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{\it Astronomy 45 problem set 5}\hfill {\it Spring 1997}\newline
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{\bf Astronomy 45.  Introduction to Astrophysics}
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\noindent\underline{Problem set 5} \hfill Due Friday, April 18
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\noindent {\bf 1. Polytropes}
\begin{itemize}
\item[a)] Solve the Lane-Emden equation analytically for a special case 
$n=0$.  Make sure you impose proper boundary conditions at $\xi=0$.
As a check, compute $\xi_1$ and $\phi^{'}_1$ and compare 
them to the values given in the table on page 131.
\item[b)] Compute the expressions for $\rho(r)$, $P (r)$, and $T(r)$ in terms 
of the total mass $M$ and radius $R$ of the star.   Based on your result, 
how would you describe an $n=0$ polytrope model?
\end{itemize}

\vskip12pt\noindent {\bf 2. White Dwarfs} 
\newline\noindent
Suppose a sequence of completely degenerate white dwarf stars has been 
constructed for different central densities for a composition of pure 
$^{12}_6$C.  Consider a new sequence of models composed of pure $^{56}_{26}$Fe.
\begin{itemize}
\item[a)] How are the physical parameters $P(r),\ \rho (r),\ m (r)$, and 
$R$ in the two sequences related?  (Compare the models corresponding to the
same central density.)  You can give either qualitative or 
quantitative  answer.
\item[b)] How would the Chandrasekhar mass change from one case to another?
For extra credit, try to compute the ratio $M_{Ch}({\rm C})/
M_{Ch} ({\rm Fe})$.
\end{itemize}

\noindent
Incidentally, no white dwarfs composed of iron exist in nature.  

\vskip12pt\noindent {\bf 3. Supernovae and Inverse $\beta$ decay}
\newline\noindent
A type II supernova can be viewed as a degenerate white dwarf core collapsing 
to form a neutron star through inverse $\beta$ decay.  The rest of the star 
forms an envelope around the core which is ejected after the collapse.
\begin{itemize}
\item[a)]  For a stellar core of mass $\sim 1.4 M_{\odot}$ consisting 
initially of 
protons and electrons calculate the total energy carried away by neutrinos
created in the reaction
$$
p^+ + e^- \rightarrow n + \nu_e, 
$$
if each neutrino has an energy $\sim$ 1 MeV.

\item[b)]  Compute the initial and final radii of the core (use a 
non-relativistic equation of state both for a white dwarf and a neutron star)
and calculate the gravitational energy lost in the collapse.  How does it
compare to the energy lost through neutrinos?

\item[c)] Assume  that the energy not carried away by neutrinos is used 
in ejecting the stellar envelope.  Estimate the resulting velocity of the 
envelope material assuming that the initial mass of the star is $10 M_{\odot}$.
Is it a significant fraction of the speed of light?

\item[d)] The expanding envelope inevitably collides with the surrounding
interstellar medium.  Since the expansion velocity is so large, a shock 
front forms, that essentially transforms kinetic energy of the expansion
into thermal energy.  Using your result from part (c), estimate the 
temperature of the material behind the shock front.  Will electrons and 
protons have the same temperature?  What frequency band should we use to 
observe such shock front?  (Keep in mind that it is primarily 
electrons that emit radiation.)
\end{itemize}

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