Stability boundaries for resonant migrating planet pairs

Abstract

Convergent migration allows pairs of planet to become trapped into mean motion resonances. Once in resonance, the planets' eccentricities grow to an equilibrium value that depends on the ratio of migration time scale to the eccentricity damping timescale, K = τ a /τ e , with higher values of equilibrium eccentricity for lower values of K. For low equilibrium eccentricities, e eq ∝ K −1/2 . Equilibrium eccentricities also depend on the distance between the planets. Resonances near the planet have lower equilibrium eccentricity. The stability of a planet pair depends on eccentricity so the system can become unstable before it reaches its equilibrium eccentricity.Using a resonant overlap criterion that takes into account the role of first and second order resonances and depends on eccentricity, we find a function K min (µ p , j) that defines the lowest value for K, as a function of the ratio of total planet mass to stellar mass (µ p ) and the period ratio of the resonance defined as P 1 /P 2 = j/(j + k), that allows two convergently migrating planets to remain stable in resonance at their equilibrium eccentricities. We scaled the functions K min for each resonance of the same order into a single function K c . The function K c for planet pairs in first order resonances is linear with increasing planet mass and quadratic for pairs in second order resonances with a coefficient depending on the relative migration rate and strongly on the planet to planet mass ratio. The linear relation continues until the mass approaches a critical mass defined by the 2/7 resonance overlap instability law and K c → ∞.We compared our analytic boundary with an observed sample of resonant two planet systems. All but one of the first order resonant planet pair systems found by radial velocity measurements are well inside the stability region estimated by this model. The one system in the instability region is well below K c but is also in the 4:3 resonance which is not explained well with smooth migration (Rein et al. 2012). We calculated K c for Kepler systems without well-constrained eccentricities and found only weak constraints on K. The Kepler systems have all have lower bounds less than K = 10 with most systems with K min < 1. c 0000 RAS

Detectable close-in planets around white dwarfs through late unpacking

“…Also, two planets which are not Hill stable may still be bounded and ordered over long timescales. This behaviour is especially connected to the overlap of mean motion resonances (Chirikov 1979;Wisdom 1980;Mardling 2008;Funk et al 2010;Mustill & Wyatt 2012;Deck et al 2013;Giuppone et al 2013;Bodman & Quillen 2014).…”

Orbital Stability of Multi-Planet Systems: Behavior at High Masses

“…This so-called cleared zone indicates the region from Figure 6. The outermost first order (solid) and second order (dashed) resonance locations and widths (Deck et al 2013;Bodman & Quillen 2014, and references therein) as a function of eccentricity and period ratio for μ = 10 −2 . The calculation of first order resonance libration widths breaks down at low eccentricities, so the resonance widths are shaded only for e >= e crit and treated as the low eccentricity case as shown in Wisdom (1980) where m e 0.5 crit .…”

On the migration-induced formation of the 9:7 mean motion resonance

“…As recently shown (Goldreich & Schlichting 2014;Deck & Batygin 2015) a numeric value of this factor may be important for the stability of the equilibrium. In the literature γ = 0 is used as well (e.g., Lee & Peale 2002;Kley et al 2004;Libert & Tsiganis 2009;Bodman & Quillen 2014). Below we study the stability of the equilibrium for different mass ratios and different models of the migration.…”