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