The Angular Distribution of High Energy Electrons in Air Showers. I. Landau Approximation

Abstract

The angular distribution of high energy (> 5 X 108 e V) electrons in air showers is calculated on a track length basis, using approximation A of Rossi and Greisen (1941) (no ionization loss) and the Landau (1940) multiple scattering approximation. We start with a discussion of the approximations used and an estimate of their validity. The basic equations are written down; qualitative results and very rough solutions are developed and applied to the Furry model of a cascade. For the Furry cascade the qualitative arguments lead directly to an Ansatz which yields an exact solution. In the actual cascade the corresponding Ansatz does not yield an exact solution. We then perform an iteration, employing a general method of Friedman. The final (iterated) solution is compared with the exact (in the Landau approximation) solution by means of their moments, and appears to be within 10 per cent. of the correct solution for E6fEs <l. Our solution compares well with earlier work on this problem. Appendices contain a short derivation of the angular moments, a general inversion formula for going from the distribution-in-projected-angle to the distribution-in-angle-with-the-showeraxis, and a derivation of the Friedman variation principle in vector space terminology.