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\def\outdate{16 Mar 2012} 
\def\duedate{Due: 23 Mar 2012}
\def\psetno{8}

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

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\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


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\def\msun{M_{\odot}}
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\vskip 0.3cm
\prob 1. {\bf Molecular Emission} 

%based on C&O 12.6
\noindent The rotational kinetic energy of a molecule is given by 
\be
E_{\rm rot} = \frac{1}{2} I \omega^2 = \frac{L^2}{2 I},
\ee
where $L$ is the molecule's angular momentum and $I$ its moment of
inertia.  The angular momentum is restricted by quantum mechanics to the 
discrete values 
\be
L= \hbar \sqrt{l (l+1)} ,
\ee  
where $l = 0, 1, 2, ...$.

\bl
\item{\bf (a)} For a diatomic molecule, 
\be
I = m_1 r_1^2 + m_2 r_2^2,
\ee 

where $m_1$ and $m_2$ are the masses of the individual atoms and $r_1$
and $r_2$ are their respective separations from the center of mass of the 
molecule.  Show that $I$ can be written as 
\be
I = \frac{m_1 m_2}{m_1+m_2} r^2, 
\ee
where $r = r_1 + r_2$ is the total separation between atoms in the molecule, 
and the first factor is the reduced mass. (Recall our discussion of the 
reduced mass in the beginning of the semester.) 
  
\item{\bf (b)} The separation between the C and O atoms in CO molecule 
is approximately $1.2\,{\rm \AA}$, and the atomic masses of $^{12}$C, $^{13}C$
and $^{16}$O are 12.000\,u, 13,003\,u and 15.995\,u, respectively ($1u = 
1.66\times 10^{-27}\kg$).
Calculate the moments of inertia for $^{12}$CO and $^{13}$CO.

\item{\bf (c)} What is the wavelength of a photon that is emitted by
$^{12}$CO during a transition between the rotational angular momentum
states $l = 3$ and $l = 4$?  Repeat you calculation for $^{13}$CO.
This is one method astronomers use to distinguish between different
isotopes in the ISM.


\el

\vskip 0.3cm
\prob 2. {\bf Dust Shrouding of Stars}

\bl
\item{\bf (a)} Consider an F0V star entirely hidden inside a roughly
  spherical dust cloud of radius $100\AU$.  Describe the spectrum of
  the emission you will see from this object, i.e. what will be its
  total luminosity and the peak emission wavelength.  You will find
  Appendix G useful.  (You can assume for the purposes of this problem 
  that the dust particles in the cloud are good blackbody emitters at 
  all frequencies. This is not really true, but will not give a wildly 
  wrong answer here.)

\item{\bf (b)} Now, let's assume that the cloud is partly transparent,
  and our F0 star is still visible in the optical, but has visual (V
  band) extinction of 5 magnitudes.  Estimate the average number
  density of dust particles in the cloud if the extinction is all due
  to dust particles with radii $a=0.2\,\mu{\rm m}$ and extinction
  coefficient in the V band $Q_V \sim 1.3$.  The extinction
  coefficient is defined as $\sigma_V = Q_V \pi a^2$.

\item{\bf (c)} The extiction coefficient depends on wavelength as
  $Q_{\lambda} \propto 1/\lambda$.  What is the extinction in the B
  band if the extinction in the V band is 5 magnitudes?  Based on your
  result, can you come up with a way to estimate extinction for any
  star with a known spectral type (which of course can be determined
  through spectroscopy)?  
\el

\vskip 0.3cm
\prob 3. {\bf Disk Formation in Protostars}
%based on C&O 12.13 

\vskip 0.3cm
\noindent The radial acceleration of a proto-stellar cloud rotating with 
angular velocity $\omega$ is given by equation
\be
\label{d2r}
\frac{d^2 r}{d t^2} = -\frac{G M_r}{r^2} + r \omega^2.
\ee
When the total acceleration is equal to 0, we get the familiar result that 
the gravitational 
force is equal to the centripetal acceleration ($r \omega^2$).

\bl
\item{\bf (a)} Use Eq. (\ref{d2r}) and the conservation of angular
momentum to show that the collapse of a cloud will stop (i.e. radial
velocity, $v_r$, is equal to 0) in the plane perpendicular to its axis
of rotation when the radius reaches
\be
r_f = \frac{\omega_0^2 r_0^4}{2 G M_r}, 
\ee
where $M_r$ is the interior mass, and $\omega_0$ and $r_0$ are the
original angular velocity and radius of the surface of the cloud
respectively.  Assume that the initial radial velocity of the cloud is
zero and that $r_f \ll r_0$.  You may also assume (incorrectly) that
the cloud rotates as a rigid body during the entire collapse.  {\em Hint:} 
Use the fact that
\be
\frac{d^2 r}{d t^2} = \frac{d v_r}{d t} =  \frac{d v_r}{d r} \frac{d r}{d t}
= v_r \frac{d v_r}{d r}.
\ee 
(Since no centripetal acceleration exists for
collapse along the rotational axis, disk formation is a consequence of
the original angular momentum of the cloud.)

\item{\bf (b)} Assume that the original cloud had a
mass of $1\msun$ and an initial radius of $0.5\pc$.  If collapse is
halted at approximately $100\AU$, find the initial angular velocity of
the cloud.

\item{\bf (c)} What was the original rotational velocity (in 
$\m\s^{-1}$) of the edge of the cloud?

\item{\bf (d)} Assuming that the moment of inertia is
approximately that of a uniform solid sphere, $I_{sphere} =
\frac{2}{5}M r^2$, when the collapse begins and a uniform disk,
$I_{disk}= \frac{1}{2} M r^2$, when it stops, determine the rotational
velocity at $100\AU$.
\el

\vskip 0.5cm 
\prob 4. {\bf Jeans Mass Revisited}

\vskip 0.3cm
\noindent In class we derived the Jeans mass limit using the virial
theorem.  Another way to understand this condition for cloud collapse
is to compare the free-fall time for the cloud with the sound crossing
time.  The free-fall time, $t_{ff}$, tells us how fast the cloud will
shrink due to its own gravity if pressure is ignored.  The sound
crossing time, $t_s$, determines how fast the pressure can respond to
counteract the pull of gravity.

\bl
\item{\bf (a)} What is the correct condition for the cloud to initiate
collapse, $t_{ff} < t_s$ or $t_{ff} > t_s$?  Justify your answer.
  
\item{\bf (b)} Consider a spherical isothermal cloud of radius $R$,
with uniform density $\rho_0$, temperature $T$ and gas pressure $P$.
The sound crossing time is typically taken to be $t_s = R/v_s$, where
$v_s$ is the sound speed, defined so that $P = \gamma \rho_0 v_s^2$.
Recall that $\gamma$ is the adiabatic index, which we can set to $5/3$ in a
cloud consisting mostly of neutral atomic hydrogen.  Use your result in part
(a) to calculate the minimum radius necessary for the onset of the collapse
as a function of $\rho$ and $T$.  The expression for free-fall timescale is 
derived in C\&O, Eq. (12.26).  (I gave a simplified derivation of $t_{ff}$ in 
lecture, but you should use the more precise textbook result in your 
calculation.)

\item{\bf (c)} Calculate the minimum mass necessary to initiate
the collapse of our cloud.  This is our new estimate of the Jeans
mass.  Compare your result to that derived in lecture and in
the text, making sure that the functional dependencies on the cloud
parameters ($\rho$ and $T$) are the same.  

\item{\bf (d)} Is your minimum mass larger or smaller than the Jeans 
mass we derived in class?  By what factor?  Can you think of a reason for 
this discrepancy.  ({\it Hint:} Think of the conditions under which the 
free-fall timescale was derived.)
\el


\end{document}