%% Galactic 1
\bigskip
{\bf 1.}~ 
Consider a globular cluster 5 kpc away, moving with a tangential
velocity of 100 km/s, and the \emph{radial} velocity dispersion of 10 km/s.
It consists of 10$^5$ stars with an average luminosity
$\langle L_* \rangle = 0.8 L_\odot$, and amazingly enough, they all
have G-type spectra.  Its apparent half-light radius is 40 arcsec.
\begin{itemize}

\item[(a)] What is the cluster's parallax?  Can it be measured with the existing
technology, or at any time soon?  (Give an estimate of the state-of-the-art
astrometric accuracy.)

\item[(b)] What is the proper motion of the cluster, in arcsec/yr?  Of its typical
stars, relative to each other? 

\item[(c)] Estimate how large time baselines in years would be needed to detect
these motions from the ground.  Assume the limiting accuracy of the proper
motions measurement similar to that of the parallaxes. 

\item[(d)] What is the unobscured, integrated apparent magnitude of the cluster
in the $V$ band?

\item[(e)] Estimate the mass of the cluster, using the virial theorem.  Is the
resulting $(M/L)$ ratio reasonable for a stellar population?  Is there an
evidence for a dark matter in the cluster?

\item[(f)] Estimate the half-mass relaxation time of the cluster.

\item[(g)] Estimate the mean time between stellar collisions in the cluster, assuming
that all stars have radii 30 times larger than the Sun.

\end{itemize}
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%% Galactic 2
\bigskip
{\bf 2.}~ Assume that the stellar halo of the Milky Way is spherically summetric, with
a radial density law $\rho(r) \sim r^{-3}$, and that it is truncated at $R = 50$ kpc;
the local number density of halo stars is $\sim 10^{-3}$ pc$^{-3}$, and their mean mass is
$\sim 0.3 ~M_\odot$.

\begin{itemize}

\item[(a)] Estimate the total mass of the stellar halo.

\item[(b)] Assume that the halo was made from disrupted dwarf galaxies with masses
$< 10^9 ~M_\odot$, and the mass distribution given by the Schechter-type function with the 
faint end slope  $\alpha = -1$.  How many dwarf galaxies were used to make the stellar halo?

\end{itemize}
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%% Extragalactic 1
\bigskip
{\bf 1.}~ 
Assume that surface brightness of galaxies is given by the Sersic formula,
$I(r) = I_0 exp \{ –b_n (r/r_e)^{1/n} \}$.

\begin{itemize}

\item[(a)] Derive the values of $b_n$ for $n = 1, 2, 4, 8$, so that $r_e$ represents the radius enclosing 1/2 of the total light.

\item[(b)] Derive the values of the total light enclosed by these profiles for $n = 1, 2, 4, 8$, expressed in the units of $I_0 r_e ^2$.  Are there values of $n$ for which this formula diverges?
Assume a circular symmetry, and feel free to do numerical integrations as needed.

\end{itemize}
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