The gas-drag effect on the orbital instability of a protoplanet system

We have investigated numerically the orbital instability of a protoplanet system while taking account of the gasdrag force due to the solar nebula. In the present work, we considered an equally spaced five-protoplanet (with the same mass of 1 x 10^-7 M_⊙ ) 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 reexamined and confirmed that, under a gas-free condition, log₁₀ T_inst can be approximately written as a linear function of the initial orbital separation distance, Δã₀, where T_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^gas_inst and found that T^gas_inst suddenly becomes large compared with T_inst, when Δã₀ is larger than a certain critical separation distance, (Δã₀)_crit. Furthermore, we showed that (Δã₀)_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 (Δã₀)_crit in the nebula as being about 10 Hill radius at 1 AU.