\documentclass[12pt]{article}
\usepackage{wrapfig,graphicx,verbatim}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0pt}
\setlength{\headheight}{0in}
\setlength{\topmargin}{-0.25in}
\setlength{\textheight}{8.75in}

\def\bl{\begin{list}{}{\setlength{\leftmargin}{0pt}}}
\def\el{\end{list}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}

\begin{document}

\def\outdate{24 Feb 2012} 
\def\duedate{Due: 2 Mar 2012}
\def\psetno{6}

 
\def\prob{\medskip\noindent\hskip-16pt }

\def\boxit#1{\vbox{\hrule\hbox{\vrule\kern3pt
       \vbox{\kern3pt#1\kern3pt}\kern3pt\vrule}\hrule}}
 
\noindent Astronomy 62 \hfill Ann Esin
 
\noindent Introduction to Astrophysics \hfill \outdate
 
\vskip0.3cm \hrule height1pt \vskip0.3cm
 
\noindent {\large Problem Set \psetno} \hfill  \duedate


\medskip
\def\lta{\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$}}
     \raise 2.0pt\hbox{$\mathchar"13C$}}}
\def\gta{\mathrel{\spose{\lower 3pt\hbox{$\mathchar"218$}}
     \raise 2.0pt\hbox{$\mathchar"13E$}}}
 
\def\ep{\epsilon}
\def\av#1{\langle#1\rangle}
\def\cm{{\rm \, cm}}
\def\pc{{\rm \, pc}}
\def\mm{{\rm \, mm}}
\def\s{{\rm \, s}}
\def\erg{{\rm \, erg}}
\def\eV{{\rm \, eV}}
\def\K{{\rm \, K}}
\def\g{{\rm \, g}}
\def\kg{{\rm \, kg}}
\def\km{{\rm \, km}}
\def\dyne{{\rm \, dyne}}
\def\Hz{{\, \rm Hz}}
\def\m{{\rm \, m}}
\def\msun{M_{\odot}}

\vskip 0.3cm
\noindent {\bf Please staple problems 1+2 and 3+4.}

\vskip 0.3cm
\prob 1. {\bf Pressure Inside the Stars} 

\vskip 0.2cm
\noindent Since photons have momentum, photons scattering off an electron 
produce a force.  With many photons trapped inside the Sun the repeated 
scatterings generate radiation pressure that acts in addition to gas 
pressure.  The expression for the pressure due to the blackbody radiation 
field is derived in chapter 9.1 of C\&O (Eq. 9.11): 
\be
P_{\rm rad} = \frac{4 \sigma T^4}{3 c}.
\ee  

\bl
\item{\bf (a)} Show that the radiation pressure inside the Sun is
negligible compared to gas pressure.  The temperature and density
inside the present day Sun are $T_{c,\odot}=1.58\times 10^7\K$, 
$\rho_{c,\odot}=1.62 \times 10^5
\kg\m^{-3}$.  Assume that the mass fraction of hydrogen is $X=0.34$ and of 
metals $Z= 0.02$.

\item{\bf (b)} For the stars on the main sequence the central density
is roughly inversely proportional to mass, i.e. $\rho_c(M) \simeq
\rho_{c,\odot} (\msun/M)$, and the central temperature is increasing
with mass as $T_c \simeq T_{c,\odot} (M/\msun)^{0.5}$.  Find the mass
of the star for which the radiation pressure in the center becomes 
comparable to gas pressure.   Assume that the composition ($X$, $Y$ and $Z$)
remains the same for all masses. 
\el

\vskip 0.3cm
\prob 2. {\bf Structure of Earth's Atmosphere}

\bl
\item{\bf (a)} Re-derive the equation for the equilibrium temperature
of the Earth, balancing the heat inflow from the Sun and heat loss
from the Earth's atmosphere.  Evaluate your result taking the albedo to be
$a = 0.3$.

\item{\bf (b)} Use the equation for hydrostatic equilibrium to derive
an equation describing how density in the Earth's atmosphere changes
with altitude.  Assume that the atmosphere is isothermal ($T$ does not
change with altitude) and can be treated as an ideal gas.  Express
your answer in terms of air density at the surface, $\rho_0$, and
temperature $T$.  ({\it Hint}: Since the thickness of the atmosphere 
is much smaller than the Earth's radius, you can assume $g$ is constant.)

\item{\bf (c)} Use your answer to parts (a) and (b) to find the
altitude at which the atmospheric density falls to $1/e$ of its
surface value.  Give both an algebraic and a numerical answer.
This special altitude is called the pressure scale height.  
\el

\vskip 0.3cm
\prob 3. {\bf Chemical Energy}

\noindent Problem 10.3 in C\&O.

\vskip 0.3cm
\prob 4. {\bf Mass Loss from the Sun}

\vskip 0.2cm
\noindent The Sun is constantly losing mass by converting it to energy
via nuclear reactions and through launching of the solar wind.  Solar
wind is a continuous stream of high-energy electrons and ions
escaping from the Sun. 
  
\bl
\item{\bf (a)} Calculate the escape speed at the surface of the Sun.
This gives us an order of magnitude estimate for the speed of the
solar wind particles, $v_{\rm sw}$.

\item{\bf (b)} Assume that the solar wind is launched isotropically.
Show that the rate at which the solar wind matter is crossing a spherical
surface of radius $r$ is equal to 
\be
\left(\frac{d M}{d t}\right)_{\rm sw} = 4 \pi r^2 \rho v_{\rm sw}. 
\ee
{\it Hint:} Start by calculating the mass contained is a thin shell
of radius $r$ and thickness $v_{\rm sw} d t$.   

Evaluate the solar wind mass loss rate if the solar wind contains mostly 
protons and their number density at $r=1\,{\rm AU}$ is $n=7\times 10^6\m^{-3}$.

\item{\bf (c)} Calculate $(d M/d t)_{\rm nuc}$, the rate at which the Sun's 
mass is decreasing due to nuclear reactions. 

\item{\bf (d)} Compare your answers parts (b) and (c).  Assuming that the 
mass loss rates remain constant, would either mass loss process significantly 
affect the total mass of the Sun over its main-sequence lifetime? 

\el
\end{document}