The Evolution of Gravitationally Unstable Protoplanetary Disks: Fragmentation and Possible Giant Planet Formation

Abstract

We carry out a large set of very high resolution, three-dimensional, smoothed particle hydrodynamics 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☉, roughly consistent with the high end of the mass distribution inferred for disks around T Tauri stars. Minimum outer temperatures range from 30 to 100 K, as expected from studies of the early protosolar nebula and suggested by the modeling of the spectra of protoplanetary disks. The mass of the central star is also varied, although it is usually assumed to be equal to that of the Sun. Overall, the initial disks span minimum Q-parameters between 0.8 and 2, with most models having Q ~ 1.4. The disks are evolved assuming either a locally isothermal equation of state or an adiabatic equation of state with varying γ. Heating by (artificial) viscosity and shocks is included when the adiabatic equation of state is used. When condensations above a specific density 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 two to three protoplanets on fairly eccentric orbits (the mean eccentricity being around 0.2) after ~103 yr. Fragmentation is not strongly dependent on whether the disk starts from a marginally unstable state or gradually achieves it; we show that if the disk is allowed to grow in mass from a very light, very stable state over tens of orbital times, it still fragments at roughly the same mass and temperature as in the standard disk models. We show that the first stages of the instability, until the appearance of the overdensities, can be understood in terms of the maximum unstable Toomre wavelength and the local Jeans length. A high mass and force resolution are needed to correctly resolve both scales and follow the fragmentation process appropriately. Varying disk mass and temperature affects such physical scales and hence the typical masses of the protoplanets that form. Objects smaller than Saturn or a couple of times bigger than Jupiter can both be produced by fragmentation. Their final masses will then depend on the subsequent interactions and mergers with other clumps and on the accretion of disk material. The accretion rate depends on the disk thermodynamics and is negligible with adiabatic conditions. After ~103 yr the masses range from just below 1MJup to more than 7MJup, well in agreement with those of detected extrasolar planets.