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.     

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.     

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.     

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.     

Closed-form expressions for the rate of growth of perturbations in zero-pressure Friedmann-Lemaître universes

The general solutions to the density perturbation equation derived by Bonnor are pressented for the zero-pressure Friedmann-Lemaître universes in terms of elliptic functions. The particular solutions already published to this equation in terms of elementary functions are exhibited as special cases arising from the general solutions. These occur when the elliptic functions simplify to either trigonometric or hyperbolic functions.     

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.     

Resonant and non-resonant gravity-gradient perturbations of a tumbling tri-axial satellite

Gravity-gradient perturbations of the attitude motion of a tumbling tri-axial satellite are investigated. The satellite center of mass is considered to be in an elliptical orbit about a spherical planet and to be tumbling at a frequency much greater than orbital rate. In determining the unperturbed (free) motion of the satellite, a canonical form for the solution of the torque-free motion of a rigid body is obtained. By casting the gravity-gradient perturbing torque in terms of a perturbing Hamiltonian, the long-term changes in the rotational motion are derived. In particular, far from resonance, there are no long-period changes in the magnitude of the rotational angular momentum and rotational energy, and the rotational angular momentum vector precesses abound the orbital angular momentum vector.At resonance, a low-order commensurability exists between the polhode frequency and tumbling frequency. Near resonance, there may be small long-period fluctuations in the rotational energy and angular momentum magnitude. Moreover, the precession of the rotational angular momentum vector about the orbital angular momentum vector now contains substantial long-period contributions superimposed on the non-resonant precession rate. By averaging certain long-period elliptic functions, the mean value near resonance for the precession of the rotational angular momentum vector is obtained in terms of initial conditions.     

Pulsation regime of the thermonuclear explosion of a star's dense carbon core

The hydrodynamical problem of nuclear explosion of a dense carbon core of a star with mass 1.40M _⊙ is solved numerically. In calculation the kinetics of carbon burning at the nuclear reaction C¹²+C¹²→M²⁴+γ rate is included. Thus the inverse effect of hydrodynamical motion on the process of thermonuclear burning is taken into account, as compared with Bruenn's (1972) calculations. The calculations show that a pulsation regime of burning is realized (actually three pulses were obtained) which evolves to the detonation regime with full combustion and disruption of the star only at the third pulse. The effects of disintegration of iron group nuclei, neutronization of matter and neutrino losses in URCA processes have not yet been considered in calculations. The influence of initial conditions (mainly the temperature distributions) and the above mentioned effects, which have not been included in calculation, on the results of the hydrodynamical problem solution are discussed. The conclusion is made on new possibilities of formation of a gravitationally bound remnant of the explosion and a neutron star.     

Burning regimes for thermonuclear supernovae and cosmological applications of SNe Ia

The method of multigroup radiation hydrodynamics is used to compute light curves for thermonuclear supernovae. Opacities are computed by taking into account spectral lines and expansion. UBVI fluxes are predicted. Our computed times of brightness rise to a maximum in B and V have been found to agree with observations better than those of other authors. The validity of our results is justified physically. The nuclear burning regime is shown to affect significantly the slope of the light curve in B and, to a slightly lesser extent, in V. If the prevailing burning regime during supernova explosions changed with age of the Universe, then the conclusion about a positive cosmological constant Λ drawn from observations of Type Ia supernovae may prove to be wrong.     

Tritium chain of the hydrogen cycle of thermonuclear reactions in the solar interior

The tritium chain of the hydrogen cycle on the Sun including the reactions ³He(e⁻, ν _e) ³H(p, γ)⁴He is considered. The flux of tritium neutrinos at a distance of 1 AU is 8.1 × 10⁴ cm⁻² s⁻¹. It exceeds the neutrino flux from the (hep)-reaction by one order of magnitude. The radial distribution of the yield of ³H neutrinos inside the Sun and their energy spectrum, which has the form of a line at an energy of 2.5–3.0 keV, have also been calculated. The flux of thermal tritium neutrinos is accompanied by a very weak flux of antineutrinos (∼10³ cm⁻² yr⁻¹) with an energy below 18.6 keV. These antineutrinos are produced in the URCA processes ³He ⇆ ³H.     

Massive star evolution: nucleosynthesis and nuclear reaction rate uncertainties

We present a nucleosynthesis calculation of a 25 M_⊙ star of solar composition that includes all relevant isotopes up to polonium. We follow the stellar evolution from hydrogen burning till iron core collapse and simulate the explosion using a ‘piston’ approach. We discuss the influence of two key nuclear reaction rates, ¹²C(α, γ)¹⁶O and ²²Ne(α, n)²⁵Mg, on stellar evolution and nucleosynthesis. The former significantly influences the resulting core sizes (iron, silicon, oxygen) and the overall presupernova structure of the star. It thus has significant consequences for the supernova explosion itself and the compact remnant formed. The later rate considerably affects the s-process in massive stars and we demonstrate the changes that different currently suggested values for this rate cause.     

Detection of the first thermonuclear X-ray burst from AX J1754.2-2754

When analyzing the archival data of the INTEGRAL observatory, we detected an intense X-ray burst recorded on April 16, 2005, by the JEM-X and IBIS/ISGRI telescopes from the weak and poorly studied source AX J1754.2-2754. Analysis of its time profiles and spectra allows this event to be attributed to type I X-ray bursts associated with thermonuclear explosions on the surfaces of neutron stars and the source itself to X-ray bursters. Peculiarities of the X-ray emission observed at the initial evolutionary phase of the burst point to a dramatic expansion and a corresponding cooling of the neutron star photosphere that took place at this time under the action of radiation pressure. Assuming the luminosity of the source at this phase to be the Eddington one, we have estimated the distance to the burst to be d = 6.6 ± 0.3 kpc (for a hydrogen atmosphere of the neutron star) and d = 9.2 ± 0.4 kpc (for a helium atmosphere).     

The rate of the reaction between CN and C2H2 at interstellar temperatures

The rate coefficient for the important interstellar reaction between CN and C2H2 has been calculated as a function of temperature between 10 and 300 K. The potential surface for this reaction has been determined through ab initio quantum chemical techniques; the potential exhibits no barrier in the entrance channel but does show a small exit channel barrier, which lies below the energy of reactants. Phase-space calculations for the reaction dynamics, which take the exit channel barrier into account, show the same unusual temperature dependence as determined by experiment, in which the rate coefficient at first increases as the temperature is reduced below room temperature and then starts to decrease as the temperature drops below 50-100 K. The agreement between theory and experiment provides strong confirmation that the reaction occurs appreciably at cool interstellar temperatures.     

Review of mathematical techniques applicable in astrophysical reaction rate theory

An overview is presented on statistical techniques for the analytic evaluation of integrals for non-resonant, non-resonant depleted, non-resonant cut-off, non-resonant sccreened, and resonant thermonuclear reaction rates. The techniques are based on statistical distribution theory and the theory of Meijer's G-function and Fox's H-function. The implementation of Meijer's G-function in Mathematica constituts an additional utility for analytic manipulations and numerical computation of thermonuclear reaction rate integrals. Recent results in the astrophysical literature related to the use of analytic thermonuclear reaction rates are incorporated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Types I and II supernovae and the neutrino mechanism of thermonuclear explosion of degenerate carbon-oxygen stellar cores

The present work studies the hydrodynamic process of thermonuclear explosion of hydrostatic equilibrium, degenerate carbon-oxygen cores withM _C=1.40M _⊙ with different values of central densityϱ _c within the interval 2 × 10⁹ <ϱ _c < 3 × 10¹⁰ g cm⁻³. The initial temperature distribution has been determined by the preceding thermal stage of explosion. The calculations successively include the kinetics of thermonuclear burning, the kinetics of β-processes, and neutrino energy losses. By considering the neutrino mechanism of heating and carbon ignition we obtained in our numerical hydrodynamic calculations two characteristic versions of the development of an explosion: (a) at 2 × 10⁹ <ϱ _c < 9 × 10⁹ g cm⁻³ there is disruption of the whole star with either complete or partial burning of the carbon and a 10⁵⁰–10⁵¹ erg kinetic energy; and (b) at 9 × 10⁹ <ϱ _c < 3 × 10¹⁰ g cm⁻³ the stellar core collapses into a neutron star with partial outburst of the outer envelope with a smaller kinetic energy of 10⁴⁹–10⁵⁰ erg. The paper proposes and details a hypothesis (the scenario of supernovae and the formation of neutron stars) on the first version of explosion, corresponding to SNII, and on the second, supplemented by some mechanism of slow energy release into the envelope expelled from the newly formed neutron star, corresponding to SNI. On the basis of the proposed hypothesis a satisfactory agreement with the observed masses and energies of the supernovae envelope, their light curves and spectra, as well as with the data on their chemical composition has been obtained. For this agreement we must assume that type I pre-supernovae are almost bare compact carbon-oxygen stellar cores, and that type II presupernovae are red supergiants. It is most probable that the evolution of type I pre-supernovae occurs in close binaries while the evolution of type II pre-supernovae seems to be very similar to the evolution of a single star.     

Numerical investigation of non-resonant and resonant scattering of charged particles with a spatially varying magnetic field

By integrating many charged particle trajectories in a magnetic field model consisting of a series of equally spaced field discontinuities with equal angular displacements, constant ¦B¦ and successive displacements oppositely directed, a parallel diffusion coefficient K _∥ is obtained. The particle gyroradius was kept sufficiently small for the interaction to be non-resonant. The diffusion coefficient is found to be in good agreement with that predicted by the known reflection properties for charged particles of individual discontinuities. However an attempt to reproduce the diffusion coefficient using the results of a recent study by Klimas and Sandri of a non-local diffusion equation applying to the non-resonant case lead to too low a value of K _∥. The computational approach was also applied to the case where the particle motion was in resonance with the wavelength of the train of discontinuities and a lower limit to K _∥ obtained. This lower limit exceeded the quasi-linear approximation value for K _∥ under resonant scattering conditions.     

The rate of the reaction between C2H and C2H2 at interstellar temperatures

The reaction between the radical C2H and the stable hydrocarbon C2H2 is one of the simplest neutral-neutral hydrocarbon reactions in chemical models of dense interstellar clouds and carbon-rich circumstellar shells. Although known to be rapid at temperatures > or = 300 K, the reaction has yet to be studied at lower temperatures. We present here ab initio calculations of the potential surface for this reaction and dynamical calculations to determine its rate at low temperature. Despite a small potential barrier in the exit channel, the calculated rate is large, showing that this reaction and, most probably, more complex analogs contribute to the formation of complex organic molecules in low-temperature sources.