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


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