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\title{Astronomy 545: Homework 6 (Extra Credit) \\
\normalsize Due: Wed. Dec. 11, 2019}

\begin{document}
\maketitle

\begin{enumerate}
	\item {\bf Electron-Scattering Opacity}\\
	Here we derive the electron scattering opacity, $\kappa_e$ including the dependence on composition.  The significance of this problem: it described the dominant high $T$ opacity in stars, and the $X$ dependence was shown to be relevant for e.g. semi-convection.
	\begin{enumerate}
		\item Express $\kappa_e$ 
		in terms of $\sigma_e$, the cross section for a photon to interact with an electron, $\mu_e$ and any constants.  Explain this simple ``derivation".
		\item For non-relativistic electrons $\sigma_e = 8 \pi r_e^2/3$ where $r_e$ is the classical electron radius.  Equate electrostatic and rest mass energies to ``derive" $r_e$.  Give the resulting $\kappa_e $ in terms of constants and $\mu_e$.
		\item Derive $\mu_e(X)$ for a mixture of (fully ionized) hydrogen and helium (and $X$ the hydrogen mass fraction).
		\item Evaluate $\kappa_e$ numerically, for general $X$, to reproduce the textbook result.
	\end{enumerate} 
	
	\item {\bf Convection}
	\begin{enumerate}
		\item Argue that if $\nabla > \delad$ implies  convective instability, that $\delrad > \delad$ is an equivalent instability criterion. 
		\item Similarly, argue that if $\nabla < \delad$ implies stability to convection, that $\delrad <  \delad$ is an equivalent stability criterion. 
		\item For a region of convective instability place $\nabla$, $\delad$, $\delrad$, $\nabla_{\rm e}$ in order, which short written justifications for each inequality.
		\item Do the above for a region of convective stability.
	\end{enumerate} 
	
	\item{\bf Helium core burning}
	\begin{enumerate}
		\item Describe the Horizontal branch (HB) and it's role in post-MS evolution.  Specifically address: what types of stars have a HB, what nuclear burning occurs, what events occur before and after the HB (i.e. what they are called and what burning happens).  Sketch and label these 3 stages (and more if desired) schematically on a HR diagram.
		\item Estimate the timescale for Helium core burning in a ``typical" HB star with $L \approx 50 L_\odot$ and a He core of $0.5 M_\odot$ at the ZAHB. Assume that $\sim 10\%$ of core $^{4}$He burns to $^{16}$O.
	\end{enumerate} 
	
	\item {\bf Fluid Equations}
	\begin{enumerate}
		\item Show that the following relation is true 
		$$ \rho \frac{Df}{Dt} = {\p (\rho f) \over \p t} +  \nabla \cdot (\rho f \vc{u})$$
		(In class we explain that this relation is useful for relating Eulerian and Lagrangian conservation laws.)
		\item For the next two subproblems start with the basic energy equation:
		$$\frac{DQ}{Dt} = -\dot{Q}_{\rm cool}$$
		where $\dot{Q}_{\rm cool}$ represents cooling per unit mass (or heating if negative).  First show that this heating equation can reproduce the standard stellar structure equation
		$$\frac{ \partial L}{\partial m} = \epsilon -\epsilon_g$$ 
		Be sure to define all terms and briefly explain all steps.
		\item Next show that for an ideal gas the fluid energy equation can be written in the form
		$$ \frac{D \ln P}{Dt} = - \gamma \nabla \cdot \vc{u} - \frac{\dot{Q}_{\rm cool}}{C_V T}$$
		where terms have their usual meaning, including  $\gamma = d \ln P/d \ln \rho|_S$.  You can use (without derivation) the following properties of an ideal gas:  $\mathcal{E} = C_V T$ (internal energy),   $\gamma = C_P/C_V$ and $C_P = C_V + \mathcal{R}/\mu \equiv C_V + k_{\rm B}/(\mu m_u)$.  Otherwise justify and briefly explain your steps.   
	\end{enumerate} 	
	
	
\end{enumerate}



\end{document}