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A General Relativistic Model of Light Propagation in the Gravitational Field of the Solar System: The Dynamical Case

A General Relativistic Model of Light Propagation in the Gravitational Field of the Solar System: The Dynamical Case
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  See discussions, stats, and author profiles for this publication at: A General Relativistic Model of Light Propagation in the Gravitational Field of theSolar System: The Dynamical Case Article   in    The Astrophysical Journal · October 2006 DOI: 10.1086/508701 · Source: arXiv CITATIONS 32 READS 21 5 authors , including: Some of the authors of this publication are also working on these related projects:  The APACHE Project   View projectAstrometric tests of General Relativity   View projectAlberto VecchiatoNational Institute of Astrophysics 102   PUBLICATIONS   757   CITATIONS   SEE PROFILE M. G. LattanziNational Institute of Astrophysics 285   PUBLICATIONS   2,938   CITATIONS   SEE PROFILE All content following this page was uploaded by Alberto Vecchiato on 30 January 2013.  The user has requested enhancement of the downloaded file.    a  r   X   i  v  :  a  s   t  r  o  -  p   h   /   0   6   0   9   0   7   3  v   1   4   S  e  p   2   0   0   6 A general relativistic model of light propagation in thegravitational field of the Solar System: the dynamical case F. de Felice 1 Department of Physics, University of Padova via Marzolo 8, 35131 Padova, Italy A. Vecchiato INAF – Turin Astronomical Observatory strada Osservatorio 20, 10125 Pino Torinese (TO), Italy M. T. Crosta INAF – Turin Astronomical Observatory strada Osservatorio 20, 10125 Pino Torinese (TO), Italy B. Bucciarelli INAF – Turin Astronomical Observatory strada Osservatorio 20, 10125 Pino Torinese (TO), Italy andM. G. Lattanzi INAF – Turin Astronomical Observatory strada Osservatorio 20, 10125 Pino Torinese (TO), Italy  1 INFN - Sezione di Padova  – 2 – ABSTRACT Modern astrometry is based on angular measurements at the micro-arcsecondlevel. At this accuracy a fully general relativistic treatment of the data reductionis required. This paper concludes a series of articles dedicated to the problemof relativistic light propagation, presenting the final microarcsecond version of arelativistic astrometric model which enable us to trace back the light path to itsemitting source throughout the non-stationary gravity field of the moving bodiesin the Solar System. The previous model is used as test-bed for numerical com-parisons to the present one. Here we also test different versions of the computercode implementing the model at different levels of complexity to start exploringthe best trade-off between numerical efficiency and the  µ as accuracy needed tobe reached. Subject headings:  astrometry — gravitation — reference systems — relativity —time 1. Introduction Modern space technology will soon provide stellar positioning with micro-arcsecond ac-curacy ( µ as). At this level one has to take into account the general relativistic effects on lightpropagation arising from metric perturbations due not only to the bulk mass but also to therotational and translational motion of the bodies of the Solar System, and to their multipolestructure (see Kopeikin and Mashhoon 2002; Klioner 2003; Le Poncin-Lafitte et al. 2004and references therein). Our aim is to develop a Relativistic Astrometric MODel (RAMOD)which enabled us to deduce, to the accuracy of one  µ as, the astrometric parameters of a starin our Galaxy from observations taken by modern space-born astrometric satellites like Gaia(Turon et al. 2005), which are fully consistent with the precepts of General Relativity.In this paper we present an astrometric model which contains an extension to the dy-namical case,  i.e.  with the inclusion of the 1 /c 3 terms, of our previous model which was onlyaccurate to 1 /c 2 (de Felice et al. 2004). We term it as RAMOD3 since it was intended as thesuccessor of two previous models (de Felice et al. 1998, 2001). Following the same scheme,we shall refer to the model described here as RAMOD4.The inclusion of terms of the order of 1 /c 3 corresponds to an accuracy of 0 . 1  µ as, at leastone order of magnitude better than the expected precision of the Gaia measurements. Here  – 3 –we require that the Solar System is isolated and source of a weak gravitational field. Thesetwo conditions imply that the velocities of the gravitational sources within the Solar Systemare very small compared to the velocity of light, typically of the order of 10km s − 1 . Underthese conditions we select a coordinate system ( x i ,x 0 ≡ ct ) ( i =1 , 2 , 3)  such that the backgroundgeometry has a post-Minkowskian form: g αβ   =  η αβ   + h αβ   + O  h 2   (1)where  η αβ   is the Minkowski metric and  h αβ   are small perturbations which describe effectsgenerated by the bodies of the Solar System. These perturbations are  small   in the sensethat  | h αβ  | ≪  1, their spatial variations are of the order of   | h αβ  |  while their time variationsare of the order of   | h αβ  | /c ; here and in what follows Greek indeces run from 0 to 3. Clearlymetric form (1) is preserved under coordinate gauge transformations of the order of   h . The h αβ  ’s contain terms of the order of at least 1 /c 2 , hence we shall keep our approximation tofirst order in  h . To the order of 1 /c 3 the time dependence of the background metric cannotbe ignored therefore the time-like vector field  η  =  ∂  0  tangent to the coordinate time axis,will not in general be a Killing field (namely an isometry for the space-time) unless onemoves to far distances from the Solar System where the metric tends to be Minkowski’s.The components of the vector field  η  are  η µ =  δ  µ 0 , η µ  =  g 0 µ  hence we easily deduce that thecongruence  C  η , namely the family of curves having the vector field  η  as tangent field 1 , willhave a non zero vorticity. From its definition ω αβ   =  P  ( η ) ασ P  ( η ) β ρ ∇ [ ρ η σ ] ,  (2)where  P  ( η ) µλ  =  δ  µλ  + η λ η µ is the operator which projects orthogonally to  η , ∇ ρ  the covariant ρ − derivative relative to the given metric and square brackets mean antisymmetrization, wehave that the only non vanishing components are, to the lowest order: ω ij  =  ∂  [  j h i ]0  + O  1 /c 4  , i,j  = 1 ,  2 ,  3 .  (3)We remind that  h 0 i  and  ∂   j h 0 i  are  ∼  O (1 /c 3 ). Condition (3) implies that the surfaces t  = constant are not orthogonal to the integral curves of   η  at least in and nearby theSolar System (de Felice et al. 2004). Nevertheless, because the time lines with tangent field η  are asymptotically Killing and vorticity free then the slices  S  ( t ) :  t  = constant allow fora non ambiguous 3 + 1 splitting of spacetime with a coordinate representation such thatspace-like coordinates are fixed within each slice. We fix the srcin of the space-like coor-dinates at the barycenter of the Solar System and assume that the spatial coordinate axes 1 These curves are also termed integral curves of the vector field  η .  – 4 –point to distant sources chosen in such a way to assure that the system is kinematically nonrotating. Adopting the IAU prescriptions (IAU 2000), this system is termed BarycentricCelestial Reference System (BCRS) and will be our main system since all coordinate ten-sorial components will be relative to it. The non stationarity of the spacetime makes theconstruction of a relativistic astrometric model much less straightforward than for the staticcase considered in de Felice et al. (2004).In section 2 these complications are handled to define the geometrical environment forlight propagation and data analysis. The spacetime metric is not diagonal since terms as h 0 i  are different than zero; they are mainly generated by the velocity of the metric sourcesrelative to the given BCRS. Then the gravitational potential at each point of the lighttrajectory depends on the sources in the Solar System at the appropriate retarded positionas specified in section 3. This will have consequences when fixing the observables and theboundary conditions for the differential equations of the light rays in section 4. Section 5shows how this model was numerically tested and presents the results of these tests.In what follows we shall use geometrized units such that  c  = 1 =  G ,  G  being thegravitational constant; with these units a mass M , expressed in kilograms, has the dimensionof a length according to the relation  M   =  G M /c 2 ; similarly the time coordinate  x 0 =  ct will be simply written as  t  and the spatial velocities are in units of   c . For sake of clarity,the velocity of light  c  will appear explicitly only when we specify the order of magnitude of terms under discussion. 2. The light trajectory We require that spacetime admits a family of hypersurfaces  S  ( t ) with  t  = constant sothat the spatial coordinates  { x i }  are fixed on each of them. As said, these surfaces areconstrained by the condition of being asymptotically orthogonal to the  time   direction  η which will also be asymptotically Killing. In the nearby of, and of course within the SolarSystem, the spatial coordinates  x i are not constant along the normals to these hypersurfaces;along them, in fact, they vary according to the shift law  δx i ∼  h 0 i δt . To the order of 1 /c 3 ,all terms proportional to  h 0 i  are in general not zero and cannot be made vanish in a gaugeinvariant way. Let us term  u  a vector field parallel to  η  and tangent to the family of timelike curves ˆ γ  , say, along which the spatial coordinates are constant; furthermore let ˆ σ  bethe parameter on these curves which makes the vector field  u  unitary, namely  u α u α  =  − 1.Clearly along each integral curve of   u  carrying the coordinates  x i , the parameter ˆ σ ( x i ,t ) is
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