Recent observations of Kepler multi-planet systems have revealed a number of systems with planets very close to second-order mean motion resonances ( MMRs , with period ratio 1 : 3 , 3 : 5 , etc . )
We present an analytic study of resonance capture and its stability for planets migrating in gaseous disks .
Resonance capture requires slow convergent migration of the planets , with sufficiently large eccentricity damping timescale T _ { e } and small pre-resonance eccentricities .
We quantify these requirements and find that they can be satisfied for super-Earths under protoplanetary disk conditions .
For planets captured into resonance , an equilibrium state can be reached , in which eccentricity excitation due to resonant planet-planet interaction balances eccentricity damping due to planet-disk interaction .
This “ captured ” equilibrium can be overstable , leading to partial or permanent escape of the planets from the resonance .
In general , the stability of the captured state depends on the inner to outer planet mass ratio q = m _ { 1 } / m _ { 2 } and the ratio of the eccentricity damping times .
The overstability growth time is of order T _ { e } , but can be much larger for systems close to the stability threshold .
For low-mass planets undergoing type I ( non-gap opening ) migration , convergent migration requires q \lesssim 1 , while the stability of the capture requires q \gtrsim 1 .
These results suggest that planet pairs stably captured into second-order MMRs have comparable masses .
This is in contrast to first-order MMRs , where a larger parameter space exists for stable resonance capture .
We confirm and extend our analytical results with N -body simulations , and show that for overstable capture , the escape time from the MMR can be comparable to the time the planets spend migrating between resonances .