On exact equilibrium states in external gravitational fields of heated,self-gravitating gas clouds cooling by conducting and radiation

We present exact analytic solutions describing the equilibrium states available to a one-dimensional, self-gravitating cloud of gas subject to an external constant gravitational acceleration due to a plane of “stars”. The gas is taken to be heated at a rate proportional to the local gas density and is cooling by both radiation and conduction. The solutions are valid for a thermal conductivity which is an arbitrary function of gas temperature, T, and for radiative cooling which is proportional to the local gas density, ?, multiplied by an arbitrary function of gas pressure, ?. Illustrations of the general spatial dependence are given for the cases where the radiative cooling is proportional to ?²T, and in which the thermal conductivity is either constant, or proportional to T^a(a > 0) in the limits of T tending zero or infinity, respectively.We show that the phenomenon of density “inversion”, reported earlier, is indeed ameliorated by the radiative cooling term, as we had speculated it might be, but is not removed. This indicates that the phenomenon of density inversion is of rugged quality, persisting under a wide variety of conditions and, therefore, of general astrophysical import. We also show that, depending on the ratios of various parameters entering the problem, there is a new phenomenon possible in which the gas temperature has a local minimum at some non-central location so that a wedge of cool gas is in equilibrium surrounded by a hot medium.We have done these calculations as an aid to understanding the complicated behavior of interstellar gas clouds in particular, and the general physical interplay between force balance and energy balance in models of gas clouds more realistic than those heretofore available.