A general model of resonance capture in planetary systems: first- and second-order resonances

Mean motion resonances are a common feature of both our own Solar system and of extrasolar planetary systems. Bodies can be trapped in resonance when their orbital semimajor axes change, for instance when they migrate through a protoplanetary disc. We use a Hamiltonian model to thoroughly investigate the capture behaviour for first- and second-order resonances. Using this method, all resonances of the same order can be described by one equation, with applications to specific resonances by appropriate scaling. We focus on the limit where one body is a massless test particle and the other a massive planet. We quantify how the probability of capture into a resonance depends on the relative migration rate of the planet and particle, and the particle's eccentricity. Resonant capture fails for high migration rates, and has decreasing probability for higher eccentricities, although for certain migration rates, capture probability peaks at a finite eccentricity. More massive planets can capture particles at higher eccentricities and migration rates. We also calculate libration amplitudes and the offset of the libration centres for captured particles, and the change in eccentricity if capture does not occur. Libration amplitudes are higher for larger initial eccentricity. The model allows for a complete description of a particle's behaviour as it successively encounters several resonances. Data files containing the integration grid output will be available online. We discuss implications for several scenarios: (i) Planet migration through gas discs trapping other planets or planetesimals in resonances: we find that, with classical prescriptions for Type I migration, capture into second-order resonances is not possible, and lower mass planets or those further from the star should trap objects in first-order resonances closer to the planet than higher mass planets or those closer to the star. For fast enough migration, a planet can trap no objects into its resonances. We suggest that the present libration amplitude of planets may be a signature of their eccentricities at the epoch of capture, with high libration amplitudes suggesting high eccentricity (e.g. HD 128311). (ii) Planet migration through a debris disc: we find the resulting dynamical structure depends strongly both on migration rate and on planetesimal eccentricity. Translating this to spatial structure, we expect clumpiness to decrease from a significant level at e less than or similar to 0.01 to non-existent at e greater than or similar to 0.1. (iii) Dust migration through Poynting-Robertson (PR) drag: we predict that Mars should have its own resonant ring of particles captured from the zodiacal cloud, and that the capture probability is less than or similar to 25 per cent that of the Earth, consistent with published upper limits for its resonant ring. To summarize, the Hamiltonian model will allow quick interpretation of the resonant properties of extrasolar planets and Kuiper Belt Objects, and will allow synthetic images of debris disc structures to be quickly generated, which will be useful for predicting and interpreting disc images made with Atacama Large Millimeter Array (ALMA), Darwin/Terrestrial Planet Finder (TPF) or similar missions.