Gaia relativistic astrometric models
I. Proper stellar direction and aberration
M. Crosta  A. Vecchiato
INAF  Astronomical Observatory of Torino, via Osservatorio 20, 10025 Pino Torinese (TO), Italy
Received 12 June 2009 / Accepted 5 October 2009
Abstract
The high accuracy achievable by modern space astrometry requires
the use of General Relativity to model the stellar light propagation
through the gravitational field encountered from a source to a given
observer inside the Solar System. The general relativistic definition
of an astrometric measurement needs an appropriate use of the concept
of reference frame, which should then be linked to the conventions of
the IAU resolutions. On the other hand, a definition of the astrometric
observables in the context of General Relativity is also essential for
finding the stellar coordinates and proper
motion uniquely, this being the main physical task of the inverse
raytracing problem. The aim of this work is to set the level of
reciprocal consistency
of two relativistic models, GREM and RAMOD (Gaia, ESA mission), in
order to guarantee a physically correct definition of
the light's local direction to a star and deduce the star coordinates
and proper motions at the level of accuracy required by these models
consistently with the IAU's adopted reference systems.
Key words: relativity  astrometry  gravitation  reference systems  methods: data analysis  techniques: high angular resolution
1 Introduction
The correct definition of a physical measurement requires identification of an appropriate reference frame. This also applies to determining the position and motion of a star from astrometric observations made from within our Solar System. Moreover, modern instruments housed in spaceborne astrometric probes like Gaia (Turon et al. 2005) and SIM (Unwin et al. 2008) aim to be accurate at the microarcsecond level, or higher as in the case of bright stars observed by SIM (0.2 microarcsecond), thus requiring that any astrometric measurement be modeled in a way that both light propagation and detection should be conceived in a general relativistic framework. One needs, in fact, to solve the relativistic equations of the null geodesic that describe the trajectory of a photon emitted by a star and detected by an observer with an assigned state of motion. The whole process takes place in a geometrical environment generated by an Nbody distribution such as for our Solar System. Essential to the solution of the above astrometric problem (inverse ray tracing from observational data) is the identification, as boundary conditions, of the local observer's lineofsight defined in a suitable reference frame (Bini et al. 2003; de Felice & Preti 2006; de Felice et al. 2006).
Summarizing from the quoted references, the astrometric problem consists of determining the astrometric parameters of a star (its coordinates, parallax, and proper motion) from a prescribed set of observational data (hereafter observables). However, while these quantities are well defined in classical (non relativistic) astrometry, in General Relativity (GR) they must be interpreted consistently with the relativistic framework of the model. Similarly, the parameters describing the attitude and the centerofmass motion of the satellite need to be defined consistently with the chosen relativistic model.
As far as Gaia is concerned, at present two conceptual frameworks are able to treat the astrometric problem at the microarcsecond level within a relativistic context. The first model, named GREM (Gaia RElativistic Model) and described in Klioner (2003), is an extension of a seminal study by Klioner & Kopeikin (1992) conducted in the framework of the postNewtonian (pN) approximation of GR. GREM has been formulated according to a parametrized postNewtonian scheme accurate to 1 microarcsecond. In this model finite dimensions and angular momentum of the bodies of the Solar System are included and linked to the motion of the observer in order to consider the effects of parallax, aberration, and proper motion, and the light path is solved using a matching technique that links the perturbed internal solution inside the near zone^{} of the Solar System with the (assumed) flat external one.
Basically, the pN approach (and postMinkowskian one, pM, as in Kopeikin & Mashhoon 2002) solves the light trajectory as a straight line (Euclidean geometry) plus integrals, containing the perturbations encountered, from a gravitating source at an arbitrary distance from an observer located within the Solar System. This allows one to transform the observed light ray in a suitable coordinate direction and to read off the aberrational terms and light deflections effects, evaluated at the point of observation. This model is considered as baseline for the Gaia data reduction.
The second model, RAMOD, is an astrometric model conceived to solve the inverse raytracing problem in a general relativistic framework not constrained by a priori approximations. RAMOD is actually a family of models of increasing intrinsic accuracy, all based on the geometry of curved manifolds (de Felice et al. 2004,2006). As in Kopeikin & Mashhoon (2002), the full development to the microarcsecond level imposes consideration of the retarded distance effects by the motion of the bodies of the Solar System. At present, the RAMOD full solution requires numerical integration of a set of coupled nonlinear differential equations (also called ``master equations''^{}), which allows the light trajectory to be traced back to the initial position of the star and which naturally entangles the contributions by the aberration and those by the curvature of the background geometry. RAMOD is formulated with a completely different methodology. This makes its comparison with the former one a difficult task.
Despite its difficulty, this comparison is a necessity, beacuse GREM and RAMOD will be used for the Gaia data reduction with the purpose of creating a catalog of one billion positions and proper motions based on measurements of absolute astrometry, so any inconsistency in the relativistic model(s) would invalidate the quality and reliability of the estimates, hence all related scientific output.
In this paper we present the first step in the theoretical comparison, showing how it is possible to isolate the aberration terms from the global RAMOD construct (which are normally entangled together with other terms such as those of the deflection) and recasting them in a GREMlike formula.
In Sect. 2 we review all the building steps of the RAMOD astrometric setup. In Sect. 3 we show the procedures used in RAMOD to define the observables and compare the quantities of GREMlike formulations by making the aberration part in the RAMOD framework explicit. Sect. 4 will comment on the results of the comparison and on what has to be addressed to proceed with the theoretical comparison of the two models. Finally, Appendix B reports the calculations of the pN/pM approaches recovering the stellar aberration.
Throughout the paper, regular bold indicates fourvector (e.g. ) and italic bold indicates threevector (e.g. ); the components of vectorial quantities are indicated with indexes (no bold symbols), where the Latin index stands for 1, 2, 3 and the Greek ones for 0, 1, 2, 3. A repeated index means Einstein summation convention and indexes are raised and lowered with the metric (in particular, n^{i} n_{i} stands for the scalar product with respect to the Euclidean metric , whereas with respect to the metric ). The speed of light is symbolized by c, notations like indicate a set of quantities (e.g. ), and or an operator projecting with respect to the observer .
2 The RAMOD frames
In order to bring out the different methodologies and mathematical constructs applied in RAMOD, this section summarizes the setup of RAMOD by focusing on the reference frames needed to define the measurements.
The setup of any astrometric model primarily implies the
identification
of the gravitational sources and of the background geometry. Then
one needs to label the spacetime points with a coordinate system.
These steps allow us to fix a reference frame with respect to
which one describes the light trajectory, the motion of the stars,
and that of the observer. The RAMOD framework is based on the
weakfield requirement for the background geometry, which in turn have
to be
specialized to the particular case one wants to model. For example,
keeping in mind a Gaialike mission, we can assume the Solar System
is the only source of gravity, i.e. a physical system gravitationally
bound, in the weak field and slow motion regime. Then, only firstorder
terms in the metric perturbation h (or equivalently in the constant G as in the pM approximation) are retained. These terms already include all of the possible (v/c)^{n}order expansions of the pN approach, but just those up to (v/c)^{3}
are needed to reach the microarcsecond accuracy required for the next
generation astrometric missions, like e.g. Gaia and SIM. With these
assumptions the background geometry is given by the following
line element
where collects all nonlinear terms in h, and the coordinates are x^{0}=ct,x^{1}=x,x^{2}=y,x^{3}=z, the origin being fixed at the barycenter of the Solar System, and is the Minkowskian metric. In the small curvature limit (Misner et al. 1973), the metric components used in RAMOD are
where , , and w and w^{i} are, respectively, the gravitational potential and the vector potential generated by all the sources inside the Solar System that can be chosen according to the IAU resolution B1.3 (Soffel et al. 2003). The metric of Eq. (1) is also adopted in GREM and the subscripts indicate the order of (v/c) (e.g. ).
2.1 The BCRS
In RAMOD, a Barycentric Celestial Reference System (BCRS, Soffel et al. 2003) is identified requiring that a smooth family of spacelike hypersurfaces exists with the equation (see de Felice et al. 2004). The function t can be taken as a time coordinate. On each of these hypersurfaces, one can choose a set of Cartesianlike coordinates centered at the barycenter of the Solar System (B) and running smoothly as parameters along spacelike curves that point to distant cosmic sources. The latter are chosen to assure that the system is kinematically nonrotating, i.e. nonrotating with respect to the reference distant sources as recommended by the IAU (Soffel et al. 2003). The parameters x, y, z, together with the time coordinate t, provide a basic coordinate representation of the spacetime according to the IAU resolutions^{}.
Any tensorial quantity will be expressed in terms of coordinate components relative to coordinate bases induced by the BCRS.
2.2 The local BCRS
As shown in detail in (de Felice et al. 2004,2006), in RAMOD at any spacetime point a unitary fourvector exists
that is tangent to the world line of a physical observer at rest with respect to the spatial grid of the BCRS defined as
The totality of these four vectors over the spacetime forms a vector field that is proportional to a timelike and asymptotically Killing vector field . In fact, to the order of accuracy required for Gaia, the congruence of curves does not admit a global family of orthogonal hypersurfaces, i.e. a restspace that covers the entire spacetime. However, the restspace of can be locally identified by a spatial triad lying on a surface, which differs from the one (see Fig. 1) in such a way that their spatial components point to the local coordinate directions as chosen by the BCRS. This frame works as a local BCRS.
Figure 1: The local observer with respect to the BCRS coordinate system. The spatial axes of the BCRS point toward distant sources. The dashed lines are the curves that are orthogonal, say , to the hypersurfaces, asymptotically orthogonal to the time direction. The restspace (green area) of locally deviates from the spacelike hypersurface with equation by terms of the order of a microarcsecond. 

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The tetrad associated to the local BCRS has spatial axes (the triad) coinciding with the local coordinate
axes, but its origin is the barycenter of the satellite. At the
,
this triad is (Bini et al. 2003)
Let us stress that is an essential prerequisite of RAMOD, because at any spacetime point and apart from a positiondependent rescaling of its time rate, it plays the role of a barycentric observer as the one located at the spatial coordinate fixed at the barycenter of the Solar System. In RAMOD any physical measurement refers to this local BCRS.
2.3 The proper reference frame for the satellite
The proper reference frame of a satellite consists of its restspace and a clock that measures the satellite proper time. The tensorial quantity that expresses a proper reference frame of a given observer is a tetrad adapted to that observer, namely a set of four unitary, mutually orthogonal fourvectors , one of which, i.e. , is the observer's fourvelocity, while the other s form a spatial triad of spacelike fourvectors (Misner et al. 1973). The physical measurements made by the observer (satellite) represented by such a tetrad are obtained by projecting the appropriate tensorial quantities on the tetrad axes.
The same measurements can also be defined by splitting the spacetime into two subspaces, as sketched in Appendix A. Essentially, this last method is useful when we do not know the solution of a tetrad, which depends on the metric, and we only need to know the moduli of the physical quantities. As far as RAMOD is concerned, given the metric (1) and in the case of a Gaialike mission, an explicit analytic expression for a tetrad adapted to the satellite fourvelocity exists and can be found in Bini et al. (2003). The spatial axes of this tetrad, named , are used to model the attitude of the satellite.
2.3.1 Satellite proper reference frame and IAU conventions
In RAMOD the satellite reference frame is obtained by successive transformations of the local BCRS tetrad
as defined in Eq. (3).
In particular, the vectors of the triad
are boosted to the satellite restframe by means of an instantaneous Lorentz transformation (Bini et al. 2003, and reference therein),
which depends on the relative spatial velocity
of the satellite
identified by the fourvelocity with respect to the local BCRS , whose Lorentz factor is given by (Jantzen et al. 1992).
The boosted tetrad
obtained in this way represents a CoMRS (CenterofMass Reference System, comoving with the satellite), similar to what is defined for Gaia (Klioner 2004; Bastian 2004). In
addition to the definition in the cited works, one of the axes is Sunlocked,
i.e. one axis points toward the Sun at any point of its Lissajous
orbit around L2. The Gaia attitude frame is finally obtained by applying the following rotations to the
Sunlocked frame: (i) by an angle
about the fourvector
which constantly points towards the Sun (where
is the
angular velocity of precession); (ii) by a fixed angle
about the image of the fourvector
after the previous rotation; and (iii) by an angle
about the image of the fourvector
after the previous two rotations (where
is now the spin angular velocity). The triad resulting from these transformations establishes the satellite attitude triad, given by
The final triad only depends on the attitude parameters of the satellite, and should be the RAMOD equivalent of the GREM Satellite Reference System (SRS) (Bastian 2004). This defines the spatial components of the reference system.
To complete the process one has to include the transformations between
the observer's proper time and the barycentric coordinate time. This can be
done using the subspace splitting technique cited in Appendix A. Let us consider the satellite's worldline in the spacetime geometry as
where are the BCRS coordinate components of the satellite velocity and ( v^{2}=v^{i}v_{i}); u_{s}^{0} can be chosen as the normalization factor. Since the satellite is a physical observer, from the unitary condition , we deduce the expansion of u_{s}^{0}in powers of (v/c) (once we use the pN potentials w and w^{i} defined by IAU resolutions, Soffel et al. 2003):
Then, by inserting the last expression (6) in Eq. (A.3), we obtain the formula which ties the running between the clock on board up to the order (v/c)^{4}and that for the origin of the BCRS:
where x^{i} is any spatial location inside or in the neighborhood of the satellite and R_{s}^{i}=x^{i}  x_{s}^{i}. It is trivial to check that, when we make a firstorder Taylor series expansion around the satellite barycenter location of the potential (and the vector potential),
Equation (7) can be transformed (Crosta 2003) in the relationships between the proper time on board the satellite and the barycentric coordinate time interval as reported in IAU resolutions B1.5. This finally completes the definition of the proper reference frame for the Gaialike satellite in the RAMOD framework and, moreover, gives proof of the compatibility of the RAMOD formalism with the IAU conventions (hence with a GREMlike approach).
3 Multistep application of the observable e in RAMOD to the aberration
The classical (non relativistic) approach of astrometry has traditionally privileged a ``multistep'' definition of the observable; i.e., the quantities that ultimately enter the ``final'' catalog and are referred to a global inertial reference system, are obtained taking into account effects such as aberration and parallax, one by one and independent from each other.
GREM reproduces this approach of classical astrometry in a relativistic framework. For this model the BCRS is the equivalent of the inertial reference system of the classical approach, while the final expression of the star direction in the BCRS is obtained after converting the observed direction into coordinate ones in several steps that divide the effects of the aberration, the gravitational deflection, the parallax, and proper motion (Klioner 2003) (see Appendix B). As is well known, stellar aberration arises from the motion of the observer relative to the BCRS origin, assumed to coincide with the center of mass of the Solar System.
In the previous section we mentioned that RAMOD relies on the
tetrad formalism for the definition of the observable. In general,
the three direction cosines that identify the local lineofsight
to the observed object are relative to a spatial triad
associated with a given observer ;
the direction cosines
with respect to the axes of this triad are defined as
where the final is a shorthand notation for and the fourvector tangent to the null geodesic connecting the star to the observer, and all the quantities are obviously computed at the event of the observation^{}. As a consequence, given the solution of the null geodesic equation and the motion and the attitude of the observer, Eq. (8) expresses a relation between the unknowns, the position and motion of the star, and the observable quantities that include all of these effects mentioned for GREM. Once the procedure for defining Eq. (8) is complete, the final measurements will naturally entangle every GR ``effect'' in a single result. In other words, RAMOD does not need to disentangle each single effect, relativistic or not. The main purpose is to keep as long as possible the physical expressions of the quantities entering Eq. (8). Therefore, the natural way to ``extract'' any of those effects in a separate formula, as in GREM, is to express the observable with a specific tetrad that makes the aberration part evident.
3.1 Attitudefree tetrad for the aberration
Whatever tetrad we consider, the expression of Eq. (8) for the
relativistic observable in the RAMOD model can also be written as (de Felice et al. 2006)
where is the spatial fourvelocity (see Eq. (4), also called as the ``physical velocity'') of the satellite relative to the local barycentric observer . The quantity was introduced in RAMOD and is a unitary fourvector that represents the local lineofsight of the photon as seen by at the moment of observation. In general, (de Felice et al. 2004,2006). Finally, is the Lorentz factor of with respect to ; that is,
where .
To retrieve the aberration effect given by the motion of the satellite
with respect to the BCRS in RAMOD, one needs to specialize Eq. (9)
to the case of a tetrad
adapted to the center of mass of the satellite assumed with no attitude
parameters. In this case, in fact, the observation equation will give
a relation between the ``aberrated'' direction represented by
the direction cosines e_{a}, as measured by the satellite
and the ``aberrationfree'' direction given by the quantity
referring to the local BCRS frame
.
The
vectors of the triad
differ from the local BCRS's
for a boost transformation with fourvelocity
.
This
means that it can be derived from Eq. (3)
using the relation (Jantzen et al. 1992)
where and are the abovementioned four velocity of the satellite and its physical velocity relative to the local BCRS, and .
3.2 Aberration at the order as function of the local lineofsight
To recover a GREMlike aberration relativistic effect in RAMOD, we have to expand Eq. (9) with respect to the (v/c) small pN parameter. From de Felice et al. (2006) and Bini et al. (2003)
it is
where v^{i} are the same as was defined in the previous sections. Now, when considering Eq. (4) one deduces that and
Expanding Eq. (11) with relations (10) and (4), one gets
Then, using Eqs. (3), (12), and (13) and expanding the scalar products to the right order, we obtain
so that the expression for the boosted tetrad finally becomes
Given Eq. (18) one can consistently recast Eq. (9) as
where are the cosines related to the tetrad , which, as said, does not contain the attitude parameters. Here and in the rest of the section, we replace with , and the symbol with to ease the notation.
After long calculations and considering the IAU metric, the first term
on the righthandside of this formula can be written as
The second term is zero since both and are zero, while the third one becomes
Finally, collecting all terms, we get
3.3 Recasting to the GREMlike aberration
Expression (22) relates the observed direction cosines with . The equivalent relation for the GREM observable is Eq. (B.6) where the aberration is expressed in terms of a vector . At first glance, it comes out that we cannot simply identify with , since the last expression shows differences in terms up to the order! In particular, the appearance of the term and of different factors at the order cannot allow a straightforward comparison, as expected, of Eq. (22) to the GREM vectorial one of Eq. (B.6). Therefore, to compare formula (22) with GREM's formula (B.6) and find a relationship between and , we need to reduce the s to their coordinate Euclidean expressions.
In GREM,
represents the ``aberrationfree'' coordinate
line of sight of the observed star at the position of the satellite
momentarily at rest. In RAMOD, as said,
represents
the normalized local lineofsight of the observed star
as seen by the local barycentric observer .
In other words,
is a fourvector that fixes the
line of sight of an object with respect to the local BCRS. Do
and
have a similar role in the
two approaches? From the physical point of view they have the same
meaning, as the observed ``aberration free'' direction
to the star. Let us start from the definition of
in
GREM:
where and pis the Euclidean norm of p^{i}, so that . This means that
On the other hand, using the definition of , it can be easily shown that its spatial components are
and, from and , we get
Finally, from Eqs. (23) and (24) one has
namely, the spatial light direction, expressed in terms of its Euclidean counterpart at the satellite location in the gravitational field of the solar system. It is worth noticing that no terms of the order of appear in (25).
Combining Eq. (22) with (25) and
setting
to ease the notation, we obtained
i.e. the righthand side of the aberration expression of RAMOD rewritten as in GREM.
Now we have to be certain that the lefthand side of
Eq. (26) can also be directly compared with GREM's
formula (B.6). Let us apply the tetrad property
to the definition of
and get
Is there a relation between the direction cosines of this equation with the spatial components of the observed vector in GREM? The crucial point stands on the definition of the coordinate system. The tetrad components of the light ray can be directly associated to CoMRS coordinates, i.e. to a coordinateinduced tetrad (as in Klioner 2004), if the boosted local BCRS tetrad coordinates are equivalent to the CoMRS ones .
In RAMOD, at the milliarcsecond level, i.e. at (v/c)^{2}, the restspace of the local barycentric observer coincides globally with the spatial hypersurfaces that foliate the spacetime and define the BCRS (de Felice et al. 2004). At microarcsecond accuracy, instead, the vorticity cannot be neglected and the geometry is affected by nondiagonal terms of the metric, meaning that the hypersurfaces do not coincide with the restspace of the local barycentric observer (de Felice et al. 2006). Then, to be consistent, at each point of observation we can only define a spatial direction measured by the local barycentric observer and then associate it to the satellite measurements via the direction cosines relative to the boosted attitude frame.
The equivalence of the two coordinate systems thus holds if the origins of the
two reference systems coincide and only locally, i.e. in a sufficiently small
neighborhood, since the tetrads are not necessarily holonomic. Under
these hypotheses and from (B.1), one can state that
and it follows that
Therefore, using the local validity of Eq. (28) and considering that and , Eq. (26) can be written as
Finally, from the relation , it is
which is formula (B.6) for the aberration in GREMlike model if we consider that , and . This result states that, limited to the case of aberration and using the appropriate definitions of the IAU recommendations, RAMOD recovers GREM at the order.
4 Conclusions
This paper compares two approaches within the context of relativistic astrometry, GREM and RAMOD, both suitable for modeling modern astrometric observations at microarcsecond accuracy. Because of the structure of GREM, the earliest stage of a theoretical comparison starts with the evaluation of the aberration ``effect'' in RAMOD.
This work presents a first analysis between two different methods in applying general relativity, the only theory of gravitation up to now, to astrometry. Understanding any difference and/or equivalence represents a valuable help to exploit the Gaia observations to their full extent and to validate data analysis in the new era of relativistic astrometry.
Indeed, the different mathematical structures of GREM and RAMOD hinder a straightforward comparison and call for a more indepth analysis of the two models. While GREM favors the direct application of the coordinate approach since the beginning, RAMOD prefers, instead, to keep the meaning of the physical quantity as far as possible, i.e. to move to the coordinates once the condition equations are solved (namely the equations linking the measurements and astrometric unknowns). This implies a certain number of differences between the two derivations that have to be taken into account to avoid misinterpretations of parallel but different quantities. Up to now, we can distinguish the following differences in how the two models use: (i) the boundary conditions; (ii) the astrometric measurements; (iii) the attitude implementation; (iv) and the definition of the proper light direction.
Of crucial importance is point (i). The light signal arriving at the local BCRS along the spatial direction satisfies the RAMOD master equations, a set of nonlinear coupled differential equations (de Felice et al. 2006). Therefore the cosines (i.e. the astrometric measurements) taken as a function of the local line of sight (the physical one), at the time of observation, allow fixing the boundary conditions needed to solve the master equations and determining the star coordinates uniquely. However, since the direction cosines are expressed in terms of the attitude, the mathematical characterization of the attitude frame is essential for completing the boundary value problem in the process of reconstructing the light trajectory. The vector in GREM, i.e. the ``aberrationfree'' counterpart of of RAMOD, is instead used to derive the aberration effect (in a coordinate language), and there is no need to connect it with a RAMODlike boundary value problem.
In RAMOD the direction cosines link the attitude of the satellite to the measurements, combining several reference frames useful to determine, as final task, the stellar coordinates: the BCRS (kinematically nonrotating global reference frame), the CoMRS (a local reference frame comoving with the satellite centre of mass), and the SRS (the attitude triad of the satellite). The coordinate transformations between BCRS/CoMRS/SRS come out naturally once the IAU conventions are adopted. A proof of this is given when we apply proper time formula (A.3) to get the relationship between the running time on board and the barycentric coordinate time (7). This is inside the conceptual framework of RAMOD, where the astrometric setup allows one to trace the light ray back to the emitting star in a curved geometry, and it is not natural to disentangle each single effect. As for the solution of the geodesic equation, RAMOD defines a complete procedure to derive the satellite attitude that as input depends only on the specific terms of the metric that describes the addressed physical problem. GREM, instead, embeds the definition of its main reference system (BCRS) within the metric, consequently each further step depends on this choice. This includes all the subsequent transformations among the reference systems that are essential for extracting the GREM observable as a function of the astrometric unknowns. On the other side, the RAMOD directly implements in the solution of the astrometric problem the relativistic algorithms of the attitude frame, assuring its consistency with GR, since by definition the origin of the tetrad system follows the observer's worldline (i.e. the center of mass of the satellite in this case). The last comment explains items (ii) and (iii) and introduces item (iv).
Because physical quantities do not depend on the coordinates, the direction cosines are a powerful tool for comparing the astrometric relativistic models: their physical meaning allows us to correctly interpret the astrometric parameters in terms of coordinate quantities. This justified the conversion of the physical stellar proper direction of RAMOD into its analogous Euclidean coordinate counterpart, which ultimately leads to the derivation of a GREMstyle aberration formula. Another point arises when the observables of RAMOD have to be identified with s^{i}, i.e. the components of the observed vector of GREM. This matching is admitted only if the origins of the boosted local BCRS tetrad in RAMOD and of the CoMRS in GREM coincide.
In conclusion, to what extent, then, is the process of star coordinate ``reconstruction'' consistent with General Relativity&Theory of Measurements? Solving the astrometric problem in practice means to compile an astrometric catalog with the same order of accuracy as the measurements. This paper shows not only that the two models give the same results, but also that particular care is needed in the interpretation of the observables and of the quantities that constitute the final catalog in order to avoid differences that already exist at the level of the aberration effect.
AcknowledgementsThe authors wish to thank Prof. Fernando de Felice and Dr. Mario G. Lattanzi for constant support and useful discussions. In particular, we thank the referee for his valuable comments and suggestions. This work is supported by the ASI grants COFIS and I/037/08/0.
Appendix A: Length and time measurements due to a spacetime splitting
An observer
carrying its laboratory is usually
represented as a world tube; in the case of a nonextended body, the world
tube can be restricted to a world line tracing the history of the observer's
barycenter in the given spacetime. At any point P along the world line of
,
and within a sufficiently small neighborhood, it is possible to
split the spacetime into a onedimensional space and a threedimensional
one (de Felice & Clarke 1990), each space being endowed with its
own metric, respectively
and
.
Clearly,
The subspace with metric is generated by lines (i.e. geodesics in a normal neighborhood) that are orthogonal to the world line of at P. This subspace defines the restspace of the observer at P and here one is allowed to measure proper lengths. The subspace with metric is generated by lines that differ from that of by a new parameterization. In this subspace one measures the observer's proper time.
As a consequence of Eq. (A.1), the invariant interval
between two events in spacetime can be written as
,
from which we are able to extract the measurements of infinitesimal
spatial distances and time intervals taken by
as, respectively,
(A.2) 
and
Appendix B: Stellar observed direction and stellar aberration in GREMlike approaches
The pN/pM approaches (Klioner 2003; Kopeikin & Mashhoon 2002; Kopeikin & Schäfer 1999)
transform the observed direction to the source ()
into
the BCRS spatial coordinate direction of the light ray at the point
of observation with coordinates
(see Fig. B.1). Now, paraphrasing Klioner (2003),
the coordinate direction to the light source at
is defined by the fourvector
,
where
,
x^{i}, and t are the BCRS coordinates. But the coordinate components
p^{i} are not directly observable quantities; the observed vector
towards the light source is the fourvector
,
defined with respect to the local inertial frame of the observer.
In the local frame:
where are the coordinates in the CoMRS. Then to deduce the spatial direction p^{i} from s^{i} an infinitesimal transformation between CoMRS and BCRS is adopted, given by the formula
Figure B.1: The vectors representing the light direction in the pM/pN approaches inside the nearzone of the solar system. 

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One can make Eq. (B.3) explicit by following the procedure reported in (Klioner & Kopeikin 1992) and adopting the IAU resolution B1.3 (Soffel et al. 2003). From the BCRS to the CoMRS ( ), the transformation between the time coordinates reads as
and between the spatial coordinates as
All the functions A, B, C, D are defined in Klioner & Kopeikin (1992) or in IAU resolutions, and are the coordinate displacements with respect to the center of mass of the satellite in the BCRS, and finally .
As reported in Klioner (2004), the attitude in GREM (SRS) is obtained by applying an orthogonal rotation matrix to in Eq. (B.5). At this stage the role of the SRS is equivalent to that of the s in Eq. (8).
If one keeps all the terms up to the order of 1 microarcsecond,
the observed stellar direction s^{i} is transformed (in the CoMRS) into the unitary
``aberrationfree'' direction
n^{i}=p^{i}/p (where
):
References
 Bastian, U. 2004, Reference System, Conventions and Notations for Gaia, Research Note GAIAARIBAS003, GAIA livelink [Google Scholar]
 Bini, D., Crosta, M. T., & de Felice, F. 2003, Class. Quantum Grav., 20, 4695 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Crosta, M. T. 2003, Methods of Relativistic Astrometry for the analysis of astrometric data in the Solar System gravitational field, Ph.D. Thesis, Università di Padova, Centro Interdipartimentale di Studi e Attività Spaziali (CISAS) ``G. Colombo'' [Google Scholar]
 de Felice, F., & Clarke, C. J. S. 1990, Relativity on curved manifolds (Cambridge University Press) [Google Scholar]
 de Felice, F., & Preti, G. 2006, Class. Quantum Grav., 23, 5467 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 de Felice, F., Crosta, M. T., Vecchiato, A., Lattanzi, M. G., & Bucciarelli, B. 2004, ApJ, 607, 580 [NASA ADS] [CrossRef] [Google Scholar]
 de Felice, F., Vecchiato, A., Crosta, M. T., Bucciarelli, B., & Lattanzi, M. G. 2006, ApJ, 653, 1552 [NASA ADS] [CrossRef] [Google Scholar]
 Jantzen, R. T., Carini, P., & Bini, D. 1992, Ann. Phys., 215, 1 [NASA ADS] [CrossRef] [Google Scholar]
 Klioner, S. A. 2003, AJ, 125, 1580 [NASA ADS] [CrossRef] [Google Scholar]
 Klioner, S. A. 2004, Phys. Rev. D, 69, 124001 [NASA ADS] [CrossRef] [Google Scholar]
 Klioner, S. A., & Kopeikin, S. M. 1992, AJ, 104, 897 [NASA ADS] [CrossRef] [Google Scholar]
 Kopeikin, S. M., & Mashhoon, B. 2002, Phys. Rev. D, 65, 64025 [NASA ADS] [CrossRef] [Google Scholar]
 Kopeikin, S. M., & Schäfer, G. 1999, Phys. Rev. D, 60, 124002 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Lattanzi, M. G., Drimmel, R., Gai, M., et al. 2006, Astrometric Verification Unit, Tech. rep., GAIAC3TNINAFML0012 [Google Scholar]
 Misner, C. W., Thorne, K. S., & Wheeler, J. A. 1973, Gravitation, (San Francisco: W.H. Freeman and Co) [Google Scholar]
 Soffel, M., Klioner, S. A., Petit, G., et al. 2003, AJ, 126, 2687 [NASA ADS] [CrossRef] [Google Scholar]
 Turon, C., O'Flaherty, K. S., & Perryman, M. A. C. 2005, The ThreeDimensional Universe with Gaia [Google Scholar]
 Unwin, S. C., Shao, M., Tanner, A. M., et al. 2008, Publ. Astron. Soc. Pac., 120, 38 [NASA ADS] [CrossRef] [Google Scholar]
Footnotes
 ... zone^{}
 The near zone of a system of bound sources, which generates no stationary gravitational field, is defined as the region of space with a size comparable to the wavelength of the gravitational radiation emitted by that system.
 ... equations''^{}
 These equations derive from the null geodesic with the appropriate projection onto the restspace of the local barycentric observer (de Felice et al. 2004,2006).
 ... resolutions^{}
 These resolutions are based on the pN approximation, which is still compatible with RAMOD, since the perturbation to the Minkowskian metric in (1) can be calculated at any desired order of approximations in inside the Solar System.
 ... observation^{}
 Also, each Eq. (8) is essential in RAMOD as it represents a boundary condition needed to uniquely solve the master equations.
All Figures
Figure 1: The local observer with respect to the BCRS coordinate system. The spatial axes of the BCRS point toward distant sources. The dashed lines are the curves that are orthogonal, say , to the hypersurfaces, asymptotically orthogonal to the time direction. The restspace (green area) of locally deviates from the spacelike hypersurface with equation by terms of the order of a microarcsecond. 

Open with DEXTER  
In the text 
Figure B.1: The vectors representing the light direction in the pM/pN approaches inside the nearzone of the solar system. 

Open with DEXTER  
In the text 
Copyright ESO 2010