This paper focuses on two-planet systems in a first-order $(q+1):q$ mean motion resonance and undergoing type-I migration in a disc. We present a detailed analysis of the resonance valid for any value of $q$. Expressions for the equilibrium eccentricities, mean motions and departure from exact resonance are derived in the case of smooth convergent migration. We show that this departure, not assumed to be small, is such that period ratio normally exceeds, but can also be less than, $ (q+1)/q.$ Departure from exact resonance as a function of time for systems starting in resonance and undergoing divergent migration is also calculated. We discuss observed systems in which two low mass planets are close to a first-order resonance. We argue that the data are consistent with only a small fraction of the systems having been captured in resonance. Furthermore, when capture does happen, it is not in general during smooth convergent migration through the disc but after the planets reach the disc inner parts. We show that although resonances may be disrupted when the inner planet enters a central cavity, this alone cannot explain the spread of observed separations. Disruption is found to result in either the system moving interior to the resonance by a few percent, or attaining another resonance. We postulate two populations of low mass planets: a small one for which extensive smooth migration has occurred, and a larger one that formed approximately in-situ with very limited migration.
Download