  The temperature of the interstellar medium (ISM) is governed by several
physical process, among which radiative cooling, external UV/cosmic ray
heating, and the mechanical work by 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. 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. We solve the Navier-Stokes equations with a
parametric cooling term in spherical coordinate and look for a self-similar
collapse solution. We present a solution which describes a cloud that is
contracting while losing energy through radiation. This yields an effective EOS
that can be generally applied to various ISM context, where the cooling
function is available from first principles, and expressed as powerlaw product
of the density and temperature. 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 cost.
