An analytically solvable model of a nuclear cascade in the atmosphere M. G. Kostyuk BNO INR RAS

1. An analytically solvable model of a nuclear cascade in the atmosphere M. G. Kostyuk BNO INR RAS Using sufficiently accurate uniform (that is, suitable for all physically significant values ​​of depths, zenith angles and energies), analytical approximations of the solutions of the equations of a simple model of a nuclear cascade, flows of particles of different types are calculated.        It has been found out that it is possible to obtain, without laborious calculations, sufficiently accurate (within the framework of the model) theoretical estimates of the fluxes of particles of secondary cosmic radiation for all depths (above sea level), energies and zenith angles.

2. Motivation of using analytical approximations of the solutions of the equations of simple models of the nuclear cascade in the atmosphere: The possibility, without laborious calculations, of obtaining sufficiently accurate (within the framework of the model) theoretical estimates of the characteristics of the cascade;         Analyze the qualitative behavior of various quantities at extreme values ​​of the parameters;          Approximations of the solutions of a simple model can be sufficiently accurate and describe the particle fluxes, etc., in a variable manner (the main theme of the report);          Sufficiently accurate approximations of solutions in comparison with experiment can determine the limits of applicability of the model used at the quantitative level;          The studies of this type are essentially auxiliary in nature and do not pretend to replace detailed calculations in the framework of more meaningful models.

3. In order to effectively perform calculations for a sufficiently broad class of the interaction models under consideration, it is necessary to have a set of simple and universal analytical or numerically-alithic methods, allowing to calculate the average characteristics and their variances for a set of components in wide intervals of variation of variables. Only after comparing the results of such calculations with all available experimental data, it is expedient for the selected models to carry out more detailed and accurate calculations (using, for example, the Monte Carlo method).

4. CASCADE EQUATIONS The flux of hadrons in the atmosphere can be described by a set of coupled integro-differential transport (cascade) equations (T. K. Gaisser, Cosmic Rays and Particle Physics, Cambridge University Press, Cambridge, UK, 1990) : is the flux of hadrons of type i with energy in the interval at an depth X: Is the flux of hadrons of type i with energy in the interval? Ei, Ei + d Ei at an

5. The second assumption is known as the hadron (or Feynman)scaling and is performed in a wide energy range. At very high energies, this assumption is violated, as well as the assumption for λi , since the cross section of the interaction of hadrons with nuclei of air are not completely independent of energy. λi , diis the characteristic length of the interaction and decay of hadrons, measured in g/cm2, for di accept: In this case is the Lorentz factor of the decaying particle in the rest frame associated with the atmosphere, - life time of the particles in the rest frame. Itisgenerallyassumedthat

6. Taking into account the assumptions made, as well as the power nature of the PCR spectrum, for the nucleon component of the cascade, one can obtain:: The approximation, which is valid for Ɵ ≤ 70 ° ("plane" atmosphere), for isothermal (T), exponentially tending to zero atmospheric density: ρ=X cos(θ*)/h(T), altitude scale : h (Т) = RT / Mg, h(Т) ≈ 6.4km. R, T, M, g are the universal gas constant, the constant temperature, molar mass, free fall acceleration respectively. The "renormalization" θ → θ * (D. Chirkin, hep-ph / 0407078) makes it possible to apply all relations for arbitrary zenith angles.

7. Here is denoted: For the meson (pion and kaon) component fluxes, the well-known expression was used (T. K. Gaisser, Cosmic Rays and Particle Physics, Cambridge University Press, Cambridge, UK, 1990), here written in terms of dimensionlessvariables: Meson fluxes

8. Meson Fluxes Distributions (moments) A weak dependence on energy is seen.

9. d1 is the position of the meson's distribution “center of gravity”: variance: It can be seen that the variance is commensurate in magnitude with the position of the meson's center of gravity, the meson distribution is asymmetrical ( the "forward" tendency).

10. T. K. Gaisser, Cosmic Rays and Particle Physics, Cambridge University Press, Cambridge, UK, 1990:

11. APPENDIX_______________________________________________________________________APPENDIX_______________________________________________________________________ For the function J(ε,y) applied an approximate analytical representation through a special approximation for the exponent under the integral sign here is the interpolation polynomial in the nodes at the zeros of the Chebyshev polynomials, the quantities (“splitting parameter " of the integration interval [0,1]) depends only on the degree of the polynomial under consideration, there is a table of values. y=X/l, The explicit form of the polynomial Table 1. The values of the splitting parameter of the interval m and the estimation o parameter f the approximation error. Table. 2. The coefficients of polynomials of different degrees.

12. Approximations for the integral J(ε,y): The error of approximating the exponent under the integral sign is not worse then , but for the integral J (ε, X) is not worse than _______________________________________________________________ Muon Fluxes The fluxes of muons are governed by the equation (in terms of the dimensionless variables defined above): It is seen that the flux of muons is a sum of the contributions of mesons of various types, below which these contributions will be analyzed separately, providing them with appropriate indices.

13. An approximate solution will be obtained within the framework of a simple version of the projection (Bubnov-Galerkin) method; the trial function ( is yet undetermined variational parameter) and the equation to which it will satisfy are the solution: Approximation and neglect of exponentially small contributions, which corresponds to the consideration of muon fluxes outside the region where meson production is localized, gives the expression:

14. The residual for our equation on the trial function under consideration is the quantity to be orthogonalized to the basic function of the form being defined in the domen of two independent variables : This equation allow as to determine the required variation parameter : Thus, an explicit simple expression for the muon flux is obtained without using the difficult multidimensional quadratures that are usual for this case.

15. CONCLUSION Simple uniform analytical expressions are obtained for the dependences of meson and muon fluxes for all energies of depths and zenith angles. All "generations" are taken into account, no decomposition into any series with a limited radius of convergence is used. Explicit analytical formulas for the moments of the meson distribution function are obtained, in particular, their almost independence from the energy is revealed, as well as the asymmetry of the indicated function with respect to the center (with a tendency to "forward"). Electromagnetic energy losses by muons are taken into account in the framework of simple analytical expressions, with the possibility of systematic refinement of the results obtained.

16. Energy-dependent quantities in the monograph (T. K. Gaisser, Cosmic Rays and Particle Physics, Cambridge University Press, Cambridge, UK, 1990)are replaced by constants estimated for a particular energy. In view of the fact that these quantities appear in explicit (“letter”) form in the approximations of this work, it is possible to evaluate them at other energies using the previous formulas in order to expand the field of suitability of the approximations.