On the evaluation of an integral connected with the thermonuclear reaction rate in closed form

The standard expression of the reaction rate for low-energy, nonresonant nuclear reactions in nondegenerate plasma contains a parameter-dependent integral which in all previous calculations with physical or astrophysical background is considered as not capable of being evaluated in a closed form. So one usually resorts to approximation methods concerning large values of the parameter. At first we point out that CONSUL (1964) has given a series representation of the integral which was identified with a MEIJER 's G-function by MATHAI (1971). Next, in view of a physically more exact determination of the reaction rate formula, especially in connection with calculations concerning stellar energy generation, we consider a more general integral containing the mentioned one as special case and give an approximation-free representation by means of MEIJER 's G-function. The G-function so obtained may be conceived as complex-valued continuation of CONSUL 's series representation of a certain class of integrals contained in the considered one. From the series we extract a small parameter approximation of the special integral.     

Generalizations of reaction rate integrals with applications to nuclear synthesis in astrophysics and some corrections to published literature

Under the Maxwell-Boltzmann approach, the study of nuclear reactions in dense astrophysical plasmas under various cases (such as resonant, non-resonant, modified non-resonant, non-resonant under electron screening, and so on) leads to a class of complicated reaction rate integrals. It is shown that this general class of integrals can be identified with an integral involving the product of twoH-functions. This latter integral is evaluated in this article, and following Buschman (1979), several similar results in the published literature are shown to be incorrect.     

On resonant thermonuclear reaction rate integrals- closed-form evaluation and approximation considerations

The central problem in the theory of stellar nucleosynthesis (and in reaction theory of controlled thermonuclear fusion as well) is the determination of thermonuclear reaction rates. Continuing our preceding paper on closed-form evaluation of nonresonant reaction rate integral (HAUBOLD and JOHN 1978) now we consider a general thermonuclear reaction rate integral involving the full BREIT -WIGNER resonance cross section. Discerning between two cases – energy-independent partial widths and energy-dependent partial width – we evaluate the arising parameter-dependent integrals in closed form by means of MEIJER 's G-functions of one and two variables. At next, by using the BUNJAKOVSKIJ -SCHWARZ inequality we derive global upper bounds for thermonuclear reaction rate integrals with arbitrary admissible values of input parameters. Finally, in case of great values of the characteristic parameter COULOMB barrier energy divided by thermal energy we give an asymptotic representation of the general resonant reaction rate integral.     

Spectral line profiles and neutron cross sections: New results concerning the analysis of Voigt functions

The Voigt functions, so important in spectroscopy and neutron physics, are represented as generalized hypergeometric functions (G-functions) of two real variables. A system of partial differential equations for the Voigt functions is derived. By applying Hölder's inequality to an integral representation of the Voigt functions apparently not known in the literature until now, lower and upper bounds are obtained. Moreover, from this representation an asymptotic expansion of Voigt functions for large values of one variable is extracted.     

Oscillations in nuclear reaction systems inside stars

Oscillations related to the particle number density in the nuclear reaction systems inside stars are investigated, which include the oscillations in the helium nuclear reaction and in the P-PI branch of proton-proton reaction under the (g - T) effect and the gas pressure effect. The investigation shows that there is a supercritical oscillating bifurcation with time in the helium nuclear reaction system. If the oscillations are excited by the pressure effect, the nuclear reaction must occur in the convecting zone with a radially reducing convective velocity; if the oscillations are excited by the (g - T) effect, the nuclear reaction may occur in the convecting zone or no-convection zone. The supercritical oscillating bifurcation with time also exists in the P-PI reaction system, which is excited only by the (g - T) effect, while the oscillation excited by the pressure effect is decaying.     

Application of the Wiener-Hopf technique for a new representation of X- and Y-equations

The application of the Wiener-Hopf technique to the coupled linear integral equation ofX- andY-equations gives rise to the Fredholm equations with simpler kernels.X-equation is expressed in terms ofY-equation and vice-versa. These are unique in representation with respect to coupled linear constraints.     

An exact solution of transfer equations for interlocked multiplets

The transfer equations for non-coherent scattering arising from interlocking of principal lines without redistribution is exactly solved in a very simple way by Laplace tranform and Wiener-Hopf technique which are easily applied by the use of the new representation ofH-functions obtained recently by the author (1977). The emergent intensity in therth line is expressed in terms of anH-function and a Cauchy type integral admitting of closed form evaluation.     

The shear modulus of neutron star matter in the free-neutron state

The lattice whose properties have been tabulated by Baymet al. (1971) is unstable against nuclear quadrupole deformation for nuclear chargeZ>Z _c, 120<Z _c<201, if the electron gas is assumed to be of uniform density. Electron screening corrections to the change in energy density with nuclear deformation are large and it is shown that their inclusion removes the instability. The expression of Fuchs (1936), given originally for monovalent metals, overestimates the shear modulus of the lattice, for largeZ, by less than a factor of two.     

Astrophysical thermonuclear functions

Stars are gravitationally stabilized fusion reactors changing their chemical composition while transforming light atomic nuclei into heavy ones. The atomic nuclei are supposed to be in thermal equilibrium with the ambient plasma. The majority of reactions among nuclei leading to a nuclear transformation are inhibited by the necessity for the charged participants to tunnel through their mutual Coulomb barrier. As theoretical knowledge and experimental verification of nuclear cross sections increases it becomes possible to refine analytic representations for nuclear reaction rates. Over the years various approaches have been made to derive closed-form representations of thermonuclear reaction rates (Critchfield, 1972; Haubold and John, 1978; Haubold, Mathai and Anderson, 1987). They show that the reaction rate contains the astrophysical cross section factor and its derivatives which has to be determined experimentally, and an integral part of the thermonuclear reaction rate independent from experimental results which can be treated by closed-form representation techniques in terms of generalized hypergeometric functions. In this paper mathematical/statistical techniques for deriving closed-form representations of thermonuclear functions, particularly the four integrals $$\begin{gathered} I_1 (z,v)\mathop = \limits^{def} \int\limits_0^\infty {y^v e^{ - y} e^{ - zy^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } dy,} \hfill \\ I_2 (z,d,v)\mathop = \limits^{def} \int\limits_0^\infty {y^v e^{ - y} e^{ - zy^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } dy,} \hfill \\ I_3 (z,t,v)\mathop = \limits^{def} \int\limits_0^\infty {y^v e^{ - y} e^{ - z(y + 1)^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } dy,} \hfill \\ I_4 (z,\delta ,b,v)\mathop = \limits^{def} \int\limits_0^\infty {y^v e^{ - y} e^{ - by^\delta } e^{ - zy^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } dy,} \hfill \\ \end{gathered} $$ will be summarized and numerical results for them will be given. The separation of thermonuclear functions from thermonuclear reaction rates is our preferred result. The purpose of the paper is also to compare numerical results for approximate and closed-form representations of thermonuclear functions. This paper completes the work of Haubold, Mathai, and Anderson (1987).     

The effect of screening factors and thermonuclear reaction rates in the pre-main sequence evolution of low mass stars

In understanding the nucleosynthesis of the elements in stars, one of the most important quantities is the reaction rate and it must be evaluated in terms of the stellar temperature T, and its determination involves the knowledge of the excitation function σ(E) of the specific nuclear reaction leading to the final nucleus. In this paper, the effect of thermonuclear reaction rates to the pre-main sequence evolution of low mass stars having masses 0.7, 0.8, 0.9 and 1M_⊙ are studied by using our modified Stellar Evolutionary Program.     

Study of the dependence of the star formation rate in the nuclear regions of 39 Kazarian galaxies on their integral parameters

A statistical study of the dependence of the star formation rate in the nuclear regions of 39 Kazarian galaxies on the integral parameters of these galaxies is carried out on the basis of spectra from SDSS DR6. The value of SFR/kpc² for our sample lies in the range 0.013÷2.04M _⨀ year⁻¹kpc⁻² (with the maximum value of 2.04 corresponding to the Kaz 98 (merger)). It is found that the surface density of the rate of star formation correlates positively with the bar structure parameter and EW(Hα), and that, for spiral galaxies of early morphological types, SFR/kpc² is greater than for the later types. It is shown that the color B-R for the galaxies and the color (u − g) _nucl for the nuclear region correlate positively with the total absorption A(Hα) in the Ha line for the nuclear region. The average value of A(Hα) for our samples is found to be A(Hα)=1.3±0.09 magnitudes. Translated from Astrofizika, Vol. 52, No. 2, pp. 211–224 (May 2009).     

Dynamic screening in solar p-p reactions: is the mean-field approach applicable in solar plasma?

Although the Salpeter approximation for static screening is widely accepted and used in stellar modeling, the question of dynamic screening has been revisited. Here we reproduce Shaviv and Shaviv’s numerical analysis of the screening energy for p-p reactions in the solar core using the techniques of molecular dynamics to directly calculate the motion of ions and electrons due to Coulomb interactions without the mean-field assumption that is inherent in the Salpeter approximation. We conclude that the effects of dynamic screening are relevant and should be included in the treatment of the plasma, especially in the computation of nuclear reaction rates.     

Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, literal analytical expressions for the coefficients of the Fourier series representation ofF will be established for anyx _i ; withn, N positive integers 1 and | _i | fori=1, 2,...n. Moreover, the recurrence formulae satisfied by these coefficients will also be established. Illustrative analytical examples and a full recursive computational algorithm, with its numerical results, are included. The applications of the recurrence formulae are also illustrated by their stencils. As a by-product of the analyses is an integral which we may call a complete elliptic integral of thenth kind, in which the known complete elliptic integrals (1st, 2nd and 3rd kinds) are special cases of it.     

An improved version of the implicit integral method to solving radiative transfer problems

Radiative transfer (RT) problems in which the source function includes a scattering-like integral are typical two-points boundary problems. Their solution via differential equations implies making hypotheses on the solution itself, namely the specific intensity I (τ; n) of the radiation field. On the contrary, integral methods require making hypotheses on the source function S(τ). It seems of course more reasonable to make hypotheses on the latter because one can expect that the run of S(τ) with depth is smoother than that of I (τ; n). In previous works we assumed a piecewise parabolic approximation for the source function, which warrants the continuity of S(τ) and its first derivative at each depth point. Here we impose the continuity of the second derivative S′′(τ). In other words, we adopt a cubic spline representation to the source function, which highly stabilizes the numerical processes.     

A numerical test of the quasi-equilibrium approximation for stellar silicon burning

A measure of the range of the validity of the nuclear quasi-equilibrium approximation, employed byBodansky, Clayton andFowler (1968) for the treatment of stellar silicon burning, is obtained by comparison of the predicted abundances with the results of a numerical solution of the equations for the time rates of change of the nuclear abundances. The results of these calculations, performed for a temperatureT=3×10⁹K, are compared with those obtained by Bodanskyet al. for a temperatureT=5×10⁹K. The time required for the realization of the quasi-equilibrium condition at high temperatures is found to comprise a more substantial fraction of the silicon burning lifetime. This behavior is found to be attributable to the relative temperature sensitivities of the nuclear photodisintegration rates which determine the silicon burning lifetime and the charged particle reaction rates (largely Ti⁴⁴(,p) V⁴⁷) which determine the rate of buildup of iron peak nuclei.     

Fluctuations in helium nuclear reaction-diffusion systems

In this paper the question about the space correlation of the fluctuation in helium nuclear reaction diffusion system has been analysed. We have introduced the diffusion term into master equation of the probability distribution function of density. The spectrum of fluctuation waves in helium nuclear reaction system have been obtained. There is a series of waves which described the propagation of the fluctuation moments of the local density. The instability of helium combustion at its beginning for the star which mass satisfyingM<2M may be related with those increasing fluctuation waves. We also use the bifurcation theory to deal with the partial differential equation of helium nuclear reaction system. The expression of supercritical ordinary bifurcation is obtained.     

Integral field spectroscopy of QSO host galaxies

We describe a project to study the state of the ISM in ∼20 low redshift (z < 0.3) QSO host galaxies observed with the PMAS integral field spectrograph. We describe the development of the method to access the stellar and gas components of the spectrum without the strong nuclear emission, in order to access the host galaxy properties in the central region. It shows that integral field spectroscopy promises to be very efficient in studying the gas distribution and its velocity field, and also the spatially resolved stellar population in the host galaxies of luminous AGN. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

恒星内部电荷屏蔽对电子俘获反应的影响

本文分析了在恒星环境下，电荷强屏蔽对原子核电子俘获反应的影响，分析中忽略了电荷屏蔽对核跃迁矩阵元的影响．作为例子，分别讨论了电荷屏蔽对核素33S、30P和57Fe的电子俘获率的影响．结果表明：在较低温度和较高密度下，由于电荷屏蔽，其电子俘获率相对于无屏蔽情形有明显降低，这可能对恒星晚期演化和超新星爆发理论带来明显的影响．     

Unified structure of analytical solutions for radiative transfer problems in plane-parallel,homogeneous media

The basic concepts for developing a system of analytic solutions for the standard problems of radiative transfer theory are discussed. These solutions, which are found using Ambartsumyan’s layer addition method in Sobolev’s probabilistic interpretation for radiative diffusion problems, are maximally compact and easily used in numerical computations. New expressions are obtained for the resolvents and the resolvent functions, as well as a unified structure for the form of an integral representation for solving different radiative transfer problems in semi-infinite media and in finite layers. Block diagrams of the sequence of stages for solving these problems are provided, where the Ambartsumyan function φ(η) (more precisely, 1/φ(η)) plays a fundamental role in the case of semi-infinite media while the functions a(η, τ₀ ) and b(η, τ₀) play an analogous role for finite layers.     

Theory of the isotropic scattering of radiation in a plane layer: Feasibility of obtaining a complete analytical solution of the problem

Analytical investigations of the method of linear nonsingular integral equations, originally proposed by É. Kh. Danielyan [Astrofizika 36,225 (1993)] for the solution of problems in the theory of radiative transport in a medium of finite optical thickness with isotropic scattering, are continued in the present article. It is shown that the solution of problems of the stated class reduce to the determination of only the functions u ^± (, ) in the general case with true absorption. Explicit expressions are obtained for these functions at =0. The feasibility of a complete analytical solution of the problem is newly formulated as the solution of a Fredholm integral equation on the semiaxis with a kernel that admits representation by a superposition of exponential functions [Eq. (25)]. The choice of an efficient procedure for determining the Ambartsumyan -function for a semiinfinite medium is discussed. In particular, a new equation is given for this function.Translated from Astrofizika, Vol. 37, No. 1, pp. 129–145, January–March, 1994.