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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. EAS Inverse Problem. Detected EAS size spectra X = d 2 F/dN e dN m.

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gamma experiment
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

eas inverse problem
EAS Inverse Problem

Detected EAS size spectra


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

Kernel function

{A,E}  X

The problem of uniqueness

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

if onlyW(E,X) g(E) dE << F(X)

g(E) - oscillating functions


Problem of uniqueness for NA>1 and

Mutually compensative pseudo solutions

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

WA(E,X) gA(E) dE =0(F)


- WA(E,X) gA(E) dE =  WA(E,X) gA(E) dE + 0(F)






number of possible combinations of pseudo functions:



at NA=5, nc=26

how can we find the domains of pseudo solutions
How can we find the domains of pseudo solutions ?

WA(E,X) gA(E) dE =0(F)


1.In general, it is an open question for mathematicians.

2.Our approach:

a) Computationof WA(E,X)

b) for given fA(E) 

 F(X)

c) Quest for | gA(,, | E) | 0from

Using 2-minimization

simulation of kascade eas spectra
Simulation of KASCADE EAS spectra

Reconstructed EAS size spectra

EAS spectra atobservation level

2D Log-normal probability density funct.




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. <1.2

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

quest for pseudo solutions
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)


i=1,…60; j=1,…45

Ne,min=4103, N,min =6.4 104

examples of pseudo solutions 1
Examples of pseudo solutions, 1

WA(E,X) gA(E) dE = 0(F)

N=7105, Em=1 PeV, 2=1.08


Examples of pseudo solutions, 2

WA(E,X) gA(E) dE = 0(F)

N=7105, Em=1 PeV, 2=1.1


Examples of pseudo solutions, 3

WA(E,X) gA(E) dE = 0(F)

P=3 PeV

=1 at E < A

=5 at E > A

N=7106 ; 2=2.01

N=7105 ;2=0.25


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 )

N=7105, Em=1 PeV, 2=1.0


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 NA>1. The significance of the pseudo

solutions in mostcases exceeds the significance of the

evaluatedprimary energyspectra.

  • 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


Just this approach was used in the GAMMA experiment.