added exercises from teh stealar structure.

tutorials/Lab.tex

@@ -85,6 +85,9 @@
 \section{Galaxies}
 \input{galaxies.tex}
 
+\section{Stars}
+\input{stars.tex}
+
 
 \bibliography{Lab}
 \end{document}

tutorials/stars.tex

+\subsection{Stellar structure}
+We discussed in the class that the structure of a spherically symmetric star is governed by the following equations, along with an equation of state $P(\rho)$: 
+\begin{eqnarray}
+\frac{dM}{dr} &=& 4\pi r^2 \rho ~~~ \mathrm{(Mass~conservation)} \\
+\frac{dP}{dr} &=& -\frac{G M \rho}{r^2} ~~~ \mathrm{(Hydrostatic~equilibrium)} \label{eq:hydro_eq}\\
+\frac{dL}{dr} &=& 4\pi r^2 \rho \epsilon ~~~ \mathrm{(Energy~generation)} \\
+\frac{dT}{dr} &=& -\frac{3 \kappa \rho L}{16\pi a c T^3 r^2} ~~~ \mathrm{(Radiative~transport)}  \label{eq:rad_eq}
+\end{eqnarray}
+
+
+
+\subsubsection{Problems}
+\begin{enumerate}
+\item Derive the equation of hydrostatic equlibrium Eq.~\eqref{eq:hydro_eq}. 
+\item Derive the radiative transfer equation Eq.~\eqref{eq:rad_eq} assuming radiative equlibrium. 
+\item Using the above set of equations, construct a numerical model of a spherically symmetric star in hydrostatic and thermal equilibrium, where energy is transported by radiative diffusion. Solve the coupled ODEs numerically using a standard ODE solver, such as scipy's \href{https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html}{solve\_ivp}. Integrate outward from small $r$ and stop when $P \to 0$. Plot $T(r), \rho(r), L(r), M(r)$. You can assume: 
+\begin{itemize}
+\item Ideal gas equation of state: $P = {\rho k_B T}/({\mu m_p})$, with $\mu = 0.6$.
+\item Kramers opacity: $\kappa = 4 \times 10^4 \rho T_6^{-3.5}~ \mathrm{cm^2/g}$. The dimensionless variable $T_6$ is the temperature in units of $10^6$K. 
+\item Energy Generation by $p-p$ chain: $\epsilon = 10^{-5} \rho T_6^4~\mathrm{erg/s/g}$.  
+\item Boundary Conditions (at $r \to 0$):
+$M(r) \approx \frac{4}{3}\pi r^3 \rho_c$, 
+$L(r) \approx \frac{4}{3}\pi r^3 \rho_c \epsilon_c$, 
+$P(0) = P_c, \quad T(0) = T_c$. 
+\item Boundary Conditions (at the surface): $P(R) \approx 0, \quad \rho(R) \approx 0$
+\item Input Parameters: 
+$T_c = 1.5 \times 10^7 \, \text{K},$
+$\rho_c = 150 \, \text{g cm}^{-3}.$
+\end{itemize}
+\item Repeat the calculation using an equation of state including both gas and radiation pressure: $P = \frac{\rho k_B T}{\mu m_p} + \frac{1}{3} a T^4$.
+\item Optional: Compte the mass-radius, mass-luminosity and surface temperature-luminosity relations by varying central density (and obtaining the corresponding central temperature consistently). How do they compare with the analytical relations that we derived in the class? 
+
+
+\end{enumerate}
+