  Determination of absolute parallaxes by means of a scanning astrometric
satellite such as Hipparcos or Gaia relies on the short-term stability of the
so-called basic angle between the two viewing directions. Uncalibrated
variations of the basic angle may produce systematic errors in the computed
parallaxes. We examine the coupling between a global parallax shift and
specific variations of the basic angle, namely those related to the satellite
attitude with respect to the Sun. The changes in observables produced by small
perturbations of the basic angle, attitude, and parallaxes are calculated
analytically. We then look for a combination of perturbations that has no net
effect on the observables. In the approximation of infinitely small fields of
view, it is shown that certain perturbations of the basic angle are
observationally indistinguishable from a global shift of the parallaxes. If
such perturbations exist, they cannot be calibrated from the astrometric
observations but will produce a global parallax bias. Numerical simulations of
the astrometric solution, using both direct and iterative methods, confirm this
theoretical result. For a given amplitude of the basic angle perturbation, the
parallax bias is smaller for a larger basic angle and a larger solar aspect
angle. In both these respects Gaia has a more favourable geometry than
Hipparcos. In the case of Gaia, internal metrology is used to monitor basic
angle variations. Additionally, Gaia has the advantage of detecting numerous
quasars, which can be used to verify the parallax zero point.
