Compact Resonant Chains of Super-Earths

• Compact resonant chains of super-Earths are planetary systems where multiple super-Earths are closely spaced, locked in mean-motion resonances through convergent migration and resonant trapping.
• Analytical models and N-body simulations show that gas disk dynamics, migration barriers, and inner disk structures critically influence resonant capture and subsequent orbital evolution.
• Observational signatures, including near-integer period ratios and system uniformity, support theoretical predictions and explain features such as the exoplanet radius valley.

Compact resonant chains of super-Earths are planetary systems where multiple super-Earth planets are closely spaced such that their orbital periods are near ratios of small integers—mean-motion resonances (MMRs)—often as a result of convergent migration and subsequent dynamical evolution. These architectures are central to understanding the dynamical processing, composition, and final orbital spacing of close-in exoplanets as observed by transit and radial velocity surveys. The underlying formation scenario typically involves the growth and migration of rocky embryos in a gaseous disk, resonant trapping into chains, and post-gas instabilities that drive their final physical and orbital properties.

1. Physical Processes Governing Formation

The assembly of compact resonant chains begins with the oligarchic growth of rocky embryos interior to the ice line. The maximum isolation mass $M_{\rm c, iso}$ attainable by an embryo is set by the local dust surface density $\Sigma_d$:

$$M_{\rm c, iso}\simeq 0.16\,\eta_{\rm ice}^{3/2}\, f_d^{3/2}\left(\frac{a}{1\,\mathrm{AU}}\right)^{3/4 - (3q_d'/2)} \left(\frac{M_\ast}{M_{\odot}}\right)^{-1/2}M_\oplus,$$

where $\eta_{\rm ice}$ accounts for the solid density variation at the ice line, and $f_d, q_d'$ parameterize disk surface density scaling and profile.

Once formed, these embryos experience Type I migration, with a timescale:

$$\tau_{\rm mig1}\simeq 5\times10^4\times 10^{-q_g'/4}\frac{1}{C_1 f_g}\left(\frac{M_{\rm c}}{M_{\oplus}}\right)^{-1}\left(\frac{a}{1\,\mathrm{AU}}\right)^{q_g'}\left(\frac{M_\ast}{M_{\odot}}\right)^{3/2}\; \mathrm{yrs},$$

where $C_1$ incorporates reduction from non-linear/turbulent effects ($<1$ for realistic disks), and $f_g, q_g'$ control gas surface density properties.

Differential migration, due to embryo mass and local disk variations, leads embryos to converge and encounter each other. Once closely spaced, mutual gravitational interactions and dissipative effects from the disk facilitate their capture into low-order MMRs—a phenomenon quantified by the spacing for resonant trapping:

$$b_{\rm trap}\simeq 0.16\left(\frac{m_i+m_j}{M_\oplus}\right)^{1/6}\left(\frac{\Delta v_{\rm mig}}{v_{\rm K}}\right)^{-1/4}r_{\rm H},$$

with $b_{\rm trap}\simeq 5 r_{\rm H}$ commonly used for typical disk conditions, where $r_{\rm H}$ is the mutual Hill radius. As a result, compact “convoys” of resonant embryos migrate inward as a unit.

2. Protoplanetary Disk Architecture and Migration Barriers

The gaseous disk plays several critical roles beyond setting the initial embryo masses:

• Migration/Damping: Disk torques drive convergent Type I migration and provide strong orbital eccentricity damping, ensuring that embryos approach and become resonantly trapped in dynamically cold, low-eccentricity states.
• Inner Edge Features: The migration of embryos is ultimately halted by inner disk structures, such as the magnetospheric cavity (or “dead zone” boundary), which creates a positive corotation torque that can overpower inward migration. The inner edge is typically set at a fixed radius, e.g., $0.04$–$0.1$ AU, ensuring that the resonant chain stalls close to the star (Ida et al., 2010).
• Disk Evolution: As the disk evolves and dissipates (often exponentially decaying over Myr timescales), the lack of damping can lead to the destabilization of previously stable resonant chains.

These features ensure that resonant convoys accumulate interior to a migration barrier, leading to an over-density of planets at small orbital radii and setting up the prerequisites for subsequent dynamical evolution.

3. Dynamical Outcomes: Resonant Chains, Instability, and Giant Impacts

The fate of the resonant chain after disk dispersal depends on system compactness and residual dynamical excitation. The end-state architecture arises through a sequence of transitions:

• Resonant Capture: In the disk phase, planetary embryos repeatedly undergo capture into low-order MMRs (2:1, 3:2, 4:3, etc.), forming very compact chains. Orbital eccentricities remain small via gas damping.
• Instability Onset: Once the disk is depleted, damping is no longer effective. Chaotic perturbations among the closely-packed planets drive a rapid growth in eccentricity; when these approach values of order the escape eccentricity,

$$e_{\rm esc}\simeq 0.28\left(\frac{m_i+m_j}{M_\oplus}\right)^{1/3}\left(\frac{a}{1\,\mathrm{AU}}\right)^{1/2},$$

orbits cross, initiating close scattering and ultimately giant impacts (Ida et al., 2010).

• Collision Outcomes: These giant impacts are often efficient mergers due to the anti-aligned periastron orientations at collision (“LRL vector cancellation”), resulting in final planets with lower eccentricities than the escape value ($e \sim 0.01$–$0.1$). Simulations find that most collisions lead to near-perfect mergers—fragmentation is rare under typical impact velocities (Goldberg et al., 2022).
• Final System Properties: The result is a system of several super-Earths with relatively tight orbital spacings (period ratios $\sim1.5$–$2.5$), moderate eccentricities, and a lack of current resonant locking, even though the dynamical “memory” of past resonance can persist in orbital architecture.

4. Theoretical and Numerical Modeling Approaches

The formation and evolution of resonant chains are studied using advanced numerical population synthesis schemes and N-body integrations:

• Simultaneous Mass Growth and Migration: Embryos’ growth and migration are integrated on a finely-spaced grid, including the local evolution of $\Sigma_d$ and mutual interactions (Ida et al., 2010).
• Analytical “Recipes” for Resonant Trapping: When pairwise separations drop below $b_{\rm trap}$, migration torques are reassigned so that embryos migrate as a resonant unit.
• Post-Gas Evolution: After gas dispersal, the excitation and diffusion of eccentricities, scattering, and mergers are handled semi-analytically or with direct numerical integration to replicate the giant impact phase (Ida et al., 2010).
• Performance and Predictions: These schemes reproduce not only the mass and semimajor-axis structure of super-Earth systems but also yield statistically robust distributions of orbital period ratios, eccentricities, and a natural explanation for the observed mixture of resonant and non-resonant systems (Izidoro et al., 2017, Goldberg et al., 2022).

Such modeling approaches reveal that stable resonant chains are expected to be rare among observed mature systems; the high observed fraction of systems without strong resonance supports ubiquitous late instabilities (“breaking the chains” scenario).

5. Observational Signatures and Implications

Observational constraints derived from Kepler and radial velocity surveys directly bear on the theory:

Architecture	Key Observed Features	Theoretical Consequence
Resonant Chains	Excess near integer period ratios; e.g., Kepler-223 shows 3:4:6:8 chain (Mills et al., 2016)	Indicates migration and resonant capture in gas-rich phase
Non-resonant Compact	Multiple planets within $\sim0.1$ AU, moderate period ratios 1.5–2.5	Consistent with post-instability spreading of original chains
Peas-in-a-Pod	Intra-system mass/radius uniformity (“peas-in-a-pod”) (Goldberg et al., 2022, 2207.13833)	Implies simultaneous or nearly simultaneous formation/migration/merger from a uniform local reservoir

Late giant impacts not only drive eccentricity and inclination excitation but can also strip planetary atmospheres, helping to explain the exoplanet “radius valley”—a dearth of planets with radii between $\sim1.4$ and $2.4$ $R_\oplus$ that separates rocky and volatile-rich planets (Izidoro et al., 2022). The mass and period distributions of observed super-Earths are also closely matched by simulation outputs.

6. Special Cases and Model Limitations

While the baseline scenario robustly explains many features of observed exoplanetary systems, some specific regimes or variations require additional consideration:

• Disk Edge Physics: The nature (sharpness/steepness) of the inner migration barrier—whether due to a dead zone or other edge effects—sets the ultimate system compactness, and may affect stability and the final number of surviving planets (Ataiee et al., 2021).
• Diversity of Disk Properties: The viscosity, opacities, and temperature profiles of disks introduce variation in resonance trapping efficiency and final planet spacing (Bitsch et al., 18 Nov 2024).
• Rare Outcomes: Occasionally, numerous mergers in the instability phase can yield an anomalously massive remnant planet (e.g., a close-in Neptune around a low-mass star), though such outcomes are rare and do not require exotic initial conditions (Liveoak et al., 9 Sep 2024).
• Comparison with Solar System: The lack of compact super-Earths in the Solar System may be attributed to a lower rocky planetesimal mass at $\sim1$ AU, preventing sufficient accretion and migration to launch a resonant chain inward (Batygin et al., 2023).

7. Synthesis and Theoretical Implications

The compact resonant chain scenario offers a rigorous, quantitative framework for the origin of compact, close-in super-Earths, integrating the following sequence:

1. Oligarchic growth and rapid migration produce rocky embryos that accumulate in resonant chains near the disk’s inner edge.
2. Gas-driven migration stalls at an inner barrier; embryos continue to accumulate, forming compact convoys.
3. Gas dispersal eliminates damping, triggering dynamical instabilities and a late phase of mergers and orbit crossing, producing the observed compact yet non-resonant (or weakly resonant) architecture and “peas-in-a-pod” uniformity.
4. This sequence quantitatively predicts observed distributions of periods, masses, eccentricities, and provides a mechanistic explanation for the radius valley, the scarcity of strict resonant locking, and variances in system compactness.

Analytical formulae for migration and resonance spacing, as well as robust numerical algorithms for simultaneous mass growth and migration coupling, form the technical backbone for detailed simulation and comparative exoplanetology. This framework is essential for both interpretation of current survey data and for modeling the diversity of multi-planet system architectures seen across exoplanetary populations.