3.1.7 Transformations of astrometric data and error propagation

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

where $𝒃\left(T_0\right)$ is the barycentric position at the initial epoch $T_0$ and $𝒗$ 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:

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

where ${𝝁}_0$ is the proper motion at the initial epoch, three unit vector constitute the normal triad $[{𝒑}_0,{𝒒}_0,{𝒓}_0]$ 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 $𝒖=𝒃/b$ is given by Equation 3.8. Squaring both sides of Equation 3.8 and making use of obvious relations ${𝒃}_0^{\prime }{𝒗}_0=b_0^2{\mu }_{r0}$ and $v_0^2=b_0^2\left({\mu }_{r0}^2+{\mu }_0^2\right)$, we find

where ${\mu }_0^2={\mu }_{\alpha *0}^2+{\mu }_{\delta 0}^2$. Introducing the distance factor

the propagation of the barycentric direction is

and the propagation of the parallax becomes

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

Direct differentiation of Equation 3.12 gives the propagated proper motion vector:

and the propagated radial proper motion is found to be

To obtain the proper motion components (${\mu }_{\alpha *}$, ${\mu }_{\delta }$) from vector $𝝁$ it is necessary to resolve the latter along the tangential vectors $𝒑$ and $𝒒$ at the propagate direction:

The tangential vectors are defined in terms of the propagated $𝒖$ or ($\alpha$, $\delta$) at the epoch $T$ according to Equation 3.14.

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 $𝑪$ 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 ${𝑪}_0$ of the initial parameters and the matrix $𝑪$ of the propagated parameters are related as

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

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

The initial covariance matrix ${𝑪}_0$ 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 ${𝑪}_0$ 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 ${𝑪}_0$, 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 ${𝑪}_0$ are

(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: