Post-Keplerian perturbations of the hyperbolic motion in the field of a massive, rotating object

The perturbations of the hyperbolic motion of a test particle due to the general relativistic gravitoelectromagnetic Schwarzschild and Lense-Thirring components of the gravitational field of a massive, rotating body are analytically worked out to the first post-Newtonian level. To the Newtonian order, the impact of the quadrupole mass moment of the source is calculated as well. The resulting analytical expressions are valid for a generic orientation in space of both the orbital plane of the probe and the spin axis of the primary, and for arbitrary values of the eccentricity. They are applied first to 'Oumuamua, an interstellar asteroid which recently visited our solar system along an unbound heliocentric orbit. While its gravitoelectric shifts occurred close to the Sun's flyby are less than some tens of milliarcseconds, those due to the solar oblateness and angular momentum are of the order of microarcseconds throughout the whole trajectory. Comparable values occur for the post-Newtonian shifts of the Near Earth Asteroid Rendezvous (NEAR) spacecraft during its flyby of the Earth, while those due to the oblateness of the latter are nominally several orders of magnitude larger. The current (formal) uncertainty in the quadrupole mass moment of the geopotential would bring the mismodeling of such classical effects below the nominal value of the predicted relativistic disturbances. The hyperbolic excess velocity is not changed by any of the post--Keplerian accelerations considered. The calculational approach developed can be straightforwardly extended to any alternative models of gravity as well.