Isothermal Fragmentation: Is there a low-mass cut-off?

The evolution of self-gravitating clouds of isothermal gas forms the basis of many star formation theories. Therefore it is important to know under what conditions such a cloud will undergo monolithic collapse into a single, massive object, or will fragment into a spectrum of smaller ones. And if it fragments, do initial conditions (e.g. Jeans mass, sonic mass) influence the mass function of the fragments, as predicted by many theories of star formation? In this paper we show that the relevant parameter separating monolithic collapse from fragmentation is not the Mach number of the initial turbulence (as suspected by many), but the infall Mach number M_infall∼ √{G M/(R c_s^2)}, equivalent to the number of Jeans masses in the initial cloud N_J. We also show that fragmenting clouds produce a power-law mass function with slopes close to the expected -2 (i.e. equal mass in all logarithmic mass intervals). However, the low-mass cut-off of this mass function is entirely numerical; the initial properties of the cloud have no effect on it. In other words, if M_infall≫ 1, fragmentation proceeds without limit to masses much smaller than the initial Jeans mass.