Satellite orbital precessions caused by the first odd zonal J 3 multipole of a non-spherical body arbitrarily oriented in space

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.