\documentstyle[11pt]{article}
\textwidth 14.0cm
\oddsidemargin 0pt
\evensidemargin 0pt
\begin{document}
\begin{large}
\begin{center}
{\bf Introduction to Astrophysics I, 29:119\\
     Fall, 2008\\
Second Homework Set...September 04, 2008.\\ Due September 11, 2008}\\
\end{center}
\end{large}
Show calculations and give reasons for your answers. Don't go around 
confused and despondent; if you do not know how to get started, ask me for help. \\

\noindent
{\bf (1)}  Problem 2.3 from textbook.\\

\noindent
{\bf (2)} Problem 2.7 \\ 

\noindent
{\bf (3)}  Problem 2.9 \\

\noindent
{\bf (4)} Problem 2.14  \\

\noindent
{\bf (5)} A {\em rotation curve} of a galaxy is a plot of orbital speed versus galactocentric distance.  Let's consider an elliptical galaxy, that can be roughly considered spherically symmetric.  Assume the density as a function of galactocentric distance $R$ can be described as follows. 
\begin{eqnarray}
\rho(R) = \rho_0; \mbox{ if } R < R_c  \\
\rho(R) = \rho_0 (\frac{R}{R_c})^{-2}; \mbox{ if } R \geq R_c
\end{eqnarray}
Write an expression for the orbital speed as a function of $r$, and graph (roughly, by hand)  the rotation curve. You may assume that the orbit is circular.  \\

\noindent
{\bf (6)} In lecture, I showed how elliptical orbits could be derived from Newton's Laws.  In the course of that derivation, a vector $\vec{D}$ emerged, defined by the equation (check class notes) 
\begin{equation}
\vec{v} \times \vec{L} = G M \mu \hat{r} + \vec{D}
\end{equation}  \\
In lecture I said that $\vec{D}$ pointed towards perihelion.  Show that this is true (i.e. give a good mathematical argument).  {\bf Suggestion:} Draw a fairly eccentric orbit and think about what $\vec{D}$ would be at some special points.\\

\noindent
{\bf (7)} A comet orbits in the solar system (and therefore obeys Kepler's Laws).  Its orbit has a semimajor axis of 10 astronomical units, and an eccentricity of 0.95.  On January 1, 2009, it is at perihelion ( polar angular coordinate $\theta=0^{\circ}$).  What will its coordinates be 10 years later?  {\bf Strong Suggestion:} This is a good one to solve with {\em Mathematica} or {\em MathCad}.  You {\em can} do it with pencil and paper, but your life will be horrible.  

\end{document}
