Parametrization of the angular distribution of Cherenkov light in air showers

The Cherenkov light produced in air showers largely contributes to the signal observed in gamma-ray and cosmic-ray observatories. Yet, no description of this phenomenon is available covering both small and large angular regions. To fill this gap, a parametrization of the angular distribution of Cherenkov photons is performed in terms of a physically motivated parametric function. Model parameters are constrained using simulated gamma-ray and proton showers with energies in the TeV to EeV region. As a result, a parametrization is obtained that overcomes in precision previous works. Results presented here can be used in the reconstruction of showers with imaging Cherenkov telescopes as well as the reconstruction of shower profiles in fluorescence detectors.


I. INTRODUCTION
Large amount of Cherenkov light is produced in atmospheric air showers [1] and several experimental techniques have been proposed to explore this signal to study astroparticles.
The generation of light in the cascade is highly dominated by electrons. The emission of Cherenkov light by relativistic electrons including geometry, intensity, and wavelength is explained by classical electrodynamics [2], what has been used as an inspiration for the development of robust detection techniques with minimum bias and reduced systematic uncertainties.
The total signal produced by all particles in air showers evolves with depth due to the multiplication of particles, to the division of the primary energy, and to changes of the atmosphere [3]. The description of this evolution is mandatory to understand the signal and to reconstruct the properties of the primary particle. In order to extract physical results from measurements, the collaborations running Imaging Atmospheric Cherenkov Telescopes (IACT), ground-based detectors, fluorescence detectors and proposing space experiments have to understand the properties of Cherenkov light production in air showers, such as the longitudinal distribution, the lateral distribution, and the angular distribution. In this paper, special attention is given to the description of the angular distribution of Cherenkov photons in air showers.
IACTs are at the very foundation of contemporary gamma-ray astronomy. The identification and the reconstruction of the primary gamma-ray is done by the interpretation of the Cherenkov light detected by telescopes at ground. Current observatories [4-6] are equipped with some (< 5) telescopes with few degrees (< 10 • ) of field of view installed a hundred meters apart from each other. The Cherenkov Telescope Array (CTA) [7] is the next generation of IACTs presently under development. The CTA baseline design calls for 118 telescopes to be installed at two sites covering areas of 0.6 km 2 in La Palma, Spain and 4 km 2 in Paranal, Chile. The angular distribution of Cherenkov photons in an air shower determines the image shape detected by IACTs and therefore is a key aspect in many reconstruction techniques [8][9][10].
Fluorescence Detectors (FD) have been long used to study Ultra-High Energy Cosmic Rays (UHERC) [11]. These telescopes have been optimized to measure the isotropic fluorescence light emitted by nitrogen molecules due to the passage of the particles in the atmosphere. The telescopes in operation [12,13] have large aperture (≈30 • ) and cover a detection area of thousands km 2 . The emitted fluorescence light spectrum lays in the same wavelength band of the transmitted Cherenkov light (300-450 nm) making it impossible for FDs to filter it out. Traditionally Cherenkov light was considered noise in the FD measurements [13,14], but recently the Cherenkov light seen by FD has been used as signal to detect showers with energies down to 2 PeV [15]. Direct Cherenkov light is also used to study UHECR with ground detectors [16,17] and is proposed as an important signal source in future space experiments [18]. The angular distribution of Cherenkov photons in an air shower is an important feature for all UHECRs experiments because it determines the lateral spread of light and the balance between fluorescence and Cherenkov signal measured by FD including large angles (>10 • ) and great distances (several km) from shower axis.
The number of Cherenkov photons produced in an air shower reaching a detector at a given distance from the shower axis can be calculated only if the angular distribution of photons is known. Reversely, the reconstruction of the primary particle properties is only possible if the measured amount of light in each detector is converted into the amount of light emitted by the particles in the shower. The angular distribution of Cherenkov photons is determined by the convolution of the longitudinal development of electrons, energy distribution of the electrons, angular distribution of the electrons, scattering of the electrons, refractive index, geomagnetic effects, and scattering of the photons [19][20][21][22][23].
Influenced by the main techniques detecting Cherenkov light (IACT and FD), the study of the angular distribution of Cherenkov photons has been divided respectively in two regimes: a) gamma-ray primaries, small angles < 10 • , and TeV energies and b) cosmic ray primaries, large angles > 10 • , and highest energies (10 17 eV). Experiments have measured the angular distribution of Cherenkov photons [24] in regime b). Since the pioneering work [20], the angular distribution was simulated for regime a) [8] and b) [23,25].
In this paper, the angular distribution of Cherenkov photons is simulated using the most updated simulations software and a new parametrization based on shower physics is proposed. The new parametrization presented here improves the precision of the angular distribution in comparison to previous proposals [8,23,25]. Besides the needed update of the parametrizations concerning the new shower models made available after the previous works, this paper aims at the improvement of the precision requested by the new generation of experiments [7,18] and at the refinement demanded by the new uses of Cherenkov light as the main signal in fluorescence telescope analysis [15]. Moreover, a unified view of the two regimes is presented for the first time.
This paper is organized as follows. In section II an exact model to compute the angular distribution of Cherenkov models is derived. This model is simplified in section III to obtain a simple form in terms of free parameters. The parameters of the model are constrained by Monte Carlo simulations in section IV. A discussion of the results and a comparison to previous works is presented in section V and some final remarks are given in section VI.

II. EXACT MODEL FOR THE CHERENKOV ANGULAR DISTRIBUTION
A mathematical description of the number of Cherenkov photons emitted in a given angular interval as a function of the shower development in the atmosphere ( d 2 Nγ dθ dX ) is presented in this section. Each element that contributes to this quantity is identified and explained below.
It is known that electrons 1 are responsible for more than 98% of the Cherenkov photon content in a shower [23], therefore it is assumed here that all photons are emitted by electrons. Figure  The number of Cherenkov photons emitted by electrons with energy E and angle θ p in a shower per interval of depth dX is given by 2 : where s is the shower age 3 and h is the emission height above sea level. N e (s) is the total number of electrons, dNe dE is the energy distribution of electrons and dNe dθp is the angular distribution of electrons. The function Y γ (E, h) represents the number of photons emitted 1 The term electrons here refer to both electrons and positrons. 2 The dependency on the primary particle energy (E 0 ) is omitted here for brevity and discussed in terms of simulated showers in the following sections. For the purpose of this section, E 0 may be regarded as fixed. 3 s = 3X/(X + 2X max ) where X max is the depth in which the shower reaches the maximum number of particles.
by one electron per depth interval (yield) and the factor of 1/ cos θ p takes into account the correction in the length of the electron track due to its inclined trajectory. Photons are uniformly distributed in φ em (factor of 1/2π). According to reference [23], Y γ (E, h) is given by: in which α ≈ 1 /137 is the fine-structure constant, n(h) is the refractive index of the medium, ρ(h) is the atmospheric density, and λ i the wavelength interval of the emitted photons. The threshold energy E thr for an electron to produce Cherenkov light is where m e is the electron rest mass.
The dependency of dN γ on the angle between the Cherenkov photon and the shower axis directions (θ) is found after a change of variable from φ em to θ (see figure 1): which leads to in which a factor of 2 was added to account for the fact that there are always two values of φ em resulting in the same value of θ (see figure 2). The Cherenkov cone emission angle (θ em ) relates to the particle velocity β by the usual relation Substitution of equation 4 into equation 1 gives: Finally, to obtain the desired angular distribution of Cherenkov photons d 2 Nγ dθ dX it is necessary to integrate equation 6 over all possible values of electron energies E and angles θ p . Integration over E must assert that relation 5 is satisfied, therefore E takes values for which E > E thr (h). Limits of the integral over electron angles θ p should take only values that contribute to θ. From figure 2 and equation 3, it is found that this interval is |θ − θ em | < θ p < θ + θ em . Thus, the exact angular distribution of Cherenkov photons is given by III. APPROXIMATED MODEL FOR THE CHERENKOV ANGULAR DISTRI-

BUTION
In this section an approximation of the above equation is going to be proposed in order to obtain a simpler yet meaningful description of the angular distributions of Cherenkov light.
The idea is to summarize the angular distribution to a minimum set of parameters allowing its parametrization.
First, note that the integration in θ p is done in a very narrow interval given that θ em < 1.5°. Therefore it is possible to consider that: 1 cos θp dNe(θp,E) dθp varies little within integration limits and, in a first approximation, can be taken as constant and calculated in the mean angle θ p of the range in between the limits of the integration: where The remaining integral over θ p is a complete elliptic integral of the first kind and can be approximated by a logarithmic function: The abbreviation below is introduced: and by noting that cos θ em = 1/βn rapidly converges to 1/n as the electron energy increases, it is reasonable to assume that cos θ em = 1/n for all electrons. With this assumption the function I(θ, θ em , E) ∼ I(θ, θ em ) = I(θ, h) becomes independent of the electron energy 4 : The validity of approximations done until here were tested using Monte Carlo simulations of air showers as shown in A.
The remaining integral over electron energies: has been studied before in references [23,25]. A parametric form to describe this quantity is proposed here: where ν, θ 1 , θ 2 , and are parameters varying with shower age, height (or refractive index), and, possibly, the primary energy. The constant C is intended to normalize equation (14) according to equation (13). In the next section the parameters of this function are going to be studied and the quality of the description is going to be tested. The approximated model is summarized as: Taking this dependency into account, the angular distribution of Cherenkov photons in a given interval with mean ages and heighth in a shower of energy E 0 can be described by: in which N (different from N e (s)) is a normalization constant that depends on the parameters of K(θ,s,h, E 0 ).
The parameters of K(θ,s,h, E 0 ) are considered to be: ν(s, n) = p 0,ν (n − 1) p 1,ν + p 2,ν log(s) , θ 1 (s, n, E 0 ) = p 0,θ 1 (n − 1) p 1,θ 1 (E 0 /TeV) p 2,θ 1 + p 3,θ 1 log(s) , θ 2 (s, n) = θ 1 (s, n) × (p 0,θ 2 + p 1,θ 2 s) , The coefficients p i,µ are the parameters of the model to be fitted. In these equations, the dependence in height (h) was changed by the dependence in the refractive index (n) to make the parametrization independent of the atmospheric model used in the simulations. showers are fit separately, as distributions strongly depend on the primary particle type in lower energies. All coefficients p i,µ are allowed to vary in the fit procedure. In the case of gamma showers, however, the energy dependency is dropped (p 2,θ 1 , p 1, , p 2, = 0). Fitted values of p i,µ and their associated confidence intervals are found in tables I and II.

V. RESULTS
The parametrization proposed in the previous section is compared to the Monte Carlo distributions and to previous works. Figure 4 shows the simulated angular distribution of Cherenkov photons in comparison to four models for one single gamma-ray and one single proton shower. The ability of the model to be adjusted to simulated data both around the peak of the distributions and at the small and large θ regions is evident.