Homework 3

AST400A - Theoretical Astrophysics - Fall 2025, Steward Observatory

Due Tue. Nov. 4, 12:30p (before class)

Relevant Chapters: HKT Ch. 5; Pols Lectures Ch. 10,11,12 here. LeBlanc Chapters 2,6. Not a complete list of the topics covered in the problem set.

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 5 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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1. Total: (50 points) - Assuming that while on the Horizontal Branch, the Sun will burn helium via the triple-\(\alpha\) reaction only and that the luminosity in the core will be constant \(L_{\rm{HB}}=100L_{\odot}\).

   • (a) - Compute the energy emitted by the triple-\(\alpha\) reaction in MeV. (10 points)

   • (b) - Compute the fraction \(f\) of mass transformed into energy for the triple-\(\alpha\) reaction. (10 points)

     • Hint: Mass–energy equivalence

   • (c) - Compute the total emitted energy on the horizontal branch assuming 10% of the total mass (\(1M_{\odot}\)) will be converted to \({^{12}\rm{C}}\) and with the efficiency found in (b). (10 points)

   • (d) - Compute the total time this star will spend on the horizontal branch (\(t_{\rm{HB}}\)). (10 points)

   • (e) - Compare this to the typical main-sequence lifetime for a Solar like star and describe physically why the \(t_{\rm{HB}}\) is shorter or longer. (10 points)

2. Total: (25 points) - Consider a homogeneous spherical cloud with temperature \(T\) and density \(\rho\). Neglecting all factors except for gravitational and thermal energy, it may be assumed that the cloud can collapse under the condition: \(-\Omega>2U\).

   • (a) - Calculate the Jeans length \(R_{\rm{J}}\) for this cloud, the length at which this condition is met. (15 points)

   • (b) - Respond in a few sentences: If a cloud has a radius larger than its \(R_{\rm{J}}\), will it collapse? What are the next steps of a cloud that has began to collapse? (10 points)

3. Total: (30 points) - Conceptual questions from Pols Chapter 10,12. Respond in a few sentences.

   • (a) - Why does the luminosity of a star increase on the main sequence? Why do low-mass stars, like the Sun, expand less during the main sequence than higher-mass stars? (10 points)

   • (b) - Explain what happens during the ‘hook’ at the end of the main sequence of stars more massive than the Sun. (5 points)

   • (c) - Explain the existence of a Hertzsprung gap in the HRD for high-mass stars. Why is there no Hertzsprung gap for low-mass stars? (10 points)

   • (d) - Explain why the timescales of the burning stages from C-burning onward are very short compared to the H- and He-burning phases. (5 points)

4. Total: (45 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 helium mass fraction lower limit of 1e-6. Sometimes a non-UA email works better.
   Note: If you have issues for your choice of mass, change to a different initial mass.
   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 15 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 core-helium burning phase (or elsewhere if using a test_suite) and

     • label where the star is convective and radiative. (15 points)

     • describe the expected evolutionary fate of the star and your reasoning. (5 points)

   • (b) - Produce an HR diagram using the history data from your model and label at least four of the applicable evolutionary phases:

     • Main-Sequence, Red Giant Branch, Asymptotic Giant Branch, Blue Loop, Hertzsprung Gap (15 points)

   • (c) - Produce a time-evolution (model_number or star_age for example) plot of the luminosity of the all helium burning \(L_{\rm{He}}\) and use this to show if the star undergoes a helium flash or not.

     • Describe in a few words your reasoning based on the results of the plot. (10 points)