Combining astrometric and radial velocity data for exoplanet detection

Abstract: Context: The Gaia satellite will be provide astrometry with micro-arcsecond accuracy. This will allow for the determination of the orbital elements of exoplanets with measurable astrometric signature. Astrometry alone is not be able to disentangle an ambiguity in the orientation of planetary orbits, just as radial velocity is only able to give the mass of orbiting planets combined with multiplicative factor. Combining astrometry and radial velocity this can be resolved. Many different approaches exist of how to parameterize and solve the problem of finding the orbital parameters. This text takes a look at one parametrization and one method for solving the problem. Aim: Define a model for the parametrization of the problem of finding the orbital elements for an exoplanet. This model should both describe the astrometric and radial velocity. Implement the model in AGISLab and determine how well it is able to retrieve the orbital elements. Method: The problem was parameterized using Thiele-Innes parameters for describing the orientation and semi-major axis of the system. Further parameters were; mass ratio of the planet to the star, time of periapsis passage, eccentricity of the orbit and period. In addition, the astrometric parameters were also included; position, parallax and proper motion. Radial motion was not included. The Levenberg-Marquardt algorithm was used for the optimization. Boundaries on the planetary parameters were introduced, in order to prevent unrealistic solutions, through transformations. The transformation of the parameters was a standard trigonometric function. To estimate the formal errors on the optimized parameters, parametric bootstrapping was performed. Result: The optimization works and provides sensible parameters, though the solution is very sensitive to the initial selection of parameters. Singular value decomposition of the matrix J|WJ, the Hessian of the merit function, indicates that the problem is ill-conditioned so bootstrapping may be a better solution for computing formal errors on the parameters than calculating them from the square root of the diagonal elements of its inverse. Bootstrapping also indicates that the formal errors of the parameters are not normally distributed. Conclusion: The method does work and allows for very easy combination of radial velocity and astrometric data. Due to the choice of parametrization and/or optimization procedure the method is unstable.