Deep-down ionization of protoplanetary discs

The possible occurrence of dead zones in protoplanetary discs subject to the magneto-rotational instability highlights the importance of disc ionization. We present a closed-form theory for the deep-down ionization by X-rays at depths below the disc surface dominated by far-ultraviolet radiation. Simple analytic solutions are given for the major ion classes, electrons, atomic ions, molecular ions and negatively charged grains. In addition to the formation of molecular ions by X-ray ionization of H2 and their destruction by dissociative recombination, several key processes that operate in this region are included, e.g. charge exchange of molecular ions and neutral atoms and destruction of ions by grains. Over much of the inner disc, the vertical decrease in ionization with depth into the disc is described by simple power laws, which can easily be included in more detailed modelling of magnetized discs. The new ionization theory is used to illustrate the non-ideal magnetohydrodynamic effects of Ohmic, Hall and Ambipolar diffusion for a magnetic model of a T Tauri star disc using the appropriate Elsasser numbers