Radiative Transport in Stellar Interiors

[The following discussion is taken from "Stellar Structure and Evolution" by R.Kippenhahn, A. Weigert, 1990]

Here we address the problem on how stars like our sun transport energy to their surfaces where it is radiated away. We know that energy is generated in a star's core primarily by means of fusing hydrogen nuclei into helium nuclei. This energy, mostly in the form of xrays must be transported out to the surface and radiated away to account for the 1033 ergs/s of energy generated by an average star like our sun. In explaining this process we will ultimately derive the equation for radiative transport given by:

-T = (-3kpL(r))/(16pacr2T3)

[Equation I.I.1]

Here we will only consider sphereically symmetric stars in static state. In which case

- = d/dr

I will also restrict the disscussion to deep stellar interiors near the core. Therefore the radiative transport of energy is due primarily to the net outward radiation flux from hot material near the core to cooler material futherout in radius. For this reason and one other reason which I will explain shortly we will treat the energy transport as a diffusion process with the particles being photons of light. In the stellar interior photons undergo what is known as a random walk. This is a process by which photons leaving the core are either scattered, or absorbed and then emitted by atoms in a random direction. The average distance that a photon will travel before a scattering or absorption/emission event is called the mean free path. At an average point inside the star the mean free path for a photon is

lph = 1/kr

[Equation I.I.2]

Where k is the absorption coefficient which ranges in value from k = 1 cm2g-1 to k = 0.4 cm2g-1. Taking the mean density of the sun to be r = 1.4 gcm-3 we obtain an approximate value for the mean free path of a photon.

lph = 2 cm

The fact that the ratio of the mean free path to the radius of a typical star is on the order of 10-11 also allows us to treat the energy transport, to a good approximation, as a diffusion process.

Now consider the diffusive flux equation. The diffusive flux is proportional to the gradient of the particle density n. This is written as,

j = -D(dn/dr)

[Equation I.I.3]

Where n is the particle density of photons and D is the coefficient of diffusion. D is proportional to the average velocity of the photons vp and the mean free path lph.

In other words the direction of the diffusive flux of particles is that in which the gradient of the particle density decreases. Which is the same as diffusion from an area of high concentration of particles to an area of lower concentration of particles.

D = (1/3)vplph

[Equation I.I.4

In order to consider energy transport we must express the particle density in terms of energy. Consider the the energy of the photons at a certain temperature. Then n becomes

U = aT4

[Equation I.I.5]

Where U is the energy density of radiation and a = 7.57 x 10-15 erg cm-3K-4 is the radiation density constant.

Taking the gradient of[I.I.5] we obtain

dU/dr = 4aT3dT/dr

[Equation I.I.6]

Now substituting [I.I.6] and [I.I.4] into [I.I.3] and replacing vp with c we obtain

F = -(4ac/3)(T3/kr)(dT/dr)

[Equation I.I.7]

Which is the flux of radiative energy. In other words it is the amount of photons leaving the core. It depends primarily on the temperature of a fluid at some radius r and the temperature gradient between that fluid and the fluid at some larger radius.

We now define a fundamental quantity called the luminosity which is the total amount of radiation energy flux F crossing the area of a sphere of radius r.

L(r) = 4pr2F

[Equation I.I.8]

Now substituting and rearranging we can see that we finnally obtain

dT/dr = -(3krL(r))/(16par2T3)

Which we see is the one dimensional version of [I.I.1].