Precession of the Orbit of a Planet around Stars Revolving along: Low-Eccentricity Orbits in a Binary System: Analytical Solution

In our previous paper (hereafter, paper I) we presented analytical results on the non-planar motion of a planet around a binary star for the cases of the circular orbits of the components of the binary. We found that the orbital plane of the planet (the plane containing the “unperturbed” elliptical orbit of the planet), in addition to precessing about the angular momentum of the binary, undergoes simultaneously the precession within the orbital plane. We demonstrated that the analytically calculated frequency of this additional precession is not the same as the frequency of the precession of the orbital plane about the angular momentum of the binary, though the frequencies of both precessions are of the same order of magnitude. In the present paper we extend the analytical results from paper I by relaxing the assumption that the binary is circular – by allowing for a relatively small eccentricity ε of the stars orbits in the binary. We obtain an additional, ε-dependent term in the effective potential for the motion of the planet. By analytical calculations we demonstrate that in the particular case of the planar geometry (where the planetary orbit is in the plane of the stars orbits), it leads to an additional contribution to the frequency of the precession of the planetary orbit. We show that this additional, ε-dependent contribution to the precession frequency of the planetary orbit can reach the same order of magnitude as the primary, ε-independent contribution to the precession frequency. Besides, we also obtain analytical results for another type of the non-planar configuration corresponding to the linear oscillatory motion of the planet along the axis of the symmetry of the circular orbits of the stars. We show that as the absolute value of the energy increases, the period of the oscillations decreases.