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

Abstract

We investigate orbital resonances expected to arise when a system of two planets, with masses in the range 1–4 M⊕, 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. For the above mass range, the planets undergo type I migration. 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. Slow convergent migration results in commensurabilities with small p such as 1 or 2. Instead, when the convergent migration is relatively rapid as tends to occur for disparate masses, higher p commensurabilities are realized such as 4:3, 5:4, 7:6 and 8:7. However, in these cases the dynamics is found to have a stochastic character with some commensurabilities showing long-term instability with the consequence that several can be visited during the course of a simulation. Furthermore, the successful attainment of commensurabilities is also a sensitive function of initial conditions. When the convergent migration is slower, such as occurs in the equal-mass case, lower p commensurabilities such as 3:2 are obtained, which show much greater stability. Resonant capture leads to a rise in eccentricities that can be predicted using a simple analytic model that assumes the resonance is isolated, constructed in this paper. We find that, once the commensurability has been established, the system with an 8:7 commensurability is fully consistent with this prediction. We find that very similar behaviour is found when the systems are modelled using an N-body code with simple prescriptions for the disc–planet interaction. Comparisons with the hydrodynamic simulations indicate reasonably good agreement with predictions for these prescriptions obtained using the existing semi-analytic theories of type I migration. We have run our hydrodynamic simulations for up to 103–104 orbits of the inner planet. Longer times could only be followed in the simpler N-body approach. Using that, we found that, on the one hand, an 8:7 resonance established in a hydrodynamic simulation could be maintained for more than 105 orbits. On the other hand, other similar cases show instability leading to another resonance and ultimately a close scattering. There is already one known example of a system with nearly equal masses in the range of several Earth masses, namely the two pulsar planets in PSR B1257+12, which are intriguingly, in view of the results obtained here, close to a 3:2 commensurability. This will be considered in a future publication. Future detection of other systems with masses in the Earth mass range that display orbital commensurabilities will give useful information on the role and nature of orbital migration in planet formation.