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

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

References

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. Vol. 8. Plovdiv, Bulgaria. p. 353. Bibcode:1977ICRC....8..353G.