Rigorous treatment of barycentric stellar motion: Perspective and light-time effects in astrometric and radial velocity data

High-precision astrometric and radial-velocity observations require accurate modelling of stellar motions in order to extrapolate measurements over long time intervals, and to detect deviations from uniform motion caused for example by unseen companions. We aim to explore the simplest possible kinematic model of stellar motions, namely that of uniform rectilinear motion relative to the Solar System Barycentre, in terms of observable quantities including error propagation. The apparent path equation for uniform rectilinear motion is solved analytically in a classical (special-relativistic) framework, leading to rigorous expressions which relate the (apparent) astrometric parameters and radial velocity to the (true) kinematic parameters of the star in the barycentric reference system. We present rigorous and explicit formulae for the transformation of stellar positions, parallaxes, proper motions, and radial velocities from one epoch to another, assuming uniform rectilinear motion and taking into account light-time effects. The Jacobian matrix of the transformation is also given, allowing accurate and reversible propagation of errors over arbitrary time intervals. The light-time effects are generally very small but exceeds 0.1 mas or 0.1 m/s over 100 yr for at least 33 stars in the Hipparcos Catalogue. For high-velocity stars within a few tens of pc from the Sun light-time effects are generally more important than the effects of the curvature of their orbits in the Galactic potential.Comment: Accepted for publication in A&

Gaia astrometry for stars with too few observations - a Bayesian approach

Gaia's astrometric solution aims to determine at least five parameters for each star, together with appropriate estimates of their uncertainties and correlations. This requires at least five distinct observations per star. In the early data reductions the number of observations may be insufficient for a five-parameter solution, and even after the full mission many stars will remain under-observed, including faint stars at the detection limit and transient objects. In such cases it is reasonable to determine only the two position parameters. Their formal uncertainties would however grossly underestimate the actual errors, due to the neglected parallax and proper motion. We aim to develop a recipe to calculate sensible formal uncertainties that can be used in all cases of under-observed stars. Prior information about the typical ranges of stellar parallaxes and proper motions is incorporated in the astrometric solution by means of Bayes' rule. Numerical simulations based on the Gaia Universe Model Snapshot (GUMS) are used to investigate how the prior influences the actual errors and formal uncertainties when different amounts of Gaia observations are available. We develop a criterion for the optimum choice of priors, apply it to a wide range of cases, and derive a global approximation of the optimum prior as a function of magnitude and galactic coordinates. The feasibility of the Bayesian approach is demonstrated through global astrometric solutions of simulated Gaia observations. With an appropriate prior it is possible to derive sensible positions with realistic error estimates for any number of available observations. Even though this recipe works also for well-observed stars it should not be used where a good five-parameter astrometric solution can be obtained without a prior. Parallaxes and proper motions from a solution using priors are always biased and should not be used.Comment: Revised version, accepted 21st of August 2015 for publication in A&

Impact of basic angle variations on the parallax zero point for a scanning astrometric satellite

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.Comment: 8 pages, 2 figures; Accepted for publication in Astronomy & Astrophysic

Proper Motion and Secular Variations of Keplerian Orbital Elements

High-precision observations require accurate modeling of secular changes in the orbital elements in order to extrapolate measurements over long time intervals, and to detect deviation from pure Keplerian motion caused, for example, by other bodies or relativistic effects. We consider the evolution of the Keplerian elements resulting from the gradual change of the apparent orbit orientation due to proper motion. We present rigorous formulae for the transformation of the orbit inclination, longitude of the ascending node and argument of the pericenter from one epoch to another, assuming uniform stellar motion and taking radial velocity into account. An approximate treatment, accurate to the second-order terms in time, is also given. The proper motion effects may be significant for long-period transiting planets. These theoretical results are applicable to the modeling of planetary transits and precise Doppler measurements as well as analysis of pulsar and eclipsing binary timing observations