The Effect of a Non-Isotropic Flux of Very High Energy Cosmic Rays on the values of Mean Shower Maxima

In our previous works we described a statistical method to interpret the results of extensive air shower simulations. For an isotropically distributed flux of cosmic rays, we used this method to deduce diagrams of mean values of shower maxima versus energy decades. To have a more realistic result, we considered the effect of a non-isotropic flux of cosmic rays at different energy ranges. This effect was considered as a weight factor deduced from a set of observed data. We discussed about the effect of this weight factor on our final resulted diagrams of mean shower maxima and for different interaction models compared the resulted distributions of very high energy cosmic ray's mass composition.


Introduction
Entering the geomagnetic field, charged particles and ultra high energy γ-rays (E γ ≥ 10 19 eV ) are affected by the Earth magnetic field. For instance, the deflection of low energy charged particles in the Earth magnetic field, cause the azimuthal asymmetry or interaction of ultrahigh energy γ-rays with geomagnetic field cause pre-showering effect. In an array of cosmic ray's detectors with a horizontal geometry, the event rate is also distorted due to azimuthal event rate modulation in the geomagnetic field.
Obviously, considering a set of simulated Extensive Air Showers (EASs), there is an statistical distribution of depth of shower maxima, X max , over zenith angle intervals [1]. Using CORSIKA (version 7.37) [2] simulation, we studied this distribution for different hadronic interaction models and re-produced the diagrams of mean depth of shower maxima versus energy decades. The resulted diagrams then were used to study the cosmic ray mass composition, in which an isotropic flux of cosmic rays were supposed [1].
Anyhow the zenith angle dependence of the arrival rate of showers of size>N, I(> N, X, θ), for most of the experiments were considered to have a form of: I(> N, X, θ) = I(> N, X, 0)cos n θ[m −2 s −1 deg −1 ] in which X is the vertical depth of experiment [3]. Though, such a phenomena has not a direct effect on the calculated [1] mean depth of shower maxima, X max ; but it may cause a tolerance on the average values of X max when the statistical fluctuations over zenith angle intervals are considered. In this paper, the order of magnitude of this possible tolerances are investigated.

The Method
For Yakutsk array, different fits to experimental data are available for the energy ranges of E < 10 18 eV , E > 10 18 eV , E > 10 19 eV and E > 4 × 10 19 eV [4,5]. For this array, it was also shown that the azimuthal effect on event rate at E > 10 17 eV is approximately the same in the whole energy range [5].
At the top of atmosphere and in the energy range of ∼ 10 18 − 5 × 10 19 eV , one may consider the event rate per degrees as: is the event rate per degree at the top of atmosphere with a vertical depth equal to 0. As a result, using data from Yakutsk array (for energies higher than 10 19 eV ) [6] the expected event rates for different zenith angle intervals are given in table 1. Simulating the total number of N showers for each zenith angle interval and dividing the atmospheric depth to intervals of 10 gr/cm 2 , the occurrence frequencies of depth of shower maxima were counted. We used the second set of formulas were presented previously [1]: in which n is the number of intervals (i.e. i=1,...,n f requency i = N ) and for energies in the range of ∼ 10 19 − 5 × 10 19 , N were chosen from table 1. Here ω i is a weight factor equivalent to ω i ≡ f requency i N . No functional fits are considered as the number of simulated showers are now different. As before, zenith angle dependency may be removed using a function of the form of: and X max has the value of ∼ a + b.
Omitting the zenith angles of lower than ∼ 10 • , it is possible to use less simulations. But as the most uncertainties occur in this range, an important part of data may be lost. So we did this study for 5 fixed energies (1 × 10 18 eV , 5 × 10 18 eV , 1 × 10 19 eV , 5 × 10 19 eV and 1 × 10 20 eV ) for p and F e primaries. Interaction models were QGSJETII (QGSJET-II-04) [7] and FLUKA [8]. Thinning parameters was set to 10 −4 . No SLANT option was considered and using of Gaisser-Hillas curves [9] were avoided.

Discussion and Results
The effect of this non-isotropic event rate is a decrease in the values of statistical errors by increasing zenith angle. In fact, a fit of formula [3] to the distribution of mean values of X max versus zenith angle now is possible by establishing a standard deviation, λ θ , for each point (figure 1) which is now related to variable values of N (θ). For the studied energies these values were ranged from 10 down to 2 for a primary proton of 5 × 10 19 eV , and 9 down to 2 for a primary proton of 1 × 10 18 eV . In comparison, the standard deviation had a constant value for an isotropic distribution of the event rates. Anyhow, in the case of a Non-Isotropic distribution of event rates; when the distribution of depth of shower maxima experiences higher fluctuations for lower zenith angle intervals, a fit of the form of formula 3 eliminates this effect and the resulted mean value of depth of shower maxima differs not much than 2% of its previous value [1]. For instance, for a 100 EeV proton using a non-isotropic distribution caused an upper value of X max by 5 gr/cm 2 which is less than 1% of its previous calculated value for an isotropic distribution.
The percentage of these differences with respect to previous calculated values [1] were presented in table 2.
This research shows that in the method we used [1] for calculating diagrams of X max versus energy decades, the use of a non-isotropic event rate of primaries may affect the results on mass composition of primaries only slightly. In fact in the energy range of 10 19 eV and above, these values (i.e. the values of X max ) may be shifted to larger values of X max by less than 8 gr/cm 2  for a proton primary and 7 gr/cm 2 for an Iron nuclei (see table I of [1]), and as these values are much less than the magnitude of experimental uncertainties [10,11,12,13,14,15,16], this has not a considerable effect on mass composition studies at the highest energies. Same result were attained for SYBILL 1.6 and SYBILL 2.1 [1,17]