An exact method for orbit determinations of an astrometric binary

Here we present an exact method for orbit determinations of an astrometric binary for which the primary star can be seen but the companion is unseen like a black hole or an extra-solar planet. Since the laws of the motion of celestial objects were discovered by Kepler in the seventeenth century, many attempts have been made to solve a fundamental problem of how to determine the orbital elements (and the mass) of a binary by measuring its positions projected onto a celestial ∗Email: [email protected] 1 sphere. It has been long believed that some approximate/iterative or numerical technique must be employed to determine the orbital elements. In the nineteenth century, the problem was solved for resolved double stars, for which the relative vector from the primary star to the secondary is observable. For astrometric binaries, however, for which this relative vector is not observable, an exact solution was very recently found by Asada, Akasaka and Kasai by assuming no observational errors. This paper extends the solution considering realistic observational data. The generalized solution is still simply expressed only by elementary functions. keywords: astrometric binary — celestial mechanics — orbit determination