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{\Large ASTR450 Homework \# 11 -- Resonances and Final Review\\
Due Tuesday, May 6}
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1. Find the places along an elliptical orbit where the angle between
the radius vector and the velocity vector is minimum and maximum.

2.  Apollo 13 is on a circular orbit 100km above the Earth's surface.
It fires its engines and produces a tangential velocity which will
take it to the Moon (assume that the spacecraft's orbit is always in
the Moon's orbital plane).\\ a) 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 does it need to escape from the Earth altogether? \\ b)
What is the minimum extra velocity that might enable the spacecraft to
get to the Moon in the three-body approximation (Earth, Moon,
spacecraft)? Hint: Use Jacobi's integral to see when a transfer orbit
first becomes possible.

3. Estimate the period of oscillations in Neptune's eccentricity
induced by Jupiter. 

4. Pluto and Neptune are linked by a 3:2 eccentricity-type resonance.
The resonant term in the disturbing function felt by Pluto (due to
Neptune) is ${\cal R}_N = M_N\beta e_P\cos(3\lambda_P-2\lambda_N
-\comega_P)$ while the term felt by Neptune (due to Pluto) is ${\cal
  R}_P = M_P\beta e_P\cos(3\lambda_P-2\lambda_N -\comega_P)$. Here
``P'' subscripts refer to Pluto, ``N'' subscripts refer to Neptune,
$M, e, \lambda,$ and $\comega$ are mass, eccentricity, mean longitude,
and longitude of pericenter, and $\beta$ is a constant.\\ a) List all
other first-order 2:1 resonant terms for Neptune's perturbations on
Pluto. List all other second-order resonant terms. \\ b) Find the time
rates of change of the orbital elements ($a,e,i,\Omega,\comega$) of
Pluto and Neptune due to disturbing functions ${\cal R}_N$ and ${\cal
  R}_P$. Remember that $d{\cal R}/d\epsilon = d{\cal R}/ d\lambda$ and
$p=a(1-e^2)$. Give your answer to lowest order in eccentricity and
inclination, and keep only the most perturbed element in each of these
sets: $(de_N/dt, di_N/dt, de_P/dt, di_P/dt)$ and ($d\Omega_N/dt,
d\comega_N/dt, d\Omega_P/dt, d\comega_P/dt)$. You can set the other
members of each set equal to zero.\\ c) Show that the sum of the
orbital energy of Pluto plus that of Neptune is conserved to lowest
order in $e,i$.  \\ 


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