CONDITION FOR CAPTURE INTO FIRST-ORDER MEAN MOTION RESONANCES AND APPLICATION TO CONSTRAINTS ON THE ORIGIN OF RESONANT SYSTEMS

Abstract

We investigate the condition for capture into first-order mean motion\nresonances using numerical simulations with a wide range of various parameters.\nIn particular, we focus on deriving the critical migration timescale for\ncapture into the 2:1 resonance; additional numerical experiments for closely\nspaced resonances (e.g., 3:2) are also performed. We find that the critical\nmigration timescale is determined by the planet-to-stellar mass ratio, and its\ndependence exhibits power-law behavior with index -4/3. This dependence is also\nsupported by simple analytic arguments. We also find that the critical\nmigration timescale for systems with equal-mass bodies is shorter than that in\nthe restricted problem; for instance, for the 2:1 resonance between two\nequal-mass bodies, the critical timescale decreases by a factor of 10. In\naddition, using the obtained formula, the origin of observed systems that\ninclude first-order commensurabilities is constrained. Assuming that pairs of\nplanets originally form well separated from each other and then undergo\nconvergent migration and are captured in resonances, it is possible that a\nnumber of exoplanets experienced rapid orbital migration. For systems in\nclosely spaced resonances, the differential migration timescale between the\nresonant pair can be constrained well; it is further suggested that several\nexoplanets underwent migration that can equal or even exceed the type I\nmigration rate predicted by the linear theory. This implies that some of them\nmay have formed in situ. Future observations and the use of our model will\nallow us to statistically determine the typical migration speed in a\nprotoplanetary disk.\n