Mutually compensative pseudo solutions of primary energy spectra in the knee region
aa r X i v : . [ a s t r o - ph ] J un Mutually compensative pseudo solutions ofprimary energy spectra in the knee region
S.V. Ter-Antonyan
Yerevan Physics Institute, 2 Alikhanyan Br. Str., 375036 Yerevan, Armenia
Abstract
The problem of the uniqueness of solutions during the evaluation of primaryenergy spectra in the knee region using an extensive air shower (EAS) data set andthe EAS inverse approach is investigated. It is shown that the unfolding of primaryenergy spectra in the knee region leads to mutually compensative pseudo solutions.These solutions may be the reason for the observed disagreements in the elementaryenergy spectra of cosmic rays in the 1-100 PeV energy range obtained from differentexperiments.
Key words:
Cosmic rays, primary energy spectra, extensive air shower, inverseproblem.
PACS:
The Extensive air shower (EAS) inverse approach to a problem of the pri-mary energy spectra reconstruction in the region of 1 −
100 PeV energies hasbeen an essential tool in the past decade [1,2,3,4,5,6,7]. Basically, it followsfrom the high accuracies of recent experiments [8,9,10,11,12] and the availabil-ity of the EAS simulation code [13], which was developed in the frameworkof contemporary interaction models in order to compute the kernel functionsof a corresponding integral equation set [6,11]. At the same time, the energyspectra of primary (
H, He and
F e ) nuclei obtained from the KASCADE ex-periment [6] using the EAS inverse approach disagree with the same data fromthe ongoing GAMMA experiment [11,12], where parameterization of the EASinverse problem is used.Below, a peculiarity of the EAS inverse problem is investigated, and one of
Email address: [email protected] (S.V. Ter-Antonyan).
Preprint submitted to Elsevier Science 8 January 2019 he possible reasons for the observed disagreements between the energy spectrain [6] and [11] is considered in the framework of the SIBYLL [14] interactionmodel.The paper is organized as follows: In Section 2 the EAS inverse approachand the definition of the problem of uniqueness is described. It is shown,that the abundance of primary nuclear species leads to pseudo solutions forunfolded primary energy spectra. The existence and significance of the pseudosolutions are shown in Section 4. The pseudo solutions for primary energyspectra were obtained on the basis of simulation of KASCADE [6] showerspectra. The EAS simulation model is presented in Section 3. In Section 5the peculiarities of the pseudo solutions are discussed in comparison with themethodical errors of the KASCADE data.
The EAS inverse problem is ill-posed by definition and the unfolding ofthe corresponding integral equations does not ensure the uniqueness of thesolutions. The regularized unfolding on the basis of a priori information onexpected solutions (smoothness, monotony and non-negativity) in some casescan redefine the inverse problem [15] and provide the appropriate solutions.However, the expected singularities (e.g. knees) in the primary energy spectraat 10 − eV may erroneously be smoothed by regularization algorithmsand vice versa, be imitated by the unavoidable oscillations [15] of the solu-tions. Furthermore, the EAS inverse problem implies evaluations of at leasttwo or more unknown primary energy spectra from the integral equation setof Fredholm kind [6,11,12]. These peculiarities have not been studied in detailand the problem of the uniqueness of solutions can limit the number of eval-uated spectra.Let f A ( E ) be the energy spectrum of a primary nucleus A over the atmo-sphere, W A ( x | E ) be the probability density function describing the transfor-mation of A and E parameters of the primary nucleus to a measurable vector x . Then the EAS inverse problem, i.e. the reconstruction of the energy spec-tra of N A primary nuclei on the basis of the detected spectra Y ( x ) of EASparameters, is defined by the integral equation Y ( x ) = A NA X A = A Z f A ( E ) W A ( x | E )E. . (1)2vidently, if f A ,...A NA ( E ) are the solutions of eq. (1), the functions f A ( E ) + g A ( E ) should also be the solutions of (1), provided equation X A Z g A ( E ) W A ( x | E )E. = 0( ± ∆ Y ) (2)is satisfied for the given measurement errors ∆ Y ( x ) and for at least one of thecombinations of the primary nuclei n C = N A X j =1 N A j ! . (3)The number of combinations (3) stems from a possibility of the existence ofa set of functions g A ( E ) ≡ g ,A ( E ) , . . . , g i,A ( E ) for each of the primary nuclei( A ), which can independently satisfy eq. (2).For example, suppose that N A = 3. Let us denote R g i,A k ( E ) W A k ( E )E. by I i,A k and, for simplicity, set the right-hand side of eq. (2) to 0. Then, followingexpression (3), we find n C = 7 independent combinations of eq. (2): I ,A k = 0for k = 1 , I ,A + I ,A = 0, I ,A + I ,A = 0, I ,A + I ,A = 0 and I ,A + I ,A + I ,A = 0 with different g i,A k ( E ) functions. The measurementerrors ± ∆ Y on the right-hand side of these equations can both increase anddecrease the domains of g i,A k ( E ) functions.One may call the set of functions g A ( E ) the pseudo functions with the cor-responding pseudo solutions (spectra) f A ( E )+ g A ( E ). The oscillating g A ( E ) ≡ g ,A ( E ) functions at j = 1 are responsible for the first N A equations R g ,A ( E ) W A ( x | E )E. =0( ± ∆ Y ), A ≡ A , . . . A N A , due to the positive-definite probability densityfunction W A ( E ). The pseudo solutions f A ( E ) + g ,A ( E ) can be avoided byusing iterative unfolding algorithms [6,15].Additional sources of the pseudo solutions originate from the mutuallycompensative effects at j ≥ − X k Z g A k ( E ) W A k ( x | E )E. ≃ X m = k Z g A m ( E ) W A m ( x | E )E. (4)inherent to eq. (2) for arbitrary groups of k and m = k primary nuclei.Since there are no limitations on the types of the pseudo functions (exceptfor f A ( E ) + g A ( E ) >
0) that would follow from expression (4), and the num-ber of possible combinations (3) rapidly increases with the number of evalu-ated primary spectra ( N A ), the problem of the uniqueness of solutions maybe insoluble for N A >
3. Moreover, the pseudo functions have to restrict theefficiency of unfolding energy spectra for N A ≃ −
3, because the unification of Z = 1 , . . . ,
28 primary nuclei spectra into 2 − W A ( E )3nd thereby also increases the domains of the pseudo functions.Notice, that the pseudo solutions will always appear in the iterative unfold-ing algorithms if the initial iterative values are varied within large intervals. Atthe same time, it is practically impossible to derive the pseudo functions fromthe unfolding of equations (1,2) due to a strong ill-posedness of the inverseproblem. However, for a given set of the measurement errors ∆ Y ( x ) and theknown kernel functions W A ( x | E ) for A ≡ A , . . . A N A primary nuclei, eq. (2)can be regularized by parametrization of the pseudo functions g A ( α, β, . . . | E ).The unknown parameters ( α, β, . . . ) can be derived from a numerical solutionof parametric eq. (2), and thereby one may also evaluate the parametrizedpseudo functions g A ( E ).Below (Section 3), an EAS simulation model for computing the kernel func-tion W A ( E ) and replicating the KASCADE [6] EAS spectral errors ∆ Y ( x ) isconsidered. The primary energy spectra obtained in the KASCADE experiment werederived on the basis of the detected 2-dimensional EAS size spectra Y ( x ) ≡ Y ( N e , N µ ) and an iterative unfolding algorithm [15] for N A = 5 primary nu-clei [6]. Evidently, whether these solutions are unique or not depends on thesignificance of the arbitrary pseudo functions | g A ( E ) | from eq. (2).We suppose that the convolution of the shower spectra W A ( N e , N µ | E ) atthe observation level and corresponding measurement errors σ ( N e ), σ ( N µ )[1] are described by 2-dimensional log-normal distributions with parameters ξ e = ln N e ( A, E ), ξ µ = ln N µ ( A, E ), σ e ( A, E ), σ µ ( A, E ) and correlation coef-ficients ρ e,µ ( A, E ) between the shower size (ln N e ) and the muon truncatedsize (ln N µ ). We tested this hypothesis by the χ goodness-of-fit test usingthe CORSIKA(NKG) EAS simulation code [13] for the SIBYLL2.1 [14] in-teraction model, 4 kinds of primary nuclei ( A ≡ p, He, O, F e ), 5 energies( E ≡ , . , , . ,
100 PeV) and simulation samples for each of E and A :5000, 3000, 2000, 1500, 1000 respectively in 0 − zenith angular interval.The values of corresponding χ ( A i , E j ) /n d.f. , ( i = 1 , . . . j = 1 , . . .
5) weredistributed randomly in the interval 0 . − . N e, min = 6 . · and N µ, min = 4 · ) and thebin sizes ∆ ln N e , ∆ ln N µ = 0 . σ e, ( A, E ) at ln N e < ξ e , σ e, ( A, E ) at ln N e > ξ e , σ µ, ( A, E ) atln N µ < ξ µ and σ µ, ( A, E ) at ln N µ > ξ µ , more precisely ( χ /n d.f. ≤ .
2) de-scribe the shower spectra W A ( N e , N µ | E ) in the tail regions.We performed an additional test of the log-normal fit of the W A spectrausing multiple correlation analysis for the shower parameters simulated by the4og-normal W A ( N e , N µ | E ) probability density functions and shower parame-ters obtained from the CORSIKA EAS simulations at power-law primary en-ergy spectra ( γ = − .
5) and equivalent abundances of primary nuclei. The cor-responding correlation coefficients were equal to ρ (ln E | ln N e , ln N µ ) = 0 . ρ (ln A | ln N e , ln N µ ) = 0 . ρ (ln A, ln N e ) = − . ± . ρ (ln A, ln N µ ) =0 . ± .
01, and were in close agreement for both methods of N e and N µ generations.We replicated the KASCADE 2-dimensional EAS size spectrum Y ( N e , N µ )(and corresponding ∆ Y ) by picking out N e and N µ randomly from the 2-dimensional shower spectra W A ( N e , N µ | E ) after randomly picking A and E parameters of a primary particle from the power-law energy spectra f A ( E ) ∝ E − . (cid:16) (cid:16) EE k (cid:17) ǫ (cid:17) − . /ǫ (5)with a rigidity-dependent knee E k = Z · T V , the sharpness parameter ǫ = 3 and normalization of the all-particle spectrum R P A f A ( E )E. = 1. Therelative abundance of nuclei was arbitrarily chosen to be 0 . , . , .
15 and0 . H, He, O and
F e nuclei respectively, which approximatelyconforms with the expected abundance from balloon and satellite data [16].The mediate values of the parameters of the probability density function W A ( N e , N m | E ) were estimated by the corresponding log-parabolic splines.The total number of simulated EAS events was set to 7 · in order toreplicate the corresponding statistical errors ∆ Y ( N e , N µ ) of the KASCADEdata. On the basis of the obtained estimations of ∆ Y ( N e , N µ ) (Section 3) forthe KASCADE experiment, we examined the uniqueness of unfolding (1) by χ -the minimization: χ = I X i =1 J X j =1 (cid:18) G ( N e,i , N µ,j )∆ Y ( N e,i , N µ,j ) (cid:19) , (6)where G ( N e,i , N µ,j ) represents the left-hand side of eq. (2) for 2 kinds of theempirical pseudo functions g A ( E ) = α A (cid:16) EE m (cid:17) − γ A , (7)5 A ( E ) = α A ((ln E − β A ) + η A ) (cid:16) EE m (cid:17) − , (8)while g A ( E )+ f A ( E ) >
0, otherwise g A ( E ) = − f A ( E ). The unknown α A , β A , γ A and η A parameters in expressions (7,8) were derived from χ minimization (6).The numbers of bins were I = 60 and J = 45 with the bin size ∆ ln N e , ∆ ln N µ ≃ . χ (6) for different representations (7,8) of thepseudo functions g A ( E ) provides a solution of the corresponding parametriceq. (2) with a zero right-hand side. To avoid the trivial solutions g A ( E ) ≡ χ (6). The mag-nitudes of the fixed parameters were empirically determined via optimizationof conditions χ /n d.f. ≃ | g A ( E ) | ∼ f A ( E ) for the pseudo spectra withthe fixed parameters.The true primary energy spectra f A ( E ) for A ≡ H, He, O, F e nuclei (5)and the all-particle energy spectrum P f A ( E ) (lines) along with the corre-sponding distorted (pseudo) spectra f A ( E ) + g A ( E ) (symbols) are presentedin Fig. 1 respectively. The parameters of the pseudo functions (7) derived for Primary energy, E [ TeV ] E . F A ( E ) [ T e V . ] f A f A +g A A - p - He - O - All - Fe Fig. 1. Primary energy spectra f A ( E ) and the all-particle spectrum P f A ( E )for A ≡ H, He, O, F e nuclei (lines) and the corresponding pseudo solutions f A ( E ) + g A ( E ) for the pseudo function (7) (symbols). χ /n d.f. = 1 .
08 ( n d.f. = 717) are presented in Table 1.The effect of the pseudo functions (8) on the resulting primary energyspectra is shown in Fig. 2. Evaluations of the corresponding parameters arepresented in Table 2 for χ /n d.f. = 1 .
1. The variations of the cubic power6 able 1Parameters α A (TeV − ) and γ A of the pseudo function (7) for different primarynuclei A and E m = 1000 TeV. A α A · γ A p . ± .
06 2 . ± . He − .
80 (fixed) 2 .
60 (fixed) O . ± .
05 2 . ± . F e − .
50 (fixed) 2 .
90 (fixed) Primary energy, E [ TeV ] E . F A ( E ) [ T e V . ] f A f A +g A A - p - He - O - All - Fe Fig. 2. The same as Fig. 1 for the pseudo function (8). indices in expression (8) in the range of 2 − g A ( E ) can be comparable and even significantly larger than the values of thetrue spectra f A ( E ). Moreover, the pseudo solutions lose both the slopes andthe intensities of the spectra. At the same time, the all-particle spectra slightlydepend on the contribution of the pseudo functions.The same results (Tables 1,2) were obtained using both the combined 2-dimensional log-normal representation of the shower spectra W A ( N e , N µ | E )(Section 3) and the 3-dimensional (ln E, ln N e , ln N µ ) parabolic interpolationsof corresponding probability density functions obtained by the CORSIKAcode. 7 able 2Parameters α A (TeV − ), γ A and η of the pseudo function (8) for different primarynuclei A and E m = 1000 TeV. A α A · β A η A p − .
00 (fixed) 7 . ± .
01 0 (fixed) He . ± .
02 13 . ± .
08 169 ± O − .
80 (fixed) 8 . ± .
05 0 . ± . F e . ± .
002 11 . ± .
14 50 (fixed)Table 3Parameters α A (TeV − ) and ε A (TeV) of the pseudo function (9) for different pri-mary nuclei A and ε H = 3000 TeV. A α A · ε A /ε H p − . He . ± .
07 1 . ± . O − . ± .
06 1 . ± . F e . ± .
02 1 . ± . Evidently, the range of relatively large measurement errors ∆ Y ( x ) expandsthe domain of the pseudo functions. Contributions of the mutually compen-sative effects (eqs. 2,4) of the pseudo functions to the domain of the pseudo so-lutions were tested using a 10 times larger EAS simulation sample ( n = 7 · )and the pseudo functions with evident singularity: g A ( E ) = α A ε − A (cid:16) Eε A (cid:17) δ , (9)where δ = − E ≤ ε A and δ = − E > ε A . The singularity of the pseudofunction (9) for A ≡ H was fixed at ε H = 3000 TeV and the scale factor α H = − .
03. The remaining parameters for primary nuclei A ≡ He, O, F e were es-timated by χ -minimization (6) and presented in Table 3 for χ /n d.f. = 2 . n d.f. = 857. The accuracies of integrations (2) were about 0 . He and O spectra. This is due to8 Primary energy, E [ TeV ] E . F A ( E ) [ T e V . ] f A f A +g A A - p - He - O - All - Fe Fig. 3. The same as Fig. 1 for the pseudo function (9) and n = 7 · simulatedshowers. both the large number ( n C = 15) of possible mutually compensative combina-tions (3) and the peculiarities of EAS development in the atmosphere (kernelfunctions W A ( E ), Section 3), which are expressed by the approximately log-linear dependences of the statistical parameters < ln N e > , < ln N µ > , σ e and σ µ of shower spectra W A ( E ) on energy (ln E ) and nucleon number (ln A )of primary nuclei [20,21]. The value of χ /n d.f. for a 10 times smaller EASsample ( n = 7 · ) was equal to 0 . The results from Figs. 1–3 show that the pseudo functions with mutuallycompensative effects exist and belong practically to all families - linear (7),non-liner (8) and even singular (9) in a logarithmic scale.The all-particle energy spectra in Figs. 1–3 are practically indifferent to thepseudo solutions of elemental spectra. This fact directly follows from eq. (2) forpseudo solutions and is well confirmed by the identity of the GAMMA [11,12]and KASCADE [6] all-particle energy spectra in spite of disagreements of theelemental ( p, He, F e ) primary energy spectra (see [11,12]).The χ minimization (6) uses mainly the nearest pseudo energy spectra withfree parameters for compensation of the pseudo spectra with fixed parameters.The significance of the pseudo functions | g A ( E ) | in most cases exceeds the sig-nificance of the evaluated primary energy spectra f A ( E ) and unfolding of (1)can not be effective for N A = 4. 9 ( f A ( E ) + g A ( E )) / f A ( E ) He E [ TeV ] Fe Fig. 4. Domains of the pseudo solutions for He and F e primary nuclei (light shadedareas) and corresponding ”methodical errors” of the KASCADE unfolding spectra(dark shaded areas) taken from [6]. The solid and dotted lines resulted from pseudofunctions (7) and the dashed lines stemmed from (8).
The unfolding of the primary energy spectra for N A = 5 will increase thenumber of possible combinations (3) of the pseudo solutions and the corre-sponding pseudo functions by a factor of two. Taking into account the largevalues of applied χ /n d.f. ≃ − N A = 5 define the uncertaintiesof the solutions intrinsic only to the given unfolding algorithms. The existenceand significance of the mutually compensative pseudo solutions follow fromeqs. (1,2) and from the peculiarities of the shower spectra W A ( x | E ) regardlessof the unfolding algorithms.Comparison of the methodical errors ( f A ( E ) + ∆ f A ( E )) /f A ( E ) for A ≡ He and A ≡ F e from [6] with corresponding errors ( f A ( E ) + g A ( E )) /f A ( E ) dueto the pseudo solutions from expressions (7,8) are shown in Fig. 4. The mag-nitudes of the fixed parameters were empirically determined by maximizing | g He ( E ) | (left panel) and | g F e ( E ) | (right panel) for a given goodness-of-fit test χ /n d.f ≃ . Conclusion
The results show that the reconstruction of primary energy spectra usingunfolding algorithms [6,15] can not be effective and the disagreement betweenthe KASCADE [6] and GAMMA [11,12] data is insignificant in comparisonwith the large domains of the mutually compensative pseudo solutions (Fig 4)of the unfolded spectra [6].Even though the oscillating pseudo solutions g ,A ( E ) (Section 2) are possi-ble to avoid using regularization algorithms [15], the mutually compensativeeffect (4) of the arbitrary pseudo functions g A ( E ) intrinsic to the expression(2) is practically impossible to avoid at N A > a priori (expected from theories [17,18,19]) known primary energy spec-tra with a set of free spectral parameters. This transforms the EAS inverseproblem into a set of equations with unknown spectral parameters, and therebythe EAS inverse problem is transmuted into a test of the given primary energyspectra using detected EAS data [4]. The reliability of the solutions can bedetermined by their stability depending on the number of spectral parame-ters, the agreement between the expected and detected EAS data sets, andthe conformity of the spectral parameters with theoretic predictions.The all-particle energy spectra (Fig. 1–3) are practically indifferent towardthe pseudo solutions for elemental spectra.The obtained results depend slightly on the spectral representations of theshower spectra W A ( E ) and the primary energy spectra f A ( E ). Acknowledgments
I thank my colleagues from the GAMMA experiment for stimulating thiswork and the anonymous referee for suggestions which considerably improvedthe paper.
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