1. 1 Ephemerides in the relativistic framework: _ time scales, spatial coordinates, astronomical constants and units Sergei A.Klioner Lohrmann Observatory, Dresden Technical University Paris Observatory, 6 April 2006

2. 2 Contents Newtonian astrometry and Newtonian equations of motion Why relativity? Coordinates, observables and the principles of relativistic modelling IAU 2000: BCRS and GCRS, metric tensors, transformations, frames Relativistic time scales and reasons for them: TCB, TCG, proper times, TT, TDB, T eph Scaled-BCRS Astronomical units in Newtonian and relativistic frameworks Do we need astronomical units? Scaled-GCRS TCB/TCG-, TT- and TDB-compatible planetary masses TCB-based or TDB-based ephemeris: notes, rules and recipes

3. 3 Accuracy of astrometric observations 1 mas 1 µas 10 µas 100 µas 10 mas 100 mas 1“ 10” 100” 1000” 1 µas 10 µas 100 µas 1 mas 10 mas 100 mas 1” 10” 100” 1000” 140015001700190020002100016001800 Ulugh Beg Wilhelm IV Tycho Brahe Hevelius Flamsteed Bradley-Bessel FK5 Hipparcos Gaia SIM ICRF GC naked eye telescopes space 140015001700190020002100016001800 Hipparchus 4.5 orders of magnitude in 2000 years further 4.5 orders in 20 years 1  as is the thickness of a sheet of paper seen from the other side of the Earth

4. 4 Modelling of astronomical observations in Newtonian physics M. C. Escher Cubic space division, 1952

5. 5 Astronomical observation physically preferred global inertial coordinates observables are directly related to the inertial coordinates

6. 6 Modelling of positional observations in Newtonian physics Scheme: aberration parallax proper motion All parameters of the model are defined in the preferred global coordinates: Newtonian equations of motion:

7. 7 Why general relativity? Newtonian models cannot describe high-accuracy observations: many relativistic effects are many orders of magnitude larger than the observational accuracy  space astrometry missions or VLBI would not work without relativistic modelling The simplest theory which successfully describes all available observational data: APPLIED RELATIVITY 

8. 8 Astronomical observation physically preferred global inertial coordinates observables are directly related to the inertial coordinates

9. 9 Astronomical observation no physically preferred coordinates observables have to be computed as coordinate independent quantities

10. 10 General relativity for astrometry Coordinate-dependent parameters Equations of signal propagation Astronomical reference frames Observational data Relativistic equations of motion Definition of observables Relativistic models of observables A relativistic reference system

11. 11 General relativity for astrometry Coordinate-dependent parameters Relativistic reference systems Equations of signal propagation Astronomical reference frames Observational data Relativistic equations of motion Definition of observables Relativistic models of observables

12. 12 The IAU 2000 framework in Manchester…

13. 13 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

14. 14 Relativistic Astronomical Reference Systems particular reference systems in the curved space-time of the Solar system One can use any but one should fix one

15. 15 The Barycentric Celestial Reference System The BCRS is suitable to model processes in the whole solar system

16. 16 N-body problem in the BCRS Equations of motion in the PPN-BCRS: Einstein-Infeld-Hoffman (EIH) equations: General relativity: Lorentz, Droste, 1916 EIH, 1936 Damour, Soffel, Xu, 1992 PPN formalism: Will, 1973; Haugan, 1979; Klioner, Soffel, 2000 used in the JPL ephemeris software (usually  =  =1)

17. 17 Geocentric Celestial Reference System The GCRS is adopted by the International Astronomical Union (2000) to model physical processes in the vicinity of the Earth… Why not BCRS?

18. 18 Geocentric Celestial Reference System Imagine a sphere (in inertial coordinates of special relativity), which is then forced to move in a circular orbit around some point… What will be the form of the sphere for an observer at rest relative to that point? Lorentz contraction deforms the shape… Direction of the velocity Additional effect due to acceleration (not a pure boost) and gravitation (general relativity, not special one)

19. 19 Geocentric Celestial Reference System The GCRS is adopted by the International Astronomical Union (2000) to model physical processes in the vicinity of the Earth: A: The gravitational field of external bodies is represented only in the form of a relativistic tidal potential. B: The internal gravitational field of the Earth coincides with the gravitational field of a corresponding isolated Earth.

20. 20 Geocentric Celestial Reference System The GCRS is adopted by the International Astronomical Union (2000) to model physical processes in the vicinity of the Earth: A: The gravitational field of external bodies is represented only in the form of a relativistic tidal potential. B: The internal gravitational field of the Earth coincides with the gravitational field of a corresponding isolated Earth.

21. 21 Geocentric Celestial Reference System The GCRS is adopted by the International Astronomical Union (2000) to model physical processes in the vicinity of the Earth: A: The gravitational field of external bodies is represented only in the form of a relativistic tidal potential. B: The internal gravitational field of the Earth coincides with the gravitational field of a corresponding isolated Earth. internal + inertial + tidal external potentials

22. 22 BCRS-GCRS transformation The coordinate transformations: with whereare the BCRS position and velocity of the Earth,and and the orientation is CHOSEN to be kinematically non-rotating: are explicit functions,

23. 23 Local reference system of an observer The version of the GCRS for a massless observer: A: The gravitational field of external bodies is represented only in the form of a relativistic tidal potential. Modelling of any local phenomena: observation, attitude, local physics (if necessary) internal + inertial + tidal external potentials observer

24. 24 Celestial Reference Frame All astrometric parameters of sources obtained from astrometric observations are defined in BCRS coordinates: positions proper motions parallaxes radial velocities orbits of (minor) planets, etc. orbits of binaries, etc. These parameters represent a realization (materialization) of the BCRS This materialization is „the goal of astrometry“ and is called Celestial Reference Frame

25. 25 Relativistic 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.

26. 26 Relativistic Time Scales: TCB and TCG Important special case gives the TCG-TCB relation at the geocenter: linear drift removed:

27. 27 Relativistic Time Scales: 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:

28. 28 Relativistic Time Scales: proper time scales Specially interesting case: an observer close to the Earth surface: But is the definition of the geoid! Therefore is the height above the geoid is the velocity relative to the rotating geoid

29. 29 Relativistic Time Scales: TT Idea: let us define a time scale linearly related to T=TCG, but which numerically coincides with the proper time of an observer on the geoid: with Then To avoid errors and changes in TT implied by changes/improvements in the geoid, the IAU (2000) has made L G 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

30. 30 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 a 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…

31. 31 Relativistic Time Scales: T eph Since the original TDB definition has been recognized to be flawed Myles Standish (1998) introduced one more time scale T eph 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! The coefficients could only be restored by some numerical procedure (Fukushima’s “Time ephemeris”) For JPL only the transformation from TT to T eph which matters… VSOP-based analytical formulas (Fairhead-Bretagnon) are used for this transformations

32. 32 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: T 0 = 2443144.5003725 exactly, JD TCB = T 0 for the event 1977 Jan 1.0 TAI at the geocenter and increases by 1.0 for each 86400s of TCB, L B = 1.550519768×10 −8 exactly, TDB 0 = −6.55 ×10 −5 s exactly. Using this “new TDB”, it is trivial to convert from TDB to TCB and back.

33. 33 Gaia needs: TCB With all this involved situation with TDB/T eph the only unambiguous way is to use TCB for all aspects of data processing: solar system ephemeris Gaia orbital data time parametrization of proper motions time parametrization of orbital solutions (asteroids and stars) … TCB was officially agreed to be the fundamental time scale for Gaia

34. 34 Scaled BCRS: not only time is scaled If one uses scaled version TCB – T eph or TDB – one effectively uses three scaling: time spatial coordinates masses (  = GM) of each body (from now on “*” refer to quantities defined in the scaled BCRS; these quantities are called TDB-compatible ones) WHY THREE SCALINGS?

35. 35 These three scalings together leave the dynamical equations unchanged: for the motion of the solar system bodies: for light propagation: Scaled BCRS

36. 36 These three scalings lead to the following: semi-major axes period mean motion the 3 rd Kepler’s law Scaled BCRS

37. 37 Quantities: numerical values and units of measurements Arbitrary quantity can be expressed by a numerical value in some given units of measurements : XX denote a name of unit or of a system of units, like SI

38. 38 Quantities: numerical values and units of measurements Consider two quantities A and B, and a relation between them: No units are involved in this formula! The formula should be used on both sides before numerical values can be discussed. In particular, is valid if and only if

39. 39 For the scaled BCRS this gives: Scaled BCRS Numerical values are scaled in the same way as quantities if and only if the same units of measurements are used.

40. 40 SI units time length mass Astronomical units time length mass Astronomical units in the Newtonian framework

41. 41 Astronomical units vs SI ones: time length mass AU is the unit of length with which the gravitational constant G takes the value AU is the semi-major axis of the [hypothetic] orbit of a massless particle which has exactly a period of in the framework of unperturbed Keplerian motion around the Sun Astronomical units in the Newtonian framework

42. 42 Values in SI and astronomical units: distance (e.g. semi-major axis) time (e.g. period) GM Astronomical units in the Newtonian framework

43. 43 Be ready for a mess! Astronomical units in the relativistic framework

44. 44 Astronomical units in the relativistic framework Let us interpret all formulas above as TCB-compatible astronomical units Now let us define a different TDB-compatible astronomical units The only constraint on the constants:

45. 45 Astronomical units in the relativistic framework Possibility I: Standish, 1995 This leads to strange scaling…

46. 46 Astronomical units in the relativistic framework Possibility II: Brumberg & Simon, 2004; Standish, 2005 This leads to The same scaling as with SI: Either k is different or the mass of the Sun is not one or both!

47. 47 From the DE405 header one gets: TDB-compatible AU: Using that (also can be found in the DE405 header!) one gets the TDB-compatible GM of the Sun expressed in SI units The TCB-compatible GM reads (this value can be found in IERS Conventions 2003) How to extract planetary masses from the DEs

48. 48 The reason to introduce astronomical units was that the angular measurements were many order of magnitude more accurate than distance measurements. Arguments against astronomical units The situation has changed crucially since that time! Solar mass is time-dependent just below current accuracy of ephemerides Complicated situations with astronomical units in relativistic framework Why not to define AU conventionally as fixed number of meters? Do you see any good reasons for astronomical units? Do we need astronomical units?

49. 49 Scaled GCRS Again three scalings (“**” denote quantities defined in the scaled GCRS; these TT-compatible quantities): time spatial coordinates masses (  =GM) of each body the scaling is fixed Note that the masses are the same in non-scaled BCRS and GCRS… Example: GM of the Earth from SLR (Ries et al.,1992; Ries, 2005) TT-compatible TCG-compatible

50. 50 GM of the Earth from SLR: TT-compatible TCG/B-compatible TDB-compatible GM of the Earth from DE: DE403 DE405 TCG/TCB-, TT- and TDB-compatible planetary masses Should the SLR mass be used for ephemerides?

51. 51 Note 1: T eph defined by a fixed relation to TT may be a source of inconsistency since newer ephemerides are not fully compatible with the T eph –TT relation used for their development (derived on the basis of VSOP87/DE200) Note 2: No good reasons to develop more accurate analytical formulas: just like with the ephemeredes too many terms… Note 3: With fixed scaling constant K=1-L B (that is, with the re-defined TDB) it is impossible to have different post-fit residuals when using TDB or TCB. The fits must be absolutely equivalent! Note 4: Once a TDB ephemeris is constructed, it is trivial to convert it to TCB and vice verse: just use the three scalings given above! TCB-based or TDB-based ephemeris?

52. 52 Iterative procedure to construct ephemeris with TCB or TDB in a fully consistent way 1.Use some apriori T(C/D)B–TT relation (based on some older ephemeris) to convert the observational data from TT to T(C/D)B 2.(Re-) Construct the new ephemeris 3.Update the T(C/D)B–TT relation by numerical integration using the new ephemeris 4.Convert the observational data from TT to T(C/B)D using the updated T(C/D)B–TT relation 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)