Analytic Rationale for the “Nodes of Lesser Uncertainty” Inversion of the Zodiacal Brightness Integral

Abstract The inversion of the zodiacal brightness integral by R. Dumont and A.-C. Levasseur-Regourd [1985a, Planet. Space Sci. 33 , 1–9; 1985b, in Properties and Interactions of Interplanetary Dust (R. H. Giese and P. Lamy, Eds.), IAU Colloq. 85, pp. 207–213, Reidel, Dordrecht] is based on the standard representation of an unknown function, the integrand, by an analytic expression. The order of the analytic function is the same as the assumed number of boundary conditions and constraints. The conditions and constraints are used to eliminate all but one of the unknown arbitrary coefficients in the expression. The result can only have definite values at the roots of the function multiplying the remaining arbitrary coefficient. The various functional forms analyzed by Dumont and Levasseur-Regourd (1985a) to invert the visible brightness integral for observations in the ecliptic plane produced almost the same roots, and the method, consequently, was labeled “nodes of lesser uncertainty.” This confluence of solutions is due to the fact that all of the functional forms investigated are, themselves, well represented by orthogonal polynomials of the same degree. It is found that the orthogonal basis functions to the order adopted by Dumont and Levasseur-Regourd are reasonable representations to the traditional, physical analytic models for the volume scattering and emission except at small scattering angles.