The Spread of the Soft Component of the Cosmic Radiation

Abstract

The fundamental diffusion equations describing the three-dimensional development of the electron-photon component of the cosmic radiation, are formulated. These equations take into account exactly ionization loss and variation of density in the medium, and enable one to determine all angular and radial moments of the distribution functions, as explicit functions of depth and energy for arbitrary initial conditions.The exact general solution of these equations obtained in this paper are readily adapted to any physical situation of interest. The method is similar to that devised by the authors in considering the three-dimensional development of the nucleon component. The only approximations involved are those inherent in the Bethe-Heitler cross sections in the full screening approximation, and the neglect of angular deflections in processes other than elastic Coulomb scattering.It is shown that all previous work is subject to very large errors on the following counts: (1) the neglect of fourth and higher angular moments for the Coulomb scattering in Landau's equation and equivalent integral equations; (2) the neglect of variation of density in the atmosphere---which alone can lead to errors as high as 5000 percent; (3) elimination of the depth dependence either by integration over all depths, or evaluation in the neighborhood of the cascade maximum; (4) miscellaneous errors introduced in the evaluation of already approximate integrals; (5) use of results due to Moliere, hitherto unpublished in detail, which involve errors of several orders of magnitude in the higher moments; and also in the distribution functions concerned.No calculation of the actual distribution functions in the atmosphere or elsewhere has yet been made on the basis of a realistic physical model. The results obtained in this paper will allow the authors to do this in the future.