Effects of turbulence on average refraction angles in occultations by planetary atmospheres

Four separable effects of atmospheric turbulence on average refraction angles in occultation experiments are derived from a simplified analysis, and related to more general formulations by B.S. Haugstad. The major contributors are shown to be due to gradients in height of the strength of the turbulence, and the sense of the resulting changes in refraction angles is explained in terms of Fermat's principle. Because the results of analyses of such gradient effects by W. B. Hubbard and J. R. Jokipii are expressed in other ways, a special effort is made to compare all of the predictions on a common basis. We conclude that there are fundamental differences, and use arguments based on energy conservation and Fermat's principle to help characterize the discrepancies.