Gaisser–Hillas function

{{Short description|Function in physics to determine particle density}}

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.{{cite book |author=Hillas, A. M. |title=Cosmic rays |publisher=Pergamon Press |location=New York |date=1972 |isbn=978-0-08-016724-4 |url=https://archive.org/details/cosmicrays0000hill|url-access=registration }} [https://books.google.com/books?id=1j8fAwAAQBAJ 2013 edition]

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

:$N(X)=\; N\_\backslash text\{max\}\backslash left(\backslash frac\{X-X\_0\}\{X\_\backslash text\{max\}-X\_0\}\backslash right)^\{\backslash frac\{X\_\backslash text\{max\}-X\_\{0\}\}\{\backslash lambda\}\}\backslash exp\backslash left(\backslash frac\{X\_\backslash text\{max\}-X\}\{\backslash lambda\}\backslash right),$

where $N\_\backslash text\{max\}$ is maximum number of particles observed at depth $X\_\backslash text\{max\}$, and $X\_0$ and $\backslash lambda$ are primary mass and energy dependent parameters.

Using substitutions

$n=\backslash frac\{N\}\{N\_\backslash text\{max\}\}$, {{spaces|5}} $x=\backslash frac\{X-X\_0\}\{\backslash lambda\}$ {{spaces|5}} and {{spaces|5}} $m=\backslash frac\{X\_\backslash text\{max\}-X\_0\}\{\backslash lambda\}$

the function can be written in an alternative one-parametric (m) form{{cite journal |title= Lambert W Function for Applications in Physics|date= 2012|author1= Darko Veberič |doi= 10.1016/j.cpc.2012.07.008 |journal= Computer Physics Communications |volume= 183 |issue= 12 |pages= 2622–2628|arxiv=1209.0735 |bibcode= 2012CoPhC.183.2622V|s2cid= 315088}} as

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