Determination of the orbit of a natural satellite with hyperbolic flybys

We show that, when a natural satellite like Titan is invisible (e.g., due t o an opaque atmosphere) its planetary orbit and its mass can be determined by tracking a spacecraft in close flybys. This is an important problem in t he Cassini mission to the Saturnian system, which will be greatly improved by a good astrometric model for all its main components in particular, an a ccuracy of a few hundred meters for the orbit of Titan is necessary to allo w a measurement of its moment of inertia. The orbit of the spacecraft is th e union of elliptical arcs,joined by short hyperbolic transitions: a proble m of singular perturbation theory, whose solution leads to a matching condi tion between the inner hyperbolic orbit and the elliptical orbital elements . Since the inner elements are given in terms of the relative position and velocity of the spacecraft, accurate Doppler measurements in both regions c an provide a satisfactory determination of Titan's position and velocity, h ence of its Keplerian elements. The errors in this determination are discus sed on the basis of the expected Allan deviation of the Doppler method; it is found that the driving errors are those in the elliptical arcs; the frac tional errors in Titan's orbital elements are expected to be approximate to 10(-7). It is also possible to measure the mass of the satellite; however, when the eccentricity e of the flybys is large, the mass and a scaling tra nsformation are highly correlated and the fractional error in the mass is e xpected to be e times worse.