Latest advances in an astrometric model based on the Time Transfer Functions formalism

Transcription

1 Latest advances in an astrometric model based on the Time Transfer Functions formalism S. Bertone M. Crosta, C. Le Poncin-Lafitte, A. Vecchiato SYRTE - Paris Observatory Astrophysical Observatory of Torino - INAF Journées Systèmes de Référence Spatio-Temporels 16 September 2013 S. Bertone et al. Latest advances in an astrometric model based on TTF JSR13 1 / 14

2 Light bending in astrometric observations Classical models for ground based and early space era mas accuracy body light bending ψ max(1 µas) (µas) (deg) Sun o Mercury 83 9 Venus o Earth o Mars Jupiter o Saturn o Uranus Neptune Telescopes: astrometry, ground based catalogs Hipparcos: astrometry Relativistic models for space era µas accuracy Gaia: global map of the Milky way GAME: tests of General Relativity S. Bertone et al. Latest advances in an astrometric model based on TTF JSR13 1 / 14

3 Relativistic model for astrometry through Time Transfer Functions Time Transfer Functions (TTF) Information about light propagation between 2 points Alternative to reconstruction of null-geodesic path t A ˆk A ˆk B t B t : coordinate time k α g αβ dx β dλ Time of flight t B t A = T e/r (x A, x B, t A/B ) Direction triple (ˆk i ) A/B = ( ki k 0 )A/B [Le Poncin-Lafitte, Linet and Teyssandier, 2004] = c T [ e/r x i 1 T e/r B t B ] 1 S. Bertone et al. Latest advances in an astrometric model based on TTF JSR13 2 / 14

4 Relativistic model for astrometry through Time Transfer Functions Astrometric observables (1) Angular distance of two sources O A Receiver WL φ Γ 1 Γ 2 Source 1 WL Source 2 WL φ: angle between two incoming light ray ( k) : light direction triple at B observation coordinates β : observer s velocity in the chosen reference system (RS) Pointing astrometry (used by GAME) Relativistic model for φ [( g00 + 2g 0k β k + g kl β k β l) g ij (ˆk i ˆk i )(ˆk j ˆk ] j ) sin 2 φ 2 = 1 4 [Teyssandier and Le Poncin-Lafitte, 2006] (1 + β mˆk m )(1 + β lˆk l ) B S. Bertone et al. Latest advances in an astrometric model based on TTF JSR13 3 / 14

5 Relativistic model for astrometry through Time Transfer Functions Astrometric observables (2) Director cosines of an observation n : light direction in the observer s reference system (RS) ( k) : light direction triple at observation coordinates λ α (µ) B : transformation matrix to the RS comoving with the observer Scanning astrometry (used by Gaia) n (i) = λ0 (i) + λj (i)ˆk j λ 0 (0) + λj (0)ˆk j = n (i) ( x B, C P ; x A ) [Bertone and Le Poncin-Lafitte, 2012] S. Bertone et al. Latest advances in an astrometric model based on TTF JSR13 4 / 14

6 Relativistic model for astrometry through Time Transfer Functions General closed form 2PM equations PM development : g µν = η µν + G i h (i) µν, T (x A, x B, t A/B ) = G i T (i) i=1 i=0 Replace in the Hamilton-Jacobi like equation derived from isotropic condition: g 00 (t B T r, x A ) + 2cg 0i (t B T r, x A ) Tr x i + c 2 g ij (t B T r, x A ) Tr T r A x i A x j = 0 A Minkowskian order + perturbation as integral along straight line z α ( 1PM, 2PM ) T r = R AB c [ ] m z α (µ); g (1) αβ, x A, t B, x B dµ [ ] n z α (µλ); g (2) αβ, g(1) αβ, g(1) αβ,γ, x A, t B, x B dλdµ ) 1 ( ki = N AB i + B m A/B [ z α (µ); g (1) αβ, g(1) αβ,γ, x A, t B, x B ] dµ n A/B [ z α (µλ); g (2) αβ, g(2) αβ,γ, g(1) αβ, g(1) αβ,γ, g(1) αβ,γδ, x A, t B, x B ] dλdµ [Hees, Bertone and Le Poncin-Lafitte, 2013] S. Bertone et al. Latest advances in an astrometric model based on TTF JSR13 5 / 14

7 Relativistic model for astrometry through Time Transfer Functions Application to Gaia Scanning the whole sky 6h period sky scanning law No observations is Sun direction First fully-relativistic astrometric mission No external verification possible at Gaia accuracy Data analysis by independent models S. Bertone et al. Latest advances in an astrometric model based on TTF JSR13 6 / 14

8 Application to Gaia Relativistic astrometric models for Gaia GSR in Gaia 339 GREM [Klioner, 2003] Figure 1. RAMOD identifies a family of astrometric models with increasing accuracies. The attitude models belonging to the project are called RAMODINO1 and RAMODINO2. The present relativistic model implemented in GSR is an adaptation of PPN-RAMOD to the Gaia-type of measurement. 4. The Gaia Sphere Reconstruction in AVU AGIS has a duplication in one of the sub-systems of the AVU called Global Sphere Reconstruction (GSR). The input of both AGIS and GSR is a set of pre-processed data from the Gaia telemetry. As said in section 2, AGIS will process the data for up to 10 8 well-behaved stars. It is presently foreseen that GSR will rather use a subset of up to 10 million of stars chosen from the AGIS dataset. To keep the two reductions as independent as possible, GSR will differ from AGIS both from the point of view of the astrometric model and for the algorithm adopted for the sphere reconstruction, i.e. for solving the system of the linearized observation equations GSR astrometric model The astrometric model of AGIS is GREM (Gaia RElativistic Model) (Klioner, 2003), which is an extension of a seminal study (Klioner & Kopeikin, 1992) conducted in the framework of the post-newtonian (pn) approximation of General Relativity. In GREM this model has been formulated according to a Parametrized Post Newtonian (PPN) scheme accurate to 1 micro-arcsecond. The astrometric model of GSR is taken from the RAMOD project, which identifies a family of astrometric models with increasing accuracies (see, e.g. Vecchiato et al., 2003, de Felice et al., 2006 and references therein) conceived to solve the inverse raytracing problem in a general relativistic framework RAMOD and to use the tetrad formalism for the description of the observer s reference system. [de Felice et al., 2004] 4.2. GSR algorithm for the sphere reconstruction S. Bertone et al. Latest advances in an astrometric model based on TTF JSR13 7 / 14

9 Application to Gaia Relating the relativistic models Aberration [Crosta and Vecchiato, 2010] Additional analytical cross-check using the TTF Coordinate time of flight and light deflection RAMOD3 (ˆk i ) B = l i B [ ] (h 00) B + O(c 3 ) l = projection of k in the rest space of the observer GREM (ˆk i ) B = σ i (1 + 2(h 00 ) B ) + ẋi B + O(c 3 ) c σ = direction at past-null infinity ẋ = gravitational deflection Relations proven for a PPN metric adapted for Gaia S. Bertone et al. Latest advances in an astrometric model based on TTF JSR13 8 / 14

10 Application to Gaia TTF-RAMOD (µas) Distance from planets (deg/10) Jupiter Saturn Mars GSR1+ : PPN-RAMOD + Gaia tetrad + corrections for planets Expected deviations (up to 500 µas) near Jupiter Sub-µas difference far from planets Future GSR3 : will implement RAMOD3 + Gaia tetrad µas accuracy S. Bertone et al. Latest advances in an astrometric model based on TTF JSR13 9 / 14

11 Application to Gaia TTF-GREM (µas) Distance from planets (deg/10) Jupiter Saturn Mars GREM (light deflection + tetrad [Klioner 2003/2004]) accurate at 1 µas Deviation under Gaia accuracy near massive planets ( 1 from Jupiter) Unexpected shape with a periodic trend and sign changes S. Bertone et al. Latest advances in an astrometric model based on TTF JSR13 10 / 14

12 Conclusions Conclusions General covariant description of an astrometric observation Sistematic development in the PM approximation of GR Application in the Gaia context with analytical comparison of light propagation equations and simulated observation angles Differences in the results of TTF and GREM based simulations. This points out the necessity of independent relativistic models Perspectives Further analysis of implemented models Sphere reconstruction from TTF model and 5 years of simulated observations S. Bertone et al. Latest advances in an astrometric model based on TTF JSR13 11 / 14

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14 Simulation of a GAME-like observation Observation of two sources at 100 AU from Earth Earth orbiting around the Sun, one of the sources on the galactic plane Maximum bending at conjunction (one source aligned with Sun) 1PM correction 1.75 as 2PM correction 2.4 mas Comparison to [Klioner and Zschocke, 2010], [Teyssandier, 2012] S. Bertone et al. Latest advances in an astrometric model based on TTF JSR13 13 / 14

15 Implementation of the GSR-TTF software Simulation of astrometric observations (at Turin Astrophysical Observatory) star coordinates Solar System ephemeris Gaia coordinates light direction ki (TTF) Gaia tetrad (RAMOD) Φ from GREM and RAMOD simulated Φ Φ : angle of observation with respect to Gaia axes S. Bertone et al. Latest advances in an astrometric model based on TTF JSR13 14 / 14