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\begin{center}
{\Large ASTR430 Homework \#7 \\
Due Wednesday, December 10, 2003}
\end{center}



1. a) Use a symmetry argument and dimensional analysis to determine
how the direction and magnitude of gravity from Saturn's wide flat
ring (modeled as a uniform-density flat sheet of infinite extent)
varies with height above the ring. 

b) Show that the undetermined constant from dimensional analysis is
$2\pi$ using Gauss' Law.

c) Find the vertical component of gravity from Saturn. Compare this
with your answer for the ring's gravity. At what height(s) are the two
equal? How does this compare to the $\approx 100$m thickness of
Saturn's rings?

d) Now imagine a ring ``atmosphere'' - gas molecules above and below
the rings of Saturn. There is some evidence that a ring atmosphere of
Oxygen, Hydrogen, Water Vapor, and OH actually exists.  Use previous
handouts and homeworks to help you derive how the density of gas
should vary as a function of height above the rings. Be sure to use
the FULL gravity (ring + vertical component of Saturn).

2. Consider the 2:1 resonance between Io and Europa which is governed
by the following approximate expressions.


\begin{displaymath}
{da_I \over dt} = 2\beta m_E a_I^2 n_I e_I \sin(\Psi)~~~~~~~~~~
{da_E \over dt} = -4\beta m_I a_E^2 n_E e_I \sin(\Psi) 
\end{displaymath}
\begin{displaymath}
{de_I \over dt} = -\beta m_E a_I n_I \sin(\Psi)~~~~~~~~~~
{de_E \over dt} = 0~~~~~~~~~~~~~~~~~~~~~~~
\end{displaymath}
\begin{displaymath}
{d\varpi_I \over dt} = -{\beta m_E a_I n_I \over e_I} \cos(\Psi)~~~~~~~~~~
{d\varpi_E \over dt} = 0~~~~~~~~~~~~~~~~~~~~~~~
\end{displaymath}

In the above expressions, subscripts $E$ and $I$ stand for Europa and
Io, respectively, $t$ is time, $\beta$ is a constant, $m$ is mass, $a$
is the semimajor axis, $n = \sqrt{(GM/a^3)}$ is the mean motion (the
satellite's average angular speed), $e$ is eccentricity, and $\varpi$
is the longitude of pericenter. The resonant angle $\Psi = \lambda_I -
2\lambda_E + \varpi_I$ where $\lambda$ is the satellite longitude and
$d\lambda/dt \approx n$.\\ a) Show that the above expressions conserve
energy. You can use $2n_E \approx n_I$.\\ b) Now, because of their
steep dependence on distance, the tides pushing Io outward dominate
the tides pushing Europa outward. To a good approximation, therefore,
we can add constant tidal terms $\dot a_{drag}> 0$ and $\dot
e_{drag}<0$ to the Io equations only ($\dot\varpi_{drag} <<
{d\varpi_I/dt}$ for small $e_I$). Set the new $da_I/dt$ equation equal
to zero (not a great approximation since in resonance Io and Europa
are still moving outward, but not too bad), solve for $\sin\Psi$ and
use it to eliminate $\sin\Psi$ in $de_I/dt$. Show that $de_I/dt$ is
positive for small $e_I$ and find the equilibrium eccentricity. The
existence of this equilibrium eccentricity allows tides to dissipate
energy in Io.

3. CHALLENGE PROBLEM (Extra Credit). Extend problem 1 from HW\#6 to
wide rings.\\a) Write down and solve integral expressions for the
total mass $M$, the orbital energy $E$, and the orbital angular
momentum $L$ of a vertically-thin ring of constant surface mass
density $\sigma$, inner radius $r_1$, and outer radius $r_2$.\\b) Use
Lagrange Multipliers to find the extrema in $E$ subject to constant
$M$ and constant $L$. Interpret and discuss your results.





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