Does “τ≈1” Terminate the Isothermal Evolution of Collapsing Clouds?

We examine when gravitationally collapsing clouds terminate their isothermal evolution. According to our previous work, the condition with which isothermality is broken down is classified into three cases, i.e., when (1) the compressional heating rate overtakes the thermal cooling rate, (2) the optical depth for thermal radiation reaches unity, or (3) the compressional heating rate becomes comparable with the energy transport rate because of radiative diffusion. In the present paper this classification is extended to more general values of the initial cloud temperature Tinit and opacity κ, and we determine the critical densities with which these conditions are satisfied. For plausible values of Tinit and κ, we find that the isothermal evolution ceases when case 1 or 3 is satisfied, and case 2 has no significance. We emphasize that the condition of "τ≈1" never terminates isothermality, but nonisothermal evolutions begin either earlier or later depending on the initial temperature and opacity. This result contrasts with the conventional idea that opaqueness breaks isothermality. On the basis of the critical density discussed above, the minimum Jeans mass for fragmentation, MF, is reconsidered. In contrast to the results by previous authors that MF is insensitive to Tinit and κ, we find that MF can be substantially larger than the typical value of ~10-2 M☉ depending on Tinit and κ. In particular, MF increases with decreasing metallicity, MF∝κ−1, for low-metal clouds. A cloud with κ=10−4 cm2 g-1 and Tinit=10 K yields MF=3.7 M☉. Finally, our critical densities would be helpful for hydrodynamic simulations that are intended to simply handle the hardening of the equation of state.