Gravitational collapse of a rotating iron stellar core and physical properties of the accompanying neutrino emission

We have ascertained an important role of rotation effects in a collapsing stellar core using a quasi-one-dimensional hydrodynamic model with a rigorous allowance for the neutrino energy losses including the neutrino opacity stage. However, the neutrino scattering processes are not considered in the neutrino emission kinetics as secondary compared to the absorption processes. The quasi-one-dimensional approximation (with averaging of the expression for the centrifugal force over the polar angle) allows numerical calculations to be performed relatively easily up to the formation of a hydrostatically equilibrium neutron star after a very long stage of collapsar cooling by neutrino emission (about 2 s). We present detailed results of our numerical solution, including the neutrino spectra, with electron neutrinos making a dominant contribution to them and the contribution from electron antineutrinos being smaller by an order of magnitude. In the model under consideration, we solve the equation of matter neutronization kinetics by taking into account the main process of nuclear reactions on free nucleons, although the contribution from iron and helium nuclei is included in the equation of state.     

Multiple compton scattering of electrons in a photon field

Consideration is given to the motion of electrons in a photon field of the monoenergetic or power-law spectrum under the conditions when the main mechanism of energy loss is the inverse Compton scattering by field photons. This process changes the primary spectrum of electrons and converts low-energy field photons to high-energy gamma-quanta for which the electron confinement region is assumed to be optically thin. The electron and gamma-ray spectra have been obtained in a wide energy interval including the Klein-Nishina and Thomson regions. A simple qualitative dependence of the solutions found on the field parameters and the primary spectrum of electrons has been established.The electron and gamma-ray spectra have been obtained by numerically solving the kinetic equation dependent on two variables: the energy of electrons and their path (or the time of motion) in a photon field. The results dramatically differ from the solution of the steady-state kinetic equation which depends only on the electron energy and is frequently used in the given problem.     

Analytic approach to auroral electron transport and energy degradation

The interaction of a beam of auroral electrons with the atmosphere is described by the linear transport equation, encompassing discrete energy loss, multiple scattering and secondary electrons. A solution to the transport equation provides the electron intensity as a function of altitude, pitch angle (with respect to the geomagnetic field) and energy. A multi-stream (discrete ordinate) approximation to the transport equation is developed. An analytic solution is obtained in this approximation. The computational scheme obtained by combining the present transport code with the energy degradation method of Swartz (1979) conserves energy identically. The theory provides a framework within which angular distributions can be easily calculated and interpreted. Thus, a detailed study of the angular distributions of “non-absorbed” electrons (i.e., electrons that have lost just a small fraction of their incident energy) reveals a systematic variation with incident angle and energy, and with penetration depth. The present approach also gives simple yet accurate solutions in low order multi-stream approximations. The accuracy of the four-stream approximation is generally within a few per cent, whereas two-stream results for backscattered mean intensities and fluxes are accurate to within 10–15%.     

A relaxation method for the computation of model stellar atmospheres

A general Monte Carlo relaxation method has been formulated for the computation of physically self-consistent model stellar atmospheres. The local physical state is obtained by solving simultaneously the equations of statistical equilibrium for the atomic and ionic level populations, the kinetic energy balance equation for the electron gas to obtain the electron temperature, and the equation of radiative transfer. Anisotropic Thomson scattering is included in the equation of transfer and radiation pressure effects are included in the hydrostatic equation. The constraints of hydrostatic and radiative equilibrium are enforced. Local thermodynamic equilibrium (L.T.E.) is assumed as a boundary condition deep in the atmosphere. Elsewhere in the atmosphere L.T.E. is not assumed.The statistical equilibrium equations are solved with no assumptions made concerning detailed balance for the bound-bound radiative processes. The source function is formulated in microscopic detail. All atomic processes contributing to the absorption and emission of radiation are included. The kinetic energy balance equation for the electron gas is formulated in detail. All atomic processes by which kinetic energy is gained and lost by the electron gas are included.The method has been applied to the computation of a model atmosphere for a pure hydrogen early-type star. An idealized model of the hydrogen atom with five bound levels and the continuum was adopted. The results of the trial calculation are discussed with reference to stability, accuracy, and convergence of the solution.Contribution No. 385 from the Kitt Peak National Observatory.Operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.     

New Boundary Integral Equation Representation for Finite Energy Force-Free Magnetic Fields in Open Space above the Sun

A boundary integral equation to describe a force-free magnetic field with finite energy content in the open space above the solar surface is found. This is a new representation for a 3-D nonlinear force-free field in terms of the boundary field and its normal gradient at the boundary. Therefore the magnetic field observed on the solar surface can be incorporated into the formulation directly and a standard numerical technique, the boundary element method, can be applied to solve the field. A numerical test case demonstrates the power of the method by recovering the analytical solution to the desired accuracy and its application to practical solar magnetic field problems is straightforward and promising.     

Analytical description of charged particle motion in a reconnecting current sheet

We have obtained an analytical solution to the equation of motion in the guiding center approximation for nonrelativistic charged particles in a reconnecting current sheet with a three-component magnetic field. Given the electric field attributable to magnetic reconnection, the solution describes stable and unstable three-dimensional particle orbits. We have found the domain of input parameters at which the motion is stable. A physical interpretation of the processes affecting the stability of the motion is given. Charge separation is shown to take place in the sheet during the motion: oppositely charged particles are localized mostly in different regions of the current sheet. A formula is derived for the particle energy in stable and unstable orbits. The results obtained by numerical and analytical methods are compared.     

Dispersion Relation of Longitudinal Oscillations in Relativistic Plasmas of Fast Electron Distribution

In the relativistic case, the dispersion equation of longitudinal oscillations in unmagnetized, collisionless and isotropic plasmas of fast electron distribution is derived from the longitudinal dielectric constant of plasma. The equation is analytically solved, and the long-wavelength and short-wavelength dispersion relations are obtained. Because of the discontinuity of the analytical dispersion curve, the dimensionless dispersion equation of longitudinal oscillations is numerically calculated to obtain the full dispersion curve of longitudinal oscillations in relativistic plasmas of fast electron distribution. Further more, by fitting the numerical solution, a simple functional expression of the dispersion curve is given in favor of applications. Finally, in the extremely relativistic case, the dispersion relation of fast electron distribution is compared with that of Maxwell distribution. It is shown that the two kinds of dispersion relations have similar properties in a certain range of wave numbers.     

Cylindrical Zakharov–Kuznestov equation for ion-acoustic waves with electrons featuring non-extensive distribution

Cylindrical Zakharov–Kuznestov equation for ion-acoustic waves comprising of ions and electrons featuring non-extensive distribution are derived from the fluid equations through reductive perturbation technique. System of first order ordinary differential equations is obtained from Zakharov–Kuznestov equation through dynamical system approach and ultimately it is solved using numerical method. It is found that the electron to positron ratio parameter and the non-extensive distributed parameter due to electron play crucial role on the solution.     

Electron acoustic soliton energy of the Kadomtsev-Petviashvili equation in the Earth’s magnetotail region at critical ion density

The nonlinear properties of small amplitude electron-acoustic solitary waves (EAWs) in a homogeneous system of unmagnetized collisionless plasma consisted of a cold electron fluid and isothermal ions with two different temperatures obeying Boltzmann type distributions have been investigated. A reductive perturbation method was employed to obtain the Kadomstev-Petviashvili (KP) equation. At the critical ion density, the KP equation is not appropriate for describing the system. Hence, a new set of stretched coordinates is considered to derive the modified KP equation. Moreover, the solitary solution, soliton energy and the associated electric field at the critical ion density were computed. The present investigation can be of relevance to the electrostatic solitary structures observed in various space plasma environments, such as Earth’s magnetotail region.     

Kadomtsev–Petviashvili equation for solitary waves in warm dense astrophysical electron-positron-ion plasmas

Theoretical investigation is carried out to understand the dynamics and stability of three dimensional ion solitary waves propagating in dense plasma comprising of ultra-relativistic degenerate electrons and positrons and warm ions. A linear dispersion relation is derived which shows a strong dependence of wave on positron concentration (through the change of density balance) and ion-to-degenerate electron temperature ratio. A nonlinear Kadomtsev-Petviashvili equation is derived by employing the reductive perturbation technique and solved analytically and the conditions for existence of stable solitary waves are found. The analysis reveals that only compressive solitary waves exist in the system. Effects of the change of density balance and Fermi temperature ratios are studied in detail, both analytically and numerically. Furthermore, the conditions for stable solitary waves are discussed by using energy consideration method. The numerical results are also presented by using the parameters consistent with the degenerate and ultrarelativistic astrophysical plasmas.     

Calculation of the perihelion advance of planets in a field approach to gravitation

It is shown that, when taking into account the self-gravity of field energy of a gravitational field, one obtains a modified field equation for the intensity of gravitational field, the solution of which, when inserted in Kepler's problem, furnishes a formula for the perihelion precession of planets which (except of a fitting numerical factor) is identical with Einstein's. In Appendix 1 we point out the significance of an analogous equation in electrodynamics, and in Appendix 2 we shall try to construct a field model within the relativistic field theory which justifies the previous perihelion-shift calculation.     

Relation between hard X-ray spectra and electron energy spectra

The relationship between hard X-ray spectra and energetic electron spectra in solar X-ray bursts is investigated, and a simplified cross-section for bremsstrahlung which is applicable to the region of mildly relativistic energies is proposed. Using the proposed cross-section, we solve an integral equation to obtain the electron energy spectrum. The validity of the proposed cross-section is checked by comparing the spectrum calculated by the exact Bethe-Heitler formula. A good agreement between two calculated spectra is obtained up to 10 MeV energy with an accuracy of 20 %.     

Effect of electron trapping and background nonextensivity on the ion-acoustic soliton energy

Weak ion-acoustic (IA) solitary wave propagation is investigated in the presence of electron trapping and background nonextensivity. A physically meaningful distribution is outlined and a Schamel-like equation is derived. The role a background electron nonextensivity may play on the energy carried by the IA soliton is then examined. It is found that nonextensivity may cause a soliton energy depletion. An increase of the amount of electron trapping leads to a net shift towards higher values of the soliton energy.     

A new simple method for the analytical solution of Kepler's equation

A new simple method for the closed-form solution of nonlinear algebraic and transcendental equations through integral formulae is proposed. This method is applied to the solution of the famous Kepler equation in the two-body problem for elliptic orbits. The resulting formulae are quite elementary and, beyond their analytical interest, they can also provide quite accurate numerical results by using Gausstype quadrature rules.     

Inverse compton scattering in a strong magnetic field

In this paper, I discuss inverse Compton scattering in a strong magnetic field. Using the standard technique in quantum electrodynamics, the solution of Dirac's equation for an electron in a uniform magnetic field and the accurate propagator of electron in a magnetic field, I have calculated the scattering matrix elements in the coordinate representation, the transition probabilities, and found the general formulae for the energy distribution of the scattered photons, and the differential and total scattering cross-sections.     

Polytropic equilibrium figures of equal mass

A numerical method for the solution of the (astrophysical) potential problem is presented. The problem is formulated as a free boundary problem for a mildly nonlinear elliptic partial differential equation and the method is obtained by combining Newton-Raphson's procedure and two different types of discretization. The performance of the method is discussed.     

Application of the three-dimensional telegraph equation to cosmic-ray transport

An analytical solution to the three-dimensional telegraph equation is presented.This equation has recently received some attention but so far the treatment has been one-dimensional.By using the structural similarity to the Klein-Gordon equation,the telegraph equation can be solved in closed form.Illustrative examples are used to discuss the qualitative differences from the diffusion solution.A comparison with a numerical test-particle simulation reveals that some features of an intensity profile can be better explained using the telegraph approach.     

Exact and numerical solutions of the kompaneets equation: Evolution of the spectrum and average frequencies

The evolution of the spectrum of isotropic uniform radiation in an infinite space filled with a homogeneous, nonrelativistic electron gas is calculated by solving the Kompaneets equation. For an infinitely narrow initial spectrum, the time dependence of the average frequency and frequency dispersion is determined in a linear approximation of the equation. Characteristic times corresponding to changes in the character of this dependence are introduced. Two schemes are proposed for the numerical solution of the nonlinear equation: a nonconservative scheme with a grid that is uniform in frequency and a conservative scheme with automatic selection of an adaptive grid in frequency and time. For the linear equation the method yields results consistent with calculations of its solutions in terms of an eigenfunction expansion of the Kompaneets operator calculated in [D. I. Nagirner and V. M. Loskutov, Astrofizika, 40, 97 (1977)]. The influence of nonlinearity on the evolution of the spectrum of initially monochromatic radiation of various intensities is traced as an example of the application of the method.     

Contribution of Higher-Order Nonlinearity to obliquely electron-acoustic solitary waves in a magnetized auroral zone plasma

Using the Viking Satellite observations data in the dayside auroral zone, a theoretical investigation is carried out for contribution of the higher-order nonlinearity to nonlinear obliquely electron-acoustic solitary waves (EASWs) in a magnetized collisionless plasma consisting of a cold electron fluid and non-thermal hot electrons obeying a non-thermal distribution, and stationary ions. A Zakharov–Kuznetsov (ZK) equation that contains the lowest-order nonlinearity and dispersion is derived from the lowest order of perturbation and a linear inhomogeneous (ZK-type) equation that accounts for the higher-order nonlinearity and dispersion is obtained. A stationary solution for equations resulting from higher-order perturbation theory has been found using the renormalization method. The effects of the external magnetic field and the obliqueness are found to significantly change the higher-order properties (viz. the amplitude, width, electric field and energy) of the EASWs. The effect of higher-order nonlinearity on the amplitude and width of the soliton are also discussed. A comparison with the Viking Satellite observations in the dayside auroral zone are taken into account.     

Quasi-similar solution of the strong shock wave problem in non-ideal gas dynamics

The quasisimilar theory is used to investigate the solution of the blast wave problem with generalized geometries in a non-ideal gas satisfying the equation of state of the Van der Waals type. Here it is assumed that the distribution of normalized velocity, pressure and density are nearly similar in the narrow range of the shock strength. A comparison between approximate analytical solution and numerical solution of the problem is presented for the cylindrical geometry. The numerical solutions are presented for the generalized geometry in a non-ideal gas. It is also assessed as to how the non-idealness of the gas affects the behavior of the flow parameters.