Fragmentation of protoplanetary disks around M-dwarfs

We investigate the conditions required for planet formation via gravitational instability (GI) and protoplanetary disk (PPD) fragmentation around M-dwarfs. Using a suite of 64 SPH simulations with $10^6$ particles, the parameter space of disk mass, temperature, and radius is explored, bracketing reasonable values based on theory and observation. Our model consists of an equilibrium, gaseous, and locally isothermal disk orbiting a central star of mass $M_*=M_{sol}/3$. Disks with a minimum Toomre $Q$ of $Q_{min} \lesssim 0.9$ will fragment and form gravitationally bound clumps. Some previous literature has found $Q_{min} < 1.3-1.5$ to be sufficient for fragmentation. Increasing disk height tends to stabilize disks, and when incorporated into $Q$ as $Q_{eff}\propto Q(H/R)^\alpha$ for $\alpha=0.18$ is sufficient to predict fragmentation. Some discrepancies in the literature regarding $Q_{crit}$ may be due to different methods of generating initial conditions (ICs). A series of 15 simulations demonstrates that perturbing ICs slightly out of equilibrium can cause disks to fragment for higher $Q$. Our method for generating ICs is presented in detail. We argue that GI likely plays a role in PPDs around M-dwarfs and that disk fragmentation at large radii is a plausible outcome for these disks.