Star formation in a quasar disk

Using a version of the ZEUS code, we carry out two-dimensional simulations of self-gravitating shearing sheets, with application to QSO accretion disks at a few thousand Schwarzschild radii, corresponding to a few hundredths of a parsec for a 100-million-solar-mass black hole. Radiation pressure and optically thick radiative cooling are implemented via vertical averages. We determine dimensionless versions of the maximum surface density, accretion rate, and effective viscosity that can be sustained by density-wave turbulence without fragmentation. Where fragments do form, we study the final masses that result. The maximum Shakura-Sunyaev viscosity parameter is approximately 0.4. Fragmentation occurs when the cooling time is less than about twice the shearing time, as found by Gammie and others, but can also occur at very long cooling times in sheets that are strongly radiation-pressure dominated. For accretion at the Eddington rate onto a 10⁸ solar-mass black hole, fragmentation occurs beyond 4000 Schwarzschild radii, r _S. Near this radius, initial fragment masses are several hundred suns, consistent with estimates from linear stability; final masses after merging increase with the size of the sheet, reaching several thousand suns in our largest simulations. With increasing black hole mass at a fixed Eddington ratio, self-gravity prevails at smaller multiples of r _S, where radiation pressure is more important and the cooling time is longer compared to the dynamical time; nevertheless, fragmentation can occur and produces larger initial fragment masses. Because the internal thermal and gravitational energies of these massive, radiation-pressure-dominated fragments nearly cancel, small errors in energy conservation can cause spurious results such as spontaneous dissolution of isolated bodies, unless special care is taken. This is likely to be a challenge for all Eulerian codes in self-gravitating regimes where radiation pressure dominates.