Air shower (physics)

Shortly after entering the atmosphere, the primary cosmic ray (which is assumed to be a proton or nucleus in the following) is scattered by a nucleus in the atmosphere and creates a shower core - a region of high-energy hadrons that develops along the extended trajectory of the primary cosmic ray, until it is fully absorbed by either the atmosphere or the ground. The interaction and decay of particles in the shower core feeds the main particle components of the shower, which are hadrons, muons, and purely electromagnetic particles. The hadronic part of the shower consists mostly of pions, and some heavier mesons, such as kaons and ${\displaystyle \varrho }$ mesons.[16][17]

Neutral pions, ${\displaystyle \pi ^{0}}$, decay by the electroweak interaction into pairs of oppositely spinning photons, which fuel the electromagnetic component of the shower. Charged pions, ${\displaystyle \pi ^{\pm }}$, preferentially decay into muons and (anti)neutrinos via the weak interaction. The same holds true for charged and neutral kaons. In addition, kaons also produce pions.[17] Neutrinos from pion and kaon decay are usually not accounted for as parts of the shower because of their very low cross-section, and are referred to as part of the invisible energy of the shower.

Sketch of the hadronic and electromagnetic sub-cascades in an air shower.

Qualitatively, the particle content of a shower can be described by a simplified model, in which all particles partaking in any interaction of the shower will equally share the available energy.[18] One can assume that in each hadronic interaction, ${\displaystyle 2N_{\text{ch}}}$ charged pions and ${\displaystyle N_{\text{ch}}}$ neutral pions are produced. The neutral pions will decay into photons, which fuel the electromagnetic part of the shower. The charged pions will then continue to interact hadronically. After ${\displaystyle n}$ interactions, the share of the primary energy ${\displaystyle E_{0}}$ deposited in the hadronic component is given by

${\displaystyle E_{\pi }=\left({\frac {2}{3}}\right)^{n}E_{0}}$,

and the electromagnetic part thus approximately carries

${\displaystyle E_{\gamma }=\left(1-\left({\frac {2}{3}}\right)^{n}\right)E_{0}}$.

A pion in the ${\displaystyle n}$th generation thus carries an energy of ${\displaystyle E_{0}/(3N_{\text{ch}}/2)^{n}}$. The reaction continues, until the pions reach a critical energy ${\displaystyle \epsilon _{\text{c}}^{\pi }\simeq 20\,{\text{GeV}}}$, at which they decay into muons. Thus, a total of

${\displaystyle n_{\text{c}}=\left\lceil {\frac {\ln \left(E_{0}/\epsilon _{\text{c}}^{\pi }\right)}{\ln \left({\tfrac {3}{2}}\,N_{\text{ch}}\right)}}\right\rceil }$

interactions are expected and a total of ${\displaystyle (N_{\text{ch}})^{n_{\text{c}}}=(E_{0}/\epsilon _{\text{c}}^{\pi })^{\beta }}$ muons is produced, with ${\displaystyle \beta =\ln N_{\text{ch}}/\ln(3N_{\text{ch}}/2)\simeq 0.95}$. The electromagnetic part of the cascade develops by bremsstrahlung and pair production. For the sake of simplicity, photons, electrons, and positrons are often treated as equivalent particles in the shower. The electromagnetic cascade continues, until the particles reach a critical energy of ${\displaystyle \epsilon _{\text{c}}^{\gamma }\simeq 87\,{\text{MeV}}}$, from which on they start losing most of their energy due to scattering with molecules in the atmosphere. A good approximation for the number of electromagnetic particles produced in a shower is ${\displaystyle N_{\gamma }\simeq E_{0}/{\text{GeV}}}$. Assuming each electromagnetic interaction occurs after the average radiation length ${\displaystyle X_{0}\simeq 37\,{\text{g}}/{\text{cm}}^{2}}$, the shower will reach its maximum at a depth of approximately

${\displaystyle X_{\text{max}}\simeq X_{1}+X_{0}\ln \left({\frac {E_{0}}{\text{GeV}}}\right)}$,

where ${\displaystyle X_{1}}$ is assumed to be the depth of the first interaction of the cosmic ray in the atmosphere. This approximation is, however, not accurate for all types of primary particles. Especially showers from heavy nuclei will reach their maximum much earlier.

The number of particles present in an air shower is approximately proportional to the calorimetric energy deposit of the shower. The energy deposit as a function of the surpassed atmospheric matter, as it can for example be seen by fluorescence detector telescopes, is known as the longitudinal profile of the shower. For the longitudinal profile of the shower, only the electromagnetic particles (electrons, positrons, and photons) are relevant, as they dominate the particle content and the contribution to the calorimetric energy deposit.

Number of particles for different primary energies as a function of the surpassed atmospheric depth.

The shower profile is characterized by a fast rise in the number of particles, before the average energy of the particles falls below ${\displaystyle \epsilon _{\text{c}}^{\gamma }}$ around the shower maximum, and a slow decay afterwards. Mathematically the profile can be well described by a slanted Gaussian, the Gaisser-Hillas function or the Greisen function.

The image shows the ideal longitudinal profile of showers using different primary energies, as a function of the surpassed atmospheric depth ${\displaystyle X}$ or, equivalently, the number of radiation lengths ${\displaystyle t}$. The shower age parameter ${\displaystyle s}$ is introduced to compare showers with different starting depths and different primary energies to highlight their universal features, as for example always ${\displaystyle s=1}$ at the shower maximum.

The longitudinal profiles of showers are particularly interesting in the context of measuring the total calorimetric energy deposit and the depth of the shower maximum, ${\displaystyle X_{\text{max}}}$, since the latter is an observable that is sensitive to type of the primary particle.