Physics: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 [math]\displaystyle{ N(X) }[/math] as a function of traversed atmospheric depth [math]\displaystyle{ X }[/math] is expressed as
- [math]\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), }[/math]
where [math]\displaystyle{ N_\text{max} }[/math] is maximum number of particles observed at depth [math]\displaystyle{ X_\text{max} }[/math], and [math]\displaystyle{ X_0 }[/math] and [math]\displaystyle{ \lambda }[/math] are primary mass and energy dependent parameters.
Using substitutions
[math]\displaystyle{ n=\frac{N}{N_\text{max}} }[/math], [math]\displaystyle{ x=\frac{X-X_0}{\lambda} }[/math] and [math]\displaystyle{ m=\frac{X_\text{max}-X_0}{\lambda} }[/math]
the function can be written in an alternative one-parametric (m) form[2] as
- [math]\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)\, . }[/math]
References
- ↑ Hillas, A. M. (1972). Cosmic rays. New York: Pergamon Press. ISBN 978-0-08-016724-4. https://archive.org/details/cosmicrays0000hill.
- ↑ Darko Veberic (2012). "Lambert W Function for Applications in Physics". Computer Physics Communications 183 (12): 2622–2628. doi:10.1016/j.cpc.2012.07.008. Bibcode: 2012CoPhC.183.2622V.
Gaisser, T.K.; Hillas, A.M. (1977). "Reliability of the method of constant intensity cuts for reconstructing the average development of vertical showers". 8. Plovdiv, Bulgaria. pp. 353. Bibcode: 1977ICRC....8..353G.
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