Optimal prediction for moment models: Crescendo diffusion and reordered equations

A direct numerical solution of the radiative transfer equation or any kinetic equation is typically expensive, since the radiative intensity depends on time, space and direction. An expansion in the direction variables yields an equivalent system of infinitely many moments. A fundamental problem is how to truncate the system. Various closures have been presented in the literature. We want to study moment closure generally within the framework of optimal prediction, a strategy to approximate the mean solution of a large system by a smaller system, for radiation moment systems. We apply this strategy to radiative transfer and show that several closures can be re-derived within this framework, e.g. $P_N$, diffusion, and diffusion correction closures. In addition, the formalism gives rise to new parabolic systems, the reordered $P_N$ equations, that are similar to the simplified $P_N$ equations. Furthermore, we propose a modification to existing closures. Although simple and with no extra cost, this newly derived crescendo diffusion yields better approximations in numerical tests.Comment: Revised version: 17 pages, 6 figures, presented at Workshop on Moment Methods in Kinetic Gas Theory, ETH Zurich, 2008 2 figures added, minor correction

A 3D radiative transfer framework IX. Time dependence

Context. Time-dependent, 3D radiation transfer calculations are important for the modeling of a variety of objects, from supernovae and novae to simulations of stellar variability and activity. Furthermore, time-dependent calculations can be used to obtain a 3D radiative equilibrium model structure via relaxation in time. Aims. We extend our 3D radiative transfer framework to include direct time dependence of the radiation field; i.e., the $\partial I/ \partial t$ terms are fully considered in the solution of radiative transfer problems. Methods. We build on the framework that we have described in previous papers in this series and develop a subvoxel method for the $\partial I/\partial t$ terms. Results. We test the implementation by comparing the 3D results to our well tested 1D time dependent radiative transfer code in spherical symmetry. A simple 3D test model is also presented. Conclusions. The 3D time dependent radiative transfer method is now included in our 3D RT framework and in PHOENIX/3D.Comment: A&A in press, 7 pages, 14 figure

Milne-Eddington Solutions for Relativistic Plane-Parallel Flows

Radiative transfer in a relativistic plane-parallel flow, e.g., an accretion disk wind, is examined in the fully special relativistic treatment. Under the assumption of a constant flow speed, for the relativistically moving atmosphere we analytically obtain generalized Milne-Eddington solutions of radiative moment equations; the radiation energy density, the radiative flux, and the radiation pressure. In the static limit these solutions reduce to the traditional Milne-Eddington ones for the plane-parallel static atmosphere, whereas the source function nearly becomes constant as the flow speed increases. Using the analytical solutions, we analytically integrate the relativistic transfer equation to obtain the specific intensity. This specific intensity also reduces to the Milne-Eddinton case in the static limit, while the emergent intensity is strongly enhanced toward the flow direction due to the Doppler and aberration effects as the flow speed increases (relativistic peaking).Comment: 1o pages, 5 figure

Analytical Models of Exoplanetary Atmospheres. II. Radiative Transfer via the Two-stream Approximation

We present a comprehensive analytical study of radiative transfer using the method of moments and include the effects of non-isotropic scattering in the coherent limit. Within this unified formalism, we derive the governing equations and solutions describing two-stream radiative transfer (which approximates the passage of radiation as a pair of outgoing and incoming fluxes), flux-limited diffusion (which describes radiative transfer in the deep interior) and solutions for the temperature-pressure profiles. Generally, the problem is mathematically under-determined unless a set of closures (Eddington coefficients) is specified. We demonstrate that the hemispheric (or hemi-isotropic) closure naturally derives from the radiative transfer equation if energy conservation is obeyed, while the Eddington closure produces spurious enhancements of both reflected light and thermal emission. We concoct recipes for implementing two-stream radiative transfer in stand-alone numerical calculations and general circulation models. We use our two-stream solutions to construct toy models of the runaway greenhouse effect. We present a new solution for temperature-pressure profiles with a non-constant optical opacity and elucidate the effects of non-isotropic scattering in the optical and infrared. We derive generalized expressions for the spherical and Bond albedos and the photon deposition depth. We demonstrate that the value of the optical depth corresponding to the photosphere is not always 2/3 (Milne's solution) and depends on a combination of stellar irradiation, internal heat and the properties of scattering both in optical and infrared. Finally, we derive generalized expressions for the total, net, outgoing and incoming fluxes in the convective regime.Comment: Accepted by ApJS. 23 pages, 11 figures, 3 tables, 158 equations. No change from previous version except for title (to match ApJS convention

Numerical Solution of the Expanding Stellar Atmosphere Problem

In this paper we discuss numerical methods and algorithms for the solution of NLTE stellar atmosphere problems involving expanding atmospheres, e.g., found in novae, supernovae and stellar winds. We show how a scheme of nested iterations can be used to reduce the high dimension of the problem to a number of problems with smaller dimensions. As examples of these sub-problems, we discuss the numerical solution of the radiative transfer equation for relativistically expanding media with spherical symmetry, the solution of the multi-level non-LTE statistical equilibrium problem for extremely large model atoms, and our temperature correction procedure. Although modern iteration schemes are very efficient, parallel algorithms are essential in making large scale calculations feasible, therefore we discuss some parallelization schemes that we have developed.Comment: JCAM, in press. 28 pages, also available at ftp://calvin.physast.uga.edu:/pub/preprints/CompAstro.ps.g

The Classical Stellar Atmosphere Problem

We introduce the classical stellar atmosphere problem and describe in detail its numerical solution. The problem consists of the solution of the radiation transfer equations under the constraints of hydrostatic, radiative and statistical equilibrium (non-LTE). We outline the basic idea of the Accelerated Lambda Iteration (ALI) technique and statistical methods which finally allow the construction of non-LTE model atmospheres considering the influence of millions of metal absorption lines. Some applications of the new models are presented.Comment: accepted for publication in The Journal of Computational and Applied Mathematics, Computational Astrophysics, eds. H. Riffert, K. Werne

3D Radiative Transfer with PHOENIX

Using the methods of general relativity Lindquist derived the radiative transfer equation that is correct to all orders in v/c. Mihalas developed a method of solution for the important case of monotonic velocity fields with spherically symmetry. We have developed the generalized atmosphere code PHOENIX, which in 1-D has used the framework of Mihalas to solve the radiative transfer equation (RTE) in 1-D moving flows. We describe our recent work including 3-D radiation transfer in PHOENIX and particularly including moving flows exactly using a novel affine method. We briefly discuss quantitative spectroscopy in supernovae.Comment: 13 pages, 9 figures, to appear in Recent Directions in Astrophysical Quantitative Spectroscopy and Radiation Hydrodynamics, Ed. I. Hubeny, American Institute of Physics (2009