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{\bf Introduction to Astrophysics I, 29:119\\
     Fall, 2006\\
Second Homework Set...September 14, 2006.\\ Due September 21, 2006}\\
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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. All problem numbers refer to the first edition of the book.\\


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{\bf (1)} 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 plot the rotation curve.  \\

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{\bf (2)}  Problem 2.9 from textbook.\\

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{\bf (3)} Problem 2.10 \\ 


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{\bf (4)}  Problem 2.11\\

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{\bf (5)} Problem 2.13 (I recommend you use {\em MathCad} in solution of this problem.)\\

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{\bf (6)} Problem 2.14 (I recommend you use {\em MathCad} in solution of this problem.)  \\

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{\bf (7)} 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.\\

 
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{\bf (8)} Here is one to let you appreciate how special the inverse square law of gravity is for our existing here in the universe.  An object is subject to a central force, and moves on an orbit given by 
\begin{equation}
r = k \theta
\end{equation}
What is the force law which would produce such an orbit?  {\bf Hint:} You need to generalize the arguments which led to the equation for $u(\theta)$ for the case of the inverse square law.  

 
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