Nonlinear outcome of gravitational instability in an irradiated protoplanetary disc

Using local three dimensional radiation hydrodynamics simulations, the nonlinear outcome of gravitational instability in an irradiated protoplanetary disc is investigated in a parameter space of the surface density $\Sigma$ and the radius $r$. Starting from laminar flow, axisymmetric self-gravitating density waves grow first. Their self-gravitating degree becomes larger when $\Sigma$ is larger or the cooling time is shorter at larger radii. The density waves eventually collapse owing to non-axisymmetric instability, which results in either fragmentation or gravito-turbulence after a transient phase. The boundaries between the two are found at $r \sim 75$ AU as well as at the $\Sigma$ that corresponds to the initial Toomre's parameter of $\sim 0.2$. The former boundary corresponds to the radius where the cooling time becomes short, approximating unity. Even when gravito-turbulence is established around the boundary radius, such a short cooling time inevitably makes the fluctuation of $\Sigma$ large enough to trigger fragmentation. On the other hand, when $\Sigma$ is beyond the latter boundary (i.e. the initial Toomre's parameter is less than $\sim 0.2$), the initial laminar flow is so unstable against self-gravity that it evolves into fragmentation regardless of the radius or, equivalently, the cooling time. Runaway collapse follows fragmentation when the mass concentration at the centre of a bound object is high enough that the temperature exceeds the H$_2$ dissociation temperature.