PRELIMINARY RESULTS OF SIMULATIONS

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PRELIMINARY RESULTS OF SIMULATIONS

• L.G. DedenkoM.V. Lomonosov Moscow State
  University,119992 Moscow, Russia

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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

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Transport equations for hadrons

• here k1,2,....m number of hadron types
• - number of hadrons k in bin
  EEdE and depth bin xxdx ?k(E)
  interaction length Bk decay constant
  Wik(E',E) energy spectra of hadrons of type k
  produced by hadrons of type i.

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The integral form

• here
• E0 energy of the primary particle Pb (E,xb)
  boundary condition xb point of interaction
  of the primary particle.

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• The decay products of neutral pions are regarded
  as a source function S?(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?

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• The basic cascade equations for electrons and
  photons can be written as follows
• where ?(E,t), P(E,t) the energy spectra of
  photons and electrons at the depth t ß the
• ionization losses µe, µ? the absorption
  coefficients Wb, Wp the bremsstrahlung and
• the pair production cross-sections Se, S? the
  source terms for electrons and photons.

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• 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.

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• Source functions for low energy electrons and
  gamma quanta
• xmin(E0E/e)

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• For the grid of energies
• Emin Ei Eth (Emin1 MeV, Eth10 GeV)
• and starting points of cascades
• 0XkX0 (X01020 gcm-2)
• simulations of 2108 cascades in the atmosphere
  with help of CORSIKA code and responses (signals)
  of the scintillator detectors using GEANT 4 code
• SIGN?(Rj,Ei,Xk)
• SIGN?(Rj,Ei,Xk)
• 10mRj2000m
• have been calculated

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• Responses of scintillator detectors at distance
  Rj from the shower core (signals S(Rj))
• Eth10 GeV
• Emin1 MeV

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• Source test function
• S?(E,x)dEdxP(E0,x)/E?dEdx
• P(E0,x) a cascade profile of a shower
• ?dx?dES?(E,x)0.8E0
• Basic formula
• E0a(S600)b

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• 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.

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Transverse impulse distribution

• 
  
  
• here p00.2 ???/?.

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The angle a

• 
  
• here hk hk(xk) production height for hadrons.

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• Direction of muon velocity is defined by
  directional cosines
• 
  
• All muons are defined in groups with bins of
  energy EiEi?E angles ajaj?aj,
• dm dm? dm and height production hk hk ?hk.
  The average values have been used , ,
  and . Number of muons and
  were regarded as some weights.

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The relativistic equation

• here mµ muon mass e charge ? lorentz
  factor t time geomagnetic field.

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The explicit 2-d order scheme

• here
• Ethr , E threshold energy and muon energy.

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CONCLUSION

• In terms of test functions
• The basic formula used for energy estimation at
  the Yakutsk array have been confirmed at energies
  of 1018 eV.
• At energies 1020 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
  31020 eV.
• ?257 for 25 d.o.f.

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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.