  We study the dynamics of a vertically thin, dispersion-dominated disk of
planetesimals with eccentricities $e$ and inclinations $i$ (normalized in Hill
units) satisfying $e >> 1$, $i << e^{-2} << 1$. This situation may be typical
for e.g. a population of protoplanetary cores in the end of the oligarchic
phase of planet formation. In this regime of orbital parameters planetesimal
scattering has an anisotropic character and strongly differs from scattering in
thick ($i ~ e$) disks. We derive analytical expressions for the planetesimal
scattering coefficients and compare them with numerical calculations. We find
significant discrepancies in the inclination scattering coefficients obtained
by the two approaches and ascribe this difference to the effects not accounted
for in the analytical calculation: multiple scattering events (temporary
captures, which may be relevant for the production of distant planetary
satellites outside the Hill sphere) and distant interaction of planetesimals
prior to their close encounter. Our calculations show that the inclination of a
thin, dispersion-dominated planetesimal disk grows exponentially on a very
short time scale implying that (1) such disks must be very short-lived and (2)
planetesimal accretion in this dynamical phase is insignificant. Our results
are also applicable to the dynamics of shear-dominated disks switching to the
dispersion-dominated regime.
