Diffusion and convection of relativistic electrons in non-thermal sources: Steady-state solutions

In this paper we study the propagation of relativistic electrons in stationary spherically-symmetric models of non-thermal sources. Electrons are injected by a collapsed object in the centre and then diffuse away in the presence of a flux of accreting matter. We calculate the electron density taking into account both the outward diffusion and the inward convection: it is assumed that the diffusion coefficient and the velocity field have a power dependence on the radial coordinate. In Section 3 the transport equation is solved for the particular case where the electron energy is constant; while Section 4 treats the case in which the particles suffer synchrotron or inverse-Compton losses. The effects due to the convection are then discussed in comparison with the purely diffusive case.     

Temperature in transition region and inner corona

The investigation of Alamet al. (1979), of the temperature distribution in spherically-symmetric transition region and inner corona, appears to be in error.     

Spherically symmetric electromagnetic mass models with cosmological parameter ⋀

An exact solution has been obtained for the Einstein-Maxwell field equation with corresponding to a spherically-symmetric charged perfect fluid distribution. Here the cosmological constant is assumed to be a scalar variable depending on the radial coordinater of the spherical system, viz., = (r). The solution set thus obtained presents an electromagnetic mass model.     

Massless scalar static spheres

It is known that the requirement of asymptotic flatness places restrictions on spherically-symmetric solutions to field equations. Here it is shown that the most general solution to the static spherically-symmetric massless scalar Einstein equations with zero cosmological constant is asymptotically flat; furthermore, the general solution is derived and shown to be identical to a solution previusly found by M. Wyman.     

Self-gravitating fluid in a conformally-flat space-time

The Einstein field equations for an irrotational perfect fluid with pressurep, equal to energy density are studied when the space-time is conformally flat. The coordinate transformation to co-moving coordinates is discussed. The energy and Hawking-Penrose inequalities are studied. Static and non-static solutions of the field equations are obtained. It is interesting to note that in the static case the only spherically-symmetric conformally flat solution for self-gravitating fluid is simply the empty flat space-time of general relativity.     

Qualitative behaviour of the trajectories of test bodies in five-dimensional spherically-symmetric gravitational field

Qualitative behaviour of the trajectories of test bodies is investigated in the five-dimensional gravitation theory for the spherically-symmetric solution. The Binnet formula is obtained for the trajectory. The region of the existence of circular trajectories as well as the regions of their stability are specially investigated using the method of numerical calculations. The values of the free parameters in the five-dimensional gravitation theory are estimated for the solar system.     

The effect of a bounded interplanetary diffusion medium on the propagation of solar flare cosmic rays

An analysis of solar flare data indicates that the graph of log(nt ^3/(2–)) deviates late in the solar event from the straight line predicted for the infinite, unbounded interplanetary medium. It is shown by mathematical analysis, utilizing a model based on the radial diffusion coefficient D = Mr , with 1, that the deviation can be ascribed to the loss of flare particles through an external boundary at about 5–6 AU from the Sun. An inner region terminating at 5–6 AU, followed by an extensive region of increasingly less resistance to the diffusion of flare particles is also feasible and it is shown that measurements taken at the Earth cannot predict the extent of this outer region. The results are applicable to either the isotropic or highly anisotropic models. The constant diffusion model is shown to be inadequate since it requires a boundary 1.5 AU from the Sun. In view of the present and previous studies of solar flare data, it is asserted that the fundamental principle governing the diffusion of solar flare particles through interplanetary space is the radial diffusion coefficient mode of propagation.     

Interplanetary magnetic field structure and solar wind parameters as inferred from solar magnetic field observations and by using a numerical 2-D MHD model

An attempt is made to infer parameters of the solar corona and the solar wind by means of a numerical, self-consistent MHD simulation. Boundary conditions for the magnetic field are given from the observations of the large-scale magnetic field at the Sun. A two-region, planar (the ecliptic plane is assumed) model for the solar wind flow is considered. Region I of transonic flow is assumed to cover the distances from the solar surface up to 10R _S (R _S is the radius of the Sun). Region II of supersonic, super-Alfvénic flow extends between 10R _S and the Earth's orbit. Treatment for region I is that for a mixed initial-boundary value problem. The solution procedure is similar to that discussed by Endler (1971) and Steinolfson, Suess, and Wu (1982): a steady-state solution is sought as a relaxation to the dynamic equilibrium of an initial state. To obtain a solution to the initial value problem in region II with the initial distribution of dependent variables at 10R _S (deduced from the solution for region I), a numerical scheme similar to that used by Pizzo (1978, 1982) is applied. Solar rotation is taken into account for region II; hence, the interaction between fast and slow solar wind streams is self-consistently treated. As a test example for the proposed formulation and numerical technique, a solution for the problem similar to that discussed by Steinolfson, Suess, and Wu (1982) is obtained. To demonstrate the applicability of our scheme to experimental data, solar magnetic field observations at Stanford University for Carrington rotation 1682 are used to prescribe boundary conditions for the magnetic field at the solar surface. The steady-state solution appropriate for the given boundary conditions was obtained for region I and then traced to the Earth's orbit through region II. We compare the calculated and spacecraft-observed solar wind velocity, radial magnetic field, and number density and find that general trends during the solar rotation are reproduced fairly well although the magnitudes of the density in comparison are vastly different.     

On the disappearance of isolating integrals in dynamical systems with more than two degrees of freedom

We continue to study the number of isolating integrals in dynamical systems with three and four degrees of freedom, using as models the measure preserving mappingsT already introduced in preceding papers (Froeschlé, 1973; Froeschlé and Scheidecker, 1973a).Thus, we use here a new numerical method which enables us to take as indicator of stochasticity the variation withn of the two (respectively three) largest eigenvalues-in absolute magnitude-of the linear tangential mappingT ⁿ * ofT ⁿ . This variation appears to be a very good tool for studying the diffusion process which occurs during the disappearance of the isolating integrals, already shown in a previous paper (Froeschlé, 1971). In the case of systems with three degrees of freedom, we define and give an estimation of the diffusion time, and show that the gambler's ruin model is an approximation of this diffusion process.     

Quasinormal modes of a black hole with quintessence-like matter and a deficit solid angle: scalar and gravitational perturbations

In the previous paper (Li et al. in Phys. Lett. B 666:125–130, 2008), we show the solutions of Einstein equations with static spherically-symmetric quintessence-like matter surrounding a global monopole. Furthermore, this monopole become a black hole with quintessence-like matter and a deficit solid angle when it is swallowed by an ordinary black hole. We study its quasinormal modes by WKB method in this paper. The numerical results show that both the real part of the quasinormal frequencies and the imaginary part decrease as the state parameter w, for scalar and gravitational perturbations. And we also show variations of quasinormal frequencies of scalar and gravitational fields via different ε (deficit solid angel parameter) and different ρ ₀ (density of static spherically-symmetric quintessence-like matter at r=1), respectively.     

The stability of multi-component gravitational systems described by a Boltzmann equation

This paper uses existing techniques developed by Ipser, Thorne, and Kandrup to examine the stability of multi-component systems, with a distribution of masses, whose evolution is described by the collisionless or collisional Boltzmann equation.The principal conclusions are as follows: (1) All static, spherically-symmetric solutions to the collisionless equations, appropriate for a star cluster, are guaranteed to be stable with respect to spherically-symmetric disturbances, provided only that the population of stars is, for fixed mass, a decreasing function of the mean field energy. (2) If, furthermore, the static configuration has an isotropic distribution of momenta, it will also be stable towards nonradial perturbations. (3) The unique static solution to the collisional equations for a spherically-symmetric, spatially truncated configuration can be stable only if that configuration is a local entropy maximum. (4) A simple expression is obtained for the Jeans length for a system with an arbitrary isotropic distribution of momenta.     

Superluminal velocity and deviation from Newton's inverse square law

Starting from the theory of gravitation in flat space-time of Petry (1981), the gravitational field of a static spherically-symmetric body is studied. Petry (1982) has shown that this field in the exterior of the body depends on a parameter which is fixed by the interior solution, i.e., it depends on the density of matter, the pressure, the equation of state, etc. If this parameter is small, the results for the well-known effects, i.e., redshift, deflection of light, perihelion shift, and radar time-delay, agree with those of general relativity. In this paper, we study these effects for larger values of this parameter. Furthermore, for sufficiently large positive values of the parameter, in the neighbourhood of the body the radial velocity of light can exceed several times the vacuum light-velocity. Therefore, the components of such an object can move away with velocities a few times greater than that of light in agreement with the observed superluminal velocities in extragalactic objects.     

Verification of maximum limit of density variation parameter λ for a stable neutron star using Reissner-Nordström interior solution

A static spherically-symmetric model, based on an exact solution of Einstein's equation, gives the permissible matter density ~2 × 10¹⁴ g cm^–3. By use of the change in radius density (i.e., central density per unit radius) minimum, Parui and Sarma (1991) have estimated the upper limit of the density variation parameter = 0.68 for a superdense star such as a neutron star withK = –2. In this paper we have verified this upper limit using the Reissner-Nordström interior solution of the Einstein-Maxwell's field equations withK = - 3.     

Conformally flat static space-time in the general scalar-tensor theory of gravitation

Explicit vacuum field equations in the general scalar-tensor theory of gravitation proposed by Nordtvedt are obtained with the aid of the most general conformally flat spherically-symmetric static space-time. It is shown that the most general conformally flat spherically-symmetric static solution of Nordtvedt-Barker vacuum field equations is simply the empty flat space-time of general relativity.     

Radial gradients and anisotropies due to galactic cosmic rays

Numerical calculations have been made of the radial gradients and the anisotropyvector atr=1 AU due to galactic cosmic-ray protons and helium nuclei. The model used assumes transport by convection and anisotropic diffusion, and includes the energy losses due to adiabatic deceleration. The present calculations are for the 1964–65 solar minimum. An important constraint applied ineach case was that the model reproduces the electron modulation known from deductions of the galactic spectrum and observations of the near-Earth spectrum; and also reproduces the near-Earth proton and helium nuclei spectra. The diffusion coefficients have been based upon those deduced from magnetic-field power spectra.The principal aim has been to provide estimates of radial gradients and anisotropies, particularly at kinetic energiesT100 MeV/nucleon, by the complete solution of realistic models. Typical values for protons, obtained with a galactic differential number density (total energy)^–2.5, atT50 MeV are: radial gradient, 25%/AU; radial anisotropy, –0.2%; azimuthal anisotropy, 0.2%. These values change markedly when the galactic spectrum is cut-off or greatly enhanced atT<150 MeV, but the intensity spectrum near Earth remains substantially unchanged.It has been shown that it is possible to obtain negative radial gradients and positive radial anisotropies atT50 MeV for galactic particles and thus to mimic solar sources. The radial gradient for 1964–65 reported by Anderson (1968) and by Krimigis and Venkatesan (1969) are shown to be consistent with the diffusion coefficient deduced from the magnetic-field power spectrum; those reported by O'Gallagher are higher than expected and that for 20T30 MeV protons appears to be inconsistent. More precise data on conditions throughout the solar cavity are required if more definitive gradients and anisotropies are to be determined.     

Time-dependent Green's functions of the cosmic ray equation of transport

Two spherically symmetric time-dependent Green's functions of the equation of transport for cosmic rays in the interplanetary region are derived by transform techniques. The solar wind velocity is assumed radial and of constant speedV. In the first model the radial diffusion coefficient =₀ r (₀ constant), and in the second solution =₀= constant. The solutions are for monoenergetic, impulsive release of particles from a fixed heliocentric radius. Integration of the solutions over timet, fromt=0 tot=, gives the steady-state Green's functions obtained previously.     

Simultaneous solutions for photometric and spectroscopic observations of TX UMa

B andV light curves for one epoch and radial velocity curves of three different epochs have been analyzed to revise the solution of TX UMa. The solution has been adjusted simultaneously in the light curves and radial velocity curves by the method of Wilson and Devinney's differential correction. The primary star's surface rotation rate to synchronous rate is determined as 1.768 from one of the radial velocity curves. The absolute dimension of the system has been deduced based on the simultaneous solution. The primary star is well fitted to the evolutionary track for a single star while the secondary star, while filling its Roche lobe, is fitted to the evolutionary track for a close binary system.     

Radial diffusion revisited

The radial diffusion of equatorially mirroring particles (J = 0) is considered for Jupiter. A steady-state phase-space density distribution is obtained for (i) source-loss-free diffusion; (ii) diffusion with synchrotron radiation losses only and (iii) diffusion with synchrotron radiation plus the resonant wave-particle interaction losses. The resonant wave-particle interaction is assumed to occur when particles are in phase with a wave propagating across the magnetic field. The interaction of particles which go through a B drift with electrostatic plasma waves is shown to alter the phase-space density which is observed byPioneer 10 and 11 flybys.     

Axial perturbations of Wyman's solution

Wyman's solution is the most general solution to the static spherically-symmetric Einstein massless scalar field equations. It is shown that it has no axial perturbation in which the scalar field is incremented, except in the case where the initial scalar field and the cross-metric increments are negligible. The one dimensional Schrödinger equation which governs axial metrix perturbations is produced.     

Solutions of the spherically-symmetric cosmic-ray transport equation in interplanetary space

Numerical solutions of the well-known spherically-symmetric transport equation for interstellar cosmic rays in the interplanetary medium are presented. It is shown: (a) why nuclei and electron modulations differ, (b) that it is easy to read more in the model than it contains, (c) that the only significant parameter that determines the level of modulation is the product of the solar wind speed and the number of diffusion mean free paths between observer and boundary, and (d) that observations do exist which cannot be explained in terms of this simple model.