  Astrophysical shocks or bursts from a photoionizing source can disturb the
typical collisional plasma found in galactic interstellar media or the
intergalactic medium. The spectrum emitted by this plasma contains diagnostics
that have been used to determine the time since the disturbing event, although
this determination becomes uncertain as the elements in the plasma return to
ionization equilibrium. A general solution for the equilibrium timescale for
each element arises from the elegant eigenvector method of solution to the
problem of a non-equilibrium plasma described by Masai (1984) and Hughes &
Helfand (1985). In general the ionization evolution of an element Z in a
constant electron temperature plasma is given by a coupled set of Z+1 first
order differential equations. However, they can be recast as Z uncoupled first
order differential equations using an eigenvector basis for the system. The
solution is then Z separate exponential functions, with the time constants
given by the eigenvalues of the rate matrix. The smallest of these eigenvalues
gives the scale of slowest return to equilibrium independent of the initial
conditions, while conversely the largest eigenvalue is the scale of the fastest
change in the ion population. These results hold for an ionizing plasma, a
recombining plasma, or even a plasma with random initial conditions, and will
allow users of these diagnostics to determine directly if their best-fit result
significantly limits the timescale since a disturbance or is so close to
equilibrium as to include an arbitrarily-long time.
