Homework 1

Homework 1#

AST400A - Theoretical Astrophysics - Fall 2025, Steward Observatory

Due Thur. Sept. 18, 12:30p (before class)

Relevant Chapters: HKT Ch. 3,4; Pols Lectures Ch. 2,3 here. 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: 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.

Total: (90 points)

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  1. A binary star system is observed, and since the separation between the two stars is much smaller than the distance of the system from the observer, it can be supposed that both stars are found at the same distance from Earth. The absolute magnitude in a given photometric band of the first star is determined to be -0.5, while its apparent magnitude is 3.5. Total: (20 points)

    • (a) - If the apparent magnitude of the second star is 4.5, what is its absolute magnitude? (10 points)

    • (b) - At what distance (in light-years) is the binary system from the observer? (10 points)

  2. Two stars have the same \(T_{\rm{eff}}\) but one of these stars is in the giant phase and has a radius of 15 times larger than the other star, which is on the main-sequence. Total: (10 points)

    • (a) - What is the numerical difference between the absolute magnitudes of these stars (10 points)

  3. At what distance would the Sun have to be to have the same apparent magnitude as a 100 W light bulb found at 100 m away? Express your answer in lightyear (ly). Total: (20 points)

  4. Calculate \(P( r)\) inside a sphere of radius \(R_\star\) with a constant density \(\rho\) assuming hydrostatic equilibrium is valid for this sphere. Total: (20 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.

  5. Calculate the total gravitational potential energy of a fictitious star with mass \(M_\star\) and radius \(R_\star\) that has a density profile \(\rho( r) = \rho_{0}(1 - r/ R_\star)\). Give your answer in terms of \(M_\star\) and \(R_\star\). Total: (20 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.