1. Signatures for showers:
Shower Characteristics
R. Vazquez, USC
Trasgo meeting, February 2010.
Santiago de Compostela

2. Extensive Air Showers iniated by Cosmic Rays have
intrinsic characteristics:
Size, timing, energies, densities, and rates are
intrinsic to the shower and must be taken into
account in the design of cosmic ray detectors.
The ability to determine physical parameters from
extensive air showers depends on the correct
interpretation of these characteristics.
Arrival direction
Energy
Chemical composition
Hadronic interactions

3. Cosmic ray showers: Heitler model
Energy Number of particles Depth
E 1 0
E/2 2 λ
E/4 4 2λ
After n steps Ec= E/2n 2n Xmax= n λ
Then : Xmax = λ log(E/Ec)
and N~E

4. This simple model works well even for realistic MC.
If the multipliticy depends on energy
µ = KE δ
Then
X max = A(1− δ ) log(E / Ec ) + B
However: Assuming perfect
scaling.
-Only forward region is
relevant

5. For a nucleus primary one may apply the superposition model
Nucleus of energy E, mass A = A nucleons of energy E/A
X max (E , A) = λ log(E /( AEc )) = λ log(E ) + B
Hadronic model dependence
Composition dependence
J. Knapp

6. Number of charged particles as a function of energy
Nmax ~ E
Differences between composition and hadronic models

7. Kascade
Auger
Argo
The altitude of the experiment determine the energy range!!!

8. Longitudinal profile
Near the maximum fluctuations
are smaller. Fluctuations do not
scale with energy
15 %
4%

9. π0 →γγ
Muonic component π± →µ ν
π0 decay instantly π± continue the cascade
N 2N N= total
π0 π± 3
3 multiplicity
N (1+ 2 N )
3 3 π0 π± ( 2N ) 2
γ 3 N2
After n steps, charged pions decay N ± = (
2N )n
E 3
Where E c = n
N N µ = ( E )1+ log(2 / 3 ) / log(N ) ∝ E β
E
c β ≅ 0.8 − 0.9
1− β β
For nucleus N µ (E , A) = AN (E / A) = A E

10. Seen in realistic
MC
QGSJET Proton
Slope ≈0.9
independent of θ

11. Casa-Mia
Data
AGASA

12. Lateral Spread of the Shower
Shower shape depends on the development stage

13. t = log(x)
y = log(E)

14. Timing
For muons timing is
well understood. It is
related to the height
production distribution <t> = 250 ns σ= 210 ns
dN/dt ~ dN/dz
But has an additional
R dependence
1019 eV Protons <t> = 700 ns σ= 350 ns

15. Muon height production depends on the composition. It could be used,
in principle, as a handle to determine composition.
However fluctuations are large.
Max = 306 gr/cms
1019 eV Shower Max = 337 gr/cm2 σ= 158 gr/cm2
@ 60 deg.
Max = 448 gr/cm2 σ= 172 gr/cm2

16. For electrons, the arrival time distribution is poorly understood
Structure
on µs
scale
E=89 EeV
Θ = 31 deg.

17. Timing II: Uncertainties Core uncertainties
induce timing uncertainties
For r ~ 1000 m
h ~ 10 km
d ~ 100 m

18. Relativistic effects
A muon with E ~ 1 GeV
has γ ~ 10 and 1-β ~ 5
10-3
Then after x = 1000 m
Same effect for relativistic electrons

19. Rates
Accidental trigger rate
The rate of accidental triggers
Assume a time window T, and a
single station accidental rate of r is R ~ r2 T
T must account for inclined shower, for instance T~ d/c
The flux of random muons is given by Φ ~ 100 1/(m2 s
sr)
Then R ~ (Φ A)2 d/c for A ~ 1 m2 d ~ 100m
R ~ 300 events/day

20. dN = KE −γ
dE
γ~ 2.7
Cosmic ray spectrum compilation

21. Rates
F ~ E-γ+1
The shower rate is given by
R ~ Flux d2
R = B E-γ+1 d2
R ~ 7.4 108 1/s (Eth/1 GeV)-γ+1
R ~ 4 103 events/day Eth = 106 GeV
R ~ 80 events/day Eth = 107 GeV