  A planetary system can undergo multiple episodes of intense dynamical
activities throughout its life, resulting in the production of star-grazing
planetesimals (or exocomets) and pollution of the host star. Such activity is
especially pronounced when giant planets interact with other small bodies
during the system's evolution. However, due to the chaotic nature of the
dynamics, it is difficult to determine the properties of the perturbing
planet(s) from the observed planetesimal-disruption activities. In this study,
we examine the outcomes of planetesimal-planet scatterings in a general
setting. We focus on one-planet systems, and determine the likelihood and
timescale of planetesimal disruption by the host star as a function of the
planet properties. We obtain a new analytical expression for the minimum
distance a scattering body can reach, extending previous results by considering
finite planet eccentricity and non-zero planetesimal mass. Through N-body
simulations, we derive the distribution of minimum distances and the likelihood
and timescales of three possible outcomes of planetesimal-planet scatterings:
collision with the planet, ejection, and disruption by the star. For
planetesimals with negligible mass, we identify four defining dimensionless
parameters (the planet eccentricity, planet-to-star mass ratio, planet radius
to semi-major axis ratio, and the stellar disruption radius to planet
semi-major axis ratio) that enable us to scale the problem and generalize our
findings to a wide range of orbital configurations. Using these results, we
explore three applications: falling evaporating bodies in the beta Pictoris
system, white dwarf pollution due to planetesimal disruption and planet
engulfment by main-sequence stars.
