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

Global m=1 asymmetries are observed in a variety of 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 centers, such as the double nucleus of M31. However, the dynamical origin and long-term stability of such modes remain poorly understood. We investigated the dynamical evolution and stability of a self-gravitating, collisionless torus orbiting a dominant central mass, with the aim of determining whether a slow non-axisymmetric (m=1) mode can arise spontaneously. We performed a suite of direct N-body simulations exploring a range of torus-to-central mass ratios and different initial particle distributions. The calculations were carried out with the high-order Hermite GPU integrator (̌arphi-GPU), enabling us to follow the long-term dynamical evolution of systems with a large number of particles. We find that a global slow m=1 mode forms spontaneously from initially axisymmetric configurations without imposed perturbations. The lopsided structure is sustained by coherent apsidal alignment of orbits and persists over secular timescales. Its maintenance requires hierarchical nonlinear coupling of low-order modes, including the participation of the m=3 component, as well as a sufficient vertical thickness of the torus, indicating that the instability is inherently three-dimensional. As a dynamical consequence of the long-lived overdensity, the central mass acquires an essential displacement with respect to the system barycenter. Our results demonstrate that a long-lived global m=1 mode can arise naturally in a geometrically thick self-gravitating torus orbiting a central mass. The mechanism identified here provides a dynamical framework for understanding eccentric nuclear disks, such as the double nuclei of M31 and NGC 4486B, as well as the molecular tori in active galactic nuclei (AGNs), and it suggests that the resulting lopsided asymmetry may produce observable offsets of the central supermassive black hole.