Random errors of star abscissae in the ROEMER space astrometry project

We discuss the propagation of random errors in the so-called great-circle reduction of the Hipparcos mission and for the proposed space astrometry project ROEMER. As a step towards the determination of stellar positions, proper motions and parallaxes, one-dimensional instantaneous relative positions of stars along fixed great circles are estimated from elementary measurements of the locations of stellar images within the instrument's field of view. The measurement errors, being dominated by photon noise, can be regarded as uncorrelated. The precision of the calculated one-dimensional positions (abscissae) depends on the precision and number of elementary measurements, the number of stars and their distribution in magnitude, and finally on the rigidity of the great-circle reduction. The rigidity quantifies how well the random measurement errors are averaged out in the least-squares solution, and is closely related to the condition number of the design matrix. We discuss the rigidity concept for idealised situations involving one, two, or several fields of view (zero, one, or more basic angles). A simple model of the error propagation is derived and used to predict the precision for a hypothetical space astrometry project such as ROEMER. It is found that the rigidity is much improved by the greater number of stars observed with ROEMER.