Analytic Representation of Maxwell—Boltzmann and Tsallis Thermonuclear Functions with Depleted Tail

Dilip Kumar,Hans Haubold

doi:10.3390/axioms10020115

Abstract

The closed forms of the non-resonant thermonuclear function in the Maxwell–Boltzmann and Tsallis case with depleted tail are obtained in generalized special functions. The results are written in terms of H-function of two variables. The importance of the results in this paper lies in the fact that the reaction rate probability integrals in Maxwell-Boltzmann and Tsallis cases are not obtained by the conventional method of approximation or by means of a single variable transform technique but by means of a two variable transform method. The behaviour of the depleted non-resonant thermonuclear functions are examined using graphs. The results in the paper are of much interest to astrophysicists and statisticians in their future work in this area.

Highlights

Thermonuclear reactions taking place in Sun-like stars has received considerable interest in the past few years
The reaction rate probability integrals were obtained in closed forms by using generalized specials functions by many authors, see for example [1,2,3,4]
If ni and n j are the number densities of particles i and j, respectively, and if the reaction cross section is denoted by σ (v) where v is the relative velocity of the particle and f (v) is the normalized velocity distribution, the thermonuclear reaction rate rij is obtained by averaging the reaction cross section over the normalized distribution function of the relative velocity of the particles given by [3,6,7]

Summary

Introduction

Thermonuclear reactions taking place in Sun-like stars has received considerable interest in the past few years. The evaluation of the reaction rates for low-energy non-resonant thermonuclear reactions in the non-degenerate case is performed using the principles of nuclear physics and kinetic theory of gases [5]. For a non-resonant nuclear reactions between two nuclei of charges zi and z j colliding at low energies below the Coulomb barrier, the reaction cross section has the form [6,8]. The reaction rate probability integral in the Maxwell–Boltzmann case is given by. Physical situations different from the ideal non-resonant Maxwell–Boltzmann case can be obtained by modification of the cross section σ ( E) for the reacting particles and/or by the modification of their energy distribution. If the thermonuclear fusion plasma is not in a thermodynamic equilibrium there is a cut-off in the high energy tail of the Maxwell–Boltmann distribution function, the thermonuclear function to be evaluated takes the form. −ρ yγ−1 e−zy− xy dy, γ ∈ C, z > 0, x > 0, d < ∞

Non-Resonant Case with Depleted Tail

Non-Resonant Case with Screening

Standard Non-Resonant Thermonuclear Functions with Depleted Tail

Extension of the Non-Resonant Thermonuclear Function with Depleted Tail

Comparison of the Extended Results with the Standard Results

Conclusions