Collisionless relaxation from near equilibrium configurations: Linear theory and application to tidal stripping

Abstract

Placed slightly out of dynamical equilibrium, an isolated stellar system quickly returns towards a steady virialized state. We study this process of collisionless relaxation using the matrix method of linear response theory. We show that the full phase space distribution of the final virialized state can be recovered directly from the disequilibrium initial conditions, without the need to compute the time evolution of the system. This shortcut allows us to determine the final virialized configuration with minimal computational effort. Complementing this result, we develop tools to model the system's full time evolution in the linear approximation. In particular, we show that moments of the velocity distribution can be efficiently computed using a generalized moment matrix. We apply our linear methods to study the relaxation of energy-truncated Hernquist spheres, mimicking the tidal stripping of a cuspy dark matter subhalo. Comparison of our linear predictions against controlled, isolated $N$-body simulations shows agreement at per cent level for the parts of the system where a linear response to the perturbation is expected. We find that relaxation generates a tangential velocity anisotropy in the intermediate regions, despite the initial disequilibrium state having isotropic kinematics. Our results also strengthen the case for relaxation depleting the amplitude of the density cusp, without affecting its asymptotic slope. Finally, we compare the linear theory against $N$-body simulation of tidal stripping on a radial orbit, confirming that the theory still accurately predicts density and velocity dispersion profiles for most of the system.