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Relation of muon flux local anisotropy with primary spectrum index

Muon hodoscope URAGAN allows to obtain the angular distribution of the muon flux. This distribution may be characterized by a vector of local anisotropy (the sum of the vectors of the particle arrival directions, normalized to the total number of muons). It was shown that annual variations in the vertical projection of the anisotropy vector AZ are not related with changes in atmospheric conditions. The dependence of AZ on the index of the primary particles spectrum γ was calculated for several zenith angle intervals with the help of simulation of generation and propagation of secondary cosmic ray particles through the atmosphere using the CORSIKA package. Experimental temporal series of the vertical projection of the local anisotropy AZ for several intervals of zenith angles were obtained for 2007-2015. According to the obtained AZ time series, annual and diurnal changes of Δγ were estimated.


Muon hodoscope URAGAN
Muon hodoscope URAGAN [1] is a wide-aperture precision muon hodoscope (Moscow, Russia, 55.7° N, 37.7° E, 173 m a.s.l.) which is used to study atmospheric and heliospheric processes responsible for variations in the muon flux at the Earth surface [2]. The hodoscope consists of separate horizontal assemblies (supermodules) with the area of 11.5 m 2 each. Three supermodules (SM) of the hodoscope are now under operation in the exposure mode. The supermodule detects muons with high spatial and angular accuracies (1 cm and 1º, respectively) over a wide range of zenith angles (0-80º). One-minute matrix of each SM contains about 70-80 thousand events. For the analysis of muon flux variations caused by extra-atmospheric processes it is necessary to introduce corrections for meteorological effects.

Corrections for basic atmospheric effects
Barometric effect is the anticorrelation of cosmic ray intensity with the pressure at the observation level. Temperature effect is caused by changes of the temperature at all altitudes of the atmosphere. Corrections for barometric and temperature effects are introduced as follows: where θ and φ are zenith and azimuth angles for matrix cell centers; M(θ,φ) is the number of reconstructed events in a cell (θ,φ) of the matrix M; ΔMT and ΔMP are corrections for temperature and pressure effects: Р is the current pressure at registration level, Р0 = 993 mbar is the averaged over a long period pressure at the registration level, В(θ) are barometric coefficients; WT(h,θ) are differential in atmospheric depth temperature coefficients [3], ΔT(h)=TSMA(h)-T(h) is the change of the temperature, h is the atmospheric depth, Δh = 0.05 atm, T(h) is the current temperature profile of the atmosphere, TSMA(h) is the temperature profile for standard model of the atmosphere [4]. Information about temperature profile of the atmosphere is obtained from meteorological balloon flights and numerical forecasting model of the atmosphere GDAS (The Global Data Assimilation System) [5].

Vector of local anisotropy
URAGAN allows to obtain the vectors of the particle arrival directions and we can use their vector sum. The summed vector normalized to the total number of muons (the vector of local anisotropy) will characterize the angular distribution of the detected particles. The projections of this vector (AS, AE and AZ) can be generally defined from the original matrix data M as follows [6]: , , , cos sin , ( , ) , , , sin sin , , , , cos , , , , .
Experimental temporal series of the vertical projection of the local anisotropy AZ for aforementioned intervals of zenith angles were obtained for 2007-2015. For AZ estimations, 1 h averaged muon matrices with corrections for barometric and temperature effects were used. As seen from figure 2, the annual variations in the vertical projection of the anisotropy vector AZ are observed, and they are not related with changes in atmospheric conditions. It was supposed that the annual variations are caused by changes in the shape of the energy spectrum of primary particles. This dependence can be estimated with a help of simulation of generation and propagation of secondary particles through the atmosphere.

Scheme of muon spectrum calculations with CORSIKA and experimental results
The direct simulation of propagation and interaction of protons of primary cosmic rays (CR) takes a long time. The method of relatively fast simulation of cosmic rays flux at the Earth's surface (with similar statistics) with a help of the CORSIKA package [7] in a wide range of zenith angles and energies was developed earlier [8]. For simulation of the flux of CR components, version 6.980 of the CORSIKA code was used. A combination of models of hadronic interactions SIBYLL + FLUKA was chosen. The differential energy spectrum of muons at the Earth surface dN/d(lgE) can be estimated by the following formulas: Here dGµ(Eµ,E)/dlgEµ is the distribution in the logarithm of the energy of muons for a fixed energy Е of primary particle. The shape of the primary spectrum: dNp/dE ~ E -γ . The integration over the energy of the primary protons is replaced by summation: Here Np1 = E1·ln10·dNp(E1)/dEp = 8.28·10 -2 (cm 2 ·s·sr) -1 is the normalization coefficient; E1 = 10 GeV is the reference energy for primary cosmic ray particles; ΔlgEp is the step in the logarithm of the energy of primary protons; γ is the index of the primary particles spectrum.
Counting rate for the cell of zenith angle θ for a current value of γ can be estimated by the next formula: is the integral intensity for zenith angle θ and threshold energy Emin; dNμ(θ,E,γ)/dEμ is the calculated differential spectrum. The value of the vertical projection of the vector of local anisotropy SIM Z () A  for different zenith angle intervals can be calculated using the following formula: