Exact Time Dependent Solutions — Violent Nonrelaxation

Abstract

We present some exact oscillating solutions to the Collisionless Boltzmann Equation (CBE). The solutions all share the property that at any instant of time, the real space density is uniform over some (in general, ellipsoidal) region of space and zero outside. The shape and size of the region change with time. Then, the gravitational force within the region is linear in the spatial coordinates, though time dependent. The equations of motion of a particle are therefore linear. Under these conditions time dependent quadratic forms in the coordinates and momenta can be found which are constants of the motion. The phase space distribution function (f) is then chosen to be a function of these and solves the CBE by Jeans’ theorem (see ref. 1). The functional form of f is so chosen that integrating over velocities gives uniform density ellipsoids.