9.3: Equation of Radiative Transfer

In this section we describe, with some generality, the flow of radiation through the outer layers of the star. We developed the formalism for this in Chapter 1 in the form of the Boltzmann transport equation. This formalism basically allows us to describe the flow of any ensemble of particles from one point to another as long as we include all mechanisms for the "creation" and "destruction" of those particles in phase space. In Chapter 1, we used the Boltzmann transport equation to describe the flow of material particles and their momentum through an arbitrary medium. Now we consider the analogous flow of photons.

For material particles, three of the phase space coordinates were velocity. But such coordinates are clearly inappropriate for photons, so we replaced those coordinates with the three components of the photon momentum. This enabled us to write equation \ref{1.2.5} in momentum coordinates so that \[\frac{\partial f}{\partial t}+\sum_{i=1}^3\left(\dot{x}_i \frac{\partial f}{\partial x_i}+\dot{p}_i \frac{\partial f}{\partial p_i}\right)=S\label{9.2.1}\]

For describing the flow of photons, \(f\) represents the density in phase space of photons while S describes their creation and destruction at a local point in phase space. However, it is traditional to describe the photon phase density in terms of a quantity called the specific intensity.

a. Specific Intensity and Its Relation to the Density of Photons in Phase Space

The specific intensity is an energy-like quantity that describes the flow of energy in a particular direction, through a differential area, into a differential solid angle, per unit frequency and time (see Figure 9.2). Remember that the momentum of a photon is just its energy divided by the speed of light: \[p=\frac{h \nu}{c}\label{9.2.2}\]

Figure 9.2 shows the differential parameters defining the specific intensity. Since \(\mathrm{d}A\) is a differential area, the end of the differential solid angle dΩ covers it and all photons passing through \(\mathrm{d}A\) in the direction of \(\hat{n}\) flow into dΩ.

We let the energy carried by photons with momentum \(p\), moving in a direction \(\hat{n}\), passing through a differential area \(\mathrm{d}A\), into a differential solid angle dΩ, in a time \(\mathrm{d}t\) and frequency interval \(\mathrm{d}v\) be \(\mathrm{d}E_\nu(p,\hat{n})\). We can then define the specific intensity as \[I_\nu(p, \hat{n}) \equiv \frac{d E_\nu}{d A \cos \theta d \Omega d v d t}\label{9.2.3}\]

Now the number of photons traveling in a direction \(\hat{n}\) and crossing \(\mathrm{d}A\) in a time \(\mathrm{d}t\) comes from a physical volume \[d V=c d A \cos \theta d t\label{9.2.4}\]

However, the number of photons occupying that volume is just \[d N=f(p, x) d V_p d V\label{9.2.5}\]

For photons in that volume, there is no preferred direction so that the differential volume of momentum space is \[d V_p=4 \pi p^2 d p\label{9.2.6}\]

[see equation \ref{1.3.6}]. Some of these photons will flow in a direction \(\hat{n}\), and into the differential solid angle dΩ, each carrying energy \(\mathrm{h}v\). Therefore, the differential energy in our definition of specific intensity becomes \[d E_\nu=h\nu d N \frac{d \Omega}{4 \pi}\label{9.2.7}\]

Combining equations \ref{9.2.2} through \ref{9.2.7}, we can relate the specific intensity to the phase space density of photons \[I_\nu(p, \hat{n})=\frac{h^4 \nu^3}{c^2} f(p, x) \quad f(p, x)=\frac{c^2}{h^4 \nu^3} I_\nu(p, \hat{n})\label{9.2.8}\]

b. General Equation of Radiative Transfer

Now let us rewrite equation \ref{9.2.1} in vector form: \[\frac{\partial f(p, \vec{r})}{\partial t}+\overrightarrow{\dot{r}} \cdot \nabla f+\overrightarrow{\dot{p}} \cdot \nabla_p f=S\label{9.2.9}\]

If we assume that the photons are moving under the influence of a strong potential gradient ∇Φ, then we can write for photons that \[\overrightarrow{\dot{r}}=c \hat{n} \quad \vec{p}=\frac{h \nu}{c} \hat{n} \quad \overrightarrow{\dot{p}}=\frac{h \nu}{c^2} \nabla \Phi \quad \nabla_p=\frac{\hat{n} c \partial}{h \partial \nu}\label{9.2.10}\]

Substitution of equations \ref{9.2.10} and \ref{9.2.8} into equation \ref{9.2.9} yields an extremely general form of the equation of radiative transfer: \[\frac{1}{c} \frac{\partial I_\nu}{\partial t}+\hat{n} \cdot \nabla I_\nu+\hat{n} \cdot \nabla \Phi\left(\frac{\nu}{c^2}\right)\left(\frac{\partial I_\nu}{\partial \nu}-\frac{3 I_\nu}{\nu}\right)=\frac{h^4 \nu^3 S}{c^3}\label{9.2.11}\]

This equation gives the correct description of the transfer of radiation in an arbitrary coordinate system, even if the boundary conditions are changing on a time scale comparable to the photon diffusion time. It is even correct if the photons are subject to energy loss by virtue of their moving through a strong gravitational field, although some care must be exercised in the choice of coordinates. However, if the propagation takes place in a dispersive medium, then \(\dot{r}\) must be replaced by \[\vec{\dot{r}}=\frac{c}{\mathrm{n}} \hat{n}\label{9.2.12}\]

and the unit of length is changed by n, where n is the index of refraction of the medium, as well.

Fortunately, in normal stellar atmospheres, the radiation field is time-independent, and the gravitational potential gradient is usually negligible so that equation \ref{9.2.11} becomes \[\hat{n} \cdot \nabla I_\nu=\frac{h^4 \nu^3 S}{c^3}\label{9.2.13}\]

The assumption of plane parallelism will simplify this even further, but first let us turn to the "creation" rate \(S\).

c. "Creation" Rate and the Source Function

The "creation" rate \(S\) is just a measure of the rate at which photons that contribute to the flow through \(\mathrm{d}A\) into \(\mathrm{d}\boldsymbol{\Omega}\) are lost to the volume \(\mathrm{d}V\mathrm{d}V_p\). Any absorption process that takes place in that phase space volume will result in the loss of a photon. However, photons can be "lost" from the volume without being destroyed. Any scattering process that changes the momentum of the photon can remove the photon from the volume. Thus, we can write the number lost to the differential volume as \[d n_l=\alpha f d V d V_p\label{9.2.14}\]

where \(\alpha\) is just the fraction of particles present that are lost due to scattering and absorption. Particles may also "appear" in the volume or be "created" by thermal emission or scattering processes. We assume that the thermal emission processes are isotropic so that the number gained in the volume and radiated into a unit solid angle is \[d n_{g t}=\frac{\epsilon d V d V_p}{4 \pi}\label{9.2.15}\]

where ε is the thermal emission per unit volume of phase space.

The situation for scattering is somewhat more complicated. Photons may appear in the volume and be scattered by matter in the volume into direction \(\hat{n}\) with the appropriate momentum. These photons appear to be created just as surely as the thermal photons do, but with a difference. The thermal emission rate depends only on the thermodynamic characteristics of the material gas, whereas the scattered photons have their origin directly in the radiation field. This dependence of the "creation" rate, and hence the specific intensity, on the radiation field itself is one of the hallmarks of radiative transfer in stellar atmospheres. It is through the scattering process that the local value of the radiation field depends on the values of the radiation field throughout the medium. This coupling of the local radiation field to the global radiation field generates mathematical problems of an entirely different character from those found in stellar interiors.

Definition of the Redistribution Function For scattering to act as a source of photons in the direction and solid angle of interest, the process must take a photon of a given momentum and change its direction and momentum to coincide with that of the beam (i.e., the direction and frequency of the specific intensity). The processes that can do this are characterized by a function known as the redistribution function. This function is essentially the probability that a photon with a initial momentum \(p'\) coming from an initial solid angle Ω' will be scattered into a solid angle Ω with final momentum \(p\). We call this probability density function \(\mathrm{R}\left(p^{\prime}, p, \boldsymbol{\Omega}^{\prime}, \boldsymbol{\Omega}\right)\) the redistribution function because it describes how interacting photons will be redistributed in momentum and direction. It is normalized so that \[\int_0^{\infty} \int_0^{\infty} \oint_{4 \pi} \oint_{4 \pi} \mathrm{R}\left(p^{\prime}, p, \Omega^{\prime}, \Omega\right) d \Omega^{\prime} d \Omega d p^{\prime} d p=1\label{9.2.16}\]

The specific nature of the redistribution function depends on the details of the physical scattering mechanism and is discussed later. At this point, it is necessary only to know that the redistribution function exists and can be calculated for specific physical processes. Since the redistribution function has been normalized in accordance with equation \ref{9.2.16}, it represents the redistribution of a scattered photon. To calculate the number of particles gained from scattering, we still must include a measure of the fraction entering the volume \(\mathrm{d}V\mathrm{d}V_\mathrm{p}\) that undergo a scattering. Therefore, the number of particles gained from scattering processes is \[d n_{g s}=\frac{\sigma^{\prime}}{4 \pi} \int_0^{\infty} \oint_{4 \pi} \mathrm{R}\left(p^{\prime}, p, \Omega^{\prime}, \Omega\right) f\left(p^{\prime}, \Omega^{\prime}\right) d \Omega^{\prime} d p^{\prime} d V d V_p\label{9.2.17}\]

where \(\sigma’\) is simply that fraction. The integrals run over all values of \(p’\) and Ω' so that photons entering the volume from all possible directions and with all possible values of momentum are included.

"Creation" Rate in Terms of Scattering and Absorption Processes The net change of particles in volume \(\mathrm{d}V\mathrm{d}V_\mathrm{p}\) is obtained by combining equations \ref{9.2.14}, \ref{9.2.15}, and \ref{9.2.17} and replacing the momentum derivatives and photon phase densities by equations \ref{9.2.2} and \ref{9.2.8}. This process yields \[\begin{aligned}
d n&=d n_{g t}+d n_{g s}+d n_l \\
&=\frac{c^2}{h^4 \nu^3}\left[\frac{h^4 \nu^3}{c^2} \frac{\varepsilon}{4 \pi}+\frac{\sigma^{\prime} h}{4 \pi c} \int_0^{\infty} \oint_{4 \pi}\left(\frac{\nu}{\nu^{\prime}}\right)^3 \mathrm{R}\left(\nu, \nu^{\prime}, \Omega^{\prime}, \Omega\right) \mathrm{I}_{\nu^{\prime}}\left(\Omega^{\prime}\right) d \Omega^{\prime} d \nu^{\prime}-\alpha \mathrm{I}_\nu(\Omega)\right] d V d V_{\mathrm{p}}
\end{aligned}\label{9.2.18}\]

Now the "creation" rate \(S\) is the number of photons created per unit phase space volume per unit time. But in deriving the transformations from phase density to specific intensity given by equations \ref{9.2.8}, we did not choose an arbitrary spatial volume \(\mathrm{d}V\) because it had a length cdt. Therefore, to relate the number of particles created in an arbitrary phase-space volume \(\mathrm{d}V\mathrm{d}V_p\) to \(S\), we must normalize by that length so that \[d n=\frac{S d V d V_p}{c}\label{9.2.19}\]

Using this and equation \ref{9.2.18} to express the "creation" rate S in terms of the physical processes taking place in the volume, we have \[\begin{aligned}
\frac{h^4 \nu^3}{c^3} S= & \rho j_\nu+\frac{\rho \sigma_\nu}{4 \pi} \int_0^{\infty} \oint_{4 \pi} R\left(\nu, \nu^{\prime}, \Omega, \Omega^{\prime}\right) I_{\nu^{\prime}}\left(\Omega^{\prime}\right) d \Omega^{\prime} d \nu^{\prime} \\
& -\left(\kappa_\nu+\sigma_\nu\right) \rho I_\nu(\Omega)
\end{aligned}\label{9.2.20}\]

where we have introduced the volume emissivity \(j_\nu\), the mass scattering coefficient \(\sigma_\nu\) and the mass absorption coefficient \(\kappa_\nu\) and replaced R from \[\begin{aligned}
& j_\nu=\frac{h^4 \nu^3 \epsilon /\left(4 \pi c^2\right)}{\rho} \quad \sigma_\nu=\frac{\sigma^{\prime}}{\rho} \\
& R\left(\nu, \nu^{\prime}, \Omega, \Omega^{\prime}\right)=\frac{h\left(\nu / \nu^{\prime}\right)^3}{c} \mathrm{R}\left(\nu, \nu^{\prime}, \Omega, \Omega^{\prime}\right) \\
& \kappa_\nu+\sigma_\nu=\frac{\alpha}{\rho}
\end{aligned}\label{9.2.21}\]

Thermal Emission For a gas that is in thermal equilibrium, the relationship between the rate of absorption and emission is not arbitrary. This is the first use of LTE. Since we are assuming that the gas is in thermal equilibrium with its surroundings (LTE), we may invoke Kirchhoff's law for the relationship between the thermal emissivity and absorptivity, namely, \[j_\nu=\kappa_\nu B_\nu(T)\label{9.2.22}\]

where \(B_\nu(\mathrm{T})\) is the Planck function which depends only on the local temperature. If we use this and equation \ref{9.2.20}, the equation of radiative transfer given by equation \ref{9.2.13} becomes \[\begin{aligned}
\hat{n} \cdot \nabla I_\nu= & \rho\left[\kappa_\nu B_\nu(T)-\left(\kappa_\nu+\sigma_\nu\right) I_\nu(\Omega)\right. \\
& \left.+\frac{\sigma_\nu}{4 \pi} \int_0^{\infty} \oint_{4 \pi} R\left(\nu, \nu^{\prime}, \Omega, \Omega^{\prime}\right) I_{\nu^{\prime}}\left(\Omega^{\prime}\right) d \Omega^{\prime} d \nu^{\prime}\right]
\end{aligned}\label{9.2.23}\]

Figure 9.3 shows the geometry of a plane-parallel atmosphere.

We may further simplify the equation of radiative transfer by invoking the plane-parallel approximation so that ∇ becomes \(\hat{x}d/dx\) (see Figure 9.3), yielding \[\begin{aligned}
\frac{\cos \theta}{-\left(\kappa_\nu+\sigma_\nu\right) \rho} \frac{d I_\nu}{d x}= & I_\nu-\frac{\kappa_\nu B_\nu}{\kappa_\nu+\sigma_\nu} \\
& -\frac{\sigma_\nu}{4 \pi\left(\kappa_\nu+\sigma_\nu\right)} \int_0^{\infty} \oint_{4 \pi} R\left(\nu, \nu^{\prime}, \Omega, \Omega^{\prime}\right) I_{\nu^{\prime}}\left(\Omega^{\prime}\right) d \Omega^{\prime} d \nu^{\prime}
\end{aligned}\label{9.2.24}\]

Optical depth The notion of a dimensionless depth parameter called optical depth is central to the study of stellar atmospheres. It is usually taken to increase inward as one moves into the star, and it can be viewed physically in the following manner. Optical depth of unity is that depth of material wherein \((1/e)\) of the photons will be scattered or absorbed while traversing the depth. In terms of the mass absorption and scattering coefficients and the differential distance parameter, it is defined as \[d \tau_\nu=-\left(\kappa_\nu+\sigma_\nu\right) \rho d x\label{9.2.25}\]

Making use of the definition of optical depth, we can write the equation of radiative transfer for a plane-parallel atmosphere as \[\mu \frac{d I_\nu\left(\mu, \tau_\nu\right)}{d \tau_\nu}=I_\nu\left(\mu, \tau_\nu\right)-S_\nu\left(\mu, \tau_\nu\right)\label{9.2.26}\]

where \[S_\nu \equiv \frac{\kappa_\nu B_\nu}{\kappa_\nu+\sigma_\nu}+\frac{\sigma_\nu}{4 \pi\left(\kappa_\nu+\sigma_\nu\right)} \int_0^{\infty} \oint_{4 \pi} R\left(\nu, \nu^{\prime}, \Omega, \Omega^{\prime}\right) I_{\nu^{\prime}}\left(\Omega^{\prime}\right) d \Omega^{\prime} d \nu^{\prime}\label{9.2.27}\]

The parameter \(S_\nu\) is known as the source function of the radiation field. Since the quantity \((\kappa_\nu+\sigma_\nu)\) appears so frequently, it is customary to call it the mass extinction coefficient. The name is reasonable as it is, indeed, a measure of the total ability of material to attenuate the flow of photons.

d. Physical Meaning of the Source Function

The source function is one of the most important concepts in the theory of radiative transfer, and it is important to have a good intuitive feeling for its meaning. As the name implies, the source function represents the local contribution to the radiation field. It is a measure of the energy contributed to the radiation field by physical processes taking place at a particular spot in the atmosphere. Consider the case where scattering is unimportant so that \(\sigma_\nu=0\). Under these conditions the expression for the source function [equation \ref{9.2.27}] becomes \[S_\nu=B_\nu\label{9.2.28}\]

and all photons locally contributed to the radiation field can be characterized by the Planck function since they arise from thermal processes. This is a consequence of the assumption of LTE which enabled us to use Kirchhoff's law to characterize the local emissivity of the gas in terms of its absorptivity. Some authors take this as a definition of LTE, but as such, it would be unduly restrictive. The presence of scattering, say by electrons will require a more complicated source function such as that given by equation \ref{9.2.27}, but the excitation and ionization characteristics of the gas may still be those expected for a gas in thermodynamic equilibrium. Thus, \(S_\nu=B_\nu\) is normally a sufficient condition for the existence of LTE, but not a necessary one.

Now consider the case when pure absorption processes are negligible and virtually all the opacity of the material arises from scattering processes. Then \[S_\nu=\frac{1}{4 \pi} \int_0^{\infty} \oint_{4 \pi} R\left(\nu, \nu^{\prime}, \Omega, \Omega^{\prime}\right) I_{\nu^{\prime}}\left(\Omega^{\prime}\right) d \Omega^{\prime} d \nu^{\prime}\label{9.2.29}\]

Here the source function depends only on the incident radiation field. Since the redistribution function is normalized to unity, the integral in equation \ref{9.2.29} simply represents some sort of average of the local specific intensity over all frequencies and angles. The factor of \(1/4\pi\) then represents that part of the average that is scattered into the differential solid angle appropriate for \(\mathrm{I}_\nu\).

Thus, under the conditions of pure scattering, the source function becomes totally independent of the local physical conditions and is completely determined by the local radiation field. If this condition were to prevail throughout the atmosphere, one would have the curious result that the radiation field would be independent of the local values of the state variables (\(\mathrm{P}\), \(\mathrm{T}\), and \(\rho\)) and depend only on the ability of particles to scatter photons and the details of how the particles do it. In some real sense, the radiation field would become decoupled from the physical properties of the gas. Indeed, one can learn little about the physical conditions that prevail in a fog by observing the light transmitted through it from say an automobile headlight. This independence of the radiation field from the state variables of the gas enables one to solve the entire problem of radiative transfer for pure scattering without knowing anything about the gas other than the redistribution function. We use this property later to discuss methods of solving the equation of radiative transfer. However, as the case of the fog illustrates, this is a two edged sword. The decoupling of the radiation field from the state variables of the gas, in the case of pure scattering, means that we can not use the radiation field to determine the run of state variables with depth.

e. Special Forms of the Redistribution Function

Since the redistribution function plays such an important role in specifying the nature of scattering in the source function, we examine some common physical situations and the corresponding redistribution functions.

Coherent Scattering The term coherent scattering refers to the case where photons are scattered in direction but not in frequency. Thomson scattering by electrons is of this form. Such processes are generally known as conservative processes because no energy is exchanged between the radiation field and the particles. While this is never strictly true, in many cases it is an excellent approximation. This is certainly true for the scattering of optical photons by the electrons present in a stellar atmosphere. Under these conditions we can write the redistribution function as \[R\left(\nu^{\prime}, \nu, \vec{\Omega}, \vec{\Omega}^{\prime}\right)=h\left(\vec{\Omega}, \vec{\Omega}^{\prime}\right) \delta\left(\nu-\nu^{\prime}\right)\label{9.2.30}\]

where \(\delta(\nu-\nu’)\) is the Dirac delta function. The delta function on frequency causes the frequency integral in equation \ref{9.2.27} to collapse, simplifying the source function considerably.

Noncoherent Scattering This phrase has come to mean considerably more than the opposite of coherent scattering. For fully noncoherent scattering, the frequency of a scattered photon is completely uncorrelated with the frequency of the incident photon. In some sense, the photon "forgets" its prior frequency. Like coherent scattering, this case also represents an approximation. Clearly, if the situation were to apply to the entire frequency range from zero to infinity, the value of the redistribution function at any specific value of ν would have to be arbitrarily small. Thus, the common use of the approximation is confined to a finite frequency range such as a spectral line. As we shall see later, very strong spectral lines often possess the property that an electron in the upper state is so perturbed by interactions with other particles of the gas that the specific value of the absorbed energy is irrelevant in determining the energy of the photon that will be emitted in the subsequent transition. Thus, over a finite frequency interval, the wavelength of the emitted photon will be totally uncorrelated with the wavelength of the absorbed photon. Under these conditions, the frequency simply does not appear in the redistribution function and \[R\left(\nu, \nu^{\prime}, \vec{\Omega}, \vec{\Omega^{\prime}}\right)=h\left(\vec{\Omega}, \vec{\Omega^{\prime}}\right)\label{9.2.31}\]

Redistribution functions of this form are often called complete redistribution functions.

Isotropic Scattering As with complete redistribution, the photon undergoing isotropic scattering suffers from "amnesia". The direction of the scattered photon is completely uncorrelated with the direction of the incident photon. Thus, the angular dependence of the redistribution function vanishes and \[R\left(\nu, \nu^{\prime}, \vec{\Omega}, \vec{\Omega}^{\prime}\right)=g\left(\nu, \nu^{\prime}\right)\label{9.2.32}\]

This also considerably simplifies the source function in equation \ref{9.2.27}. If the radiation field were isotropic, the integral over the solid angle merely produces a factor of \(4\pi\), which cancels the corresponding factor in front of the integral. In general, this is also an approximation. Although it is far from obvious, we shall see that it is an excellent approximation for electron scattering of optical photons in a stellar atmosphere. So great is the simplification introduced by the assumption of isotropic scattering that there is a tendency to invoke it even when it is totally inappropriate. Later, we shall see what sorts of methods can be used to incorporate the full redistribution function in the solution of the equation of radiative transfer. Such cases are often called partially coherent anisotropic scattering, and their solution poses one of the most difficult problems in radiative transfer. However, before we consider these formidable problems, we must understand how to approach the solution of more basic problems. The dominant form of scattering in normal stellar atmospheres is Thomson scattering by electrons, and for purposes of determining the atmospheric structure it is an excellent approximation to assume that such scattering is isotropic. Under the assumption of coherent isotropic scattering, the source function given by equation \[S_\nu=\frac{\kappa_\nu B_\nu}{\kappa_\nu+\sigma_\nu}+\frac{\sigma_\nu}{4 \pi\left(\kappa_\nu+\sigma_\nu\right)} \oint_{4 \pi} I_\nu\left(\Omega^{\prime}\right) d \Omega^{\prime}\label{9.2.33}\]