Building a Model Atmosphere - Solving the Equation of Hydrostatic Equilibrium with Mathematica
Note: This exercise is for advanced students.

Note: What follows is a take home exercise used in 82-304 (Methods in Astrophysics) here at UW Oshkosh. I can provide solutions to interested parties!
(This will also be written up for publication at a later date)

Part #1

For part 1, you are asked to construct a model atmosphere approriate to an early B main-sequence star (Teff=25,000K) composed entirely of hydrogen. The equation of hydrostatic equilibrium:

dP/dt = g/k

(where P is pressure, t is Rosseland mean optical depth, g is acceleration due to gravity, in this case log g = 4.0, and k is the Rosseland mean opacity)

requires us to know k, which is a function of pressure and temperature. A Mathematica function appropriate for log k with X=1.0, Y=0.0, Z=0.0 and a variety of pressures and temperatures is avaiable here:

The last line of the above program essentially performs a two dimensional 5th order pollynomial fit to the log Rosseland mean opacity table opacityx10. Thus, references to fitx10[t6,Log[10,r6]] will return log(k).

What? I know that's what you are saying...The the log Rosseland mean opacity table gives log(k) as a function of temperature and density, not temperature and pressure. Moreover, the temperature is expressed in terms of t6, which is t/10^6, and the log of r6, where r6 is rho/t6^3 (yes, that is density in gm/cm^3 divided by t6^3). These are units which are convienient for stellar interior work.

However, this is not a problem because of our assumption of Local Thermodynamic Equilibrium - we have a relation between pressure, temperature and denisty (rho). The only trick part is the mean molecular weight, which I want you to assume to be constant and equal to the case of no ionization for these exercises (although you can imagine how to solve for mu as a function of temperature and pressure - which in this case is the total gas pressure).

The procedure for doing this problem is:

1. Express the Rosseland Mean Opacity as a function of temperature(optical depth) and pressure.
2. Define the realtionship betweem Mean Optical Depth and temperature.
3. Integrate the equation of Hydrostatic Equilibrium over a range in mean optical depth.

One last problem - the Rosseland Mean Opacity table does not extend to the case of P=0, so we have to limit how close we can get to the surface when solving the equations. In this case, an upper limit of tau = 0.001 is about as far as you want to go. This also gives us trouble when setting our boundary condition. Ideally, we'd want P[tau=0]==0 (i.e.) the surface pressure to be zero. Of course we can't do this because of the limits on our table. So, try a boundary considiton of P[tau=0.006]==10.

You should try different boundary conditions and see if these have a large effect on the model structure. Do they?

Once you have solved the equation of Hydrostatic Equilibrium, plot pressure as a function of depth from the surface to tau=10.

Part #2

Now that we have P[tau], and k[P,T(tau)], we are in a position to find x as a function of optical depth. We can get this simply from the definition of mean optical depth:

dx/dt = -1/(k rho)

(where x is physical depth, t is mean optical depth, k is the Rosseland Mean Opacity, and rho is the denisty)

From Part 1, we know what all of these are, i.e. we have kappa[P,t] and rho[P,t], so all we need to do is integrate the above equation.

Part #3

Part #4

Part #5

Redo parts 1 & 2 for the case of a star with the above composition. How is this star different from the case of pure H?