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.2013 edition
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The number of particles $N(X)$ as a function of traversed atmospheric depth $X$ is expressed as

:$N(X)= N_\text\left(\frac\right)^\exp\left(\frac\right),$

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

Using substitutions

$n=\frac$, $x=\frac$ and $m=\frac$

the function can be written in an alternative one-parametric (''m'') form as

:$n(x)=\left(\frac\right)^m\exp(m-x)=\frac=\exp\left(m\,(\ln x-\ln m)-(x-m)\right)\, .$

References

Cosmic rays

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