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.^[1]

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

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

where ${\displaystyle N_{\text{max}}}$ is maximum number of particles observed at depth ${\displaystyle X_{\text{max}}}$, and ${\displaystyle X_{0}}$ and ${\displaystyle \lambda }$ are primary mass and energy dependent parameters.

Using substitutions

${\displaystyle n={\frac {N}{N_{\text{max}}}}}$,       ${\displaystyle x={\frac {X-X_{0}}{\lambda }}}$       and       ${\displaystyle m={\frac {X_{\text{max}}-X_{0}}{\lambda }}}$

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

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