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PRELIMINARY RESULTS OF SIMULATIONS. L.G. Dedenko M.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 - PowerPoint PPT Presentation
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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.