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% Homework #1

\title{Astronomy 545: Homework 3}

\begin{document}

You may collaborate on all problems this week.  Since they are
computational, everyone should write/run their own code.

\begin{enumerate}
\item Solve the Lane-Emden equation numerically for $n=1.5$ and $n=3$.  Write
your own numerical integration routine for this, using a higher-order
midpoint technique (e.g., 4th-order Runge-Kutta). 
Your solution should include your code, and plots or tables showing the behavior of $\theta$ versus 
$\xi$.  
\begin{enumerate}
\item 
List $\xi_1$, $(d\theta/d\xi)_{\xi_1}$, and $M(\xi_1)=M(R_{\ast})$ for
each of your solutions.  You can compare your values to those in the book to make sure
you have done this correctly.
\item 
Derive the Chandrasekhar mass, using the equation of state for
relativistic degenerate matter.  This will require you to make an
assumption about the mean molecular weight per electron, $\mu_e$.  To
compute this, you may assume the star is made of pure, fully-ionized He.
\end{enumerate}

\item
In this problem, we will compare our semi-analytic polytropic
solutions to the numerical solutions we obtain from MESA.
\begin{enumerate}
\item Start with the $n=3$ polytropic model you generated in Problem
  1.  Assume $M=1$ M$_{\odot}$, $R=1$ R$_{\odot}$.
\begin{itemize}
\item Compute the central density.  
\item Compute the central pressure ($P_{\rm c}$) and
  temperature ($T_{\rm c}$). You may assume an ideal gas equation of state.
\item Plot $P/P_{\rm c}$ and $T/T_{\rm c}$ for your polytrope model.
\end{itemize}
\item Repeat (a) for a $n=3/2$ polytrope.
\item Generate a model of a zero-age main-sequence star with $M=1$
  M$_{\odot}$, $R=1$ R$_{\odot}$ using MESA.  Compare the central
  density, pressure, and temperature to your ploytrope models.  Then
  compare $P/P_{\rm c}$ and $T/T_{\rm c}$ for your MESA and polytrope
  models.  Which polytropic model seems like a better representation
  of the ``actual'' (i.e., numerically-determined) structure of this object?
\item Compute $d\log P / d\log \rho$ for your MESA model.  Note that
  this quantity is equal to $1+1/n = \gamma$ for a polytrope.  Comment
  on what physical processes are responsible for the fact that 
  different regions of the star appear to be governed by different
  values of this quantity.
\end{enumerate}
\end{enumerate}

\end{document}
