Global m=1 slow mode in near-Keplerian self-gravitating torus: applications to stellar nuclear disks and AGN molecular tori

Original Abstract

Global m=1 asymmetries are observed in many self-gravitating astrophysical systems and are often interpreted as large-scale slow modes in near-Keplerian potentials. Prominent examples include eccentric nuclear disks in galactic centres, such as the double nucleus of M31. However, the origin and long-term stability of such modes remain unclear. We investigate the evolution and stability of a collisionless, self-gravitating torus orbiting a dominant central mass, aiming to determine whether a slow non-axisymmetric (m=1) mode can arise spontaneously. We perform direct N-body simulations exploring different torus-to-central mass ratios and initial conditions. The calculations use the high-order Hermite GPU integrator (φ-GPU), allowing us to follow long-term evolution with many particles. We find that a global slow m=1 mode forms spontaneously from initially axisymmetric configurations. The lopsided structure is sustained by coherent apsidal alignment and persists over secular timescales. Its maintenance requires nonlinear coupling of low-order modes, including the m=3 component, as well as a sufficient vertical thickness of the torus. As a result of the long-lived overdensity, the central mass is displaced from the system barycenter. These results provide a framework for understanding eccentric nuclear disks, such as those in M31 and NGC4486B, as well as molecular tori in AGNs, and suggest that such asymmetries may produce observable offsets of the central supermassive black hole.