15.4: Non-LTE Transfer of Radiation and the Redistribution Function

While we did indicate how departures of the populations of the energy levels from their LTE values could be included in the construction of a model atmosphere so that any structural effects are included, the major emphasis of the effects of non-LTE has been on the strengths and shapes of spectral lines. During the discussion of the two level atom, we saw that the form of the source function was somewhat different from what we discussed in Chapter 10. Indeed, the equation of transfer [equation \ref{15.2.6}] for complete redistribution appears in a form somewhat different from the customary plane-parallel equation of transfer. Therefore, it should not be surprising to find that the effects of non-LTE can modify the profile of a spectral line. The extent and nature of this modification will depend on the nature of the redistribution function as well as on the magnitude of the departures from LTE. Since we already introduced the case of complete redistribution [equations \ref{15.2.1} and \ref{15.2.2}, we begin by looking for a radiative transfer solution for the case where the emitted and absorbed photons within a spectral line are completely uncorrelated.

a. Complete Redistribution

In Chapter 14, we devoted a great deal of effort to developing expressions for the atomic absorption coefficient for spectral lines that were broadened by a number of phenomena. However, we dealt tacitly with absorption and emission processes as if no energy were exchanged with the gas between the absorption and reemission of the photon. Actually this connection was not necessary for the calculation of the atomic line absorption coefficient, but this connection is required for calculating the radiative transfer of the line radiation. Again, for the case of pure absorption there is no relationship between absorbed and emitted photons. However, in the case of scattering, as with the Schuster-Schwarzschild atmosphere, the relationship between the absorbed and reemitted photons was assumed to be perfect. That is, the scattering was assumed to be completely coherent. In a stellar atmosphere, this is rarely the case because micro-perturbations occurring between the atoms and surrounding particles will result in small exchanges of energy, so that the electron can be viewed as undergoing transitions within the broadened energy level. If those transitions are numerous during the lifetime of the excited state, then the energy of the photon that is emitted will be uncorrelated with that of the absorbed photon. In some sense the electron will "lose all memory" of the details of the transition that brought it to the excited state. The absorbed radiation will then be completely redistributed throughout the line. This is the situation that was described by equations \ref{15.2.1} through \ref{15.2.3}, and led to the equation of transfer \ref{15.2.6} for complete redistribution of line radiation.

Although this equation has a slightly different form from what we are used to, it can be put into a familiar form by letting \[d \tau_x=\phi_x d \tau\label{15.3.1}\]

It now takes on the form of equation \ref{10.1.1}, and by using the classical solution discussed in Chapter 10, we can obtain an integral equation for the mean intensity in the line in terms of the source function. \[J\left(\tau_x\right)=\frac{1}{2} \int_0^{\infty} S_{\ell}(t) E_1\left|\int_{\tau_x}^t \phi_x\left(t^{\prime}\right) d t^{\prime}\right| \phi_x(t) d t\label{15.3.2}\]

This can then be substituted into equation \ref{15.2.19} to obtain an integral equation for the source function in the line. \[\begin{aligned}
S_{\ell}\left(\tau_x\right)= & \epsilon B_\nu(T)+\frac{1}{2}(1-\epsilon) \\
& \times \int_0^{\infty} S_{\ell}(t)\left[\int_{-\infty}^{+\infty} \phi_x\left(\tau_x\right) \phi_x(t) E_1\left|\int_{\tau_x}^t \phi_x\left(t^{\prime}\right) d t^{\prime}\right| d x\right] d t
\end{aligned}\label{15.3.3}\]

The integral over x results from the integral of the mean intensity over all frequencies in the line [see equation \ref{15.2.19}]. Note the similarity between this result and the integral equation for the source function in the case of coherent scattering [equation \ref{9.1.14}]. Only the kernel of the integral has been modified by what is essentially a moment in frequency space weighted by the line profile function \(\phi_\mathrm{x}\mathrm{(t)}\). This is clearly seen if we write the kernel as \[K\left(\tau_x, t\right)=\int_{-\infty}^{+\infty} \phi_x\left(\tau_x\right) \phi_x(t) E_1\left|\int_{\tau_x}^t \phi_x\left(t^{\prime}\right) d t^{\prime}\right| d x\label{15.3.4}\]

so that the source function equation becomes a Schwarzschild-Milne equation of the form \[S_{\ell}\left(t_x\right)=\epsilon B_\nu(T)+\frac{1}{2}(1-\epsilon) \int_{-\infty}^{+\infty} S_{\ell}(t) K\left(\tau_x, t\right) d t\label{15.3.5}\]

Since \[\left|\int_{\tau_x}^t \phi_x\left(t^{\prime}\right) d t^{\prime}\right|=\left|\Phi(t)-\Phi\left(\tau_x\right)\right|\label{15.3.6}\]

the kernel is symmetric in \(\tau_\mathrm{x}\) and \(\mathrm{t}\), so that \(K\left(\tau_{\mathrm{x}}, \mathrm{t}\right)=K\left(\mathrm{t}, \tau_{\mathrm{x}}\right)\). This is the same symmetry property as the exponential integral \(\mathrm{E}_1|\tau-\mathrm{t}|\) in equation 10.1.14. Unfortunately, for an arbitrary depth dependence of \(\phi_\mathrm{x}\mathrm{(t)}\), equations \ref{15.3.4} and \ref{15.3.5} must be solved numerically. Fortunately, all the methods for the solution of Schwarzschild-Milne equations discussed in Chapter 10 are applicable to the solution of this integral equation.

While it is possible to obtain some insight into the behavior of the solution for the case where \(\phi_{\mathrm{x}}(\mathrm{t}) \neq f(\mathrm{t})\) (see Mihalas³, pp.366-369), the insight is of dubious value because it is the solution for a special case of a special case. However, a property of such solutions, and of noncoherent scattering in general, is that the core of the line profile is somewhat filled in at the expense of the wings. As we saw for the two-level atom with continuum, the source function takes on a more complicated form. Thus we turn to the more general situation involving partial redistribution.

b. Hummer Redistribution Functions

The advent of swift computers has made it practical to model the more complete description of the redistribution of photons in spectral lines. However, the attempts to describe this phenomenon quantitatively go back to L.Henyey⁵ who carried out detailed balancing within an energy level to describe the way in which photons are actually redistributed across a spectral line. Unfortunately, the computing power of the time was not up to the task, and this approach to the problem has gone virtually unnoticed. More recently, D.Hummer⁶ has classified the problem of redistribution into four main categories which are widely used today. For these cases, the energy levels are characterized by Lorentz profiles which are appropriate for a wide range of lines. Regrettably, for the strong lines of hydrogen, many helium lines as well as most strong resonance lines, this characterization is inappropriate (see Chapter 14) and an entirely different analysis must be undertaken. This remains one of the current nagging problems of stellar astrophysics. However, the Hummer classification and analysis provides considerable insight into the problems of partial redistribution and enables rather complete analyses of many lines with Lorentz profiles produced by the impact phase-shift theory of collisional broadening.

Let us begin the discussion of the Hummer redistribution functions with a few definitions. Let \(p\left(\xi^{\prime}, \xi\right) \mathrm{d} \xi\) be the probability that an absorbed photon having frequency \(\xi’\) is scattered into the frequency interval \(\xi \rightarrow \xi+d \xi\). Furthermore, let the probability density function \(p\left(\xi^{\prime}, \xi\right)\) be normalized so that \(\int \mathrm{p}\left(\xi^{\prime}, \xi\right) \mathrm{d} \xi^{\prime}=1\). That is, the absorbed photon must go somewhere. If this is not an appropriate result for the description of some lines, the probability of scattering can be absorbed in the scattering coefficient (see Section 9.2). In addition, let \(g\left(\hat{\mathrm{n}}^{\prime}, \hat{\mathrm{n}}\right)\) be the probability density function describing scattering from a direction \(\hat{n}’\) into \(\hat{n}\), also normalized so that the integral over all solid angles, \(\left[\int g\left(\hat{\mathrm{n}}^{\prime}, \hat{\mathrm{n}}\right) \mathrm{d} \Omega\right] /(4 \pi)=1\). For isotropic scattering, \(g\left(\hat{\mathrm{n}}^{\prime}, \hat{\mathrm{n}}\right)=1\), while in the case of Rayleigh Scattering \(g\left(\hat{\mathrm{n}}^{\prime}, \hat{\mathrm{n}}\right)=3\left[1+\left(\hat{\mathrm{n}}^{\prime} \cdot \hat{\mathrm{n}}\right)^2\right] / 4\). We further define \(f\left(\xi^{\prime}\right) \mathrm{d} \xi^{\prime}\) as the relative [that is, \(\int f\left(\xi^{\prime}\right) \mathrm{d} \xi^{\prime}=1\)] probability that a photon with frequency \(\xi’\) is absorbed. These probability density functions can be used to describe the redistribution function introduced in Chapter 9.

In choosing to represent the redistribution function in this manner, it is tacitly assumed that the redistribution of photons in frequency is independent of the direction of scattering. This is clearly not the case for atoms in motion, but for an observer located in the rest frame of the atom it is usually a reasonable assumption. The problem of Doppler shifts is largely geometry and can be handled separately. Thus, the probability that a photon will be absorbed at frequency \(\xi’\) and reemitted at a frequency \(\xi\) is \[R\left(\xi^{\prime}, \xi, \hat{n}^{\prime}, \hat{n}\right) d \xi^{\prime} d \xi \frac{d \omega^{\prime}}{4 \pi} \frac{d \omega}{4 \pi}=f\left(\xi^{\prime}\right) p\left(\xi^{\prime}, \xi\right) g\left(\hat{n}^{\prime}, \hat{n}\right) d \xi^{\prime} d \xi \frac{d \omega^{\prime}}{4 \pi} \frac{d \omega}{4 \pi}\label{15.3.7}\]

David Hummer⁶ has considered a number of cases where \(f\), \(p\), and \(g\), take on special values which characterize the energy levels and represent common conditions that are satisfied by many atomic lines.

Emission and Absorption Probability Density Functions for the Four Cases Considered by Hummer Consider first the case of coherent scattering where both energy levels are infinitely sharp. Then the absorption and reemission probability density functions will be given by \[\begin{aligned}
&\begin{aligned}
f\left(\xi^{\prime}\right) & =\delta\left(\xi^{\prime}-\nu_0\right) \\
p\left(\xi^{\prime}, \xi\right) & =\delta\left(\xi^{\prime}-\xi\right)
\end{aligned}\\
&\text { Hummer's case I }
\end{aligned}\label{15.3.8}\]

If the lower level is broadened by collisional radiation damping but the upper level remains sharp, then the absorption probability density function is a Lorentz profile while the reemission probability density function remains a delta function, so that \[\begin{aligned}
f\left(\xi^{\prime}\right) & =\frac{\gamma / \pi}{\left(\xi^{\prime}-\nu_0\right)^2+\gamma^2} \\
p\left(\xi^{\prime}, \xi\right) & =\delta\left(\xi^{\prime}-\xi\right)\\
& \text { Hummer's case II }
\end{aligned}\label{15.3.9}\]

If the lower level is sharp but the upper level is broadened by collisional radiation damping, then both probability density functions are given Lorentz profiles since the transitions into and out of the upper level are from a broadened state. Thus, \[\begin{aligned}
f\left(\xi^{\prime}\right) & =\frac{\gamma_u / \pi}{\left(\xi^{\prime}-\nu_0\right)^2+\gamma_u^2} \\
p\left(\xi^{\prime}, \xi\right) & =\frac{\gamma_u / \pi}{\left(\xi-\nu_0\right)^2+\gamma_u^2}\\
&\text { Hummer's case III }
\end{aligned}\label{15.3.10}\]

Since \(\xi’\) and \(\xi\) are uncorrelated, this case represents a case of complete redistribution of noncoherent scattering. Hummer gives the joint probability of transitions from a broadened lower level to a broadened upper level and back again as \[f\left(\xi^{\prime}\right) p\left(\xi^{\prime}, \xi\right)=\frac{\gamma_u \gamma_l / \pi^2}{\left[\left(\xi^{\prime}-\xi\right)^2+\gamma_l^2\right]\left[\left(\xi-\nu_0\right)^2+\gamma_u^2\right]} \quad \text { Hummer's case IV }\label{15.3.11}\]

This probability must be calculated as a unit since \(\xi’\) is the same for both \(f\) and \(p\). Unfortunately a careful analysis of this function shows that the lower level is considered to be sharp for the reemitted photon and therefore is inconsistent with the assumption made about the absorption. Therefore, it will not satisfy detailed balancing in an environment that presupposes LTE. A correct quantum mechanical analysis⁷ gives \[\begin{aligned}
& f\left(\xi^{\prime}\right) p\left(\xi^{\prime}, \xi\right)= \\
& \quad \frac{\gamma_u\left(2 \gamma_l+\gamma_u\right) \gamma_l / \pi^2}{\left\{\left(\xi^{\prime}-\nu_0\right)^2+\left[\left(\gamma_l+\gamma_u\right) / 2\right]^2\right\}\left\{\left(\xi^{-} \nu_0\right)^2+\left[\left(\gamma_l+\gamma_u\right) / 2\right]^2\right\}\left[\left(\xi^{\prime}-\xi\right)^2+\gamma_l^2\right]} \\
& \quad+\frac{\gamma_l \gamma_u}{\left\{\left(\xi^{\prime}-\nu_0\right)^2+\left[\left(\gamma_u+\gamma_l\right) / 2\right]^2\right\}\left[\left(\xi^{\prime}-\xi\right)^2+\gamma_l^2\right]} \\
& \quad+\frac{\gamma_l \gamma_u}{\left\{\left(\xi-\nu_0\right)^2+\left[\left(\gamma_l+\gamma_u\right) / 2\right]^2\right\}\left[\left(\xi^{\prime}-\xi\right)^2+\gamma_l^2\right]} \\
& \quad+\frac{\gamma_l^2}{\left\{\left(\xi-\nu_0\right)^2+\left[\left(\gamma_u+\gamma_l\right) / 2\right]^2\right\}\left\{\left(\xi^{\prime}-\nu_0\right)^2+\left[\left(\gamma_u+\gamma_l\right) / 2\right]^2\right\}}
\end{aligned}\label{15.3.12}\]

A little inspection of this rather messy result shows that it is symmetric in \(\xi’\) and \(\xi\), which it must be if it is to obey detailed balancing. In addition, the function has two relative maxima at \(\xi’=\xi\) and \(\xi’=\nu_0\). Since the center of the two energy levels represent a very likely transition, transitions from the middle of the lower level and back again will be quite common. Under these conditions \(\xi’=\xi\) and the scattering is fully coherent. On the other hand, transitions from the exact center of the lower level \((\xi’=\nu_0)\) will also be very common. However, the return transition can be to any place in the lower level with frequency \(\xi\). Since the function is symmetric in \(\xi'\) and \(\xi\), the reverse process can also happen. Both these processes are fully noncoherent so that the relative maxima occur for the cases of fully coherent and noncoherent scattering with the partially coherent photons being represented by the remainder of the joint probability distribution function.

Effects of Doppler Motion on the Redistribution Functions Consider an atom in motion relative to some fixed reference frame with a velocity \(\vec{\mathrm{v}}\). If a photon has a frequency (\xi'\) as seen by the atom, the corresponding frequency in the rest frame is \[\nu^{\prime}=\xi^{\prime}+\frac{\nu_0}{c} \vec{v} \cdot \hat{n}^{\prime}\label{15.3.13}\]

Similarly, the photon emitted by the atom will be seen in the rest frame, Doppler-shifted from its atomic value by \[\nu=\xi+\frac{\nu_0}{c} \vec{v} \cdot \hat{n}\label{15.3.14}\]

Thus the redistribution function that is seen by an observer in the rest frame is \[R\left(\nu^{\prime}, \nu, \hat{n}^{\prime}, \hat{n}\right)=f\left(\nu^{\prime}-\frac{\nu_0 \vec{v} \cdot \hat{n}^{\prime}}{c}\right) p\left(\nu^{\prime}-\frac{\nu_0 \vec{v} \cdot \hat{n}^{\prime}}{c}, \nu-\frac{\nu_0 \vec{v} \cdot \hat{n}}{c}\right) g\left(\hat{n}^{\prime}, \hat{n}\right)\label{15.3.15}\]

We need now to relate the scattering angle determined from \(\hat{\mathrm{n}}^{\prime} \cdot \hat{\mathrm{n}}\) to the angle between the atomic velocity and the directions of the incoming and outgoing scattered photon. Consider a coordinate frame chosen so that the x-y plane is the scattering plane and the x axis lies in the scattering plane midway between the incoming and outgoing photon (see Figure 15.2). In this coordinate frame, the directional unit vectors \(\hat{\mathrm{n}}\) and \(\hat{\mathrm{n}}’\) have Cartesian components given by \[\hat{n}^{\prime}=\cos \left(\frac{\psi}{2}\right) \hat{i}-\sin \left(\frac{\psi}{2}\right) \hat{j} \quad \hat{n}=\cos \left(\frac{\psi}{2}\right) \hat{i}+\sin \left(\frac{\psi}{2}\right) \hat{j}\label{15.3.16}\]

Figure 15.2 displays a Cartesian coordinate frame where the x-axis bisects the angle between the incoming and outgoing photon and the x-y plane is the scattering plane.

Now if we assume that the atoms have a maxwellian velocity distribution \[P(\vec{v})=\left(\frac{m}{2 \pi k T}\right)^{3 / 2} e^{-[m /(2 k T)]\left(v_x^2+v_y^2+v_x^2\right)}\label{15.3.17}\]

we can obtain the behavior of an ensemble of atoms by averaging equation \ref{15.3.15} over all velocity. First it is convenient to make the variable transformations \[\begin{aligned}
& \vec{u} \equiv \sqrt{\frac{m}{2 k T}} \vec{v}=\frac{\vec{v}}{v_{\mathrm{th}}} \\
& \alpha \equiv \cos \frac{\psi}{2} \quad \tilde{\alpha} \equiv \cos \psi \\
& \beta \equiv \sin \frac{\psi}{2} \quad \tilde{\beta} \equiv \sin \psi \\
& w \equiv \frac{\nu_0 v_{\mathrm{th}}}{c}=\frac{\nu_0}{c} \sqrt{\frac{2 k T}{m}}
\end{aligned}\label{15.3.18}\]

so that the components of the particle's velocity projected along the directions of the photon's path become \[\vec{v} \cdot \hat{n}^{\prime}=\sqrt{\frac{2 k T}{m}}\left(\alpha u_x-\beta u_y\right) \quad \vec{v} \cdot \hat{n}=\sqrt{\frac{2 k T}{m}}\left(\alpha u_x+\beta u_y\right)\label{15.3.19}\]

and the velocity distribution is \[P(\vec{u}) d \vec{u}=\pi^{-3 / 2} e^{-\vec{u} \cdot \vec{u}} d \vec{u}\label{15.3.20}\]

The symbol \(\mathrm{d\vec{u}}\) means \(\mathrm{du_{x}du_{y}du_{z}}\).

We define the ensemble average over the velocity of the redistribution function as \[\left\langle R\left(\nu^{\prime}, \nu, \hat{n}^{\prime}, \hat{n}\right)\right\rangle \equiv \int_{\vec{v}} R\left(\nu^{\prime}, \nu, \hat{n}^{\prime}, \hat{n}, \vec{v}\right) P(\vec{v}) d \vec{v}\label{15.3.21}\]

or \[\begin{aligned}
\left\langle R\left(\nu^{\prime}, \nu, \hat{n}^{\prime}, \hat{n}\right)\right\rangle= & \frac{g\left(\hat{n}^{\prime}, \hat{n}\right)}{\pi} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} e^{-\left(u_x^2+u_y^2\right)} f\left[\nu^{\prime}-w\left(\alpha u_x-\beta u_y\right)\right] \\
& \times p\left[\nu^{\prime}-w\left(\alpha u_x-\beta u_y\right), \nu-w\left(\alpha u_x+\beta u_y\right)\right] d u_x d u_y
\end{aligned}\label{15.3.22}\]

A coordinate rotation by y/2 about the y axis so that \(\hat{\mathrm{n}}’\) is aligned with \(\hat{\mathrm{x}}\) (see Figure 15.2) leads to the equivalent, but useful, form \[\begin{aligned}
\left\langle R\left(\nu, \nu^{\prime}, \hat{n}^{\prime}, \hat{n}\right)\right\rangle= & \frac{g\left(\hat{n}^{\prime}, \hat{n}\right)}{\pi}\left[\int_{-\infty}^{+\infty} e^{-u_x^2} f\left(\nu^{\prime}-w u_x\right) d u_x\right. \\
& \times\left\{\int_{-\infty}^{+\infty} e^{-u_y^2} p\left[\left(\nu^{\prime}-w u_x\right), \nu-w\left(\tilde{\alpha} u_x+\tilde{\beta} u_y\right)\right] d u_y\right\}
\end{aligned}\label{15.3.23}\]

We are now in a position to evaluate the effects of thermal Doppler motion on the four cases given by Hummer⁶, represented by equations \ref{15.3.8} through \ref{15.3.12}. The substitution of these forms of \(\mathrm{f}(\xi’)\) and \(\mathrm{p}(\xi’,\xi)\) into equation \ref{15.3.22} or equation \ref{15.3.23} will yield the desired result. The frequencies \(\xi’\) and \(\xi\) must be Doppler shifted according to equations \ref{15.3.13} and \ref{15.3.14} and some difficulty may be encountered for the case of direct forward or back scattering (that is, β = 0) and when one of the distribution functions is a delta function (i.e., for a sharp energy level). The fact that β = 0 for these cases should be invoked before any variable transformations are made for the purposes of evaluating the integrals.

Making a final transformation to a set of dimensionless frequencies \[x \equiv \frac{\nu-\nu_0}{w} \quad x^{\prime} \equiv \frac{\nu^{\prime}-\nu_0}{w}\label{15.3.24}\]

we can obtain the following result for Hummer's case I: \[\left\langle R_{\mathrm{I}}\left(x^{\prime}, x, \hat{n}^{\prime}, \hat{n}\right)\right\rangle=\frac{g\left(\hat{n}^{\prime}, \hat{n}\right)}{\pi \sin \psi} e^{-\left[x^2+\left(x^{\prime}-x \cos \psi\right)^2 / \sin ^2 \psi\right]}\label{15.3.25}\]

Consider what the emitted radiation would look like for an ensemble of atoms illuminated by an isotropic uniform radiation field \(\mathrm{I_0}\). Substitution of such a radiation field into equation \ref{9.2.29} would yield \[S_x=\frac{1}{4 \pi} \int_{4 \pi} \int_{-\infty}^{+\infty} I_0\left\langle R_{\mathrm{I}}\left(x^{\prime}, x, \hat{n}^{\prime}, \hat{n}\right)\right\rangle d x^{\prime} d \omega^{\prime}\label{15.3.26}\]

which after some algebra gives \[S_x=\frac{I_0 e^{-x^2}}{\sqrt{\pi}}=I_0 \phi(x)\label{15.3.27}\]

This implies that the emission of the radiation would have exactly the same form as the absorption profile. But this was our definition of complete redistribution [see equation \ref{15.2.3}]. Thus, although a single atom behaves coherently, an ensemble of thermally moving atoms will produce a line profile that is equivalent to one suffering complete redistribution of the radiation over the Doppler core. Perhaps this is not too surprising since the motion of the atoms is totally uncorrelated so that the Doppler shifts produced by the various motions will mimic complete redistribution.

As one proceeds with the progressively more complicated cases, the results become correspondingly more complicated to derive and express. Hummer's cases II and III yield \[\begin{aligned}
\left\langle R_{\mathrm{II}}\left(x^{\prime}, x, \hat{n}^{\prime}, \hat{n}\right)\right\rangle= & \frac{g\left(\hat{n}^{\prime}, \hat{n}\right)}{\pi \sin \psi} H\left(a \sec \frac{\psi}{2}, \frac{x+x^{\prime}}{2} \sec \frac{\psi}{2}\right) \\
& \times \exp -\left(\frac{x-x^{\prime}}{\sqrt{2}} \csc \frac{\psi}{2}\right)^2
\end{aligned}\label{15.3.28}\]

and \[\begin{aligned}
\left\langle R_{\mathrm{III}}\left(x^{\prime}, x, \hat{n}^{\prime}, \hat{n}\right)\right\rangle= & \frac{g\left(\hat{n}^{\prime}, \hat{n}\right)}{\pi}\left(\frac{a \csc \psi}{\pi}\right) \\
& \times \int_{-\infty}^{+\infty} \frac{e^{-u^2} H[a \csc \psi,(x \csc \psi-u \cot \psi)] d u}{\left(x^{\prime}-u\right)^2+a^2}
\end{aligned}\label{15.3.29}\]

respectively. There is little point in giving the result for case IV as given by equation \ref{15.3.11}. But the result for the correct case IV (sometimes called case V) that is obtained from equation \ref{15.3.12} is of some interest and is given by McKenna⁸ as \[\begin{aligned}
\left\langle R_{\mathrm{IV}}\left(x^{\prime}, x, \hat{n}^{\prime}, \hat{n}\right)\right\rangle= & \frac{g\left(\hat{n}^{\prime}, \hat{n}\right) \gamma_u}{w^2 \pi^3} \int_{-\infty}^{+\infty}\left(\frac{e^{-u^2}}{\left(x^{\prime}-x-2 \beta u\right)^2+\left(\gamma_u / \pi\right)^2}\right. \\
& \times\left\{\frac{\gamma_u}{\alpha^4}\left[\frac{\left(\gamma_u+\gamma_1\right)^2}{\pi^2}+\left(x^{\prime}-x-2 \beta u\right)^2\right]\right. \\
& \times\Upsilon \left(\frac{\gamma_u+\gamma_1}{4 \pi}, \frac{x^{\prime}-\beta u}{\alpha}, \frac{\gamma_u+\gamma_1}{\alpha}, \frac{a+\beta u}{\alpha}\right)+\frac{\pi \gamma_l}{\alpha\left(\gamma_u+\gamma_l\right)} \\
& \left.\left.\times\left[H\left(\frac{\gamma_u+\gamma_1}{4 \pi \alpha}, \frac{x^{\prime}-\beta u}{\alpha}\right)+H\left(\frac{\gamma_u+\gamma_1}{4 \pi \alpha}, \frac{x-\beta u}{\alpha}\right)\right]\right\}\right) d u
\end{aligned}\label{15.3.30}\]

where \[\begin{aligned}
\Upsilon(a, x, b, y) & \equiv \int_{-\infty}^{+\infty} \frac{e^{-u^2} d u}{\left[(x-u)^2+a^2\right]\left[(y-u)^2+b^2\right]} \\
& =\frac{\pi}{2 a^3}\left[\left(1-2 a^2\right) H(a, x)-2 a x K(a, x)+\frac{2 a}{\sqrt{\pi}}\right]
\end{aligned}\label{15.3.31}\]

and the function \(K(a,x)\), which is known as the shifted Voigt function is defined by \[K(a, x) \equiv \frac{1}{\pi} \int_{-\infty}^{+\infty} \frac{(x-u) e^{-u^2} d u}{(x-u)^2+a^2}\label{15.3.32}\]

Unfortunately, all these redistribution functions contain the scattering angle ψ explicitly and so by themselves are difficult to use for the calculation of line profiles. Not only does the scattering angle appear in the part of the redistribution function resulting from the effects of the Doppler motion, but also the scattering angle is contained in the phase function \(g(\hat{\mathrm{n}}’,\hat{\mathrm{n}})\). Thus, the Doppler motion can be viewed as merely complicating the phase function. While there are methods for dealing with the angle dependence of the redistribution function (see McKenna⁹), they are difficult and beyond the present scope of this discussion. They are, however, of considerable importance to those interested in the state of polarization of the line radiation. For most cases, the phase function is assumed to be isotropic, and we may remove the angle dependence introduced by the Doppler motion by averaging the redistribution function over all angles, as we did with velocity. These averaged forms for the redistribution functions can then be inserted directly into the equation of radiative transfer. As long as the radiation field is nearly isotropic and the angular scattering dependence (phase function) is also isotropic, this approximation is quite accurate. However, always remember that it is indeed an approximation.

Angle-Averaged Redistribution Functions We should remember from Chapter 13 [equation \ref{13.2.14}], and the meaning of the redistribution function [see equation \ref{9.2.29}], that the equation of transfer for line radiation can be written as \[\mu \frac{d I_\nu}{d \tau_\nu}=I_\nu-\mathscr{L}_\nu B_\nu-\frac{\left(1-\mathscr{L}_\nu\right)}{4 \pi} \int_0^{\infty} \int_{4 \pi} I_{\nu^{\prime}}\left(\mu^{\prime}\right) R\left(\nu^{\prime}, \nu, \mu^{\prime}, \mu\right) d \omega^{\prime} d \nu^{\prime}\label{15.3.33}\]

Here the parameter \(\mathscr{L_\nu}\) is not to be considered constant with depth as it was for the Milne-Eddington atmosphere. If we assume that the radiation field is nearly isotropic, then we can integrate the equation of radiative transfer over \(\mu\) and write \[\frac{d F_\nu}{d \tau_\nu}=4\left(J_\nu-\mathscr{L}_\nu B_\nu\right)-2\left(1-\mathscr{L}_\nu\right) \int_0^{\infty} J_{\nu^{\prime}} \oint_{4 \pi} \oint_{4 \pi}\left\langle R\left(\nu^{\prime}, \nu, \mu^{\prime}, \mu\right)\right\rangle d \omega^{\prime} d \omega d \nu^{\prime}\label{15.3.34}\]

If we define the angle-averaged redistribution function as \[R_A\left(\nu^{\prime}, \nu\right) \equiv \frac{1}{(4 \pi)^2} \oint_{4 \pi} \oint_{4 \pi} R\left(\nu^{\prime}, \nu, \hat{n}^{\prime}, \hat{n}\right) d \Omega^{\prime} d \Omega\label{15.3.35}\]

then in terms of the absorption and reemission probabilities \(\mathrm{f}(\xi’)\) and \(\mathrm{p}(\xi’,\xi)\) it becomes \[\begin{aligned}
R_A\left(\nu^{\prime}, \nu\right) \equiv & \frac{1}{(4 \pi)^2} \int_0^{2 \pi} \int_0^{2 \pi} \int_{-1}^{+1} \int_{-1}^{+1} f\left(\nu^{\prime}-w u \mu^{\prime}\right) p\left(\nu^{\prime}-w u \mu^{\prime}, \nu-w u \mu\right) \\
& \times g\left(\hat{n}^{\prime}, \hat{n}\right) d \mu^{\prime} d \mu d \phi^{\prime} d \phi
\end{aligned}\label{15.3.36}\]

The phase function \(g(\hat{\mathrm{n}}’,\hat{\mathrm{n}})\) must be expressed in the coordinate frame of the observer, that is, in terms of the incoming and outgoing angles that the photon makes with the line of sight (see Figure 15.3).

Figure 15.3 describes the scattering event as seen in the coordinate frame of the observer. The \(\mathrm{\hat{k}}\) - axis points along the normal to the atmosphere or the observer's line-of-sight. The angle θ is the angle between the scattered photon and the observer's line-of-sight, while the angle θ' is the corresponding angle of the incoming photon. The quantities µ and µ' are just the cosines of these respective angles.

We may write the phase function \(g(\hat{\mathrm{n}}’,\hat{\mathrm{n}})\) as \[g\left(\mu^{\prime}, \mu\right)=\frac{1}{4 \pi} \int_0^{2 \pi} g\left(\mu^{\prime}, \mu, \phi^{\prime}\right) d \phi^{\prime}\label{15.3.37}\]

so that the angle-averaged redistribution function becomes \[R_A\left(\nu^{\prime}, \nu\right)=\frac{1}{2} \int_{-1}^{+1} f\left(\nu^{\prime}-w u \mu^{\prime}\right) \int_{-1}^{+1} p\left(\nu^{\prime}-w u \mu^{\prime}, \nu-w u \mu\right) g\left(\mu^{\prime}, \mu\right) d \mu d \mu^{\prime}\label{15.3.38}\]

The two most common types of phase functions are isotropic scattering and Rayleigh scattering. Although the latter occurs more frequently in nature, the former is used more often because of its simplicity. Evaluating these phase functions in terms of the observer's coordinate frame yields \[g_{\text {iso }}\left(\mu^{\prime}, \mu\right)=\frac{1}{2} \quad g_{\text {Ray }}\left(\mu^{\prime}, \mu\right)=\frac{3\left(3-\mu^2-\mu^{\prime 2}+3 \mu^2 \mu^{\prime 2}\right)}{16}\label{15.3.39}\]

In general, the appropriate procedure for calculating the angle-averaged redistribution functions involves carrying out the integrals in equation \ref{15.3.38} and then applying the effects of Doppler broadening so as to obtain a redistribution function for the four cases described by Hummer. For the first two cases, the delta function representing the upper and lower levels requires that some care be used in the evaluation of the integrals (see Mihalas⁴, pp. 422-433). In terms of the normalized frequency x, the results of all that algebra are, for case I \[\begin{aligned}
\left\langle R_{\mathrm{I}, A}\left(x^{\prime}, x\right)\right\rangle & =\frac{1}{\sqrt{\pi}} \int_{x_m}^{\infty} e^{-x^2} d x=\frac{1}{2} \operatorname{erfc}\left(x_m\right) \\
x_m & \equiv \operatorname{Max}\left(|x|,\left|x^{\prime}\right|\right)
\end{aligned}label{15.3.40}\]

For case II the result is somewhat more complicated where \[\begin{aligned}
\left\langle R_{\mathrm{II}, A}\left(x^{\prime}, x\right)\right\rangle & =\frac{1}{\pi^{3 / 2}} \int_{\left|x-x^{\prime}\right| / 2}^{\infty} e^{-u^2}\left(\operatorname{Tan}^{-1} \frac{\underline{x}+u}{a}-\operatorname{Tan}^{-1} \frac{\bar{x}-u}{a}\right) d u \\
\bar{x} & \equiv \operatorname{Max}\left(x, x^{\prime}\right) \quad \underline{x} \equiv \operatorname{Min}\left(x, x^{\prime}\right)
\end{aligned}\label{15.3.41}\]

while for case III it is more complex still: \[\begin{aligned}
\left\langle R_{\text {III,}A}\left(x^{\prime}, x\right)\right\rangle= & \frac{1}{\pi^{5 / 2}} \int_0^{\infty} e^{-u^2}\left(\operatorname{Tan}^{-1} \frac{x^{\prime}+u}{a}-\operatorname{Tan}^{-1} \frac{x^{\prime}-u}{a}\right) \\
& \times\left(\operatorname{Tan}^{-1} \frac{x+u}{a}-\operatorname{Tan}^{-1} \frac{x-u}{a}\right) d u
\end{aligned}\label{15.3.42}\]

Note that for all these cases the redistribution function is symmetric in x and x'. From equations \ref{15.2.1} through \ref{15.2.3}, it is clear that the angle-averaged redistribution functions will yield a complete redistribution profile in spite of the fact that case I is completely coherent.

To demonstrate the effect introduced by an anisotropic phase function, we give the results for redistribution by electrons. Although we have always considered electron scattering to be fully coherent in the atom's coordinate frame, the effect of Doppler motion can introduce frequency shifts that will broaden a spectral line. This is a negligible effect when we are calculating the flow of radiation in the continuum, but it can introduce significant broadening of spectral lines. If we assume that the scattering function for electrons is isotropic, then the appropriate angle-averaged redistribution function has the form \[\left\langle R_{e, A}\left(x^{\prime}, x\right)\right\rangle=\operatorname{ierfc}\left|\frac{x^{\prime}-x}{2}\right| \quad \operatorname{ierfc}(z) \equiv \pi^{-1 / 2} e^{-z^2}-z \operatorname{erfc}(z)\label{15.3.43}\]

However, the correct phase function for electron scattering is the Rayleigh phase function given in the observer's coordinate frame by the second of equations \ref{15.3.39}. The angle-averaged redistribution function for this case has been computed by Hummer and Mihalas¹⁰ and is \[\begin{aligned}
\left\langle R_{e, B}\left(x^{\prime}, x\right)\right\rangle & =\frac{\left(11+4 \beta^2+\frac{1}{2} \beta^4\right) e^{-\beta^2 / 4}-\frac{1}{2} \beta \sqrt{\pi}\left(15+5 \beta^2+\frac{1}{2} \beta^4\right) \operatorname{erfc}(\beta / 2)}{10 \sqrt{\pi}} \\
\beta & \equiv\left|\left(x^{\prime}-x\right) / 2\right|
\end{aligned}\label{15.3.44}\]

Clearly the use of the correct phase function causes a significant increase in the complexity of the angle-averaged redistribution function. Since the angle-averaged redistribution function itself represents an approximation requiring an isotropic radiation field, one cannot help but wonder if the effort is justified.

We must also remember that the entire discussion of the four Hummer cases relied on the absorption and reemission profiles being given by Lorentz profiles in the more complicated cases. While considerable effort has been put into calculating the Voigt functions and functions related to them that arise in the generation of the redistribution functions¹¹, some of the most interesting lines in stellar astrophysics are poorly described by Lorentz profiles. Perhaps the most notable example is the lines of hydrogen. At present, there is no quantitative representation of the redistribution function for any of the hydrogen lines. While noncoherent scattering is probably appropriate for the cores of these lines, it most certainly is not for the wings. Since a great deal of astrophysical information rests on matching theoretical line profiles of the Balmer lines to those of stars, greater effort should be made on the correct modeling of these lines, including the appropriate redistribution functions.

The situation is even worse when one tries to estimate the polarization to be expected within a spectral line. It is a common myth in astrophysics that the radiation in a spectral line should be locally unpolarized. Hence, the global observation of spectral lines should show no net polarization. While this is true for simple lines that result only from pure absorption, it is not true for lines that result from resonant scattering. The phase function for a line undergoing resonant scattering is essentially the same as that for electron scattering - the Rayleigh phase function. While noncoherent scattering processes will tend to destroy the polarization information, those parts of the line not subject to complete redistribution will produce strong local polarization. If the source of the radiation does not exhibit symmetry about the line of sight, then the sum of the local net polarization will not average to zero as seen by the observer. Thus there should be a very strong wavelength polarization through such a line which, while difficult to model, has the potential of placing very tight constraints on the nature of the source. Recently McKenna¹² has shown that this polarization, known to exist in the specific intensity profiles of the sun, can be successfully modeled by proper treatment of the redistribution function and a careful analysis of the transfer of polarized radiation. So it is clear that the opportunity is there remaining to be exploited. The existence of modern computers now makes this feasible.