  In this work we derive the minimum allowed orbital periods of H-rich bodies
ranging in mass from Saturn's mass to 1 $M_{\odot}$, emphasizing gas giants and
brown dwarfs over the range $0.0003 - 0.074 \, M_\odot$. Analytic fitting
formulae for $P_{\rm min}$ as a function of the mass of the body and as a
function of the mean density are presented. We assume that the density of the
host star is sufficiently high so as not to limit the minimum period. In many
instances this implies that the host star is a white dwarf. This work is aimed,
in part, toward distinguishing brown dwarfs from planets that are found
transiting the host white dwarf without recourse to near infrared or radial
velocity measurements. In particular, orbital periods of $\lesssim 100$ minutes
are very likely to be brown dwarfs. The overall minimum period over this entire
mass range is $\simeq 37$ minutes.
