Quantifying Departures from Equilibrium with the Spherical Jeans
  Equation
  Proper motions of collisionless pointlike objects in a spherically symmetric
system, for example stars in a galaxy, can be used to test whether that system
is in equilibrium, with no assumptions regarding isotropy. In particular, the
fourth order spherical Jeans equation yields expressions for two observable
quantities characterizing the departure from equilibrium, both of which can be
expressed in terms of time derivatives of first and third moments of the
velocities. As illustrations, we compute these quantities for tracer
distributions drawn from an exact equilibrium configuration and also from near
equilibrium configurations generated using the N-body code GALIC.
