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A General Relativistic Model of Light Propagation in the Gravitational Field of theSolar System: The Dynamical Case
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The Astrophysical Journal · October 2006
DOI: 10.1086/508701 · Source: arXiv
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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
fernando.defelice@pd.infn.it
A. Vecchiato
INAF – Turin Astronomical Observatory strada Osservatorio 20, 10125 Pino Torinese (TO), Italy
vecchiato@to.astro.it
M. T. Crosta
INAF – Turin Astronomical Observatory strada Osservatorio 20, 10125 Pino Torinese (TO), Italy
crosta@to.astro.it
B. Bucciarelli
INAF – Turin Astronomical Observatory strada Osservatorio 20, 10125 Pino Torinese (TO), Italy
bucciarelli@to.astro.it
andM. G. Lattanzi
INAF – Turin Astronomical Observatory strada Osservatorio 20, 10125 Pino Torinese (TO), Italy
1
INFN  Sezione di Padova
– 2 –
lattanzi@to.astro.it
ABSTRACT
Modern astrometry is based on angular measurements at the microarcsecondlevel. 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 nonstationary gravity ﬁeld of the moving bodiesin the Solar System. The previous model is used as testbed for numerical comparisons 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 tradeoﬀ 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 microarcsecond accuracy (
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 PoncinLaﬁtte 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 spaceborn 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 dynamical 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 postMinkowskian 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 timelike vector ﬁeld
η
=
∂
0
tangent to the coordinate time axis,will not in general be a Killing ﬁeld (namely an isometry for the spacetime) 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 thatspacelike coordinates are ﬁxed within each slice. We ﬁx the srcin of the spacelike coordinates 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 tensorial 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