Statistical Thermodynamics of Surface-Bounded Exospheres

Surface-bounded gravitationally-bound exospheres are expected on large airless bodies, such as the Moon and Mercury, where thermal desorption occurs at speeds below the escape speed and molecules are expected to undergo a sequence of ballistic hops. Through this lateral transport, a water exosphere may supply cold traps. Exospheres above dense atmospheres have long been investigated theoretically, but these results do not necessarily apply to surface-bounded exospheres.

An exosphere in thermodynamic equilibrium with the surface is described by the temperature of the surface, irrespective of the velocity distribution of the gas or the absence of collisions above the surface, which are not required to define a temperature. Within this theoretical framework, the cold trapping temperature can be interpreted as the solid-gas phase transition, as long as the long-term average of the pressure is taken.

Knudsen’s Cosine Law established that the velocity distribution for an ideal gas near a wall is the “Maxwell-Boltzmann Flux" (MBF) distribution, which is the Maxwell-Boltzmann distribution with an extra factor of the cosine of the angle. The direction in which a molecule leaves a wall is uncorrelated with the incident one, as expected from thermal accommodation. Oscillations in a crystal lead to an equivalent law for desorption velocities known as “Armand distribution".

The vertical density profile of a surface-bounded exosphere can be calculated using thermodynamic averages of an ensemble of ballistic trajectories. When the initial velocities follow the MBF distribution, the resulting density profile is the same as for exobase-bounded exospheres. For the simple case of constant gravitational acceleration and a scale height much smaller than the radius of the body, the density profile is exponential.

For molecules desorbed from a vertical wall, the vertical density profile is no longer exponential and diverges near the body’s surface. For a rough surface the density profile is therefore expected to have a ground-hugging population. Populations interpreted as a superposition of a hot and a cold population may in fact be consistent with a single population at a single temperature.