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Dynamical evolution of thin dispersion-dominated planetesimal disks

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.