On the migration-induced resonances in a system of two planets with masses in the Earth mass range

We investigate orbital resonances expected to arise when a system of two planets, with masses in the range 1-4 Earth masses, undergoes convergent migration while embedded in a section of gaseous disc where the flow is laminar. We consider surface densities corresponding to 0.5-4 times that expected for a minimum mass solar nebula at 5.2 AU. Using hydrodynamic simulations we find that when the configuration is such that convergent migration occurs the planets can become locked in a first order commensurability for which the period ratio is (p+1)/p with p being an integer and migrate together maintaining it for many orbits. Relatively rapid convergent migration as tends to occur for disparate masses, results in commensurabilities with p larger than 2. However, in these cases the dynamics is found to have a stochastic character. When the convergent migration is slower, such as occurs in the equal mass case, lower p commensurabilities such as 3:2 are attained which show much greater stability. In one already known example of a system with nearly equal masses in the several Earth mass range (planets around pulsar PSR B1257+12) the two largest planets are intriguingly close to a 3:2 commensurability. A very similar behaviour is obtained when the systems are modeled using an N body code with simple prescriptions for the disc planet interaction. Using that, we found that an 8:7 resonance established in a hydrodynamic simulation run for 10-100 thousand orbits could be maintained for more than million orbits. Resonant capture leads to a rise in eccentricities that can be predicted using a simple analytic model constructed in this paper. We find that the system with the 8:7 commensurability is fully consistent with this prediction.