The Dipole Instability in Gravitational $N$-body Systems: A Natural Explanation for Lopsidedness and Off-Centered Nuclei in Galaxies

We explore the stability of isotropic, spherical, self-gravitating systems with a double-power law density profile. Systems with rapid transitions between the inner and outer slopes are shown to have an inflection in their isotropic distribution function (DF), where ${\rm d} f/{\rm d} E > 0$, thereby violating Antonov's stability criterion. Using high-resolution $N$-body simulations, we show that the resulting instability causes the growth of a rotating dipole (or $l=1$) mode. The inflection feature in the DF responds to the mode by promoting its growth, driving the instability. The growth of the dipole results in a torque that dislodges the original cusp from its central location, and sets it in motion throughout the central region. Once the mode goes non-linear, it saturates, together with the cusp, into a long-lived soliton (the $l=1$ equivalent of a bar in a disk galaxy), which maintains its sloshing motion through the center of the halo along a slowly precessing, elliptical orbit. Concurrently, the soliton traps increasingly more particles into libration, and the exchange of energy and angular momentum with these trapped particles works towards eroding the bump in the distribution function. We point out similarities between the dipole mode and the bump-on-tail instability in electrostatic plasmas, and highlight a potential connection with core stalling and dynamical buoyancy in systems with a cored density profile. Finally, we discuss the astrophysical implications in terms of lopsidedness and off-center nuclei in galaxies.