  We present an analysis of the stability spectrum for all stationary periodic
solutions to the sine-Gordon equation. An analytical expression for the
spectrum is given. From this expression, various quantitative and qualitative
results about the spectrum are derived. Specifically, the solution parameter
space is shown to be split into regions of distinct qualitative behavior of the
spectrum, in one of which the solutions are stable. Additional results on the
stability of solutions with respect to perturbations of an integer multiple of
the solution period are given.
