Time-independent Equation Of Radiative Transfer Research Articles

3D radiative transfer: Continuum and line scattering in non-spherical winds from OB stars

Context. State of the art quantitative spectroscopy utilizes synthetic spectra to extract information from observations. For hot, massive stars, these synthetic spectra are calculated by means of 1D, spherically symmetric, NLTE atmosphere and spectrum-synthesis codes. Certain stellar atmospheres, however, show strong deviations from spherical symmetry, and need to be treated in 3D. Aims. We present and test a newly developed 3D radiative transfer code, tailored to the solution of the radiation field in rapidly expanding stellar atmospheres. We apply our code to the continuum transfer in wind-ablation models, and to the UV resonance line formation in magnetic winds. Methods. We have used a 3D finite-volume method for the solution of the time-independent equation of radiative transfer, to study continuum- and line-scattering problems, currently approximated by a two-level-atom. Convergence has been accelerated by coupling the formal solver to a non-local approximate Λ-iteration scheme. Particular emphasis has been put on careful tests, by comparing with alternative solutions for 1D, spherically symmetric model atmospheres. These tests allowed us to understand certain shortcomings of the methods, and to estimate limiting cases that can actually be calculated. Results. The typical errors of the converged source functions, when compared to 1D solutions, are of the order of 10–20%, and rapidly increase for optically thick (τ ≳ 10) continua, mostly due to the order of accuracy of our solution scheme. In circumstellar discs, the radiation temperatures in the (optically thin) transition region from wind to disc are quite similar to corresponding values in the wind. For MHD simulations of dynamical magnetospheres, the line profiles, calculated with our new 3D code, agree well with previous solutions using a 3D-SEI method. When compared with profiles resulting from the so-called analytic dynamical magnetosphere (ADM) model, however, significant differences become apparent. Conclusions. Due to similar radiation temperatures in the wind and the transition region to the disc, the same line-strength distribution can be applied within radiation hydrodynamic calculations for optically thick circumstellar discs in “accreting high-mass stars”. To properly describe the UV line formation in dynamical magnetospheres, the ADM model needs to be further developed, at least in a large part of the outer wind.

Simultaneous solution of Kompaneets equation and radiative transfer equation in the photon energy range 1–125 keV

Radiative transfer equation in plane parallel geometry and Kompaneets equation is solved simultaneously to obtain theoretical spectrum of 1–125 keV photon energy range. Diffuse radiation field are calculated using time-independent radiative transfer equation in plane parallel geometry, which is developed using discrete space theory (DST) of radiative transfer in a homogeneous medium for different optical depths. We assumed free-free emission and absorption and emission due to electron gas to be operating in the medium. The three terms n, n 2 and ( ∂ n / ∂ x k ) where n is photon phase density and x k = ( h ν / kT e ) , in Kompaneets equation and those due to free-free emission are utilized to calculate the change in the photon phase density in a hot electron gas. Two types of incident radiation are considered: (1) isotropic radiation with the modified black body radiation I MB [1] and (2) anisotropic radiation which is angle dependent. The emergent radiation at τ = 0 and reflected radiation τ = τ max are calculated by using the diffuse radiation from the medium. The emergent and reflected radiation contain the free-free emission and emission from the hot electron gas. Kompaneets equation gives the changes in photon phase densities in different types of media. Although the initial spectrum is angle dependent, the Kompaneets equation gives a spectrum which is angle independent after several Compton scattering times.

Three-dimensional optical tomography with the equation of radiative transfer

We report on the derivation and implementation of the first three-dimensional optical tomographic image reconstruction scheme that is based on the time-independent equation of radiative transfer (ERT) and allows for arbitrarily shaped medium boundaries and arbitrary spatial material distributions. The scheme builds on the concept of model-based iterative image reconstruction, in which a forward model provides prediction of detector readings, and a gradient-based updating scheme minimizes an appropriately de- fined objective function. The forward model is solved by using an even-parity formulation of the ERT, which lends itself to a finite- element discretization method. The finite-element technique pro- vides the suitable framework for predicting light propagation in arbi- trarily shaped three-dimensional media. For an efficient way of calculating the gradient of the objective function we have imple- mented an adjoint differentiation scheme. Initial reconstruction re- sults using synthetic data from simple media and a three- dimensional mesh of the human forehead illustrate the performance of the code. © 2003 SPIE and IS&T. (DOI: 10.1117/1.1587730)

Optical tomography using the time-independent equation of radiative transfer — Part 1: forward model

The presented study consists of two parts. The overall goal is to introduce and experimentally test a novel optical tomographic imaging algorithm that is based on the equation of radiative transfer. Using the equation of radiative transfer rather than the diffusion equation permits the consideration of highly scattering media that contain void-like regions that have very low absorption and scattering coefficients. In part I we concentrate on the detailed description and evaluation of a numerical forward model that accurately describes photon propagation in such media. In part II we focus on the inclusion of this forward model into a model-based iterative image reconstruction (MOBIIR) scheme. Using the MOBIIR scheme one can determine the spatial distribution of optical properties inside highly scattering media from measurements acquired on the surface of the medium. The mathematical and numerical background for the reconstruction algorithm, especially the adjoint differentiation scheme for the gradient calculation, will be presented. The code is tested with experimental data from tissue-phantoms that contain water-filled, void-like regions. The forward model to be described in part I is based on an upwind-difference discrete-ordinate formulation of the time-independent equation of radiative transfer. The upwind-difference representation has the advantage that it provides a convenient mathematical framework for calculating the derivative of the fluence with respect to the optical parameters using an adjoint differentiation technique, to be described in part II. The performance of the forward model is tested with experimental data obtained from homogeneous tissue-phantoms and from phantoms that contain void-like regions. We find good agreement between experimental measurements and theoretical predictions of the measurements.

Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues

We analyse the limits of the diffusion approximation to the time-independent equation of radiative transfer for homogeneous and heterogeneous biological media. Analytical calculations and finite-difference simulations based on diffusion theory are compared with discrete-ordinate, finite-difference transport calculations. The influence of the ratio of absorption and transport scattering coefficient on the accuracy of the diffusion approximation are quantified and different definitions for the diffusion coefficient, D, are discussed. We also address effects caused by void-like heterogeneities in which absorption and scattering are very small compared with the surrounding medium. Based on results for simple homogeneous and heterogeneous systems, we analyse diffusion and transport calculation of light propagation in the human brain. For these simulations we convert density maps obtained from magnetic resonance imaging (MRI) to optical-parameter maps ( and ) of the brain. We show that diffusion theory fails to describe accurately light propagation in highly absorbing regions, such as haematoma, and void-like spaces, such as the ventricles and the subarachnoid space.

Iterative method for treating multiple scattering in fogs

The scalar time-independent equation of radiative transfer is solved in an iterative manner for the forward propagating radiance (specific intensity) distribution function. In the small-angle appro...