Der gegenwártige Stand der Theorie und der analytischen Auswertung von nichtresonanten thermonuklearen Reaktionsraten

Concerning the analytic evaluation of reaction rates we propose to use the integration theory of special functions and show for the special case of thermonuclear reactions in non-degenerate, nonrelativistic plasma that the non-resonant reaction rate can be evaluated in closed form by means of generalized hypergeometric functions. In connection with these closed-form results one can give differential and functional relations and series expansions. Also approximation considerations can be performed.     

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.     

Closed-form evaluation and approximation considerations of the non-resonant thermonuclear reaction rate

By use of a highly efficient theory of generalized hypergeometric functions, we show that the non-resonant thermonuclear reaction rate can be evaluated in closed form. This new approach is advantageous to operate analytically with reaction rates The numerical computation of the rate may be performed with the aid of associated differential equation, a convergent series expansion for small values of the characteristic parameter and an asymptotic expansion for large values of the characteristic parameter.     

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.     

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).     

Analytic results for screened non-resonant nuclear reaction rates

This paper deals with the closed-form representation of the non-resonant nuclear reaction rate taking into account electron screening effects for the reacting particles. The basic physical principles concerning nuclear reactions in dense astrophysical plasmas are applied to derive the representation of the screened nuclear reaction rate integral. Taking advantage of the theory of Meijer'sG-function it is shown that both the reaction rate integrals with and without screening corrections can be represented in closed form. Mathematical approximations for the reaction rate integral, till now considered as inevitable in the literature, are avoided and exact computable representations given.     

Transformation of the Extended Gamma Function Γ>_{0,2}^{2,0} [(b,x)] with Applications to Astrophysical Thermonuclear Functions dos Campos, Brazil, 10–14 November 1997.

Two representations of the extended gamma functions [(b,x)] are proved. These representations are exploited to find a transformation relation between two Fox's H-functions. These results are used to solve Fox's H-function in terms of Meijer's G-function for certain values of the parameters. A closed form representation of the kernel of the Bessel type integral transform is also proved.     

Approximate Analytical Theory of Satellite Orbit Prediction Presented in Spherical Coordinates Frame

A comparative review of analytic theories for the motion of Earth satellites in quasi-circular orbits written in the spherical coordinate frame is presented. The theory of motion is developed for satellites in quasi-circular and quasi-equatorial orbits subjected to geopotential, luni-solar and solar radiation pressure force perturbations. The intermediate orbit is Keplerian and the equations of motion are solved by the Lyapunov–Poincaré small parameter method. Both resonant and non-resonant cases are considered. The results can be useful for the development of a complete theory of weakly eccentric orbits.     

Reaction rates for neutrino processes

Some integrals involved in neutrino processes are evaluated by transformation to a special system of reference—usually to the center of mass system (CM). Rather simple analytic expressions are obtained for reaction rates and, though less simple, for moments. An interesting result thus obtained is for an isotropic interaction (in CM) of a neutrino with a monoenergetic isotropic gas of extreme relativistic electrons: it is found that the probability of the scattered neutrino to have energy in a certain range is independent of this energy.     

On Thermonuclear Reaction Rates

Nuclear reactions govern major aspects of the chemical evolution of galaxies and stars. Analytic study of the reaction rates and reaction probability integrals is attempted here. Exact expressions for the reaction rates and reaction probability integrals for nuclear reactions in the cases of nonresonant, modified nonresonant, screened nonresonant and resonant cases are given. These are expressed in terms of H-functions, G-functions and in computable series forms. Computational aspects are also discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

The criterion of appearance of unstable thermonuclear reaction in the helium burning shell of AGB stars

A criterion of the appearance of unstable thermonuclear reaction in the helium burning shell of thermal pulsating AGB (TP-AGB) star is established. The new criterion contains abundant physical information. It involves not only the geometric parameters of the helium burning shell, but also its mechanical, thermal and chemical parameters.The following mechanism of the occurrence and disappearance of unstable thermonuclear reaction in the helium burning shell of TP-AGB star is proposed: The appearance of a region of unstable convection in the helium burning shell of the TP-AGB star triggers unstable thermonuclear reaction which will promote a rapid expansion and a rapid geometric deformation of the shell, thereby removing the unstable thermonuclear reaction.Using the improved program of stellar evolution of Kippenhahn, the evolution of a 5Mo star is followed from the main sequence to the TP-AGB stage. The results show that the new criterion can well reflect the status of the thermonuclear reaction in the helium burning shell of the star. Besides, it is revealed that in the sixth period of thermal pulsation of the star the elements that are dredged up to the surface of the star, are synthesized mainly by thermonuclear reaction under the conditions, temperature lgT₂/K < 8.155 and density 4.0 < lg P2 /9 . CM-3 < 4.6.     

Analytic solar structure

In this paper an attempt is made to derive an analytic solar model by assuming a one-parameter family of density distributions. The analytic representation of the solar interior is derived for hydrostatic equilibrium and energy conservation. The mathematics involved in equating the solar luminosity to the thermonuclear energy generation near the center is illustrated in terms of special functions.     

Analytical study of thermonuclear reaction probability integrals

An analytic study of the reaction probability integralscorresponding to the various forms of the slowly varyingcross-section factor S(E) is attempted. Exact expressions forreaction probability integrals are expressed in terms ofthe extended gamma functions.     

Stability of thermonuclear reaction affected by gravitational force

A reaction-diffusion equation describing thermonuclear reaction system affected by gravitational force and temperature gradient is obtained by using the theory of nonequilibrium thermodynamics. We take CNO cycle as an example to investigate the effects of the gravitional force and temperature gradient on the stability of nuclear reaction inside the star. The stability criterion of the stellar structure has been analysed and some attractive results have also been discussed.     

大行星内部能源问题

根据宇宙飞船等对巨行星观测结果,我们首次发现大行星(包括巨行星和地球)的光度随其质量存在有规律地变化。我们利用线性回归律,求出大行星的质-光关系式。它能为大行星能源探索提供某些重要线索和限制条件。其次,我们讨论了目前解释此能源的各种理论及其存在困难。最后,着重介绍一种新的热核反应机制。它不仅对超高温的而且对低温的聚变等离子体都适用。原来老的热核反应机制只是它的一个特例。它不仅能说明大行星内部能源问题,而且能解释实验室中和天体上观察到许多异常核聚变现象。我们发现:虽然这些异常现象在自然界表现形式各式各样,但是它们的基本原理都是相同的,即都可用新的热核反应机制来解释它们。     

Saturated magnetic field amplification at supernova shocks

Cosmic ray streaming instabilities at supernova shocks are discussed in the quasi-linear diffusion formalism which takes into account the feedback effect of wave growth on the cosmic ray streaming motion. In particular, the non-resonant instability that leads to magnetic field amplification in the short wavelength regime is considered. The linear growth rate is calculated using kinetic theory for a streaming distribution. We show that the non-resonant instability is actually driven by a compensating current in the background plasma. The non-resonant instability can develop into a non-linear regime generating turbulence. The saturation of the amplified magnetic fields due to particle diffusion in the turbulence is derived analytically. It is shown that the evolution of parallel and perpendicular cosmic ray pressures is predominantly determined by non-resonant diffusion. However, the saturation is determined by resonant diffusion which tends to reduce the streaming motion through pitch angle scattering. The saturated level can exceed the mean background magnetic field.     

Carbon-oxygen plasma; Parameter domain for a strong screening effect

A critical study relating to the enhancement factor of nuclear reaction rates in the carbon-oxygen plasma is made. Plots are given of the various dimensionless parameters in the temperature-density plane after the recent analysis of Itohet al. (1977). The enhancement factor of thermonuclear reaction rates critically depends on 3/. The enhancement factors also have been calculated here for the carbon-oxygen plasma studied by Iben (1978).     

Studies on the evolved carbon cores

Physical conditions prevalent in a degenerate carbon plasma lead to the enhancement of the carbon-carbon thermonuclear reaction rates. Nuclear energy generation rate in the carbon core is thereby augmented. The possible dissipation of energy due to pair-annihilation neutrinos, plasma neutrinos and neutrino bremsstrahlung are considered. Neutral current contribution to these weak processes are also taken into account. It is suggested that the enhanced nuclear gene-ration rate in the evolved core might halt the core collapse for a time, thus necessitating a reassessment of the phenomenon of core collapse as a precursor of carbon detonation supernova events.     

An explanation for the solar neutrino problem

A new mechanism of thermonuclear reaction is briefly introduced. It shows that a certain amount of thermonuclear reaction can take place in dense, low-temperature (T < 1 × 10⁵ K) plasmas. At a super-high temperature, it is in agreement with the old thermonuclear reaction mechanism first presented by Gamow and Bethe. As most regions in the Sun are at moderate and low temperature, a sufficient amount of fusion energy is generated therein. Therefore the solar central temperature in standard models must be decreased to about 13.8 × 10⁶ K. That would make the flux of high-energy neutrinos conform with the observational results.     

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.