:

0 0( ) t

- This can be postulated for one clock, but for different clock the values must be consistent with each other.

40

One-way synchronization

Observed:

Given:

Calculated:

Result:

0 , ( ), ( )i ia bt x t x t

0 1, a b

1t

1 1( ) b bt

Clock a

Clock b

1 0

0 1

1,

( ) ( )

gr

i ia b

t t Rc

R x t x t

41

Two-way synchronization

Observed:

Given:

Calculated:

Result:

0 1 2, , a b a

0 , ( ), ( )i ia bt x t x t

1t

1 1( ) b bt

Clock a

Clock b

2 0 2 0

1 0 2 0

2 1

,

1,

21

2

a a

gr

t t

t t t t

R Rc

42

Clock-transport synchronization

Observed:

Given:

Calculated:

Result:

0 0 1 1, , , a c c b

0 , ( ), ( ), ( )i i ia b ct x t x t x t

1 1( ) b bt

1t 1 0 1 0 c c t t

43

Clock-transport synchronization: experiments

Hafele & Keating(1972):

comparison withground-based clocks

eastward flight:

tg + tv =

+144 ns -184 ns = - 40 ns

westward flight:

+179 ns + 96 ns = + 275 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 observationsas 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

http://astro.geo.tu-dresden.de/RIFA

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+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

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 clockare fitted to get the calibration of model of the OBT.

61

Clock calibration: how to go from OBT to TG

OBTPosition, velocity

Gaia: OBT recording event

site on the Earth:OBT packet reception event

UTCPosition, velocity

TCB of the OBT recording event

TG of the OBT recording event

OB

T

calibratio

n

Gaia orbit, hardware calibration

Signal propagation

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

• 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: 6.926 ×10 –10

sec

days

Relation between TG and TT

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

65

TT-TDB: DE200 vs. SOFA

1700 1800 1900 2000 2100

-40

-30

-20

-10

0

10

1700 1800 1900 2000 2100

-10

-5

0

5

10

182.773 ns 2.60 10 t

ns

ns