Milne Problem Research Articles

General theory of radiative transfer in a magnetized atmosphere with scattering by electrons

ABSTRACT Here, we consider the general radiative transfer theory in a magnetized atmosphere for any value of parameter $x=\omega _B/\omega \simeq 0.933\times 10^{-8}\lambda (\mu \mathrm{m}) B(\text{G})$, where ωB is the cyclotron frequency of electron rotation and ω is the angular frequency of considered monochromatic radiation. The main term of the radiative transfer equations $\textbf {J}_{\alpha \beta }$ for the Stokes parameters I, V, U, and Q describes the scattering of radiation coming from all directions and distances. All Stokes parameters of the incident radiation mutually transform into each other along their path due to interference and different cross-sections for them. To find this transformation of the Stokes parameters one has to solve the complex system of transfer equations without the sources and term $\textbf {J}_{\alpha \beta }$. This is done in our paper. First, we present the general solution and then give the solution for the case of a homogeneous magnetic field, where the formulas have clear algebraic form. We note that for small parameter x our formulas describe the known Faraday rotation. Our formulas allow us to derive an integral equation for the density of polarized radiation, multiple scattered in a magnetized atmosphere for any values of the parameter x. The obtained correct radiation transfer equation allows us to calculate the Stokes parameters of radiation emerging from an atmosphere, in particular, for the Milne problem.

Stability of the Nonlinear Milne Problem for Radiative Heat Transfer System

Stability of the Nonlinear Milne Problem for Radiative Heat Transfer System

Study of Forward and Backward Scattering in Two-Region Milne Problem for Non-absorbing Medium Using a Synthetic Kernel

In this work, the effect of an extremely anisotropic scattering function on the two-region Milne problem is studied. This scattering function divides the neutrons resulting from the collisions into three parts; part ( which moves backward, part (m) which moves forward, and part (n) which emerge isotropically from the collisions, where + m + n = 1. The integral version of the transport equation is solved using trial functions based on Case’s eigenmodes and exponential integral function. Hence, the solution to the Milne problem is formulated in terms of characteristic quantities such as the extrapolation length and the fractional scalar flux discontinuity. Numerical results for the analytically evaluated quantities are presented. Some of our numerical results are compared with the available published results.

Machine Learning Applications to the One-speed Neutron Transport Problems

Machine learning is a branch of artificial intelligence and computer science. The purpose of machine learning is to predict new data by using the existing data. In this study, two different machine learning methods which are Polynomial Regression (PR) and Artificial Neural Network (ANN) are applied to the neutron transport problems which are albedo problem, the Milne problem, and the criticality problem. ANN applications contain two different activation functions, Leaky Relu and Elu. The training data set is calculated by using the HN method. PR and ANN results are compared with the literature data. The study is only based on the existing data; therefore, the study could be thought only data mining on the one-speed neutron transport problems for isotropic scattering.

The Milne Problem with the Anlı-Güngör Scattering

Recently developed the Anlı-Güngör scattering function is applied to the Milne problem in this study. It is the function of Legendre polynomials with t parameter which can be called as scattering parameter. The extrapolation distance values are calculated for varying t parameters and varying secondary neutron numbers with the HN method. The numerical results are calculated by 40th approximation in order to get high precision results. The calculated extrapolation distance results are very convergence. Since there is no a similar study for the Milne problem with the Anlı-Güngör scattering in the literature, the interpolation technique is applied to the HN method results. Thus, the isotropic scattering results could be determined by t → 0 limit.

The Milne Problem for Linear-Triplet Anisotropic Scattering with HN Method

The Milne problem is investigated for the linear-triplet anisotropic scattering with HN method in this study. The scattering function is the linear combination of linear anisotropic scattering and triplet anisotropic scattering in Mika’s scattering. The positivity condition is needed to find physical results since the scattering function defines the scattering probabilities. It defines the relationship between the scattering coefficients. HN method is based on using the Case method. Therefore, properties of the Case method should be derived for the scattering function. The linear anisotropic scattering is the dominant scattering according to the calculated results.

Construction and optimization of numerically-statistical projection algorithms for solving integral equations

Abstract The problem of minimizing the root-mean-square error of the numerical-statistical projection estimation of the solution to an integral equation is solved. It is shown that the optimal estimator in this sense can be obtained by equalizing deterministic and stochastic components of the error in the case when the norm of the remainder of the utilized decomposition decreases inversely proportional to its length. As a test, the Milne problem of radiation transfer in a semi-infinite layer of matter is solved using Laguerre polynomials. To solve such a problem in the case of a finite layer, a special regularized projection algorithm is used.

The Milne problem with chaotic magnetic field

ABSTRACT We consider the semi-infinite plane-parallel electron atmosphere with chaotic magnetic field B′ ≤ 1010 G, when the parameter $x^{\prime 2}=(\omega _{B^{\prime }}/\omega)^2 \simeq 0.87 \times 10^{-16}\lambda ^2(\mu \mathrm{ m}) B^{\prime 2}(G)$ can be both ≪1 and ≫1. Regular magnetic field $\boldsymbol{B}_0$ is absent. All magnetic fluctuations are assumed as Gaussian type and isotropic. The isotropic magnetic fluctuations $\boldsymbol{B}^{\prime }$ give rise to the same extinction for all Stokes parameters and the additional extinction factor h for parameters Q and U, which arises due to chaotic Faraday rotations. We consider the Milne problem, which is described by the system of transfer equations for intensity I and parameter Q. It is shown that the phase matrix depends on parameter G(B′), which is equal to 3 for x′2 ≪ 1 and to 27 for x′2 ≫ 1. The calculations demonstrate that the polarization of outgoing radiation strongly depends on parameter h and degree of absorption ε.

Boundary layer of the Boltzmann equation in 2-dimensional convex domains

Consider the stationary Boltzmann equation in 2D convex domains with diffusive boundary condition. In this paper, we establish the hydrodynamic limits while the boundary layers are present, and derive the steady Navier-Stokes-Fourier system with non-slip boundary conditions. Our contribution focuses on novel weighted $W^{1,\infty}$ estimates for the Milne problem with geometric correction. Also, we develop stronger remainder estimates based on an $L^{2m}-L^{\infty}$ framework.

The radiative transfer model for the greenhouse effect

Radiative transfer is at the heart of the mechanism to explain the greenhouse effect based on the partial infrared opacity of carbon dioxide, methane and other greenhouse gases in the atmosphere. In absence of thermal diffusion, the mathematical model consists of a first order integro-differential equation coupled with an integral equation for the light intensity and the temperature, in the atmosphere. We revisit this much studied system from a mathematical and numerical point of view. Existence and uniqueness and implicit solutions of the Milne problem for grey atmospheres are stated. Numerical simulations are given for grey and non-grey atmospheres and applied to calculate the effect of greenhouse gases. In the context of a transparent atmosphere for sunlight, it is found that by doubling the absorption coefficient in the infrared absorption range of texttt {CO}_2 the temperature decreases by 2%. On the other hand, the same changes but in the low infrared range of the sunlight leads to an increase of temperature in the atmosphere. Several computer codes were written to cross-validate the results. The authors conclude that the radiative transfer model without thermal diffusion for an atmosphere transparent to the incident sunlight is not capable of explaining the greenhouse effect due to the greenhouse gases. A decreasing temperature due to an increasing proportion of texttt {CO}_2 has been observed in the high atmosphere (D.W.J. Thomson et al, nature11579). In the lower atmosphere thermal diffusion and convection cannot be neglected and since the absorption coefficient are highly dependent on the temperature, a full ocean–atmosphere–biosphere climate model is required. Hence, driving conclusions from this study on climate change should be cautiously avoided and a review of the hypothesis of the radiative transfer argument commonly found in textbooks should be revisited.

Monte Carlo and Milne-like solutions for temporal correlation function in a wide temporal range

The Milne problem for the temporal correlation function of radiation backscattered from a half space is solved analytically. The angular and temporal dependencies calculated with the Monte Carlo simulations coincide with high precision with the Milne-like solution for a near isotropic scattering; for a highly anisotropic scattering the Monte Carlo results turn to be in a fair agreement with the known measurement, while the Milne-like solution greatly disagree with the experiment. Effects of finite thickness and non-zero absorption are also considered.

Chaotically magnetized atmosphere with dust

ABSTRACT We consider the influence of absorbing dust particles on the intensity and polarization of radiation in chaotically magnetized atmospheres. The Milne problem is considered. It is known that the existence of absorbing dust particles leads to more elongated intensity distribution along the normal to the atmosphere and increasing of polarization degree near the direction perpendicular to the normal. The chaotic Faraday rotations practically do not change the intensity distribution, but they decrease the polarization degree. How looks the picture when the atmosphere is chaotically magnetized and contains the absorbing dust particles? Usually one assumes that such atmospheres exist in active galactic nuclei in Seyfert galaxies, both in accretion discs and gas–dusty toruses. In particular, we establish when both mechanisms cancel one another and the resulting polarization is near to ordinary atmosphere without the magnetic field fluctuations and absorbing particles. Of course, the angular distribution mostly corresponds to the level of existing dust particles, because it is practically independent of polarization.

Milne problem for the linear and linearized Boltzmann equations relevant to a binary gas mixture

A stationary boundary-value problem for the Boltzmann equation in a half space is considered for a binary mixture of gases when the indata on the boundary are given for the both species. Under the assumption that one of the species is dominant and close to equilibrium but the density of the other is small, the problem is decomposed into two half-space problems: the so-called Milne problem for the linearized Boltzmann equation with a source term for the dominant species and that for the linear Boltzmann equation for the low-density species. The two problems are coupled in such a way that the source term in the former is specified by the solution of the latter. The existence and uniqueness of the solutions to the two problems are proved, and their accurate asymptotic behavior in the far field is obtained. In particular, the precise rate of approach of the solution to the state at infinity is expressed in terms of the decay rate in the molecular velocity of the boundary data for both species.

Boundary Layer of Transport Equation with In-Flow Boundary

Consider the steady neutron transport equation in two dimensional convex domains with an in-flow boundary condition. We establish the diffusive limit while the boundary layers are present. Our contribution relies on a delicate decomposition of boundary data to separate the regular and singular boundary layers, novel weighted $$W^{1,\infty }$$ estimates for the Milne problem with geometric correction in convex domains, and an $$L^{2m}-L^{\infty }$$ framework which yields stronger remainder estimates.

Resonance line in rotating accretion disc

We study the resonance line emission from the rotating plane optically thick accretion disc, consisting of free electrons and resonant atoms. We use the standard assumption that the source of continuum radiation is located near central plane of the accretion disc, where the temperature is the highest. This corresponds to the Milne problem consideration for continuum. We shortly discuss the impossibility of the Milne problem for the resonance radiation. We assume that the resonant atoms are located in a thin layer of an accretion disc near the surface. In this case the resonance line emission arises due to scattering of a continuum on the resonant atoms. In thin layer we can neglect the multiple scattering of the resonance radiation on the resonant atoms. We consider the axially symmetric problems, where the Stokes parameter U =0. We take into account the Doppler effect for the frequencies of the resonance line. The three types of the resonant atom sources are considered (see Figs.1-3). The first source is the axially symmetric continuous distribution of the resonant atoms along the circular orbit. The second spot-like source rotates in the orbit. The third type presents two spot-like sources located in the orbit contrary one to another. In the first and third cases the shape of the emitting resonance line is symmetric, i.e. the right and left wings have the similar shapes. In the second case the resonance line has asymmetric shape. The shape of the emerging line depends significantly on the ratio of the rotation velocity value to the velocity, characterizing the Doppler width. It also depends on the ratio of the electron number density to the number density of resonant atoms. The results of the calculations characterize the different observational effects of H$\alpha$ radiation in the accretion discs and can be used for estimations of the parameters mentioned above.

Radiation intensity and polarization from accretion disks with progressive increasing height

AbstractThis article considers the optically thick accretion disks with the progressive increasing height. The surface is assumed to be the conical form. The radiation with considered wavelength emerges from a ring on the cone and is described by the Milne problem for the intensity and polarization. The inclination angles of the rings are taken 15 and 30 grad. The inclination angles between the line of sight and the normal to the central accretion disk plane are taken 30, 45, and 60 grad for every value of the ring inclination. For the continuum radiation, the polarization of the emerging light is less than that in the case of the plane accretion disk. The polarization position angles of radiation emerging from the right and left parts of the ring are opposite to one another. They are determined by the geometry of the problem. The position angle of the observed continuum radiation is parallel to the central plane of the accretion disk. Our theory gives the new explanation of that the position angles in “red” and “blue” wings of a spectral line are opposite to one another. This behavior exists in many Seyfert galaxies.

A complete probabilistic solution for a stochastic Milne problem of radiative transfer using KLE-RVT technique

A novel technique, named (KLE-RVT), is constructed to find a full probabilistic solution of a finite Milne problem of radiative transfer in spatially stochastic atmosphere. This technique is a combination between the random variable transformation (RVT) technique and the Karhunen–Loève expansion (KLE) of the input stochastic process (the total cross section of the medium) to find the probability density function of the solution stochastic process. The RVT technique is applicable only if the probability density function of the input random variable (process) is known in a closed form. To overcome this obstacle, KLE is applied to represent the spatially continuous random cross section, defined only by its mean and covariance function, in terms of a finite number of uncorrelated random variables with known probability density functions. By this technique, the probability density function of the solution process is evaluated dealing with the input process itself instead the integral transformation of it. This solution is general and valid for any input second order stochastic process. Numerical results of our findings are presented to realize this work.

Modified FN solution of the neutron transport equation for the Milne problem with FBIS kernel

The solution of the Milne problem is studied by one-speed neutron transport equation in plane geometry with Inonu’s scattering kernel, which is known as a linear combination of the forward, backward and isotropic scattering kernel (FBIS kernel). The solution of the neutron transport equation with Inonu’s scattering kernel can be written in terms of the solution of the neutron transport equation for isotropic scattering case. The extrapolation distance is calculated with modified FN (or $$H_{\mathrm{N}}$$ ) method. The numerical values of the extrapolation distance are obtained depending on the secondary neutron numbers and anisotropy coefficients and compared with the available data in the literature values.

Radiation intensity and polarization in an atmosphere with a chaotic magnetic field

We consider the influence of Faraday rotation in chaotic magnetic field on the intensity and linear polarization of multiple scattered radiation. All fluctuations are assumed as Gaussian type and isotropic. The values of magnetic field are less than $10^5$Gauss. In this case the parameter $ \omega_B/\omega=(eB/m_ec)/\omega\simeq 0.93\cdot 10^{-8}\lambda$($\mu$m)$B($G$)$ is small and the radiation scatterings are the usual scatterings on free electrons. Only the Faraday rotation influences on the intensity and polarization of radiation. We consider the Milne problem in the electron magnetized atmosphere without the mean magnetic field. We found that chaotic magnetic field does not increase the number of H-functions, describing the Milne problem. There are four H-functions as in the Milne problem for non-magnetized case. The calculations demonstrate that the polarization of outgoing radiation diminishes strongly with the increasing of the level of magnetic fluctuations. The calculations can be used for estimations of inclination angle $i$ of accretion discs and the level of magnetic fluctuations.

The Discontinuous Asymptotic Telegrapher’s Equation (P1) Approximation

Modeling the propagation of radiative heat waves in optically thick material using a diffusive approximation is a well-known problem. In optically thin material, classic methods, such as classic diffusion or classic , yield the wrong heat wave propagation behavior, and higher-order approximation might be required, making the solution more difficult to obtain. The asymptotic approximation [Heizler, Nucl. Sci. Eng., Vol. 166, p. 17 (2010)] yields the correct particle velocity but fails to model the correct behavior in highly anisotropic media, such as problems that involve a sharp boundary between media or strong sources. However, the solution for the two-region Milne problem of two adjacent half-spaces divided by a sharp boundary yields a discontinuity in the asymptotic solutions that makes it possible to solve steady-state problems, especially in neutronics. In this work we expand the time-dependent asymptotic approximation to a highly anisotropic medium using the discontinuity jump conditions of the energy density, yielding a modified discontinuous equation in general geometry. We introduce numerical solutions for two fundamental benchmarks in plane symmetry. The results thus obtained are more accurate than those attained by other methods, such as Flux Limiters or Variable Eddington Factors.