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, $α$, 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 $r_{\rm{c}}\approx 1-4$ 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.

Figures (5)

• Figure 1: Schematic view of the Kepler-36 planetary system. A $\sim4\,M_{\oplus}$ super-Earth (planet b) and a $\sim8\,M_{\oplus}$ mini-Neptune (planet c) orbit a $\sim1.1\,M_{\odot}$ subgiant host star at orbital periods of approximately 14 and 16 days. The planets’ semi-major axes differ by only $\sim 0.01\,$AU, placing them near the 7:6 mean-motion commensurability.
• Figure 2: Numerically computed resonance capture map. The $x$-axis shows the dimensionless product of the semi-major axis convergence timescale and the inner orbital frequency, while the $y$-axes show the ratios of eccentricity-to-semi-major-axis-damping times for the inner ($\mathcal{K}_{1} = (\tau_{a}/\tau_{e_1})$, left) and outer ($\mathcal{K}_{2} = (\tau_{a}/\tau_{e_2})$, right) planets. The masses of the bodies are fixed to the observed values of the Kepler-36 system. In the compact limit, this ratio directly reflects the disk aspect ratio $h/r$, also indicated on the right-hand scale (computed with $s=3/5$). Gray lines denote theoretical capture boundaries: vertical lines mark the adiabaticity threshold, while slanted lines show the dissipative stability criterion. Each point represents a distinct $N$-body calculation and the color bands correspond to regions of successful numerical capture into the labeled first-order resonance, while the white domain indicates unstable configurations. The overstability boundary for the 7:6 resonance is shown with a yellow dashed line.
• Figure 3: The orbital radius for resonance capture, $r_{\rm{c}}$, as a function of the product $\alpha \, f_\mu$, where $\alpha$ is the disk viscosity parameter and $f_\mu$ is the dust-to-gas ratio of opacity-contributing small grains. The two black curves denote dissipative stability-based capture criteria for the 7:6 and 6:5 resonances, bracketing the admissible parameter range. Qualitatively, the left-hand-side of the diagram (with $r_{\rm{c}}\sim1-4\,$AU) corresponds to a parameter regime of a mature, relatively quiescent disk, where the small-grain dust-to-gas ratio has been dramatically reduced due to particle coagulation and planetesimal formation. The right-hand-side of the diagram denotes the opposite regime of a young, highly turbulent disk, where the majority of the solid budget of the nebula is in the form of small grains. The quantities annotating the dotted lines denote the minimal surface density, $\Sigma$ (horizontal labels) and aspect ratio, $h/r$ (vertical labels) -- both evaluated at $r_{\rm{c}}$ -- necessary for capture to occur interior to the water-ice sublimation line, so as to match the rocky composition of the inner planet. Regions of parameter space away from the smaller end of $r_{\rm c}$ and $\alpha f_\mu$ are disfavored: in this regime (marked as a hatched region), the implied surface densities and disk aspect ratios rise to unphysical values, and the corresponding resonant configurations are expected to be dynamically unstable due to overstability and turbulent disruption.
• Figure 4: Minimal disk accretion rate consistent with resonance capture within the nebular ice line. The thin lines show the resonance capture radius $r_{\rm{c}}$, which depends on the product $\alpha\,f_\mu$. The thick lines show the minimal requisite value of $\dot{M}$ such that, at the corresponding value of $r_{\rm{c}}$, the disk mid-plane temperature exceeds the water-ice sublimation temperature, $T_{\rm{ice}}$. Typical accretion rates in class-II disks are approximately bounded from above by $\dot{M} \lesssim 10^{-8} M_\odot/$yr, suggesting capture radii that correspond to the lower end of the admissible range (i.e., $1-2\,$AU, possibly up to $4\,$AU).
• Figure 5: Constraints on disk structure from resonance capture. Panel A shows the disk aspect ratio $h/r$ as a function of capture radius $r_{\rm{c}}$ for a range of the normalized accretion parameter $X=\dot{M}/\dot{M}_{\rm{min}}$, with the overstability threshold (equation \ref{['eqn:overstability']}) marked by a dotted line. Panel B depicts the corresponding surface density $\Sigma$, with a reference MMSN-like active disk profile plotted for comparison. Panel C shows an analytic stability map informed by turbulent disruption, projected onto the $\alpha - f_\mu$ plane. Thick lines denote limits beyond which turbulent disruption prohibits resonance maintenance for various values of $X$, while the dotted line denotes the onset of overstability. Together, these constraints favor resonance capture at $r_{\rm{c}} \lesssim 4\,$AU under moderately massive disk conditions.