\documentstyle[11pt]{article}
\textwidth 14.0cm
\oddsidemargin 0pt
\evensidemargin 0pt
\begin{document}
\begin{large}
\begin{center}
{\bf Introduction to Astrophysics II, 29:120\\
     Spring, 2009\\
Ninth Homework Set...April 9, 2009. Due April 16, 2009}\\
\end{center}
\end{large}


\noindent
{\bf (1)} Derive equations 24.41 and 24.42 in the textbook from 24.37 and 24.38   \\

\noindent
{\bf (2)} Consider a star in the vicinity of the Sun (say, Vega, Altair, or Arcturus). It is moving on an orbit such that the perigalacticon is 4.8 kpc, and the apogalacticon is 12.8 kpc (Use the parameters for the solar orbit which are adopted in the book, which correspond to International Astronomical Union recommendations).    \\
(a) What are the components of its velocity $(\Pi, \Theta, Z)$?\\
(b) What is its peculiar velocity $(u,v,w)$?\\
(c) Based on the peculiar velocity, how would you categorize this star, as part of the young, intermediate, or old population? \\
{\em Mathematica} would come in handy in doing the calculations.  \\

\noindent
{\bf (3)} A star is in a circular orbit with radius $R_0$ (i.e. the same value as the Sun), but the orbit is inclined to the galactic plane at an angle $\iota = 70^{\circ}$.  The Sun is on the line of nodes of the orbit.  Calculate the space velocity $(\Pi, \Theta, Z)$ and peculiar velocity $(u,v,w)$ for this star.   \\ 

\noindent
{\bf (4)} Show that for observations in the galactic plane, at a galactic longitude $l < 90^{\circ}$, there is a distance ambiguity, i.e. that there are two distances along the line of sight that have the same radial velocity.   \\

\noindent
{\bf (5)} Show that if $\Theta(R)$ is constant with galactocentric distance, the radial velocity $v_r$ maximizes at the {\em tangent point}. The tangent point is that point along the line of sight at which the line of sight is tangent to an orbit of radius $R$.    \\

\noindent
{\bf (6)} {\em Here's a problem that is best done using Mathematica, particularly when it comes to plotting. } Assume we make observations in the galactic plane at $l=30^{\circ}$ and $l=50^{\circ}$.  Calculate the radial velocity as a function of distance (from the Sun along the line of sight) for each of these two longitudes, and plot up the results. You will need the data in Figure 24.25 of the book.  Note that inversion of the relationship you produce is the basis of measuring distances in the Galaxy for objects which are too distant for parallax measurements.  {\em Helpful suggestion:} It will help if you familiarize yourself with the Table and List functions in Mathematica.   \\

\noindent
{\bf (7)} Neutral hydrogen gas is distributed fairly uniformly throughout the Galactic disk, and is detected from the 21cm line.  Recall the material earlier in the semester.  Given what you have learned from the above problems, think about a way of determining the rotation curve $\Theta(R)$ utilizing only 21 cm observations.  Be sure and adequately describe your method. \\

\noindent
{\bf (8)} Problem 24.21 from the textbook.   \\

\end{document}
