Opacity

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\begin{document} \section*{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.

Flux1.jpg


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 Transfer Equation where there is assumed to be no emission.

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

In general, opacity has a power law dependence on both the density of the gas as well as its temperature.

$$ \kappa \simeq \rho^a T^b $$

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.

\section*{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:

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

or

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

\subsection*{Deriving Kramer's Opacity}

An example of a Rosseland mean opacity is Kramer's Opacity. For this opacity we assume that we are dominated by free-free absorption, which occurs when the temperature in the plasma is hot enough to ionize most electrons while still having a temperature low enough for those electrons to be pulled in by the proton's electron potential well. We begin by solving the numerator of the above fraction using the Planck function and the absorption coefficient for free-free absorption (see Thermal Bremsstrahlung).

$$ \begin{aligned} \int_{0}^{\infty} \frac{1}{\alpha_\nu} \frac{dB_\nu(T)}{dT} d\nu &= \int_{0}^{\infty} \frac{1}{\alpha_\nu} \frac{2 h^2 \nu^4}{c^2 k T^2} \frac{\exp(h\nu/kT)}{(\exp(h\nu/kT)-1)^2} d\nu \\ &\propto \int_{0}^{\infty} \frac{1}{\rho^2 \nu^{-3} T^{-1/2} (1-\exp(-h\nu/kT))} \frac{2 h^2 \nu^4}{c^2 k T^2} \frac{\exp(h\nu/kT)}{(\exp(h\nu/kT)-1)^2} d\nu \\ &\propto \rho^{-2} T^{-3/2} \int_{0}^{\infty} \nu^7 \frac{\exp(2h\nu/kT)}{(\exp(h\nu/kT) - 1)^3} d\nu \\ \end{aligned} $$

We set $h\nu/kT = x$ to get the form:

$$ \begin{aligned} &\propto \rho^{-2} T^{-3/2} \int_{0}^{\infty} T^8 x^7 \frac{\exp(2x)}{(\exp(x-1)^3} dx \\ \end{aligned} $$

Next we move on to the denominator, using the Stefan-Boltzmann law to express the integrated power emission across all frequencies of the blackbody.

$$ \begin{aligned} \int_{0}^{\infty} \frac{dB_\nu(T)}{dT} d\nu &= \frac{d}{dT} \int_{0}^{\infty} B_\nu d\nu \\ &= \frac{d(\sigma T^4 / \pi}{dT} \\ &= 4 \sigma T^3 / \pi \\ \end{aligned} $$

We place the above numerator and denominator together.

$$ \begin{aligned} \frac{1}\alpha_{R,ff} &\propto \rho^{-2} T^{7/2} \int_{0}^{\infty} x^7 \frac{\exp(2x)}{(\exp(x-1)^3} dx \\ \end{aligned} $$

This integral is the Str{\"o}mgren Integral and evaluates to a constant value of $\approx 5020$. This evaluation, along with carrying along the constants, allows us to find a final form for the absorption coefficient and consequently Kramer's Opacity.

$$ \begin{aligned} \kappa_{R,ff} &= \frac{\alpha_{R,ff}}{\rho} &\simeq 10^{-23} \rho T^{-7/2} cm^2 g^{-1} \\ \end{aligned} $$

\section*{Opacity Table}

The Rosseland opacity can be calculated for a plasma where the collision timescale is small enough when compared to the dynamical timescale for the plasma to come into thermodynamic equilibrium. The Rosseland opacity in such a plasma will take on different scaling relationships with temperature and density depending on the electronic state of the atoms. Pictured here is a radiative opacity table for a hydrogen gas (with solar metallicity) calculated by the OPAL code developed at the Lawrence Livermore National Laboratory.

The opacity dependence on temperature can be roughly broken into three temperature ranges.

\subsection*{H$^-$ opacity}

At low temperatures $(\log T (K) \lesssim 4)$, the gas is partially but not fully ionized. Therefore free electrons can fall onto a neutral hydrogen atom forming a H$^-$ ion. The lower the temperature the less free electrons are present in the gas available to absorb photons attempting to pass through the gas. Therefore H$^-$ opacities decrease at lower temperatures. The general power law form for the H$^-$ opacity has $a = 0.5$ and $b = 9$:

$$ \kappa \propto \rho^{0.5} T^{9}$$

Notice the steep temperature dependence in this section of the opacity table.

\subsection*{Free-free and bound-free opacity}

At higher temperatures $(4 \lesssim \log T (K) \lesssim 8)$, the gas is partially ionized with more energized electrons. Inverse Thermal Bremsstrahlung (free-free) and Radiative Recombination (bound-free) are the dominate contributions to the opacity of the plasma. Both of these processes can be represented with Kramer's Opacity, which has the general power law form for the opacity with $a = 1$ and $b = -3.5$:

$$ \kappa \propto \rho T^{-3.5}$$

The increasing temperature of the plasma gives the free electrons enough energy to break out of the electric potential energy wells of the protons. Photons are less affected by the fast streaming electrons and as a result there is a decrease in the opacity.

\subsection*{Thomson (electron) scattering}

At still higher temperatures $(\log T (K) \gtrsim 8)$, free electrons in the plasma will absorb and subsequently reemit incident photons via Thomson Scattering. This is a coherent scattering process (or 'grey scattering') resulting in no shift to the incident photon's energy. However the incident photon can be reemitted in a different direction, resulting in a decrease in the amount of radiation along a given line of sight. Thompson scattering is also independent of both temperature and density ($a = 0$ and $b = 0$). All values of density in the OPAL opacity table approach a constant value of $\kappa_{Th} = 0.2 (1 + X)$ where $X$ is the hydrogen mass fraction of the plasma, at sufficiently high temperatures.

Thomson Scattering is the low energy approximation of Compton Scattering, during which photons decrease in energy as they scatter off of electrons.

\end{document}