Effective equation of state of a radiatively cooling gas Self-similar solution of spherical collapse

Context. The temperature of the interstellar medium (ISM) is governed by several physical processes, including radiative cooling, external UV/cosmic-ray heating, and mechanical work due to compression and expansion. In regimes where the dynamical effect is important, the temperature deviates from that derived by simply balancing the heating and cooling functions. This renders the expression of the gas energy evolution with a simple equation of state (EOS) less straightforward. Aims. Given a cooling function, the behavior of the gas is subject to the combined effect of dynamical compression and radiative cooling. The goal of the present work is to derive the effective EOS of a collapsing gas within a full fluid solution. Methods. We solved the Navier-Stokes equations with a parametric cooling term in spherical coordinates, and looked for a self-similar collapse solution. Results. We present a solution that describes a cloud that is contracting while losing energy through radiation. This yields an effective EOS that can be generally applied to various ISM contexts, where the cooling function is available from first principles and is expressed as a power-law product of the density and temperature. Conclusions. Our findings suggest that a radiatively cooling gas under self-gravitating collapse can easily manifest an effective polytropic EOS, even isothermal in many scenarios. The present model provides theoretical justification for the simplifying isothermal assumptions of simulations at various scales, and can also provide a more realistic thermal recipe without additional computation costs.