The Ray Tracing Analytical Solution within the RAMOD framework. The case of a Gaia-like observer

2.5 MB·English

by  Mariateresa Crosta

The Ray Tracing Analytical Solution within the
RAMOD framework. The case of a Gaia-like
observer.
M Crosta1, A Vecchiato1 F de Felice2, M G Lattanzi1
5
1 1 AstrophysicalObservatoryofTorino,INAF,viaOsservatorio20,I-10025Pino
0 Torinese(TO),Italy
2 2 DepartmentofPhysicsandAstronomy,UniversityofPadova,viaMarzolo8,
I-35131Padova,Italy
n
a E-mail: [email protected]
J
October2014
5
1
Abstract. Thispaperpresentstheanalyticalsolutionoftheinverseraytracing

] problemforphotonsemittedbyastarandcollectedbyanobserverlocatedinthe
c
gravitationalfieldoftheSolarSystem. Thissolutionhasbeenconceivedtosuitthe
q
accuracyachievablebytheESAGaiasatellite(launchedonDecember19, 2013)
-
consistentlywiththemeasurementprotocolinGeneralrelativityadoptedwithin
r
g the RAMOD framework. Aim of this study is to provide a general relativistic
[ toolforthescienceexploitationofsucharevolutionarymission,whosemaingoal
istotracebackstardirectionsfromwithinourlocalcurvedspace-time,therefore

1 providing a three-dimensional map of our Galaxy. The results are useful for a
v thoroughcomparisonandcross-checkingvalidationofwhatalreadyexistsinthe
5 fieldofRelativisticAstrometry. Moreover,theanalyticalsolutionspresentedhere
0 can be extended to model other measurements that require the same order of
7 accuracyexpectedforGaia.
3
0
.
1 PACSnumbers: 0.4,95.30.Sf,95.10.Jk,04.20.Cv,04.25.-g
0
5
1
: Submitted to: Class. Quantum Grav.
v
i
X
r
a
Relativistic Astrometry for a Gaia-like observer 2
1. Introduction
To fully exploit the science of the Gaia mission (ESA, [1]), a relativistic astrometric
model is needed able to cope with an accuracy of few µas 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 three-dimensional 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 Sun-Earth system,
the measurements of Gaia are performed in a weak gravitational regime and the
solution of Einstein equation, i.e the space-time metric, has the general form
g =η +h +O(cid:0)h2(cid:1), (1)
αβ αβ αβ
where |h |(cid:28)1 and |∂ h |(cid:28)1 can be treated as perturbations of a flat space-time
αβ i αβ
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 weak-field case, h
αβ
can always be expanded in powers of a given smallness parameter (cid:15), as
h =h +h +h +···,
αβ αβ αβ αβ
(1) (2) (3)
where the underscript n indicates the order of (cid:15). This expansion is usually made
in power of the gravitational constant G (post-Minkowskian approach) or in power of
1/c(post-Newtonianapproach)bothapproachescoincidinginsidethenearzoneofthe
SolarSystem[2]. Whilethepost-Minkowskianformalismisbettersuitableoutsidethe
near-zoneoftheSolarSystem,theestimatesperformedinsidethiszonearesufficiently
wellsupportedbyanapproximationtotherequiredorderin(v/c)∼(cid:15)whichamounts
toabout10−4 forthetypicalvelocitiesofourplanet. Moreover,forthepropagationof
thelightinsidetheSolarSystem, thesourcesofgravityshouldbeconsideredtogether
with their internal structure and geometrical shape. This is particularly true when
the light passes close to the giant planets. Inother circumstances it is an unnecessary
complication to consider the planets different from point-like 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. (v/c)3 ∼ (cid:15)3, the contribution to the
metriccoefficientsofthemotionandtheinternalstructureofthegiantplanetsshould
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 [3; 4]. RAMOD uses a 3+1 description of the
space-time in order to measure physical effects along the proper time and in the rest-
space of a set of fiducial observers according to the following measurement protocol
[5]:
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 space-time splitting into
the observer’s space and time;
Relativistic Astrometry for a Gaia-like observer 3
v) understand the locality properties of the measurement under consideration
(namely whether it is local or non-local 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)verifythedegreeoftheresidualambiguity,ifany,intheinterpretationofthe
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 [6].
Solvingtheastrometricproblemturnsouttocompileanastrometriccatalogwith
the accuracy of the measurement model. Indeed there exist several models conceived
for the above task and formulated in different and independent ways ([7–9] and
referencestherein). Theiravailabilitymustnotbeconsideredasan“oversizedtoolbox”
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
gravitationalsourcevelocitiesandretardedtimeeffects. Fromtheexperimentalpoint
ofview, infact,modernspaceastrometryisgoingtocastourknowledgeintoawidely
unknown territory. Such a huge push-forward will not only come from high-precision
measurements, which call for a suitable relativistic modeling, but also in form of
absolute results which need be validated by independent, ground-based observations.
In this regard, it is of capital importance to have different, and cross-checked 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, [7]) 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.
Section2liststhenotationusedinthispaper;section3isdevotedtothedefinition
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 non-spherical 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.
Relativistic Astrometry for a Gaia-like observer 4
2. Notations
• Greek indeces run from 0 to 3, whereas Latin indeces from 1 to 3;
• ” ”: partial derivative with respect to the α coordinate;
,α
• ”·”: scalar product with respect to the euclidean metric δ ;
ij
• ”×”: cross product with respect to the euclidean metric δ ;
ij
• tilde-ed symbols ”˜” refer to quantities related to the gravitational sources;
• repeated indeces like u vα for any four vectors uα,vα means summation over
α
their range of values;
• ”[αβ]”: antisymmetrization of the indeces α,β
• ∇: covariant derivative;
• ∇ : α-component of the covariant derivative;
α
• ∇(f): spatial gradient of a function f
• we use geometrized unit, namely G=1 and, c=1.
3. The RAMOD equations for Gaia
The basic unknown of the RAMOD method is the space-like four-vector (cid:96)¯α , which
is the projection of the tangent to the null geodesic into the rest-space of the local
barycentric observer, namely the one locally at rest with respect to the barycenter
of the Solar System. Physically, such a four-vector identifies the line of sight of the
incoming photon relative to that observer.
Once defined (cid:96)¯α, the equations of the null geodesic takes a form which we shall
refer to as master-equations. Neglecting all the O(h2) terms, these read [4; 10]:
d(cid:96)¯0 1
−(cid:96)¯i(cid:96)¯jh − h =0, (2)
dσ 0j,i 2 00,0
d(cid:96)¯k 1 (cid:18) 1 (cid:19)
− (cid:96)¯k(cid:96)¯i((cid:96)¯jh −h )+(cid:96)¯i(cid:96)¯j h − h (3)
dσ 2 ij,0 00,i kj,i 2 ij,k
1
+(cid:96)¯i(h +h −h )− h −(cid:96)¯k(cid:96)¯ih +h =0.
k0,i ki,0 0i,k 2 00,k 0i,0 k0,0
Here σ is the affine parameter of the geodesic and
dσ =dt+O(h), (4)
whereas t is the coordinate time.
In order to solve for the master equations one should define appropriate metric
coefficients. To the order of (cid:15)3, 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 a-th gravity source at the appropriate retarded
timetogetherwiththedynamicalcontributiontothebackgroundmetricbytherelative
motion of the gravitational sources. More specifically ri is the retarded distance
(a)
defined as
ri (σ,σ˜(cid:48))=xi(σ)−x˜i(σ˜(cid:48)), (5)
(a) a
where σ˜ is the parameter of the (a)-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
Relativistic Astrometry for a Gaia-like observer 5
point on the photon trajectory. However, the retarded time t(cid:48) = t−r and the
(a)
retarded distance r are intertwined in an implicit relation which would prevent us
(a)
to solve the geodesic equations. Nonetheless, we show that it is possible to write an
approximateformofthemetricwhichretainstherequiredorderofaccuracyof(cid:15)3,but
where the dependence from the retarded contribution is simplified.
By using the Taylor expansion around any σ˜(cid:48) to the first order in (cid:15), we get for
each source:
xi(σ˜)≈x˜i(σ˜(cid:48))+v˜i(σ˜(cid:48))(σ˜−σ˜(cid:48)), (6)
which allows to rewrite the retarted distance as
ri =xi(σ)−x˜i(σ˜(cid:48))≈xi(σ)−x˜i(σ˜)+v˜i(σ˜(cid:48))(σ˜(cid:48)−σ˜),
(a)
i.e.
ri =ri(σ,σ˜)+r˜i(σ˜,σ˜(cid:48))+O(v˜2), (7)
(a)
where we set r˜i(σ˜,σ˜(cid:48)) = x˜i(σ˜)−x˜i(σ˜(cid:48)). Nevertheless (σ˜(cid:48) −σ˜) is again proportional
to the retarded distance as measured along the source world line. In fact, considering
the tangent four-vector of the source world line
u˜α =−(u u˜β)(uα+v˜α),
β
where v˜α is the α-component of the spatial four-velocity of the source relative to the
origin of the coordinate system and defined in the rest frame of the local barycentric
observeru,theintervalelapsedfromthepositionofthesourceatthetimet(cid:48) andthat
at t is
σ˜(cid:48)−σ˜ =∆T =−u˜ ∆xα
u˜ α
= −η u˜α∆xβ +O(h)
αβ
≈∆x0−δ v˜i∆xj +O(h), (8)
ij
where∆xα =xα(σ˜(t(cid:48)))−xα(σ˜(t)). Tothefirstorderinv˜,wehavealongthegenerator
of the light cone
(r(σ,σ˜)·˜r(σ˜,σ˜(cid:48)))
∆x0 =r(σ,σ˜)+ +O(v˜2), (9)
r(σ,σ˜)
then we get the following approximate expression for (5):
ri(σ,σ˜(cid:48)) ≈ri(σ,σ˜)−v˜i(σ˜(cid:48))r(σ,σ˜) (10)
(a)
or
ri(σ,σ˜(cid:48)) ≈r(σ,σ˜)[ni(σ,σ˜)−v˜i(σ˜(cid:48))], (11)
(a)
where ni(σ,σ˜)=ri/r. This is equivalent, to first order in v˜, to the distance found in
[11] and entering the expression of the metric, i.e.
r(σ,σ˜(cid:48)) ≈r−r·v˜. (12)
(a)
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 h terms is
(cid:15)2 and the present space astrometry accuracy does not exceed the (cid:15)3 level.
Relativistic Astrometry for a Gaia-like observer 6
Then, for our purpose, a standard suitable solution of Einstein’s equations in
terms of a retarded tensor potential [4; 12], which can be further specialized as the
Li´enard-Wiechert potentials [11] is
h = (cid:88)2M(a) +O(cid:0)(cid:15)4(cid:1)
00 r
a (a)
h = −(cid:88)4M(a)β˜ +O(cid:0)(cid:15)5(cid:1) (13)
0i r i(a)
a (a)
h = (cid:88)2M(a)δ +O(cid:0)(cid:15)4(cid:1),
ij r ij
a (a)
where M is the mass of the ath gravity source, r is the position vector of the
(a) (a)
photonwithrespecttothesource,β˜j =x˜j =(1−h /2)v˜i(σ˜)+O(h2)isthecoordinate
,0 00
spatial velocity of the gravity source.
Note that the time component of the tangent vector to the source’s worldline [4]
is
u˜0 = dt =1+ h00 + v˜2 +O(h2)+O(cid:0)(cid:15)4(cid:1) (14)
dσ˜ 2 2c2
while that of the local barycentric observer is
dt h
u0 = =1+ 00 +O(h2). (15)
dσ 2
Then from (14) and (15) we derive the following relationships in the linear
approximation
dσ˜ =dt(cid:18)1− h00 − v˜2(cid:19)+O(h2)+O(cid:0)(cid:15)4(cid:1), (16)
2 2
and
(cid:18) (cid:19)
dσ =dt 1− h00 +O(h2)+O(cid:0)(cid:15)4(cid:1). (17)
2
Within the approximation (12) the perturbation of the metric transforms as
h =2(cid:88)M(a)(1+n ·v˜ )+O(cid:0)(cid:15)4(cid:1)
00 r (a) (a)
a (a)
h = −4(cid:88)M(a)v +O(cid:0)(cid:15)5(cid:1) (18)
0i r i(a)
a (a)
h =2(cid:88)M(a)(1+n ·v˜ )δ +O(cid:0)(cid:15)4(cid:1)
ij r (a) (a) ij
a (a)
or, by simplifing the notation
(cid:88) (cid:88) 1
h =h(cid:39) h =2 M (1+n ·v˜ )+O((cid:15)4)
00 (a) (a)r (a) (a)
a a (a)
(cid:88)
h = −2hv˜ (cid:39)−2 h v˜ +O((cid:15)3) (19)
0i i (a) i(a)
a
h (cid:39)hδ +O((cid:15)4),
ij ij
In the follows, r , unless explicitly expressed, will indicate a function with
(a)
arguments σ and σ˜. Moreover, to ease notation we drop the index (a) wherever it
is not necessary.
Relativistic Astrometry for a Gaia-like observer 7
3.1. The n-bodies spherical case
Let us consider a space-time splitting with respect to the congruence of fiducial
observers u in the gravitational field of the Solar System [3]. The field equations
can be rewritten in terms of the shear, expansion and vorticity of the congruence u
[see 12]. For our purpose it is enough to consider only the expansion term and the
vorticity [see 10].
The master equations (2) and (3) are obtained by retaining the vorticity term at
leasttotheorderofO(h ),andtheexpansiontotheorderofO(∂ h )andO(∂ h ).
0i 0 00 0 0i
In the case of a vorticity and expansion-free geometry, the RAMOD master equations
are named RAMOD3 master equation [3]
d(cid:96)¯k +(cid:96)¯i(cid:96)¯j(cid:18)h − 1h (cid:19)+ 1(cid:96)¯k(cid:96)¯ih − 1h +O(cid:0)h2(cid:1)=0, (20)
dσ kj,i 2 ij,k 2 00,i 2 00,k
where (cid:96)¯0 = 0. Taking into account that (cid:96)¯i(cid:96)¯jδ = 1+O(cid:0)(cid:15)2(cid:1), equations (2), (3), and
ij
(20) can be reduced respectively to:
d(cid:96)¯0 1
=−2((cid:96)¯·v˜)((cid:96)¯ih )+ h +O(h2) (21)
dσ ,i 2 ,0
d(cid:96)¯k 3 1
=− (cid:96)¯k((cid:96)¯ih )+h − (cid:96)¯kh −2((cid:96)¯·v˜)h +2v˜k((cid:96)¯ih )−(cid:96)¯k(cid:96)¯i(v˜h) (22)
dσ 2 ,i ,k 2 ,0 ,k ,i i ,0
+(v˜kh) +O(h2) (23)
,0
and for the static case (h =0 and v˜=0)
αβ,0
d(cid:96)¯k 3
+ (cid:96)¯k((cid:96)¯ih )−h +O(h2)=0. (24)
dσ 2 ,i ,k
In order to fully accomplish the precepts of the measurement protocol, it would
beuseful,toisolatethecontributionsfromthederivativesofthemetricatthedifferent
retained orders.
This allows us to classify the master equation as follows:
• RAMOD3a (R3a), the spatial derivatives of the metric are considered while h
0i
are neglected
d(cid:96)¯k 3
=− (cid:96)¯k((cid:96)¯ih )+h +O(h2)≡(R3a)k (25)
dσ 2 ,i ,k
• RAMOD3b (R3b), the spatial and time derivatives of the metric are considered
while h are neglected
0i
d(cid:96)¯0 1
= h +O(h2)≡(R3b)0 (26)
dσ 2 ,0
d(cid:96)¯k 1
=(R3a)k− (cid:96)¯kh +O(h2)≡(R3b)k (27)
dσ 2 ,0
• RAMOD4a (R4a), the spatial derivatives of the metric are considered including
h
0i
d(cid:96)¯0
=(R3b)0−2((cid:96)¯·v˜)((cid:96)¯ih )+O(h2)≡(R4a)0 (28)
dσ ,i
Relativistic Astrometry for a Gaia-like observer 8
d(cid:96)¯k
=(R3b)k−2((cid:96)¯·v˜)h +2v˜k((cid:96)¯ih )
dσ ,k ,i
+ O(h2)≡(R4a)k (29)
• RAMOD4b (R4b), the spatial and time derivatives of the metric are considered
including h
0i
d(cid:96)¯0
=(R4a)0+O(h2)≡(R4b)0 (30)
dσ
d(cid:96)¯k
=(R4a)k−2(cid:96)¯k(cid:96)¯i(v˜h) +2(v˜kh) +O(h2)≡(R4b)k (31)
dσ i ,0 ,0
The implementation of RAMOD models and the need of testing them through
a self-consistency 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 m:
(r)m =mri(r)m−2, (r)m =−m(β˜·r)(r)m−2. (32)
,i ,0
From the last computation one could expect that in the case of mapped
trajectories for RAMOD3-like model [see 3; 10] the term xi should be retained, since
,0
each mapped spatial coordinate depends on the σ value of the local one-parameter
diffeormorphism. In this respect, note that the null geodesic crosses each slice S(t) at
a point with coordinates xi(σ(t)), but this point also belongs to the unique normal
to the slice S(t) crossing it with a value σ = σ(xi,t) which runs differently for any
spatial coordinate and therefore does not coincide with the proper time of the local
barycentric observer. Therefore
∂ xi =(∂ xi(σ))(∂ σ(xi(λ),t))=0.
0 σ 0
Now, by using the retarted time approximation we get:
(r)−m(σ,σ˜(cid:48))= −mr−m−1(1+n ·v˜)m−1[ni (1+2n ·v˜)−v˜i]+O(h) (33)
,i (a) (a) (a) (a)
and
(r)−m(σ,σ˜(cid:48))=mr−m−1(n ·β˜)(1+n ·β˜)m+O(h). (34)
,0 (a) (a) (a)
Let us indicate the photon impact parameter with respect to the source position
as:
dk =rk −(cid:96)¯k(r ·(cid:96)¯) (35)
(a) (a) (a)
and, for sake of convenience, let us denote also:
dk =v˜k−(cid:96)¯k(v˜·(cid:96)¯). (36)
v(a)
Finally, according to the previous derivatives, making them explicit, and denoting
ni = ri/r, the master equations assume the following expressions, valid up to the (cid:15)3
order:
• RAMOD3a:
d(cid:96)¯k =2(cid:88)M(a) (cid:26)(cid:18)1(cid:96)¯k(n ·(cid:96)¯)−dk(cid:19)(1+2n ·v˜)− 1(cid:96)¯k((cid:96)¯·v˜)+dk(cid:27)
dσ r2 2 (a) (a) 2 v
a (a)
+O(v˜2)+O(h2), (37)
where in case of zero velocity we recover the static RAMOD recorded as follows
Relativistic Astrometry for a Gaia-like observer 9
• RAMOD3s(R3s):
d(cid:96)¯k =2(cid:88)M(a) (cid:26)1(cid:96)¯k(n ·(cid:96)¯)−dk(cid:27)+O(h2). (38)
dσ r2 2 (a)
a (a)
Similarly for the other classification items we have:
• RAMOD3b:
d(cid:96)¯0 =(cid:88)M(a)(n ·v˜)+O(v˜2)+O(h2) (39)
dσ r2 (a)
a (a)
d(cid:96)¯k =(R3a)k−(cid:96)¯k(cid:88)M(a)(n ·v˜)+O(v˜2)+O(h2) (40)
dσ r2 (a)
a (a)
• RAMOD4a
d(cid:96)¯0 =(R3b)0+4(cid:88)M(a) (cid:2)((cid:96)¯·v˜)((cid:96)¯·n )(cid:3)+O(v˜2)+O(h2) (41)
dσ r2 (a)
a (a)
d(cid:96)¯k =(R3b)k+4(cid:88)M(a) (cid:2)(cid:96)¯×(n ×v˜)(cid:3)k+O(v˜2)+O(h2) (42)
dσ r2 (a)
a (a)
Itisclearthat,totheorderof(cid:15)3,wedonotneedtoincludethetimederivativeof
h since these are at least of the order of (cid:15)4 and should be neglected. Therefore, we
0k
donotconsiderthepartoftheRAMOD4equationswhichcontainthetimederivative
of h and the second order velocity contributions.
0k
3.2. The case for an oblate body
Now, let us consider the a-th source and define
h =2M h¯ ≡2M 1 (cid:34)1− (cid:88)∞ J (cid:18)R(a)(cid:19)mP (cosθ )(cid:35), (43)
(a) (a) (a) (a)r m r m (a)
(a) (a)
m=2
whichmeanstotakeintoaccountthemassmultipolestructureofthea-thbodywhere
P are the Legendre polynomials, M the mass of the body, R its equatorial radius,
m a a
θ the co-latitude, and J the coefficients of the mass multipole moments. With
(a) m
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 [13].
A rigorous treatment of a n-body multipolar expansion should take into account
the different orientation of its axis of symmetry. However, this contribution decrease
soquicklythatatanyaccuracycurrentlyattainableitturnsouttobeanunnecessary
complication, since just one planet at a time would give a detectable effect.
Thederivativesofthemetriccoefficients,withretardedtimeapproximation,have
the following expressions:
h¯ =(cid:32)−nk(a)(1+2n ·v˜)+ v˜k (cid:33)(cid:34)1− (cid:88)∞ J (cid:18)R(a)(cid:19)mP (cosθ )(cid:35)
(a),k r2 (a) r2 m r m (a)
(a) (a) m=2 (a)
+ 1 (1+n ·v˜)(cid:40)(cid:88)∞ J (cid:18)R(a)(cid:19)m(cid:2)m(1+n ·v˜)mP (cosθ )
r (a) m r (a) m (a)
(a) (a)
m=2
(cid:32)nk v˜k (cid:33) (cid:35)(cid:41)
(a) − (1+δ ni v˜j)−1 −P (cosθ ) , (44)
r r ij (a) m (a) ,k
(a) (a)
Relativistic Astrometry for a Gaia-like observer 10
h¯ =(cid:32)n(a)·β˜(cid:33)(1+n ·β˜ )(cid:34)1− (cid:88)∞ J (cid:18)R(a)(cid:19)mP (cosθ )(cid:35)
(a),0 r2 (a) (a) m r m (a)
(a) m=2 (a)
1 (cid:34) (cid:88)∞ (cid:18)R(a)(cid:19)m(cid:0)
+ (1+n ·v˜) − J P (cosθ )
r (a) m r m (a) ,0
(a) (a)
m=2
n ·β˜ (cid:33)(cid:35)
+ mP (cosθ ) (a) (1+n ·v˜)m , (45)
m (a) r2 (a)
(a)
where ni = ri/r . A general n-body solution should include the multipolar structure
ofthesources. Nevertheless, accordingtothecurrentastrometricaccuracyandforan
oblate body, the quadrupole approximation can be considered enough [see 7]. If we
omit the higher multipole moments and restrict ourselves only to m=2, denoting by
sk the axis of the sources which is normal to the source equatorial plane, P (cosθ )
m (a)
is approximated as:
3(s ·r(σ,σ˜(cid:48)))2 1 3 1
P (cosθ )= (a) − ≈ (s ·n −s ·v˜)2(1+n ·v˜)2− ,(46)
2 (a) 2r(σ,σ˜(cid:48))2 2 2 (a) (a) (a) (a) 2
which to first order in v˜i becomes
3 1
P (cosθ )= (s ·n )2(1+2n ·v˜)−3(s ·n )(s ·v˜)− +O(v˜2); (47)
2 (a) 2 (a) (a) (a) (a) (a) (a) 2
therefore
nk v˜k (cid:40)3nk (cid:20)5(s ·n )2
h¯ =− (a)(1+2n ·v˜)+ +J R2 (a) (a) (a) +9(s ·n )2
(a),k r2 (a) r2 2 (a) r4 2 (a) (a)
(a) (a) (a)
(cid:21)
1 2
(n ·v˜)− − 3(s ·n )(s ·v˜)− (n ·v˜)
(a) 2 (a) (a) (a) 3 (a)
+ v˜k (cid:2)1−3(s ·n )2(cid:3)+ 3sk(a)[(s ·v˜)−(s ·n )(1+3n ·v˜)](cid:41)
2r4 (a) (a) r4 (a) (a) (a) (a)
(a) (a)
+O(v˜2), (48)
and
h¯ = n(a)·v˜ + 3J2R(2a) (cid:20)n(a)·v˜ − 5(n ·v˜)(s ·n )2(cid:21)+O(v˜2). (49)
(a),0 r2 r4 2 2 (a) (a) (a)
(a) (a)
By taking into account the target accuracy of Gaia [see 7; 14], the velocity
contributionsforanoblatebodyshouldbeneglected. However,forsakeofconsistency
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
willsurelydismissmanyofthem. ProbablyaGaia-likemissionthatachievesfewsub-
microarsecond in accuracy will benefit of these analytical contributions, especially in
regards of a cross-checking comparison between different approaches. In this case it
would be better to consider a metric which properly contemplate all the complexities
of a non-spherical gravitational body; that, at the moment, is out of the scope of the
present paper and deserves a dedicated work (see, e.g., [13]).
Therefore, the RAMOD reduced master equations which take into account the
quadrupole structure for the a-th single source finally become: