Multi–pressure Polytropes as Models for the Structure and Stability of Molecular Clouds. I. Theory

We present a theoretical formalism for determining the structure of molecular clouds and the precollapse conditions in star–forming regions. The model consists of a pressure-bounded, self-gravitating sphere of an ideal gas that is supported by several distinct pressures. Since each pressure component is assumed to obey a polytropic law Pi(r) ∝ ρ γpi , we refer to these models as multi–pressure polytropes. We treat the case without rotation. The time evolution of one of these polytropes depends additionally on the adiabatic index γi of each component, which is modified to account for the effects of any thermal coupling to the environment of the cloud. We derive structure equations, as well as perturbation equations for performing a linear stability analysis. Special attention is given to properly representing the significant pressure components in molecular clouds: thermal motions, static magnetic fields, and turbulence. The fundamental approximation in our treatment is that the effects of turbulent motions in supporting a cloud against gravity can be approximated by a polytropic pressure component. In particular, we approximate the turbulent motions as a superposition of Alfvén waves. We generalize the standard treatment of the stability of polytropes to allow for the flow of entropy in response to a perturbation, as expected for the entropy associated with wave pressure. In contrast to the pressure components within stars, the pressure components within interstellar clouds are “soft”, with polytropic indexes γpi ≤ 4/3 and (except for Alfvén waves) adiabatic indexes γi ≤ 4/3. This paper focuses on the characteristics of adiabatic polytropes with a single pressure component that are near the brink of gravitational instability as a function of γpi and γi for γpi ≤ 4/3. The properties of such polytropes are generally 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 γp ≤ 1, regardless of the value of γ. The central–to–surface density and pressure drops in isentropic clouds (γi = γpi) are also limited, but they can become quite large (as observed) in non–isentropic clouds, which have γi > γpi. We find that the motions associated with Alfvén waves are somewhat less effective in supporting clouds than are the kinetic motions in an isothermal gas. Subject headings: ISM: Clouds — ISM: molecules — ISM: structure — Stars: formation