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\begin{document}

\section*{Stars \& galaxies, multiple-choice questions}
[$\msun =1.989\times 10^{30}$ kg, $m_p=1.67\times 10^{-27}$ kg,
$k_B=1.38\times 10^{-23}$ J K$^{-1}$, $1\,\pc=3.09\times 10^{16}$ m,
$G=6.67\times 10^{-11}$ N m$^2$ kg$^{-2}$]\\

\noindent{\bf Problem 1.}\\

The mass of the Andromeda galaxy is dominated by a dark halo with
density profile $\rho(r)\propto 1/r^2$, mass $M=2.0\times
10^{12}\,M_\odot$, and extent $r_\star=200\,\kpc$.

\begin{enumerate}

\item The circular velocity at distance $r=300\,\kpc$ is

\begin{itemize}
\item[(a)] 170 km s$^{-1}$
\item[(b)] 207 km s$^{-1}$
\item[(c)] 220 km s$^{-1}$
\end{itemize}

\item The circular velocity at $r=100\,\kpc$ is

\begin{itemize}
\item[(a)] 170 km s$^{-1}$
\item[(b)] 207 km s$^{-1}$
\item[(c)] 293 km s$^{-1}$
\end{itemize}

\item The galaxy in addition has a uniform density bulge with mass
$M_b=2.0\times 10^{10}\msun$ and radius $r_b=4.0\,\kpc$. Combining dark halo and bulge,
the circular velocity at $r=2.0\,\kpc$ is

\begin{itemize}
\item[(a)]  73  km s$^{-1}$
\item[(b)] 207  km s$^{-1}$
\item[(c)] 220  km s$^{-1}$
\end{itemize}

\item Compute the equivalent of Oort's constant $A$ at $r=2~\kpc$ from
  the centre of Andromeda (using its halo and bulge parameters from above).

\item[(a)] 14.4 \kms \kpc $^{-1}$
\item[(b)] 48.9 \kms \kpc $^{-1}$
\item[(c)] 55.0 \kms \kpc $^{-1}$


\end{enumerate}

\noindent{\bf Problem 2.}\\

Doppler motions show that the line-of-sight velocity dispersion of
galaxies in a cluster of galaxies is $\sigma_{\rm 1D}=800\,\kms$. 

\begin{enumerate}

\item  Assuming isotropic orbits, the 3D velocity dispersion,
  $\sigma_{\rm 3D}$, is

\begin{itemize}
\item[(a)]  800 km s$^{-1}$
\item[(b)] 1386 km s$^{-1}$
\item[(c)] 2400 km s$^{-1}$
\end{itemize}

\item The cluster's radius $R=0.5$~Mpc. The virial mass of the cluster is

\begin{itemize}
\item[(a)] $M=7.4\times 10^{13}\msun$.
\item[(b)] $M=2.2\times 10^{14}\msun$.
\item[(c)] $M=6.7\times 10^{14}\msun$.
\end{itemize}

\item Another cluster has mass $M=1.0\times 10^{15}M_\odot$ and radius $R=1.0$~Mpc
Compute its virial temperature T, assuming pure hydrogen
gas. (The binding energy of the hydrogen atom is 13.6eV.)

\begin{itemize}
\item[(a)] $T=1.7\times 10^8$~K and the gas is mostly ionised
\item[(b)] $T=3.5\times 10^8$~K and the gas is mostly ionised
\item[(c)] $T=1.7\times 10^8$~K and the gas is mostly neutral
\item[(d)] $T=3.5\times 10^8$~K and the gas is mostly neutral
\end{itemize}

\item Compute the 1D velocity dispersion, $\sigma_{\rm 1D}$, of
  galaxies in this cluster. Use this to compute the cluster's crossing
  time, $t_{\rm cr}=R/\sigma_{\rm 1D}$.

\begin{itemize}
\item[(a)] $t_{\rm cr} = 470$ Myr
\item[(b)] $t_{\rm cr} = 820$ Myr
\item[(c)] $t_{\rm cr} = 1200$ Myr
\end{itemize}

\end{enumerate}
\end{document}
