Analytical and numerical study of the ground-track resonances of Dawn orbiting Vesta

The aim of Dawn mission is the acquisition of data from orbits around two bodies, (4)~Vesta and (1)~Ceres, the two most massive asteroids. Due to the low thrust propulsion, Dawn will slowly cross and transit through ground-track resonances, where the perturbations on Dawn orbit may be significant. In this context, to safety go the Dawn mission from the approach orbit to the lowest science orbit, it is essential to know the properties of the crossed resonances. This paper analytically investigates the properties of the major ground-track resonances (1:1, 1:2, 2:3 and 3:2) appearing for Vesta orbiters: location of the equilibria, aperture of the resonances and period at the stable equilibria. We develop a general method using an averaged Hamiltonian formulation with a spherical harmonic approximation of the gravity field. If the values of the gravity field coefficient change, our method stays correct and applicable. We also discuss the effect of one error on the C_20 and C_22 coefficients on the properties of the 1:1 resonance. These results are checked by numerical tests. We determine that the increase of the eccentricity appearing in the 2:3 resonance is due to the J_{22} coefficient. Finally we numerically study the probability of the capture in resonance 1:1. Our paper reproduces, explains and completes the results of Tricarico and Sykes (2010).