4.3 Processing steps

Author(s): Lennart Lindegren

4.3.1 The use of prior information in AGIS

Author(s): Daniel Michalik

Gaia DR1 uses four different priors, which are incorporated in the astrometric solution as described in Section 4.2.3.

  • Tycho-2 positions: this prior applies to all Tycho-2 (non-Hipparcos) stars seen by Gaia. We use their Tycho-2 position as a prior, together with the individually assigned uncertainties and correlation coefficients. We take this prior at an epoch of observation computed individually for each Tycho-2 entry, i.e., the average of the observation epoch in RA and DEC. Proper motions from Tycho-2 are only used for the epoch propagation, but not as prior information.

  • Hipparcos positions: here we use the J1991.25 positions from the Hipparcos catalogue (van Leeuwen 2007) as prior information, together with uncertainties and correlation coefficients. We use parallax and proper motion information for the epoch propagation, but not as prior in the solution.

  • Quasar proper motions: The third prior type applies to all quasars in the Gaia DR1 primary data set. This prior leaves the position and parallax unrestricted to allow their independent determination, but limits the proper motions to 0±10μas (Michalik and Lindegren 2016).

  • For all stars in the secondary update a generic prior on parallax and proper motion is applied. This prior ensures good astrometric properties of the solution and suitable individual formal uncertainties. It is discussed in detail in Michalik et al. (2015). As described in the reference, the attitude and calibration uncertainties in short data sets at the beginning of the mission require the use of a relaxation factor of the prior to ensure good properties of the results. For Gaia DR1 we multiplied the prior uncertainties by 10, as verified through the simulations in the quoted paper. This prior is always centred on zero and thus will not skew the solution when being applied to extra-Galactic objects.

The type of prior used is also given in the form of an enumeration value for each individual source:

  • 0: No prior used (n/a to Gaia DR1);

  • 1: Galaxy Bayesian Prior for parallax and proper motion (n/a to Gaia DR1);

  • 2: Galaxy Bayesian Prior for parallax and proper motion relaxed by a factor of 10;

  • 3: Hipparcos prior for position;

  • 4: Hipparcos prior for position and proper motion (n/a to Gaia DR1);

  • 5: Tycho-2 prior for position;

  • 6: Quasar prior for proper motion.

4.3.2 Epoch propagation of prior information

Author(s): Alexey Butkevich

This section contains the description of general procedure for the transformation of the source parameters from one the initial epoch T0 to the arbitrary epoch T. The rigorous treatment of the epoch propagation including the effects of light-travel time was developed by Butkevich and Lindegren (2014). However, for the propagation of the prior information to the Gaia reference epoch, it is sufficient to use the simplified treatment, which was employed in the reduction procedures used to construct the Hipparcos and Tycho catalogues, since the light-time effects are negligible at milli-arcsecond accuracy (ESA 1997, Vol. 1, Sect. 1.5.5).

The epoch propagation is based on the standard astrometric model assuming the uniform rectilinear motion with respect to the solar-system barycentre. In the framework of this model, the barycentric position of a source at the epoch T is

𝒃(T)=𝒃(T0)+(T-T0)𝒗. (4.2)

where 𝒃(T0) is the barycentric position at the initial epoch T0 and 𝒗 the constant space velocity. To simplify the expressions, we use subscript 0 to denote quantities at T0 and the corresponding un-subscripted variables when they refer to the epoch T. Furthermore, the epoch difference t=T-T0 is used as the time argument:

𝒃=𝒃0+t𝒗. (4.3)

The expression for the space velocity in terms of the source parameters reads

𝒗=AVϖ0(𝝁0+𝒓0μr0)=AVϖ0(𝒑0μα*0+𝒒0μδ0+𝒓0μr0), (4.4)

where 𝝁0 is the proper motion at the initial epoch, three unit vector constitute the normal triad [𝒑0,𝒒0,𝒓0] and AV=4.740 470 446 equals the astronomical unit expressed in kmyrs-1. This relation implies that the parallax and proper motions are expressed in compatible units, for instance, mas and mas yr-1, respectively.

Propagation of the source parameters

The propagation of the barycentric direction 𝒖=𝒃/b is given by Equation 4.3. Squaring both sides of Equation 4.3 and making use of obvious relations 𝒃0𝒗0=b02μr0 and v02=b02(μr02+μ02), we find

b2=b02(1+2μr0t+(μ02+μr02)t2), (4.5)

where μ02=μα*02+μδ02. Introducing the distance factor

f=b0/b=[1+2μr0t+(μ02+μr02)t2]-1/2, (4.6)

the propagation of the barycentric direction is

𝒖=[𝒓0(1+μr0t)+𝝁0t]f (4.7)

and the propagation of the parallax becomes

ϖ=ϖ0f. (4.8)

The celestial coordinates (α, δ) at epoch T are obtained from 𝒖 in the usual manner and the normal triad associated with the propagated direction is

[𝒑,𝒒,𝒓]=[-sinα-sinδcosαcosδcosαcosα-sinδsinαcosδsinα0cosδsinδ]. (4.9)

Direct differentiation of Equation 4.7 gives the propagated proper motion vector:

𝝁=d𝒖dt=[𝝁0(1+μr0t)-𝒓0μ02t]f3, (4.10)

and the propagated radial proper motion is found to be

μr=dbdtϖA=[μr0+(μ02+μr02)t]f2. (4.11)

To obtain the proper motion components (μα*, μδ) from vector 𝝁 it is necessary to resolve the latter along the tangential vectors 𝒑 and 𝒒 at the propagate direction:

μα*=𝒑𝝁,μδ=𝒒𝝁. (4.12)

The tangential vectors are defined in terms of the propagated 𝒖 or (α, δ) at the epoch T according to Equation 4.9.

The above formulae describe the complete transformation of (α0, δ0, ϖ0, μα*0, μδ0, μr0) at epoch T0 into (α, δ, ϖ, μα*, μδ, μr) at the arbitrary epoch T=T0+t. The transformation is rigorously reversible: a second transformation from T to T0 recovers the original six parameters.

Propagation of errors (covariances)

The uncertainties in the source parameters α, δ, ϖ, μα*, μδ, μr and correlations between them are quantified by means of the 6×6 covariance matrix 𝑪 in which the rows and columns correspond to the parameters taken in the order given above. The general principle of (linearised) error propagation is well known and briefly summarized below. The covariance matrix 𝑪0 of the initial parameters and the matrix 𝑪 of the propagated parameters are related as

𝑪=𝑱𝑪0𝑱, (4.13)

where 𝑱 is the Jacobian matrix of the source parameter transformation:

𝑱=(α,δ,ϖ,μα*,μδ,μr)(α0,δ0,ϖ0,μα*0,μδ0,μr0) (4.14)

Thus, the propagation of the covariances requires the calculation of all 36 partial derivatives constituting the Jacobian 𝑱.

Initialization of 𝑪0:

The initial covariance matrix 𝑪0 must be specified in order to calculate the covariance matrix of the propagated astrometric parameters 𝑪. Available astrometric catalogues seldom give the correlations between the parameters, nor do they usually contain radial velocities. Absence of the correlations does not create any problems for the error propagation since all the off-diagonal elements of 𝑪0 are just set to zero, but the radial velocity is crucial for the rigorous propagation. While the Hipparcos and Tycho catalogues provide the complete first five rows and columns of 𝑪0, this matrix must therefore be augmented with a sixth row and column related to the initial radial proper motion μr0. If the initial radial velocity vr0 has the standard error σvr0 and is assumed to be statistically independent of the astrometric parameters in the catalogue, then the required additional elements in 𝑪0 are

[C0]i6=[C0]6i=[C0]i3(vr0/A),i=15, (4.15)

(Michalik et al. 2014). If the radial velocity is not known, it is recommended that vr0=0 is used, together with an appropriately large value of σvr0 (set to, for example, the expected velocity dispersion of the stellar type in question), in which case [C0]66 in general is still positive. This means that the unknown perspective acceleration is accounted for in the uncertainty of the propagated astrometric parameters. It should be noted that strict reversal of the transformation (from T to T0), according to the standard model of stellar motion, is only possible if the full six-dimensional parameter vector and covariance is considered.

The elements of the Jacobian matrix are given hereafter:

J11 =α*α*0=𝒑𝒑0(1+μr0t)f-𝒑𝒓0μα*0tf (4.16)
J12 =α*δ0=𝒑𝒒0(1+μr0t)f-𝒑𝒓0μδ0tf (4.17)
J13 =α*ϖ0=0 (4.18)
J14 =α*μα*0=𝒑𝒑0tf (4.19)
J15 =α*μδ0=𝒑𝒒0tf (4.20)
J16 =α*μr0=-μα*t2 (4.21)
J21 =δα*0=𝒒𝒑0(1+μr0t)f-𝒒𝒓0μα*0tf (4.22)
J22 =δδ0=𝒒𝒒0(1+μr0t)f-𝒒𝒓0μδ0tf (4.23)
J23 =δϖ0=0 (4.24)
J24 =δμα*0=𝒒𝒑0tf (4.25)
J25 =δμδ0=𝒒𝒒0tf (4.26)
J26 =δμr0=-μδt2 (4.27)
J31 =ϖα*0=0 (4.28)
J32 =ϖδ0=0 (4.29)
J33 =ϖϖ0=f (4.30)
J34 =ϖμα*0=-ϖμα*0t2f2 (4.31)
J35 =ϖμδ0=-ϖμδ0t2f2 (4.32)
J36 =ϖμr0=-ϖ(1+μr0t)tf2 (4.33)
J41 =μα*α*0=-𝒑𝒑0μ02tf3-𝒑𝒓0μα*0(1+μr0t)f3 (4.34)
J42 =μα*δ0=-𝒑𝒒0μ02tf3-𝒑𝒓0μδ0(1+μr0t)f3 (4.35)
J43 =μα*ϖ0=0 (4.36)
J44 =μα*μα*0=𝒑𝒑0(1+μr0t)f3-2𝒑𝒓0μα*0tf3-3μα*μα*0t2f2 (4.37)
J45 =μα*μδ0=𝒑𝒒0(1+μr0t)f3-2𝒑𝒓0μδ0tf3-3μα*μδ0t2f2 (4.38)
J46 =μα*μr0=𝒑[𝝁0f-3𝝁(1+μr0t)]tf2 (4.39)
J51 =μδα*0=-𝒒𝒑0μ02tf3-𝒒𝒓0μα*0(1+μr0t)f3 (4.40)
J52 =μδδ0=-𝒒𝒒0μ02tf3-𝒒𝒓0μδ0(1+μr0t)f3 (4.41)
J53 =μδϖ0=0 (4.42)
J54 =μδμα*0=𝒒𝒑0(1+μr0t)f3-2𝒒𝒓0μα*0tf3-3μδμα*0t2f2 (4.43)
J55 =μδμδ0=𝒒𝒒0(1+μr0t)f3-2𝒒𝒓0μδ0tf3-3μδμδ0t2f2 (4.44)
J56 =μδμr0=𝒒[𝝁0f-3𝝁(1+μr0t)]tf2 (4.45)
J61 =μrα*0=0 (4.46)
J62 =μrδ0=0 (4.47)
J63 =μrϖ0=0 (4.48)
J64 =μrμα*0=2μα*0(1+μr0t)tf4 (4.49)
J65 =μrμδ0=2μδ0(1+μr0t)tf4 (4.50)
J66 =μrμr0=[(1+μr0t)2-μ02t2]f4 (4.51)