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\section*{Stars \& galaxies, example class 3}
\noindent{\bf Question 1.}\\
\begin{enumerate}
\item 1) From observations of the X-ray gas, assuming hydrostatic
equilibrium, (2) from galaxy motions, assuming virial equilibrium
\item Assume that the ratio dark matter/gas in cluster is universal
value. Get the mean baryon density from Deuterium abundance and Big
Bang nucleo-synthesis. Combine these to get mean dark matter density.
\item TF relates luminosity $L$ with circular velocity $V_c$ of spiral
galaxy, $L=\alpha V_c^4$. To use as standard candle: first determine
$\alpha$ from nearby spirals with known distance. For spiral with
unknown distance: measure $V_c$, use relation to get $L$, from apparent
magnitude then determine distance.
\end{enumerate}

\noindent{\bf Question 2.}\\
\noindent 1) The 3D velocity dispersion $\sigma=3^{1/2}\,\sigma_{1D}$
(assuming isotropy). Virial theorem, Eq. (9.4), then gives $M=R\sigma^2/G=
1.5\times 10^{15}\msun$.\\

\noindent 2) Eq. (9.7) gives $T=(2/3) (\mu m_p/k_B)\, (GM/R)=8.75\times
10^7$K=7.54 keV, for $\mu = 0.5$. Temperature is much higher than H
binding energy, hence fully ionised, hence $\mu=0.5$ for hydrogen
gas.\\

\noindent 3) Total gas mass $M_g=M/7=2.16\times 10^{14}\msun$, mean
electron density $n_e=M_g/(4\pi\,R^3 m_p/3) = 0.77\times 10^{-4}$ electrons
per cm$^3$.\\

\noindent 4) $L_x = 1.42\times 10^{-27} n_e^2 T^{1/2}\,(4\pi
R^3/3)=2.64\times 10^{44}$ erg s$^{-1}$.\\

\noindent 5) Energy of 1 photon $E_x=10$ keV = $1.6\times 10^{-8}$
erg, hence total photon flux is $N_x = L_x/E_x$. Count rate in Chandra
= $C=N_x* (S/4\pi d^2) = 13.6$ photons per second.\\

\noindent 6) Metal mass $M_z=Z_\odot \times M_g/3=1.44\times 10^{12}\msun$,
where $Z_\odot=0.02$ is the metal abundance of the Sun (e.g. page 95,
footnote 3). For yield $y=0.02$, mass
once in stars, $M_\star=M_z/y = Z_\odot M_g/3 / y = M_g/3=7.20\times
10^{13}\msun$,{\em i.e.} 1/3 of the total gas mass.\\

\noindent 7) Number of SNe $N=M_\star/100\msun =7.20\times 10^{11}$,
currently $N/t_0=55$ per year.\\

\noindent 8) Total SN energy is $E_{\rm SN}=N\times 10^{51}$
erg=$7.20\times 10^{62}$ erg. Thermal energy of gas per unit mass is
$u=(3/2) (\boltz T/\mu\mh)=(GM/R)$. Hence total thermal energy is
$U=u\,M_g=9.31\times 10^{63}$ erg, hence $E_{\rm SN}/U=7.7$\%, so SNe
may have had a small effect on heating the gas.

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