Conditions of Gravitational Instability in Protoplanetary Disks

Abstract

Abstract Gravitational instability is one of the considerable mechanisms to explain the formation of giant planets. We have studied the gravitational stability in protoplanetary disks around a protostar. The temperature and Toomre's $ Q$ value were calculated by assuming local equilibrium between viscous heating and radiative cooling (local thermal equilibrium). We assumed a constant $ \alpha$ viscosity and used a cooling function with realistic opacity. Then, we derived the critical surface density, $ \Sigma_{\rm {c}}$ , that is needed in order for a disk to become gravitationally unstable as a function of $ r$ . This critical surface density, $ \Sigma _{\rm c}$ , is strongly affected by the temperature dependence of the opacity. At a radius of $ r_{\rm c}$$ \sim$ 20 AU, where ices form, the value of $ \Sigma _{\rm c}$ changes discontinuously by one order of magnitude. This $ \Sigma _{\rm c}$ is determined only by a local thermal process and the criterion of gravitational instability. By comparing a given surface density profile with $ \Sigma _{\rm c}$ , one can discuss the gravitational instability of protoplanetary disks. As an example, we discuss the gravitational instability of two semianalytic models for protoplanetary disks. One is a steady state accretion disk, which is realized after viscous evolution. The other is a disk that has the same angular-momentum distribution as its parent cloud core, which corresponds to the disk that has just formed. As a result, it is found that the disk tends to become gravitationally unstable for $ r$$ \ge$$ r_{\rm c}$ because ices make the disk temperature low. In a region closer to the protostar than $ r_{\rm c}$ , it is difficult for a typical protoplanetary disk to fragment because of the high temperature and the large Coriolis force. Based on this result, we conclude that fragmentation near the central star is possible, but difficult.