  The resonant perturbations from planets are able to halt the drag-induced
migration, and capture the inwardly drifting planetesimals into mean motion
resonances. The equilibrium eccentricity of planetesimals in resonances, and
the minimum size of planetesimal that can trigger resonance trapping, have been
analyzed and formulated. However, the analytical works based on the assumption
that the disk is axisymmetric, which is violated by the asymmetric structures
developed by planets. We perform long-term 2D hydrodynamic simulations to study
the dynamics of planetesimals in the $j:(j+1)$ first-order exterior resonances,
and reexamine the theoretical expressions. We find the expression of
equilibrium eccentricity underestimates the values for resonances with $j < 5$,
in particular the 1:2 resonance that the underestimation can be $30 - 40\%$.
Within the parameter space we explored, we find the equilibrium eccentricity
and the minimum size are reduced in an asymmetric disk. The amount of
discrepancy in eccentricity depends on the degree of asymmetric structures. For
cases of Earth-sized planets, where the disk is less disturbed, the
planetesimal's eccentricity can reach to the values predicted by our modified
expression. For gaseous planets, however, the eccentricity can be $0.01 - 0.02$
smaller in value. We find the minimum size is 10 times smaller, and the factor
seems to be independent of the planet's mass. The influences of asymmetric
profiles on the eccentricity and the minimum size could affect the outcome of
collisions between resonant and nonresonant planetesimals, and the amount of
planetesimals migrated into the planet's feeding zone.
