add some exercises in the galaxy section.

tutorials/Lab.tex

@@ -82,5 +82,9 @@
 \section{Cosmology}
 \input{cosmology.tex}
 
+\section{Galaxies}
+\input{galaxies.tex}
+
+
 \bibliography{Lab}
 \end{document}

tutorials/galaxies.tex

+\subsection{Collissionless Boltzmann Equation}
+Given the distribution function $f(\mathbf{p}, \mathbf{q}, t)$ in the phase space [with $\int d^3\mathbf{p}  \, d^3\mathbf{q} \, f(\mathbf{p},  \mathbf{q}, t) = 1]$, and the Hamiltonian $H$, the Collissionless Boltzmann Equation (CBE) is given by 
+\begin{equation} 
+\frac{\partial f}{\partial t} + \left[f, H \right] = 0.
+\end{equation} 
+
+\subsubsection{Problems}
+\begin{enumerate}
+\item Derive the explicit forms of the CBE in spherical polar coordinates assuming a spherically symmetric gravitational potential $\Phi(r, t)$. 
+\item Derive the explicit forms of the CBE in cylindrical polar coordinates assuming a cylindrically symmetric gravitational potential $\Phi(R, z, t)$. 
+\item Assuming an NFW density profile $\rho(r)$ with isotropic velocity dispersion $\sigma^2_r(r) = \sigma^2_\theta(r) = \sigma^2(r)$, we derived the following relationship using Jean's equation. 
+\begin{equation}
+\rho(r) = \frac{\rho_s}{(r/r_s)(1+r/r_s)^2}~~, ~~~~~~~ \sigma^2(r) = \frac{1}{\rho(r)} \int_r^{\infty} ds ~ \rho(s) \frac{d\Phi(s)}{ds},
+\end{equation}
+where $\Phi(r)$ is the gravitational potential sourced by the density $\rho(r)$, while $r_s$ and $\rho_s$ are the parameters of NFW model. Plot $\sigma(r)$ for a Milky-Way type dark matter halo ($r_s \simeq 15 \mathrm{kpc}$ and $\rho_s \simeq 10^{-2} M_\odot \mathrm{pc^{-3}}$).
+\end{enumerate}
\ No newline at end of file