Introduction to Astrophysics II, 29:120
Spring, 2009
Seventh Homework Set...March 23, 2009. Due March 27, 2009

My apologies for getting this posted a bit late. I will postpone the due date until noon on Friday, March 27.

(1) Consider the effective potential for a 3rd body moving in the gravitational potential of 2 stars with masses $M_1$ and $M_2$. The coordinate system chosen is the usual one, in which the coordinate system rotates with a period equal to the binary revolution period. The x axis is defined by a line passing through the centers of the stars (the coordinate system discussed in class). Here's the question: Obtain an expression for the effective potential which consists of a dimensional term multiplying a dimensionless function of suitably dimensionless variables. Clearly identify your scaling, i.e. the transformation between dimensional and nondimensional variables.

(2) Use Mathematica (or Mathcad) to generate a contour plot of the effective potential for the case of $M_1 = 3 M_{\odot}$ and $M_2 = 1 M_{\odot}$. Plot only the dimensionless function referred to in #1 above, which gives you all the physical information. I don't want to see huge, uninterpretable numbers in your homework paper. The Mathematica function ContourPlot will be useful. You need to hand in the plot with your papers.

(3) Assume $M_1$ (we are still talking about the system in #1 and #2) is a degenerate object and $M_2$ is a main sequence star. What is the distance between the stars such that $M_2$ just begins to overflow its Roche lobe. Use the plot you generated in #2.

(4) Calculate the orbital period of the system in #3. Then compare your system with those described in Chapter 16 of the book. Are similar binary stars (or at least binary star periods) known in astronomy?

(5) Problem 18.2 from the textbook. You can do it with pencil and paper, or you can use your Mathematica notebook.

(6) Problem 18.5 from the textbook.

(7) Problem 18.7 from the textbook.

(8) Problem 18.9 from the textbook.