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\begin{center}
{\Large ASTR430 Homework \#6 \\
Due Wednesday, November 26, 2003}
\end{center}

1. In class, we showed that a spherical distribution of material
orbiting a primary will flatten out into a ring or disk. The key
physical point is that collisions will decrease the orbital energy
while keeping the angular momentum constant.  Here you will use the
same physical argument to see if flat rings (oir disks) will spread
out or shrink in the radial direction. Our model for this process will
be one thin ring splitting into two, or two thin rings merging into
one.

a) Write down expressions for the total orbital energy and the total
angular momentum contained in a very thin circular ring of mass 
$m_{ring}$ at distance $r$ from the planet. 

b) Calculate the total orbital energy and angular momentum contained
in two very thin rings, each of mass $m_{ring}/2$, at distances $r_1$
and $r_2$ from the planet.

c) Kepler Shear. Take $r_{particle}= 10$m to be the typical size of a
Saturnian ring particle and imagine two of these particles in the same
ringlet, but on very slightly different uninclined circular orbits.
What is the maximum speed at which they can collide (ignore ring
self-gravity)? Evaluate your answer numerically for ring particles
with $a \approx 2R_S$. How this compare with the escape velocity from
a ring particle?

d) Assume that states a) and b) occur at different times in the
evolution of a ring system. Argue that the angular momentum must be
the same for each state. Now, a collisional system (you know that the
system is collisional from your answer to part c) will evolve toward
the state with lower orbital energy. Why does this happen?  Use
Lagrange Multipliers to find extrema in Energy that have constant
Angular Momentum. Determine which of your extrema are maxima and
minima. Do rings spread or contract?

e) The above calculation is strictly applicable only to narrow rings. What
weaknesses does the calculation have? What general conclusions can you
draw from it?


2. Structure of an exosphere. The derivation of an atmosphere's scale
height discussed in class assumed that gravity and temperature are
both constant with height. It is not difficult to relax these
assumptions. Work out how the density $\rho$ in the exosphere (distant
part of the atmosphere) drops off with height $z$ using $g= GM/r^2$
for gravity (assume T=const). Start with the expression for pressure
derived in class ($dP/dz = -\rho g$), derive an expression for
$d\rho/dz$, and integrate. Put your final answer in terms of $\rho_0$,
the density of the atmosphere at Earth's surface, and $H_0 = kT/mg_0$,
the scale height at the Earth's surface.

3. You are circling the Earth in your spacecraft at an altitude of
100km and have just received clearance from Houston to fire the
spacecraft engines to take you to the Moon (assume that the
spacecraft's orbit is always in the Moon's orbital plane). In a
two-body approximation (Earth, spacecraft) how much extra velocity
must be added to put the spacecraft onto an orbit whose apocenter is
at the Moon? How much extra velocity would you need to escape from the
Earth altogether?  Give answers in km/s and, since your life depends
on your calculations, be sure to check your results!



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