1. PRELIMINARY RESULTS OF SIMULATIONS L.G. Dedenko M.V. Lomonosov Moscow State University, 119992 Moscow, Russia

2. CONTENT Introduction 5-level scheme - Monte-Carlo for leading particles - Transport equations for hadrons - Transport equations for electrons and gamma quanta - Monte-Carlo for low energy particles in the real atmosphere - Responses of scintillator detectors The basic formula for estimation of energy Lateral distribution function A group method for muons The relativistic equation for a group Results for the giant inclined shower detected at the Yakutsk array Cherenkov radiation Conclusion

4. Transport equations for hadrons: here k=1,2,....m – number of hadron types; - number of hadrons k in bin E÷E+dE and depth bin x÷x+dx; λ k (E) – interaction length; B k – decay constant; W ik (E′,E) – energy spectra of hadrons of type k produced by hadrons of type i.

5. The integral form: here E 0 – energy of the primary particle; P b (E,x b ) – boundary condition; x b – point of interaction of the primary particle.

6. The decay products of neutral pions are regarded as a source function S γ (E,x) of gamma quanta which give origins of electron-photon cascades in the atmosphere: Here – a number of neutral pions decayed at depth x+ dx with energies E΄+dE΄

7. The basic cascade equations for electrons and photons can be written as follows: where Г(E,t), P(E,t) – the energy spectra of photons and electrons at the depth t; β – the ionization losses; μ e, μ γ – the absorption coefficients; W b, W p – the bremsstrahlung and the pair production cross-sections; S e, S γ – the source terms for electrons and photons.

8. The integral form: where At last the solution of equations can be found by the method of subsequent approximations. It is possible to take into account the Compton effect and other physical processes.

9. Source functions for low energy electrons and gamma quanta x=min(E 0 ;E/ε)

10. For the grid of energies E min ≤ E i ≤ E th (E min =1 MeV, E th =10 GeV) and starting points of cascades 0≤X k ≤X 0 (X 0 =1020 g∙cm -2 ) simulations of ~ 2·10 8 cascades in the atmosphere with help of CORSIKA code and responses (signals) of the scintillator detectors using GEANT 4 code SIGNγ(Rj,Ei,Xk) 10m≤Rj≤2000m have been calculated

11. Responses of scintillator detectors at distance R j from the shower core (signals S(R j )) E th =10 GeV E min =1 MeV

17. Source test function: S γ (E,x)dEdx=P(E 0,x)/E γ dEdx P(E 0,x) – a cascade profile of a shower ∫dx∫dES γ (E,x)=0.8E 0 Basic formula: E 0 =a·(S 600 ) b

19. Number of muons in a group with h k (x k ) and E i : here P(E,x) from equations for hadrons; D(E,E μ ) – decay function; limits E min (E μ ), E max (E μ ); W(E μ,E thr,x,x 0 ) – probability to survive.

20. here p 0 =0.2 ГэВ/с. Transverse impulse distribution:

21. here h k = h k (x k ) – production height for hadrons. The angle α:

22. Direction of muon velocity is defined by directional cosines: All muons are defined in groups with bins of energy E i ÷E i +ΔE; angles α j ÷α j +Δα j, δ m ÷ δ m +Δ δ m and height production h k ÷ h k +Δh k. The average values have been used:,, and. Number of muons and were regarded as some weights.

23. The relativistic equation: here m μ – muon mass; e – charge; γ – lorentz factor; t – time; – geomagnetic field.

24. The explicit 2-d order scheme: here ; E thr, E – threshold energy and muon energy.

27. el_ed.jpg

28. ga_ed.jpg

29. pos_ed.jpg

66. CONCLUSION In terms of test functions: The basic formula used for energy estimation at the Yakutsk array have been confirmed at energies of 10 18 eV. At energies ~ 10 20 eV simulations display larger energies than this formula shows supporting the Greizen-Zatsepin-Kuzmin enigma. Lateral distribution function of signal used at the Yakutsk array have been confirmed by simulations. Estimate of energy of the giant air shower detected at the Yakutsk array is not less than 3·10 20 eV. χ 2 =57 for 25 d.o.f.

67. Acknowledgements We thank G.T. Zatsepin for useful discussions, the RFFI (grant 03-02-16290), INTAS (grant 03-51-5112) and LSS- 1782.2003.2 for financial support.