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 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 x 10(-7) M-circle dot) 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, log(10)T(inst) can be approximately written as a linear function of the initial orbital separation distance, Delta(a) over tilde (0), where T-inst is the time of the orbital instability (i.e., the time of the first orbital crossing between any two protoplanets). Nest, we investigated the instability time under the gas-drag effect, T-inst(gas) and found that T-inst(gas) suddenly becomes large compared with T-inst, when Delta(a) over tilde (0) is larger than a certain critical separation distance, (Delta(a) over tilde (0))(crit). Furthermore, we showed that (Delta(a) over tilde (0))(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(a) over tilde (0))(crit) in the nebula as being about 10 Hill radius at I AU.