Solar structure in terms of Gauss' hypergeometric function

Hydrostatic equilibrium and energy conservation determine the conditions in the gravitationally stabilized solar fusion reactor. We assume a matter density distribution varying non-linearly through the central region of the Sun. The analytic solutions of the differential equations of mass conservation, hydrostatic equilibrium, and energy conservation, together with the equation of state of the perfect gas and a nuclear energy generation rate = ₀ ⁿ T ⁿT ^m ,are given in terms of Gauss' hypergeometric function. This model for the structure of the Sun gives the run of density, mass, pressure, temperature, and nuclear energy generation through the central region of the Sun. Because of the assumption of a matter density distribution, the conditions of hydrostatic equilibrium and energy conservation are separated from the mode of energy transport in the Sun.     

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.     

Reaction-Diffusion Systems and Nonlinear Waves

The authors investigate the solution of a nonlinear reaction-diffusion equation connected with nonlinear waves. The equation discussed is more general than the one discussed recently by Manne et al. (2000). The results are presented in a compact and elegant form in terms of Mittag-Leffler functions and generalized Mittag-Leffler functions, which are suitable for numerical computation. The importance of the derived results lies in the fact that numerous results on fractional reaction, fractional diffusion, anomalous diffusion problems, and fractional telegraph equations scattered in the literature can be derived, as special cases, of the results investigated in this article.     

Geotechnical Aspects of Clayey Sediments Off Badagara on the Kerala Coast,India

Offshore geotechnical surveys form part of an integrated investigation to rejuvenate a decrepit minor port at Badagara, Kerala on the southwestern coast of India. The sediments typify a fluvio-marine milieu ranging from silty clay, sand, silty sand, sandy silt and clayey silt. Geotechnical and sedimentological studies of shallow cores reveal the geotechnical aspects besides the depositional history of the sediments. Downcore geotechnical variations and regressive coefficients based on their inter-relationships highlight diverse factorial inferences. X-Ray Diffraction data indicate the prominent clay type.

A comparative evaluation of the geotechnical characteristics of clayey sediments off Badagara, with similar studies along various sectors of the Kerala coast, both on land as well as in the near shore, is broadly attempted. Geotechnical studies carried out earlier on the uplifted Cochin marine clays provide comparative data for evaluating the possible variations between present day marine clayey sediments occurring along the Kerala coast and uplifted marine clays which, besides their gross variations in levels with respect to the present sea-level, also obviously relate to a much older depositional environment and provenance during probable Holocene times.     

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.     

A heuristic remark on the periodic variation in the number of solar neutrinos detected on Earth

Four operating neutrino observatories confirm the long standing discrepancy between detected and predicted solar neutrino flux. Among these four experiments the Homestake experiment is taking data for almost 25 years. The reliability of the radiochemical method for detecting solar neutrinos has been tested recently by the GALLEX experiment. All efforts to solve the solar neutrino problem by improving solar, nuclear, and neutrino physics have failed so far. This may also mean that the average solar neutrino flux extracted from the four experiments may not be the proper quantity to explain the production of neutrinos in the deep interior of the Sun. Occasionally it has been emphasized that the solar neutrino flux may vary over time. In this paper we do address relations among specific neutrino fluxes produced in the proton-proton chain that are imposed by the coupled systems of nonlinear partial differential equations of solar structure and kinetic equations by focusing our attention on a statistical interpretation of selected kinetic equations of PPII/PPIII branch reactions of the protonproton chain. A fresh look at the statistical implications for the outcome of kinetic equations for nuclear reactions may shed light on recent claims that the⁷ Be-neutrino flux of the Sun is suppressed in comparison to the pp- and⁸B neutrino fluxes and may hint at that the solar neutrino flux is indeed varying over time as shown by the Homestake experiment.     

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.     

Evolution of Archaean Southern Indian lithospheric mantle: a geochemical study of Proterozoic Agali — Coimbatore dykes

 Agali–Coimbatore dolerite dykes constitute an important Proterozoic magmatic event that affected the south Indian shield. Rb-Sr whole rock isotope data yield an “errorchron” of 2369±400 Ma (2σ error) which is within error of the reported 2030±65 Ma K-Ar age. The dyke magmas were evolved Fe-rich tholeiitic melts produced by fractionation of clinopyroxene, orthopyroxene and olivine in the initial stages. Plagioclase became a fractionation phase during the latter stages of crystallization. The dykes characteristically have high ⁸⁷Sr/⁸⁶Sr_i (0.703–0.706) and are enriched in large-ion lithophile and light rare earth elements relative to primordial mantle values and show negative Nb anomalies. These compositional characteristics are interpreted as source mantle characteristics whereas some crustal effects are visible in some samples with high initial ⁸⁷Sr/⁸⁶Sr. Peridotite with minor hydrous metasomatic phases like amphibole (and phlogopite) within the shallow lithospheric mantle could be a potential source material for the dykes. However, at this stage we cannot convincingly differentiate whether the source of the parent magmas is solely lithospheric or a product of asthenosphere-lithosphere mixing. The δ¹⁸O values of the dykes range from +5.2 to +7.2 per mil (vs standard mean oceanic water). Initial Nd isotope values at the time of dyke intrusion (ɛ_Nd at t=2.0 Ga) range from −2.3 to −4.8. Whole rocks define a correlation on an Sm-Nd isochron plot with a slope equivalent to an age of 3.15±0.53 Ga (2σ error); Sm-Nd crustal residence ages average at 2.87 Ga. The isochron age does not appear to be the result of systematic mixing with an older crustal component. These results together with trace element geochemistry suggest that the south Indian mantle lithosphere developed by addition of enriched melts/fluids at about 3.0 Ga synchronously with major crustal gene- ration in the south Indian shield. Received 20 June 1994/Accepted: 17 May 1995     

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 fractional kinetic equation and thermonuclear functions

The paper discusses the solution of a simple kinetic equation of thetype used for the computation of the change of the chemical compositionin stars like the Sun. Starting from the standard form of the kineticequation it is generalized to a fractional kinetic equation and itssolutions in terms of H-functions are obtained. The role of thermonuclearfunctions, which are also represented in terms of G- and H-functions,in such a fractional kinetic equation is emphasized. Results containedin this paper are related to recent investigations of possibleastrophysical solutions of the solar neutrino problem.