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{\bf Introduction to Astrophysics II, 29:120\\
     Spring, 2009\\
Tenth Homework Set...April 17, 2009. Due April 23, 2009}\\
{\em Nunc et in umbrosis, Fauno decet immolare lucis \ldots Horace}
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{\bf (1)} Starting with the mathematical definition of the epicyclic frequency $\kappa$, derive equation 25.38, relating $\kappa$ to the Oort constants.  \\

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{\bf (2)} Calculate the period of revolution for a star at the galactocentric distance of the Sun (the ``galactic year''), and use your results from \# 1 to calculate the period corresponding to the epicyclic frequency. Are the two periods equal?  Are they low-order commensurable?  \\

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{\bf (3)} (a) What is the rotational period of the angular speed $\Omega_{lp}$, corresponding to the $(n,m)=(1,2)$ commensurability of galactic orbits at the galactocentric radius of the Sun? (b) What is the time difference between successive passages of a spiral arm by the Sun?  Assume that the Milky Way is a two-armed spiral at the radial position of the Sun.  \\ 

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{\bf (4)} Prove that for a $r^{-1}$ attractive gravitational potential (like we have in the solar system) the epicyclic frequency and frequency of revolution are the same.  What is the signficance of this equality?   \\

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{\bf (5)} Derive equation 25.36 in the textbook.    \\

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{\bf (6)} Assume a star moves on a galactic orbit with $R_m = R_0 = 8.5$ kpc, and $A_r = 2$ kpc.  Use Mathematica to plot the orbit over 3 galactic revolutions.   \\

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{\bf (7)} Problem 25.14 from the book.\\

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{\bf (8)} Problem 25.18 from the book.   \\

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