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
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:
Pk ( E , x)
  Pk ( E , x) / k ( E )  Bk Pk ( E , x) /( E  x) 
x
m
  dE P ( E , x)W
i 1
i
ik
( E , E ) / i ( E ),
here k=1,2,....m – number of hadron types;
Pk ( E , x)dEdx - 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.
The integral form:
Pk ( E , x)  Pb ( E , xb )  exp( ( x  xb ) /  ( E )  ( B / E ) ln( x / xb )) 
x
  d  exp( ( x   ) /  ( E )  ( B /E ) ln( x /  ))  f ( E ,  ),
xb
here
f ( E, ) 
m
E0
  dE P ( E ,  )W
i 1
i
ik
( E , E ) /  i ( E )
E
E0 – energy of the primary particle; Pb (E,xb) –
boundary condition; xb – point of interaction of the
primary particle.
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:
E0
S  ( E , x)  2  P 0 ( E , x)E )dE  / E 
E
S e ( E , x)  0.
Here P ( E , x)dE 
– a number of neutral pions
decayed at depth x+ dx with energies E΄+dE΄
0
The basic cascade equations for electrons and photons
can be written as follows:
P / t    e P  P / E  S e   PWb dE    ГW p dE 
Г / t     S   PWb dE '
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.
The integral form:
t
P( E , t )  P[ E   (t  t 0 ), t 0 ] exp(    e [ E   (t   )]d ) 
t
t
t0

t0
  d exp(    e [ E   (t  t )]dt [ S e [ E   (t   ),  ]  Ae  Be ]
Г ( E , t )  Г ( E , t 0 ) exp(    ( E )(t  t 0 )] 
t
  d exp[   ( E )(t   )][ S  ( E ,  )   dE P( E ,  )Wb ( E , E )],
t0
where
Ae   P( E ,  )Wb [ E , E   (t   )]dE ,
Be   Г ( E ,  )W p [ E , E   (t   )]dE 
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.
Source functions for low energy electrons and gamma
quanta
x
S ( E , t )   dE P( E , t )Wb ( E , E )
E
E0
S e ( E , t )   dE ( P( E , t )Wb ( E , E )  ( E , t )W p ( E , E ))
E
x=min(E0;E/ε)
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
1,0
6
17
E0=10 eV
Ethr=0.00005 GeV
0,9
0,8
1
=30
o
5
0,7
3
E/E0
0,6
0,5
4
0,4
2
0,3
0,2
0,1
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
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
1,0
6
18
E0=10 eV
Ethr=0.00005 GeV
0,9
0,8
1
=30
o
5
0,7
3
E/E0
0,6
0,5
0,4
2
0,3
0,2
4
0,1
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
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
1,0
6
20
E0=10 eV
Ethr=0.00005 GeV
0,9
0,8
1
=30
o
5
0,7
E/E0
0,6
3
0,5
0,4
2
0,3
0,2
4
0,1
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
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
1,0
6
20
0,9
E0=3*10 eV
Ethr=0.00005 GeV
0,8
=30
0,7
5
O
1
3
E/E0
0,6
0,5
0,4
2
0,3
0,2
4
0,1
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
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
1,0
17
6
E0=10 eV
Ethr=0.001 GeV
0,9
0,8
1
=30
o
5
0,7
4
E/E0
0,6
3
0,5
2
0,4
0,3
0,2
0,1
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
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
1,0
18
6
E0=10 eV
Ethr=0.001 GeV
0,9
0,8
1
=30
o
5
0,7
E/E0
0,6
3
0,5
2
0,4
4
0,3
0,2
0,1
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
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
1,0
20
6
E0=10 eV
Ethr=0.001 GeV
0,9
0,8
1
=30
o
5
0,7
E/E0
0,6
3
0,5
2
0,4
0,3
4
0,2
0,1
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
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
1,0
20
6
E0=3*10 eV
Ethr=0.001 GeV
0,9
0,8
1
=30
o
5
0,7
E/E0
0,6
3
0,5
2
0,4
0,3
4
0,2
0,1
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
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
1,0
6
19
E0=10 eV
Ethr=0.00005 GeV
0,9
0,8
1
=30
o
5
0,7
E/E0
0,6
0,5
3
0,4
2
0,3
4
0,2
0,1
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
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
1,0
19
6
E0=10 eV
Ethr=0.001 GeV
0,9
0,8
1
=30
o
5
0,7
E/E0
0,6
0,5
3
0,4
2
0,3
4
0,2
0,1
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
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
1,0
6
19
E0=10 eV
Ethr=0.01 GeV
0,9
0,8
1
=30
5
o
0,7
3
E/E0
0,6
0,5
2
0,4
0,3
0,2
4
0,1
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
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
1,0
6
19
E0=10 eV
Ethr=0.5 GeV
0,9
0,8
1
=30
o
5
4
0,7
3
E/E0
0,6
0,5
2
0,4
0,3
0,2
0,1
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
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
1,0
6
19
E0=10 eV
Ethr=10 GeV
0,9
0,8
1
=30
o
5
4
0,7
3
E/E0
0,6
0,5
2
0,4
0,3
0,2
0,1
0,0
0
200
400
800
600
X, g/cm
2
1000
1200
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
1,0
6
0,9
19
E0=10 eV
Ethr=0.001 GeV
0,8
=60
0,7
5
o
4
1
E/E0
0,6
3
0,5
0,4
2
0,3
0,2
0,1
0,0
0
200
400
600
800
1000
X, g/cm
1200
2
1400
1600
1800
2000
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
1,0
6
0,9
19
E0=10 eV
Ethr=0.00005 GeV
0,8
=60
0,7
5
o
4
1
E/E0
0,6
3
0,5
0,4
2
0,3
0,2
0,1
0,0
0
200
400
600
800
1000
X, g/cm
1200
2
1400
1600
1800
2000
Energy by under threshold: 1 - by electrons; 2 - by photons;
3 - by pair; 4 - sum of 1, 2, 3
0,65
4
17
0,55
E0=10 eV
Ethr=0.001 GeV
0,50
=30
0,60
O
0,45
E/E0
0,40
0,35
1
0,30
0,25
0,20
2
0,15
3
0,10
0,05
0,00
0
200
400
600
X, g/cm
800
2
1000
1200
Energy by under threshold: 1 - by electrons; 2 - by photons;
3 - by pair; 4 - sum of 1, 2, 3
0,50
4
18
0,45
E0=10 eV
Ethr=0.001 GeV
0,40
=30
O
0,35
E/E0
0,30
1
0,25
0,20
0,15
2
3
0,10
0,05
0,00
0
200
400
600
X, g/cm
800
2
1000
1200
Energy by under threshold: 1 - by electrons; 2 - by photons;
3 - by pair; 4 - sum of 1, 2, 3
1,0
19
E0=10 eV
Ethr=0.001 GeV
0,8
=60
4
O
E/E0
0,6
1
0,4
2
0,2
3
0,0
0
200
400
600
800
1000
X, g/cm
1200
2
1400
1600
1800
2000
Energy by under threshold: 1 - by electrons; 2 - by photons;
3 - by pair; 4 - sum of 1, 2, 3
0,25
4
20
E0=10 eV
Ethr=0.001 GeV
0,20
=30
O
0,15
E/E0
1
0,10
2
3
0,05
0,00
0
200
400
600
X, g/cm
800
2
1000
1200
Energy by under threshold: 1 - by electrons; 2 - by photons;
3 - by pair; 4 - sum of 1, 2, 3
0,25
20
E0=3*10 eV
Ethr=0.001 GeV
0,20
=30
4
O
E/E0
0,15
1
0,10
2
3
0,05
0,00
0
200
400
600
X, g/cm
800
2
1000
1200
Energy by under threshold: 1 - by electrons; 2 - by photons;
3 - by pair; 4 - sum of 1, 2, 3
0,35
4
18
E0=10 eV
Ethr=0.00005 GeV
0,30
=30
O
0,25
E/E0
0,20
1
0,15
0,10
2
3
0,05
0,00
0
200
400
600
X, g/cm
800
2
1000
1200
Energy by under threshold: 1 - by electrons; 2 - by photons;
3 - by pair; 4 - sum of 1, 2, 3
0,25
19
E0=10 eV
Ethr=0.001 GeV
0,20
=30
4
O
E/E0
0,15
1
0,10
2
0,05
3
0,00
0
200
400
600
X, g/cm
800
2
1000
1200
Energy by under threshold: 1 - by electrons; 2 - by photons;
3 - by pair; 4 - sum of 1, 2, 3
1,0
19
E0=10 eV
Ethr=0.01 GeV
0,8
=30
O
E/E0
0,6
4
0,4
1
0,2
2
3
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
Energy by under threshold: 1 - by electrons; 2 - by photons;
3 - by pair; 4 - sum of 1, 2, 3
1,0
19
E0=10 eV
Ethr=0.5 GeV
0,8
=30
O
4
E/E0
0,6
0,4
1
2
0,2
3
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
Energy by under threshold: 1 - by electrons; 2 - by photons;
3 - by pair; 4 - sum of 1, 2, 3
1,0
19
E0=10 eV
Ethr=10 GeV
0,8
=30
4
O
E/E0
0,6
1
0,4
2
3
0,2
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
Energy by under threshold: 1 - by electrons; 2 - by photons;
3 - by pair; 4 - sum of 1, 2, 3
1,0
19
E0=10 eV
Ethr=0.00005 GeV
0,8
=60
4
O
E/E0
0,6
1
0,4
2
3
0,2
0,0
0
200
400
600
800
1000
X, g/cm
1200
2
1400
1600
1800
2000
Energy by under threshold: 1 - by electrons; 2 - by photons;
3 - by pair; 4 - sum of 1, 2, 3
1,0
4
18
E0=10 eV
Ethr=10 GeV
0,8
=0
O
E/E0
0,6
1
0,4
2
0,2
3
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
Energy by under threshold: 1 - by electrons; 2 - by photons;
3 - by pair; 4 - sum of 1, 2, 3
0,8
4
19
E0=10 eV
Ethr=10 GeV
=0
E/E0
0,6
O
1
0,4
2
0,2
3
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
Energy by under threshold: 1 - by electrons; 2 - by photons;
3 - by pair; 4 - sum of 1, 2, 3
0,8
4
20
E0=10 eV
Ethr=10 GeV
E/E0
0,6
=0
O
0,4
1
2
0,2
3
0,0
0
200
400
600
X, g/cm
800
2
1000
1200
B-H SHOWERS
12
10
11
10
10
10
9
10
20
E0=10 Ev
8
10
7
N(X)
10
6
10
5
10
4
10
3
10
2
10
1
10
0
10
0
200
400
600
X, g/cm
2
800
1000

Cascade curves:
- NKG;

- LPM; lines - individual LPM curves
12
10
11
20
10
E0=10 eV
cos=1.
10
10
9
10
8
10
7
10
6
N(X)
10
5
10
4
10
3
10
2
10
1
10
0
10
-1
10
-2
10
0
200
400
600
X, g/cm
2
800
1000
Cascade curves:
______ - NKG;
______ - LPM
11
10
20
E0=10 eV
cos=1.
10
10
9
10
8
10
7
10
N(X)
6
10
5
10
4
10
3
10
2
10
1
10
0
10
-1
10
0
200
400
600
X, g/cm
2
800
1000
Cascade curves:

- NKG;

- LPM; lines - individual LPM curves
12
10
11
10
10
10
9
10
8
10
7
10
N(X)
6
10
5
10
20
E0=10 eV
cos=0.6
4
10
3
10
2
10
1
10
0
10
-1
10
0
200
400
600
800
1000
X, g/cm
2
1200
1400
1600
1800
Cascade curves:
______ - NKG;
______ - LPM
12
10
11
10
10
10
9
10
8
10
7
N(X)
10
6
10
20
E0=10 eV
cos=0.6
5
10
4
10
3
10
2
10
1
10
0
10
0
200
400
600
800
1000
2
X, g/CM
1200
1400
1600
1800
Cascade curves:
______ - NKG;
______ - LPM
12
10
11
10
10
10
9
10
8
10
7
N(X)
10
6
10
20
5
10
E0=10 eV
cos=0.5
4
10
3
10
2
10
1
10
0
10
0
500
1000
X, g/cm
1500
2
2000
Cascade curves:
_____ - NKG;
______ - LPM
11
10
10
N(X)
10
20
E0=10 eV
cos=0.5
9
10
8
10
500
1000
X, g/cm
1500
2
Muon density in gamma-induced showers:
______ - BH; ______ - LPM; ■ – Plyasheshnikov, Aharonian;
- our individual points
2
LogN(1000) (1/m )

-1
10
-2
10
-3
10
1E18
1E19
LogE0 (eV)
1E20
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))
x0
Eth
xb
Emin
S ( R j )   d  dE (S ( E, )SIGN ( R j , E, ) Se ( E, )SIGNe ( R j , E, ))
Eth=10 GeV
Emin=1 MeV
Source test function:
Sγ(E,x)dEdx=P(E0,x)/EγdEdx
P(E0,x) – a cascade profile of a shower
(x  C) 2
P( E0 , x)  K 0 exp( 
)
2
A( x  C )  2 B
∫dx∫dESγ(E,x)=0.8E0
Basic formula:
E0=a·(S600)b
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:
0.98
E0  4.8  1017  s600
• Test source function with γ=1
Coefficient: 4.8/3.2=1.5
1.08
E0  3.2  1017  s600
• Source function from CORSIKA
Coefficient: 4.8/3=1.6
0.988
E0  3  1017  s600
• Thinning by CORSIKA (10-6)
Coefficient: 4.8/2.6=1.8
0.99
E0  2.6  1017  s600
Direction of muon velocity is defined by directional
cosines:
sin   cos   cos   cos   cos   sin   cos   sin   sin   sin  ;
sin   sin   cos   cos   sin   sin   cos   cos   sin   sin  ;
 cos   cos   sin   sin   cos 
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: E ,  j ,  m and hk .



N
Number of muons N  and
 were regarded as
some weights.
The relativistic equation:

 
dV
m
 eV  B,
dt
here mμ – muon mass; e – charge; γ –
lorentz factor; t – time;  – geomagnetic
B
field.
The explicit 2-d order scheme:
Vx
n 1 / 2
 Vx  CHE n (V y Bz  Vz B y )  (0.5  ht );
Vy
n 1 / 2
 V y  CHE n (Vz Bx  Vx Bz )  (0.5  ht );
Vz
n 1 / 2
 Vz  CHE n (Vx B y  V y Bx )  (0.5  ht );
n
n
x n1 / 2  x n  Vx  (0.5  ht )
n
n
n
y n1 / 2  y n  V y  (0.5  ht )
n
n
Vx
n 1
Vy
n 1
 V y  CHE
Vz
n 1
 Vz  CHE n1 / 2 (Vx
n
 Vx  CHE n1 / 2 (V y
n
n
n
n
n 1 / 2
n
n 1 / 2
(Vz
Bz  Vz
n 1 / 2
n 1 / 2
n 1 / 2
Bx  Vx
By  Vy
B y )  ht ;
n
z n1 / 2  z n  Vz  (0.5  ht )
x n1  x n  Vx
n 1 / 2
Bz )  ht ;
y n1  y n  V y
Bx )  ht ;
z n1  z n  Vz
n 1 / 2
n 1 / 2
n
n 1 / 2
n 1 / 2
here CHE  e  ( Ethr / E ) ;
Ethr , E – threshold energy and muon energy.
 ht
 ht
 ht ,
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
usahuentemeyer-P-abs2-he15oral
Altitude dependence
4,3
fluorescence yield (photons/m)
4,1
3,9
3,7
3,5
3,3
Keilhauer with Ulrich cross sections, only 10
wavelengths as for Nagano
Keilhauer with Ulrich cross sections, all 19
wavelengths
Nagano (2004)
3,1
2,9
Kakimoto (1996)
2,7
0
2
4
6
8
10
12
altitude (km a.s.l.)
14
16
18
20
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 LSS5573.2006.2 for financial support.