Cascade equations in a hybrid approach

Air showers are commonly simulated by tracking every single particle in the cascade and simulating each interaction explicitly. At high energies, this is not feasible anymore, as the number of particles is so high that it would take months to simulate them. To resolve this problem people commonly apply the thinning algorithm, where, below a certain energy threshold, one tracks only a part of the secondary particles while attributing them a higher weight. This can produce useful results, but has the disadvantage of introducing artificial fluctuations while still being relatively slow in computation time.

Figure 1: Schematic illustration of the hybrid approach.

The SENECA model employs cascade equations to solve the problem where it can be treated in a one-dimensional way. The cascade equations are a set of transport equations which describe propagation and interactions of the relevant particles:

$$\frac{\partial h_{n}(E,X)}{\partial X} = -h_{n}(E,X)\left[\frac{1}{\lambda_{n}(E)}+\frac{B_{n}}{EX}\right]$$ (1)

$$+\sum_{m}\int_{E}^{E_{\mathrm{max}}^{\mathrm{had}}}h_{m}(E',X)\le... ...n}(E',E)}{\lambda_{m}(E')}\right.\left.+\frac{B_{m}D_{mn}(E',E)}{E'X}\right]dE'$$

where $h_{n}(E,X)dE$ is the number of particles of type $n$ at height $X$ and energy range $]E,E+dE]$. The numerical solution of this equation is described in detail in references [1,2]. In the low energy regime, where the lateral spread of particles is important, particles are created from the source function:

$$\frac{\partial h_{n}^{\mathrm{source}}(E,X)}{\partial X} = \sum_{m}\int_{E_{\mathrm{min}}^{\mathrm{had}}}^{E_{\mathrm{max}}^... ...n}(E',E)}{\lambda_{m}(E')}\right.+\left.\frac{B_{m}D_{mn}(E',E)}{E'X}\right]dE'$$

These particles are followed in traditional Monte Carlo method.

Figure 2: Flow-chart for the SENECA model.

The hybrid approach consists of taking advantage of the cascade equations where it is possible, because the numerical solution is quite fast. The Monte Carlo method has to be employed for the first interactions, because of the natural fluctuations, and in the low energy regime, because of the lateral spread of secondaries. A schematic illustration is shown in figure 1. The translation of this scheme into a computer flow chart is shown in Figure 2. One can see how particles are treated depending on their type and energy. Particles are either followed in a Monte Carlo method or filled into hadronic or electromagnetic cascade equations. Once all particles are processed, the cascade equations are solved, and low energy particles are produced from the source function. These, in turn, are tracked in a Monte Carlo approach.

The angular condition for low energy particles might deserve some special attention. In the initial steps of the program, particles can be produced in the energy range $E<E_{\mathrm{min}}^{\mathrm{had}}$. These should be followed further by Monte Carlo, since the energy is by definition too low to fill the initial condition for the cascade equations, and technically they would just fill instead the source function. It is efficient to do so anyway, because this gives the user more control over the amount of low energy particles produced, and therefore more control over the necessary CPU time. But one can do so only if the angle of the particle momentum with respect to the shower axis is not large. Therefore, another condition appears in the flow chart in Figure 2.