Author(s): Lennart Lindegren

Author(s): Lennart Lindegren

The Hipparcos data used in the TGAS solution for Gaia DR1 were taken from the new reduction of the raw data by van Leeuwen (2007, 2007) as retrieved from CDS (VizieR catalogue I/311). For the interpretation of covariances the recipe in Appendix B of Michalik et al. (2014) was used. The astrometric parameters at the Hipparcos epoch J1991.25 were propagated to the Gaia DR1 epoch J2015.0 as described in Section 4.3.2.

Author(s): Lennart Lindegren

Author(s): Daniel Michalik

As described in Section 3.4.2, the reduction of astrometric data in AGIS is done using a least-squares solution, i.e., solving a linear system of normal equations $\bm{N}\bm{x}=\bm{b}$. Here, $\bm{x}$ is the vector of astrometric source parameters, $\bm{N}$ the normal equations matrix constructed from the observations, and $\bm{b}$ a vector constructed from the residuals of the problem. The covariance $\bm{C}$ of the solution $\widehat{\bm{x}}={\bm{N}}^{-1}\bm{b}$ is formally given by $\bm{C}={\bm{N}}^{-1}$.

In AGIS the observations of all well-behaved stars (‘primary sources’) are considered together in a single large least-squares problem, in order to allow the simultaneous determination of the spacecraft attitude and instrument geometry. This requires an iterative solution. However, for the discussion of the incorporation of prior information it is sufficient to consider one star at a time. We ignore attitude and calibration, since prior information is independent of them and limit the discussion to the determination of the astrometric source parameters. A brief discussion of the practical implications of priors on the AGIS algorithm is given in Michalik et al. (2014, Sect. 2.7).

On the assumption that the adopted kinematic model for the Gaia astrometric data processing (Michalik et al. 2014, Sects. 2.1 and 2.2) is valid for the historic observations of a particular star, the matrix $\bm{N}$ and vector $\bm{b}$ encapsulate the essential information on the astrometric parameters, as determined by the least-squares solution. Thus, in order to use the Tycho-2 or Hipparcos data for a given star there is no need to consider the individual observations of that star: all data is contained in the ‘information array’ $[\bm{N}\bm{b}]$. In Michalik et al. (2014, Sect. 2.6) it is shown how this array can be reconstructed from the published Hipparcos and Tycho-2 catalogues.

Let $[{\bm{N}}_{1}{\bm{b}}_{1}]$ and $[{\bm{N}}_{2}{\bm{b}}_{2}]$ be the information arrays for the same star as given by two independent astrometric catalogues. From the way the normal equations are calculated from observational data it is clear that the information arrays are additive, so that $[{\bm{N}}_{1}{\bm{b}}_{1}]+[{\bm{N}}_{2}{\bm{b}}_{2}]$ is the information array that would have resulted from processing the two data sets together. This is discussed in Michalik et al. (2012), Michalik et al. (2014, Sect. 2.4), and Michalik et al. (2015, Sect. 2.3), and is ultimately a direct result of applying Bayes’ rule to combining two sets of individual catalogue information a priori, before solving the least-squares solution. The result,

$\widehat{\bm{x}}={({\bm{N}}_{1}+{\bm{N}}_{2})}^{-1}({\bm{b}}_{1}+{\bm{b}}_{2}),$ | (4.1) |

is the joint solution of the astrometric parameters with covariance $\widehat{\bm{C}}={({\bm{N}}_{1}+{\bm{N}}_{2})}^{-1}$. The two catalogue entries for the star must use the same reference epoch and the same SMOK (Michalik et al. 2014, Appendix A) comparison point. In the case of TGAS we propagated the historic data from their reference epoch around 1991.25 to the TGAS reference epoch J2015.0 using the full covariance matrix from the Hipparcos/Tycho-2 catalogues (see Section 4.3.2). It is interesting to note that the reference epoch of the joint solution can be arbitrarily chosen. In practise the Gaia data are much better than the prior data, therefore the optimal reference epoch would always be very close to the epoch of the Gaia data alone.

The joint solution method has some advantages over a conventional a posteriori combination methods, as discussed in Michalik et al. (2014, Sect. 2.3). This is true in particular if one were to use the full information arrays from historical catalogues. For the primary data set in Gaia DR1 we however decided to use only the positional information from Hipparcos and Tycho-2, which avoids correlations in the derived mean parallaxes and proper motions. This is discussed (for the Tycho-2 stars) in Michalik et al. (2015, Sect. 2.2). We ultimately applied the exact same principle of using only a positional prior also for the Hipparcos stars in Gaia DR1 (see Section 4.3.1). This allows us to derive independent parallaxes for the Hipparcos stars in the TGAS data set, and thus gives us the possibility of a comparison with the Hipparcos parallaxes. It also allows us to derive independent proper motions in TGAS and subsequently an unbiased comparison with the Hipparcos and Tycho-2 proper motions.

Finally, we also used the joint solution method for all non-Tycho-2 and non-Hipparcos stars in order to obtain the best astrometric solution possible, and to ensure that the formal uncertainties correctly characterize the actual errors in the solution. There we incorporate a generic prior based on model assumptions, further details are given in Section 4.3.1.

Author(s): Lennart Lindegren

For the final AGIS solution of Gaia the reference frame will be established by means of quasars, both by linking to the optical counterparts of radio (VLBI) sources defining the orientation of the International Celestial Reference Frame, and by using the zero proper motion of quasars to determine a non-rotating frame. The apparent proper motion of quasars due to the Galactocentric acceleration is expected to have an amplitude of $\sim $4 $\mu $as yr${}^{-1}$ and is taken into account when determining the spin of the reference frame. This can also be done for earlier Gaia data releases, at least for the orientation part, while the shorter time span will limit the determination of the spin. It is desirable to rotate the TGAS results into the same reference frame as used for the first Gaia data release. This must be done in two steps. First, a provisional TGAS must be computed in the Tycho-2 frame (as it will be when the Tycho-2 data are used as prior, see Section 4.2.3, without imposing any other constraints on the frame. This solution will contain (many) non-Tycho-2 stars with only Gaia observations which include a multitude of quasars. Their positions and proper motions are used in a second step to correct the provisional TGAS (and other data in the same solution) for the estimated orientation and spin. Since the TGAS solution is integrated in AGIS, the estimation and correction of the frame can be accomplished using the procedures and tools developed for TGAS (Lindegren et al. (2012), Sect. 6.1).