The ionization fraction in alpha models of protoplanetary discs

We calculate the ionization fraction of protostellar alpha discs, taking into account vertical temperature structure and the possible presence of trace metal atoms. Both thermal and X-ray ionization are considered. Previous investigations of layered discs used radial power-law models with isothermal vertical structure. But alpha models are used to model accretion, and the present work is a step towards a self-consistent treatment. The extent of the magnetically uncoupled ('dead') zone depends sensitively on alpha, on the assumed accretion rate and on the critical magnetic Reynolds number, below which magnetohydrodynamic (MHD) turbulence cannot be self-sustained. Its extent is extremely model-dependent. It is also shown that a tiny fraction of the cosmic abundance of metal atoms can dramatically affect the ionization balance. Gravitational instabilities are an unpromising source of transport, except in the early stages of disc formation.