On the Resonant Orbit of a Solar Companion Star in the Gravitational Field of the Galaxy

Abstract

For the motion of a weakly bound solar companion star, we have investigated the resonance effects that arise when the Keplerian period of the system is half the period with which the system oscillates through the galactic plane. This resonance could be important in the motion of the putative solar companion star Nemesis. The resonant perturbation of the Keplerian motion is a dynamic tidal compression of the Sun-Nemesis system by the galactic field. We have expressed the Hamiltonian in terms of the action-angle variables that describe the unperturbed Keplerian motion and the oscillation through the galactic plane. By averaging the Hamiltonian over the fast angle variables, we have reduced the problem to a motion in two degrees of freedom. Our study of the averaged motion includes the isolation of the periodic orbits, the determination of the stability of the periodic orbits, and numerical solutions for the nonperiodic orbits. We have delineated the resonance phenomena with the aid of surfaces of section and other diagnostic tools. We have tested the orbits derived from the averaged Hamiltonian with the said of numerical solutions of the equations of motion derived from the "exact" Hamiltonian of the system. The resonance is strong enough to produce a well-defined population of resonant orbits in which the Keplerian motion of Nemesis locks onto the oscillation of the system through the galactic plane. Orbits of large inclinations relative to the galactic plane tend to be chaotic. Only a small population of resonant and nonresonant orbits with low inclinations would penetrate the Oort cloud and trigger comet showers at every perihelion passage.