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1.
The comoving-frame equation of radiative transfer and moment equations are derived in orthogonal, curvilinear coordinates, inclusive of terms of orderv/c. The equation of radiative transfer, which contains the terms due to the effect of curvature of coordinate lines explicitly as well as those of Doppler shift and aberration, is the generalization of Castor's equation for spherical symmetry and of Buchler's equation for Cartesian coordinates. The moment equations agree with Buchler's.  相似文献   

2.
A method of analysis is presented for solving the radiative transfer problem in an absorbing, emitting, inhomogeneous, and anisotropically scattering plane-parallel medium with specular and diffuse reflecting boundaries and internal source (problem 1). Exact relations for the radiation heat flux at the boundaries of problem 1 are obtained in terms of the radiation density and albedos of the corresponding source-free medium with specular reflecting boundaries (problem 2). Two coupled integral equations for the radiation density and the second moment of the radiation intensity for problem 2 with Rayleigh phase functions are obtained. The Galerkin method is used to solve these equations. Albedos of problem 2 are compared with theF n method. Numerical results for radiation heat fluxes at the boundaries of problem 1 are tabulated for different forms of the internal source.  相似文献   

3.
It is shown that the equation of motion Du j/Ds = (e/mc 2)F ji u i , a natural generalization to the curved spacetime of the Heaviside-Lorentz law of ponderomotive force, is equivalent to the metric independent and covariant Van Dantzig's equations of motion dx j [jpi] = 0 or L v p i = 0, where p i is the conjugate momentum 4-vector and v a vector determined by the condition p i v i, only with respect to holonomic coordinates. With respect to an anholonomic system, the Heaviside-Lorentz equation is a particular case of the VD equations valid for a privileged class of anholonomic frames, those consisting of orthogonal unit vectors.  相似文献   

4.
On the largest scales there is evidence of discrete structure, examples of this are superclusters and voids and also by redshift taking discrete values. In this paper it is proposed that discrete redshift can be explained by using the spherical harmonic integer l; this occurs both in the metric or density perturbations and also in the solution of wave equations in Robertson-Walker spacetime. It is argued that the near conservation of energy implies that l varies regularly for wave equations in Robertson-Walker spacetime, whereas for density perturbations l cannot vary regularly. Once this is assumed then perhaps the observed value of discrete redshift provides the only observational or experimental data that directly requires an explanation using both gravitational and quantum theory. In principle a model using this data could predict the scale factor R (or equivalently the deceleration parameter q). Solutions of the Klein-Gordon equation in Robertson-Walker spacetimes are used to devise models which have redshift taking discrete values, but they predict a microscopic value for R. A model in which the stress of the Klein-Gordon equation induces a metrical perturbation of Robertson-Walker spacetime is devised. Calculations based upon this model predict that the Universe is closed with 2 q0 - 1=10-4. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

5.
The coupled set of equations of hydrodynamics and radiative transfer is derived for small disturbances in a plane, grey atmosphere. Only radiative transfer is taken into account in the energy equation; dynamical effects of radiation are ignored. A mean stationary radiative flux through the photosphere is taken into account. The radiative transfer equation is used by assuming the Eddington approximation, moreover, an exponential height profile of the temperature and an analytical opacity formula are supposed. For this model we obtained an asymptotic solution for plane nonadiabatic acoustic waves and radiation waves. The approach provides a detailed discussion of the interaction of nonadiabatic p‐modes and radiation waves in a realistic model of the photosphere of a solar‐like star.  相似文献   

6.
Radiation interacts with matter via exchange of energy and momentum. When matter is moving with a relativistic velocity or when the background space–time is strongly curved, rigorous relativistic treatment of hydrodynamics and radiative transfer is required. Here, we derive fully general relativistic radiation hydrodynamic equations from a covariant tensor formalism. The equations can be applied to any three-dimensional problems and are rather straightforward to understand compared to the comoving frame-based equations. The current approach is applicable to any space–time or coordinates, but in this work we specifically choose the Schwarzschild space–time to show explicitly how the hydrodynamic and the radiation moment equations are derived. Some important aspects of relativistic radiation hydrodynamics and the difficulty with the radiation moment formalism are discussed as well.  相似文献   

7.
We investigate the spacetime of anisotropic stars admitting conformal motion. The Einstein field equations are solved using different ansatz of the surface tension. In this investigation, we study two cases in details with the anisotropy as: (1) p t =n p r , (2) -p_r=(+c_2)p_{t}-p_{r}=\frac{1}{8\pi}(\frac{c_{1}}{r^{2}}+c_{2}) where, n, c 1 and c 2 are arbitrary constants. The solutions yield expressions of the physical quantities like pressure gradients and the mass.  相似文献   

8.
A most general exact solution to the Einstein-Maxwell equations for static charged perfect fluid is sought in terms of pressure. Subsequently, metrics (e λ and e υ ), matter density and electric intensity are expressible in terms of pressure. Consequently, Pressure is found to be an invertible arbitrary function of ω(=c 1+c 2 r 2), where c 1 and c 2(≠0) are arbitrary constants, and r is the radius of star, i.e. p=p(ω). We present a general solution for charged pressure fluid in terms for ω. We list and discuss some old and new solutions which fall in this category.  相似文献   

9.
The key equation which commonly appears for radiative transfer in a finite stellar atmosphere having ground reflection according to Lambert's law is considered in this paper. The exact solution of this equation is obtained for surface quantities in terms of theX-Y equations of Chandrasekhar by the method of Laplace transform and linear singular operators. This exact method is widely applicable for obtaining the solution for surface quantities in a finite atmosphere.  相似文献   

10.
S. I. Grachev 《Astrophysics》2001,44(4):505-517
A new method is proposed for the numerical solution of nonsteady problems in the theory of radiative transfer. In this method, if the solution at some time t (such as the initial time) is known, then by representing the radiation intensity and all time-dependent quantities (level populations, kinetic temperature, etc.) in the form of Taylor series expansions in the vicinity of t, one can, from the transfer equation and the equations accompanying it (population equations, energy-balance equation, etc.), find all derivatives of that solution at the given time from certain recursive equations. From the Taylor series one can then calculate the solution at some later time t + t, and so forth. The method enables one to analyze nonsteady tradiative transfer both in stationary media and in media with characteristics that vary with time in a given way. This method can also be used to solve nonlinear problems, i.e., those in which the radiation field significantly affects the characteristics of the medium. No iterations are used for this: everything comes down to calculations based on recursive equations. Several problems, both linear and nonlinear, are solved as examples.  相似文献   

11.
The polarization-free (POF) approximation (Trujillo Bueno and Landi Degl'Innocenti, 1996) is capable of accounting for the approximate influence of the magnetic field on the statistical equilibrium, without actually solving the full Stokes vector radiative transfer equation. The method introduces the Zeeman splitting or broadening of the line absorption profile I in the scalar radiative transfer equation, but the coupling between Stokes I and the other Stokes parameters is neglected. The expected influence of the magnetic field is largest for strongly-split strong lines and the effect is greatly enhanced by gradients in the magnetic field strength. Formally the interaction with the other Stokes parameters may not be neglected for strongly-split strong lines, but it turns out that the error in Stokes I obtained through the POF approximation to a large extent cancels the neglect of interaction with the other Stokes parameters, so that the resulting line source functions and line opacities are more accurate than those obtained with the field-free approach. Although its merits have so far only been tested for a two-level atom, we apply the POF approximation to multi-level non-LTE radiative transfer problems on the premise that there is no essential difference between these two cases. Final verification of its validity in multi-level cases still awaits the completion of a non-LTE Stokes vector transfer code.For two realistic multi-level cases (CaII and MgI in the solar atmosphere) it is demonstrated that the POF method leads to small changes, with respect to the field-free method, in the line source functions and emergent Stokes vector profiles (much smaller than for a two-level atom). Real atoms are dominated by strong ultraviolet lines (only weakly split) and continua, and most lines with large magnetic splitting (in the red and the infrared) are at higher excitation energies, i.e. they are relatively weak and unable to produce significant changes in the statistical equilibrium. We find that it is generally unpredictable by how much the POF results will differ from the field-free results, so that it is nearly always necessary to confirm predictions by actual computations.The POF approximation provides more reliable results than the field-free approximation without significantly complicating the radiative transfer problem, i.e. without solving any extra equations and without excessive computational resource requirements, so that it is to be preferred over the field-free approximation.  相似文献   

12.
We have solved the equation of radiative transfer in spherical symmetry with scattering and absorbing medium. We have set the albedo for single scattering to be equal to 0.5. We have set the Planck function constant throughout the medium in one case and in another case the Planck function has been set to vary asr –2. The geometrical extension of the spherical shell has been taken as large as one stellar radius. Two kinds of variations of the optical depth are employed (1) that remains constant with radius and (2) that varies asr –2. In all these cases the internal source vectors and specific intensities change depending upon the type of physics we have employed in each case.  相似文献   

13.
We use moment formalism of relativistic radiation hydrodynamics to obtain equations of motion of radial jets and solve them using polytropic equation of state of the relativistic gas. We consider curved space-time around black holes and obtain jets with moderately relativistic terminal speeds. In addition, the radiation field from the accretion disc, is able to induce internal shocks in the jet close to the horizon. Under combined effect of thermal as well as radiative driving, terminal speeds up to 0.75 (units of light speed) are obtained.  相似文献   

14.
We present a numerical treatment of radiative transfer in three dimensions. The radiation is modeled by the grey moment M 1 system. The introduced scheme is able to compute accurate numerical solutions over a broad class of regimes from the transport to the diffusive limit. We discuss numerical issues concerning the resolution and the parallelization of this scheme for multi-dimensional simulations. Several numerical results are then presented, which show that this approach is robust and have the correct behavior in both the diffusive and free-streaming limits. We also present a comparison in two dimensions of our code with a Monte-Carlo transfer code.  相似文献   

15.
Quantum theory in Robertson – Walker spacetime suggests the existence of a minimal energy ε of the order of 10−45 erg. Reasonable forms for ε give the expansion factor R=R(t)(t= the cosmic time) with no need of gravitational field equations.Einstein's theory should be modified in gravitational fields of strength less than ε c/ħ ∼ 10−8 cm/s2 where c is the speed of light and ħ is the reduced Planck constant. The cosmological term λ is expected to decrease as the universe expands.In the Appendix, ε is derived from a big bang – big crunch Newtonian cosmology.  相似文献   

16.
Earlier models of compressible, rotating, and homogeneous ellipsoids with gas pressure are generalized to include the presence of radiation pressure. Under the assumptions of a linear velocity field of the fluid and a bounded ellipsoidal surface, the dynamical behaviour of these models can be described by ordinary differential equations. These equations are used to study the finite oscillations of massive radiative models with masses 10M and 30M in which the effects of radiation pressure are expected to be important.Models with two different degrees of equilibrium are chosen: an equilibrium (i.e., dynamically stable) model with an initial asymmetric inward velocity, and a nonequilibrium model with a nonequilibrium central temperature and which falls inwards from rest. For each of these two degrees of equilibrium, two initial configurations are considered: rotating spheroidal and nonrotating spherical models.From the numerical integration of the differential equations for these models, we obtain the time evolution of their principal semi-diametersa 1 anda 3, and of their central temperatures, which are graphically displayed by making plots of the trajectories in the (a 1,a 3) phase space, and of botha 1 and the total central pressureP c against time.It is found that in all the equilibrium radiative models (in which radiation pressure is taken into account), the periods of the oscillations of botha 1 andP c are longer than those of the corresponding nonradiative models, while the reverse is true for the nonequilibrium radiative models. The envelopes of thea 1 oscillations of the equilibrium radiative models also have much longer periods; this result also holds for the nonequilibrium models whenever the envelope is well defined. Further, as compared to the nonradiative models, almost all the radiative models collapse to smaller volumes before rebouncing, with the more massive model undergoing a larger collapse and attaining a correspondingly larger peakP c.When the mass is increased, the dynamical behavior of the radiative model generally becomes more nonperiodic. The ratio of the central radiation pressure to the central gas pressure, which is small for low mass models, increases with mass, and at the center of the more massive model, the radiation pressure can be comparable in magnitude to the gas pressure. In all the radiative models, the average periods as well as the average amplitudes of both thea 1 andP c oscillations also increase with mass.When either rotation or radiation pressure effects or both are included in the equilibrium nonradiative model, the period of the envelope of thea 1 oscillations is increased. The presence of rotation in the equilibrium radiative model, however, decreases this period.Some astrophysical implications of this work are briefly discussed.  相似文献   

17.
The Vinti problem, motion about an oblate spheroid, is formulated using the extended phase space method. The new independent variable, similar to the true anomaly, decouples the radius and latitude equations into two perturbed harmonic oscillators whose solutions toO(J 2 4 ) are obtained using Lindstedt's method. From these solutions and the solution to the Hamilton-Jacobi equation suitable angle variables, their canonical conjugates and the new Hamiltonian are obtained. The new Hamiltonian, accurate toO(J 2 4 ) is function of only the momenta.  相似文献   

18.
Using slab model atmospheres that are irradiated from both sides by photospheric, chromospheric, and coronal radiation fields we have determined the ionization and excitation equilibrium for hydrogen.The model atom consists of two bound levels (n = 1 and n = 2) and a continuum. Ly- was assumed to be optically thick with the transition in detailed radiative balance. The Balmer continuum was assumed to be optically thin with the associated radiative ionization dominated by the photospheric radiation field (T rad = 5940 K). The ionization equilibrium was determined from an exact treatment of the radiative transfer problem for the internally generated Ly-c field and the impressed chromospheric and coronal field (characterized by T rad = 6500K).Our calculations corroborate the hypothesis that N2, the n = 2 population density, is uniquely determined by the electron density N e. We also present ionization curves for 6000K, 7500K, and 10000K models ranging in total hydrogen density from 1 × 1010/cm3 to 3 × 1012/cm3. Using these curves it is possible to obtain the total hydrogen density from the n = 2 population density in prominences and spicules.The National Center for Atmospheric Research is sponsored by the National Science Foundation.  相似文献   

19.
We formulate rate equations for the reaction network coupling H, H–, H+, H2, and H2 +. We attempt to systematize the notation, and to write the equations in a form suitable for modern computational methods of handling the coupled rate equations and radiative transfer equations, for both dynamical and static atmospheres. We have accounted for more processes than are generally considered in most current work; some of these may have an impact on the equilibrium of H– (hence its opacity) and on charge conservation (hence the proton density) in the atmospheres of solar-type stars.Operated by the Association of Universities for Research in Astronomy, Inc. under Contract AST 78-17292 with the National Science Foundation.The National Center for Atmospheric Research is sponsored by the National Science Foundation.  相似文献   

20.
In radiative transfer, the intensities of radiation from the bounding faces of a scattering atmosphere of finite optical thickness can be expressed in terms of Chandrasekhar’s X- and Y-functions. The nonlinear nonhomogeneous coupled integral equations which the X- and Y-functions satisfy in the real plane are meromorphically extended to the complex plane to frame linear nonhomogeneous coupled singular integral equations. These singular integral equations are then transformed into nonhomogeneous Riemann–Hilbert problems using Plemelj’s formulae. Solutions of those Riemann–Hilbert problems are obtained using the theory of linear singular integral equations. New forms of linear nonhomogeneous decoupled expressions are derived for X- and Y-functions in the complex plane and real plane. Solutions of these two expressions are obtained in terms of one known N-function and two new unknown functions N 1- and N 2- in the complex plane for both nonconservative and conservative cases. The N 1- and N 2-functions are expressed in terms of the known N-function using the theory of contour integration. The unknown constants are derived from the solutions of Fredholm integral equations of the second kind uniquely using the new linear decoupled constraints. The expressions for the H-function for a semi-infinite atmosphere are obtained as a limiting case.  相似文献   

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