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{\bf Introduction to Astrophysics II, 29:120\\
     Spring, 2009\\
Seventh Homework Set...March 23, 2009. Due March 27, 2009}\\
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My apologies for getting this posted a bit late.  I will postpone the due date until noon on Friday, March 27. \\ 

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{\bf (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).  {\bf 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.   \\

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{\bf (2)} Use {\em Mathematica} (or Mathcad) to generate a contour plot of the effective potential for the case of $M_1 = 3 M_{\odot}$ and $M_2 = 1 M_{\odot}$.  {\em 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 {\em Mathematica} function ContourPlot will be useful.  You need to hand in the plot with your papers.     \\

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{\bf (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.     \\ 


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{\bf (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? \\

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

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{\bf (6)} Problem 18.5 from the textbook. \\

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{\bf (7)} Problem 18.7 from the textbook. \\

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{\bf (8)} Problem 18.9 from the textbook. 


 
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