Three-fluid plasmas in star formation - I. Magneto-hydrodynamic equations

Context. Interstellar magnetic fields influence all stages of the process of star formation, from the collapse of molecular cloud cores to the formation and evolution of circumstellar disks and protostellar jets. This requires us to have a full understanding of the physical properties of magnetized plasmas of different degrees of ionization for a wide range of densities and temperatures. Aims. We derive general equations governing the magneto-hydrodynamic evolution of a three-fluid medium of arbitrary ionization, also including the possibility of charged dust grains as the main charge carriers. In a companion paper ( Pinto & Galli 2008, A&A, 484, 17), we complement this analysis computing accurate expressions of the collisional coupling coefficients for a variety of gas mixtures relevant for the process of star formation. Methods. Over spatial and temporal scales larger than the so-called large-scale plasma limit and the collision-dominated plasma limit, and for non-relativistic fluid speeds, the electric field, the electric current and the ion-neutral drift have their instantaneous values determined by the evolution of the magnetic field, which obeys an advection-diffusion equation. The validity of the approximations made is discussed critically. Results. We derive the general expressions for the resistivities, the diffusion timescales and the heating rates in a three-fluid medium and we use them to estimate the evolution of the magnetic field in molecular clouds and protostellar jets. Collisions between charged particles significantly increase the value of the Ohmic resistivity during the process of cloud collapse, possibly affecting the decoupling of matter and magnetic field and enhancing the rate of energy dissipation. The Hall resistivity can take larger values than previously found when the negative charge is mostly carried by dust grains. In weakly- or mildy-ionized protostellar jets, ambipolar diffusion is found to occur on a time scale comparable to the dynamical time scale, limiting the validity of steady-state and nondissipative models to study the jet's structure.