Pairwise Tidal Equilibrium States and the Architecture of Extrasolar Planetary Systems

Current observations indicate that the planet formation process often produces multiple planet systems with nearly circular orbits, regular spacing, a narrow range of inclination angles, and similar planetary masses of order $m_{\rm p}\sim10M_\oplus$. Motivated by the observational sample, this paper determines the tidal equilibrium states for this class of extrasolar planetary systems. We start by considering two planet systems with fixed orbital spacing and variable mass ratios. The basic conjecture explored in this paper is that the planet formation process will act to distribute planetary masses in order to achieve a minimum energy state. The resulting minimum energy configuration --- subject to the constraint of constant angular momentum --- corresponds to circular orbits confined to a plane, with nearly equal planetary masses (as observed). We then generalize the treatment to include multiple planet systems, where each adjacent pair of planets attains its (local) tidal equilibrium state. The properties of observed planetary systems are close to those expected from this pairwise equilibrium configuration. In contrast, observed systems do not reside in a global minimum energy state. Both the equilibrium states of this paper and observed multi-planet systems, with planets of nearly equal mass on regularly spaced orbits, have an effective surface density of the form $\sigma\propto r^{-2}$, much steeper than most disk models.