Homework 1

Homework 1#

Total: (80 points) - Each sub question worth 10 points.

Submitting your work: Turn into D2L as a PDF. In most cases, solutions will be found by hand, then written up in LaTeX or Markdown and exported as a final PDF.

Due Before Class Wednesday, January 29, 2025#

  1. HKT Section 1.6 describes the notion of homologous stars. Show that:

    • Eq. 1.65 follows from 1.2

    • Eq. 1.66 (with a factor of G included) is a good order of magnitude estimate for the central pressure (e.g. using the equation of HSE).

    • Demonstrate that Eq. 1.74 is the correct matrix equation for homologous stars where heat transport is dominated by radiative diffusion.

  2. HKT Exercise 1.3 - A useful (albeit not terribly realistic) model for a homogeneous composition star may be obtained by assuming that the density is a linear function of radius. (See Stein, 1966.) Thus assume that \(\rho = \rho_{c} [1-(r/R)]\) where where \(\rho_{c}\) is the central density and \(R\) is the total radius where zero boundary conditions, \( P ( R ) = T ( R ) = 0\) apply.

    • Find an expression for the central density in terms of R and M. (You will have to use the mass equation.)

    • Use the equation of hydrostatic equilibrium and zero boundary conditions to find pressure as a function of radius. Your answer will be of the form \(P ( R )= P_{c} \times \)(polynomial in \(r/R\)). What is \(P_c\) in terms of \(M\) and \(R\)? (It should be proportional to GM\(^2\)/R\(^4\).) Express \(P_c\) numerically with \(M\) and \(R\) in solar units.

    • In this model, what is the central temperature, \(T_c\)? (Assume an ideal gas.) Compare this result to that obtained for the constant-density model. Why is the central pressure higher for the linear model whereas the central temperature is lower?

    • Verify that the virial theorem is satisfied and write down an explicit expression for \(\Omega\) (i.e., what is \(q\) of Eq. 1.7?).

  3. Using MESA, run the 1M_pre_ms_to_wd test suite and export a pgstar image at the end of the simulation of an HR diagram when the stellar model has began cooling as a white dwarf. You can embed the pgstar image into your solutions PDF or include it separately.

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