Ambipolar diffusion in self-gravitating isothermal layers

Quasi-magnetohydrostatic and local ionization equilibria are assumed in the present formulation and solution of the drift of magnetic field and ions embedded in a self-gravitating layer of neutral isothermal gas. The introduction of Lagrangian coordinates referred to the neutral gas allows this problem to be reduced to a nonlinear diffusion equation for the magnetic field, whose dimensionless form involves no parameters other than those introduced by the initial values. In the shape-invariant solution, the magnetic field at each surface density point in the neutral fluid decays as the inverse square root of the elapsed time. Explicit estimates are given, as a function of the initial magnetic to neutral gas pressure in a natural family of cases, for the amount of time that must pass before the shape-invariant solution becomes a good approximation for actual behavior. The results obtained are interpreted physically.