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{\bf Introduction to Astrophysics II, 29:120\\
     Winter, 2006-2007\\
Second Homework Set...January 26, 2007. Due February 1, 2007}\\
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Show calculations and give reasons for your answers. Don't go around 
confused and despondent; if you do not know how to get started, ask me for help. The purpose
of problem sets is to promote thinking and lead to understanding, not 
produce a {\em confiteor} of revealed truth.\\


\noindent
{\bf (1)} Problem 10.16 from textbook \\

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{\bf (2)} Assume that the pressure {\em everywhere} in a star is given by 
\begin{equation}
p=K\rho^{\gamma}
\end{equation}\\
where $\rho$ is the density.  Show that the equations of stellar structure yield a non-coupled differential equation for $\rho(r)$, where $r$ is the radial coordinate.  By ``non-coupled'', I mean an equation for $\rho$ alone as a function of $r$.  Derive that equation. You are not required to solve it.  {\bf Hint:} In this case, you don't need all of the equations of stellar structure to answer this problem. \\  

For the following problems, consult Appendix I, p A-54, which gives a stellar model for a $1 M_{\odot}$ star.  Use finite differences to approximate the derivatives which appear in the equations of stellar structure. \\ 

\noindent
{\bf (3)} Use the luminosity equation (from the equations of stellar structure), plus information in the table, to obtain the energy generation rate $\epsilon$ at a radial distance $r=4.90 \times 10^9$ cm.  \\ 

\noindent
{\bf (4)} Using equations in the textbook (and discussed in lecture), calculate $\epsilon_{pp}$ (the energy generation rate due to the proton-proton cycle) at $r=4.90 \times 10^9$.  Compare it with your answer from \# 3.  \\

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{\bf (5)} Show that hydrostatic equilibrium is satisfied by the model at $r=9.17 \times 10^9$ cm, by calculating both sides of the equation of stellar structure that describes hydrostatic equilibrium, and showing that they are equal. \\

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{\bf (6)} At $r=9.17 \times 10^9$ cm, is the star radiative or convective?  Show your work. \\

\noindent
{\bf (7)} Calculate $\frac{d \rho}{dr}$ at $r=1.06 \times 10^{10}$ cm.  Use this number to calculate the radius of the star, if the density continued to decrease linearly with $r$.  Compare this number with the true radius of a $1 M_{\odot}$ star. \\

\noindent
{\bf (8)} Use the data from the table to calculate $\frac{d^2 \rho}{dr^2}$ at 
$r=1.06 \times 10^{10}$ cm.  Use this number, plus $\frac{d \rho}{dr}$ from problem \#7, to get a 2nd estimate for the radius of the star.  Discuss the degree of improvement (or detriment) that occurs.  \\


 
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