4.3: Radiative Transport and the Radiative Temperature Gradient

Although all forms of energy transport may be present at any given place in a star, we will see that their relative efficiency is such that generally only one form will be important for describing the flow of energy. The transport of energy by radiation is essentially the radiative diffusion of photons through the stellar material. It is the opacity of the material that opposes this flow. To establish the interplay between thermodynamics and radiative opacity, we assume that all the energy is flowing by this process.

a. Radiative Equilibrium

Since we are assuming that all the energy is flowing outward by means of radiative diffusion, the entire energy produced by the star within a sphere of radius r can be characterized by a local luminosity L(r) which is entirely made up of photons. When this is the case, we may describe this flow of photons locally by defining the radiative flux as \[\mathrm{F(r)=L(r) / 4 \pi r^2}\label{4.2.1}\]

When these conditions prevail, the entire flow of energy is carried by photons and the star is said to be in radiative equilibrium.

b. Thermodynamic Equilibrium and Net Flux

In Chapter 1 we developed an elegant formalism to describe the flow of particles through space. In a later chapter we shall use this to produce an extremely general equation of radiative transfer which describes the flow in momentum space as well as physical space. But at this point, we are dealing with a gas in STE, and that fixes many properties of the gas. For example, we know that the phase density f that appears in the Boltzmann transport equation will be the Planck function since we have shown that to be the equilibrium distribution function for photons in STE. We also know that while there is a net flow of photons, the energy involved in that flow must be small compared to the local energy density; otherwise, the photon gas could not be considered in equilibrium.

Another way of visualizing this is to observe that any system said to be in thermodynamic equilibrium cannot have temperature gradients. If it did, there would be a flow of energy driven by the temperature gradient. In a star we must have such a flow, or the star will not shine. What is important is the relative size of the temperature radiant through some volume for which the system is to be considered in equilibrium. In the case of the sun, this typical length would be the distance a photon travels before it encounters an atom. From the opacity calculations of Chapter 3 and our knowledge of the conditions within the sun, we would calculate that the mean free path for a photon in the center of the sun is less than a centimeter. Thus, as a measure of the extent to which STE is met in the sun, let us calculate \[\left(\frac{\nabla T}{T}\right) l \approx\left(\frac{T_c / R}{T_c}\right) l \approx \frac{l}{R} \approx 10^{-11}\label{4.2.2}\]

In other words, the change in the local temperature over a scale length appropriate for the photon gas is about 1 part in 10¹¹. There are few gaseous structures in the universe where the conditions for STE are met better than this. Small as this relative temperature gradient is, it drives the luminous flux of the sun, and so we must estimate its dependence on the state variables.

c. Photon Transport and the Radiative Gradient

Since we know so much about the nature of the photons in the star, we need not resort to the basic Boltzmann transport equation in order to describe how photons flow. Instead, consider the Euler-Lagrange equations of hydrodynamic flow. Since they were derived under fairly general conditions, they should be adequate to describe the flow of photons. Equation \ref{1.2.27} provides a reasonably simple description of this process. But we are interested in a steady-state description, so all explicit time dependence in that equation must vanish. Thus, equation \ref{1.2.27} becomes \[(\vec{u} \cdot \nabla) \vec{u}=-\nabla \Phi-\left(\frac{1}{\rho}\right) \nabla P\label{4.2.3}\]

However, in deriving equation \ref{1.2.27}, we averaged the local particle phase density over velocity space. For photons traveling at the velocity of light, this does not make much sense. Instead, the moment generation which led to the Euler-Lagrange equations of hydrodynamic flow should be carried out over momentum space, or photon frequency. Since this expression is for photons, P is the local radiation pressure due to photons and \(\vec{\mathrm{u}}\) is the mean flow velocity, or diffusion velocity of those photons. However, in equation \ref{1.2.27}, p is the local mass density. For photons, this translates to the local energy density. In addition, the influence of gravity on the photons throughout the star can be estimated by the gravitational red shift that photons will experience which is \[\frac{\Delta h \nu}{h \nu} \approx \frac{G M}{R c^2} \approx 10^{-6}\label{4.2.4}\]

Since the change in the photon energy resulting from moving through the gravitational potential is about 1 part in a million, we may safely neglect the influence of \(\nabla \Phi\). In spherical coordinates, all spatial operators in equation \ref{4.2.3} simply become derivatives with respect to the radial coordinate, so that equation \ref{4.2.3} becomes \[\vec{u}_r \cdot\left(\frac{d \vec{u}_r}{d r}\right) \hat{r}=-\left(\frac{1}{\rho_e}\right) \nabla P_r\label{4.2.5}\]

where \[\begin{aligned}
& \rho_e=\langle h \nu\rangle \int_0^{\infty} f(p) \frac{d \vec{p}}{c^2} \\
& \vec{u}_r=\left(\frac{\langle h \nu\rangle}{c^2}\right)^{-1} \frac{\int_0^{\infty} \vec{p} f(p) d \vec{p}}{\int_0^{\infty} f(p) d \vec{p}}
\end{aligned}\label{4.2.6}\]

Here \(\langle\mathrm{h\nu}\rangle\) is the average photon energy, and \(\mathrm{d\vec{p}}\) indicates integration over all momentum coordinates which, in the absence of a strong potential gradient, can be represented by the differential spherical momentum volume \(\mathrm{4\pi p^2dp}\). We can then write equation \ref{4.2.5} as \[\nabla P_r=-\left(\int_0^{\infty} \vec{p} f(p) d \vec{p}\right) \cdot\left[\frac{d}{d r} \int_0^{\infty} \frac{\vec{p} f(p) d \vec{p}}{\rho_e}\right]\label{4.2.7}\]

The first term on the right-hand side represents the net flow of momentum and can be related to the flow of radiant energy by \[\int_0^{\infty} \vec{p} f(p) d \vec{p}=\frac{1}{c} \int_0^{\infty} h \nu f(p) \hat{r}\left(4 \pi p^2\right) d p=\frac{\vec{F}(r)}{c}\label{4.2.8}\]

and equation \ref{4.2.7} becomes \[\nabla P_r=\frac{-\vec{F}(r)}{c} \cdot\left\{\frac{d}{d r}\left[\frac{F(r)}{\rho_e c}\right]\right\}\label{4.2.9}\]

The quantity \(\left[F(\mathrm{r}) / \mathrm{c} \rho_{\mathrm{e}}\right]\) is the fraction of photons which are participating in the net flow of energy. Thus the radial derivative represents the change in the fraction with \(\mathrm{r}\). The only reason for this fraction to change is the interaction of the flowing photons with matter. If we define the volume absorption coefficient \(\mathrm{\alpha_v}\), to be the "collision" cross section per unit volume, then the probability per unit length that a photon will be absorbed in passing through that volume is just \(\mathrm{\alpha_v}\). However, the probability that one photon will be absorbed per unit length is equal to the fraction of \(n\) photons that will be absorbed in that same unit length. Thus, the second term on the right hand side of equation \ref{4.2.9} becomes \[\frac{d}{d r}\left[\frac{F(r) / c}{\rho_e}\right]=\bar{\alpha}_{\nu}=\bar{\kappa} \rho\label{4.2.10}\]

The radiation pressure gradient is now \[\nabla P_r=-\frac{\bar{\kappa} \rho F(r)}{c}=\frac{-\bar{\kappa} \rho L(r)}{4 \pi c r^2}\label{4.2.11}\]

But in STE the radiation pressure depends on only a single parameter and is given by \[P_r=\frac{a T^4}{3}\label{4.2.12}\]

This implies we can write the radiation pressure gradient in terms of the temperature as \[\frac{d P_r}{d r}=\frac{4}{3} a T^3 \frac{d T}{d r}\label{4.2.13}\]

Equating this to the magnitude of the radiation pressure gradient from equation \ref{4.2.11}, we finally obtain an expression for the radiative temperature gradient: \[\frac{d T}{d r}=-\frac{3 \bar{\kappa} \rho L(r)}{16 \pi a c r^2 T^3}\label{4.2.14}\]

This relationship specifies how the temperature must change if the energy is being carried by radiative diffusion and the specification is made in terms of the state variables and parameters that we have already determined characterize the problem.

d. Conservation of Energy and the Luminosity

With the advent of the radiant flux F(r), we have introduced a new variable into the problem. Relating the flux to the total luminosity [equation \ref{4.2.1}] only transfers the source of the problem to the luminosity L(r). That such a parameter is important should surprise no one, for the luminosity of a star is perhaps its most obvious characteristic. However, it is only with the transport of energy that we are faced with the internal energy, stored or produced, arriving at the surface and leaking into space. As far as the structure of the star is concerned, this is a "second-order" effect. It is only a small part of the internal energy that is lost during a dynamical time interval. However, for the proper understanding of the star as an object in steady state, it is a central condition which must be met, for in steady state the energy lost must be matched by the energy produced.

Fortunately, we have an additional fundamental constraint that must be met by any physical system which we have not yet imposed - the conservation of energy. This is completely analogous to the conservation of mass which we invoked in Chapter 2 [equations \ref{2.1.7} and \ref{2.1.8}] only now it is the total energy interior to r which must pass through r per unit time is called L(r). Thus, \[L(r)=\int_0^r 4 \pi r^2 \rho \epsilon d r\label{4.2.15}\]

The corresponding differential form is \[\frac{d L(r)}{d r}=4 \pi r^2 \rho \epsilon\label{4.2.16}\]