Orbital stability of a protoplanet system under a drag force proportional to the random velocity

We investigated the effects of a tidal interaction with a gas disk and the dynamical friction with a planetesimal disk on the orbital instability of a protoplanet system. Both effects are expressed as the drag force, which is proportional to the random velocity of a protoplanet. We calculated numerically the orbits of 5 protoplanets with the same separation distance under the drag-force effect and examined the orbital instability time under the drag force, T_inst^df. We found that T_inst^df can become much larger than the instability time under the drag-free condition, and that the onset of the orbital instability is prevented when the separation distance exceeds a critical value. We obtained a relation between the critical separation distance and the surface density of the gas or planetesimal disk. By applying this relation, we found that, for the formation of terrestrial planets from a protoplanet system with a typical orbital separation (i.e., ∼ 10 Hill radii), the surface density of the nebular gas must be reduced to about one-thousandth of that in the minimum-mass nebula model. Terrestrial planets would be formed after such a depletion of the solar nebula.