Nonhomologous contraction and equilibria of self-gravitating, magnetic interstellar clouds embedded in an intercloud medium: star formation. I. Formulation of the problem and method of solution

The discovery that flux-freezing can be built into the magnetohydrostatic equations, thus removing all arbitrary functions, enables us to construct equilibrium states for interstellar clouds that can be reached from nonequilibrium configurations, characterized by the same mass-to-flux ratio, through continuous deformations of the field lines; it becomes unnecessary to solve a timedependent problem. In this paper we formulate the equilibrium problem for isothermal clouds with a frozen-in magnetic field, and we describe a method for its solution. Self-gravity and the pressure of the hot and tenuous intercloud medium bind a cloud against the disruptive effects of its internal pressure and magnetic stresses. The surface of a cloud is a free boundary determined by the requirement that there exist pressure balance across it. The formulation of the problem is general enough to account for self-gravitation in the medium surrounding a cloud. Although we consider isothermal clouds only, our formalism may be extended to nonisothermal equations of state in a manner analogous to that described in a previous paper. The equilibrium problem is characterized by two free parameters and a free function, which describes the amount of mass in each of the flux tubes of the system. In principle, this function can be obtained from high-resolution observations of dense clouds. If one assumes that an interstellar cloud contracted nonhomologously from some initial uniform state while the magnetic field remained frozen in the matter, then the free function reduces to a single parameter. We shall employ such an assumption in a later paper, in which we shall present specific solutions and we shall discuss their relevance to star formation. Subject headings: hydromagnetics - interstellar: matter - magnetic fields - plasmas - stars: formation