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

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

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 mkmk 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:

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

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.  ;  2 /n.d.f. <1.2  (E|LnN e,LnN  )=0.97;  (LnA|LnN e,LnN  )=0.71

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

8/28/20157 N=7  10 5, E m =1 PeV,  2 =1.08 Examples of pseudo solutions, 1 A  10 4 [TeV] -1 AA P 1.10   0.04 He-1.80 (fixed)2.6 (fixed) O 0.97   0.04 Fe-0.50 (fixed)2.9 (fixed)  W A (E,X) g A (E) dE = 0(  F)

8/28/20158 N=7  10 5, E m =1 PeV,  2 =1.1 Examples of pseudo solutions, 2 A  10 4 [TeV] -1 AA AA P-9.00 (fixed) 7.76  (fixed) He    98 O-0.8 (fixed) 8.47   0.2 Fe 0.01   (fixed)  W A (E,X) g A (E) dE = 0(  F)

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   0.01 O   0.03 Fe 0.15   0.10  P =3 PeV  =1 at E <  A  =5 at E >  A  W A (E,X) g A (E) dE = 0(  F)

8/28/ Domain of pseudo solutions and KASCADE spectral errors

8/28/ 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 )

8/28/ 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.