Photometric functions for photoclinometry and other applications

Least-squared fits to the brightness profiles across a disk or “limb darkening” described by Hapke's photometric function are found for the simpler Minnaert and lunar-Lambert functions. The simpler functions are needed to reduce the number of unknown parameters in photoclinometry, especially to distinguish the brightness variations of the surface materials from that due to the resolved topography. The limb darkening varies with the Hapke parameters for macroscopic roughness (θ)">(θ), the single-scattering albedo (w), and the asymmetry factor of the particle phase function (g). Both of the simpler functions generally provide good matches to the limb darkening described by Hapke's function, but the lunar-Lambert function is superior when viewing angles are high and when (θ)">(θ) is less than 30°. Although a nonunique solution for the Minnaert function at high phase angles has been described for smooth surfaces, the discrepancy decreases with increasing (θ)">(θ) and virtually disappears when (θ)">(θ) reaches 30° to 40°. The variation in limb darkening with w and g, pronounced for smooth surfaces, is reduced or eliminated when the Hapke parameters are in the range typical of most planetary surfaces; this result simplifies the problem of photoclinometry across terrains with variable surface materials. The Minnaert or lunar-Lambert fits to published Hapke models will give photoclinometric solutions that are very similar (>1° slope discrepancy) to the Hapke-function solutions for nearly all of the bodies and terrains thus far modeled by Hapke's function.