1 / 52

550 likes | 737 Vues

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. Newtonian astrometry and Newtonian equations of motion Why relativity?

Télécharger la présentation
## Sergei A.Klioner

**An Image/Link below is provided (as is) to download presentation**
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.
Content is provided to you AS IS for your information and personal use only.
Download presentation by click this link.
While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

**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**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, Teph • 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 Contents**naked eye**telescopes space 0 1400 1500 1600 1700 1800 1900 2000 2100 Hipparchus Ulugh Beg 1000” 1000” 100” 100” Wilhelm IV Tycho Brahe Flamsteed 10” Hevelius 10” Bradley-Bessel 1“ 1” GC 100 mas 100 mas FK5 10 mas 10 mas Hipparcos further 4.5 orders in 20 years 1 mas 1 mas ICRF 100 µas 100 µas 10 µas Gaia 10 µas 1 µas SIM 1 µas 0 1400 1500 1600 1700 1800 1900 2000 2100 Accuracy of astrometric observations 4.5 orders of magnitude in 2000 years 1 as is the thickness of a sheet of paper seen from the other side of the Earth**Modelling of astronomical observations in Newtonian physics**M. C. Escher Cubic space division, 1952**Astronomical observation**physically preferred global inertial coordinates observables are directly related to the inertial coordinates**Scheme:**• aberration • parallax • proper motion • All parameters of the model are defined • in the preferred global coordinates: • Newtonian equations of motion: Modelling of positional observations in Newtonian physics**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 Why general relativity? **Astronomical observation**physically preferred global inertial coordinates observables are directly related to the inertial coordinates**Astronomical observation**no physically preferred coordinates observables have to be computed as coordinate independent quantities**A relativistic reference**system Relativistic equations of motion Equations of signal propagation Definition of observables Relativistic models of observables Coordinate-dependent parameters Observational data Astronomical reference frames General relativity for astrometry**Relativistic reference**systems Relativistic equations of motion Equations of signal propagation Definition of observables Relativistic models of observables Coordinate-dependent parameters Observational data Astronomical reference frames General relativity for astrometry**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**particular reference systems in the curved space-time of the**Solar system Relativistic Astronomical Reference Systems • One can • use any • but one • should • fix one**The BCRS is suitable to model processes in the whole solar**system The Barycentric Celestial Reference System**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 N-body problem in the BCRS used in the JPL ephemeris software (usually ==1)**The GCRS is adopted by the International Astronomical Union**(2000) to model physical processes in the vicinity of the Earth… Why not BCRS? Geocentric Celestial Reference System**Direction ofthe velocity**• 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… Geocentric Celestial Reference System Additional effect due to acceleration (not a pure boost) and gravitation (general relativity, not special one)**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. 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. 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. Geocentric Celestial Reference System internal + inertial + tidal external potentials**with**where and are the BCRS position and velocity of the Earth, • The coordinate transformations: BCRS-GCRS transformation are explicit functions, and the orientation is CHOSEN to be kinematically non-rotating:**internal + inertial + tidal external potentials**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. Local reference system of an observer observer • Modelling of any local phenomena: • observation, • attitude, • local physics (if necessary)**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 Celestial Reference Frame**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. Relativistic Time Scales: TCB and TCG**Important special case gives the**TCG-TCB relation • at the geocenter: Relativistic Time Scales: TCB and TCG linear drift removed:** 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: Relativistic Time Scales: proper time scales**is the height above the geoid**is the velocity relative to the rotating geoid • Specially interesting case: an observer close to the Earth surface: Relativistic Time Scales: proper time scales • But is the definition of the geoid! • Therefore**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 Relativistic Time Scales: TT • Then • 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**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… Relativistic Time Scales: TDB-1**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! • The coefficients could only be restored by some numerical procedure (Fukushima’s “Time ephemeris”) • For JPL only the transformation from TT to Teph which matters… • VSOP-based analytical formulas (Fairhead-Bretagnon) are used for this • transformations**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 exactly, • TDB0 = −6.55 ×10−5 s exactly. • Using this “new TDB”, it is trivial to convert from TDB to TCB and back. Relativistic Time Scales: TDB-2**With all this involved situation with TDB/Teph 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 Gaia needs: TCB**If one uses scaled version TCB – Teph 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? Scaled BCRS: not only time is scaled**These three scalings together leave the dynamical**equations unchanged: • for the motion of the solar system bodies: • for light propagation: Scaled BCRS**These three scalings lead to the following:**• semi-major axes • period • mean motion • the 3rd Kepler’s law Scaled BCRS**Arbitrary quantity can be expressed by a**numerical value • in some given units of measurements : Quantities: numerical values and units of measurements • XX denote a name of unit or of a system of units, like SI**Consider two quantities A and B, and a relation between**them: Quantities: numerical values and units of measurements • 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**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.**SI units**• time • length • mass • Astronomical units • time • length • mass Astronomical units in the Newtonian framework**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**Values in SI and astronomical units:**• distance (e.g. semi-major axis) • time (e.g. period) • GM Astronomical units in the Newtonian framework**Astronomical units in the relativistic framework**Be ready for a mess!**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: Astronomical units in the relativistic framework**strange scaling…**• Possibility I: Standish, 1995 • This leads to Astronomical units in the relativistic framework**The same scaling as with SI:**• Possibility II: Brumberg & Simon, 2004; Standish, 2005 • This leads to Astronomical units in the relativistic framework Either k is differentor the mass of the Sunis not one or both!**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**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?**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 Scaled GCRS**Should the SLR mass beused for ephemerides?**• 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

More Related