Kramer's Opacity

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<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{Kramers' Opacityy}

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}

Recall that we defined the coefficient for thermal free-free absorption (inverse bremsstrahlung) as: \def\anff{\alpha_{\nu,ff}} $$\anff\equiv{\jnff\over B_{\nu}}$$ We can also define the opacity: $$\knff={\anff\over\rho}\propto{n_en_p\nu^{-3}(e^{h\nu\over kT}-1)\over \rho\sqrt{T}e^{h\nu\over kT}}$$ where $\,\rho$ is the total density. For a rough scaling, we can cancel the exponential terms and since most photons have an energy of order $h\nu\,\sim\, kT$, the frequency is proportional to the temperature: $$\knff\propto{n_en_p\over\rho}{T^{-3}\over\sqrt{T}}$$ Additionally, typically the electron \# density is proportional to the total density, $n_e\propto\rho$, and the proton \# density is proportional to the total density times $Z$, $n_p\propto Z\cdot\rho$, so: $$\boxed{\knff\propto\rho T^{-3.5}}$$ Including the constant of proportionality, we get: $$\knff={\rho \over T^{3.5}\,3\cdot10^{23}}\,cm^2\, g^{-1}$$ This is known as Kramer's opacity and describes the scaling of the opacity of a plasma with density and temperature assuming absorption due to Inverse Bremsstrahlung. It is important to remember the assumptions we made to get here: \begin{itemize} \item A Maxwellian velocity distribution, which should be valid since the collision time scale is very small, so the system should relax to a Maxwellian distribution quickly. \item A non-relativistic regime, so $T\le{m_ec^2\over k}\sim7\cdot 10^9K$, which is an okay assumption for most plasmas. \end{itemize} Some examples of Thermal Bremsstrahlung are HII regions, and the diffuse IGM (which contains nuclei and H).