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 coorbital 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 c , and show that it depends only on the product of the disk viscosity parameter, α , and the opacity-contributing small-grain mass fraction, f μ . 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 c ≈ 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.