# Kramer's Opacity

### 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}}

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\title{Kramers' Opacity}

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 due to 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. Setting the energy of all photons equal to the kinetic energy scale removes the frequency dependence from the opcaity and instead transforms it into a flux-weighted mean opacity. $$\kappa_{ff}\propto{n_en_p\over\rho}{T^{-3}\over\sqrt{T}}$$ 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: $$\kappa_{ff}={\rho \over T^{3.5}\,3\cdot10^{23}}\,cm^2\, g^{-1}$$ 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).

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