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{\it Astronomy 45 problem set 4}\hfill {\it Spring 1997}\newline
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{\bf Astronomy 45.  Introduction to Astrophysics}
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\noindent\underline{Problem set 4} \hfill Due Friday, April 4
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\noindent {\bf 1. Virial Theorem for Elliptical Orbits}  
\begin{itemize}
\item[a)] Show that the time average of the potential energy over one orbital 
period $\tau$ can be reduced to the form
$$
\langle V \rangle = \frac{1}{\tau} \int_0^{\tau} {\left(\frac{G M \mu}{r}
\right) d t} = A \int_0^{2 \pi}{r d \theta},
$$
where $A$ is a constant factor. 
\item[b)] Evaluate the integral from part (a), using the fact that
$$
\int_0^{2 \pi} {\frac{d x}{1+a \cos{x}}} = \frac{2 \pi}{\sqrt{1-a^2}}.
$$
Show that the resulting expression is equal to $\langle 2E \rangle$,
where $E$ is the total orbital energy.
\item[c)] If $\langle V \rangle = \langle 2 E \rangle$,
 show that the virial theorem is satisfied.
\end{itemize}

\vskip12pt\noindent {\bf 2. Disruption by Tides} 
\begin{itemize}
\item[a)]  Find the separation $d$ between two small bodies, each of mass $m$,
lined up with a large body of mass $M$ and a distance $R$ from the nearest of
the small bodies, such that the graviational attraction between the small
bodies is just equal to the differential graviational force between them 
caused by their attraction to the larger body.  Ignore any relative motions.
\item[b)]  Imagine now that a small body of mass $m$ and radius $r$ is 
falling radially ($L=0$) towards a large body of mass $M$.  Approximately
at what distance 
between them, the smaller mass will start to be torn apart by tidal 
forces?  (Hint: this problem is very closely related to part (a).)
\item[c)]  Based on your result for part (b), could any of the solar system 
planets come close enough to the Sun to be tidally disrupted?  What if 
the Sun had the same mass, but 100 times smaller radius (typical white
dwarf star)?  The planetary radii and masses range from $0.0017 R_{\odot}$ and
$6.6\times 10^{-9} M_{\odot}$ (Pluto) to
$0.10 R_{\odot}$ and $9.5\times 10^{-4}M_{\odot}$ (Jupiter).
\item[d)]  One of the possible explanations for quasars (very far away 
and very bright objects) involves a giant black hole in the center of a 
galaxy much like our Milky Way, tidally disrupting stars that happen to 
pass near by.  What 
is the {\em minimum} mass black hole can have to disrupt a Sun-like star?

We can observe this disruption
only if it happens outside the black hole horizon (radius inside which 
even light cannot escape the hole's gravity), which is given by the 
expression
$$
R_h = \frac{2 G M_{BH}}{c^2},
$$
where $c$ is the speed of light.  Calculate the {\em maximum} mass a black hole
can have for us to observe the disruption of stars like our Sun.  Are larger 
mass black holes better or poorer at disrupting stars?  
\end{itemize}

\vskip12pt\noindent {\bf 3. Energy Lost During Accretion} 
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Suppose that two stars are bound together in a binary system, and that one
of these stars (the {\em primary}) is losing mass to the other 
(the {\em secondary}) at a rate $\dot{m}$.  Take the radius of the secondary star to be $R_2$ and its mass to be $M_2$.
\begin{itemize}
\item[a)] Show that the rate of heating of the secondary because of conversion
of the kinetic energy of the infalling matter to heat is
$$
L = \alpha \frac{G M_2 \dot{m}}{R_2},
$$
where $\alpha=1$ if the infalling matter is assumed to be falling in from rest
at infinite distance.  In a real binary system, what processes occur to make 
$\alpha < 1$?
\item[b)] In an equilibrium situation this energy is then lost through
radiation.  Assuming that the emerging radiation is a blackbody, show
that the surface temperature of the secondary star is
$$
T_2 = \left(\frac{\alpha G M_2 \dot{m}}{4 \pi \sigma R_2^3}\right)^{1/4}.
$$ 
Assume that the accreting material is distributed uniformly over the surface 
of the secondary and the internal thermonuclear energy production of the 
secondary is negligible.

\item[c)] If the secondary star resembles our Sun (i.e. has mass $M_{\odot}$
and radius $R_{\odot}$), and is accreting material at a rate $\dot{m} =
10^{-8} M_{\odot}/{\rm yr}$, what would be the observed luminosity 
and characteristic frequency?   Compare the accretion luminosity and the 
ordinary solar luminosity; is the rate of energy input 
from accretion significant compared with the internal energy generation rate
of such a star?

\item[d)] More interesting cases arise when the primary is a white
dwarf (of mass $\sim 1 M_{\odot}$ and radius $\sim 10^9\,\cm$) or a neutron
star (of mass $\sim 1.4 M_{\odot}$ and radius $\sim 10^6\,\cm$).  In what 
wavebands (radio, IR, optical, UV, X-ray, or $\gamma$-ray) should 
observations be made to find such systems, if the accretion rate is 
the same as in part (c)?  How luminous (in units of $L_{\odot}$)  are
these systems?
\end{itemize}

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