Gravitationally unstable protoplanetary disks

A simultaneous formation of a star and planets around it is considered. A gravitational Jeans-type instability of small gravity perturbations (e.g., those produced by a spontaneous disturbance or, in rare cases, a companion system) in a protoplanetary disk of gas and dust is studied using a hydrodynamical formulation. The processes of angular momentum transport, heating and cooling processes in the disk and—in a speculative manner—the formation of turbulence are discussed. An equation is derived to describe the collective torque exerted by the growing nonaxisymmetric gravity perturbations on the disk. It is shown that the torque effectively redistributes both the mass and the angular momentum of a spatially inhomogeneous disk, giving the disk a source of internal viscosity. During the early evolution of such Jeans-unstable systems mass is transported both inward to the growing star and outward while angular momentum is transferred outward. The outward transfer of orbital momentum allows the central parts of disks to contract without breaking up, and the remnant disk of gas is the reservoir for forming planets. Equations are also derived to describe the turbulent heating of the disk that results from the buildup of nonresonant Jeans instability. The considerable growth in the amplitude of the perturbation by a time of instability may act as a source of local turbulence. The turbulence that may arise as a result of instability is related to stochastic motions of gas elements. It is shown that the heating and the angular momentum redistribution bring the disk toward stability—unless some cooling mechanism is available, e.g., by radiation—against all perturbations, including the most unstable nonaxisymmetric ones. As cooling process always exists in the actual systems, the Jeans instability can be considered to be a long-term ∼ 10 6 yr generating mechanism for fresh, unstable density waves, thereby leading to recurrent short-lived ∼ 10 4 yr arc-and-lump or spiral patterns in the protostellar and protoplanetary disks.