Satellite orbital precessions caused by the first odd zonal J3 multipole of a non-spherical body arbitrarily oriented in space

Abstract

An astronomical body of mass M and radius R which is non-spherically symmetric generates a free space potential U which can be expanded in multipoles. As such, the trajectory of a test particle orbiting it is not a Keplerian ellipse fixed in the inertial space. The zonal harmonic coefficients J2,J3,… of the multipolar expansion of the potential cause cumulative orbital perturbations which can be either harmonic or secular over time scales larger than the unperturbed Keplerian orbital period T. Here, I calculate the averaged rates of change of the osculating Keplerian orbital elements due to the odd zonal harmonic J3 by assuming an arbitrary orientation of the body’s spin axis \(\hat{\boldsymbol{k}}\). I use the Lagrange planetary equations, and I make a first-order calculation in J3. I do not make a-priori assumptions concerning the eccentricity e and the inclination i of the satellite’s orbit.