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GAMMA ExperimentGAMMA Experiment

Samvel Ter-Antonyan

Yerevan Physics Institute

Mutually compensative pseudo solutions of the primary energy spectra in the knee region

Astroparticle Physics 28, 3 (2007) 321

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EAS Inverse Problem

A

AA dEEfXEWXF )(),()(

if only W(E,X) g(E) dE << F(X)

g(E) - oscillating functions

Detected EAS size spectra

X=d2F/dNedN

Unknown primary energy spectra; A H, He,…,Fe

Let NA=1 and f(E) is a solution. Then f(E)+g(E) is also a solution

Kernel function{A,E} X

The problem of uniqueness

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WA(E,X) gA(E) dE = 0(F)A

- WA(E,X) gA(E) dE = WA(E,X) gA(E) dE + 0(F)k mk

Problem of uniqueness for NA>1 andMutually compensative pseudo solutions

nc=Cj=2

NA

j

NA

at NA=5, nc=26

for NA > 1 the pseudo solutions fA(E)+gA(E) exist if only

number of possible combinations of pseudo functions:

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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) A

XFdEEAfXEAW )()(),(

c) Quest for | gA(,, | E) | 0 from

F(X)

Using 2-minimization

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Simulation of KASCADE EAS spectra

****** ),,(),(),( dNdNNNNNENNwENNW eeeeAeA

Reconstructed EAS size spectra

EAS spectra at observation level

2D Log-normal probability density funct.

),,,,(),( , eeeeA ENNW

e(A,E)=<Ln(Ne)>

(A,E)=<Ln(N)>

e(A,E), (A,E) (Ne,N|A,E)

CORSIKA, NKG, SIBYLL2.1

E 1, 3.16, 10, 31.6, 100 PeV; A p,He,O,Fe

n 5000, 3000, 2000, 1500, 10002/n.d.f. 0.4-1.4; 2/n.d.f. <1.2

(E|LnNe,LnN)=0.97; (LnA|LnNe,LnN)=0.71

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Quest for pseudo solutions

/5.07.1 ))(1()( k

A E

EEEf

A

AA XFdEEfXEW )()(),(

WA(E,X) gA(E) dE = 0(F)A

2

,

,,2 ),(

i j ji

jie

F

NNG i=1,…60; j=1,…45

Ne,min=4103, N,min =6.4 104

Abundance of nuclei: 0.35; 0.4; 0.15; 0.1

Monte-Carlo method

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A

mAA E

E)E(g

N=7105, Em=1 PeV, 2=1.08

Examples of pseudo solutions, 1

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)

WA(E,X) gA(E) dE = 0(F)

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33ln)(

mE

EAAEAEAg ])[(

N=7105, Em=1 PeV, 2=1.1

Examples of pseudo solutions, 2

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)

WA(E,X) gA(E) dE = 0(F)

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A

1AAA

E)E(g

N=7106 ; 2=2.01N=7105 ; 2=0.25

Examples of pseudo solutions, 3

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

P=3 PeV=1 at E < A

=5 at E > A

WA(E,X) gA(E) dE = 0(F)

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Domain of pseudo solutions and KASCADE spectral errors

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33ln)(

mE

EAAEAEAg ])[(

N=7105, Em=1 PeV, 2=1.0

Examples of pseudo solutions, 4:

WLight(E,X) gLight(E) dE = WHeavy(E,X) gHeavy(E) dE 0(F)

Light and Heavy components

A p, He ( Light )A O, Fe ( Heavy )

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CONCLUSION

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

GAMMA ExperimentGAMMA 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 NA>1. 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.