Origins of Compact Mean-Motion Resonances: Evidence for Long-Range Migration and the Case of Kepler-36

Abstract

The observed census of resonant extrasolar planets spans a tantalizing display of orbital architectures, ranging from familiar 2:1 and 3:2 mean-motion commensurabilities to nearly co-orbital configurations characterized by period ratios close to unity. While mean-motion resonances are widely recognized as signposts of convergent disk-driven migration, the process through which the most compact systems are established remains puzzling, since resonance capture must repeatedly fail at a series of first-order commensurabilities before finally succeeding at a high resonant index. Motivated by this discrepancy, here we develop an analytic theory that fuses the stability-based resonance capture criterion with the conventional paradigm of active accretion disks and the standard model of type-I migration. Within this framework, we derive an expression for the stellocentric radius of resonance capture, $r_{\rm{c}}$, and show that it depends only on the product of the disk viscosity parameter, $\alpha$, and the opacity-contributing small-grain mass fraction, $f_\mu$. Applying this formalism to Kepler-36 - the most compact known resonant system with a 7:6 period ratio - we find that resonance locking could not have been established near the disk's inner edge. Instead, capture must have occurred at $r_{\rm{c}}\approx 1-4$ AU, implying orbital decay of the planetary pair by approximately an order of magnitude. Viewed in this light, compact resonant architectures provide the clearest evidence for long-range migration among sub-Jovian planets. Moreover, the emerging picture is fully consistent with formation models in which super-Earths accrete within localized rings of planetesimals at orbital distances comparable to those that gave rise to the terrestrial planets of the Solar System.