Blog #15 - Radiative Energy Transport

Radiative Energy Transport 

Stars generate energy in their cores, where nuclear fusion is taking place. The energy generated is eventually radiated out at the star's surface. As a result, there is a gradient in energy density from the center (high) to the surface (low). However, thermodynamic systems tend towards "equilibrium". We therefore need to determine how energy flows through the star.

a) Inside the star, consider a mass shell of width \(\Delta r\) and radius r. This mass shell has an energy density \(u + \Delta u \), and the next mass shell out at radius \(r + \Delta r\) will have an energy density u. Both shells can be thought to behave as blackbodies. The net outwards flow of energy, L(r), must equal the total excess energy in the inner shell divided by the amount of time needed to cross the shell's width \(\Delta r\). Use this to derive an expression for L(r) in terms of the energy density profile \(\frac{du}{dr}\). This is the diffusion equation describing the outward flow of energy.

We know that the outward flow of energy L(r) has units of energy/time so we begin with equation \[L(r) = \frac{E}{\tau} \]We can then solve for E as the energy density over volume or \[E = du*V\] But the to find the change in energy density over the change in radius between the two shells \(\frac{du}{dr}\) comes out to an energy difference of \(- \Delta u\) and a radius difference of \(\Delta r\). This gives us an energy equation of \[E = du*V = - \Delta u * (4 \pi r^2) * \Delta r \] Using this E equation in our L(r) formula we get \[L(r) = \frac{E}{\tau} = \frac{- \Delta u * (4 \pi r^2) * \Delta r}{\frac{\Delta r^2 k \rho}{c}}\]The equation for \(\tau\) comes from the random walk equation, where c is the speed of light, \(\rho\) the density of the star, k an optical density. Simplifying this equation we get \[L(r) = \frac{- \Delta u}{\Delta r} * \frac{4 \pi r^2 c}{\rho c}\] Which gives us our final differential form \[L(r) = \frac{-4 \pi r^2 c}{\rho c} \frac{du}{dr}\]

b) From the diffusion equation, use the fact that the energy density of a blackbody is \(u[T(r)] = a T^4 \) to derive the equation for Radiative Energy Transport.

We begin by solving for the \(\frac{du}{dr}\) of the given energy density by taking a derivative of u with respect to r. \[\frac{du}{dr} = \frac{d}{dr}(u[T(r)]) = 4 a T(r)^3 \frac{dT}{dr} \] With an equation for \(\frac{du}{dr} \) We plug into the differential from part (a) and get \[L(r) = \frac{-4 \pi r^2 c}{\rho c} \frac{du}{dr} = \frac{-16 \pi r^2 c a T(r)^3}{\rho k} \frac{dT}{dr} \] Which rearranges to the expected equation for radiative energy transport \[\frac{dT}{dr} = \frac{-L \rho k}{16 \pi r^2 c a T(r)^3}\]