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\begin{center}
{\Large ASTR450 Homework \# 9 -- Post Midterm Exam\\
Due Tuesday, November 26}
\end{center}


1. Checking Limits. \\For parts a) and b), assume a flat Earth with
gravity independent of height. For parts c)-f), assume a spherical
Earth with gravity proportional to the inverse square of distance from
the center. \\a) Solve for the maximum height $H$ and downrange
distance $D$ for a projectile launched with velocity $v$ at an angle
$\theta$ from the horizontal in a uniform gravity field $g$. \\ b)
Check limits and units to show that your answers make sense! \\ c) Do
problem \#170 from the ``Physical Intuition'' handout (the final
problem from the midterm).  Gravity is {\bf not} constant in this
problem.  Do your limits carefully ... part a) may help! \\ d) Solve
for the maximum height exactly and check your intuition from part c).
\\ e) Now solve for the maximum height $H$ and downrange distance $D$
of a projectile exactly when it is launched at an angle $\theta$ from
the horizontal. Start by arguing that trajectories will be segments of
conic sections.  What conservation laws can you use?  Your answer will
be algebraically messy - simplify it as much as you can (using the
dimensionless parameter $X$ from part c) will help!). \\ f) Do your
answers make sense?  Consider all of the limiting cases that you can
think of as tests -- including your answers from a) and d)!  Impress
me with your thoroughness in testing and write down all of your
checks. Bonus points for anyone who thinks of one that I don't!

2. Three-Body Motion. \\Consider a special case of the circular
restricted three body problem where $M_1 >> M_2 >> m$ (mass
hierarchy). We'll consider motion near $M_2$, which is a special case
of Hill's Problem. \\ a) Equate forces to find $x$, the
distance between $M_2$ and $L_1$, the middle Lagrange point, treating
$m$ as a test particle. You will need to assume $R >> x >> \mu R$,
where $R$ is the distance from $M_1$ to $M_2$ and $\mu =
M_2/(M_1+M_2)$.  Your answer should be accurate to
lowest-order in $\mu$.\\
b) Argue that a small displacement from $L_1$ in the
$\hat z$ direction will result in small vertical oscillations through
$L_1$, and find the oscillation frequency in terms of the mean motion
$n = \sqrt{G(M_1+M_2)/R^3}$ of $M_2$ about $M_1$.

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