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\begin{document}

\def\outdate{28 Apr 2011} 
\def\duedate{(Optional) Due: 6 May 2011}
\def\psetno{12}

 
\def\prob{\medskip\noindent\hskip-16pt }

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\noindent Astronomy 62 \hfill Ann Esin
 
\noindent Introduction to Astrophysics \hfill \outdate
 
\vskip0.3cm \hrule height1pt \vskip0.3cm
 
\noindent {\large Problem Set \psetno} \hfill  \duedate


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\vskip 0.3cm
\prob 1. {\bf Growth of Supermassive Black Holes by Accretion}

\vskip 0.2cm
\noindent Let's assume that all supermassive black holes found in the
centers of galaxies started out as $10\msun$ objects very early in the
history of the Universe, which subsequently grew by feeding on the
interstellar gas and, later on, stars.  But the black hole luminosity,
and therefore its accretion rate, is fundamentally limited by the
Eddington value -- when the luminosity exceeds $L_{\rm Edd}$, the
accreting gas will simply be blown away by radiation pressure.  In HW\#11
we showed that the mass accretion rate corresponding to the Eddington 
luminosity, $L_{\rm Edd}$, is given by 
\be
\dot M_{\rm Edd} = \frac{L_{\rm Edd}}{\eta c^2} = 
\frac{4 \pi G M m_p}{\eta c \sigma_T} = 1.8\times 10^{15} 
\left(\frac{M}{\msun}\right)\kg \s^{-1},
\ee
where $\eta$ is the efficiency with which the black hole is converting 
accreting matter into radiation.  For a non-rotating black hole we set
$\eta \simeq 0.08$.  
 
\bl
\item{\bf (a)} Assuming that black holes accrete matter at the Eddington  
rate, write down a differential equation describing the time
evolution of the black hole mass. (Hint: Don't forget that $\dot M = d
M/d t$.)  Solve this equation to obtain the black hole mass as a function of
time, $M(t)$.

\item{\bf (b)} How long will take to assemble the Sgr A$^{\star}$ black hole
with a mass $3.7\times 10^6\msun$?  Comment on your result.

\item{\bf (c)} Repeat your calculation for a $10^{9}\msun$ black hole.
Such black holes are believed to be powering the most luminous quasars.
The quasar with the highest redshift found so far has $z=6.42$. Can 
its black hole be assembled in time through accretion?  (In your calculations, 
assume that the Universe is flat and ignore the cosmological constant.)
\el

\medskip 
\prob 2. {\bf Measuring a Distance to NGC 2639}

\vskip 0.3cm
\noindent NGC 2639 is an Sa galaxy with a measured maximum rotational
velocity of $324\km\s^{-1}$ and an apparent magnitude of $B=12.22\,{\rm
mag}$ (after making corrections for extinction).  

\bl 
\item{\bf (a)} Determine the distance to NGC 2639 from the appropriate
Tully-Fisher relation (Eq. 25.5 in C\&O).

\item{\bf (b)} The H$_{\alpha}$ line in the spectrum of NGC 2639 
appears at $\lambda = 666.8\nm$.  Calculate the corresponding redshift and 
estimate the Hubble constant based on this measurement.

\item{\bf (c)} Even a Hubble telescope cannot produce reliable measurements
for stars dimmer than about $V=28\,{\rm mag}$.  Use Figure 14.5 in C\&O to 
determine whether we can measure the distance to NGC 2639 using 
classical Cepheid variables.
\el

\medskip
\prob 3. {\bf Hot Gas in Clusters of Galaxies}

\vskip 0.2cm
\noindent   
X-ray observations in the late 1970's showed that clusters of galaxies
are filled with hot ionized gas that emits copious amounts of
high-energy radiation.  The total X-ray luminosity of the Virgo
cluster (the nearest cluster to the Milky Way Galaxy) is $L_x =
1.5\times 10^{36}\J\s^{-1}$.  Assume that this cluster has radius $R
\approx 1.5\,{\rm Mpc}$ and radial velocity dispersion $\sigma_r = 666 
\km\s^{-1}$.

\bl
\item{\bf (a)} Estimate the total mass of the Virgo cluster (in solar
units).  Calculate its characteristic mass-to-light ratio if its
visual luminosity is $L_V = 1.2\times 10^{12}\lsun$.  Comment on your result.

\item{\bf (b)} Estimate the temperature of the gas filling the Virgo
cluster, assuming that the average kinetic energy of each particle (and 
therefore its thermal energy)
obeys the virial theorem in the gravitational potential of the
cluster.  Take $R/2$ as a characteristic distance of a particle from
the cluster center and assume that the corresponding enclosed mass is
a quarter of the total cluster mass.  You will find that electrons and
protons have different {\it virial} temperatures, but since they are
thermally coupled, a good estimate of the gas temperature is an
average of your two results.

\item{\bf (c)} The X-ray emission is due to bremsstrahlung radiation
(free electrons accelerating due to the Coulomb forces from the protons),
which has the luminosity density (energy emitted per unit time per unit 
volume, integrated over all frequencies) given by the equation: 
\be
\label{Lvol}
{\cal L}_{\rm vol} = 1.42 \times 10^{-40} n_e n_p T^{1/2} (\J\s^{-1}\m^{-3}), 
\ee
where $n_e$ and $n_p$ are electron and proton number densities (in units 
of m$^{-3}$) and $T$ is the temperature of the gas.  ({\bf Note}: Eq. 27.19 in 
C\&O is incorrect!)

Assuming that the gas is mostly ionized hydrogen and the emission is
optically thin, estimate the number density and the total mass of the
hot gas in the Virgo cluster.  Compare your answer to the virial mass of the 
cluster.  Is there still need for dark matter? 

\item{\bf (d)}  Assuming that the gas has no energy source and that it is 
losing energy via thermal bremsstrahlung at a constant rate $L_x$, estimate 
how long it will take for the gas to lose all of its energy.  Compare 
your answer to the Hubble time $t_{\rm H} = 1/H_0$ and comment of the result.  
\el

\prob 4. {\bf The Evolution of $\Lambda$ Universe}

\bl
\item{\bf (a)} Starting with the energy conservation equation appropriate for a
Universe with non-zero cosmological constant $\Lambda$,
\be
\frac{m v^2}{2} - \frac{G M_r m}{r} - \frac{1}{6} \Lambda m c^2 r^2 = 
- \frac{1}{2} k [r(t_0)]^2 m c^2,
\ee
show that the 
evolution of the scale factor $R(t)$ is given by the equation 
\be
\label{Rlambda}
\left(\frac{d R}{d t}\right)^2 - \frac{8}{3} \pi G \rho R^2 - 
\frac{1}{3} \Lambda c^2 R^2 = - k c^2. 
\ee

\item{\bf (b)} Argue that for a constant and positive $\Lambda$, the
potential energy term due to normal matter will be larger than the
term containing the cosmological constant at early times, but will
eventually become negligible at late times.  

\item{\bf (c)} Assume that our Universe is flat ($k=0$) and solve
Eq. (\ref{Rlambda}) in the regime when the cosmological constant
dominates the potential energy of the Universe, and so the matter term
can be ignored.  Comment on your result.

\el
\end{document}