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8/28/20151 GAMMA 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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8/28/20152 EAS Inverse Problem if only W(E,X) g(E) dE << F(X) g(E) - oscillating functions Detected EAS size spectra X=d 2 F/dN e dN Unknown primary energy spectra; A H, He,…,Fe Let N A =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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8/28/20153 W A (E,X) g A (E) dE = 0( F) A - W A (E,X) g A (E) dE = W A (E,X) g A (E) dE + 0( F) k mkmk Problem of uniqueness for N A >1 and Mutually compensative pseudo solutions nc=Cnc=C j=2 NANA j NANA at N A =5, n c =26 for N A > 1 the pseudo solutions f A (E)+g A (E) exist if only number of possible combinations of pseudo functions:
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8/28/20154 How can we find the domains of pseudo solutions ? W A (E,X) g A (E) dE = 0( F) A 1. In general, it is an open question for mathematicians. 2. Our approach: a) Computation of W A (E,X) b) for given f A (E) c) Quest for | g A ( , , | E) | 0 from F(X) Using 2 -minimization
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8/28/20155 Simulation of KASCADE EAS spectra Reconstructed EAS size spectra EAS spectra at observation level 2D Log-normal probability density funct. e (A,E)= (A,E)= e (A,E), (A,E) (N e,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, 1000 2 / n.d.f. 0.4-1.4; 2 /n.d.f. <1.2 (E|LnN e,LnN )=0.97; (LnA|LnN e,LnN )=0.71
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8/28/20156 Quest for pseudo solutions W A (E,X) g A (E) dE = 0( F) A i=1,…60; j= 1,…45 N e,min =4 10 3, N ,min =6.4 10 4 Abundance of nuclei: 0.35; 0.4; 0.15; 0.1 Monte-Carlo method
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8/28/20157 N=7 10 5, E m =1 PeV, 2 =1.08 Examples of pseudo solutions, 1 A 10 4 [TeV] -1 AA P 1.10 0.062.71 0.04 He-1.80 (fixed)2.6 (fixed) O 0.97 0.052.65 0.04 Fe-0.50 (fixed)2.9 (fixed) W A (E,X) g A (E) dE = 0( F)
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8/28/20158 N=7 10 5, E m =1 PeV, 2 =1.1 Examples of pseudo solutions, 2 A 10 4 [TeV] -1 AA AA P-9.00 (fixed) 7.76 0.01 0 (fixed) He 0.044 0.0213.2 1.08169 98 O-0.8 (fixed) 8.47 0.050.94 0.2 Fe 0.01 0.00211.4 0.14 50 (fixed) W A (E,X) g A (E) dE = 0( F)
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8/28/20159 N=7 10 6 ; 2 =2.01 N=7 10 5 ; 2 =0.25 Examples of pseudo solutions, 3 A 100 [TeV] -1 A / P P-3.0 (fixed)1 (fixed) He 3.05 0.071.03 0.01 O -0.84 0.061.08 0.03 Fe 0.15 0.021.29 0.10 P =3 PeV =1 at E < A =5 at E > A W A (E,X) g A (E) dE = 0( F)
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8/28/201510 Domain of pseudo solutions and KASCADE spectral errors
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8/28/201511 N=7 10 5, E m =1 PeV, 2 =1.0 Examples of pseudo solutions, 4: W Light (E,X) g Light (E) dE = W Heavy (E,X) g Heavy (E) dE 0( F) Light and Heavy components A p, He ( Light ) A O, Fe ( Heavy )
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8/28/201512 CONCLUSION 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. 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 N A >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.
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