Relativistic satellite astrometry at arcsec precision
and the measurement of the stellar aberration
(Research Note)
G. Preti^{1,2} - F. de Felice^{1,2}
1 - Dipartimento di Fisica ``Galileo Galilei'', Università degli Studi di Padova, Italy
2 - INFN - Sezione di Padova, Italy
Received 27 October 2009 / Accepted 11 February 2010
Abstract
Analysis and interpretation of the arcsec
precision astrometric data shortly to be provided by the ESA-GAIA
satellite has prompted the development of a fully general-Relativistic
Astrometric MODel (RAMOD), which is able to account for all the
general-relativistic corrections up to the required order of precision
.
We obtain the covariant expression for the general relativistic
aberration effect and provide the analytical module for determining the
stellar aberration from the GAIA observables (the direction cosines of
the incoming light-ray trajectory) at this same
order of precision.
Key words: gravitation - relativity - methods: analytical - astrometry - reference systems
1 Introduction
Stellar aberration is a well-known astronomical effect, discovered by Bradley (1728); it consists of the variation of the apparent line of sight caused by the motion of the observer, and is a consequence of the finiteness of the velocity of light, first measured by Römer (1676). As is well known, in the context of special relativity the aberration formula can be obtained from Lorentz-transforming the coordinate components of the velocity of light, and expressing these components in terms of the angles measured by two different observers between the line of sight of the star and the apex of their motion. The case of uniformly accelerated observers has been recently dealt with in Beig & Heinzle (2008). In the context of general relativity, a covariant formula for the stellar aberration can be directly derived from the general-Relativistic Astrometric MODel (RAMOD)^{}, which was conceived and developed in de Felice et al. (1998, 2001), Bini & de Felice (2003), Bini et al. (2003), de Felice et al. (2004, 2006), de Felice & Preti (2006, 2008), Crosta & Vecchiato (2010), to reconstruct the trajectory of a light ray from the event of observation back to the star position in terms of the observational data represented by the direction cosines of the incoming light ray with respect to a given spatial frame. These data, which will soon be provided by the GAIA satellite (Turon et al. 2005), will allow us to obtain a map of the celestial sphere with a arcsec accuracy in the measurements of angles. This accuracy is assured by retaining terms of the order of c^{-3} - namely, - in the general-relativistic astrometric model. The problem of fixing the boundary conditions for the inverse ray-tracing problem was analytically discussed and fully solved in a series of works (Bini & de Felice 2003; Bini et al. 2003; de Felice & Preti 2006, 2008); in this paper we derive the covariant formula for the general relativistic aberration effect from the basic equation of the boundary conditions and provide a full analytical model for the direct determination of the stellar aberration angle in terms of the GAIA observables.
Notation and conventions: metric signature is +2; Greek (spacetime) indices run from 0 to 3, while Latin (space) indices run from 1 to 3; hatted indices identify tetrad quantities; expressions like for any and indicate scalar products relative to the given metric.
2 The aberration formula
Let our Galaxy be contained in a coordinate neighbourhood of a
spacetime manifold, which we assume to be generated by the solar system
alone. Let
be a coordinate system centred at the baricentre of the solar system; the spatial coordinates
span spacelike hypersurfaces, which we require to exist at t= const., t
being a time coordinate. On each of these hypersurfaces, the spatial
coordinate axes are chosen to identify a kinematically nonrotating
frame, termed barycentric celestial reference system (BCRS). This
coordinate system generates a field of coordinate bases for the tangent
spaces on the manifold; all tensor components referred to in this paper
are relative to those bases. The background metric is a solution of
Einstein's equations with the solar system as the only source;
its form is therefore dictated by the weak-field, slow-motion
approximation, with metric components reading
g_{00} | = | ||
g_{ij} | = |
where, to the lowest order, the metric perturbations are , , . This choice was recommended by the IAU (2000).
Since the following discussion will deal primarily with the measurement of angles, it is essential to identify the observers and the frames of reference chosen to perform these measurements. We define
- The locally baricentric observers.
They are a family of observers who are static with respect to the
spatial axes of the BCRS, and are described at each spacetime point by
a four-vector
with coordinate components
,
where
A(x)=(-g_{00})^{-1/2} is a scalar function, that assures unitarity of
with respect to the given metric:
.
A tetrad frame adapted to a locally baricentric observer is termed
``Sun-locked'' if one of the vectors of its spatial triad is identified
as stably pointing to the geometrical centre of the Sun. Below, the
spatial triad of the tetrad frame adapted to the BCRS observer will be
indicated with
.
- The boosted observer.
This observer is at rest with respect to the centre of mass of the
satellite and carries a triad, which is obtained from the local
baricentric Sun-locked frame by boosting
to the rest space of the satellite.
The boosted observer is described by a unitary four-vector field
with coordinate components
,
where
and
is a scalar function, that assures unitarity of
with respect to the given metric:
.
The vector
is
tangent to the trajectory of the centre of mass of the satellite.
The spatial triad of the tetrad frame adapted to this observer will below be indicated with
.
- The satellite observer. This observer is at rest with respect to the centre of mass of the satellite, but is also at rest with respect to the spatial rotations as established by the satellite attitude (Turon et al. 2005). This observer is characterized by the same unitary four-vector field , tangent to the trajectory of the centre of mass of the satellite considered above, but it carries a spatial triad that rotates with respect to , as stated. The spatial triad of the tetrad frame adapted to this observer will be indicated below with . Note that both translates and rotates with respect to .
The instantaneous physical (``spatial'') velocity of the satellite observer
with respect to the local barycentric observer
is given by the modulus of the four-vector
the components of which are given by
where the tensor is the projection operator on the rest space of the local baricentric observer, and is the relative Lorentz factor between and (de Felice & Clarke 1990). Similarly, if we indicate with the tangent field to the geodesic trajectory of a light signal emitted by a given star, we can define the line of sight of the local baricentric observer at the point of observation as the spatial four-vector , the components of which are given by
(2) |
note that
The astrometric observables are represented by the direction cosines between the th spatial direction fixed by the satellite attitude vector and the direction of the light ray in the rest frame of the observer ; these observables are given by
(de Felice & Clarke 1990; Brumberg 1991), where the tensor is the projection operator on the rest space of the satellite. In order to calculate the aberration effect due to the translational motion of the satellite with respect to the local BCRS, we need two angles:
- The angle
,
which the local baricentric observer would measure between the direction of
and the corresponding th direction of the triad
:
- The angle
,
which the boosted observer would measure between the direction of
and the corresponding th direction of the triad
:
As far as Eq. (6) is regarded, first we note that
as it follows from Eqs. (1) and (3); then we observe that
and that
Hence we finally obtain from Eq. (6)
This expression can be rewritten as a function of as given by Eq. (7), thus taking a more suggestive aspect, as we are going to see. To begin with, we note that the triad associated with the boosted observer, determined in Bini et al. (2003), can be rewritten in the following convenient form
where
is the local BCRS Sun-locked triad, and the boost is provided by
(17) |
where
(18) |
and
In the above expressions the are the coordinate components of the three-velocity of the satellite with respect to the local BCRS, and . The angles and are given by
where and are the BCRS coordinates of the Sun and of the satellite, respectively.
Using Eq. (9), and recalling Eq. (7), we now see that Eq. (8) can be rewritten in the following form
which, clearly reminiscent of the form characterising the well-known special relativistic formula for the aberration effect, provides the generalization of the latter to the curved spacetime case, with arbitrary direction of the relative motion of the two observers and .
The consistency of Eq. (24)
can be checked by deducing from it, as a special case, the well-known
text-book special relativistic expression for the aberration effect. To
this end, we have to neglect all the gravitational potentials and
consider the simple case of the two observers
and
being in inertial relative motion along the coordinate
x axis.
The general expression of a boosted vector - a vector of the triad in our case - reads
(Jantzen et al. 1992); hence, setting , where , , which implies from (1) that and , hence that , we get
From Eq. (25), recalling Eqs. (7) and (9), and observing that , where is the angle the local line-of-sight forms with the coordinate x-direction as seen by the local BCRS observer, the numerator N and the denominator D of Eq. (24) in the simple case under scrutiny become
From these espressions, requiring the vectors of the triad to coincide with the local coordinate directions - namely - and then considering in Eq. (24) the case (namely, the direction of the relative motion between the two inertial observers we now consider), we finally recover the well-known special relativistic aberration formula
valid at all orders of .
3 The aberration angle
Since aberration is an effect caused by the state of motion of the
observer with respect to the star, it arises at the event of
observation and is therefore accounted for - although implicitly - in
the process of fixing the boundary conditions.
For RAMOD, aberration is a correction of the observed line of sight due
to the motion of the satellite with respect to the local baricentric
observer.
The angular correction (``aberration angle'') due to translational
motion is given by
,
namely
In order to obtain the aberration angle directly from the satellite observables, we need to reexpress the director cosines appearing in Eq. (26) in terms of the actually observed data, namely the director cosines defined in Eq. (4). Resting on the results of Bini et al. (2003), we can derive the link between the non-rotating boosted triad and the triad adapted to the satellite observer, namely
where the matrix elements explicitly read
in terms of the attitude parameters of the satellite. From Eqs. (8) and (27) it therefore follows that
in terms of the GAIA observables, defined in Eq. (4). Hence, using Eqs. (7) and (37) with Eq. (26) we can write
where all the quantities appearing in the rhs are known. In fact, the are the observables, the are deducible from the observables themselves as shown in de Felice & Preti (2006), the of Eqs. (10)-(16) are fixed by the metric and the BCRS coordinates of the Sun and of the satellite (Eq. (22)), and the components of Eqs. (28)-(36) are fixed by the attitude parameters of the satellite. Equation (38) therefore provides the analytical tool for computing the stellar aberration angle directly from the GAIA observational data.
4 Conclusions
We showed how the stellar aberration issue can be tackled from the general relativistic point of view, resting on the theoretical framework provided by the Relativistic Astrometric MODel (RAMOD), which was conceived and developed to provide an all-inclusive general relativistic analysis of the inverse ray-tracing problem at the order of precision, which is the order required for a consistent dealing with the high-precision astrometric data, that will soon be provided by the ESA-GAIA mission. The results presented here can be employed as an independent module that can be attached to the main body of RAMOD, enabling us to single out and quantify the stellar aberration effect directly from the GAIA observational data.
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Footnotes
- ... (RAMOD)^{}
- For other approaches see, e.g., Bolos 2006 and Teyssandier & Le Poncin-Lafitte 2006).
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