The methods here are not specific to the Gaia mission,
but the parameters of the functions and their units are specified
in a form that is convenient for use with Gaia data,
in particular the gaia_source
catalogue available from
http://gea.esac.esa.int/archive/
and copies or mirrors.
There are currently two main sets of functions here, distance estimation from parallaxes, and astrometry propagation to different epochs.
Distance estimation
Gaia measures parallaxes, but some scientific use cases require the radial distance instead. While distance in parsec is in principle the reciprocal of parallax in arcsec, in the presence of non-negligable errors on measured parallax, this inversion does not give a good estimate of distance. A thorough discussion of this topic and approaches to estimating distances for Gaia-like data can be found in the papers
The functions provided here correspond to calculations from Astraatmadja & Bailer-Jones, "Estimating Distances from Parallaxes. III. Distances of Two Million Stars in the Gaia DR1 Catalogue", ApJ 833, a119 (2016) 2016ApJ...833..119A based on the Exponentially Decreasing Space Density prior defined therein. This implementation was written with reference to the Java implementation by Enrique Utrilla (DPAC).
These functions are parameterised by a length scale L
that defines the exponential decay (the mode of the prior PDF is at
r=2L).
Some value for this length scale, specified in parsec, must be supplied
to the functions as the lpc
parameter.
Epoch Propagation
The Gaia source catalogue provides, for at least some sources, the six-parameter astrometric solution (Right Ascension, Declination, Parallax, Proper motion in RA and Dec, and Radial Velocity), along with errors on these values and correlations between these errors. While a crude estimate of the position at an earlier or later epoch than that of the measurement can be made by multiplying the proper motion components by epoch difference and adding to the measured position, a more careful treatment is required for accurate propagation between epochs of the astrometric parameters, and if required their errors and correlations. The expressions for this are set out in section 1.5.5 (Volume 1) of The Hipparcos and Tycho Catalogues, ESA SP-1200 (1997) (but see below), and the code is based on an implementation by Alexey Butkevich and Daniel Michalik (DPAC). A correction is applied to the SP-1200 treatment of radial velocity uncertainty following Michalik et al. 2014 2014A&A...571A..85M because of their better handling of small radial velocities or parallaxes.
The calculations give the same results, though not exactly in the same form, as the epoch propagation functions available in the Gaia archive service.