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 Michael Hillas.

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 as

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