A simple method of the vertical muon energy spectrum simulations have been suggested. These calculations have been carried out in terms of various models of hadronic interactions. The most energetic $ pi^pm $-mesons and K$^pm $-mesons produced in had
ron interactions contribute mainly in to this energy spectrum of muons due to the very steep energy spectrum of the primary particles. So, some constraints on the hadronic interaction models may be set from a comparison of calculated results with the cosmic data on the vertical muon energy spectrum. This comparison showed that the most energetic secondary particles production is too high in case of the QGSJET II-04 model and rather low in case of the QGSJET II-03 model. These conclusion have been supported by the LHC data.
Here the results of calculations of pulses in Cherenkov light detectors for the Yakutsk array are presented. As long as the Vavilov-Cherenkov light is used to calibrate signals in scintillation detectors at the Yakutsk array it is vital for these mea
surements to be precise. The validity of measurements of the signal Q(400) used as the estimation parameter at the Yakutsk array has been confirmed. Our calculations show that the width of time pulses increases from nearly 50 ns at a distance of 100 m from the shower axis up to 700 ns at 1000 m.
The fluxes of electrons, positrons, gammas, Cherenkov photons and muons in individual extensive air showers induced by the primary protons and helium, oxygen and iron nuclei at the level of observation have been estimated with help of the code CORSIC
A 6.616. The comparison show that the values of the function Xi**2 per one degree of freedom changes from 1.1 for iron nuclei to 0.9 for primary protons. As this difference is small all readings of detectors of the Vavilov-Cherenkov radiation have been used. At last, readings of underground detectors of muons with energies above 1 GeV have been exploited to make definite conclusion about chemical composition.
The energies of the most energetic extensive air showers observed at the Yakutsk array have been estimated with help of the all detectors readings instead of using of the standard procedure with a parameter s(600). The energy of the most energetic ex
tensive air shower observed at the Yakutsk array happened to be 200, 200, 180 and 165 EeV with the values of the Xi**2 function per one degree of freedom 0.9, 1., 0.9 and 1.1 for the primary protons and helium, oxygen and iron nuclei accordingly.
The primary cosmic rays particles with energies above 10**20 eV have been observed at many extensive air shower arrays since the beginning of observations over 40 years ago. The validity of measurements of signal s(600) used as energy estimation para
meter at the Yakutsk array has been confirmed. Our calculations show that the width of the time pulses increases from nearly 100 ns at a distance of 100 m from the shower axis up to 4 - 5 $mu$s at 1500 m. The calculated estimate of energy of extensive air shower is ~ 1.7 times smaller than the experimental estimate for the same value s(600). The pointing directions of extensive air showers observed at the Pierre Auger Observatory were fitted within +-3.1**o with positions of the nearby active galactic nuclei from the Veron-Cetty and P. Veron catalog. The cosmic ray luminosity of the active galactic nuclei which happened to be a source of the particular cosmic ray event constitutes a fraction ~10**-4 of the optical one if only cosmic ray particles with energies above 6*10**19 eV are produced. If produced cosmic ray particles have a spectrum up to ~ 100 GeV then the cosmic ray luminosity of the active galactic nuclei should be much higher than the optical one.
The pointing directions of extensive air showers observed at the Pierre Auger Observatory were fitted within 3.1 degree with positions of the nearby active galactic nuclei from the Veron-Cetty and P. Veron catalog. The cosmic ray luminosity of the ac
tive galactic nuclei which happened to be a source of the particular cosmic ray event constitutes a fraction ~0.0001 of the optical one if only cosmic ray particles with energies above 60 EeV are produced. If produced cosmic ray particles have a spectrum dE/E^3 up to ~100 GeV then the cosmic ray luminosity would be much higher than the optical one of the active galactic nuclei.
