Global Simulations of Gravitational Instability in Protostellar Disks with Full Radiation Transport. I. Stochastic Fragmentation with Optical-depth-dependent Rate and Universal Fragment Mass

Fragmentation in a gravitationally unstable accretion disk can be an important pathway for forming stellar/planetary companions. To characterize quantitatively the condition for and outcome of fragmentation under realistic thermodynamics, we perform global 3D simulations of gravitationally unstable disks at various cooling rates and cooling types, including the first global simulations of gravitational instability that employ full radiation transport. We find that fragmentation is a stochastic process, with the fragment generation rate per disk area $p_{\rm frag}$ showing an exponential dependence on the parameter $\beta\equiv\Omega_K t_{cool}$, where $\Omega_K$ is the Keplerian rotation frequency and $t_{cool}$ is the average cooling timescale. Compared to a prescribed constant $\beta$, radiative cooling in the optically thin/thick regime makes $p_{\rm frag}$ decrease slower/faster in $\beta$; the critical $\beta$ corresponding to $\sim 1$ fragment per orbit is $\approx$3, 5, 2 for constant $\beta$, optically thin, and optically thick cooling, respectively. The distribution function of the initial fragment mass is remarkably insensitive to disk thermodynamics. Regardless of cooling rate and optical depth, the typical initial fragment mass is $m_{frag} \approx 40 M_{tot}h^3$, with $M_{tot}$ being the total (star+disk) mass and $h=H/R$ being the disk aspect ratio. Applying this result to typical Class 0/I protostellar disks, we find $m_{frag}\sim 20 M_J$, suggesting that fragmentation more likely forms brown dwarfs. Given the finite width of the $m_{frag}$ distribution, forming massive planets is also possible.