Analytical solution for the three-dimensional motion of a circumbinary planet around a binary star

By using the method of separating rapid and slow subsystem, we obtain an analytical solution for a stable three-dimensional motion of a circumbinary planet around a binary star. We show that the motion of the planet is more complicated than it was obtained for this situation analytically by Farago and Laskar (2010). Namely, in addition to the precession of the orbital plane of the planet around the angular momentum of the binary (found by Farago and Laskar (2010)), there is simultaneously the precession of the orbital plane of the planet within the orbital plane. We show that the frequency of this additional precession is different from the frequency of the precession of the orbital plane around the angular momentum of the binary. We demonstrate that this problem is mathematically equivalent both to the problem of the motion of a satellite around an oblate planet and to the problem of a hydrogen Rydberg atom in the field of a high-frequency linearly-polarized laser radiation, thus discovering yet another connection between astrophysics and atomic physics. We point out that all of the above physical systems have a higher than geometrical symmetry, which is a counterintuitive result. In particular, it is manifested by the fact that, while the elliptical orbit of the circumbinary planet (around a binary star) or of the satellite (around an oblate planet) or of the Rydberg electron (in the laser field) undergoes simultaneously two types of the precession, the shape of the orbit does not change. The fact that a system, consisting of a circumbinary planet around a binary star, possesses the hidden symmetry should be of a general physical interest. Our analytical results could be used for benchmarking future simulations.