Ambipolar diffusion, cloud cores, and star formation: Two-dimensional, cylindrically symmetric contraction. I. The issues, formulation of the problem, and method of solution

The role of ambipolar diffusion in the formation of molecular cloud cores and protostars is examined critically. The origin and physical meaning of a criterion for quasistatic or dynamic core contraction in otherwise magnetically supported clouds is explained briefly on the basis of analytical considerations. The relative magnitude of three natural length scales, which are unavoidably present in (realistic, three-dimensional) molecular clouds, determines the typical mass that can go into a protostar (∼ 1 M_⊙). We formulate the problem of the self-initiated contraction (due to ambipolar diffusion) of cylindrically symmetric, self-gravitating, isothermal, magnetic clouds embedded in a medium of constant thermal and magnetic pressures. If it were not for ambipolar diffusion, these model clouds would exist in exact equilibrium states indefinitely. The equations themselves contain three dimensionless free parameters: the ratio α_c of magnetic and thermal pressures in the core of the initial equilibrium state; the ratio v_ff of the initial free-fall and neutral-ion collision times (divided by π^1/2) in the core; and the exponent k in the parametrization n_i ∝ n_n^k of the ion density in terms of the neutral density. The boundary conditions introduce, in general, two additional free parameters, namely, the ratio of the initial surface and central neutral densities, and the ratio of the initial surface and central magnetic field strengths. The initial conditions introduce no new free parameters in the problem. In fact, they remove one free parameter if α is taken to be constant in the initial equilibrium state. The numerical method developed to follow the evolution of these model clouds, which involves an adaptive grid, is characterized by a fractional error ≃10^-5 in the approximation of the forces everywhere in a model cloud except at the surface, where the error increases to ≃10^-2 without degrading the accuracy anywhere else in the interior; a maximum relative error of the implicit time-integrator one to two orders of magnitude smaller than that introduced by spatial discretization; and a numerical diffusion of magnetic flux, introduced by the advection scheme, typically a few ×10^-5. The results, including an extensive parameter study, as they relate to the formation of cores and protostars are described in a following paper.