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\begin{document}

\section*{Stars \& galaxies, weekly problems}

\noindent{\bf Problem 1.}\\

\noindent The expression for the Jeans mass in a gas of temperature $T$ and
density $\rho$ is

\begin{equation}
M_J=\left({3k_BT\over 2m_p\,G}\right)^{3/2}\left({3\over 4\pi
\rho}\right)^{1/2}\,,\nonumber
\end{equation}


where $\boltz=1.38\times 10^{-23}{\rm J}\,K^{-1}$ is Boltzmann's constant,
$m_p=1.67\times 10^{-27}$kg is the proton mass and $G=6.67\times 10^{-11}$
m$^3$ kg$^{-1}$ s$^{-2}$ Newton's constant.\\

\noindent Show that
\begin{equation}
M_J=1.5\times 10^4\,M_\odot (n/{\rm cm}^{-3})^{-1/2}\,\nonumber
(T/100K)^{3/2}\,,
\end{equation}

when $n=\rho/m_p$ is the particle density, and $M_\odot=1.99\times
10^{30}$ kg is the solar mass. [2 marks]\\

\noindent Suppose two clouds with density $n=1{\rm cm}^{-3}$, $T=100{\rm K}$ and
$M=10^4M_\odot$ collide. Due to the shock, the density $n$ increases by
a factor of four. Compute the maximum $T$ below which the new cloud
will be Jeans unstable. [4 marks]\\

\noindent Determine whether the cloud is stable if the gas is isothermal. [2 marks]\\

\noindent Determine whether the cloud is stable if the gas behaves
adiabatically, ($T\propto\rho^{2/3}$). [2 marks]\\

\end{document}



