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\section{Stars \& Galaxies, Galaxies example class 2, answers}
\noindent{\bf Question 1.}
\begin{enumerate}
\item Rotation curve is the circular velocity, $V_c$, as function of
  radius, $R$. On a circular orbit, $V_c^2=GM(<R)/R$. For Keplerian
  motion, $M(<R)=M$ is a constant, hence $V_c=\sqrt{GM/R}$.

\item Measurements of Oort's constants, obtained from observations of
  the motions of stars near the sun, yield $dV_c/dR\approx 0$, implying
  $V_c$ constant. Also H{\sc I} 21-cm measurements show flat rotation
  curves.

\end{enumerate}

\noindent{\bf Question 2.}

\begin{enumerate}
\item Take derivative with respect to $R$ of $R\,V_c^2=GM(<R)$ yields
  $\rho(R)=V_c^2/4\pi\,GR^2$, so $\rho(R)\propto 1/R^2$.

\item Substitute the numbers in previous expression yields
  $\rho(R=8.5\,{\rm kpc})=0.012\,M_\odot\,{\rm pc}^{-3}=8.4\times
  10^{-22}$ kg m$^{-3}$. 

\item From above the ratio dark matter over stars is 0.012.
\end{enumerate}

\noindent{\bf Question 3.}

\begin{enumerate}
\item Eq.(4.4) in the notes gives $R_S=((3\dot N)/(4\pi\alpha
  n^2))^{1/3}=30$~pc for the Str\"omgren radius.

\item If $R$ is the current position of the ionisation front, then
  there are $(4\pi/3)\alpha\,n^2R^3$ recombination per unit time. To
  advance $R$ to $R+\dot R\,dt$ in a time $dt$ requires ionsing another
  $4\pi\,nR^2\dot R\,dt$ neutral atoms. Equating the number of required
  ionisations per unit time, $(4\pi/3)\alpha\,n^2R^3+4\pi\,nR^2\dot R$,
  with the number of photons emitted by the source, $\dot N$, yields
  the required relation.

\item The derivative is 
\begin{eqnarray}
\dot R &=& {R_0\over 3t_0}\,{\exp(-t/t_0)/(1-\exp(-t/t_0))^{2/3}}\nonumber\\
        &=& {R_0\over 3t_0}\, {1-(1-\exp(-t/t_0))\over (R/R_0)^2}\nonumber\\
        &=&-{1\over 3t_0}\,R + {R_0^3\over 3t_0}\,{1\over R^2}\nonumber\\
        &=& -{\alpha n\over 3}\,R + {\dot N\over 4\pi n}\,{1\over R^2}\,.
\end{eqnarray}

Comparing the last two lines shows that the solution requires
\begin{eqnarray}
t_c &=&{1\over\alpha n}\,\nonumber\\
R_0^3 &=& {3\dot N\over 4\pi\alpha n^2}\,.
\end{eqnarray}

It shows that $R_0$ is the Str\"ogren radius. For small times, $t\ll
t_0$, the solution is $(R/R_0)^3=(t/t_0)$, hence the speed of the front
is $\dot R=(R_0^3/3R^2)\,{1/t_0}=\dot N/4\pi\,nR^2$


\end{enumerate}

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