1. First energy estimates of giant air showers with help of the hybrid scheme 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 - The LPM showers - The primary photons - Monte-Carlo for low energy particles in the real atmosphere - Responses of scintillator detectors The basic formula for estimation of energy The relativistic equation for a group of muons Results for the giant inclined shower detected at the Yakutsk array Conclusion

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. Balance of energy by 1 - the primary photon; 2 - electrons; 3 - photons and 4 - under threshold in e-ph shower; 5 - sum of 1,2,3; 6 - total sum

27. Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3

41. B-H SHOWERS

42. Cascade curves: - NKG; - LPM; lines - individual LPM curves

43. Cascade curves: ______ - NKG; ______ - LPM

44. Cascade curves: - NKG; - LPM; lines - individual LPM curves

45. Cascade curves: ______ - NKG; ______ - LPM

47. Cascade curves: _____ - NKG; ______ - LPM

48. Muon density in gamma-induced showers: ______ - BH; ______ - LPM; ■ – Plyasheshnikov, Aharonian; - our individual points

49. Muon density in gamma-induced showers: 1 - AGASA; 2 - Homola et al.; 3 - BH; 4 - Plyasheshnikov, Aharonian; 5, 6 - our calculations; 7 - LPM

50. 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

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

61. 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

62. Energy spectrum of electrons

63. Energy spectrum of photons

64. Estimates of energy with test functions

66. AGASA simulation

67. Model of detector

68. Detector response for gammas

69. Detector response for electrons

70. Detector response for positrons

71. Detector response for muons

73. Comparison of various estimates of energy Experimental data: Test source function with γ=1 Coefficient: 4.8/3.2=1.5 Source function from CORSIKA Coefficient: 4.8/3=1.6 Thinning by CORSIKA (10-6) Coefficient: 4.8/2.6=1.8

81. 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.

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

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

87. assuming aerosol-free air … more typical air => E ≈ 200 EeV (atmospheric monitoring not yet routine in early 2004 …)

88. Summary: Air Fluorescence Yield Measurements Kakimoto et al., NIM A372 (1996) Nagano et al., Astroparticle Physics 20 (2003) Belz et al., submitted to Astroparticle Physics 2005; astro-ph/0506741 Huentemeyer et al., proceedings of this conference usa- huentemeyer-P-abs2-he15- oral

89. Altitude dependence

90. Lateral width of shower image in the Auger fluorescence detector. Figure 1. Image of two showers in the photomultiplier camera. The reconstructed energy of both showers is 2.2 EeV. The shower on the left had a core 10.5 km from the telescope, while that on the right landed 4.5 km away. Note the number of pixels and the lateral spread in the image in each shower.

91. Figure 2. FD energy vs. ground parameter S 38. These are hybrid events that were recorded when there were contemporaneous aerosol measurements, whose FD longitudinal profiles include shower maximum in a measured range of at least 350 g cm -2, and in which there is less than 10% Cherenkov contamination.

92. CONCLUSION In terms of the hybrid scheme with help of CORSIKA The energy estimates for the Yakutsk array are a factor of 1.5-1.8 may be lower. The energy estimates for the AGASA array have been confirmed. Estimates of energy of the most giant air shower detected at the Yakutsk array should be checked. The LPM showers have a very small muon content.

93. Acknowledgements We thank G.T. Zatsepin for useful discussions, the RFFI (grant 03-02-16290), INTAS (grant 03-51-5112) and LSS- 5573.2006.2 for financial support.