Secular dynamics of a test particle perturbed by a Miyamoto–Nagai disc

Abstract

ABSTRACT We study the secular behaviour of a test particle orbiting a dominant central body and perturbed by a Miyamoto–Nagai (MN) disc. We derive a quadrupole-level secular Hamiltonian of this system, which involves a dimensionless parameter η that is used to characterize the flattening of MN disc. (The smaller η, the flatter disc; and η = 0 gives the infinitely thin Kuzmin disc.) We find that, in the quadrupole approximation, the perturbation of the MN disc can give rise to the von Zeipel–Lidov–Kozai (ZLK)-like dynamics and depending on the value of η the dynamics has two different manifestations: (i) when η < 1/3, the test particle’s behaviour is similar to that described in the classical ZLK problem. In particular, as η increases from 0 to 1/3, the critical inclination for the large eccentricity oscillations decreases from $26{_{.}^{\circ}}56$ to 0°, ; (ii) when η > 1/3, the orbital evolution of the test particle and the phase-space morphology are opposite to the classical ZLK case. This leads to a striking result that the test particle cannot remain on a near-coplanar orbit if its eccentricity is sufficiently large. However, as η increases further the ZLK-like dynamics would be gradually suppressed by the spherical term in the Hamiltonian. We also survey the global secular dynamics numerically in which the quadrupole approximation is no longer valid.