Stellar Interiors 

Remember what controls the interior of a star: So far we have talked about how these things work inside the Sun; now we generalize and look at how they work in different stars.


Hydrostatic Equilibrium

Remember the equation of hydrostatic equilibrium:
where P is pressure, rho is density, M is mass, g is the gravitational acceleration, G is the gravitational constant, and r is the radial distance from the center of the star.
 

Equation of State

For stars on the main sequence, the equation of state is that of an ideal gas. We have typically written the ideal gas law like this, assuming hydrogen gas:
But we will not assume a pure hydrogen gas any longer! The gas has some chemical composition, so the equation of state becomes
Where mbar is the average mass of the particles (atoms, ions, or molecules) in the gas.

Let's introduce a quantity called the mean molecular weight (mu), defined as

which then lets us write the equation of state as

Question: What is the mean molecular weight of

Okay, what is the mean molecular weight of the gas inside the sun?


The chemical composition of the Sun

We characterize the composition of the Sun (and other stars) in terms of the mass fraction of different elements: Z is called the metallicity of the star, and X+Y+Z=1
 
 
Chemical Composition of the Sun's Atmosphere
Element
Atomic Number
Log(Abundance)
Hydrogen
1
12.00
Helium
2
10.99
Oxygen
8
8.93
Carbon
6
8.60
Neon
10
8.09
Nitrogen
7
8.00
Iron
26
7.67
Magnesium
12
7.58
Silicon
14
7.55
Sulfur
16
7.21
Adding this all up, for the Sun,
X=0.7     Y=0.28     Z=0.02

From X, Y, and Z, you can calculate the mean molecular weight. For the Sun, we have

So what? Big deal.

Well, what is happening inside the center of the sun? How is this changing mu? How does this change the central pressure under the ideal gas law?


 Radiation Pressure

Atoms and ions aren't the only thing running around in the center of stars. There are also plenty of photons, and they exert a pressure as well.

Remember that photons carry momentum

So that they can give momentum to particles which absorb them. Integrating over the Planck function, we can get an expression for the pressure:

So that our total equation of state is

For the sun, radiation pressure is negligible. For other stars it is not (which kinds of stars?).