PRELIMINARY RESULTS OF SIMULATIONS

1. PRELIMINARY RESULTS OF SIMULATIONS L.G. DedenkoM.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; λk(E) – interaction length; Bk – decay constant; Wik(E′,E) – energy spectra of hadrons of type k produced by hadrons of type i.

5. The integral form: here E0 – energy of the primary particle; Pb (E,xb) – boundary condition; xb– point of interaction of the primary particle.

6. The decay products of neutral pions are regarded as a source function Sγ(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; μe, μγ – the absorption coefficients; Wb, Wp – the bremsstrahlung and the pair production cross-sections; Se, Sγ– 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(E0;E/ε)

10. For the grid of energies Emin≤ Ei ≤ Eth (Emin=1 MeV, Eth=10 GeV) and starting points of cascades 0≤Xk≤X0 (X0=1020 g∙cm-2) simulations of ~ 2·108 cascades in the atmosphere with help of CORSIKA code and responses (signals) of the scintillator detectors using GEANT 4 code SIGNγ(Rj,Ei,Xk) SIGNγ(Rj,Ei,Xk) 10m≤Rj≤2000m have been calculated

11. Responses of scintillator detectors at distance Rj from the shower core (signals S(Rj)) Eth=10 GeV Emin=1 MeV

17. Source test function: Sγ(E,x)dEdx=P(E0,x)/EγdEdx P(E0,x) – a cascade profile of a shower ∫dx∫dESγ(E,x)=0.8E0 Basic formula: E0=a·(S600)b

19. Number of muons in a group with hk(xk) and Ei : here P(E,x) from equations for hadrons; D(E,Eμ) – decay function; limits Emin(Eμ), Emax(Eμ);W(Eμ,Ethr,x,x0) – probability to survive.

20. Transverse impulse distribution: here p0=0.2 ГэВ/с.

21. The angle α: here hk= hk(xk) – production height for hadrons.

22. Direction of muon velocity is defined by directional cosines: All muons are defined in groups with bins of energy Ei÷Ei+ΔE; angles αj÷αj+Δαj, δm÷ δm+Δδm and height production hk÷ hk +Δhk. The average values have been used: , , and . Number of muons and were regarded as some weights.

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

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

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