The Ray Tracing Analytical Solution within the RAMOD framework. The case of a Gaialike observer.
Abstract
This paper presents the analytical solution of the inverse ray tracing problem for photons emitted by a star and collected by an observer located in the gravitational field of the Solar System. This solution has been conceived to suit the accuracy achievable by the ESA Gaia satellite (launched on December 19, 2013) consistently with the measurement protocol in General relativity adopted within the RAMOD framework. Aim of this study is to provide a general relativistic tool for the science exploitation of such a revolutionary mission, whose main goal is to trace back star directions from within our local curved spacetime, therefore providing a threedimensional map of our Galaxy. The results are useful for a thorough comparison and crosschecking validation of what already exists in the field of Relativistic Astrometry. Moreover, the analytical solutions presented here can be extended to model other measurements that require the same order of accuracy expected for Gaia.
pacs:
0.4,95.30.Sf, 95.10.Jk, 04.20.Cv, 04.25.g
October 2014
1 Introduction
To fully exploit the science of the Gaia mission (ESA, [2005tdug.conf…..T, ]), a relativistic astrometric model is needed able to cope with an accuracy of few for observations within the Solar System.
Gaia acts as a celestial compass, measuring arches among stars with the purpose to determine their position via the absolute parallax method. The main goal is to construct a threedimensional map of the Milky Way and unravel its structure, dynamics, and evolutional history. This task is accomplished through a complete census, to a given brightness limit, of about one billion individual stellar objects.
Since the satellite is positioned at Lagrangian point L2 of the SunEarth system, the measurements of Gaia are performed in a weak gravitational regime and the solution of Einstein equation, i.e the spacetime metric, has the general form
(1) 
where and can be treated as perturbations of a flat spacetime and represent all the Solar System contributions to the gravitational field. Their explicit expression, however, can be described in different ways according to the physical situation we are considering. This means that, for the weakfield case, can always be expanded in powers of a given smallness parameter , as
where the underscript indicates the order of . This expansion is usually made in power of the gravitational constant G (postMinkowskian approach) or in power of (postNewtonian approach) both approaches coinciding inside the near zone of the Solar System [1987thyg.book…..H, ]. While the postMinkowskian formalism is better suitable outside the nearzone of the Solar System, the estimates performed inside this zone are sufficiently well supported by an approximation to the required order in which amounts to about for the typical velocities of our planet. Moreover, for the propagation of the light inside the Solar System, the sources of gravity should be considered together with their internal structure and geometrical shape. This is particularly true when the light passes close to the giant planets. In other circumstances it is an unnecessary complication to consider the planets different from pointlike objects especially when the model is devoted to the reconstruction of the stellar positions in a global sense. At the microarcsecond level of accuracy, i.e. , the contribution to the metric coefficients of the motion and the internal structure of the giant planets should be taken into account, in particular if one wants to measure specific light deflection effects, as for example, those due to the quadrupolar terms.
The scope of this paper is to present an analytical solution for a null geodesic of the metric (1) consistently with the requirements of Gaia’s astrometric mission and according to the RAMOD framework [2004ApJ…607..580D, ; 2006ApJ…653.1552D, ]. RAMOD uses a 3+1 description of the spacetime in order to measure physical effects along the proper time and in the restspace of a set of fiducial observers according to the following measurement protocol [2010ToM.book…..D, ]:
i) specify the phenomenon under investigation;
ii) identify the covariant equations which describe the above effect;
iii) identify the observer who makes the measurements;
iv) chose a frame adapted to that observer allowing the spacetime splitting into the observer’s space and time;
v) understand the locality properties of the measurement under consideration (namely whether it is local or nonlocal with respect to the background curvature);
vi) identify the frame components of the quantities which are the observational targets;
vii) find a physical interpretation of the above components following a suitable criterium;
viii) verify the degree of the residual ambiguity, if any, in the interpretation of the measurements and decide the strategy to evaluate it (i.e. comparing to what already known).
The main procedure of the RAMOD approach is to express the null geodesic in terms of the physical quantities which enter the process of measurement, in order to entangle the entire light trajectory with the background geometry at the required approximations. Then, the solution is adapted to the relevant IAU resolutions considered for Gaia 2010A&A…509A..37C [].
Solving the astrometric problem turns out to compile an astrometric catalog with the accuracy of the measurement model. Indeed there exist several models conceived for the above task and formulated in different and independent ways ([2003AJ….125.1580K, ; 2012CQGra..29x5010T, ; 2014PhRvD..89f4045H, ] and references therein). Their availability must not be considered as an “oversized toolbox” provided by the theoretical physicists. Quite the contrary, they are needed to put the future experimental results on solid grounds, especially if one needs to implement gravitational source velocities and retarded time effects. From the experimental point of view, in fact, modern space astrometry is going to cast our knowledge into a widely unknown territory. Such a huge pushforward will not only come from highprecision measurements, which call for a suitable relativistic modeling, but also in form of absolute results which need be validated by independent, groundbased observations. In this regard, it is of capital importance to have different, and crosschecked models which exploit different solutions to interpret these experimental data.
For the reason above, inside the Consortium constituted for the Gaia data reduction (Gaia CU3, Core Processing, DPAC), two models have been developed: i) GREM (Gaia RElativistic Model, 2003AJ….125.1580K []) baselined for the Astrometric Global Iterative Solution for Gaia (AGIS), and ii) RAMOD (Relativistic Astrometric MODel) implemented in the Global Sphere Reconstruction (GSR) of the Astrometric Verification Unit at the Italian data center (DPCT, the only system, together with the DPC of Madrid, able to perform the calibration of positions, parallaxes and proper motions of the Gaia data). RAMOD was originated to satisfy the validation requirement and, indeed, the procedure developed can be conceived to all physical measurements which imply light propagation.
Section 2 lists the notation used in this paper; section 3 is devoted to the definition of the mathematical environment needed to make the null geodesic explicit at the desired accuracy. In particular, in order to fully accomplish the precepts of the measurement protocol and to isolate the contributions from the derivative of the metric terms at the different retained orders, we introduce a suitable classification of the RAMOD equations. In section 4 we set the appropriate approximations which permit the analytical solution of the astrometric problem. In section 5, we show the specific solution for the light deflection by spherical and nonspherical gravitational sources. In section 6, finally, we deduce the analytical solution of the trajectories of the light signal emitted by the stars and propagating through the gravitational field within the Solar System. In the last section we summarize the conclusions.
2 Notations

Greek indeces run from 0 to 3, whereas Latin indeces from 1 to 3;

””: partial derivative with respect to the coordinate;

””: scalar product with respect to the euclidean metric ;

””: cross product with respect to the euclidean metric ;

tildeed symbols ”” refer to quantities related to the gravitational sources;

repeated indeces like for any four vectors means summation over their range of values;

”: antisymmetrization of the indeces

: covariant derivative;

: component of the covariant derivative;

: spatial gradient of a function

we use geometrized unit, namely and, .
3 The RAMOD equations for Gaia
The basic unknown of the RAMOD method is the spacelike fourvector , which is the projection of the tangent to the null geodesic into the restspace of the local barycentric observer, namely the one locally at rest with respect to the barycenter of the Solar System. Physically, such a fourvector identifies the line of sight of the incoming photon relative to that observer.
Once defined , the equations of the null geodesic takes a form which we shall refer to as masterequations. Neglecting all the terms, these read [2006ApJ…653.1552D, ; 2011CQGra..28w5013C, ]:
Here is the affine parameter of the geodesic and
(4) 
whereas is the coordinate time.
In order to solve for the master equations one should define appropriate metric coefficients. To the order of , which is what is required for the accuracy targeted for Gaia, one has to take into account the distance between the points on the photon trajectory and the barycenter of the ath gravity source at the appropriate retarded time together with the dynamical contribution to the background metric by the relative motion of the gravitational sources. More specifically is the retarded distance defined as
(5) 
where is the parameter of the th source’s world line. The retarded position of the source is fixed by the intercept of its worldline with the past light cone at any point on the photon trajectory. However, the retarded time and the retarded distance are intertwined in an implicit relation which would prevent us to solve the geodesic equations. Nonetheless, we show that it is possible to write an approximate form of the metric which retains the required order of accuracy of , but where the dependence from the retarded contribution is simplified.
By using the Taylor expansion around any to the first order in , we get for each source:
(6) 
which allows to rewrite the retarted distance as
i.e.
(7) 
where we set . Nevertheless is again proportional to the retarded distance as measured along the source world line. In fact, considering the tangent fourvector of the source world line
where is the component of the spatial fourvelocity of the source relative to the origin of the coordinate system and defined in the rest frame of the local barycentric observer , the interval elapsed from the position of the source at the time and that at is
(8)  
where . To the first order in , we have along the generator of the light cone
(9) 
then we get the following approximate expression for (5):
(10) 
or
(11) 
where . This is equivalent, to first order in , to the distance found in 1999PhRvD..60l4002K [] and entering the expression of the metric, i.e.
(12) 
The choice for the perturbation term of the metric has to match the adopted retarded distance approximation and the fact that the lowest order of the terms is and the present space astrometry accuracy does not exceed the level.
Then, for our purpose, a standard suitable solution of Einstein’s equations in terms of a retarded tensor potential [1990recm.book…..D, ; 2006ApJ…653.1552D, ], which can be further specialized as the LiénardWiechert potentials [1999PhRvD..60l4002K, ] is
(13)  
where is the mass of the th gravity source, is the position vector of the photon with respect to the source, is the coordinate spatial velocity of the gravity source.
Note that the time component of the tangent vector to the source’s worldline [2006ApJ…653.1552D, ] is
(14) 
while that of the local barycentric observer is
(15) 
Then from (14) and (15) we derive the following relationships in the linear approximation
(16) 
and
(17) 
Within the approximation (12) the perturbation of the metric transforms as
(18)  
or, by simplifing the notation
(19)  
In the follows, , unless explicitly expressed, will indicate a function with arguments and . Moreover, to ease notation we drop the index (a) wherever it is not necessary.
3.1 The nbodies spherical case
Let us consider a spacetime splitting with respect to the congruence of fiducial observers in the gravitational field of the Solar System [2004ApJ…607..580D, ]. The field equations can be rewritten in terms of the shear, expansion and vorticity of the congruence [see 1990recm.book…..D, ]. For our purpose it is enough to consider only the expansion term and the vorticity [see 2011CQGra..28w5013C, ].
The master equations (3) and (3) are obtained by retaining the vorticity term at least to the order of , and the expansion to the order of and . In the case of a vorticity and expansionfree geometry, the RAMOD master equations are named RAMOD3 master equation [2004ApJ…607..580D, ]
(20) 
where . Taking into account that , equations (3), (3), and (20) can be reduced respectively to:
(21)  
(22)  
(23) 
and for the static case ( and )
(24) 
In order to fully accomplish the precepts of the measurement protocol, it would be useful, to isolate the contributions from the derivatives of the metric at the different retained orders.
This allows us to classify the master equation as follows:

RAMOD3a (R3a), the spatial derivatives of the metric are considered while are neglected
(25) 
RAMOD3b (R3b), the spatial and time derivatives of the metric are considered while are neglected
(27) 
RAMOD4a (R4a), the spatial derivatives of the metric are considered including
(29) 
RAMOD4b (R4b), the spatial and time derivatives of the metric are considered including
(31)
The implementation of RAMOD models and the need of testing them through a selfconsistency check at different levels of accuracy, will benefit form this explicit classification. Beside this new classification of the RAMOD master equations, it is clear that the solutions call for an explicit expression of the metric terms. In general, for any integer :
(32) 
From the last computation one could expect that in the case of mapped trajectories for RAMOD3like model [see 2004ApJ…607..580D, ; 2011CQGra..28w5013C, ] the term should be retained, since each mapped spatial coordinate depends on the value of the local oneparameter diffeormorphism. In this respect, note that the null geodesic crosses each slice at a point with coordinates , but this point also belongs to the unique normal to the slice crossing it with a value which runs differently for any spatial coordinate and therefore does not coincide with the proper time of the local barycentric observer. Therefore
Now, by using the retarted time approximation we get:
(33) 
and
(34) 
Let us indicate the photon impact parameter with respect to the source position as:
(35) 
and, for sake of convenience, let us denote also:
(36) 
Finally, according to the previous derivatives, making them explicit, and denoting , the master equations assume the following expressions, valid up to the order:

RAMOD3a:
(37) where in case of zero velocity we recover the static RAMOD recorded as follows

RAMOD3s(R3s):
(38) Similarly for the other classification items we have:

RAMOD3b:
(39) (40) 
RAMOD4a
(41) (42)
It is clear that, to the order of , we do not need to include the time derivative of since these are at least of the order of and should be neglected. Therefore, we do not consider the part of the RAMOD4 equations which contain the time derivative of and the second order velocity contributions.
3.2 The case for an oblate body
Now, let us consider the th source and define
(43) 
which means to take into account the mass multipole structure of the ath body where are the Legendre polynomials, the mass of the body, its equatorial radius, the colatitude, and the coefficients of the mass multipole moments. With this choice our considerations are confined to the case in which the object’ ellipsoid of inertia is an ellipsoid of revolution and the directions of the spatial coordinate axes coincide with those of the principal axes of inertia [2013CQGra..30d5009B, ].
A rigorous treatment of a nbody multipolar expansion should take into account the different orientation of its axis of symmetry. However, this contribution decrease so quickly that at any accuracy currently attainable it turns out to be an unnecessary complication, since just one planet at a time would give a detectable effect.
The derivatives of the metric coefficients, with retarded time approximation, have the following expressions:
(44) 
(45) 
where . A general nbody solution should include the multipolar structure of the sources. Nevertheless, according to the current astrometric accuracy and for an oblate body, the quadrupole approximation can be considered enough [see 2003AJ….125.1580K, ]. If we omit the higher multipole moments and restrict ourselves only to , denoting by the axis of the sources which is normal to the source equatorial plane, is approximated as:
(46) 
which to first order in becomes
(47) 
therefore
(48) 
and
(49) 
By taking into account the target accuracy of Gaia [see 2003AJ….125.1580K, ; 2008PhRvD..77d4029L, ], the velocity contributions for an oblate body should be neglected. However, for sake of consistency and completeness with the assumptions adopted in this work, neglecting, a priori, terms which are part of the solution is not justified, even if the application to Gaia will surely dismiss many of them. Probably a Gaialike mission that achieves few submicroarsecond in accuracy will benefit of these analytical contributions, especially in regards of a crosschecking comparison between different approaches. In this case it would be better to consider a metric which properly contemplate all the complexities of a nonspherical gravitational body; that, at the moment, is out of the scope of the present paper and deserves a dedicated work (see, e.g., 2013CQGra..30d5009B []).
Therefore, the RAMOD reduced master equations which take into account the quadrupole structure for the th single source finally become:

RAMOD3aQ (R3aQ)
(50) where .
In case of velocity equal to zero, RAMOD equations become:
(51) 
RAMOD3bQ (R3bQ)
(52) (53) 
RAMOD4aQ (R4aQ)
(54)