  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.
