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\section*{Stars \& galaxies, example class 2}
\noindent{\bf Question 1.} Short questions

\begin{enumerate}
\item What is the \lq rotation curve\rq\, of a spiral galaxy? Derive
the expression for the rotation curve in a Keplerian disk.

\item Describe two key observations that suggest the presence of dark
matter in the Milky Way.


\end{enumerate}

\noindent{\bf Question 2.} Galaxy rotation curves
\begin{enumerate}
\item Show that the density profile that correspondents to a flat
rotation curve is $\rho(r)\propto 1/r^2$.

\item The circular velocity for the Milky Way is $V_c=220$\kms. Compute
the (dark) matter density at the position of the Sun, $r=8.5$\kpc.

\item The density of stars in the solar neighbourhood is around
  $n_0=1$~pc$^{-3}$. Compute the ratio of dark matter to stars near the
  sun, assuming all stars have mass $1~M_\odot$.




\end{enumerate}

\noindent{\bf Question 3.} H{\sc II} regions

\begin{enumerate}
\item A gas cloud has uniform density of 1 hydrogen atoms per
  cm$^{3}$. A central O stars emits $\dot N=10^{48}$ ionising photons
  per second. Compute the Str\"omgren radius in this cloud. Use
  $\alpha=3.1\times 10^{-13}$ cm$^{-3}$ s$^{-1}$ for the recombination
  coefficient.

\item In the discussion of the H{\rm II} region we neglected
  recombinations. Demonstrate that recombinations change Eq.~(4.2) to
\begin{equation}
\dot N=4\pi R^2\,n\,\dot R + (4\pi/3)\,\alpha\,n^2\,R^3\,.\nonumber
\end{equation}

\item Show that $R=R_0\,(1-\exp(-t/t_0))^{1/3}$ is the general solution
  to this equation. Show it reverts to the solution in Eq.(4.2) for
  $t\ll t_0$. Obtain expressions for $t_0$ and $R_0$.

\end{enumerate}

\centerline{ 1 pc = $3.09\times 10^{16}$~m, $M_\odot=2.0\times
10^{30}$~kg, $G=6.67\times 10^{-11}$ m$^3$ kg$^{-1}$ s$^{-2}$}


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