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{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 Compton 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.


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

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}

For plasmas in thermodynamic equilibrium, we can take a flux-weighted mean opacity called the Rosseland mean opacity. Often this opacity is represented with it's absorption coefficient counter-part:

$$ \alpha_R = \kappa_R \rho $$

The Rosseland mean opacity (absorption coefficient) is defined as:

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


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

\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{Kramer's Opacity: Rosseland Mean Opacity}

\section{Kramer's Opacity: 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).