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\begin{center}
{\Large ASTR430 Homework \#7 \\
Due Thursday, May 10, 2001}
\end{center}



1. a) Use a symmetry argument 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, just above the ring, the ring's gravity dominates over
the vertical component of gravity from Saturn. The undetermined
constant from part 1a) is $2\pi$.

c) 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. 

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 $\Psi = 2l_E - l_I
+ \varpi_I$ where $l$ is the satellite longitude and $dl/dt \approx n$.\\ 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
($\dot\varpi_{drag}$ is also negligible because of the $e_I$ in the
denominator of the corresponding resonant term). Set the new
$da_I/dt$ equation equal to zero (not a great approximation, but not
too bad), solve for $\sin\Psi$ and use it to eliminate $\sin\Psi$
in $de_I/dt$. Solve for the equilibrium eccentricity. The existence of
this equilibrium eccentricity allows tides to dissipate energy in Io.



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