Worksheet 11.2, Problem 1: Flow of Energy

The problem: Stars generate their energy in their cores, where nuclear fusion is taking place. The energy generated is eventually radiated out at the star’s surface. Therefore, there exists a gradient in energy density from the center (high) to the surface (low), but thermodynamic systems tend towards ‘equilibrium.’ In the following sections we will determine how energy flows through the star.

Part A: Inside the star, consider a mass shell of width ∆r, at a radius r. This mass shell has an energy density u + ∆u, and the next mass shell out (at radius r + ∆r) will have an energy density u. Both shells 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 ∆r. Use this to derive an expression for L(r) in terms of du/dr , the energy density profile. This is the diffusion equation describing the outward flow of energy.

Once again, an image from the textbook can help us visualize the situation.

Source

If we keep in mind the answer needs to be in terms of du/dr we can solve: $$L(r) = \frac{dE}{dt} = \frac{dE}{du}\frac{du}{dt} = \frac{dE}{dt}\frac{du}{dr}\frac{dr}{dt}$$ We can simplify this because you know $$\frac{dE}{du} = 4 \pi r^2 dr$$ and $$\frac{dr}{dt} = v$$ and $$v = \frac{c}{\rho \kappa \Delta r}$$ $$L(r) = 4 \pi r^2 \frac{c}{\rho \kappa} \frac{du}{dr}$$

Part B: From the diffusion equation, use the fact that the energy density of a blackbody is u(T(r)) = aT^4 to derive the differential equation: $$\frac{dT(r)}{ dr} \propto - \frac{ L(r) \kappa \rho (r)}{ πr^2acT^3}$$ where a is the radiation constant. You just derived the equation for radiative energy transport!

$$u(T(r)) = aT^4$$ $$du(T(r)) = 4aT^3 d(T(r))$$ From part a: $$du = \frac{L(r) \rho \kappa dr}{4 \pi r^2 c }$$ So $$\frac{L(r) \rho \kappa dr}{4 \pi r^2 c } =  4aT^3 d(T(r))$$ Which simplifies to our answer!   $$\frac{dT(r)}{ dr} \propto - \frac{ L(r) \kappa \rho (r)}{ πr^2acT^3}$$

Acknowledgements: I worked on this worksheet with Sean, April, and Barra.