FROZEN ORBITS FOR SATELLITES CLOSE TO AN EARTH-LIKE

We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in.the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficients J2 through J9. The model problem is averaged-over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity - not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J2 only). In particular, we examine the manner in which the odd zonal J3 breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vinti's problem (J4 + J2(2) = 0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vinti's problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions.