PRELIMINARY RESULTS OF SIMULATIONS

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

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

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.

),(/),(),(

)/(),()(/),(),(

1

EEEWxEPEd

xExEPBExEPx

xEP

iik

m

ii

kkkkk

dEdxxEPk ),(

The integral form:

here

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

),,())/ln()/()(/)(exp(

))/ln()/()(/)(exp(),(),(

EfxEBExd

xxEBExxxEPxEPx

x

bbbbk

b

0

)(/),(),(),(1

E

E

iiki

m

i

EEEWEPEdEf

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΄

.0),(

/)),(2),(0

0

xES

EEdExEPxES

e

E

E

EdxEP ),(0

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.

EdГWEdPWSEPPtP pbee //

'/ dEPWStГ b

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.

)])((exp(),(),( 00 ttEtEГtEГ

,)],(),(),()][)(([exp0

EEWEPEdEStEd b

t

t

,)](,[),( EdtEEWEPA be

t

t

e dtEtttEPtEP0

))]([exp(]),([),( 00

t

t

t

eeee BAtEStdttEd0

]]),([[)]([exp(

EdtEEWEГB pe )](,[),(

Source functions for low energy electrons and gamma quanta

x=min(E0;E/ε)

)),(),(),(),((),(

),(),(),(

0

EEWtEEEWtEPEdtES

EEWtEPEdtES

p

E

E

be

x

E

b

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

Responses of scintillator detectors at distance Rj from the shower core (signals S(Rj))

Eth=10 GeV

Emin=1 MeV

)),,(),(),,(),(()(min

0

ERSIGNESERSIGNESdEdRS jee

E

E

j

x

x

j

th

b

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

)2)(

)(exp(),(

2

2

00 BCxA

CxKxEP

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.

1 1 max

min

)(

)(

0 ),,(),(),,,(k

k

i

i

x

x

E

E

EE

EE

thr xEPEEDE

dExxEEWdE

x

dxN

here p0=0.2 ГэВ/с.

,/)/exp()( 200 pdppppdppf

Transverse impulse distribution:

here hk= hk(xk) – production height for hadrons.

,// Ecphrtg kjj

The angle α:

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+Δ δm and height production hk÷ hk +Δhk. The average values have been used: , , and . Number of muons and were regarded as some weights.

cossinsincoscos

;sinsincoscossinsincoscossinsin

;sinsinsincossincoscoscoscossin

E jm kh

N N

The relativistic equation:

here mμ – muon mass; e – charge; γ – lorentz factor; t – time; – geomagnetic field.

,BVedt

Vdm

B

The explicit 2-d order scheme:

here ;

Ethr , E – threshold energy and muon energy.

);5.0()(2/1ty

nzz

ny

nnx

nx hBVBVCHEVV

)5.0(2/1t

nx

nn hVxx

);5.0()(2/1tz

nxx

nz

nny

ny hBVBVCHEVV

);5.0()(2/1tx

nyy

nx

nnz

nz hBVBVCHEVV

;)( 2/12/12/11ty

nzz

ny

nnx

nx hBVBVCHEVV

;)( 2/12/12/11tz

nxx

nz

nny

ny hBVBVCHEVV

;)( 2/12/12/11tx

nyy

nx

nnz

nz hBVBVCHEVV

)5.0(2/1t

ny

nn hVyy

)5.0(2/1t

nz

nn hVzz

tn

xnn hVxx 2/11

tn

ynn hVyy 2/11

,2/11t

nz

nn hVzz

)/( EEeCHE thr

el_ed.jpg

ga_ed.jpg

pos_ed.jpg

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 3·1020 eV.

χ2=57 for 25 d.o.f.

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.