General Relativistic Astrometry: the RAMOD project as a tool for highly accurate observations in space Maria Teresa Crosta (P) + , D. Bini ∗ , B. Bucciarelli + , F. de Felice § , M.G. Lattanzi + , A.Vecchiato + + INAF-Astronomical Observatory of Turin, Pino Torinese, Italy; ∗ ICRA International Center for Relativistic Astrophysics, University of Rome, Italy; § Physics Departement G.Galilei , University of Padova, Italy Firenze, 28-30/09/2006 – p.1/16
RAMOD&Relativistic Astrometry: from the observer to the star 1. RAMOD1: a static non-perturbative model in the Schwarzschild metric of the Sun (de Felice at al., 1998,A&A, 332,1133) 2. RAMOD2: a dynamical extension of Several Relativistic Astrometric RAMOD1 (parallaxes and proper motions, de MODels of increasing intrinsic Felice at al., 2001,A&A, 373,336) accuracy (up to 0.1 µ as) and adapted 3. PPN-RAMOD: recasting RAMOD2 in the to many different satellite setting PPN Schwarzschild metric of the Sun (Vec- (including software engineering) chiato et al, 2003, A&A, 399,337) 4. RAMOD3: a perturbative model of the light propagations in the static field of the Solar 4 R D A System (1/ c 2 , de Felice et al., 2004, ApJ, O M R M A O M A D O R I 3 D D P P N N 607,580 ) O - R I A N O M M O 2 A M O A D R O 2 R 1 D 5. RAMOD4: the extension of RAMOD3 to the 1/ c 3 level of accuracy (1/ c 3 ≡ 0 . 1 µ as, de Fe- RAMOD1 lice et al., 2006, ApJ, in press) 6. RAMODINO1-2: satellite-observer model for Gaia (Bini et al., 2003, Class. Quantum Grav.,20,2251/4695) Firenze, 28-30/09/2006 – p.2/16
The astrometric observable as a physical measurement Modelling the Gaia observable requires to P (u’) α β solve the inverse problem of light ray trac- u’ E { } ing, which connects the satellite to the emit- a l obs ting star. The astrometric observable ≡ an- gles that the incoming light ray forms with the axes of the spatial attitude triad E ˆ a in the rest frame of the satellite: E β P ( u ′ ) αβ k α E E u’ a ,ℓ obs ) ≡ e ˆ a ˆ cos ψ ( E ˆ a = ( P ( u ′ ) αβ k α k β ) 1 / 2 l t o o where P ( u ′ ) αβ = g αβ + u ′ α u ′ u β . The in- coming light ray k α is the solution of the k null geodesic considering the full gravita- l tional field of the Solar System presented in t GAIA photon RAMOD4 (de Felice et al., ApJ, in press, u astro-ph/0609073). Firenze, 28-30/09/2006 – p.3/16
The satellite attitude triad modelling satellite optics focal palne and CCDs tetrad attitude tetrad Satellite barycentric motion local reference attitude system scanning global reference law system Attitude/Tetrad Firenze, 28-30/09/2006 – p.4/16
The astrometric set-up (1) The background geometry felt by the satellite: g αβ = ( η αβ + h αβ + O ( h 2 )) → g 00 = − 1+ h 2 00 + O (4) , g 0 a = 3 0 a + O (5) , h g ab = 1 + h 2 00 δ ab + O (4) (compatible with retarded potential solutions and/or the IAU resolution B1.3, 2000) u ′ = (2) satellite’s trajectory: u ′ u ′ T s ( ∂ ∂ t + β 1 ∂ ∂ x + β 2 ∂ ∂ y + β 3 ∂ ∂ z ) ∂ ∂ ∂ ∂ time-like, unitary four-vector ∂ ∂ α ’s ≡ coordinate basis vectors ∂ relative to the barycentric coordinate system (BCRS) β i ≡ BCRS coordinate compo- nents of the satellite three-velocity Firenze, 28-30/09/2006 – p.5/16
(3) Global BCRS : BCRS is identified by three spatial axes at the barycenter of the Solar System ( B ) and pointing to distant cosmic sources (kinematically non-rotanting); the axes define a Carthesian-like coordinate system ( x, y, z ) and there exist space-like hypersurfaces with equation t ( x, y, z ) = constant → the function t is chosen as coordinate time. u u’ i x B t (x, y, z) = const. u GAIA t u u (the satellite world line with respect to the Carthesian like coordinate system ( x i ) and the space-like hypersurface) Firenze, 28-30/09/2006 – p.6/16
(4) Local BCRS : at any point in space-time there exists an observer at rest relative to the BCRS; world-line of B u’ u = ( g tt ) − 1 / 2 ∂ local barycentric observer u u ∂ t = (1 + U ) ∂ ∂ ∂ ∂ t + O (4) t → local triad of space-like vectors which point to u the local coordinate directions ( U gravitational potential); u the proper time of u u is the barycentric proper-time u GAIA t , since t photon u α = dx α u dσ = ( − g 00 ) 1 / 2 δ α 0 σ ( x i , t ) is the world line parameter of u u u Firenze, 28-30/09/2006 – p.7/16
(5) Light trajectory : The light signal arriving at the local BCRS is ℓ = P ( u ) ρσ k σ ( local line-of-sight , ℓ ℓ P ( u ) ρσ operator which projects into the u’ u ), which is the solution of rest space of u u k l dℓ α o dσ = F α ( ∂ β h ( x, y, z, t ) , ℓ i ( σ ( x ))) t o u A general solution is: k ℓ i ( σ ) = f i ( σ, ℓ k obs ) GAIA l which links the parameters of the star to the physical measurements ( condition t photon u equation )-> the mathematical character- ization of Gaia’s attitude triad is essen- tial to solve the boundary value problem in the process of reconstructing the light trajectory Firenze, 28-30/09/2006 – p.8/16
The mathematical rest frame: the tetrad The rest-frame of an observer consists of a clock (satellite proper-time) + a space (triad of orthonormal axes). The mathematical quantity which defines a rest-frame of a given observer is the tetrad adapted to that observer : g µν λ µ α λ ν β = η ˆ α ˆ ˆ ˆ β 0 ≡ u ′ (space-time history of the observer in a given space-time) λ ˆ λ ˆ a ≡ spatial triad of space-like vectors There are many possible spaces to be fixed within a satellite → which is the actual attitude frame for Gaia? Firenze, 28-30/09/2006 – p.9/16
The attitude frame for Gaia λ 3 y λ z 2 First step: we need to λ s 1 GAIA identify the spatial direc- λ SUN θ s 1 B tion to the geometrical L2 φ s center of the Sun as seen EARTH from within the satellite w.r.t. the local BCRS de- L2 fined at each point of the satellite’s trajectory; x L2 ⇓ u ’ u u the new triad adapted to the ob- u is server u u B t GAIA λ λ λ a = R 2 ( θ s ) R 3 ( φ s ) λ λ ˆ λ s ˆ a u Firenze, 28-30/09/2006 – p.10/16
The boosted triad Second step: we boost the vectors of the triad λ λ λ 1 along the satellite relative motion s ˆ ⇓ » γ ”– γ + 1 ν σ “ α a = P ( u ′ ) ασ σ ν ρ λ λ λ λ λ a − ˆ ˆ s ρ ˆ s a bs a =1 , 2 , 3 ˆ (Jantzen, Carini and Bini, 1992, Annals of Physics 215 and references therein) ν α = 1 ′ α − γu α ) relative spatial four-velocity of u u ′ w.r.t. u γ ( u u u u ′ α u α relative Lorentz factor γ = − u The vector λ λ λ 1 identifies the direction to the Sun as seen by the satellite as a Sun-locked frame bs ˆ Firenze, 28-30/09/2006 – p.11/16
The Gaia attitude frame Final steps in order to obtain the Gaia attitude frame: λ i) rotate the Sun-locked triad by an angle ω p t about the vector λ λ 1 which constantly bs ˆ points to the Sun; ω p is the angular velocity of precession, ii) rotate the resulting triad by a fixed angle α = 50 ◦ about the image of the vector λ λ λ bs ˆ 2 under rotation i), and iii) rotate the triad obtained after step ii) by an angle ω r t about the image of the vector λ λ λ 1 under the previous two rotations; ω r is the angular velocity of the satellite spin. bs ˆ ⇓ E ˆ λ E E a = R 1 ( ω r t ) R 2 ( α ) R 1 ( ω p t ) λ λ ˆ a = 1 , 2 , 3 bs ˆ a Gaia attitude triad Firenze, 28-30/09/2006 – p.12/16
Explicit coordinate components of the Gaia attitude triad E α cos αλ α − sin α cos( ω p t ) λ α − sin α cos( ω p t ) λ α = (1) ˆ ˆ ˆ ˆ 1 1 2 3 bs bs bs E α − sin α sin( ω r t ) λ α = + ˆ ˆ 2 1 bs +[cos( ω r t ) cos( ω p t ) − sin( ω r t ) sin( ω p t ) cos α ] λ α (2) ˆ 2 bs +[cos( ω r t ) sin( ω p t ) + sin( ω r t ) cos( ω p t ) cos α ] λ α ˆ 3 bs E α − sin α cos( ω r t ) λ α = ˆ ˆ 3 1 bs − [sin( ω r t ) cos( ω p t ) + cos( ω r t ) sin( ω p t ) cos α ] λ α (3) ˆ 2 bs +[ − sin( ω r t ) sin( ω p t ) + cos( ω r t ) cos( ω p t ) cos α ] λ α ˆ 3 bs (Bini, Crosta and de Felice, 2003, Class. Quantum Grav. 20 4695) Firenze, 28-30/09/2006 – p.13/16
The clock on board of Gaia Time interval between two events in space-time dT = − 1 ′ α dx β c g αβ u interval of proper time of an observer on board of the satellite (Crosta et al., Proper frames and time scan for Gaia-like satellites , 2004, ESA livelink, tech.note); if we adopt the IAU metric »„ v 2 « – dt − c − 2 + v i dr i 2 + w ( x , t ) dT ≈ »„ w 2 ( x , t ) − v 4 8 − 3 v 2 w ( x , t ) « + c − 4 + 4 w i ( x , t ) v i dt 2 2 3 w ( x , t ) + v 2 „ « – +4 w i ( x , t ) dr i − v i dr i , 2 (IAU resolution B1.5, Second Recommendation) Firenze, 28-30/09/2006 – p.14/16
Summary 1. RAMOD is a well-established framework of general relativistic astrometric models which can be extended to whatever accuracy and physical requirements ( i.e. the metric); 2. RAMOD is fully operational from the theoretical stand-point, and ready to be implemented in an end-to-end simulation of the Gaia Mission ( i.e. , estimation of the astrometric parameters of celestial objects from a well-defined set of relativistically measured quantities) Firenze, 28-30/09/2006 – p.15/16
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