homework_4.tex

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\title{Astronomy 400B Homework \#4}
\author{Please Show Your Work for Full Credit}
\date{Due April 9, 2015 by 9:35am}


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\section{Sparke \& Gallagher Problem 7.1}

Suppose that gas atoms and galaxies in a group move at the same average random speed $\sigma$ along each direction. At temperature $T$, the average energy of gas particle is $3k_B T/2$, where $k_B$ is Boltzmann's constant.  If the gas is mainly ionized hydrogen, these particles are protons and electrons; show that, if the atom's kinetic energy $(3 m_p/2)\sigma^2$ is shared equally
between them, then
\begin{equation}
T \approx \frac{(m_p/2)\sigma^2}{k_B} \approx 5 \times 10^6 \left( \frac{\sigma}{300~\km~\s^{-1}}\right)^2~\K.
\end{equation}
\noindent
Hot gas in a group or cluster is usually close to this {\it virial temperature}.

\section{Sparke \& Gallagher Problem 7.13}

If the lens $L$ is an object of mass $M_{\odot}$ at a distance $d_{\mathrm{Lens}}$ from
us, show that the Einstein radius for a star at distance $d_S = 2 d_{\mathrm{Lens}}$ is
\begin{equation}
\theta_{E} = \sqrt{\frac{R_s}{d_{\mathrm{Lens}}}} \approx 2 \times 10^{-3} \sqrt{\frac{1~\kpc}{d_{\mathrm{Lens}}}}~\mathrm{arcsec}.
\end{equation}

\section{Sparke \& Gallagher Problem 7.17}

If a lens at distance $d_{\mathrm{Lens}}$ bends the light of a much more
distant galaxy, so that $d_S$ and $d_{LS}\gg d_{\mathrm{Lens}}$, show that the
critical density is
\begin{equation}
\Sigma_{\mathrm{crit}} \approx 2 \times 10^4 \left( \frac{100~\Mpc}{d_{\mathrm{Lens}}}\right)~M_{\odot}~\pc^{-2},
\end{equation}
\noindent
and that the mass projected within angle $\theta_E$ of the center is
\begin{equation}
M(<\theta_E) \approx \left(\frac{d_{\mathrm{Lens}}}{100~\Mpc}\right)\left(\frac{\theta_E}{1~\mathrm{arcsec}}\right)^2 ~ 10^{10}~M_{\odot}.
\end{equation}

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