Satellite Orbital Precessions Caused by the Octupolar Mass Moment of a Non-Spherical Body Arbitrarily Oriented in Space

I consider a satellite moving around a non-spherical body of mass M and equatorial radius R, and calculate its orbital precessions caused by the body’s octupolar mass moment J4. I consider only the effects averaged over one orbital period T of the satellite. I give exact formulas, not restricted to any special values of either the eccentricity e or the inclination i of the satellite’s orbit. I do not assume any preferential orientation for the body’s spin axis \({\boldsymbol{\hat{{\mathbf{k}}}}}\) because in many cases of potential interest (exoplanets, neutron stars, black holes) it is poorly known or unknown at all.