Orbital Stability of Protoplanetary Systems in Nebular Gas and Implications for Terrestrial Planet Formation

Abstract

The oligarchic growth model of planetary accretion predicts the formation of protoplanets of similar sizes, and the final stage of terrestrial planet formation involves long-term orbital instability and mutual collision of protoplanets through their gravitational interaction. However, the mass of the protoplanets formed by oligarchic growth depends on the initial surface density of the protoplanetary disk, as well as the distance from the central star. In order to better understand the final stage of terrestrial planet formation in disks with various profiles, we perform orbital integration for systems of five protoplanets and examine the dependence of the orbital instability timescale of the systems on the mass of the protoplanets, Mp, both with and without nebular gas. Previous studies have shown that the timescale for orbital instability in the absence of nebular gas increases exponentially with the initial semimajor axis difference between the protoplanets. We find that the exponential constant is independent of the mass of the protoplanets, as long as the initial separation is measured in units of ~M, which is very close to the previously derived mass dependence of the critical separation for the onset of chaotic behavior in the restricted circular three-body problem. In the presence of nebular gas, on the other hand, protoplanetary systems are stabilized by gas drag and experience no orbital instability when the initial semimajor axis difference is larger than a certain critical value. Using these results, we discuss the orbital stability of protoplanetary systems in disks with various surface densities.