Inversion formula for determining the parameters of an astrometric binary

Abstract

It is believed that some numerical technique must be employed for determining the system parameters of a visual binary or a star with a planet, because the relevant equations are not only highly nonlinear, but also transcendental owing to Kepler's equation. Such common sense, however, is not true; we have discovered an analytic inversion formula, in which the original orbital parameters are expressed as elementary functions of the observable quantities, such as the location of four observed points and the time interval between these points. The key thing is that we use the time interval, but not the time of each observation, in order to avoid treating Kepler's equation. The present formula can be applied even in cases where the observations cover a short arc of the orbit during less than one period. Thus, the formula will be useful in future astrometric missions, such as SIM, GAIA, and JASMINE.