Two-dimensional description of surface-bounded exospheres with application to the migration of water molecules on the Moon

On the Moon, water molecules and other volatiles are thought to migrate along ballistic trajectories. Here, this migration process is described in terms of a two-dimensional partial differential equation for the surface concentration, based on the probability distribution of thermal ballistic hops. A random-walk model, a corresponding diffusion coefficient, and a continuum description are provided. In other words, a surface-bounded exosphere is described purely in terms of quantities on the surface, which can provide computational and conceptual advantages. The derived continuum equation can be used to calculate the steady-state distribution of the surface concentration of volatile water molecules. An analytic steady-state solution is obtained for an equatorial ring; it reveals the width and mass of the pileup of molecules at the morning terminator.

Figure 1

Results of a Monte Carlo model calculation of water molecules hopping on the lunar surface. Black molecules reside on the surface and blue molecules are in flight. Color contours represent surface temperature. The globe spins counterclockwise. The concentration of water molecules is enhanced near the poles and near the morning terminator. Water molecules are generated in the subsolar region, destroyed by photodissociation during flight, and some are trapped near the poles. The model is described in Ref. [11].

Figure 2

Root-mean-square distance traveled after a time $t$, based on (a) probabilistic time steps, Eqs. (1, 2, 3), (b) time steps of fixed size (4) and duration (5) but random direction, and (c) the simple square-root law (12) with diffusion coefficient (11). For the calculation of (a) and (b), 2000 random walks each were averaged, which retains small wiggles in the behavior. For case (a), times were rebinned for this plot.

Figure 3

Analytic solution to the simplified one-dimensional problem as a function of local time (or geographic longitude). The surface density of ${\mathrm{H}}_2\mathrm{O},\sigma$, is normalized to the average density.