Radiative Transfer Codes

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<latex> \documentclass[11pt]{article} \def\inv#1Template:1 \over \def\ddtTemplate:D \over dt \def\mean#1{\left\langle {#1}\right\rangle} \def\sigot{\sigma_{12}} \def\sigto{\sigma_{21}} \def\eval#1{\big|_{#1}} \def\tr{\nabla} \def\dce{\vec\tr\times\vec E} \def\dcb{\vec\tr\times\vec B} \def\wz{\omega_0} \def\ef{\vec E} \def\ato{{A_{21}}} \def\bto{{B_{21}}} \def\bot{{B_{12}}} \def\bfieldTemplate:\vec B \def\apTemplate:A^\prime \def\xp{{x^{\prime}}} \def\yp{{y^{\prime}}} \def\zp{{z^{\prime}}} \def\tp{{t^{\prime}}} \def\upxTemplate:U x^\prime \def\upyTemplate:U y^\prime \def\e#1{\cdot10^{#1}} \def\hf{\frac12} \def\^{\hat } \def\.{\dot } \usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document} \subsection*{ Importance of Radiative Transfer } Radiation has a very prominent role in astrophysics, both as the only observable and also as a dominant mechanism for energy transfer within and out of astrophysical systems. Consequently, its transport through a medium is one of the most fundamental processes that needs to be considered. Analyzing the radiation received from an object provides us with useful information not only about the radiative source itself but also the medium in between and surrounding the object and the observer. The radiative transfer problem is formally defined by the non-steady state radiative transfer equation (See lecture on radiative transport (Th, Sep. 5) for detailed description).

$$\frac{dI_{\nu}(x,n,t)}{ds} + \frac{dI_{\nu}(x,n,t)}{cdt} = j_{\nu}(n,t) - (\kappa_{\nu,abs}(n,t)+\kappa_{\nu,scat}(n,t))\rho(x,t)I_{\nu}(n,t) +\kappa_{\nu,scat}(n,t)\rho(x,t)\int_{4\pi}\Phi(n,n',x,\nu)I_{\nu}(x,n')d\Omega' $$ where s is the distance along the path defined by a position \textit{\textbf{x}} and propagation direction \textit{\textbf{n}} and $\kappa (s,\lambda)$ is the mass extinction coefficient ($\alpha(s,\lambda) = \kappa(s,\lambda)\rho(s)$). $\Phi(n,n',x,\nu)$ is the scattering phase function which describes the probability that a photon originally propagating in direction \textit{\textbf{n'}} and scattered at position \textbf{x}, will have \textit{\textbf{n}} as its new propagation direction after the scattering event.

The challenging aspect of radiative transfer is that it is a six-dimensional problem (3 spatial + 2 directions ($\theta$,$\phi$) + frequency) with additional time dependence. Additionally, significant complexity is introduced as a consequence of the ability of radiation to affect the state of the medium which is the source of the radiation itself. Thus, to know the state of the matter, we need to know the radiation field. To know the radiation field, we need to know the photon emission, absorption and scattering rates. However, to estimate these rates, we in turn again need to know the state of the matter. Furthermore, the RT problem is nonlocal in space (photons propagate within the entire domain of interest), direction (scattering and absorption/re-emission can change the direction of photons), and wavelength (absorption at a particular wavelength gets re-emitted at another wavelength). As a consequence of these challenges, radiative transfer in astrophysics is frequently calculated using radiative transfer codes which can also account for 3D geometry and non-linear affects due to dust properties (See Lecture on dust grains).

\subsection*{A numerical algorithm for integrating the formal transfer equation} \textbf{Note : To understand what follows, a basic knowledge of numerical methods is useful. If you are completely new to numerical methods, "Numerical Recipes" by Press, Teukolsky, Vetterling and Flannery is a good reference source.}

The primary aspects of numerical radiative transfer can be illustrated using a 1D plane parallel atmosphere. In this model, the gas density, gas temperature etc in the atmosphere is assumed to depend only on height (z). Consequently, we only require one directional variable ($\mu = cos(\theta)$), with the steady state radiative transfer equation (RTE) reducing to the following form (overall 3D problem - spatial + direction + wavelength) -

$\mu\frac{I_{\nu}(z,\mu)}{dz} = j_{\nu}(z) - \alpha_{\nu}(z)I_{\nu}(z,\mu)$

The first step in numerical solution techniques is to discretize the solution vector or the physical properties in the RTE. The quantities requiring discretization are the spatial coordinates, the directional coordinates, the wavelengths, and/or the dust properties. Let us divide up 'z' into cells with indices i $=$ 1,2,3,··· ,Nz where Nz is the chosen number of grid cells. The cell walls, which separate the cells, also have indices, which we will give half-numbers: i $=$ 1/2, 3/2, 5/2,··· ,Nz + 1/2 (See Figure). \end{document} <\latex>