Introduction to Astrophysics II, 29:120
Spring, 2009
Tenth Homework Set...April 17, 2009. Due April 23, 2009
Nunc et in umbrosis, Fauno decet immolare lucis ...Horace

(1) Starting with the mathematical definition of the epicyclic frequency $\kappa$, derive equation 25.38, relating $\kappa$ to the Oort constants.

(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?

(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.

(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?

(5) Derive equation 25.36 in the textbook.

(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.

(7) Problem 25.14 from the book.

(8) Problem 25.18 from the book.