Hadronic Showers in the Atmosphere

The Heitler model, describing electromagnetic showers, has been adapted by Matthews to hadronic showers induced by protons or nuclei.

Matthews, 2005 describes the model, as well as Letessier-Selvon & Stanev, 2011 and Engel, Heck & Pierog, 2011, the paper by J.M.C. Montanus.

From the indicated review paper, the toy model for an electromagnetic shower is compared to a proton/nucleus hadronic interaction.

In the case of hadronic showers, the relevant parameter is the hadronic interaction length $\lambda_I$ . This quantity was calculated in Particle Physics elements for protons and heavier nuclei, and depend on the inelastic cross section.

At each step of thickness $\lambda_I ln2$ it is assumed that hadronic interactions produce $2N \pi$ charged pions and $N_\pi$ neutral ones (with total number of new particles in an interaction of $n_{tot} = 3N_\pi$ ). While $\pi^0$ decays immediately (rest lifetime $c\tau = 25 \, \rm nm$ ) and feed the electromagnetic part of the shower, $\pi^\pm$ ( $c\tau = 7.8 \rm \, m$ ) interact further. In this simplified model the pion interaction length and multiplicity ( $n_{tot}$ ) are energy independent and the energy equally shared between secondaries. Approximately, for $E_\pi \sim 1 \rm \, GeV \div 10 \rm \, TeV$ a charged multiplicity of $N_\pi = 5$ can be used (10 total charged pions). The hadronic cascade and induced electromagnetic part grow until charged pions reach an energy where decay is more likely than a new interaction. On average, one third of the available energy goes into the electromagnetic component, while the remaining 2/3 goes in the hadronic part:

$$E_{had} = \left(\frac{2}{3} \right)^n E_0; \, E_{EM} = \left[1- \left(\frac{2}{3}\right)^n\right]$$

where n is the number of generations. With $n\sim 6$ , about 90% of the initial shower energy is carried by EM particles and deposited as ionization energy in the atmosphere.

The critical energy for pions is $E_{c, \pi}=$ 20 GeV in air, then they will decay into muons. Because of the density profile of the atmosphere, $E_{c,\pi}$ is larger at high altitude than at see level, hence deep showers will produce fewer muons. In addition, primaries with higher interaction cross section with air will have a larger muon to electron ratio at ground. The average longitudinal and lateral development for showers induced by $10^{19} \rm eV$ protons are shown in the figure below from Engel, Heck and Pierog, 2011.

The lateral and longitudinal development of component particles of showers

$$X^{had}_{max}(E_0) \sim \lambda_I + X_{max}^{EM}\left[E_0/(2n_{tot})\right] \sim \lambda_I + X_0 \ln\left(\frac{E_0}{2n_{tot}E_c}\right)$$

As a matter of fact, the depth of the shower maximum of a hadronic shower is determined by the EM particles that outnumber all the other contributions. In the expression above we consider only the first hadronic interaction.

To obtain the number of muons in the shower one simply assumes that all pions decay into muons when they reach the critical energy: $N_{\mu} = (2N_\pi)^{n}$ and the energy equals the energy at which pions decay into muons: $E_{c,\pi} = \frac{E_0}{(n_{tot})^n} = \frac{E_0}{(3N_\pi)^n} \Rightarrow n \ln(3N_\pi) = \ln \frac{E_0}{E_{c,\pi}}$ $\Rightarrow n = \ln (E_0/E_{c,\pi})/\ln(3N_\pi)$ is the number of steps needed for the pions to reach $E_{c,\pi}$.

Introducing $\beta = \ln(2N_\pi)/\ln(3N_\pi)$ (= 0.85 for $N_\pi = 5$) we have:

$$N_\mu = (E_0/E_{c,\pi})^\beta$$

Unlike the electron number, the muon multiplicity does not grow linearly with the primary energy but at a slower rate. The precise value of β depends on the average pion multiplicity used. It also depends on the inelasticity of the hadronic interactions. Assuming that only half of the available energy goes into the pions at each step (rather than all of it as done above) would lead to $\beta = 0.93$.

Notice that $n_{ch}$ is measured at 8.16 TeV in the centre of mass system by the ALICE experiment of LHC and by CMS, despite unfortunately not on N or O relevant for the atmosphere. They are compared to hadronic models also present in the most used MonteCarlo simulation of atmospheric showers CORSIKA.

The model can be extended to EAS from heavy nuclei, which develop higher and faster (hence with less shower to shower fluctuations) than showers initiated by lighter nuclei. Everything can be understood applying the superposition model (see Basic Concepts and Notations). The faster development implies that pions in the hadronic cascade reach their critical energy (where they decay dominates over interaction) sooner and therefore the relative number of muons with respect to the electromagnetic component is higher. It can be seen that:

1) Showers induced by nuclei with atomic number A will develop higher in the atmosphere. The offset with respect to proton showers is simply :

$$X_{max}^A = X_{max}^p - \lambda_I \ln A$$

2) Showers initiated by nuclei with atomic number A will have a larger muon number than proton initiated ones :

$$N_{\mu}^A = N_{\mu}^p A^{1-\beta}$$

3) The evolution of the primary cross section and multiplicity with energy for nuclei is similar to proton ones. This shows up as parallel lines in an Xmax vs energy plot. See the simulated shower evolutions of $X_{max}$ for various hadronic models in the figure below from Letessier-Selvon & Stanev, 2011.

4) The fluctuation of the position of $X_{max}$ from one shower to another is smaller for heavy nuclei than for light ones. This can be seen in Extra-galactic Cosmic Rays.

The number of muon plot as a function of the electron number are correlated and are evidently separated for different primary compositions (from the Review of Engel, Heck and Pierog, 2011).