Does "tau approximate to 1" terminate the isothermal evolution of collapsing clouds?

We examine when gravitationally collapsing clouds terminate their isotherma l evolution. According to our previous work, the condition with which isoth ermality is broken down is classified into three cases, i.e., when (1) the compressional heating rate overtakes the thermal cooling rate, (2) the opti cal depth for thermal radiation reaches unity, or (3) the compressional hea ting rate becomes comparable with the energy transport rate because of radi ative diffusion. In the present paper this classification is extended to mo re general values of the initial cloud temperature T-init and opacity kappa , and we determine the critical densities with which these conditions are s atisfied. For plausible values of T-init and kappa,we find that the isother mal evolution ceases when case 1 or 3 is satisfied, and case 2 has no signi ficance. 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 con trasts with the conventionaI idea that opaqueness breaks isothermality. On the basis of the critical density discussed above, the minimum Jeans mass f or fragmentation, M-F, is reconsidered. In contrast to the results by previ ous authors that M-F is insensitive to T-init and kappa, we find that M-F c an be substantially larger than the typical value of similar to 10(-2) M. d epending on T-init and kappa. In particular, M,increases with decreasing me tallicity, M-F proportional to kappa(-1), for low-metal clouds. A cloud wit h 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.