Effect of Stellar Encounters on Comet Cloud Formation

We have investigated the effect of stellar encounters on the formation and disruption of the Oort cloud using the classical impulse approximation. We calculate the evolution of a planetesimal disk into a spherical Oort cloud due to the perturbation from passing stars for 10 Gyr. We obtain the empirical fits of the $e$-folding time for the number of Oort cloud comets using the standard exponential and Kohlrausch formulae as functions of the stellar parameters and the initial semimajor axes of planetesimals. The $e$-folding time and the evolution timescales of the orbital elements are also analytically derived. In some calculations, the effect of the Galactic tide is additionally considered. We also show the radial variations of the $e$-folding times to the Oort cloud. From these timescales, we show that if the initial planetesimal disk has the semimajor axes distribution ${\rm d}n/{\rm d}a\propto a^{-2}$, which is produced by planetary scattering (Higuchi et al. 2006), the $e$-folding time for planetesimals in the Oort cloud is $\sim$10 Gyr at any heliocentric distance $r$. This uniform $e$-folding time over the Oort cloud means that the supply of comets from the inner Oort cloud to the outer Oort cloud is sufficiently effective to keep the comet distribution as ${\rm d}n/{\rm d}r\propto r^{-2}$. We also show that the final distribution of the semimajor axes in the Oort cloud is approximately proportional to $a^{-2}$ for any initial distribution.