Gaisser–Hillas function

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 M. Hillas.^[1]

The number of particles $N(X)$ as a function of traversed atmospheric depth $X$ is expressed as

$N(X)= N_\text{max}\left(\frac{X-X_0}{X_\text{max}-X_0}\right)^{\frac{X_\text{max}-X_{0}}{\lambda}}\exp\left(\frac{X_\text{max}-X}{\lambda}\right),$

where $N_\text{max}$ is maximum number of particles observed at depth $X_\text{max}$, and $X_0$ and $\lambda$ are primary mass and energy dependent parameters.

Using substitutions

$n=\frac{N}{N_\text{max}}$,       $x=\frac{X-X_0}{\lambda}$       and       $m=\frac{X_\text{max}-X_0}{\lambda}$

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

$n(x)=\left[\frac{x}{m}\right]^m\exp(m-x)=\frac{x^m \, e^{-x}}{m^m \, e^{-m}}=\exp[m(\ln x-\ln m)-(x-m)].$

References

1. Hillas, A. M. (1972). Cosmic rays. New York: Pergamon Press. ISBN 0-08-016724-1.
2. Darko Veberic (2012). "Lambert W Function for Applications in Physics". Computer Physics Communications 183 (12): 2622–2628. arXiv:1209.0735. doi:10.1016/j.cpc.2012.07.008.

Gaisser, T.K.; Hillas, A.M. (1977). "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. p. 353.