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\begin{document}

\def\outdate{4 April 2008} 
\def\duedate{Due: 11 April 2008}
\def\psetno{10}

 
\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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\prob 1. {\bf Stellar Clusters} 
\vskip 0.2cm
\noindent 
Download the photometry for two clusters, 47 Tuc and M45 (Pleiades). 

\bl
\item{\bf (a)} Use the data provided to construct the color-magnitude
diagrams of the two clusters.  Please include your figures in you homework
solution.

\item{\bf (b)} Estimate the ages of the two clusters.  Use
the main-sequence lifetime equation we derived in HW\#7, problem 4.

\item{\bf (c)} If the distance to M45 is 135\pc, use the technique of 
main-sequence fitting to estimate the distance to 47 Tuc.
\el


\prob 2. {\bf 40 Eridani} (C\&O, problem 16.1)

\vskip 0.3cm 
\noindent The most easily observed white dwarf in the sky is in the
constellation of Eridanus. Three stars comprise the 40 Eridani
system: 40 Eri A is a 4th-magnitude star similar to the Sun; 40 Eri B
is a 10th-magnitude white dwarf; and 40 Eri C is an 11th-magnitude red
M5 star.  This problem deals only with the latter two stars, which are 
in orbit around each other and are separated from 40 Eri A by $400\AU$.

\bl
\item{\bf (a)} The period of the 40 Eri B and C system
is 247.9 years.  The system's measured trigonometric parallax is
$0.201^{\prime\prime}$ and the true angular extent of the semi-major
axis of the reduced mass is $6.89^{\prime\prime}$.  The ratio of the
distances of 40 Eri B and C from the center of mass is $a_B/a_C =
0.37$.  Find the masses of 40 Eri B and C in terms of the mass of the
Sun.

\item{\bf (b)} The absolute bolometric magnitude of 40
Eri B is 9.6. Determine its luminosity in terms of the luminosity of
the Sun.
 
\item{\bf (c)} The effective temperature of 40 Eri B is $16,900\K$.
Calculate its radius, and compare your answer to the radii of the Sun,
Earth and Sirius B (radius of the latter is $0.008 R_{\odot}$).

\item{\bf (d)} Calculate the average density of 40 Eri B, and compare
your result with the average density of Sirius B (with a mass
$1.05\msun$ the average density of Sirius B is $3\times
10^9\kg\m^{-3}$). Which is more dense, and why?

\item{\bf (e)} Calculate the product of the mass and
volume of both 40 Eri B and Sirius B.  Is there a departure from the
mass-volume relation?  What might be the cause?
\el

\newpage
\prob 3. {\bf Relativistic White Dwarfs} (based on C\&O, problem 16.6)

\vskip 0.3cm 
\noindent In the extreme relativistic limit, when a particle's kinetic energy 
is much greater than its rest mass energy, the kinetic energy of an electron 
is given by the equation
\be
K = \sqrt{p^2 c^2 + m^2 c^4} - m c^2 \simeq p c,
\ee
where $p$ is the electron's momentum.
The corresponding pressure is still proportional to the product of 
characteristic kinetic energy and number density of electrons, just like 
in the non-relativistic case.
Use this to show that in this limit the degeneracy pressure has the form:
$P \propto \hbar c n_e^{4/3}$.

\prob 4. {\bf Supernova Light Curves}
\vskip 0.2cm

\begin{wrapfigure}{r}{10cm}
\vskip -1cm
\includegraphics[width=10cm]{SNlt}
\caption{The bolometric lightcurve of SN 1987A.  The dashed lines show
the contributions expected from the radioactive isotopes produced by
the shock wave. (Figure from Suntzeff et al., 1992. Also Fig 15.12 in C\&O.)}
\vskip 0.2cm
\end{wrapfigure}

The emission from a regular supernova is dominated for several months
by the radioactive decay of heavy elements created in the explosion,
as a the blast wave moves through the stellar envelope
(e.g. $^{56}_{27}$Co or $^{56}_{27}$Ni).

\noindent {\bf (a)} Suppose we start with $N_0$ atoms of some radiactive
material with a half-life of $\tau_{1/2}$.  Starting with Eq. (15.9) in C\&O, 
show that the amount of radioactive material remaining after time $t$ is given 
by equation:
\be
N(t) = N_0 e^{-\lambda t},
\ee
where $\lambda$ is a constant, given by 
\be
\lambda = \frac{\ln{2}}{\tau_{1/2}}.
\ee

\noindent {\bf (b)} Assuming that the light curve of a supernova is
dominated by the energy released in the radioactive decay of an
isotope that has a decay constant $\lambda$, show that the slope of
the light curve is given by equation:
\be
\frac{d}{d t}\left(\log_{10}{L}\right) = -0.434 \lambda.
\ee
Thus, by measuring the slope of the light curve, astronomers can verify the
presence of large quantities of a specific radioactive isotope.

Calculate the expected light curve slope from the decay of
$^{56}_{27}$Co with a half-life of 77.7 days and compare your answer
to the data shown in Figure 1

\noindent {\bf (c)} The energy released during the decay of one
$^{56}_{27}$Co atom is $3.72\,{\rm MeV}$.  If 0.075$\msun$ of cobalt
was produced by the decay of $^{56}_{28}$Ni following the explosion of
SN 1987A, estimate the amount of energy released per second through
the radioactive decay of cobalt one year after the explosion and compare
your answer to the data shown in Figure 1.


\end{document}

\prob 3. {\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)} Estimate the total amount of energy this pulsar is putting 
out into its environment, based on its measured deceleration.

\item{\bf (b)} Take $P_0$ to be Geminga's rotational period at birth.  Assuming
that $P \gg P_0$, and that the magnetic field stays constant, 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