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 T-init and opacity kappa, and we determine the critical densities with which these conditions are satisfied. For plausible values of T-init and kappa,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 " tau approximate to 1 " never terminates isothermality, but nonisothermal evolutions begin either earlier or later depending on the initial temperature and opacity. This result contrasts with the conventionaI idea that opaqueness breaks isothermality. On the basis of the critical density discussed above, the minimum Jeans mass for fragmentation, M-F, is reconsidered. In contrast to the results by previous authors that M-F is insensitive to T-init and kappa, we find that M-F can be substantially larger than the typical value of similar to 10(-2) M. depending on T-init and kappa. In particular, M,increases with decreasing metallicity, M-F proportional to kappa(-1), for low-metal clouds. A cloud with kappa = 10(-4) cm(2) g(-1) and T-init = 10 K yields M-F = 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.