### Summary

#### Study of the fragmentation of self-gravitating disks

See Brucy & Hennebelle 2021 for more details.

#### Abstract:

Self-gravitating disks are believed to play an important role in astrophysics in particular regarding the star and planet formation process. In this context, disks subject to an idealized cooling process, characterized by a cooling timescale $\beta$ expressed in unit of orbital timescale, have been extensively studied. We take advantage of the Riemann solver and the 3D Godunov scheme implemented in the code Ramses to perform high resolution simulations, complementing previous studies that have used Smoothed Particle Hydrodynamics (SPH) or 2D grid codes.

#### Description of the simulations:

We simulate a disk of gas undergoing purely hydrodynamics forces, its own gravity and the $\beta$-cooling. The simulation is ran with the 3D-grid code Ramses (Teyssier 2002) which uses a Godunov scheme. The flux between each cell is computed with the HLLC Riemann solver. The gravity potential is updated at each timestep with a Poisson solver, and a source term is added to the energy equation to implement the $\beta$-cooling.

The $\beta$-cooling consists in removing internal energy from the gas with a cooling time: $t_\text{cool} = \beta \Omega^{-1}$ with $\Omega$ the rotation frequency.

We use the same initial conditions as in Meru & Bate (2012) to allow comparison. The specific disk setup for Ramses was inspired by Hennebelle et al. (2017). The disk is initially close to equilibrium with an initial column density profile $\Sigma \propto r^{-1}$ and a temperature profile $T \propto r^{-1/2}$ where $r$ is the cylindrical radius. The disk has a radius $r_d = 0.25$ (code units), after which the density is divided by 100. The density and temperature at the disk radius $r_d$ are chosen so that the mass of the disk $M_d = 0.1 M_\star$, where $M_\star$ is the mass of the central object, and the initial value of the Toomre parameter at the disk radius is $Q_{0,d} = 2$. The adiabatic index of the gas is $\gamma = 5/3$.

The simulation is run within a cube of size $L=2$. Although the problem has a cylindrical symmetry, we use Cartesian coordinates. This prevents having a singularity at the centre of the box. One caveat is the poor resolution on the centre of the cube but this is mitigated by the use of the adaptive mesh refinement (AMR). Another caveat is that having a cubic box may introduce spurious reflection at the border of the simulation. To avoid this, we maintain a dead zone over a radius of $0.875$ (in code units) where all variables are replaced by their initial value at each timestep. This method has been used in Hennebelle et al. (2017) and has proven to be efficient.

The simulations presented here were run for several values of $\beta$ and several resolutions. To reduce the computation time, we use the Ramses's Adaptative Mesh Refinement (AMR). Only the parts of the simulation which are prone to form fragments are simulated with full resolution. Each cell is refined until the Jeans's length is covered by at least 20 cells or it reaches the maximum level of refinement $l_{\max}$. The level of refinement of a cell is the number of times the simulation box must be divided in eight equal part to get the cell. Thus, the resolution of a simulation is given by the value of $l_{\max}$. A first set of simulations with $l_{\max} = 11$ to $l_{\max} = 12$ are run until about 5 Outer Rotation Periods (ORP), that is that the gas at the border of the disk had 5 orbits around the star. A second set of simulations, labelled tic, for Turbulent Initial Condition, were run from relaxed initial conditions for $l_{\max} = 12$ and $l_{\max} = 13$. More precisely, they were restarted from a simulation at $\beta = 20$ and $l_{\max} = 12$ for which the whole disk reached a gravito-turbulent state (after 2 ORPs). According to Paardekooper et al. (2011) and Clarke et al. (2007), departing from such turbulent condition should reduce spurious fragmentation.

### Available simulations

Group byName | lmax | $\beta$ | tic |
---|---|---|---|

11 | 4 | ||

11 | 5 | ||

11 | 6 | ||

11 | 7 | ||

11 | 8 | ||

12 | 2 | ||

12 | 3 | ||

12 | 4 | ||

12 | 5 | ||

12 | 6 | ||

12 | 7 | ||

12 | 8 | ||

12 | 9 | ||

12 | 10 | ||

12 | 11 | ||

12 | 12 | ||

12 | 14 | ||

12 | 16 | ||

12 | 18 | ||

12 | 6 | ||

12 | 7 | ||

12 | 8 | ||

12 | 9 | ||

13 | 10 | ||

13 | 12 | ||

13 | 4 | ||

13 | 6 | ||

13 | 7 | ||

13 | 8 | ||

13 | 9 | ||

13 | 10 | ||

13 | 12 | ||

13 | 14 | ||

13 | 16 |