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\begin{document}

\def\outdate{2 April 2012} 
\def\duedate{Due: 6 April 2012}
\def\psetno{10}

 
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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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\noindent {\large Please staple problems 1 and 2 separately.}
\medskip

\prob 1. {\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 2. {\bf Relativistic White Dwarfs} (based on C\&O, problem 16.6)

\vskip 0.3cm 
\noindent When relativity cannot be ignored, kinetic energy of an electron 
is given by the equation
\be
K = E - m c^2 = \sqrt{p^2 c^2 + m^2 c^4} - m c^2,
\ee
where $E$ is the total energy and $p$ is the momentum of an electron.
The electron 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 the extreme relativistic limit (i.e. when $p \gg m c$) 
the degeneracy pressure has the form:
$P \propto \hbar c n_e^{4/3}$.

\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