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\begin{document}

\def\outdate{14 Apr 2012} 
\def\duedate{Due: 23 Apr 2012}
\def\psetno{11}

 
\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


\medskip
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\noindent {\bf Please staple problems 1+2 and 3+4.}
\vskip 0.3cm

\vskip 0.3cm
\prob 1. {\bf Redshifted Blackbody}
\vskip 0.3cm

\noindent Consider a spherical blackbody of constant temperature and
mass $M$, whose radius is $R$.  An observer located at the surface of
the sphere and a distant observer both measure the radiation
given off by the sphere.  Gravitational redshift preserves the
characteristic shape of a blackbody spectrum, so both observers will
see a blackbody spectrum.

\bl
\item{\bf (a)}  If the observer at the surface measures the luminosity of the  
blackbody to be $L(R)$, use the
gravitational redshift and time dilation formulas, to show that the
observer at infinity measures
\be
L_{\infty} = L(R) \left(1-\frac{2 G M}{R c^2}\right).
\ee
{\it Hint:} Think of the luminosity as number of photons emitted per unit 
time, times the average photon energy. 

\item{\bf (b)} Both observers use Wien's law, to determine 
the blackbody's temperature.  Show that a distant observer will measure 
temperature $T_{\infty}$ given by  
\be
T_{\infty} = T(R) \sqrt{1-\frac{2 G M}{R c^2}}.
\ee

\item{\bf (c)}Both observers use the Stefan-Boltzmann law to determine the 
radius of the spherical blackbody.  Show that
\be
R_{\infty} = \frac{R}{\sqrt{1- 2 G M/R c^2}}.
\ee
Thus, using the Stefan-Boltzmann law without including the effects of
general relativity will lead to an overestimate of the size of a
compact blackbody.

\item{\bf (d)} Calculate how much of an error one would make in inferring
the radius of a neutron star from its emission properties, if General
Relativistic effects are ignored.   
\el

\begin{comment}
\prob 2. {\bf Pulsar Properties}

\vskip 0.3cm
\noindent The Geminga pulsar has a period of $P=0.237\s$ and a period
derivative of $\dot P = 1.1\times 10^{-14}$.
\bl
\item{\bf (a)} Derive the expression for the total amount of
  rotational energy this pulsar is losing, based on its measured
  deceleration.  This energy ultimately goes into heating its
  environment.

\item{\bf (b)} Recall that the rate at which a pulsar is losing its
  rotational energy is equal to the energy loss by the rotating
  magnetic dipole.  Taking the magnetic field of a pulsar to be
  constant, integrate the resulting differential equation to obtain
  the expression for $P(t)$. Take $P_0$ to be the pulsar's rotational
  period at birth ($t=0$).  Assuming that $P \gg P_0$, derive an
  analytical expression for the age of a pulsar in terms of its $P$
  and $\dot P$, and evaluate it for Geminga.

\item{\bf (c)} Using the formula you derived in part (c), estimate the
  age of the Crab pulsar, which has $P = 0.0333\s$ and $\dot P =
  4.21\times 10^{-13}$.  Compare this with what we think is the real
  age of this pulsar. If all pulsars are born with similar rotational
  periods, explain why your estimate for the age of Geminga is
  probably more reliable.  
\el
\end{comment}

\medskip
\prob 2. {\bf Accreting neutron stars}

\vskip 0.3cm
\noindent Suppose that two stars are bound together in a binary
system, and that one of these stars (the {\em secondary}) is losing
mass to the other (the {\em primary}) at a rate $\dot M = d M/d t$.  Take the
radius of the primary star to be $R_1$ and its mass to be $M_1$.

\bl
\item{\bf (a)} For simplicity, let's assume that all the energy released 
during accretion is lost through
radiation from the surface of the primary (note that here we ignore 
the energy lost in the accretion disk).  Assuming that the emerging 
radiation is a blackbody, show that the surface temperature of the primary 
star is 
\be
T_1 = \left(\frac{G M_1 \dot M}{4 \pi \sigma R_1^3}\right)^{1/4}.
\ee
Assume that the accreting material is distributed uniformly over the
surface of the primary and the internal thermonuclear energy
production of the primary is negligible.

\item{\bf (b)}  Particularly interesting cases arise when the primary is a
white dwarf or a neutron star.  Calculate the efficiency of 
rest mass energy release through accretion, i.e. find $\eta$ such that 
\be
\label{eff}
L = \eta \dot M c^2.
\ee
Estimate $\eta$ for a neutron star.

\item{\bf (c)} The accretion luminosity must always be below the
Eddington limit:
\be
L_{\rm Edd} = \frac{4 \pi G M m_p c}{\sigma_T}.
\ee
Calculate the maximum mass accretion rate (in units of g/s) possible
for a neutron star and its corresponding surface
temperature.  In what waveband (radio, IR, optical,
UV, X-ray or $\gamma$-ray) should we observe these systems?  How
luminous (in solar units, $\L_{\odot}$) are they?

\noindent (Note that according to this simplistic analysis, accreting
black holes will not produce any emission, since they do not have a
``surface''.  In reality, a lot of the emission will be produced on
route to the primary, in the accretion disk, and the emission
characteristics of accreting black holes and neutron stars are
actually very similar.)  

\item{\bf (d)} X-ray bursters are accreting neutron stars that
  periodically undergo explosive nuclear burning of He accumulated on
  their surface.  If a burster releases $10^{32}\J$ every 5 hours,
  what is the mass accretion rate onto the neutron star?  Assume that
  all the accreted H is continuously converted to He on the neutron
  star surface, and only He is burned during the burst.

\el

\prob 3. {\bf Black Hole Tides} 
\vskip 0.3cm 

\noindent One way massive isolated black holes can feed is by ripping apart 
passing stars due to their strong tidal forces.  In this problem you will 
explore this effect.

\bl
\item{\bf (a)} Suppose we have a body of mass $m$ and radius $r$
located a distance $a$ away from a black hole of mass $M$.  To
estimate the tidal force on the body produced by the proximity of the
black hole let's treat it as two halves, each of mass $m/2$, separated
by a distance $r$.  The difference between the gravitational forces on
the half closer to the black hole and the half farther from the black
hole is called the tidal force.  In the limit when $r \ll a$, show that this 
force is equal to 
\be
F_T \simeq \frac{G M m r}{a^3}.
\ee
  
\item{\bf (b)} Find the analytical expression for the tidal force
at the black hole horizon.  Use your result to estimate the tidal force
on a person passing through the horizon of a $10\msun$ black hole.
Repeat your calculation for a $10^9\msun$ black hole.  What are your
conclusions?

\item{\bf (c)} Typically, black holes catch stars rather than
people. Find the distance from the black hole at which a star
of mass $m$ and radius $r$ will be disrupted by tidal forces, assuming
that the two halves of the star are held together by gravity.  Express
your answer in terms of the back hole mass and the density of the star.
Use your result to check whether a $1\msun$ star will
get disrupted before it plunges through the horizon of a $10^8\msun$
black hole.  (If the disruption occurs inside the horizon, there will
be no visible effect, since any emission produced during this event
will not make it out of the black hole.)
\el

\medskip
\prob 4. {\bf The Central Black Hole -- Sgr A*}

\vskip 0.3cm
\noindent Assume that the mass distribution in the center of our
Galaxy is a combination of the constant density Galactic bulge and the 
central black hole.  Then the mass interior to radius $r$ is simply 
\be
M_r = \rho_0 r^3 + M_{\rm BH},
\ee
where $\rho_0$ is the mass density of the bulge and $M_{\rm BH} = 
4\times 10^6 \msun$ is the mass of the black hole.

\bl
\item{\bf (a)} Show that the velocity of a star in a circular orbit 
around the Galactic center is given by 
\be 
v = \left[G \left(\rho_0 r^2 + \frac{M_{\rm BH}}{r}\right)\right]^{1/2}.
\ee
Calculate $\rho_0$, in units of solar masses per cubic parsec, if the orbital 
velocity is 250\,km/s at $r=300\pc$.  You can assume that is this distance, 
black hole influence is negligible.

\item{\bf (b)} Plot $v$ as a function of $\log_{10}{r}$ over the range
$0.01\pc < r < 1000\pc$.  Express $v$ in km/s and $r$ is parsecs.  Use
your graph to estimate the distance at which the black hole begins to
affect the stellar motions.

\item{\bf (c)} Use your result for \#3(c) to estimate the
distance from the black hole at which a $1\msun$ main-sequence star
will get tidally disrupted.  Compare the result to your answer in part
(b) and to the Schwarzschild radius of the central black hole.

\el
\end{document}

\medskip
\prob 3. {\bf Galactic Mass Distribution}

\vskip 0.3cm
\noindent 
We showed in class that the Galactic rotation curve can be used to
determine the mass distribution in our Galaxy.  In the bulge, the data
shows that the rotational velocity is proportional to the radial distance 
from the Galactic center, i.e. $v \propto r$, while in the
disk the rotational velocity stays constant (see Figure 24.25 in C\&O).
\bl
\item{\bf (a)} Show the rotation curve implies that in the central
bulge (say at $r < 0.5\kpc$), the angular velocity and the mass density
of stars are independent of $r$.

\item{\bf (b)} Argue that the density of the dark matter halo
given by equation 
\be
\label{density}
\rho(r) = \frac{C_0}{a^2 + r^2},
\ee
where $a = 2.8\kpc$ and $C_0$ is a constant that sets the total mass of the 
halo, is consistent with the observed Galactic rotation curve.  

\item{\bf (c)} Show that according to Eq. (\ref{density}), the amount of dark 
matter interior to $r$ is given by the expression
\be
M_r = 4 \pi C_0 \left[r - a \tan^{-1}{\left(\frac{r}{a}\right)}\right].
\ee

\item{\bf (d)} If $5.4\times 10^{11}\msun$ of dark matter is located within 
$50\kpc$ of the Galactic center, determine $C_0$ in units of $\msun\kpc^{-1}$.

\item{\bf (e)} Taking $50\kpc$ to be the edge of the dark matter halo
use the data in Table 24.1 in C\&O  to calculate the
mass-to-light ratio (in B filter) for our Galaxy.  What is the
mass-to-light ratio if the dark matter halo extends to $100\kpc$?

\el
\end{document}