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%% MATHY SYMBOLS
\newcommand{\p}{\partial}
\newcommand{\nab}{\vc{\nabla}}
\newcommand{\xv}{\vc{x}}
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\def \mesa {\texttt{MESA\,}}
\def \delad {\nabla_{\rm ad}}


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\title{Astronomy : Homework 5 (revised) \\
\normalsize Due: Tuesday, Nov. 19, 2019}

\begin{document}
\maketitle

\begin{enumerate}
	\item {\bf Nuclear Statistical Equilibrium} \\
	Do problem 6.7 in HKT.  
		
	\item {\bf Scale Heights}\\
	  To order-of-magnitude, calculate the ratio of $H/r$ near the center and near the surface of a star, where $H$ is the pressure scale-height and $r$ the location radius.  Eliminate all quantities except $T_c$ and $T_{\rm eff}$.
	
	\item {\bf Alternate Hayashi} \\
	For a $\gamma = 2$ (i.e. $n = 1$) interior adiabat, derive the Hayashi line
	$$ T_{\rm eff} \propto L^A M^B$$
	calculating $A$ and $B$ for general $\kappa \propto P^a T^b$. Then evaluate numerically for H$^-$ opacity.
	
	
	\item {\bf Radiation Pressure Dominated Stars}\\
	 This problem  will show that a radiation pressure dominated star has $\gamma_{\rm ad} =4/3$.  
	 	\begin{enumerate}
		\item Briefly explain the relevance of adiabats for massive stars with strong radiation pressure.  Also what is  special about $\gamma= 4/3$?
		\item Simply  argue that $\nabla = \delad = 1/4$ when radiation pressure dominates.  
		\item Starting from the general form of the EOS:
			$$ \frac{d\rho}{\rho} = \alpha \frac{dP}{P} - \delta \frac{dT}{T}$$
			derive the general relation between $\gamma_{\rm ad}$ and $\delad$.
		\item Derive $\alpha$ and $\delta$ for an ideal gas and radiation mixture, in terms of $\beta = P_{\rm gas}/P$.
		\item Apply the preliminary results to determine $\gamma_{\rm ad}$  in the limit of strong radiation pressure ($\beta \rightarrow ~\text{what?}$).
	\end{enumerate} 
	
	\item {\bf ZAMS} \\
	Use \mesa to compute main sequence model of a mass corresponding to your last name.  In solar units: [Am. = 55., Bo = 40., Br. = 29., Cha. = 21., Che. = 16,  He =   11., Hv. =  8.3,  Jo. = 6.0 , Ka. =  4.4, Pe. = 3.2, Ra. =  2.3, Sc. =  1.7,  Sh. = 1.2, So. = 0.91, Su. = 0.66,  Te. =  0.48,  Wa. = 0.35, Wh. =  0.26,  Wo. = 0.19, Xu =  0.14 , Zh. = 
        0.1       ].
        
        The lowest mass stars may crash before reaching the ZAMS.  To fix,  change the low $T$ opacities by adding \texttt{kappa\_lowT\_prefix = 'lowT\_Freedman11'} to the \texttt{\&star\_job} section of your inlist.
         
        \begin{enumerate}
		\item   For the ZAMS model please submit plots of (1) $\log(T)$ vs. $\log(\rho)$ and (2) the temperature gradients $\nabla_{\rm rad}$, $\delad$ and $\nabla$ vs.\ $T$.  Also please submit  (3) a PMS (pre-main sequence) HR diagram.  For (3) you should trim the early evolution (at least 100-1000 years) and you can use the KH timescale vs. lifetime to motivate your choice.
		\item Calculate the following quantities and enter them in the google sheet at \url{https://tinyurl.com/astr545zams}
		\begin{align*}
			&[M, R, L, \texttt{log\_Teff}, \texttt{log\_center\_T},\texttt{log\_center\_rho}, \\
			&\texttt{mass\_conv\_core}, \texttt{mx1\_top}, \texttt{mx1\_bottom} ,  \texttt{mx2\_top}, \texttt{mx2\_bottom}  ]
		\end{align*} 
		with $M, R, L$ in Solar units and the other quantities as directly output by \mesa. [Note that \texttt{mass\_conv\_core} is in solar units, while the mass coordinate for the largest convective zone (\texttt{mx1\_top}, \texttt{mx1\_bottom} ) are relative to the total ($m/M$).]  Check if your values are consistent with the figures in Kippenhahn+ for your stellar mass. Comment if you notice any issues.  We will use these values to reconstruct, as a class, the properties of the main sequence.
	\end{enumerate} 
	\item For the class, what topic was/is confusing and/or a topic you would have liked covered in greater detail?  Explain as best you can what aspects you found unclear and/or want more information about.   Please add what parts of the texts (or other resources) you have read/consulted about these topics, if relevant.  This problem will be graded only for completion, and your replies will only be shared by the instructors.
	
\end{enumerate}

\end{document}