Exact solution of the diffusion equation of cosmic-ray nucleons with rising cross section

Portella, H. M.; Castro, F. M. Oliveira; Arata, N.

doi:10.1088/0305-4616/14/8/018

Cited by 6 publications

(6 citation statements)

References 14 publications

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“…where B is the normalization energy of the mean free path. Instead of introducing mapping operators to the two terms on the right-hand side of equation ( 3) to solve it formally in real space [3], we proceed to use the Mellin transform defined by…”

Section: The Nucleon Casementioning

confidence: 99%

“…Hadron fluxes play a very important role in deriving the lepton fluxes at different atmospheric depths, in understanding the emulsion chamber data at mountain altitudes, and also in analysing some exotic events (Halo Events, Centauro Events) detected at Mt Chacaltaya by the Brazil-Japan Emulsion Chamber Collaboration. Due to the complex structure, these integro-differential equations are often solved formally in operator forms either in real space [3] or in integral transform space [4][5][6][7]. However, according to these analytical theories, the calculated fluxes in real space are often evaluated with very limited considerations on interaction mean free paths λ(E), inelasticity coefficients K(E), and energy distribution of secondary hadrons (E, E ).…”

Section: Introductionmentioning

confidence: 99%

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Hadron cascade by the method of characteristics

Tsui

Portella

Navia

et al. 2005

J. Phys. G: Nucl. Part. Phys.

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Hadron diffusion equations with energy-dependent interaction mean free paths and inelasticities are solved using the Mellin transform. Instead of using operators on the finite difference terms, the Mellin transformed equations are expanded in a Taylor series to a first-order partial differential equation in atmospheric depth t and in transform parameter s. These equations are then solved by the method of characteristics. The hadron fluxes (nucleon and pion) in real space are evaluated by the method of residues. For the case of a regularized power law primary spectrum, these hadron fluxes are given by simple residues and one, never before mentioned, essential residue. Comparisons of our solutions are made with the nucleon flux measured at sea level and with the hadron fluxes measured at t = 840 g cm−2 and at sea level. The integral hadron flux is also compared with experimental data from the Mt Fuji Collaboration (t = 650 g cm−2). The agreement between all of them is very good in general, better than 90%.

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Section: The Nucleon Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hadron cascade by the method of characteristics

Tsui

Portella

Navia

et al. 2005

J. Phys. G: Nucl. Part. Phys.

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“…We have used the expression λ N (E) = λ N ( E B ) −β with B = 1 TeV , λ N = 83 g/cm 2 and β = 0.056 which are obtained from accelerator and EAS data in the region 1 TeV ≤ E lab ≤ 1000 TeV [25]. For the pion mean-free path, we assume that λ π /λ N 1.4, and that it has the same energy dependence like the nucleon case [32].…”

Section: Numerical Resultsmentioning

confidence: 99%

Integral energy spectra of hadrons induced by one-single nucleon by the method of characteristics

Tsui¹,

Portella²,

Gomes³

et al. 2007

Braz. J. Phys.

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Hadron energy spectra induced by one-single nucleon are obtained solving diffusion equations by the method of characteristics. Our solutions are a generalization of earlier papers that allow us to calculate hadron fluxes including the energy dependence of the interaction lengths and inelasticities. A comparison with the integral hadron spectra of the so-called "halo events" detected by the Brazil-Japan Collaboration at Mt. Chacaltaya is made, in order to test our solutions. A reasonable agreement between then is obtained, considering the rising with energy of the nucleon inelasticity coefficient.

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“…where λ(E) is the energy dependent mean-free path, η = E/E ′ < 1 is the elasticity, u(E/E ′ ) is the elasticity distribution. Modelling the mean-free path by a power index β [25],…”

Section: Faltung Formulation Of Nucleon Diffusionmentioning

confidence: 99%

“…Chacaltaya by the Brazil-Japan Collaboration. In order to make numerical calculations about the integral hadron flux with our model and to compare with the detected events, we take the mean-free path normalization energy B = 1 T eV , λ N = 80 g/cm 2 and β = 0.06 which are obtained from accelerator and EAS data in the region 1 T eV ≤ E lab ≤ 1000 T eV [25]. For the pion meanfree path, we assume that λ N /λ π = 5/7, and that it has the same energy dependence like the nucleon case, Eq.…”

Section: Calculus Of Residuesmentioning

confidence: 99%

Faltung formulation of hadron halo event cascade at Mt. Chacaltaya

Tsui,

Portella,

Navia

et al. 2007

Preprint

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It is shown that the fundamental standard hadron cascade diffusion equation in the Mellin transform space is not rigorously correct because of the inconsistent double energy integral evaluation which generates the function < η s > with its associated parametizations. To ensure an exact basic working equation, the Faltung integral representation is introduced which has the elasticity distribution function u(η) as the only fundamental input function and < η s > is just the Mellin transform of u(η). This Faltung representation eliminates standard phenomenological parameters which serve only to mislead the physics of cascade. The exact flux transform equation is solved by the method of characteristics, and the hadron flux in real space is obtained by the inverse transform in terms of the simple and essential residues. Since the essential residues are given by the singularities in the elasticity distribution and particle production transforms that appear in the exponentials, these functions should not be parametized arbitrarily to avoid introducing non-physical essential residues. This Faltung formulation is applied to the charged hadron integrated energy spectra of halo events detected at Mt. Chacaltaya by the Brazil-Japan Collaboration, where the incoming flux at the atmospheric boundary is a single energetic nucleon.

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Exact solution of the diffusion equation of cosmic-ray nucleons with rising cross section

Cited by 6 publications

References 14 publications

Hadron cascade by the method of characteristics

Hadron cascade by the method of characteristics

Integral energy spectra of hadrons induced by one-single nucleon by the method of characteristics

Faltung formulation of hadron halo event cascade at Mt. Chacaltaya

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