Astronomy 421 -- Stellar Interior Modeling

I. Introduction

When working with our model of the opacities in the atmosphere of a solar-type star, we were caught up in the details of just getting the program to run. The understanding of how this small part fit into a total model atmosphere followed. With our modeling of the interior of a star, we will accept a program (actually, a package of a great number of subroutines in FORTRAN) and work with it to generate Zero-Age-Main Sequence stars (ZAMS) with a range of stellar masses. We will be examining various properties such as temperature, density, convection with respect to the mass at a given radius. With this information in hand we should be able to better understand what is going on in the interiors of stars.

The program we will be using is one that comes with the book Stellar Interiors Physical Principles, Structure, and Evolution by Hansen, Kawaler, and Trimble, 2004 (HKT). Chapter 2, Section 2.15 and Chapter 7 contain background information for the modeling. This text is directed towards upper-level undergraduate and first-year graduate students; however, to fully understand what considerations were made within the model would take a fair amount of research and studying, and so is not required here. However, we will be summarizing the material that is critical for this project.

We will first get the program set up and running for practice. There are a few considerations that you must take into account, otherwise when you are finished (or so you think) with all of your models, you may find that all you have done is clobber previous files. We will be taking a look at the main FORTRAN program to get some insight as to how it works; there are some questions to help you along with this task.

After you’ve had a chance to examine the results from the model for a range in stellar masses, you will be answering a number of questions concerning the equations of stellar structure, assumptions made in the model, convection, numerical modeling, properties of polytropes, some of the integration routines, what is meant by the “equation of state,” and how opacities enter into a model of a stellar interior.

II. Getting the FORTRAN program set up and running and examining results

A. Copy over the necessary files and preliminary practice

You should probably set up a dedicated directory for this project. Do so and then copy from

In the directory where you saved the files, compile the program at the Linux prompt: g77 -o bzams3 bzams3.f

This will give you an executable file called bzams3. Go ahead and run the program by typing bzams3 at the Linux prompt. Take a look at the bzams.out file produced from the run.

As mentioned above, if you do not have your Linux environment set so that files are not clobbered, it is easy within FORTRAN to do so.The simple c-shell script, bzams.csh, moves each output file to one having a unique number related to the mass of the model you are running. It also prompts you to find out if you've appropriately changed the parameter file to reflect the new model. Take a look at the contents of bzams.csh. Make it executable by typing

$ chmod +x bzams.csh.

Give it a try, adding a number representing the star's mass in the process by typing

bzams.csh 3

B. Getting acquainted with input and output files

Within the directory /astro/net/projects/Astro_421/BZAMS3/ you will also see a large list of files representing the input parameters and output models for stellar masses 0.5, 1, 2, 3, 5, 10, and 20 Msun. Rather than have you all spend time rerunning these programs and producing graphs using SM, IDL, or Excel etc., I went ahead and produced a set of models for a range of stellar masses for us all to use. I also graphed various quantities against the mass mesh points based on the output; we’ll be discussing those graphs.

Take a look at the files bzamsum.1, bzams.par.1, and bzams.1 (ignore the bzams.puls and first.mod files).

bzamsum.1

The quantities represent M/Msun, log10(L/Lsun), log10(teff), log10(R/Rsun), logs of central pressure, central temp, and central density; fractional mass where convective instability exists, another measure of convection, X, Z, and alpha – the mixing length parameter.

bzams.par.1

There is a table at the bottom of the original bzams.par file with workable values. Getting convergence with this program is extremely sensitive to the input values. If we were working on this for a PhD thesis, we would have to include a program that would iterate over a range of starting values so that we weren’t just firing randomly.

bzams.1

Carroll and Ostlie (2007) derive all of the fundamental differential equations of stellar structure in Ch. 10, and summarize them on p. 330 (eqns. 10.6, 10.7, and 10.36):

; ; .

We consider only the total energy released per gram per second by all nuclear reactions: . You should definitely review these equations and their derivations.

A. Transport of Energy

The transport of energy can be by radiation or by convection. The temperature gradient for radiative transport is

. (Eqn. 10.68 of C&O)

Examine this equation closely: if the flux, opacity, or density increases, or if the temperature decreases, the temperature gradient must become steeper (more negative) if radiation is to transport all of the required energy outward. If the temperature gradient becomes too steep, the region becomes unstable to convection, and convective transport sets in to efficiently transport the energy. Here is the reason you studied thermodynamics. (This information is also important for our study of variable stars later in the quarter.) Follow through on your reading of Sec. 10.4 of C&O. On page 354 they discuss the parameter γ, which is defined to be the ratio of specific heats at constant pressure and volume, or

. (Eqn. 10.80)

The parameter γ = 5/3 for a monatomic ideal gas. The temperature gradient for adiabatic convection is expressed as

. (Eqn. 10.89)

For convection to occur, assuming that the convective temperature gradient is purely adiabatic,

. (Eqn. 10.95)

In order to interpret the convective zones of the stellar interior models from BZAMS, we must define a few more quantities. The reciprocal of the above (Eqn. 10.95) derivative is usually defined as a quantity, ∇, called “del.” The implication is that ∇ represents the actual run, or logarithmic slope, of local temperature versus pressure in the star (HKT, p. 201).

(Eqns. 4.28, 7.11 of HKT)

(Eqn. 4.30 of HKT)

In these equations, a is the radiation constant, c the speed of light, and k is a measure of the opacity. We define a third “del” and that is ∇_ad (“delad”):

. (Eqn. 3.94 of HKT)

These definitions are hardly satisfying with such limited explanation. Chapters 3, 4, 5, and 7 of HKT go into much greater detail, and we will be discussing convection to a greater depth next week.

Given the above definition of the “dels,” we can set criteria for what the modes of heat transfer are. If we go with the mixing-length theory (Böhm-Vitense, 1958) that allows only adiabatic convection, we test various conditions. We set ∇ = ∇_rad if ∇_rad ≤ ∇_ad for pure radiative transfer or conduction.

When adiabatic convection is present locally, we set ∇ = ∇_ad if ∇_rad ≥ ∇_ad. The test for convective instability is checking whether ∇ > ∇_ad in the interior model. From HKT:

The criteria (5.8-5.10) are also equivalent to the statement that if entropy decreases outward at some point (dS/dr < 0), then the fluid is convectively unstable. Put another way, convection does not take place in hydrostatic stars where the entropy increases outward. It will turn out that in regions where convection is very efficient, ∇ is only very slightly greater than ∇_ad. In such regions the entropy is very nearly constant with height.

This added information on convection is include because C & O does not go into as great an extent as HKT as to what these gradients mean and the tests that the BZAMS program makes.

Assignment (answer on a separate sheet)

Part 1 (30 pts): due date -______________

1. <2> Do the set up of the FORTRAN program in one of your subdirectories, run the program. Try the c- shell script as well. Show your instructor the results and have her initial here: _________

2. The quantity mean molecular weight is used throughout Ch. 10 of C & O, and must enter into any interior code in order to calculate the gas pressure at a given density and temperature and composition.

Relevant equations from Carroll & Ostlie

(a) <1> Using Eqns. 10.14 & 10.16 of C & O, confirm that for X = 0.70, Y = 0.28, and Z = 0.02, μ_n = 1.30 and μ_i = 0.62.

(b) <3> One explanation for a star expanding after it finishes fusing H to He in its core is the presence of a discontinuity in the mean molecular weight between the core and the overlying radiative zone. Could this work? Here’s a simple (also known as “back of the envelope”) way to estimate the effect on the pressure in these two zones, using Eqn. 10.11 and 10.14 & 10.16 of C&O, and assuming the gas is completely ionized:

Zone	X	Y	Z	ρ	T
Core (helium)	0.10	0.88	0.02	150,000 kg/m³	1.36 × 107
Surrounding zone	0.70	0.28	0.02	10,000 kg/m³	8.0 × 106

Assuming that the discontinuity is abrupt, what is the ratio of the gas pressure in the helium core to the gas pressure just outside the core? Comment.

(c) <1> The mean molecular weight depends on the composition of the gas as well as on the state of ionization of each species. How would you go about calculating the “state of ionization of each species”?

3. <5> Define or explain the following in your own words:

entropy
equations of state
boundary conditions
Vogt-Russell theorem
polytrope

4. <3> Express these equations of stellar structure:

, , and

in terms of dM_r:

.

[NOTE: The graphs all have the various quantities expressed as a function of m/M that is, in terms of a fractional mass of M, the total mass of the star. [Two of the boundary conditions are: (Mr = 0) at the center, and (Mr=M) at the surface.]

Understanding the basics of the model

We very briefly summarized the nuances of programming in FORTRAN in class. As you go through the code “bzams3.f” find the following calls to other subroutines. You will need the full code to answer the corresponding questions. [NOTE: Sometimes the program uses all caps, sometimes all lower letters. When “lessing” a file and searching, this is a nuisance. Example was RKCK and rkck. Or, even, SUBROUTINE versus subroutine.]

5. <2> There is a subroutine called NEWT that is a convergence program. In a few lines, summarize how this program works.

6. <2> Find the subroutine LOAD2. This one calls other subroutines – EOS, XMASS, OPACITY, AND BNUKE. What assumptions are in EOS? What nuclear reactions are considered in BNUKE?

7. <2> Find the subroutine FILHC, which does a lot. It works with a number of arrays. Summarize very briefly your interpretation of what it does.

8. <4> The subroutine rkck uses an integrating program of a class “Runge-Kutta” integrators. You can even Google “runge-kutta” and find out how they work. Explain basically how a Runge-Kutta integrator works. You may also choose to reference Ch. 7 of HKT.

9. <5> Read through HKT, Sec. 7.1 through 7.2.5. Summarize the code in terms of is assumptions and approximations and its range of applicability. Where is skepticism warranted? What would be needed to get more realistic models?

Part 2 (45 pts): Write up and answers due - _____________

10. <10 pts> Except for the convective instability graphs, examine each graph and state what is being plotted and in what form. Quantitatively compare the stars whose masses are 2 M_sun and greater to the 1 and 0.5 Msun models. How do the central temperature, density, energy generation rate, and effective temperature change according to the mass of the star? Interpret these results. For now, what struck you as being expected and unexpected in each graph?

11. <5 pts> Examine the different temperature gradients and the graphs of “delad” and “del” versus “mass” and explain how to identify the convective zones in the models. Use the information given in the output files as support.

12. <5 pts> Use linear, polynomial, or exponential fits to determine the mass-radius and mass-luminosity relations. The file bzams_sums has this information already tallied. Compare your results to published numbers. Also using the information in bzams_sums, make a plot of the log10(L/L_sun) vs log10(T_eff). This will be an H-R Diagram containing a zero-age main sequence.

13. <5 pts> Are these models for Population I stars or Population II stars? Explain your answer.

14. <20 pts> Comment on the difficulties you had in interpreting these graphs. You have total access to the data. Compare a couple of the quantities in a way that is clearer to you, adding at least 2 more graphs to our analysis. For example, maybe you’d prefer having the star’s radius as the independent variable, and graph density or log density against the fractional radius. In looking at the graphs I’ve provided, you may have noticed that one gets a pretty good picture of how the interiors of stars “work” by just comparing a 0.5, 1.0, and 5.0 solar mass stars, so you do not need to have 7 different relationships as a function of mass on 1 graph. Explain why you chose the quantities you did to compare, and what your interpretation is of your results. Back up your statements with formulae or references.