Author(s): Lennart Lindegren
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 $0\pm 10\mu as$ (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.
Author(s): Alexey Butkevich
This section contains the description of general procedure for the transformation of the source parameters from one the initial epoch ${T}_{0}$ 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
$$\bm{b}(T)=\bm{b}({T}_{0})+(T-{T}_{0})\bm{v}.$$ | (4.2) |
where $\bm{b}\left({T}_{0}\right)$ is the barycentric position at the initial epoch ${T}_{0}$ and $\bm{v}$ the constant space velocity. To simplify the expressions, we use subscript 0 to denote quantities at ${T}_{0}$ and the corresponding un-subscripted variables when they refer to the epoch $T$. Furthermore, the epoch difference $t=T-{T}_{0}$ is used as the time argument:
$$\bm{b}={\bm{b}}_{0}+t\bm{v}.$$ | (4.3) |
The expression for the space velocity in terms of the source parameters reads
$$\bm{v}=\frac{{A}_{\mathrm{V}}}{{\varpi}_{0}}\left({\bm{\mu}}_{0}+{\bm{r}}_{0}{\mu}_{r0}\right)=\frac{{A}_{\mathrm{V}}}{{\varpi}_{0}}\left({\bm{p}}_{0}{\mu}_{\alpha *0}+{\bm{q}}_{0}{\mu}_{\delta 0}+{\bm{r}}_{0}{\mu}_{r0}\right),$$ | (4.4) |
where ${\bm{\mu}}_{0}$ is the proper motion at the initial epoch, three unit vector constitute the normal triad $[{\bm{p}}_{0},{\bm{q}}_{0},{\bm{r}}_{0}]$ and ${A}_{\mathrm{V}}=\mathrm{4.740\hspace{0.17em}470\hspace{0.17em}446}$ equals the astronomical unit expressed in $\mathrm{km}\mathrm{yr}{\mathrm{s}}^{-1}$. This relation implies that the parallax and proper motions are expressed in compatible units, for instance, mas and mas yr${}^{-1}$, respectively.
The propagation of the barycentric direction $\bm{u}=\bm{b}/b$ is given by Equation 4.3. Squaring both sides of Equation 4.3 and making use of obvious relations ${\bm{b}}_{0}^{\prime}{\bm{v}}_{0}={b}_{0}^{2}{\mu}_{r0}$ and ${v}_{0}^{2}={b}_{0}^{2}\left({\mu}_{r0}^{2}+{\mu}_{0}^{2}\right)$, we find
$${b}^{2}={b}_{0}^{2}\left(1+2{\mu}_{r0}t+\left({\mu}_{0}^{2}+{\mu}_{r0}^{2}\right){t}^{2}\right),$$ | (4.5) |
where ${\mu}_{0}^{2}={\mu}_{\alpha *0}^{2}+{\mu}_{\delta 0}^{2}$. Introducing the distance factor
$$f={b}_{0}/b={\left[1+2{\mu}_{r0}t+\left({\mu}_{0}^{2}+{\mu}_{r0}^{2}\right){t}^{2}\right]}^{-1/2},$$ | (4.6) |
the propagation of the barycentric direction is
$$\bm{u}=\left[{\bm{r}}_{0}\left(1+{\mu}_{r0}t\right)+{\bm{\mu}}_{0}t\right]f$$ | (4.7) |
and the propagation of the parallax becomes
$$\varpi ={\varpi}_{0}f.$$ | (4.8) |
The celestial coordinates ($\alpha $, $\delta $) at epoch $T$ are obtained from $\bm{u}$ in the usual manner and the normal triad associated with the propagated direction is
$$[\bm{p},\bm{q},\bm{r}]=\left[\begin{array}{ccc}\hfill -\mathrm{sin}\alpha \hfill & \hfill -\mathrm{sin}\delta \mathrm{cos}\alpha \hfill & \hfill \mathrm{cos}\delta \mathrm{cos}\alpha \hfill \\ \hfill \mathrm{cos}\alpha \hfill & \hfill -\mathrm{sin}\delta \mathrm{sin}\alpha \hfill & \hfill \mathrm{cos}\delta \mathrm{sin}\alpha \hfill \\ \hfill 0\hfill & \hfill \mathrm{cos}\delta \hfill & \hfill \mathrm{sin}\delta \hfill \end{array}\right].$$ | (4.9) |
Direct differentiation of Equation 4.7 gives the propagated proper motion vector:
$$\bm{\mu}=\frac{\mathrm{d}\bm{u}}{\mathrm{d}t}=\left[{\bm{\mu}}_{0}\left(1+{\mu}_{r0}t\right)-{\bm{r}}_{0}{\mu}_{0}^{2}t\right]{f}^{3},$$ | (4.10) |
and the propagated radial proper motion is found to be
$${\mu}_{r}=\frac{\mathrm{d}b}{\mathrm{d}t}\frac{\varpi}{A}=\left[{\mu}_{r0}+\left({\mu}_{0}^{2}+{\mu}_{r0}^{2}\right)t\right]{f}^{2}.$$ | (4.11) |
To obtain the proper motion components (${\mu}_{\alpha *}$, ${\mu}_{\delta}$) from vector $\bm{\mu}$ it is necessary to resolve the latter along the tangential vectors $\bm{p}$ and $\bm{q}$ at the propagate direction:
$${\mu}_{\alpha *}={\bm{p}}^{\prime}\bm{\mu},{\mu}_{\delta}={\bm{q}}^{\prime}\bm{\mu}.$$ | (4.12) |
The tangential vectors are defined in terms of the propagated $\bm{u}$ or ($\alpha $, $\delta $) at the epoch $T$ according to Equation 4.9.
The above formulae describe the complete transformation of (${\alpha}_{0}$, ${\delta}_{0}$, ${\varpi}_{0}$, ${\mu}_{\alpha *0}$, ${\mu}_{\delta 0}$, ${\mu}_{r0}$) at epoch ${T}_{0}$ into ($\alpha $, $\delta $, $\varpi $, ${\mu}_{\alpha *}$, ${\mu}_{\delta}$, ${\mu}_{r}$) at the arbitrary epoch $T={T}_{0}+t$. The transformation is rigorously reversible: a second transformation from $T$ to ${T}_{0}$ recovers the original six parameters.
The uncertainties in the source parameters $\alpha $, $\delta $, $\varpi $, ${\mu}_{\alpha *}$, ${\mu}_{\delta}$, ${\mu}_{r}$ and correlations between them are quantified by means of the $6\times 6$ covariance matrix $\bm{C}$ 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 ${\bm{C}}_{0}$ of the initial parameters and the matrix $\bm{C}$ of the propagated parameters are related as
$$\bm{C}=\bm{J}{\bm{C}}_{0}{\bm{J}}^{\prime},$$ | (4.13) |
where $\bm{J}$ is the Jacobian matrix of the source parameter transformation:
$$\bm{J}=\frac{\partial (\alpha ,\delta ,\varpi ,{\mu}_{\alpha *},{\mu}_{\delta},{\mu}_{r})}{\partial ({\alpha}_{0},{\delta}_{0},{\varpi}_{0},{\mu}_{\alpha *0},{\mu}_{\delta 0},{\mu}_{r0})}$$ | (4.14) |
Thus, the propagation of the covariances requires the calculation of all 36 partial derivatives constituting the Jacobian $\bm{J}$.
The initial covariance matrix ${\bm{C}}_{0}$ must be specified in order to calculate the covariance matrix of the propagated astrometric parameters $\bm{C}$. 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 ${\bm{C}}_{0}$ 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 ${\bm{C}}_{0}$, this matrix must therefore be augmented with a sixth row and column related to the initial radial proper motion ${\mu}_{r0}$. If the initial radial velocity ${v}_{r0}$ has the standard error ${\sigma}_{{v}_{r0}}$ and is assumed to be statistically independent of the astrometric parameters in the catalogue, then the required additional elements in ${\bm{C}}_{0}$ are
${\left[{C}_{0}\right]}_{i6}={\left[{C}_{0}\right]}_{6i}={\left[{C}_{0}\right]}_{i3}\left({v}_{r0}/A\right),i=1\mathrm{\dots}5,$ | (4.15) | ||
${\left[{C}_{0}\right]}_{66}={\left[{C}_{0}\right]}_{33}\left({v}_{r0}^{2}+{\sigma}_{{v}_{r0}}^{2}\right)/{A}^{2}+{\left({\varpi}_{0}{\sigma}_{{v}_{r0}}/A\right)}^{2}$ |
(Michalik et al. 2014). If the radial velocity is not known, it is recommended that ${v}_{r0}=0$ is used, together with an appropriately large value of ${\sigma}_{{v}_{r0}}$ (set to, for example, the expected velocity dispersion of the stellar type in question), in which case ${\left[{C}_{0}\right]}_{66}$ 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 ${T}_{0}$), 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:
${J}_{11}$ | $={\displaystyle \frac{\partial \alpha *}{\partial \alpha {*}_{0}}}={\bm{p}}^{\prime}{\bm{p}}_{0}\left(1+{\mu}_{r0}t\right)f-{\bm{p}}^{\prime}{\bm{r}}_{0}{\mu}_{\alpha *0}tf$ | (4.16) | ||
${J}_{12}$ | $={\displaystyle \frac{\partial \alpha *}{\partial {\delta}_{0}}}={\bm{p}}^{\prime}{\bm{q}}_{0}\left(1+{\mu}_{r0}t\right)f-{\bm{p}}^{\prime}{\bm{r}}_{0}{\mu}_{\delta 0}tf$ | (4.17) | ||
${J}_{13}$ | $={\displaystyle \frac{\partial \alpha *}{\partial {\varpi}_{0}}}=0$ | (4.18) | ||
${J}_{14}$ | $={\displaystyle \frac{\partial \alpha *}{\partial {\mu}_{\alpha *0}}}={\bm{p}}^{\prime}{\bm{p}}_{0}tf$ | (4.19) | ||
${J}_{15}$ | $={\displaystyle \frac{\partial \alpha *}{\partial {\mu}_{\delta 0}}}={\bm{p}}^{\prime}{\bm{q}}_{0}tf$ | (4.20) | ||
${J}_{16}$ | $={\displaystyle \frac{\partial \alpha *}{\partial {\mu}_{r0}}}=-{\mu}_{\alpha *}{t}^{2}$ | (4.21) | ||
${J}_{21}$ | $={\displaystyle \frac{\partial \delta}{\partial \alpha {*}_{0}}}={\bm{q}}^{\prime}{\bm{p}}_{0}\left(1+{\mu}_{r0}t\right)f-{\bm{q}}^{\prime}{\bm{r}}_{0}{\mu}_{\alpha *0}tf$ | (4.22) | ||
${J}_{22}$ | $={\displaystyle \frac{\partial \delta}{\partial {\delta}_{0}}}={\bm{q}}^{\prime}{\bm{q}}_{0}\left(1+{\mu}_{r0}t\right)f-{\bm{q}}^{\prime}{\bm{r}}_{0}{\mu}_{\delta 0}tf$ | (4.23) | ||
${J}_{23}$ | $={\displaystyle \frac{\partial \delta}{\partial {\varpi}_{0}}}=0$ | (4.24) | ||
${J}_{24}$ | $={\displaystyle \frac{\partial \delta}{\partial {\mu}_{\alpha *0}}}={\bm{q}}^{\prime}{\bm{p}}_{0}tf$ | (4.25) | ||
${J}_{25}$ | $={\displaystyle \frac{\partial \delta}{\partial {\mu}_{\delta 0}}}={\bm{q}}^{\prime}{\bm{q}}_{0}tf$ | (4.26) | ||
${J}_{26}$ | $={\displaystyle \frac{\partial \delta}{\partial {\mu}_{r0}}}=-{\mu}_{\delta}{t}^{2}$ | (4.27) | ||
${J}_{31}$ | $={\displaystyle \frac{\partial \varpi}{\partial \alpha {*}_{0}}}=0$ | (4.28) | ||
${J}_{32}$ | $={\displaystyle \frac{\partial \varpi}{\partial {\delta}_{0}}}=0$ | (4.29) | ||
${J}_{33}$ | $={\displaystyle \frac{\partial \varpi}{\partial {\varpi}_{0}}}=f$ | (4.30) | ||
${J}_{34}$ | $={\displaystyle \frac{\partial \varpi}{\partial {\mu}_{\alpha *0}}}=-\varpi {\mu}_{\alpha *0}{t}^{2}{f}^{2}$ | (4.31) | ||
${J}_{35}$ | $={\displaystyle \frac{\partial \varpi}{\partial {\mu}_{\delta 0}}}=-\varpi {\mu}_{\delta 0}{t}^{2}{f}^{2}$ | (4.32) | ||
${J}_{36}$ | $={\displaystyle \frac{\partial \varpi}{\partial {\mu}_{r0}}}=-\varpi \left(1+{\mu}_{r0}t\right)t{f}^{2}$ | (4.33) |
${J}_{41}$ | $={\displaystyle \frac{\partial {\mu}_{\alpha *}}{\partial \alpha {*}_{0}}}=-{\bm{p}}^{\prime}{\bm{p}}_{0}{\mu}_{0}^{2}t{f}^{3}-{\bm{p}}^{\prime}{\bm{r}}_{0}{\mu}_{\alpha *0}\left(1+{\mu}_{r0}t\right){f}^{3}$ | (4.34) | ||
${J}_{42}$ | $={\displaystyle \frac{\partial {\mu}_{\alpha *}}{\partial {\delta}_{0}}}=-{\bm{p}}^{\prime}{\bm{q}}_{0}{\mu}_{0}^{2}t{f}^{3}-{\bm{p}}^{\prime}{\bm{r}}_{0}{\mu}_{\delta 0}\left(1+{\mu}_{r0}t\right){f}^{3}$ | (4.35) | ||
${J}_{43}$ | $={\displaystyle \frac{\partial {\mu}_{\alpha *}}{\partial {\varpi}_{0}}}=0$ | (4.36) | ||
${J}_{44}$ | $={\displaystyle \frac{\partial {\mu}_{\alpha *}}{\partial {\mu}_{\alpha *0}}}={\bm{p}}^{\prime}{\bm{p}}_{0}\left(1+{\mu}_{r0}t\right){f}^{3}-2{\bm{p}}^{\prime}{\bm{r}}_{0}{\mu}_{\alpha *0}t{f}^{3}-3{\mu}_{\alpha *}{\mu}_{\alpha *0}{t}^{2}{f}^{2}$ | (4.37) | ||
${J}_{45}$ | $={\displaystyle \frac{\partial {\mu}_{\alpha *}}{\partial {\mu}_{\delta 0}}}={\bm{p}}^{\prime}{\bm{q}}_{0}\left(1+{\mu}_{r0}t\right){f}^{3}-2{\bm{p}}^{\prime}{\bm{r}}_{0}{\mu}_{\delta 0}t{f}^{3}-3{\mu}_{\alpha *}{\mu}_{\delta 0}{t}^{2}{f}^{2}$ | (4.38) | ||
${J}_{46}$ | $={\displaystyle \frac{\partial {\mu}_{\alpha *}}{\partial {\mu}_{r0}}}={\bm{p}}^{\prime}\left[{\bm{\mu}}_{0}f-3\bm{\mu}\left(1+{\mu}_{r0}t\right)\right]t{f}^{2}$ | (4.39) | ||
${J}_{51}$ | $={\displaystyle \frac{\partial {\mu}_{\delta}}{\partial \alpha {*}_{0}}}=-{\bm{q}}^{\prime}{\bm{p}}_{0}{\mu}_{0}^{2}t{f}^{3}-{\bm{q}}^{\prime}{\bm{r}}_{0}{\mu}_{\alpha *0}\left(1+{\mu}_{r0}t\right){f}^{3}$ | (4.40) | ||
${J}_{52}$ | $={\displaystyle \frac{\partial {\mu}_{\delta}}{\partial {\delta}_{0}}}=-{\bm{q}}^{\prime}{\bm{q}}_{0}{\mu}_{0}^{2}t{f}^{3}-{\bm{q}}^{\prime}{\bm{r}}_{0}{\mu}_{\delta 0}\left(1+{\mu}_{r0}t\right){f}^{3}$ | (4.41) | ||
${J}_{53}$ | $={\displaystyle \frac{\partial {\mu}_{\delta}}{\partial {\varpi}_{0}}}=0$ | (4.42) | ||
${J}_{54}$ | $={\displaystyle \frac{\partial {\mu}_{\delta}}{\partial {\mu}_{\alpha *0}}}={\bm{q}}^{\prime}{\bm{p}}_{0}\left(1+{\mu}_{r0}t\right){f}^{3}-2{\bm{q}}^{\prime}{\bm{r}}_{0}{\mu}_{\alpha *0}t{f}^{3}-3{\mu}_{\delta}{\mu}_{\alpha *0}{t}^{2}{f}^{2}$ | (4.43) | ||
${J}_{55}$ | $={\displaystyle \frac{\partial {\mu}_{\delta}}{\partial {\mu}_{\delta 0}}}={\bm{q}}^{\prime}{\bm{q}}_{0}\left(1+{\mu}_{r0}t\right){f}^{3}-2{\bm{q}}^{\prime}{\bm{r}}_{0}{\mu}_{\delta 0}t{f}^{3}-3{\mu}_{\delta}{\mu}_{\delta 0}{t}^{2}{f}^{2}$ | (4.44) | ||
${J}_{56}$ | $={\displaystyle \frac{\partial {\mu}_{\delta}}{\partial {\mu}_{r0}}}={\bm{q}}^{\prime}\left[{\bm{\mu}}_{0}f-3\bm{\mu}\left(1+{\mu}_{r0}t\right)\right]t{f}^{2}$ | (4.45) | ||
${J}_{61}$ | $={\displaystyle \frac{\partial {\mu}_{r}}{\partial \alpha {*}_{0}}}=0$ | (4.46) | ||
${J}_{62}$ | $={\displaystyle \frac{\partial {\mu}_{r}}{\partial {\delta}_{0}}}=0$ | (4.47) | ||
${J}_{63}$ | $={\displaystyle \frac{\partial {\mu}_{r}}{\partial {\varpi}_{0}}}=0$ | (4.48) | ||
${J}_{64}$ | $={\displaystyle \frac{\partial {\mu}_{r}}{\partial {\mu}_{\alpha *0}}}=2{\mu}_{\alpha *0}\left(1+{\mu}_{r0}t\right)t{f}^{4}$ | (4.49) | ||
${J}_{65}$ | $={\displaystyle \frac{\partial {\mu}_{r}}{\partial {\mu}_{\delta 0}}}=2{\mu}_{\delta 0}\left(1+{\mu}_{r0}t\right)t{f}^{4}$ | (4.50) | ||
${J}_{66}$ | $={\displaystyle \frac{\partial {\mu}_{r}}{\partial {\mu}_{r0}}}=\left[{\left(1+{\mu}_{r0}t\right)}^{2}-{\mu}_{0}^{2}{t}^{2}\right]{f}^{4}$ | (4.51) |