Relativistic time scales and relativistic time synchronization

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

4. BCRS GCRS Local RS of an observer • 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 The IAU 2000 framework

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. 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: Coordinate Time Scales: TCB and TCG

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

8.  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: Proper Time Scales

9. 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 Proper Time Scales

10. 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 Local Positional Invariance

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

12. can be neglected in many cases is the height above the geoid is 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: Coordinate Time Scales: TT • 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 32.184 s) • Older name TDT (introduced by IAU 1976): fully equivalent to TT

13. 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… Relativistic Time Scales: TDB-1

14. 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: Relativistic Time Scales: Teph • 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. 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 = 2443144.5003725 exactly, • JDTCB = T0 for the event 1977Jan1.0 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. Relativistic Time Scales: TDB-2

16. Linear drifts between time scales

17. Scaled BCRS and GCRS

18. 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? Scaled BCRS: not only time is scaled

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

20. 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… Scaled GCRS

21. 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 Scaled masses

22. 4-dimensional ephemerides

23. 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! Time scales important for ephemerides

24. a priori TCB–TT relation (from an old ephemeris) Iterative procedure to construct ephemeris with TCB or TDB in a fully consistent way 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. 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.) Notes on the iterative procedure

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

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

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

31. TT-TDB: DE405 vs. SOFA for 1960-2020 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. space non-simultaneous * * * simultaneous * absolute time t Clock synchronization: Newtonian physics • Newtonian physics: absolute time means absolute synchronization • two clocks are synchronized when they “beat” simultaneously

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 timet

38. Clock synchronization: special relativity Einstein synchronizationfor 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. Clock a Clock b Observed: Given: Calculated: Result: One-way synchronization 

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

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

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 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. Created by the IAU in August 2006 • President: S.Klioner • Vice-president: G.Petit • http://astro.geo.tu-dresden.de/RIFA IAU Commission 52 “Relativity in Fundamental Astronomy”

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