## 3 Extensive Air Showers: The Heitler-Matthews Model
### Hadronic Showers  
Air showers initiated by protons have been modeled by different authors (see, for example, the the Ref. [38; 39; 40]) following the [PERSON] approach [33], similar to the [PERSON] one. The main differences with the e.m. cascades are  
Figure 4: Schematic view of an e.m. cascades (**a**), and of a hadronic shower (**b**). In the hadron shower, dashed lines show \(\pi^{0}\) which do not re-interact but decay, producing e.m. sub-showers. in the hadronic interactions a large number of secondary particles are produced. At high energy, the total multiplicity of particles per collision, \(N_{\rm tot}\), reaches values of several tens with the consequence that although the hadronic interaction lengths are larger than the e.m. radiation one (at PeV energies \(\lambda_{p\,-\,{\rm air}}\simeq 80\) g cm\({}^{-2}\)), the showers develop faster than in the e.m. case;
2. in a hadronic interaction only a fraction of the energy E\({}_{0}\) is available for secondary particle production. A single leading particle, the highest energy secondary produced in the interaction, carries a fraction \((1-\kappa)E_{0}\) deep in the atmosphere, where \(\kappa\) is the so-called _inelasticity_. Therefore, a fraction \(\frac{2}{3}\kappa E_{0}\) is used to produce \(N_{ch}\) charged pions, and a fraction \(\frac{1}{3}\kappa E_{0}\) goes via neutral pions into the e.m. component;
3. the critical energy \(\varepsilon_{c}^{\pi}\) is defined as the energy at which the decay and the hadronic interaction probabilities are equal and further particle production by \(\pi^{\pm}\) ceases. According to the Ref. [33], constant values \(N_{ch}\) = 10, corresponding to an energy of about 100 GeV, and \(\varepsilon_{c}^{\pi}\) = 20 GeV are adopted in the following. Protons travel one interaction length and interact producing \(N_{tot}\) pions, all having equal energies, \(N_{ch}\) are charged and \(\frac{N_{tot}}{3}=\frac{1}{2}\cdot N_{ch}\) neutral, which immediately decay into photons, initiating e.m. showers. As for the e.m. cascade, we assume equal division of energy during particle production. In turn, the charged pions can decay in muons and neutrinos and hence, as long as their decay length remains larger than their interaction length, they will re-interact rather than decay. This happens for \(\gamma c\tau_{\pi}>\lambda_{\pi\,-\,{\rm air}}/\rho_{\rm air}\), with the Lorentz factor \(\gamma=E_{\pi}/m_{\pi}\), the charged pion lifetime \(\tau_{\pi}\simeq 26\) ns and \(\lambda_{\pi\,-\,{\rm air}}\simeq 1.5A_{p\,-\,{\rm air}}\simeq 120\) g cm\({}^{-2}\) (since the \(\pi p\) cross-section is about 2/3 the \(pp\) cross-section). This implies that pions will re-interact as long as their energy satisfies \(E>E_{\rm d}\simeq 100\) GeV(\(10^{-4}\) g cm\({}^{-3}/\rho_{\rm air}\)). Hence, at the heights above 10 km, where the initial development of the shower takes place, \(\pi^{\pm}\) will re-interact for energies greater than \(\sim\)20-30 GeV [40]. After \(n\) interactions, the \(N_{\pi}=(N_{ch})^{n}\) charged pions produced carry a total energy of \((\frac{2}{3})^{n}\cdot E_{0}\). The energy per charged pion after \(n\) interactions is then \(E_{\pi^{\pm}}=\frac{E_{0}}{(3/2N_{ch})^{n}}\). The remainder of the primary energy goes into the e.m. component from \(\pi^{0}\) decays  
\[E_{\rm em}\simeq E_{0}\bigg{[}1-\bigg{(}\frac{2}{3}\bigg{)}^{n}\bigg{]}. \tag{5}\]  
After only six interactions, about 90% of the initial energy is transferred to the e.m. component of the shower, with the remaining 10% being essentially the muons and neutrinos from the charged pion decays. As a consequence, most of the energy of an air shower can be observed in its e.m. component. This is the so-called _calorimetric energy_ which allows to estimate the primary energy with good accuracy to detectors able to observe the longitudinal air shower development.