1. Stellar structure equations
Lecture 9
Stellar structure equations

2. Review: The Eddington approximation
This is the Eddington-Barbier relation: the surface flux is determined by the value of the source function at a vertical optical depth of 2/3

3. The Eddington Approximation
Approximation #5: Local thermodynamic equilibrium
In this case the source function is equal to the blackbody function.

4. The Eddington Approximation
Approximation #5: Local thermodynamic equilibrium
In this case the source function is equal to the blackbody function.
Recall: So

5. The Eddington Approximation
Note that T=Te when tv=2/3
Thus the photons we see (that give us the luminosity we use to define the effective temperature) originate at an optical depth of 2/3, not 0. Te

6. Limb darkening
The solar disk is darker at the edge (limb) than at the centre
The light rays that we see from the edge of the Sun must originate from higher in the atmosphere (since otherwise they would have to travel through a greater optical depth to reach us).

7. Limb darkening
Now we will again assume a plane-parallel atmosphere, but we will not assume a gray atmosphere, LTE, or the Eddington approximation
We need to find the intensity as a function of angle, q
Recall the definition of the vertical optical depth:
So we can rewrite the general solution as:

8. Limb darkening
We can compare this with observations, and the agreement is pretty good. This doesn’t prove our numerous assumptions are correct, but does show that they do not produce a result that is in strong conflict with the data.

9. Break

10. Hydrostatic equilibrium
The force of gravity is always directed toward the centre of the star. Why does it not collapse?
The opposing force is the gas pressure. As the star collapses, the pressure increases, pushing the gas back out.
How must pressure vary with depth to remain in equilibrium?

11. Hydrostatic equilibrium
Consider a small cylinder at distance r from the centre of a spherical star.
Pressure acts on both the top and bottom of the cylinder.
By symmetry the pressure on the sides cancels out dr A dm
FP,b
FP,t

12. Hydrostatic equilibrium
If we now assume the gas is static, the acceleration must be zero. This gives us the equation of hydrostatic equilibrium (HSE).
It is the pressure gradient that supports the star against gravity
The derivative is always negative. Pressure must get stronger toward the centre

13. Mass Conservation
The second fundamental equation of stellar structure is a simple one relating the enclosed mass to the density.
Consider a shell of mass dMr and thickness dr, in a spherically symmetric star
Rearranging we get the equation of mass conservation

14. Example Imagine a sphere of gas with density profile
What is the total mass and average density within radius r?

15. Example
Make a crude estimate of the central solar pressure, assuming the density is constant.
This is a big underestimate because the density increases strongly toward the centre.
The accepted value is

16. Pressure equation of state
We now need to assume something about the source of pressure in the star.
We require an equation of state to relate the pressure to macroscopic properties of the gas (i.e. temperature and density)
Consider the ideal gas approximation:
Gas is composed of point particles, each of mass m, that interact only through perfectly elastic collisions
Note the particle mass does not enter into this equation
The momentum of each collision depends on mass, but lighter particles are moving faster in a way that exactly cancels out
Thus, tiny electrons contribute as much to the pressure as massive protons

17. Mean molecular weight
We want to relate the particle number density to the mass density of the gas.
The two quantities are related by the average particle mass:
Define the mean molecular weight:
i.e. this is the average mass of a free particle, in units of the mass of hydrogen
So we can express the ideal gas law in terms of density:

18. Mean molecular weight
The mean molecular weight is an important quantity, because the pressure support against gravity depends on the number of free particles
Sudden changes in the ionization state or chemical composition of the star can lead to sudden changes in the pressure.
In general, the value of m requires solving the Saha equation to determine the ionization state of every atom.
We can derive two useful expressions for the cases of fully neutral or fully ionized gasses:
Define: X,Y,Z are the mass fractions of H, He and metals, respectively.
Neutral:
Ionized:
For solar abundances,

19. Example
By how much does the pressure increase following complete ionization, for a neutral gas with the following composition (typical of young stars):

20. Radiation pressure The equation of state then becomes
We earlier derived an expression for the radiation pressure of a blackbody:
The equation of state then becomes
In a standard solar model, the central density and temperature are
Calculate the gas pressure and radiation pressure. Assume complete ionization, so m=0.62.

21. Non-ideal gases
The ideal gas law neglects both special relativity and quantum mechanics. It therefore breaks down at high velocities (temperatures) and at high densities.
The Fermi-Dirac distribution is a modification of the Maxwell-Boltzmann distribution, accounting for the Heisenberg uncertainty principle and the Pauli exclusion principle
Essentially, no two fermions (e.g. electrons and protons) can occupy the same quantum state. This provides an extra source of pressure when densities get high

22. Non-ideal gases
The ideal gas law neglects both special relativity and quantum mechanics. It therefore breaks down at high velocities (temperatures) and at high densities.
The Bose-Einstein distribution function applies to bosons (such as photons). In this case, the presence of a particle in a quantum state enhances the probability that another particle will occupy the same state
Both the Fermi-Dirac and Bose-Einstein distribution functions approach the Maxwell-Boltzmann distribution at high energies