On exact polytropic equilibria of self-gravitating gaseous and radiative systems: their application to molecular cloud condensation

We propose a novel mathematical method to construct an exact polytropic sphere in self-gravitating hydrostatic equilibrium, improving the non-linear Poisson equation. The central boundary condition for the present equation requires a ratio of gas pressure to total one at the centre, which is uniquely identified by the whole mass and molecular weight of the system. The special solution derived from the Lane-Emden equation can be reproduced. This scheme is now available for modelling the molecular cloud cores in interstellar media. The mass-radius relation of the first core is found to be consistent with the recent results of radiation hydrodynamic simulations.