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\section*{Stars \& galaxies, example class 4}
\noindent{\bf Short questions}
\begin{enumerate}
\item Discuss three lines of evidence that suggest AGN are powered by
  a super-massive black hole.
\item What is the observational evidence the the Milky Way contains a
  super-massive central black hole?
\item How can you use gravitational lensing to determine the mass of
  a cluster of galaxies?

\end{enumerate}

\noindent{\bf Question 1.}\\
The initial mass function (or IMF), $N(M)dM$, is the number of stars born with
mass between $M$ and $M+dM$. Assume stars are born with the IMF
\begin{equation}
N(M)dM = N_0\,(M/M_\odot)^{-2.35}\,dM/M_\odot\,
\end{equation}
for $0.1<M/M_\odot<30$. 

\begin{itemize}
\item A star cluster has just formed $10^6M_\odot$ solar masses of
  stars. Determine the proportionality constant, then determine the
  number of SNe stars that will go off. (Assume stars with mass
  $>8M_\odot$ go super nova).
\item The luminosity of main sequence stars depends strongly on
  mass. Assume this relation is $L/L_\odot=(M/M_\odot)^2$. Compute the
  mean mass-to-light ratio of the stars in the cluster.
\item The massive stars in the cluster contribute a lot to the light,
  however these stars do not live for very long. Compute the mean
  mass-to-light ratio when the cluster is old, and all stars with mass
  $>1M_\odot$ have died.
\end{itemize}

\noindent{\bf Question 2.}\\

A $10^{13}\msun$ elliptical galaxy at distance 500\Mpc, lenses a spiral
galaxy a distance $500\Mpc$ behind it. For exact alignment, compute the
angular radius of the Einstein ring in arc seconds. Compare the radius
of the elliptical, 20\kpc,  with the size of the ring.\\

\noindent{\bf Question 3.}\\
\noindent The density distribution in a galaxy is 
\begin{eqnarray}
\rho(r) &=& \rho_1\, {r_1\over r} \,\,\,\hbox{for $r\le r_1$}\nonumber\\
        &=& \rho_1\, \left({r_1\over r}\right)^2 \,\,\,\hbox{for $r_1\le
        r\le r_2$}\nonumber\\
        &=& 0 \,\,\,\hbox{for $r> r_2$}\,.
\end{eqnarray}
In its centre, it has a super massive black hole with mass
$M_0=10^6\msun$. Assume $r_1=1\kpc$ and $r_2=100\kpc$.
\begin{itemize}
\item compute the total mass $M(r)$ enclosed by a sphere with radius
$r$.
\item compute the circular velocity $V_c(r)$.
\item determine $\rho_1$ such that $V_c=220\kms$ at $r=8.5\kpc$.
\item for this value of $\rho_1$, sketch $V_c(r)$.
\item suppose this galaxy is at a distance $d=5\Mpc$ from us. To find
dynamical evidence for the presence of the black hole, estimate the
required angular resolution of the observations.
\end{itemize}

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