11.5: Definitions and Origins of Mean Opacities

In Chapter 4, we averaged the frequency-dependent opacity over wavelength in order to obtain the Rosseland mean opacity. We claimed that this was indeed the correct opacity to use in the case of STE. That such a mean should exist in the case of STE seemed reasonable since the fundamental parameters governing the structure of the gas should depend on only the temperature. That the appropriate average should be the Rosseland mean is less obvious. In the early days of the study of stellar atmospheres a considerable effort was devoted to reducing the nongray problem of radiative transfer to the gray problem, because the gray atmosphere had been well studied and there were numerous methods for its description. The general idea was that there should exist some "mean" opacity that would reduce the problem to the gray problem or perhaps one that was nearly gray. We know now that such a mean does not exist, but the arguments used in the search are useful to review if for no other reason than many of the mean opacities that were proposed can still be found in the literature and often have some utility in describing the properties of an atmosphere. They are often used for calculating optical depths which label various depth points in the atmosphere. For a particularly good discussion of mean opacities, see Mihalas⁴.

Consider the equation of radiative transfer for a plane-parallel atmosphere and its first two spatial moments. Table 11.1 contains these expressions for both the gray and nongray case. Ideally, we would like to define a mean opacity so that all the moment equations for the nongray case look mathematically like those for the gray case.

a. Flux-Weighted (Chandrasekhar) Mean Opacity

Suppose we choose a mean so that the last of the moment equations takes on the gray form. We can obtain such a mean by \[-\frac{1}{\rho} \frac{d}{d x} \int_0^{\infty} K_\nu d\nu=-\frac{1}{\rho} \frac{d K}{d x}=\frac{1}{4} \int_0^{\infty} \kappa_\nu F_\nu d\nu \equiv \frac{1}{4}\left\langle\kappa_\nu\right\rangle_F F\label{11.4.1}\]

so that the mean is defined by \[\left\langle\kappa_\nu\right\rangle_F=\int_0^{\infty} \frac{\kappa_\nu F_\nu d\nu}{F}\label{11.4.2}\]

The use of such a mean will indeed reduce the nongray equation for the radiation pressure gradient (3) to the gray form. Such a mean opacity is often referred to as a flux-weighted mean, or the Chandrasekhar mean opacity. Unfortunately, such a mean will not reduce either of the other two moment equations from the nongray to the gray case. However, it does yield a simple expression for the radiation pressure gradient which is useful for high-temperature atmospheres where the radiation pressure contributes significantly to the support of the atmosphere, namely, \[\frac{d P_r}{d \tau_F}=\frac{\sigma T_{\mathrm{eff}}^4}{c} \quad d \tau_F=-\left\langle\kappa_\nu\right\rangle_F \rho d x\label{11.4.3}\]

b. Rosseland Mean Opacity

Suppose we require of the third equation in Table 11.1 that \[\int_0^{\infty} F_\nu d\nu=F\label{11.4.4}\]

This quite reasonable requirement means that the nongray moment equation will become \[\int_0^{\infty} \frac{1}{\rho \kappa_\nu} \frac{d K_\nu}{d x} d\nu=-\frac{1}{4} \int_0^{\infty} F_\nu d\nu=-\frac{F}{4} \equiv \frac{1}{\rho\left\langle\kappa_\nu\right\rangle_R} \int_0^{\infty} \frac{d K_\nu}{d x} d\nu\label{11.4.5}\]

implies that the mean opacity has the following form: \[\frac{1}{\left\langle\kappa_\nu\right\rangle}=\frac{\int_0^{\infty}\left(1 / \kappa_\nu\right)\left(d K_\nu / d x\right) d\nu}{\int_0^{\infty}\left(d K_\nu / d x\right) d\nu}\label{11.4.6}\]

Table 11.1 Equation of Radiative Transfer for a Plane-Parallel Atmosphere

As one moves more deeply into the star, the near isotropy of the radiation field and approach to STE will require that \[K_\nu \rightarrow \frac{J_\nu}{3} \rightarrow \frac{B_\nu(t)}{3}\label{11.4.7}\]

Since the Planck function depends on the temperature alone, \[\frac{d B_\nu(T)}{d x}=\frac{\partial B_\nu}{\partial T} \frac{d T}{d x}\label{11.4.8}\]

Substitution of equations \ref{11.4.7} and \ref{11.4.8} into equation \ref{11.4.6} yields \[\frac{1}{\left\langle\kappa_\nu\right\rangle_R} \equiv \frac{\int_0^{\infty}\left(1 / \kappa_\nu\right) \partial B_\nu(T) / \partial T d\nu}{\int_0^{\infty}\left[\partial B_\nu(T) / \partial T\right] d\nu}\label{11.4.9}\]

which is identical to equation \ref{4.1.18} for the appropriate mean for stellar interiors. Thus, under the conditions of STE, we find that the Rosseland mean opacity does indeed remove the frequency dependence from the radiative transfer problem. However, in the stellar atmosphere the conditions required by equation \ref{11.4.7} do not apply, so that the Rosseland mean will not fulfill the same function for the theory of stellar atmospheres as it does in the theory of stellar interiors.

c. Planck Mean Opacity

Finally, let us consider a mean opacity that will yield a correct value for the thermal emission. Thus, \[\left\langle\kappa_\nu\right\rangle_P \equiv \frac{\int_0^{\infty} \kappa_\nu B_\nu(T) d\nu}{\int_0^{\infty} B_{v}(T) d\nu}\label{11.4.10}\]

To appreciate the utility of this mean, we develop the condition of radiative equilibrium for the nongray case and see how that approaches the gray result. Let us see what condition would be placed on a mean opacity in order to bring radiative equilibrium in line with the gray case. Consider a mean opacity such that \[\int_0^{\infty}\left(\kappa_\nu-\bar{\kappa}_\nu\right)\left[J_\nu\left(\tau_\nu\right)-B_\nu\left(\tau_\nu\right)\right] d\nu=0\label{11.4.11}\]

This is just a condition that expresses the difference between the nongray condition for radiative equilibrium and the gray condition, both of which should be zero. Now from equation \ref{10.1.13}, we can write \(\mathrm{J}_\nu\) as \[J_\nu\left(\tau_\nu\right)=\frac{1}{2} \int_0^{\infty} S_\nu(t) E_1\left|t-\tau_\nu\right| d t\label{11.4.12}\]

Under the condition that \(S_\nu(\mathrm{t})=\mathrm{B}_\nu[T(\mathrm{t})]\), we can expand the source function in equation \ref{11.4.12} in a Taylor series about \(\tau_\nu\) and integrate term by term, to obtain \[J_\nu\left(\tau_\nu\right) \approx \frac{1}{2} B_\nu\left(\tau_\nu\right)\left[2-E_2\left(\tau_\nu\right)\right]+\cdots+\label{11.4.13}\]

Thus, near the surface of the atmosphere \[J_\nu\left(\tau_\nu\right)-B_\nu\left(\tau_\nu\right) \approx-\frac{1}{2} B_\nu\left(\tau_\nu\right) \quad \tau_\nu \ll 1\label{11.4.14}\]

Substitution of equation \ref{11.4.14} into equation \ref{11.4.11} yields \[\bar{\kappa}_\nu=\frac{\int_0^{\infty} \kappa_\nu B_\nu(T) d\nu}{\int_0^{\infty} B_\nu(T) d\nu} \equiv\left\langle\kappa_\nu\right\rangle_P\label{11.4.15}\]

Thus, the Planck mean opacity is the most physically relevant mean for satisfying radiative equilibrium near the surface of the atmosphere.

These various means all have their regions of validity and utility, but none can fulfill the promise of reducing the nongray atmosphere problem to that of a gray atmosphere. Indeed, clearly such a mean does not exist. Three separate moment equations are listed in table 11.1 and a mean opacity represents only one parameter available to bring them all into conformity with their gray analogs. For an arbitrary behavior for \(\kappa_\nu\), this is impossible. Thus, we must be content with solving the nongray radiative transfer problem at all frequencies for which a significant amount of flux flows through the atmosphere.