Gaia's very accurate astrometric measurements will allow the International Celestial Reference Frame (ICRF) to be improved by a few orders of magnitude in the optical. Several sets of quasars are used to define a kinematically stable non-rotating reference frame with the barycentre of the Solar System as its origin. Gaia will also observe a large number of galaxies which could obtain accurate positions and proper motions although they are not point-like. The optical stability of the quasars is critical and we investigate how accurately the reference frame can be recovered. Various proper motion patterns are also present in the data, the best known is caused by the acceleration of the Solar System Barycentre, presumably, towards the Galactic centre. We review some other less-well-known effects that are not part of standard astrometric models. We model quasars and galaxies using realistic sky distributions, magnitudes and redshifts. Position variability is introduced using a Markov chain model. The reference frame is determined using the algorithm developed for the Gaia mission which also determines the acceleration of the Solar System. We also test a method to measure the velocity of the Solar System barycentre in a cosmological frame. We simulate the recovery of the reference frame and the acceleration of the Solar System and conclude that they are not significantly disturbed in the presence of quasar variability which is statistically averaged. However, the effect of a non-uniform sky distribution of the quasars can result in a correlation between the reference frame and acceleration which degrades the solution. Our results suggest that an attempt should be made to astrometrically determine the redshift dependent apparent drift of galaxies due to our velocity relative to the CMB, which in principle could allow the determination of the Hubble parameter.
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