GAMMA Experiment

1
GAMMA Experiment
Mutually compensative pseudo solutions of the
primary energy spectra in the knee region
Samvel Ter-Antonyan
Yerevan Physics Institute
Astroparticle Physics 28, 3 (2007) 321
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EAS Inverse Problem
Detected EAS size spectra Xd2F/dNedNm
Unknown primary energy spectra A ? H, He,,Fe
Kernel function A,E ? X
The problem of uniqueness
Let NA1 and f(E) is a solution. Then
f(E)g(E) is also a solution
if only ?W(E,X) g(E) dE ltlt ?F(X)
g(E) - oscillating functions
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Problem of uniqueness for NAgt1 and Mutually
compensative pseudo solutions
for NA gt 1 the pseudo solutions fA(E)gA(E)
exist if only
??WA(E,X) gA(E) dE 0(?F)
A
-? ?WA(E,X) gA(E) dE ? ?WA(E,X) gA(E) dE 0(?F)
k
m?k
NA
nc??C
NA
number of possible combinations of pseudo
functions
j
j2
at NA5, nc26
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How can we find the domains of pseudo solutions ?
??WA(E,X) gA(E) dE 0(?F)
A
1. In general, it is an open question for
mathematicians.
2. Our approach
a) Computation of WA(E,X)
b) for given fA(E) ?
? ??F(X)
c) Quest for gA(??,?,? E) ? 0 from
Using ??2-minimization
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Simulation of KASCADE EAS spectra
Reconstructed EAS size spectra
EAS spectra at observation level
2D Log-normal probability density funct.
CORSIKA, NKG, SIBYLL2.1
?e(A,E)ltLn(Ne)gt ??(A,E)ltLn(N?)gt ?e(A,E),
??(A,E) ?(Ne,N?A,E)
E ? 1, 3.16, 10, 31.6, 100 PeV A ? p,He,O,Fe n
? 5000, 3000, 2000, 1500, 1000
?2/n.d.f. ? 0.4-1.4 ?2/n.d.f. lt1.2
?(ELnNe,LnN?)0.97 ?(LnALnNe,LnN?)0.71
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Quest for pseudo solutions
Monte-Carlo method
Abundance of nuclei 0.35 0.4 0.15 0.1
??WA(E,X) gA(E) dE 0(?F)
A
i1,60 j1,45 Ne,min4?103, N?,min 6.4 ?104
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Examples of pseudo solutions, 1
??WA(E,X) gA(E) dE 0(?F)
A ??104 TeV-1 ?A
P 1.10 ? 0.06 2.71 ? 0.04
He -1.80 (fixed) 2.6 (fixed)
O 0.97 ? 0.05 2.65 ? 0.04
Fe -0.50 (fixed) 2.9 (fixed)
N7?105, Em1 PeV, ?21.08
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Examples of pseudo solutions, 2
??WA(E,X) gA(E) dE 0(?F)
A ??104 TeV-1 ?A ?A
P -9.00 (fixed) 7.76 ? 0.01 0 (fixed)
He 0.044 ? 0.02 13.2 ? 1.08 169 ? 98
O -0.8 (fixed) 8.47 ? 0.05 0.94 ? 0.2
Fe 0.01 ? 0.002 11.4 ? 0.14 50 (fixed)
N7?105, Em1 PeV, ?21.1
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Examples of pseudo solutions, 3
??WA(E,X) gA(E) dE 0(?F)
?P3 PeV ?1 at E lt ?A ?5 at E gt ?A
A ??100 TeV-1 ?A / ?P
P -3.0 (fixed) 1 (fixed)
He 3.05 ? 0.07 1.03 ? 0.01
O -0.84 ? 0.06 1.08 ? 0.03
Fe 0.15 ? 0.02 1.29 ? 0.10
N7?106 ?22.01 N7?105 ?20.25
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Domain of pseudo solutions and KASCADE spectral
errors
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Examples of pseudo solutions, 4
Light and Heavy components
?? ?WLight(E,X) gLight(E) dE ?WHeavy(E,X)
gHeavy(E) dE ? 0(?F)
A?? p, He ( Light ) A?? O, Fe ( Heavy )
N7?105, Em1 PeV, ?21.0
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CONCLUSION
GAMMA Experiment

• The results show that the pseudo solutions with
  mutually
• compensative effects exist and belong to all
  families linear,
• non-linear and even singular in logarithmic
  scale.
• The mutually compensative pseudo solutions is
  practically
• impossible to avoid at NAgt1. The significance of
  the pseudo
• solutions in most cases exceeds the significance
  of the
• evaluated primary energy spectra.
• All-particle energy spectrum are indifferent
  toward the
• pseudo solutions of elemental spectra.

To decrease the contributions of the mutually
compensative pseudo solutions one may apply a
parameterization of EAS inverse problem using a
priori (expected from theories) known primary
energy spectra with a set of free spectral
parameters. Just this approach was used in the
GAMMA experiment.