Gaisser–Hillas function explained

The Gaisser–Hillas function is used in astroparticle physics. It parameterizes the longitudinal particle density in a cosmic ray air shower. The function was proposed in 1977 by Thomas K. Gaisser and Anthony Michael Hillas.^[1]

The number of particles

N(X) as a function of traversed atmospheric depth X is expressed asN(X) = N _{max \left( (X - X _{0})/(X _{max-X} _{0})} \right) ^{(X _{max-X} _{0})/(λ)} \exp\left( (X _{max-X})/(λ) \right),

where

N _{max} is maximum number of particles observed at depth X _{max} , and X _{0} and λ are primary mass and energy dependent parameters.

Using substitutions

n = (N)/(N _{max}) , x = (X - X _{0})/(λ) and m = (X _{max-X} _{0})/(λ)

the function can be written in an alternative one-parametric (m) form^[2] as

n(x) = \left( (x)/(m) \right) ^{m\exp(m - x) = (x ^{m} e ^{-x})/(m ^{m} e ^{-m})} = \exp\left(m( ln x -ln m) - (x - m)\right) .

References

Gaisser . T.K. . Hillas, A.M. . Reliability of the method of constant intensity cuts for reconstructing the average development of vertical showers . Proc. of 15th Int. Cosmic Ray Conf., 13–26 Aug 1977 . Plovdiv, Bulgaria . 1977 . 8 . 353 . 1977ICRC....8..353G .

Notes and References

1. Book: Hillas, A. M. . Cosmic rays . . New York . 1972 . 978-0-08-016724-4 . registration .
2. Lambert W Function for Applications in Physics. 2012. Darko Veberic . 10.1016/j.cpc.2012.07.008 . Computer Physics Communications . 183 . 12 . 2622–2628. 1209.0735 . 2012CoPhC.183.2622V. 315088.