An SPH simulation for cooling and self-gravitating protoplanetary disks

We investigate the effects of the cooling function in the formation of clumps of protoplanetary disks using two-dimensional smoothed particle hydrodynamic simulations. We use a simple prescription for the cooling rate of the flow, du/dt = -u/tau(cool), where u and tau(cool) are the internal energy and cooling timescale, respectively. We assume the ratio of local cooling to dynamical timescale, Omega tau(cool) = beta, to be a constant and also a function of the local temperature. We found that for the constant beta and gamma = 5/3, fragmentation occurs only for beta less than or similar to 7. However, in the case of beta having temperature dependence and gamma = 5/3, fragmentation can also occur for larger values of beta. By increasing the temperature dependence of the cooling timescale, the mass accretion rate decreases, the population of clumps/fragments increases, and the clumps/fragments can also form in the smaller radii. Moreover, we found that the clumps can form even in a low mass accretion rate, less than or similar to 10(-7) M-circle dot yr(-1), in the case of temperature-dependent beta. However, clumps form with a larger mass accretion rate, > 10(-7) M-circle dot yr(-1), in the case of constant beta.