1. The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies L.G. Dedenko M.V. Lomonosov Moscow State University, 119992 Moscow, Russia

2. Content Introduction Hybrid multilevel scheme The 5-level scheme for the atmosphere Examples Conclusion

3. GOALS Simulations of cascades at ultra-high energies Acoustical (radio) signals production Transport of acoustical (radio) signals in the real matter Detections of signals

4. ENERGY SCALE

5. SPACE SCALE

6. 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.

7. 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.

8. 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΄

9. 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.

10. 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.

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

12. For the various 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

14. SIGNAL ESTIMATION

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

16. ENERGY DEPOSITION

17. POSITIVE CHARGE (GEANT4)

18. NEGATIVE CHARGE (GEANT4)

19. FOR HADRON CASCADES FLUCTUATIONS ARE OF IMPORTANCE

20. CHARGE EXCESS (GEANT4)

21. THIS FUNCTIONS SHOULD BE ESTIMATED WITH THE GEANT4 CODE WITH STATISTICS OF 10**6

22. FOR E=10**12 GEV NEARLY 10**12 PARTICLES SHOULD BE TAKEN INTO ACCOUNT

23. FOR ELECTRON-PHOTON CASCADES FLUCTUATIONS ARE VERY IMPORTANT DUE TO THE LPM-EFFECT

24. EXAMPLES or

25. The Poisson formulae I.C.: It is possible at time because

26. Energy deposition Q=dE/dV in water

27. Energy deposition in water

30. ENERGY DEPOSITION IN WATER

36. Charge excess

38. Lateral distributions of gammas, electrons and positrons

39. ENERGY DEPOSITION in detector

40. Energy distributions of gammas, electrons, positrons

41. Ratio of a signal to a charge particle density

43. el_ed.jpg

44. ga_ed.jpg

45. pos_ed.jpg

46. Conclusion The hybrid multilevel scheme has been suggested to estimate acoustical (radio) signals produced by eγ and eh cascades in dense medium.

47. 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.

50. 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.

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

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

53. 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.

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

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

58. Ratio with to without magnetic field