Adventures in Boundary Conditions

Permeating the World in a Particularly Unsubtle Way

Optically Thick and Optically Thin Limits

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Within a stellar medium, internal energy may be transported independently of the flow of the medium, such as through photons generated through electron transitions. A star itself is so optically thick that that a photon is absorbed, initiating a transition of yet another electron from a lower energy state to a higher one. Photon scattering is also possible, resulting in the photon changing direction, possibly changing the frequency (but not necessary). As a result, it becomes necessary to find the limits of optically thick and optically thin material, as this greatly influences the rate at which photon retransmission occur.

1 Interior of a Star

Within a star, the local thermodynamic equilibrium is exactly what it sounds like, it is the thermodynamic equilibrium at which most conditions occur. For photons, the absorption coefficient and density are so high that the mean free path of a photon is approximately \(1-10 nm\), which is dwarfed by the radius of the star (usually \(10^{9} m\)), resulting in the photons being absorbed and re-emitted an enormous amount of times before the photons reach the surface of the star from the interior. We can define the radiation intensity (\(I_v\)) and the mass efficient coefficient (\(j_v\)) as:

\[\begin{gathered} I_{v} \approx B_v(T) \qquad j_v = \kappa_v B_v(T) \end{gathered}\]

where \(\kappa_v\) is the mass absorption coefficient and \(B_v(T)\) is the Planck function. While the Planck equation is inherently isotropic, the existance of a temperature gradient within a star results in an isotropy of the radiation.

With this assumption in mind, the absorption processes must dominate over the scattering process, resulting in the matter becoming tightly coupled to the radiation field. From this, we can rewrite the moment of the Boltzmann equation for photons as:

\[\begin{gathered} H_v = - \frac{1}{\kappa_v \rho} \nabla \cdot \textbf{K}_v = - \frac{1}{3 \kappa_v \rho} \frac{\partial B_v}{\partial T} \nabla T \end{gathered}\]

Following this, we define \(F_v \equiv 4 \pi H_v\) and \(F \equiv \int_{0}^{\infty} F_v d v\) and by integrating over all frequencies we obtain:

\[\begin{gathered} \textbf{F} = - \frac{4 \pi}{3\kappa_R \rho} \nabla T \int_{0}^{\infty} \frac{\partial B_v}{\partial T} dv \end{gathered}\]

with \(K_r\) as the Rosseland mean opacity as defined by:

\[\begin{gathered} \frac{1}{K_R} \equiv \frac{\int_v^{\infty} \frac{1}{K_v} \frac{d B_v}{d T} dv}{\int_0^{\infty} \frac{d B_v}{d T} d v} \end{gathered}\]

As a result, we can express the radiative flux as:

\[\begin{gathered} \textbf{F} = - \frac{c}{3\kappa_R \rho} \nabla (a T^4) \end{gathered}\]

where \(a\) is the radiation density constant as \(4\sigma_b / c\), \(\sigma_b\) as the Stefan-Boltzmann constant. This equation is more commonly know as the "diffusion approximation" equation and most often used for calculations of stellar evolution. For opacity, it becomes clear that if scattering is present, the equation must be corrected, and the Rosseland mean is the function of local density, temperature, and chemical composition, allowing it to be computed independently of the radiation field of a specific problem.

For an extremely diffuse medium, with the observer following the radiation from a central point source, a optically thin medium will result in the photon streaming in the radial direction, resulting in the intensity becoming a delta function, thus allowing it to be computed similar to a black-body radiation pattern.