On the Stability of Extrasolar Planetary Systems and other Closely Orbiting Pairs

Abstract

This paper considers the stability of tidal equilibria for planetary systems in which stellar rotation provides a significant contribution to the angular momentum budget. We begin by applying classic stability considerations for two bodies to planetary systems --- where one mass is much smaller than the other. The application of these stability criteria to a subset of the Kepler sample indicates that the majority of the systems are not in a stable equilibrium state. Motivated by this finding, we generalize the stability calculation to include the quadrupole moment for the host star. In general, a stable equilibrium requires that the total system angular momentum exceeds a minimum value (denoted here as $L_X$) and that the orbital angular momentum of the planet exceeds a minimum fraction of the total. Most, but not all, of the observed planetary systems in the sample have enough total angular momentum to allow an equilibrium state. Even with the generalizations of this paper, however, most systems have too little orbital angular momentum (relative to the total) and are not in an equilibrium configuration. Finally, we consider the time evolution of these planetary systems; the results constrain the tidal quality factor of the stars and suggest that $10^6\le{Q_\ast}\le10^7$.