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

Relativistic astronomical time scales

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

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 RS of an observer

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

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:

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

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:

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

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

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

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 32.184 s) 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…

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

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 = 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.

Linear drifts between time scales Pair Drift per year (seconds) Difference at J2007 TT-TCG 0.021993 0.65979 TDB-TCB 0.489307 14.67921 TCB-TCG @ geocenter 0.467313 14.01939

Scaled BCRS and GCRS

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?

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

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…

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

4-dimensional ephemerides

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!

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)

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.)

How to compute TT(TDB) from an ephemeris 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) :

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

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…

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

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

TT-TDB: DE405 vs. DE200 ns ns

TT-TDB: DE405 vs. DE403 ns ns

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 

Clock synchronization

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

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

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

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.

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

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

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

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

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

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.

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.)

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

Backup slides

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

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

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

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

Transformations between TCB and TCG 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

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

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

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

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…

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

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.

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

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

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…

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: 6.926 ×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

TT-TDB: DE200 vs. SOFA ns ns