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{\bf 29:119 Introduction to Astrophysics I\\
     Fall 2008\\
     Final Exam \ldots December 17, 2008}
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Start each question on a new page.  It allows me to make comments and generally keeps me in a better mood.  Write legibly. Explain what ideas you are using and what you are trying to do. There are 8 questions.   Good luck and no whining.\\

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(1) (a) Describe the physical conditions at the center of the Sun.  Give approximate numbers for the main physical parameters. (b) How do we deduce these numbers, i.e. what physical principles allow us to calculate them?  (c) How do we observationally confirm our conclusions?  What experiments or observations allow us to verify our conclusions about the center of the Sun. \\
  
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(2) What physical characteristic of the solar corona is surprising and unexpected, given the physical properties of the photosphere?  What mechanism or mechanisms are {\em hypothesized} to be responsible for this unusual characteristic?  \\

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(3) Make the physically unreasonable assumption that a certain type of star has a constant density $\rho_0$.  Use the equations of stellar structure to obtain an exact expression for the pressure as a function of the radial coordinate $r$.\\

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(4) Define the Russell-Vogt theorem. \\

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(5) Write down the equation for mass conservation from the equations of stellar evolution. Now derive the dimensionless form of this equation, which is used in numerical solutions. Show your work and explain your reasoning. \\

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(6) Assume the following simplified description of a highly evolved, solar-type star.  (a) The core contains 20 \% of the mass of the star, and the core has completely converted  hydrogen to $^4$He.  (b) The luminosity in the red giant phase is 100 $L_{\odot}$.  

Assuming the conditions in the core are sufficient to permit the Triple Alpha Process, how long could Triple Alpha reactions provide the luminosity requirements of the star?\footnote{In reality, a solar mass red giant gets its luminosity from proton-proton reactions in a shell surrounding the helium core}.  {\bf Helpful information:} The mass of a $^4$He nucleus is 4.002603 amu, the mass of a $^{12}$C nucleus is 12.000 amu, 1 amu = 
$1.6605402 \times 10^{-24}$  grams.   \\

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(7) In the last homework set, we calculated a stellar model.  (a) Describe what is meant by a stellar model, in words or in a combination of words and equations. (b) The stellar model was solved automatically by a function in {\em Mathematica}.  How would you go about demonstrating, in the sense of a spot check, that your solution is correct?  {\bf Nota Bene:} This last question is not out of the book or lecture; you have to do some thinking on your own. \\

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(8) A star has density dependent on the radial coordinate of the following form. 
\begin{eqnarray}
\rho(r) = \rho_0 \left[ 1-(r/R_{\ast})^2 \right] \mbox{  if } r \leq R_{\ast} \\ \nonumber
\rho(r) = 0 \mbox{  if } r >  R_{\ast}
\end{eqnarray}  
(a) What is $M(r)$ ? \\
(b) What is the total mass of the star? \\
(c) Compare the total mass with that of a star of uniform density.  \\



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