Program available at: http://www.imcce.fr/hosted_sites/naroo/program.htmlInternational audienceGaia is due for launch in the third quarter of 2013 and will open a new page in fundamental astronomy. The astrometric accuracy achievable of about 20 muas/yr at G=15 to below 10 muas/yr at the bright end will permit to know the positions of millions stars with mas accuracy almost 100 years back in time. I will present the current expectations regarding the accuracy of Gaia astrometry at the end of the mission and explain how to propagate the stellar positions and their covariance matrix several decades away from the Gaia epoch without introducing modeling error. I will show also diagrams illustrating the positional accuracy down to 1900 as a function of magnitude and sky position

Mignard, François

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Gaia astrometric accuracy in the pastFranc¸ois MignardTo cite this version:Franc¸ois Mignard. Gaia astrometric accuracy in the past. IMCCE. International WorkshopNAROO-GAIA ”A new reduction of old observations in the Gaia era”, Paris Observatory, Jun2012, Paris, France. pp. 77-82, 2013. <hal-00758167>HAL Id: hal-00758167http://hal.upmc.fr/hal-00758167Submitted on 13 May 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.77F. MignardUniversity of Nice Sophia-AntipolisCNRS, UMR LagrangeObservatoire de la Coˆte d’AzurAbstractThe use of the Gaia astrometric catalogue in the past needs some care to propagatethe positions and derive their accuracy from the published values at the Catalogueepoch. The basic propagation model is presented and the accuracy in the past isprovided as a function of magnitude and elapsed time.1 IntroductionThe ESA space astrometry mission, due for launch in late 2013, will survey the sky downto the 20th magnitude with an unprecedented astrometric accuracy of 25 muas at 15 mag,carrying out simultaneously multi-epoch photometry and spectroscopy. The mission isoptimised to observe stellar sources to produce a stereoscopic and kinematic census ofabout one billion stars in our Galaxy enabling to probe the formation and evolution ofthe Milky Way. The expected astrometric accuracy is shown in Fig. 1 as a function ofthe G magnitude (very similar to R band for most stars). More precisely this gives theend-of-mission accuracy in astrometric parameters applicable to single stars at the meanepoch of the Catalogue, typically around 2016. More complex astrometric models will beapplied for multiple systems and the accuracy should degrade a little, depending on thecomplexity of the dynamical model. Thanks to the simultaneous observation of several10,000s quasars, the Gaia frame will be kinematically non rotating and aligned to the bestversion of the ICRF available at Gaia completion.2 Propagation of Gaia AstrometryTo use the Gaia catalogue to reprocess old observations would require that a Gaia positionat mean epoch is propagated in the past, over a timespan that could reach one century.Gaia astrometric accuracy in the past78Figure 1: Gaia expected ﬁnal astrometric accuracy at mean epoch as a function of themagnitude.Since proper motions components are available this seems to be a trivial exercice to getthe barycentric position with a linear transformation like,(α(t)− α(t0))cos δ0 = μα(t− t0) (1a)δ(t)− δ(t0) = μδ(t− t0) (1b)which amounts to identifying the time derivatives of the coordinates, or better the com-ponents of the velocity on the plane of the sky, to the ﬁrst diﬀerences.But there are modelling errors in Eqs. 1 due to the neglect of terms of higher order inμα, μδ in the right-hand-side. Given the accuracy of Gaia, it can be shown that, dependingon the star distance, they are not negligible when the propagation extends over several10s years. A more general model is applicable to single stars for which the 3D motion canbe considered rectilinear with constant velocity for several centuries. While this is truefor the space motion, this is not for the projected displacement on the plane of the skywhich is far from uniform at the Gaia accuracy.792.1 The vectorial propagation equationBased on this rectilinear motion we have the very simple propagation model,OM(t) = OM0 +V t (2)relating the initial star position vector at the initial epoch (here t0 = 0) to the position atany time t. Putting|OM0| = r (3)for the distance at the ﬁrst epoch and using respectively u0 and u for the unit vectors inthe direction of the star at t0 = 0 and t, one hasOMr= u0 +Vrt (4)and then with W = V/r, to express the velocity in angular unit,OMr= u0 +W t (5)and ﬁnally with W = |W|,u(t) =u0 +W t|u0 +W t|= (u0 +W t)(1 + 2u0 ·W t + W 2 t2)−1/2 (6)Eq. (6) is the exact form of the propagation model for a single star giving the unit directionvector at time t without approximation, beyond the fact that the 3-D motion is uniform ona straight line. One can see that it depends on the three components of the velocity scaledby the distance, that is to say on the three components of the angular proper motion,including that in the radial direction. Numerically all the computations can be done inrectangular coordinates and transformed at the end in spherical coordinates, namely theright-ascension and declination at epoch. One must notice that the radial velocity vr isrequired above the ﬁrst order approximation.2.2 The coordinate propagation equationsAlthough Eq. 6 solves the practical numerical problem, it is useful to express the resultdirectly in spherical coordinates to extend the validity of Eq. 1 beyond the linear approx-imation. There are several methods to derive the more exact expressions, for example byexpanding Eq. 6 in power of t and projecting on the plane of the sky. I propose below analternative and more direct method based on the covariant derivative of the velocity.80Consider again the basic kinematical model with a uniform rectilinear motion in the Eu-clidean space. It is characterized by the fact that the velocity vector is parallel transportedalong the geodesic of the underlying space, i.e. along a straight line. This means that the3-D acceleration d2r/dt2 = 0 and more generally in any moving coordinate system thatthe covariant derivative is identically zero,D2r/Dt2 = 0 (7)Expressing this fact in curvilinear (here spherical) coordinates yi one hasγi =d2yidt2+ Γijkdyjdtdykdt(8)where the Christoﬀel symbols Γijk are computed from the metric of R3 in spherical coor-dinates,ds2 = dr2 + r2 cos2 δ dα2 + r2 dδ2 (9)using y1 = r, y2 = α, y3 = δ, which gives in the local frame (normalised with unit vectors)er, eα, eδ the non-zero Christoﬀel symbols,Γ122 = −r cos2 δ Γ133 = −r Γ212 = 1/r (10a)Γ223 = − tan δ Γ313 = 1/r Γ322 = sin δ cos δ (10b)Then with γi = 0 for a uniform rectilinear motion one has,r¨ = r cos2 δ α˙2 + r δ˙2 (11a)α¨ = −2rr˙ α˙ + 2 tan δ α˙δ˙ (11b)δ¨ = −2rr˙δ˙ − sin δ cos δ α˙2 (11c)Eqs. 11b-11c are the general expressions for the second derivatives of the right ascensionand declination of a star moving on a straight line at constant speed in space. There aresupplementary terms of the same order of magnitude which are diﬀerent from zero evenin the case of a purely tangential motion.To write down the model to third order one needs also the third time derivatives of α, δ,which after a some algebra reduce to,d3yidt3=[−∂Γijk∂yl+ 2Γimj Γmkl]y˙j y˙k y˙l (12)and with the substitution of (10) one gets,d3αdt3= 6r˙2r2α˙− 12 tan δ r˙rα˙δ˙ + 6 tan2 δ α˙δ˙2 − 2α˙3 (13a)d3δdt3= 6r˙2r2δ˙ + 3 sin 2δr˙rα˙2 − 3 α˙2 δ˙ − 2δ˙3 (13b)81Now one can write the propagation model as,α(t) = α0 + α˙ t + α¨t22+...αt36(14a)δ(t) = δ0 + δ˙ t + δ¨t22+...δt36(14b)or with Equations (11, 13, 14) and denoting μα = α˙ cos δ0, μδ = δ˙, μr = vr/r, where r isthe distance of the star at initial epoch,Δα cos δ0 = μα t− [μr μα − tan δ0 μα μδ] t2+[μ2r μα − 2 tan δ0 μr μα μδ + tan2 δ μα μ2δ −μ3α3 cos2 δ0]t3(15)Δδ = μδ t−[μr μδ +tan δ02μ2α]t2+[μ2r μδ + tan δ0 μr μ2α −μ2α μδ2 cos2 δ0− μ3δ3]t3(16)Equations (15)-(16) give the propagation model to third order of the proper motions. Onesees the presence of the radial velocity in second and third order terms.Solving now this system for μα and μδ, or equivalently inverting the series to the sameorder gives the proper motion components from the positions at two epochs. The processleads to the ﬁnal transformation,μα t = a(1 + μr t)− tan δ0 a d + 3 cos2 δ0 − 16 cos2 δ0a3 − tan δ0 a dμrt (17a)μδ t = d (1 + μr t) +12tan δ0 a2 +2 cos2 δ0 − 12 cos2 δ0a2 d +12tan δ0 a2 μrt +d33(17b)where a = Δα cos δ0, and d = Δδ.2.3 The propagation of the covariance matrixFor each star, the Gaia catalogue will provide simultaneously the 5 astrometric param-eters (position, proper motion components and parallax) together with their covariancematrix in equatorial coordinates. The diagonal terms will give the standard deviation atepoch, that is to say an estimate of the uncertainty. This is the most compact statisticalsynthesis of the noisy observations based on the single star astrometric model. As we havepropagated the position in the past, one must do the same for the estimated uncertainty.Basically if we start from the linear propagation model given in Eq. 1, one has essentiallysomething close to,σ2α(t) ≈ σ2α0 + σ2μα(t− t0)2 (18a)σ2δ (t) ≈ σ2δ0 + σ2μδ(t− t0)2 (18b)82epochmasGaia Catalogue: Positional accuracy1890 1910 1930 1950 1970 1990 2010 2030 2050 2070 20900.0050.010.020.050.10.20.5125102050G = 20G = 18G = 16G = 14G = 121 mas10 mas100 μas2016Figure 2: Evolution of the accuracy of the Gaia positional stellar catalogue with time dueto the uncertainty in the annual proper motions of stars.where correlations at t0 have been neglected and a simple linear propagation model hasbeen used. One should however note that α, μα and δ, μδ become quickly highly corre-lated and in applications using both positions and proper motions, extreme care must beexercised in the statistical inference. The propagation of the covariance matrix is muchmore complex and involves the Jacobian matrix of the general transformation between theastrometric parameters at epoch t0 and t. More precisely the covariance matrices at twoepochs are related by,C(t) = JC(t0)JT (19)For the position the transformation is the generalisation of Eq. 1 given by Eqs. 15-16 andby their derivatives for the proper motion components, while Eq. 19 generalises Eqs. 18.Without entering into the details, this has been implemented in a computer program andallowed expressing the accuracy of the Gaia propagated astrometry in the past (and future)as a function the magnitude, using the expected accuracy at the Catalogue Epoch. Theresults are shown in Fig. 2 for the sky-averaged accuracy between 1890-2090. One mustnote the ∼ mas accuracy for the faint stars of the Carte du Ciel around the beginning ofthe program.

Gaia astrometric accuracy in the past

HAL-UNICE

Gaia astrometric accuracy in the past
Franc¸ois Mignard
To cite this version:
Franc¸ois Mignard. Gaia astrometric accuracy in the past. IMCCE. International Workshop
NAROO-GAIA ”A new reduction of old observations in the Gaia era”, Paris Observatory, Jun
2012, Paris, France. pp. 77-82, 2013. <hal-00758167>
HAL Id: hal-00758167
http://hal.upmc.fr/hal-00758167
Submitted on 13 May 2013
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destine´e au de´poˆt et a` la diffusion de documents
scientifiques de niveau recherche, publie´s ou non,
e´manant des e´tablissements d’enseignement et de
recherche franc¸ais ou e´trangers, des laboratoires
publics ou prive´s.
77
F. Mignard
University of Nice Sophia-Antipolis
CNRS, UMR Lagrange
Observatoire de la Coˆte d’Azur
Abstract
The use of the Gaia astrometric catalogue in the past needs some care to propagate
the positions and derive their accuracy from the published values at the Catalogue
epoch. The basic propagation model is presented and the accuracy in the past is
provided as a function of magnitude and elapsed time.
1 Introduction
The ESA space astrometry mission, due for launch in late 2013, will survey the sky down
to the 20th magnitude with an unprecedented astrometric accuracy of 25 muas at 15 mag,
carrying out simultaneously multi-epoch photometry and spectroscopy. The mission is
optimised to observe stellar sources to produce a stereoscopic and kinematic census of
about one billion stars in our Galaxy enabling to probe the formation and evolution of
the Milky Way. The expected astrometric accuracy is shown in Fig. 1 as a function of
the G magnitude (very similar to R band for most stars). More precisely this gives the
end-of-mission accuracy in astrometric parameters applicable to single stars at the mean
epoch of the Catalogue, typically around 2016. More complex astrometric models will be
applied for multiple systems and the accuracy should degrade a little, depending on the
complexity of the dynamical model. Thanks to the simultaneous observation of several
10,000s quasars, the Gaia frame will be kinematically non rotating and aligned to the best
version of the ICRF available at Gaia completion.
2 Propagation of Gaia Astrometry
To use the Gaia catalogue to reprocess old observations would require that a Gaia position
at mean epoch is propagated in the past, over a timespan that could reach one century.
Gaia astrometric accuracy in the past
78
Figure 1: Gaia expected ﬁnal astrometric accuracy at mean epoch as a function of the
magnitude.
Since proper motions components are available this seems to be a trivial exercice to get
the barycentric position with a linear transformation like,
(
α(t)− α(t0)
)
cos δ0 = μα(t− t0) (1a)
δ(t)− δ(t0) = μδ(t− t0) (1b)
which amounts to identifying the time derivatives of the coordinates, or better the com-
ponents of the velocity on the plane of the sky, to the ﬁrst diﬀerences.
But there are modelling errors in Eqs. 1 due to the neglect of terms of higher order in
μα, μδ in the right-hand-side. Given the accuracy of Gaia, it can be shown that, depending
on the star distance, they are not negligible when the propagation extends over several
10s years. A more general model is applicable to single stars for which the 3D motion can
be considered rectilinear with constant velocity for several centuries. While this is true
for the space motion, this is not for the projected displacement on the plane of the sky
which is far from uniform at the Gaia accuracy.
79
2.1 The vectorial propagation equation
Based on this rectilinear motion we have the very simple propagation model,
OM(t) = OM0 +V t (2)
relating the initial star position vector at the initial epoch (here t0 = 0) to the position at
any time t. Putting
|OM0| = r (3)
for the distance at the ﬁrst epoch and using respectively u0 and u for the unit vectors in
the direction of the star at t0 = 0 and t, one has
OM
r
= u0 +
V
r
t (4)
and then with W = V/r, to express the velocity in angular unit,
OM
r
= u0 +W t (5)
and ﬁnally with W = |W|,
u(t) =
u0 +W t
|u0 +W t|
= (u0 +W t)
(
1 + 2u0 ·W t + W 2 t2
)−1/2 (6)
Eq. (6) is the exact form of the propagation model for a single star giving the unit direction
vector at time t without approximation, beyond the fact that the 3-D motion is uniform on
a straight line. One can see that it depends on the three components of the velocity scaled
by the distance, that is to say on the three components of the angular proper motion,
including that in the radial direction. Numerically all the computations can be done in
rectangular coordinates and transformed at the end in spherical coordinates, namely the
right-ascension and declination at epoch. One must notice that the radial velocity vr is
required above the ﬁrst order approximation.
2.2 The coordinate propagation equations
Although Eq. 6 solves the practical numerical problem, it is useful to express the result
directly in spherical coordinates to extend the validity of Eq. 1 beyond the linear approx-
imation. There are several methods to derive the more exact expressions, for example by
expanding Eq. 6 in power of t and projecting on the plane of the sky. I propose below an
alternative and more direct method based on the covariant derivative of the velocity.
80
Consider again the basic kinematical model with a uniform rectilinear motion in the Eu-
clidean space. It is characterized by the fact that the velocity vector is parallel transported
along the geodesic of the underlying space, i.e. along a straight line. This means that the
3-D acceleration d2r/dt2 = 0 and more generally in any moving coordinate system that
the covariant derivative is identically zero,
D2r/Dt2 = 0 (7)
Expressing this fact in curvilinear (here spherical) coordinates yi one has
γi =
d2yi
dt2
+ Γijk
dyj
dt
dyk
dt
(8)
where the Christoﬀel symbols Γijk are computed from the metric of R3 in spherical coor-
dinates,
ds2 = dr2 + r2 cos2 δ dα2 + r2 dδ2 (9)
using y1 = r, y2 = α, y3 = δ, which gives in the local frame (normalised with unit vectors)
er, eα, eδ the non-zero Christoﬀel symbols,
Γ122 = −r cos2 δ Γ133 = −r Γ212 = 1/r (10a)
Γ223 = − tan δ Γ313 = 1/r Γ322 = sin δ cos δ (10b)
Then with γi = 0 for a uniform rectilinear motion one has,
r¨ = r cos2 δ α˙2 + r δ˙2 (11a)
α¨ = −2
r
r˙ α˙ + 2 tan δ α˙δ˙ (11b)
δ¨ = −2
r
r˙δ˙ − sin δ cos δ α˙2 (11c)
Eqs. 11b-11c are the general expressions for the second derivatives of the right ascension
and declination of a star moving on a straight line at constant speed in space. There are
supplementary terms of the same order of magnitude which are diﬀerent from zero even
in the case of a purely tangential motion.
To write down the model to third order one needs also the third time derivatives of α, δ,
which after a some algebra reduce to,
d3yi
dt3
=
[
−∂Γ
i
jk
∂yl
+ 2Γimj Γ
m
kl
]
y˙j y˙k y˙l (12)
and with the substitution of (10) one gets,
d3α
dt3
= 6
r˙2
r2
α˙− 12 tan δ r˙
r
α˙δ˙ + 6 tan2 δ α˙δ˙2 − 2α˙3 (13a)
d3δ
dt3
= 6
r˙2
r2
δ˙ + 3 sin 2δ
r˙
r
α˙2 − 3 α˙2 δ˙ − 2δ˙3 (13b)
81
Now one can write the propagation model as,
α(t) = α0 + α˙ t + α¨
t2
2
+
...
α
t3
6
(14a)
δ(t) = δ0 + δ˙ t + δ¨
t2
2
+
...
δ
t3
6
(14b)
or with Equations (11, 13, 14) and denoting μα = α˙ cos δ0, μδ = δ˙, μr = vr/r, where r is
the distance of the star at initial epoch,
Δα cos δ0 = μα t− [μr μα − tan δ0 μα μδ] t2
+
[
μ2r μα − 2 tan δ0 μr μα μδ + tan2 δ μα μ2δ −
μ3α
3 cos2 δ0
]
t3
(15)
Δδ = μδ t−
[
μr μδ +
tan δ0
2
μ2α
]
t2
+
[
μ2r μδ + tan δ0 μr μ
2
α −
μ2α μδ
2 cos2 δ0
− μ
3
δ
3
]
t3
(16)
Equations (15)-(16) give the propagation model to third order of the proper motions. One
sees the presence of the radial velocity in second and third order terms.
Solving now this system for μα and μδ, or equivalently inverting the series to the same
order gives the proper motion components from the positions at two epochs. The process
leads to the ﬁnal transformation,
μα t = a(1 + μr t)− tan δ0 a d + 3 cos
2 δ0 − 1
6 cos2 δ0
a3 − tan δ0 a dμrt (17a)
μδ t = d (1 + μr t) +
1
2
tan δ0 a2 +
2 cos2 δ0 − 1
2 cos2 δ0
a2 d +
1
2
tan δ0 a2 μrt +
d3
3
(17b)
where a = Δα cos δ0, and d = Δδ.
2.3 The propagation of the covariance matrix
For each star, the Gaia catalogue will provide simultaneously the 5 astrometric param-
eters (position, proper motion components and parallax) together with their covariance
matrix in equatorial coordinates. The diagonal terms will give the standard deviation at
epoch, that is to say an estimate of the uncertainty. This is the most compact statistical
synthesis of the noisy observations based on the single star astrometric model. As we have
propagated the position in the past, one must do the same for the estimated uncertainty.
Basically if we start from the linear propagation model given in Eq. 1, one has essentially
something close to,
σ2α(t) ≈ σ2α0 + σ2μα(t− t0)2 (18a)
σ2δ (t) ≈ σ2δ0 + σ2μδ(t− t0)2 (18b)
82
epoch
m
as
Gaia Catalogue: Positional accuracy
1890 1910 1930 1950 1970 1990 2010 2030 2050 2070 2090
0.005
0.01
0.02
0.05
0.1
0.2
0.5
1
2
5
10
20
50
G = 20
G = 18
G = 16
G = 14
G = 12
1 mas
10 mas
100 μas
2016
Figure 2: Evolution of the accuracy of the Gaia positional stellar catalogue with time due
to the uncertainty in the annual proper motions of stars.
where correlations at t0 have been neglected and a simple linear propagation model has
been used. One should however note that α, μα and δ, μδ become quickly highly corre-
lated and in applications using both positions and proper motions, extreme care must be
exercised in the statistical inference. The propagation of the covariance matrix is much
more complex and involves the Jacobian matrix of the general transformation between the
astrometric parameters at epoch t0 and t. More precisely the covariance matrices at two
epochs are related by,
C(t) = JC(t0)JT (19)
For the position the transformation is the generalisation of Eq. 1 given by Eqs. 15-16 and
by their derivatives for the proper motion components, while Eq. 19 generalises Eqs. 18.
Without entering into the details, this has been implemented in a computer program and
allowed expressing the accuracy of the Gaia propagated astrometry in the past (and future)
as a function the magnitude, using the expected accuracy at the Catalogue Epoch. The
results are shown in Fig. 2 for the sky-averaged accuracy between 1890-2090. One must
note the ∼ mas accuracy for the faint stars of the Carte du Ciel around the beginning of
the program.


Gaia astrometric accuracy in the past

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