Minimum Orbital Periods of H-rich Bodies

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.