Dynamics of Circumstellar Disks

We present a series of 2-dimensional hydrodynamic simulations of massive disks around protostars. We simulate the same physical problem using both a ‘Piecewise Parabolic Method’ (PPM) code and a ‘Smoothed Particle Hydrodynamic’ (SPH) code, and analyze their differences. The disks studied here range in mass from 0.05M∗ to 1.0M∗ and in initial minimum Toomre Q value from 1.1 to 3.0. We adopt simple power laws for the initial density and temperature in the disk with an isothermal (γ = 1) equation of state. The disks are locally isothermal. We allow the central star to move freely in response to growing perturbations. The simulations using each code are compared to discover differences due to error in the methods used. For this problem, the strengths of the codes overlap only in a limited fashion, but similarities exist in their predictions, including spiral arm pattern speeds and morphological features. Our results represent limiting cases (i.e. systems evolved isothermally) rather than true physical systems. Disks become active from the inner regions outward. From the earliest times, their evolution is a strongly dynamic process rather than a smooth progression toward eventual nonlinear behavior. Processes that occur in both the extreme inner and outer radial regions affect the growth of instabilities over the entire disk. Effects important for the global morphology of the system can originate at quite small distances from the star. We calculate approximate growth rates for the spiral patterns; the one-armed (m = 1) spiral arm is not the fastest growing pattern of most disks. Nonetheless, it plays a significant role due to factors which can excite it more quickly than other patterns. A marked change in the character of spiral structure occurs with varying disk mass. Low mass disks form filimentary spiral structures with many arms while high mass disks form grand design spiral structures with few arms. In our SPH simulations, disks with initial minimum Q = 1.5 or lower break up into proto-binary or proto-planetary clumps. However, these simulations cannot follow the physics important for the flow and must be terminated before the system has completely evolved. At their termination, PPM simulations with similar initial conditions show uneven mass distributions within spiral arms, suggesting that clumping behavior might result if they were carried further. Simulations of tori, for which SPH and PPM are directly comparable, do show clumping in both codes. Concern that the point-like nature of SPH exaggerates clumping, that our representation of the gravitational potential in PPM is too coarse, and that our physics assumptions are too simple, suggest caution in interpretation of the clumping in both the disk and torus simulations.