Penetration of cosmic ray muons into the Earth

I present a new analytic solution to the integro-differential equation that describes the underground propagation of cosmic ray muons. The exact solution is given in the form of an infinite series in inverse powers of the muon energy. Convergence is proved for sufficiently high energies. ries is shown to be summable in closed form, in certain approximations. The closed forms provides analytic continuations to low energies of the series solution. One approximation resembles a well-known solution that ignores discrete energy loss, but this approximation introduces additional constants. I apply the approximate solution using an expression for the surface muon flux, derived from the primary flux, as a boundary condition. The result predicts the underground muon vertical intensity over seven orders of magnitude (10 km depth), compares favorably with published Monte Carlo calculations, and can be performed in seconds on a personal computer. application, the same approximation predicts the “catastrophic” energy-loss event rate at Soudan II.