The Gas-Drag Effect on the Orbital Instability of a Protoplanet System

Abstract

We have investigated numerically the orbital instability of a protoplanet system while taking account of the gas-drag force due to the solar nebula. In the present work, we considered an equally spaced five-protoplanet (with the same mass of $1 \times 10^{-7} {{{M}_{\odot}}}$) system, in which their initial orbits are coplanar and circular, and assumed that the gas-drag force is proportional to the square of the relative velocity between the gas and a protoplanet. We first re-examined and confirmed that, under a gas-free condition, $\log_{10} T_{\mathrm{inst}}$ can be approximately written as a linear function of the initial orbital separation distance, $\Delta \tilde{a}_{0}$, where $T_{\mathrm{inst}}$ is the time of the orbital instability (i.e., the time of the first orbital crossing between any two protoplanets). Next, we investigated the instability time under the gas-drag effect, $T_{\mathrm{inst}}^{\mathrm{gas}}$, and found that $T_{\mathrm{inst}}^{\mathrm{gas}}$ suddenly becomes large compared with $T_{\mathrm{inst}}$, when ${\Delta \tilde{a}}_{0}$ is larger than a certain critical separation distance, ${({\Delta \tilde{a}}_{0})}_{\mathrm{crit}}$. Furthermore, we showed that ${({\Delta \tilde{a}}_{0})}_{\mathrm{crit}}$ can be described semi-analytically as a function of the gaseous density. From a function extrapolated with a density in the minimum mass nebula model, we estimated ${({\Delta \tilde{a}}_{0})}_{\mathrm{crit}}$ in the nebula as being about 10 Hill radius at 1 AU.