Stability of polar frosts in spherical bowl-shaped craters on the Moon, Mercury, and Mars

Following Svitek (Martian Water Frost: Control of Global Distribution by Small-Scale Processes, Ph.D Thesis, California Institute of Technology, 1992), analytic solutions are presented for the effective albedo, the effective emissivity, and the radiative equilibrium temperature in the shadowed portions of a spherical bowl-shaped crater. The model assumes that the surface is a Lambert scatterer with visual albedo and infrared emissivity each independent of wavelength across their respective spectral ranges. Absorption, emission, and multiple scattering from the walls of the crater are treated rigorously to all orders. For airless bodies whose surfaces are in radiative equilibrium, all shadowed portions of any individual crater have the same temperature, whose value depends on four quantities: the insolation (product of the solar constant and the sine of the solar elevation angle), the depth/diameter ratio of the crater, the visual albedo, and the infrared emissivity. As long as the crater is deep enough to have shadows, the lowest temperatures are for the shallowest crater—those with the smallest depth/diameter ratio. The model is applied first to the Moon and Mercury using a depth/diameter ratio of 0.2, which is typical of the lunar highlands according to Pike (Geophys. Res. Lett. 1, 291–294 (1974); in Impact and Explosion Cratering (Roddy et al., Eds.), pp. 489–509, Pergamon, New York, 1977). For Mercury and the Moon, temperatures in shadows in polar craters are below 102 K, so the sublimation rate of water ice calculated according to the model of Watson et al. (J. Geophys. Res. 66, 3033-3015 (1961)) is less than 1 cm per byr. The latitudinal extent of the cold zone on the Moon is greater than that on Mercury, although temperatures at the poles of the two planets are similar. The other application is to polar frosts on Mars. Illuminated water frosts in radiative equilibrium grow rougher, because the average temperature of a depression is greater than that of flat ground. Subliming CO_2 frosts, which are always at the same temperature, grow rougher at low solar elevation angles because the heat flux absorbed by a depression is greater than that for a flat surface. At high insolation rates (high Sun near perihelion) the average heat flux to a depression is less than for a flat surface. The latter evaporates faster, which makes the average surface smoother and leads to a high average albedo. This behavior helps explain the fact that the south CO_2 cap, which receives its greatest insolation near perihelion, has a higher effective albedo and therefore can survive the summer, whereas the north CO_2 cap has a lower effective albedo and disappears each year around summer solstice