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 _ { \sun } / 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 .