9.5: Moments of the Equation of Radiative Transfer

In Chapter 1 we saw that much useful information could be obtained about the gas by taking moments of the Boltzmann transport equation. The process always generated moments of phase density that were of one order higher than that used to generate the equation itself. Thus, to be useful, a relation between the higher-order moment and one of lower order had to be found. If this could be done, a self-consistent set of moment equations could be found and solved, yielding the values of those moments throughout the configuration. A similar set of circumstances will exist for the equation of radiative transfer.

To maintain a high level of generality, let us consider the general equation of radiative transfer given by equation \ref{9.2.11} but with the "creation rate" replaced by the source function and the potential gradient taken to be zero. Thus \[\frac{1}{c} \frac{\partial I_\nu}{\partial t}+\hat{n} \cdot \nabla I_\nu=-\rho\left(\kappa_\nu+\sigma_\nu\right)\left(I_\nu-S_\nu\right)\label{9.4.1}\]

Furthermore, assume that the scattering is isotropic and coherent so that the source function in equation \ref{9.2.27} becomes \[S_\nu=\frac{\kappa_\nu B_\nu}{\kappa_\nu+\sigma_\nu}+\frac{\sigma_\nu J_\nu}{\kappa_\nu+\sigma_\nu}\label{9.4.2}\]

Now we integrate equation \ref{9.4.1} over all solid angles, using the form of the source function given by equation \ref{9.4.2}, and get \[\frac{1}{c} \frac{\partial J_\nu}{\partial t}+\frac{1}{4} \nabla \cdot \vec{F}_\nu=\kappa_\nu \rho\left(B_\nu-J_\nu\right)\label{9.4.3}\]

This is the equation of radiative equilibrium and describes how the radiative flux flows through the atmosphere. Note that the effects of scattering have disappeared from this equation. This is an expression of the conservative nature of scattering. Since no energy is gained or lost in each individual scattering event, the average can contribute nothing to the energy balance for the radiative flux and so all scattering terms must vanish.

a. Radiative Equilibrium and Zeroth Moment of the Equation of Radiative Transfer

Consider the right-hand side of equation \ref{9.4.3}. This is essentially the right-hand side of the Boltzmann transport equation, which denotes the creation and destruction of particles in phase space, suitably averaged over direction. Thus, if there is no net production of photons in the atmosphere, this term, integrated over frequency, must be zero. Therefore, integrating equation \ref{9.4.3} over all frequencies, we get \[\frac{1}{c} \frac{\partial}{\partial t} \int_0^{\infty} J_\nu d\nu+\frac{1}{4} \nabla \cdot \int_0^{\infty} \vec{F}_\nu d\nu=0=\int_0^{\infty} \kappa_\nu \rho\left(B_\nu-J_\nu\right) d\nu\label{9.4.4}\]

This is a very general statement of radiative equilibrium, and either side of this equation is an equivalent statement of it. If we let \(F=\int_0^{\infty} F_\nu d\nu\), then for a static plane-parallel atmosphere \[\frac{d F}{d x}=0 \quad \text { or } \quad F=\text { const }=\frac{\sigma T_e^4}{\pi}\label{9.4.5}\]

This will serve as a definition of the local effective temperature \(T_e\).

b. First Moment of the Equation of Radiative Transfer and the Diffusion Approximation

We multiply equation \ref{9.4.1} [with the source function of equation \ref{9.4.2} replacing the "creation rate"] by a unit vector \(\hat{n}\) pointing in the direction of flow of radiant energy and integrate over all solid angles to obtain \[\frac{1}{c} \frac{\partial \vec{F}_\nu}{\partial t}+4 \nabla \cdot \mathbf{K}_\nu=-\rho\left(\kappa_\nu+\sigma_\nu\right) \vec{F}_\nu\label{9.4.6}\]

Now we make the approximation of near isotropy for the radiation field that was done in equation \ref{9.3.12} and evaluate \(J_\nu(\tau_\nu)\) from its definition [equation \ref{9.3.1}] to get \[J_\nu\left(\tau_\nu\right)=\sum_{i=0}^{\infty} \frac{I_{2 i}\left(\mu, \tau_\nu\right)}{2 i+1} \approx I_0\label{9.4.7}\]

We have already shown that under similar assumptions \(K_\nu(\tau_\nu)=\mathrm{I}_0/3\), so for conditions of near isotropy \[K_\nu\left(\tau_\nu\right) \approx \frac{J_\nu\left(\tau_\nu\right)}{3}\label{9.4.8}\]

This is known as the diffusion approximation and it can be used to close the moment equation \ref{9.4.6}, yielding \[\frac{1}{c} \frac{\partial \vec{F}_\nu}{\partial t}+\frac{4}{3} \nabla J_\nu=-\rho\left(\kappa_\nu+\sigma_\nu\right) \vec{F}_\nu\label{9.4.9}\]

Now equations \ref{9.4.3} and \ref{9.4.9} can be combined, by utilizing radiative equilibrium [equation \ref{9.4.4}], to produce a "wave equation" for the radiative flux \(F\) \[\frac{1}{3} \nabla(\nabla \cdot \vec{F})-\frac{1}{c} \frac{\partial}{\partial t}\left[\rho \int_0^{\infty}\left(\kappa_\nu+\sigma_\nu\right) \vec{F}_\nu d\nu\right]-\frac{1}{c^2} \frac{\partial^2 \vec{F}}{\partial t^2}=0\label{9.4.10}\]

which has all the properties of the usual wave equation. Such an equation is useful in solving problems in radiative transfer when the boundary conditions change on a time scale comparable to the photon diffusion time through the medium. Such situations may occur in some nebulae, novae and supernovae, or possibly quasars. For stellar atmospheres, the time-independent solutions will generally be sufficient. For a plane-parallel atmosphere in which the radiation field can be viewed as static, equations \ref{9.4.3} and \ref{9.4.9} become, respectively, \[\frac{d F_\nu}{d x}=4 \kappa_\nu \rho\left(B_\nu-J_\nu\right) \quad \frac{d J_\nu}{d x}=-\frac{3}{4}\left(\kappa_\nu+\sigma_\nu\right) \rho F_\nu\label{9.4.11}\]

That the static equations will be appropriate for normal stellar atmospheres becomes apparent when we consider that the diffusion time for a photon through a stellar atmosphere is only a few orders of magnitude times the light travel time. An atmosphere is a place from which photons escape after perhaps a few dozen interactions. Normal stars do not change on so short a time scale.

c. Eddington Approximation

Although the diffusion approximation provides a method for closing the moment equations of the equation of radiative transfer, it does not allow the complete solution of the problem. The moment equations are, after all, differential equations and are subject to boundary conditions. Specification of these boundary conditions will provide a complete and unique solution for the radiation field. Sir Arthur Stanley Eddington suggested an additional approximation, inspired by the diffusion approximation that allows for the sufficient specification of boundary conditions to permit the solution of equations \ref{9.4.11}.

We consider the situation at the surface, and we assume the emergent radiation field to be isotropic. Since there is generally no incident radiation at the surface of a star, and using the condition of near isotropy given by equation \ref{9.3.12} we get \[\begin{aligned}
& J_\nu(0)=\frac{1}{2} \int_0^1 I(\mu, 0) d \mu=\frac{1}{2} \sum_{i=0}^{\infty} \frac{I_i(0)}{i+1} \approx \frac{1}{2} I_0(0) \\
& F_\nu(0)=2 \int_0^1 I(\mu, 0) \mu d \mu=2 \sum_{i=0}^{\infty} \frac{I_i(0)}{i+2} \approx I_0(0)
\end{aligned}\label{9.4.12}\]

Hence, \[J_\nu(0)=\frac{1}{2} F_\nu(0)\label{9.4.13}\]

This and the condition of radiative equilibrium given by equation \ref{9.4.4} provide the two additional constraints necessary to solve equations \ref{9.4.11}. For the case of the gray atmosphere (see Section 10.2) a particularly simple solution is given by equation \ref{10.2.15}. Although the emergent radiation field is only approximately isotropic, it is the genius of this approximation that the errors introduced by the surface approximation are somewhat offset by the errors incurred by the assumption of the diffusion approximation. Thus, as we shall see later, the Eddington approximation produces solutions for the radiation field that are usually accurate to about 10 percent. As a result, the Eddington approximation is frequently used to solve problems in radiative transfer. To do better, we shall have to do a great deal more.

We have seen that it is possible to describe the flow of radiation through a stellar atmosphere. The derivation involves the same formalisms that we developed in Chapter 1 to describe the flow of matter. The resulting description of this flow is known as the equation of radiative transfer and it differs significantly from the simple result developed for the study of stellar interiors. The differences point up one of the central differences between stellar interiors and stellar atmospheres. Deep inside a star, the structure of the gas and radiation field is fully determined by the local values of the state variables of the gas. This is not the case in the stellar atmosphere. At any given point in the atmosphere, the local radiation field is composed of photons which originated in an environment that differed significantly from the local environment. Thus, the solution for the equation of radiative transfer locally will depend on the solution everywhere. This global nature of radiative transfer in a stellar atmosphere is one of the central differences between the interior and the outer layers of a star. We now turn our attention to solving the equation of radiative transfer.