Explaining the characteristics of lateral shower age of cosmic ray extensive air showers

A simple analytical argument is proposed for a possible explanation of the characteristics of the lateral shower age ($s$) of proton ($p$)/nuclei-initiated showers. The analytical argument states that lateral density distribution (LDD) of electrons of a $p$-initiated shower is due to superposition of several electromagnetic (EM) sub-showers developed at a very early stage in the atmosphere from the decay of neutral pions ($\pi^{0}$s). Thanks to the superposition property of the electron LDD in a $p$ shower, a plausible analytical parametrization has been worked out by giving well represented analytic function for the electron LDDs of $p$- and $\pi^{0}$-initiated showers. Based on cosmic ray extensive air shower simulations, we have validated how the various characteristics of $s$ can be understood in the context of the present analytical argument. The $s$ parameter of a $p$ shower and its correlations with the shower ages of electron- and $\pi^{0}$-initiated showers supports the idea that the result of superposition of several EM sub-showers initiated by $\pi^{0}$s with varied energies at a very early stage might produce the LDD of electrons of a $p$ shower. It is also noticed with the simulated data that the stated feature still persists concerning the notion of the local shower age parameter.