Protostar Formation in Magnetic Molecular Clouds beyond Ion Detachment. I. Formulation of the Problem and Method of Solution

We formulate the problem of the formation of magnetically supercritical coresin magnetically subcritical parent molecular clouds, and the subsequentcollapse of the cores to high densities, past the detachment of ions frommagnetic field lines and into the opaque regime. We employ the six-fluid MHDequations, accounting for the effects of grains (negative, positive andneutral) including their inelastic collisions with other species. We do notassume that the magnetic flux is frozen in any of the charged species. Wederive a generalized Ohm's law that explicitly distinguishes between fluxadvection (and the associated process of ambipolar diffusion) and Ohmicdissipation, in order to assess the contribution of each mechanism to theincrease of the mass-to-flux ratio of the central parts of a collapsing coreand possibly to the resolution of the magnetic flux problem of star formation.We show how our formulation is related to and can be transformed into thetraditional, directional formulation of the generalized Ohm's law, and wederive formulae for the perpendicular, parallel and Hall conductivitiesentering the latter, which include, for the first time, the effect of inelasticcollisions between grains. In addition, we present a general (valid in anygeometry) solution for the velocities of charged species as functions of thevelocity of the neutrals and of the effective flux velocity (which can in turnbe calculated from the dynamics of the system and Faraday's law). The last twosets of formulae can be adapted for use in any general non-ideal MHD code tostudy phenomena beyond star formation in magnetic clouds. The results,including a detailed parameter study, are presented in two accompanying papers.Comment: 17 pages, emulateapj; accepted for publication in the Astrophysical Journa