Homework 2#
AST400A - Theoretical Astrophysics - Fall 2025, Steward Observatory
Due Tue. Oct. 14, 12:30p (before class)
Relevant Chapters: HKT Ch. 4,5; Pols Lectures Ch. 3,5 here. LeBlanc Chapters 5,6. Not a complete list of the topics covered in the problem set. You are encouraged to work together on the problem sets but you must submit your own work.
Submitting your work: You are encouraged to work in groups, but your final solutions should be your own work! Turn into D2L as a PDF. In most cases, solutions will be found by hand, then written up in LaTeX/Markdown/Word and exported as a final PDF. If you have not worked with LaTeX before consider starting from one of the Overleaf Homework Templates here.
Extra credit: HW assignments submitted that were prepared using LaTeX will earn 10 points extra credit. If you used LaTeX to prepare your solutions, make a note of this in D2L textbox.
Total: (150 points)
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Total: (30 points) - The Planck function
(a) - Show that \( \frac{dB_{\nu}}{dT} = \frac{2 k_{\rm{B}}^3 T^2}{h^2 c^2} \left [ \frac{x^4 e^x}{(e^x-1)^2} \right ]~.\) (10 points)
(b) - Compute the maximum of the bracketed term using derivatives. This will require a numerical solution. (10 points)
(c) - Plug the maximum back into your variable substitution to compute the favored photon energy. In a few sentences compare your result to that from Pols 5.24 and what this suggests for the Rosseland mean opacity. (10 points)
Total: (35 points) - Assume a star of radius \(R_\star\) having a density profile equal to \(\rho( r)=\rho_{c} \left ( 1 - \frac{r}{R_\star} \right )\) and a nuclear production rate per unit mass equal to \(\epsilon( r) = \epsilon_{c} \left (1 - \frac{r}{0.2 R_\star} \right ) \ \ \textrm{for} \ r~\leq 0.2~R_\star\) and \(\epsilon( r) = 0 \ \ \textrm{for} \ r~\gt 0.2~R_\star~.\)
(a) - Compute the luminosity of the star at its surface in terms \(R_\star\), \(\rho_{c},\) and \(\epsilon_{c}\) (15 points)
(b) - Plug in present day solar values, you can use HKT 9.2.3, and compute a numerical value for the luminosity. (10 points)
(c) - Compute the nuclear timescale for this star and compare it to the nuclear timescale of the Sun. (10 points) For this problem, you may solve by hand or using SymPy. If using sympy, upload the PDF output of your notebook as part of the solution.
Total: (20 points) - Assume that 10 eV of energy per atom found in the Sun is emitted during some chemical reaction taking place. Also assume that the Sun is composed of pure hydrogen.
(a) - Calculate the total energy emitted by this chemical process (\(E_{\rm{chem}}\)). (5 points)
(b) - Compute the nuclear timescale for this star assuming a solar luminosity. (5 points)
(c) - In a few sentences describe if it is then possible that the energy source of the Sun is chemical in nature? Why or why not? (10 points)
Total: (25 points) - Conceptual questions from Pols Chapter 3. Respond in a few sentences.
(a) - What do we mean by local thermodynamic equilibrium (LTE)? Why is this a good assumption for stellar interiors? What is the difference between LTE and thermal equilibrium (as treated in Ch. 2)? (10 points)
(b) - In what type of stars does degeneracy become important? Is it important in main-sequence stars? Is it more important in high mass or low mass MS stars? (5 points)
(c) - Explain qualitatively why for degenerate matter, the pressure increases with the density. (5 points)
(d) - In the central region of a star we find free electrons and ions. Why do the electrons become degenerate first? Why do the ions never become degenerate in practice? (5 points)
Total: (40 points) - Produce a stellar model using MESA-Web here of initial mass between 0.5 \(M_{\odot}\) to 30 \(M_{\odot}\). Set your stopping condition to central hydrogen mass fraction lower limit of 1e-6. We will focus on the evolution of the star to the end of the main-sequence.
Note: If you have issues for your choice of mass, change to a lower initial mass, the defaults are set for a 1 \(M_{\odot}\) model to evolve to a white dwarf.
Extra Credit: If you download and install MESA (instructions here) to your laptop or on UA HPC (account creation info here) then produce a model using one of the MESA Test Suites you can earn 25 points extra credit.(a) - Produce a profile plot of \(\nabla_{\rm{ad}}\) and \(\nabla_{\rm{rad}}\) as a function of mass (\(m/M_{\odot}\)) or radius (\(r/R_{\odot}\)) during the main-sequence (or elsewhere if using a
test_suite) and label where the star is convective and radiative. (15 points)(b) - Produce an HRD using the history data from your model and label the
main-sequence. On the same plot, plot the present location of the Sun. (15 points)(c) - Produce a time-evolution plot of the luminosity of the
ppandcnoburning categories and determine which is the dominant burning category for your model. If there is a crossover point, identify the approximate central \(T_6\) this occurs at. (10 points)