  It is believed that some numerical technique must be employed for the
determination of 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 the Kepler's equation. Such a common sense, however, is
not true; we discover 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 the 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 the future astrometric missions such as SIM, GAIA and JASMINE.

Astrophysics