# Opacity

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### Reference Material

<latex> \documentclass[]{article} \usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry} \usepackage{amsmath} \usepackage{graphicx} \usepackage{natbib} \def\knff{\kappa_{\nu,ff}} \def\jnff{j_{\nu,ff}}

\begin{document} \title{Opacity}

Opacity measures how effectively a microscopic process of absorption or scattering reduces radiation traveling along a line of sight. Absorption is the destruction of photons when they converted into other forms of energy. Scattering is the absorption of photons coming from one direction that are then re-emitted into other directions, reducing the amount of radiation along a given line of sight. Scattering can either by a continuum process, such as Thompson scattering and Rayleigh scattering, or a line process, such as photo-excitation of electrons from a ground state to an excited state that then fall back into that same ground state.

We consider the radiative flux incident on a slab of gas of a given density and how the opacity reduces the outgoing flux along the line of sight. The amount of flux exiting the slab depends on the width of the slab, the density, and the effectiveness of the absorption/scattering which we will call the opacity. Each of these factors reduces the amount of radiation that can flow through the slab and therefore the flux is reduced.

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\begin{aligned} dF &= -\kappa \rho F ds \\ \frac{dF}{ds} &= - \kappa \rho F \end{aligned}

Many absorption and emission processes are frequency dependent and anisotropic, and therefore we transform this flux into a specific intensity. We also can equate the product of opacity and density to the absorption coefficient to obtain a similar form as the radiative transport equation where there is assumed to be no emission.

\begin{aligned} \frac{dI}{ds} &= - \alpha_\nu I \\ \alpha_\nu &= \kappa_\nu \rho \end{aligned}

The opacity is a macroscopic parallel to the microscopic absorption coefficient, where the density of the material is divided out. Like the absorption coefficient, this opacity is frequency dependent and handling line processes require that the frequency dependence of the opacity be considered. However some applications, such as stellar interiors, require the consideration of a frequency averaged opacity.

\title{Rosseland mean opacity}

In plasmas in thermodynamic equilibrium, we can take a flux-weighted mean opacity called the Rosseland mean opacity. It is defined as:

\begin{aligned} \frac{1}\alpha_R \int_{0}^{\infty} \frac{dB_\nu(T)}{dT} d\nu &\equiv \int_{0}^{\infty} \frac{1}{\alpha_\nu} \frac{dB_\nu(T)}{dT} \end{aligned}

Kramers' opacity describes how the opacity of a plasma scales with density and temperature, assuming free-free absorption (inverse bremsstrahlung) is dominant. It is most useful for understanding and modeling stellar atmospheres.

$$\knff={\rho \over T^{3.5}\,3\cdot10^{23}}\,cm^2\, g^{-1}$$

\section{Opacity of Thermal Inverse Bremsstrahlung}

\end{document}