9.4: Moments of the Radiation Field

In Chapter 1 we saw that a good deal of information was gleaned and simplification achieved by taking moments of the phase density of the particles that made up the gas in question. By such methods we were able to obtain equations for the continuity of matter and momentum and eventually to develop expressions for the hydrodynamic flow of a gas and hydrostatic equilibrium. The basic approach was to throw away information contained in the phase density by averaging it over some appropriate part of the phase space volume. That part of the volume was generally taken to be described by coordinates for which we did not require specific knowledge of the phase density. Since we were to invoke STE for the gas, we knew that the details of the velocity distribution could be ignored since in thermodynamic systems the velocity distributions are specified by a single parameter (the temperature) which is related to the mean velocity. Thus, averaging the phase density over velocity or momentum space made good sense.

We may expect the same sort of benefits by taking moments of the radiation field and particularly the specific intensity, for there is a simple relation [equation \ref{9.2.8}] between the specific intensity and the phase density of photons. However, here we must be careful because it is the momentum distribution of photons in which we are interested so that averaging over momentum space would remove the very information we seek. We must look to other coordinates of phase space to find those which can be considered unimportant.

One of our initial assumptions is the atmosphere is well approximated by a plane-parallel slab. By symmetry, the radiation flow through such a slab will be isotropic about the normal to the slab. Hence, no important information will be contained in the azimuthal coordinate (see Figure 9.3). In addition, we might expect that information in the polar angle θ will not play a central role in the interaction of the radiation field with matter. It is this interaction that determines the emergent spectrum and the atmospheric structure. For these reasons, we can expect that the angular coordinates of phase space may prove expendable and that averages of the radiation field over these coordinates could prove useful in describing the flow of radiation through the atmosphere. Thus, we shall average over two of the three spatial coordinates, choosing the third to represent the direction of net energy flow. In the case of the plane-parallel atmosphere, this clearly is the direction of the atmosphere normal. Also, because of the simple transformation between the specific intensity and the photon phase density, the quantity to be averaged should be the specific intensity itself.

In addition, the higher-order moments should involve the spatial coordinates just as the higher moments in Chapter 1 involved the velocity itself. Such angular moments will then describe various aspects of the net flow of energy.

a. Mean Intensity

Averaging over the angular coordinates described in Figure 9.3 is equivalent to averaging over all solid angles, so with some generality we can define the lowest-order moment of the radiation field as \[J_\nu\left(\tau_\nu\right) \equiv \frac{\oint_{4 \pi} I_\nu\left(\tau_\nu, \theta, \phi\right) d \Omega}{\oint_{4 \pi} d \Omega}=\frac{1}{4 \pi} \oint_{4 \pi} I_\nu\left(\tau_\nu, \theta, \phi\right) d \Omega\label{9.3.1}\]

For a plane-parallel atmosphere, where the intensity has no \(\phi\) dependence and \(\cos\theta\) is replaced by \(\mu\), equation \ref{9.3.1} is equivalent to \[J_\nu\left(\tau_\nu\right)=\frac{1}{2} \int_{-1}^{+1} I_\nu\left(\mu, \tau_\nu\right) d \mu\label{9.3.2}\]

This quantity, known as the mean intensity, is analogous to the particle density of Chapter 1 and differs from the photon energy density by a factor of \(4\pi/\mathrm{c}\).

b. Flux

The next-highest-order moment is related to the net flow of energy in a specific direction \(\hat{n}\), and it is defined, in a manner analogous to that for the mean intensity \(\mathrm{J}_\nu\), as follows: \[\vec{H}_\nu\left(\tau_\nu\right)=\frac{\oint_{4 \pi} I_\nu\left(\tau_\nu, \theta, \phi\right) \hat{n} d \Omega}{\oint_{4 \pi} d \Omega}=\frac{1}{4 \pi} \oint_{4 \pi} I_\nu\left(\tau_\nu, \theta, \phi\right) \hat{n} d \Omega\label{9.3.3}\]

If we break \(\hat{n}\) into its components, then for the axis-symmetric case of a plane parallel atmosphere, this becomes \[\vec{H}_\nu\left(\tau_\nu\right)=\frac{\hat{n}}{2} \int_{-1}^{+1} I_\nu\left(\tau_\nu, \mu\right) \mu d \mu\label{9.3.4}\]

where \(\hat{n}\) points along the normal to the atmosphere. Indeed, it is fair to describe the flux as an intensity-weighted unit vector pointing in the direction of the flow of energy. Although the flux as defined here is a vector quantity, it is common to drop the vector properties since they are generally obvious from the geometry of the atmosphere. However, the vector nature does point to the similarity with the moments of defined in Chapter 1 where the first moment of the phase density was the mean flow velocity. This definition of the first moment of the radiation field is sometimes known as the Harvard flux because it is heavily employed by the ATLAS atmosphere computer code developed at Harvard University, where the analogy to the mean intensity was deemed more important than the physical interpretation.

The actual energy crossing a differential area \(\mathrm{d}A\) in the direction \(\hat{n}\) is \[\mathrm{F}_\nu\left(\tau_\nu\right)=\oint_{4 \pi} I_\nu\left(\tau_\nu, \theta, \phi\right) \hat{n} d \Omega=4 \pi H_\nu\left(\tau_\nu\right)\label{9.3.5}\]

The quantity \(\mathbf{F}_\nu\) is often called the physical flux because it represents the actual flow of energy. For a plane-parallel atmosphere this reduces to \[\mathbf{F}_\nu\left(\tau_\nu\right)=2 \pi \int_{-1}^{+1} I_\nu\left(\tau_\nu, \mu\right) \mu d \mu\label{9.3.6}\]

The quantity \(\pi\) appears so regularly that many early authors, who were primarily concerned with plane-parallel atmospheres, defined a third form of the flux as \[F_\nu\left(\tau_\nu\right) \equiv 2 \int_{-1}^{+1} I_\nu\left(\tau_\nu, \mu\right) \mu d \mu=\frac{\mathbf{F}_\nu\left(\tau_\nu\right)}{\pi}=4 H_\nu\left(\tau_\nu\right)\label{9.3.7}\]

This has become known as the radiative flux and it neither represents a physical quantity directly nor is analogous to the mean intensity. However, it is the most widely used definition of the first moment of the radiation field, so the student is to be warned to determine which definition of the flux a particular author is using or else all sorts of confusion may result. Throughout this book, we use all three definitions, but we try to be quite clear as to which is which and why a specific choice is made.

c. Radiation Pressure

The analogy between this moment and the pressure tensor in Chapter 1 is very close, and the formal definition has the same normalization properties as \(\mathrm{J}_\nu\). So \[\mathbf{K}_\nu\left(\tau_\nu\right)=\frac{\oint_{4 \pi} I_\nu\left(\tau_\nu, \theta, \phi\right)(\hat{n} \hat{n}) d \Omega}{\oint_{4 \pi} d \Omega}=\frac{1}{4 \pi} \oint_{4 \pi}(\hat{n} \hat{n}) I_\nu\left(\tau_\nu, \theta, \phi\right) d \Omega\label{9.3.8}\]

In a manner similar to the physical flux \(\mathbf{F}_\nu\), \(\mathbf{K}_\nu\) can be regarded as an intensity-weighted unit dyadic (not to be confused with the unit tensor 1 that has components \(\delta_\mathrm{ij}\). Now \(\mathbf{K}_\nu\) is known as the radiation pressure tensor and is completely analogous to the pressure tensor P that we obtained in Chapter 1 [equation \ref{1.2.25}]. The meaning of the unit dyadic (in this case the vector outer product of a unit vector with itself) can be seen by writing out the various Cartesian components of \(\mathbf{K}_\nu\) in spherical coordinates: \[\begin{aligned}
& \mathrm{K}_\nu=\frac{1}{4 \pi} \int_0^\pi \int_0^{2 \pi}\left[\begin{array}{lll}
\hat{i} \hat{i} \sin ^2 \theta \cos ^2 \phi & \hat{\mathrm{i}}\hat{\mathrm{j}} \sin ^2 \theta \cos \phi \sin \phi & \hat{\mathrm{i}} \hat{\mathrm{k}} \sin \theta \cos \theta \cos \phi \\
\hat{\mathrm{j}} \hat{\mathrm{i}} \sin ^2 \theta \sin \phi \cos \phi & \hat{\mathrm{j}}\hat{\mathrm{j}} \sin ^2 \theta \sin \phi & \hat{\mathrm{j}} \hat{\mathrm{k}} \sin \theta \cos \theta \sin \phi \\
\hat{\mathrm{k}} \hat{\mathrm{i}} \sin \theta \cos \theta \cos \phi & \hat{\mathrm{k}} \hat{\mathrm{j}} \sin \theta \cos \theta \sin \phi & \hat{\mathrm{k}} \hat{\mathrm{k}} \cos ^2 \theta
\end{array}\right] \\
& \times \mathrm{I}_\nu\left(\tau_\nu\right) \sin \theta d \theta d \phi
\end{aligned}\label{9.3.9}\]

For the axis-symmetric case this becomes \[J_\nu\left(\tau_\nu\right) \equiv \frac{\oint_{4 \pi} I_\nu\left(\tau_\nu, \theta, \phi\right) d \Omega}{\oint_{4 \pi} d \Omega}=\frac{1}{4 \pi} \oint_{4 \pi} I_\nu\left(\tau_\nu, \theta, \phi\right) d \Omega\label{9.3.10}\]

or \[\mathbf{K}_\nu\left(\tau_\nu\right)=\frac{1}{2} \int_{-1}^{+1}\left[\frac{\hat{i}\hat{i}\left(1-\mu^2\right)}{2}, \frac{\hat{j} \hat{j}\left(1-\mu^2\right)}{2}, \hat{k} \hat{k} \mu^2\right] I_\nu\left(\tau_\nu, \mu\right) d \mu\label{9.3.11}\]

Now consider the case where the radiation field is nearly isotropic so that we may expand \(\mathrm{I}_\nu(\tau_\nu,\mu)\) in a rapidly converging series as \[I_\nu\left(\tau_\nu, \mu\right)=\sum_{i=0}^{\infty} I_i\left(\tau_\nu\right) \mu^i\label{9.3.12}\]

where the lead term \(\mathrm{I}_0(\tau_\nu)\) is the dominant term. The components of the radiation pressure tensor then become \[K_\nu\left(\tau_\nu\right)=\left[\begin{array}{l}
\frac{\hat{i}\hat{i} \sum_{i=0}^{\infty} I_{2 i}\left(\tau_\nu\right)}{(2 i+1)(2 i+3)} \\
\frac{\hat{j}\hat{j} \sum_{i=0}^{\infty} I_{2 i}\left(\tau_\nu\right)}{(2 i+1)(2 i+3)} \\
\frac{\hat{k} \hat{k} \sum_{i=0}^{\infty} I_{2 i}\left(\tau_\nu\right)}{2 i+3}
\end{array}\right] \approx \frac{1}{3} I_0\left(\tau_\nu\right) \mathbf{1}\label{9.3.13}\]

Define the scalar moment \(K_\nu(\tau_\nu)\) so that \[K_\nu\left(\tau_\nu\right)=\frac{1}{2} \int_{-1}^{+1} I_\nu\left(\tau_\nu, \mu\right) \mu^2 d \mu \approx \frac{I_0\left(\tau_\nu\right)}{3}\label{9.3.14}\]

The identity of this moment to the magnitude of the radiation pressure tensor in the case of near isotropy ensures that \[\nabla \cdot K_\nu\left(\tau_\nu\right)=\nabla K_\nu\left(\tau_\nu\right)\label{9.3.15}\]

The isotropy condition was required in Chapter 1 in order for the divergence of the pressure tensor to be replaced by the gradient of the scalar pressure. Thus, in every sense of the word \(K_\nu(\tau_\nu)\) may be considered to be related to the pressure of radiation. There remains only the problem of units. Since P represents the transfer of momentum across a surface, the exact relationship is \[\boldsymbol{K}_\nu\left(\tau_\nu\right)=\frac{c \mathbf{P}_\nu\left(\tau_\nu\right)}{4 \pi} \quad P_r\left(\tau_\nu\right)=\frac{4 \pi K_\nu\left(\tau_\nu\right)}{c}\label{9.3.16}\]

Although these expressions give the correct formulation of the radiation pressure in terms of moments of the radiation field, it is important to remember that the radiation pressure is not identical to the force per unit area exerted by photons. That will involve the opacity, for to exert a force the photon must interact with the matter. In the stellar interior, this was no problem because the mean free path was so short as to guarantee that all photons would interact in a short distance. However, in a stellar atmosphere, this is no longer true for some of the photons escape. We return to this point when we consider the forces acting on the gas.