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04/19/23 1
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.
