The opacity limit
The opacity limit is an important concept in star formation: isothermal collapse cannot proceed without limit, because eventually cooling radiation is trapped and the temperature rises quasi-adiabatically, setting a minimum Jeans mass $M_{\rm J}^{\rm min}$. Various works have considered this scenario and derived expressions for $M_{\rm J}^{\rm min}$, generally $\sim 10^{-3}-10^{-2}M_\odot$ in normal star-forming conditions, but with conflicting results about the scaling with ambient conditions and material properties. We derive expressions for the thermal evolution of dust-cooled collapsing gas clumps in various limiting cases, given a general ambient radiation field ($u_{\rm rad}$, $T_{\rm rad}$) and a general power-law dust opacity law $σ_{\rm d} = A_{\rm d} T^β$. By accounting for temperature evolution self-consistently we rule out a previously-proposed regime in which the adiabatic transition occurs while the core is still optically-thin. If the radiation field is weak or dust opacity is small, $M_{\rm J}^{\rm min}$ is insensitive to dust properties/abundance ($\sim A_{\rm d}^{-\frac{1}{11}}-A_{\rm d}^{-\frac{1}{15}}$), but if the radiation field is strong and dust is abundant it scales $\propto A_{\rm d}^{1/3}$. This could make the IMF less bottom-heavy in dust-rich and/or radiation-dense environments, e.g. galactic centers, starburst galaxies, massive high-$z$ galaxies, and proto-star clusters that are already luminous.