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First energy estimates of giant air showers
with help of the hybrid scheme 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 - 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
ENERGY SCALE
SPACE SCALE
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
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
0 200 400 600 800 1000 12000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E0=1017 eV
Ethr
=0.00005 GeV
=30o
6
5
4
3
2
1
E/E
0
X, g/cm2
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
0 200 400 600 800 1000 12000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E0=1018 eV
Ethr
=0.00005 GeV
=30o
6
5
4
3
2
1
E/E
0
X, g/cm2
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
0 200 400 600 800 1000 12000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E0=1020 eV
Ethr
=0.00005 GeV
=30o
6
5
4
3
2
1
E/E
0
X, g/cm2
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
0 200 400 600 800 1000 12000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
5
6
E0=3*1020 eV
Ethr
=0.00005 GeV
=30O
4
3
2
1
E/E
0
X, g/cm2
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
0 200 400 600 800 1000 12000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E0=1017 eV
Ethr
=0.001 GeV
=30o
6
5
43
2
1
E/E
0
X, g/cm2
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
0 200 400 600 800 1000 12000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E0=1018 eV
Ethr
=0.001 GeV
=30o
6
5
4
3
2
1
E/E
0
X, g/cm2
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
0 200 400 600 800 1000 12000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E0=1020 eV
Ethr
=0.001 GeV
=30o
6
5
4
3
2
1
E/E
0
X, g/cm2
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
0 200 400 600 800 1000 12000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E0=3*1020 eV
Ethr
=0.001 GeV
=30o
6
5
4
3
2
1
E/E
0
X, g/cm2
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
0 200 400 600 800 1000 12000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E0=1019 eV
Ethr
=0.00005 GeV
=30o
6
5
4
3
2
1
E/E
0
X, g/cm2
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
0 200 400 600 800 1000 12000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E0=1019 eV
Ethr
=0.001 GeV
=30o
6
5
4
3
2
1
E/E
0
X, g/cm2
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
0 200 400 600 800 1000 12000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E0=1019 eV
Ethr
=0.01 GeV
=30o
6
5
4
3
2
1
E/E
0
X, g/cm2
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
0 200 400 600 800 1000 12000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E0=1019 eV
Ethr
=0.5 GeV
=30o
6
54
3
2
1
E/E
0
X, g/cm2
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
0 200 400 600 800 1000 12000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E0=1019 eV
Ethr
=10 GeV
=30o
6
54
3
2
1
E/E
0
X, g/cm2
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
0 200 400 600 800 1000 1200 1400 1600 1800 20000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E0=1019 eV
Ethr
=0.001 GeV
=60o
6
5 4
3
2
1
E/E
0
X, g/cm2
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
0 200 400 600 800 1000 1200 1400 1600 1800 20000,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E0=1019 eV
Ethr
=0.00005 GeV
=60o
6
5
4
3
2
1
E/E
0
X, g/cm2
Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3
0 200 400 600 800 1000 12000,00
0,05
0,10
0,15
0,20
0,25
0,30
0,35
0,40
0,45
0,50
0,55
0,60
0,65
E0=1017 eV
Ethr
=0.001 GeV
=30O
4
3
2
1
E/E
0
X, g/cm2
Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3
0 200 400 600 800 1000 12000,00
0,05
0,10
0,15
0,20
0,25
0,30
0,35
0,40
0,45
0,50
E0=1018 eV
Ethr
=0.001 GeV
=30O
4
32
1
E/E
0
X, g/cm2
Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3
0 200 400 600 800 1000 1200 1400 1600 1800 20000,0
0,2
0,4
0,6
0,8
1,0
E0=1019 eV
Ethr
=0.001 GeV
=60O
4
3
2
1
E/E
0
X, g/cm2
Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3
0 200 400 600 800 1000 12000,00
0,05
0,10
0,15
0,20
0,25
E0=1020 eV
Ethr
=0.001 GeV
=30O
4
32
1
E/E
0
X, g/cm2
Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3
0 200 400 600 800 1000 12000,00
0,05
0,10
0,15
0,20
0,25
E0=3*1020 eV
Ethr
=0.001 GeV
=30O 4
32
1
E/E
0
X, g/cm2
Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3
0 200 400 600 800 1000 12000,00
0,05
0,10
0,15
0,20
0,25
0,30
0,35
E0=1018 eV
Ethr
=0.00005 GeV
=30O
4
3
2
1
E/E
0
X, g/cm2
Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3
0 200 400 600 800 1000 12000,00
0,05
0,10
0,15
0,20
0,25
E0=1019 eV
Ethr
=0.001 GeV
=30O
4
3
2
1
E/E
0
X, g/cm2
Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3
0 200 400 600 800 1000 12000,0
0,2
0,4
0,6
0,8
1,0
E0=1019 eV
Ethr
=0.01 GeV
=30O
4
3
2
1
E/E
0
X, g/cm2
Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3
0 200 400 600 800 1000 12000,0
0,2
0,4
0,6
0,8
1,0
E0=1019 eV
Ethr
=0.5 GeV
=30O
4
3
2
1
E/E
0
X, g/cm2
Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3
0 200 400 600 800 1000 12000,0
0,2
0,4
0,6
0,8
1,0
E0=1019 eV
Ethr
=10 GeV
=30O
4
32
1
E/E
0
X, g/cm2
Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3
0 200 400 600 800 1000 1200 1400 1600 1800 20000,0
0,2
0,4
0,6
0,8
1,0
E0=1019 eV
Ethr
=0.00005 GeV
=60O
4
3
2
1
E/E
0
X, g/cm2
Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3
0 200 400 600 800 1000 12000,0
0,2
0,4
0,6
0,8
1,0
E0=1018 eV
Ethr
=10 GeV
=0O
3
2
1
4
E/E
0
X, g/cm2
Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3
0 200 400 600 800 1000 12000,0
0,2
0,4
0,6
0,8
E0=1019 eV
Ethr
=10 GeV
=0O
3
2
1
4
E/E
0
X, g/cm2
Energy by under threshold: 1 - by electrons; 2 - by photons; 3 - by pair; 4 - sum of 1, 2, 3
0 200 400 600 800 1000 12000,0
0,2
0,4
0,6
0,8
E0=1020 eV
Ethr
=10 GeV
=0O
3
2
1
4
E/E
0
X, g/cm2
B-H SHOWERS
0 200 400 600 800 1000100
101
102
103
104
105
106
107
108
109
1010
1011
1012
E0=1020 Ev
N(X
)
X, g/cm2
Cascade curves: - NKG; - LPM; lines - individual LPM curves
0 200 400 600 800 100010-2
10-1
100
101
102
103
104
105
106
107
108
109
1010
1011
1012
E0=1020eV
cos=1.
N(X
)
X, g/cm2
Cascade curves:______ - NKG; ______ - LPM
0 200 400 600 800 100010-1
100
101
102
103
104
105
106
107
108
109
1010
1011
E0=1020eV
cos=1.
N(X
)
X, g/cm2
Cascade curves: - NKG; - LPM; lines - individual LPM curves
0 200 400 600 800 1000 1200 1400 1600 180010-1
100
101
102
103
104
105
106
107
108
109
1010
1011
1012
E0=1020eV
cos=0.6
N(X
)
X, g/cm2
Cascade curves: ______ - NKG; ______ - LPM
0 200 400 600 800 1000 1200 1400 1600 1800100
101
102
103
104
105
106
107
108
109
1010
1011
1012
E0=1020eV
cos=0.6
N
(X)
X, g/CM2
Cascade curves: ______ - NKG; ______ - LPM
0 500 1000 1500 2000100
101
102
103
104
105
106
107
108
109
1010
1011
1012
E0=1020eV
cos=0.5
N(X
)
X, g/cm2
Cascade curves:_____ - NKG; ______ - LPM
500 1000 1500108
109
1010
1011
E0=1020eV
cos=0.5
N(X
)
X, g/cm2
Muon density in gamma-induced showers:______ - BH; ______ - LPM; ■ – Plyasheshnikov, Aharonian;
- our individual points
1E18 1E19 1E20
10-3
10-2
10-1
Lo
gN
(1
00
0)
(1/m
2 )
LogE0 (eV)
Muon density in gamma-induced showers:1 - AGASA; 2 - Homola et al.; 3 - BH; 4 - Plyasheshnikov, Aharonian;
5, 6 - our calculations; 7 - LPM
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
Energy spectrum of electrons
Energy spectrum of photons
Estimates of energy with test functions
AGASA simulation
Model of detector
Detector response for gammas
Detector response for electrons
Detector response for positrons
Detector response for muons
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
98.0600
170 108.4 sE
08.1600
170 102.3 sE
988.0600
170 103 sE
99.0600
170 106.2 sE
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
assuming aerosol-free air … more typical air => E ≈ 200 EeV
(atmospheric monitoring not yet routine in early 2004 …)
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
2,7
2,9
3,1
3,3
3,5
3,7
3,9
4,1
4,3
0 2 4 6 8 10 12 14 16 18 20
altitude (km a.s.l.)
flu
ore
scen
ce y
ield
(p
ho
ton
s/m
)
Keilhauer with Ulrich cross sections, only 10wavelengths as for NaganoKeilhauer with Ulrich cross sections, all 19wavelengthsNagano (2004)
Kakimoto (1996)
Altitude dependence
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.
Figure 2. FD energy vs. ground parameter S38. 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.
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.
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.
