The evolution of gravitationally unstable protoplanetary disks: fragmentation and possible giant planet formation

We carry out a large set of very high resolution, three dimensional smoothed particle hydrodynamics (SPH) simulations describing the evolution of gravitationally unstable gaseous protoplanetary disks. We consider a broad range of initial disk parameters. Disk masses out to 20 AU range from 0.075 to 0.125 $M_{\odot}$, roughly consistent with the high-end of the mass distribution inferred for disks around T Tauri stars. The initial disks span minimum $Q$ parameters between 0.8 and 2, with most models being around $\sim 1.4$. The disks are evolved assuming either a locally isothermal equation of state or an adiabatic equation of state with varying $\gamma$. When overdensities above a specific threshold appear as a result of gravitational instability in a locally isothermal calculation, the equation of state is switched to adiabatic to account for the increased optical depth. We show that when a disk has a minimum $Q$ parameter less than 1.4 strong trailing spiral instabilities, typically three or four armed modes, form and grow until fragmentation occurs along the arms after about 5 mean disk orbital times. The resulting clumps contract quickly to densities several orders of magnitude higher than the initial disk density, and the densest of them survive even under adiabatic conditions. These clumps are stable to tidal disruption and merge quickly, leaving 2-3 protoplanets on fairly eccentric orbits (the mean eccentricity being around 0.2) with masses between 0.7 and more than $7 M_{Jup}$, well in agreement with those of detected extrasolar planets.after $\sim 10^3$ years. Fragmentation is not strongly dependent on whether the disk starts from a marginally unstable state or gradually achieves it, as shown by a test where the disk gradually achieves the critical $Q$ by growing in mass.