Isodensitometric Measurements of Lunar Slopes from the Ranger Photographs

Abstract

The aim of the present communication will be to obtain by photometric methods the elevation profiles for seven fields of the Moon photographed in March 1965 by Ranger 9. We will assume, first, that the sunlight scattered by the lunar surface obeys a photometric law of the form I= I ₀ f(α, ε, i), where I ₀ denotes the albedo of the respective element of the lunar surface; α, the phase angle; i and ε are the angles of incidence and reflection. In order to minimise the number of unknowns in our problem, it will hereafter be assumed that the albedo of all surface elements under consideration remains constant. The angle of incidence i can also be considered constant for each frame of the Ranger photographs, in the last stage of its flight since its variation over the field of view of the respective camera is very small. Within the framework of these approximations, the distribution of brightness in the field of view should depend only on the angle ε of reflection which, in turn, depends on the slope of the surface. Therefore, if the angle of incidence i can be deduced from the observations by photometric measures, and if the respective positions of the Sun, the Moon and the spacecraft are known at the time of observation, we should be in a position to determine the direction of a normal to the respective element of the lunar surface at each point of the field, and thus its inclination to the horizontal. A numerical integration of these derivatives of the lunar surface curves, if sufficiently densely spaced, can then enable us to determine the actual warping of the local surface.