Multipressure polytropes as models for the structure and stability of molecular clouds. I. Theory

We present a theoretical formalism for determining the structure of molecul ar clouds and the precollapse conditions in star-forming regions. The model consists of a pressure-bounded, self-gravitating sphere of an ideal gas th at is supported by several distinct pressures. Since each pressure componen t is assumed to obey a polytropic law P-i(r) proportional to rho(gamma pi), refer to these models as "multipressure polytropes." We treat the case wit hout rotation. The time evolution of one of these polytropes depends additi onally on the adiabatic index yi of each component, which is modified to ac count for the effects of any thermal coupling to the environment of the clo ud. We derive structure equations as well as perturbation equations for per forming a linear stability analysis. Special attention is given to represen ting properly the significant pressure components in molecular clouds: ther mal motions, static magnetic fields, and turbulence. The fundamental approx imation in our treatment is that the effects of turbulent motions in suppor ting a cloud against gravity can be approximated by a polytropic pressure c omponent. In particular, we approximate the turbulent motions as a superpos ition of Alfven waves. We generalize the standard treatment of the stabilit y of polytropes to allow for the flow of entropy in response to a perturbat ion, as expected for the entropy associated with wave pressure. In contrast to the pressure components within stars, the pressure components within in terstellar clouds are "soft," with polytropic indices gamma(pi) less than o r equal to 4/3 and (except for Alfven waves) adiabatic indices gamma(i) les s than or equal to 4/3. This paper focuses on the characteristics of adiaba tic polytropes with a single pressure component that are near the brink of gravitational instability as a function of gamma(pi) and gamma(i) for gamma (pi) less than or equal to 4/3. The properties of such polytropes are gener ally governed by the conditions at the surface. We obtain upper limits for the mass and size of polytropes in terms of the density and sound speed at the surface. The mean-to-surface density and pressure drops are limited to less than a factor 4 for gamma(p) less than or equal to 1, regardless of th e value of gamma. The central-to-surface density and pressure drops in isen tropic clouds (gamma(i) = gamma(pi)) are also limited, but they can become quite large (as observed) in nonisentropic clouds, which have gamma(i) > ga mma(pi). We find that the motions associated with Alfven waves are somewhat less effective in supporting clouds than are the kinetic motions in an iso thermal gas.