Abstract 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/τcool, where u and τcool are the internal energy and cooling timescale, respectively. We assume the ratio of local cooling to dynamical timescale, Ωτcool = β, to be a constant and also a function of the local temperature. We found that for the constant β and γ = 5/3, fragmentation occurs only for β ≲ 7. However, in the case of β having temperature dependence and γ = 5/3, fragmentation can also occur for larger values of β. 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, ≲ 10−7M⊙ yr−1, in the case of temperature-dependent β. However, clumps form with a larger mass accretion rate, > 10−7M⊙ yr−1 , in the case of constant β.

Keywords accretion, accretion disks — planetary systems: protoplanetary disks — planetary systems: formation

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