% \documentstyle[12pt]{report}
\documentclass[12pt]{article}
\usepackage{exscale}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{graphics}
%
\newcommand{\g}{{\hbox{\rm g}}}
\newcommand{\K}{{\hbox{\rm K}}}
\newcommand{\km}{{\hbox{\rm km}}}
\newcommand{\J}{{\hbox{\rm J}}}
\newcommand{\ps}{{\hbox{\rm$^{1}$}}}
\newcommand{\yr}{{\hbox{\rm yr}}}
\newcommand{\Myr}{{\hbox{\rm Myr}}}
\newcommand{\Gyr}{{\hbox{\rm Gyr}}}
\newcommand{\s}{{\hbox{\rm s}}}
\newcommand{\cm}{{\hbox{\rm cm}}}
\newcommand{\pc}{{\hbox{\rm pc}}}
\newcommand{\kms}{{\hbox{\rm km s$^{-1}$}}}
\newcommand{\kpc}{{\hbox{\rm kpc}}}
\newcommand{\Mpc}{{\hbox{\rm Mpc}}}
\newcommand{\Mpci}{{\hbox{\rm Mpc$^{-1}$}}}
\newcommand{\msun}{\hbox{$M_\odot$}}
\newcommand{\rsun}{\hbox{$R_\odot$}}
\newcommand{\lsun}{\hbox{$L_\odot$}}
\newcommand{\mh}{\hbox{$m_{\rm h}$}}
\renewcommand{\mp}{\hbox{$m_{\sc\rm p}$}}
\newcommand{\me}{\hbox{$m_{\sc\rm e}$}}
% ions
\renewcommand{\H}{\hbox{\rm H}}
\newcommand{\He}{\hbox{\rm He}}
\newcommand{\C}{\hbox{\rm C}}
\newcommand{\N}{\hbox{\rm N}}
\renewcommand{\O}{\hbox{\rm O}}
\newcommand{\Be}{\hbox{\rm Be}}
\newcommand{\Li}{\hbox{\rm Li}}
\newcommand{\B}{\hbox{\rm B}}
\newcommand{\F}{\hbox{\rm F}}
\newcommand{\Fe}{\hbox{\rm Fe}}
%
\newcommand{\boltz}{\hbox{${\rm k}_{\sc\rm B}$}}
\renewcommand{\r}{\hbox{${\cal R}$}}
\newcommand{\planck}{\hbox{${\rm h}_{\sc\rm P}$}}

\def\d{{\rm d}}
\def\dtwod#1#2{\frac{\d^2 #1}{\d #2^2}}
\def\bfO{{\bf \Omega}}
\def\pdd#1#2{\frac{\partial #1}{\partial #2}}
\def\dd#1#2{\frac{\d #1}{\d #2}}
\def\vibar{\overline{v_i}}
\def\vijbar{\overline{v_iv_j}}
\def\vjbar{\overline{v_j}}
\def\vkbar{\overline{v_k}}
\def\sigij2{\sigma_{ij}^2}
%
%
\def \gtsima{\mbox{$\; \buildrel > \over \sim \;$}} 
\def \ltsima{\mbox{$\; \buildrel < \over \sim \;$}} 
\def \simgt{\lower.5ex\hbox{\gtsima}}            % > over ~
%
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\ba}{\begin{eqnarray}}
\newcommand{\ea}{\end{eqnarray}}
\newcommand{\nn}{\nonumber}
\newcommand{\arcsecsq}{{\hbox{\rm arcsec$^{-2}$}}}
\newcommand{\arcsec}{{\hbox{\rm arcsec}}}
\newcommand{\U}{{\hbox{\sc U}}}
\renewcommand{\B}{{\hbox{\sc B}}}
\newcommand{\V}{{\hbox{\sc V}}}
\newcommand{\R}{{\hbox{\sc R}}}
\newcommand{\I}{{\hbox{\sc I}}}
\newcommand{\KB}{{\hbox{\sc K}}}
\newcommand{\AU}{{\hbox{AU}}}
%
\newcommand{\mH}{{\hbox{m$_{\rm H}$}}}
%
\newcommand{\bfr}{\hbox{$\bf r$}}
\newcommand{\bfL}{\hbox{$\bf L$}}
\newcommand{\bfq}{\hbox{$\bf q$}}
\newcommand{\bfv}{\hbox{$\bf v$}}
\newcommand{\bfvp}{\hbox{$\bf v_{\rm p}$}}
\newcommand{\bfk}{\hbox{$\bf k$}}
\newcommand{\bfx}{\hbox{$\bf x$}}
\newcommand{\bfy}{\hbox{$\bf y$}}
\newcommand{\bfw}{\hbox{$\bf w$}}
\newcommand{\bfxp}{\hbox{$\bf x'$}}
\newcommand{\bfF}{\hbox{$\bf F$}}
\newcommand{\bfa}{\hbox{$\bf a$}}
\newcommand{\bfd}{\hbox{$\bf d$}}
\newcommand{\bfl}{\hbox{$\bf l$}}
%
\newcommand{\dex}{\hbox{\rm dex}}
\newcommand{\mol}{\hbox{[M/L]}}

\textheight24truecm
\begin{document}

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

\noindent{\bf Problem 2.}\\

\noindent{\bf Short questions}
\begin{enumerate}
\item A neutral hydrogen atoms contains a proton and an electron, both
  of which have a quantum mechanical property called spin. When these
  are parallel, the energy of the atom is higher than when they are
  anti-parallel. When the spin changes from parallel to anti-parallel,
  the atom emits a 21-cm photon. To measure a rotation curve, determine
  the wavelength $\lambda$ of the 21-cm photon as function of distance
  to the centre $R$, and use the equation for Doppler shift, to derive
  speed from $\lambda$.

\item If a spiral arm were a material structure, it would wind-up very
  quickly because of the differential rotation of the disk, making all
  spiral structures very tightly wound, which is not observed. This
  means that spiral arms are not material structures, and stars move in
  and out of them.
\end{enumerate}

\noindent{\bf Long question}
\begin{enumerate}
\item Taking the derivative with respect to $R$ of both sides of $RV_c^2=G(M<R)$
  yields $V_c^2=G\,dM/dR=4\pi\,G\rho(R) R^2$ when $V_c=V_{c,0}$ is
  constant. Re-arranging yields $\rho(r)=V_{c,0}^2/4\pi G
  R^2$

\item Using cylindrical coordinates, the enclosed mass
  $M_d(<R)=\int_0^R \sigma(R) 2\pi\,RdR =
  2\pi\,\sigma_0\,R_0^2\,(1-(1+x)\exp(-x))$, where $x\equiv R/R_0$.

\item The rotation speed due to dark halo and disk combined, is
$V_c^2(R)=V_{c,0}^2+V^2_d(R)$, where $V^2_d(R)=GM_d(R)/R$ is the
enclosed mass due to the disk. Substituting the numbers yields
$V_c(R=1{\rm kpc})=280$~km s$^{-1}$ and $V_c(R=10{\rm kpc})=210$~km
s$^{-1}$.
\end{enumerate}

\end{document}



