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The post-Newtonian orbital effects induced by the mass quadrupole andspin-octupole moments of an isolated, oblate spheroid of constant density thatis rigidly and uniformly rotating on the motion of a test particle areanalytically worked out for an arbitrary orbital configuration and without anypreferred orientation of the body's spin axis. The resulting expressions arespecialized to the cases of (a) equatorial and (b) polar orbits. Theopportunity offered by a hypothetical new spacecraft moving around Jupiteralong a Juno-like highly elliptical, polar orbit to measure them ispreliminarily studied. Although more difficult to be practically implemented,also the case of a less elliptical orbit is considered since it yields muchlarger figures for the relativistic effects of interest. The possibility ofusing the S stars orbiting the supermassive black hole in Sgr A$^\ast$ at theGalactic Center as probes to potentially constrain some parameters of thepredicted extended mass distribution surrounding the hole by means of theaforementioned orbital effects is briefly examined.

Post-Newtonian orbital effects induced by the mass quadrupole and spin-octupole moments of an axisymmetric body