Non-monotonic Lunar Plasma Sheaths

We argue that the electrostatic field profile observed by ARTEMIS in the neighbourhood of the lunar surface is sustained by singular electron an ion velocity distribution functions. These are singular solutions of the steady state, two species Vlasov-Poisson equations. The energy distributions of the hot, finite mass, mobile ions is assumed to be log singular at the position of the electric potential's minimum. We show that the electron energy distributions on opposite sides of this minimum are not equal. This leads to a jump discontinuity of the electron distribution across its separatrix. A simple relation exists between the difference of these two electron distributions and that of the ions. The distributions of both species are given in terms of elementary functions and they meet smooth boundary conditions at one plasma end. Simple, finite amplitude profiles of the electric potential result from Poisson equation, which are smoothly, but non monotonically and non symmetrically distributed in space. Three such solutions are investigated in detail as appropriate for non monotonic double layers and for a plasma of semi-infinite extent bounded by a surface.