Stability boundaries for resonant migrating planet pairs

Convergent migration allows pairs of coplanar planets to become trapped into mean motion resonances. Once in resonance, the planets' eccentricities grow to equilibrium values that depends inversely on the ratio of migration time-scale to the eccentricity damping time-scale, K = τ a /τ e . The s...

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Bibliographic Details
Published in:Monthly notices of the Royal Astronomical Society 2014-05, Vol.440 (2), p.1753-1762
Main Authors: Bodman, Eva H. L., Quillen, Alice C.
Format: Article
Language:English
Subjects:
Equilibrium
Planets
Satellites
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Summary:Convergent migration allows pairs of coplanar planets to become trapped into mean motion resonances. Once in resonance, the planets' eccentricities grow to equilibrium values that depends inversely on the ratio of migration time-scale to the eccentricity damping time-scale, K = τ a /τ e . The stability of a planet pair depends on eccentricity so the pair can become unstable before reaching the equilibrium eccentricities. Using a resonant overlap criterion that depends on eccentricity up to second order, we find a function K min that defines the lowest value for K, as a function of the ratio of total planet mass to stellar mass that allows two convergently migrating planets to remain stable in resonance. We found that for first-order resonance, K min is linear with increasing planet mass and quadratic for second-order resonances. The linear relation continues until the mass approaches a critical mass defined by the 2/7 resonance overlap instability law and K min → ∞. Comparing our analytic boundary with an observed sample of two-planet systems, all but one of the systems with measured eccentricities are well inside the stability region estimated by this model. We calculated K min for Kepler systems without well-constrained eccentricities and found only weak constraints on K.
ISSN:0035-8711
1365-2966
DOI:10.1093/mnras/stu385