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{\Large ASTR450 Homework \# 8 -- The 3-Body Problem\\
Due Tuesday, November 19}
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1. Class Research Project. What if the Moon's orbit were different?
Use the Planetary Satellite Integrator to investigate distant orbits
around the Earth. Use the default settings of the form for ``What if
the Moon's Orbit were twice as Large?'', but change the satellite's
initial conditions, the integration time, and the accuracy parameter
as needed. In each case, start the orbit above the positive X-axis
(the reference direction) at its maximum height above the XY plane
(the reference plane). See the Help File for comments on the
coordinate system, and check to see that you've got the initial
conditions right by viewing the Orbital Elements vs. Time plot or the
Cartesian Coordinates vs. Time Plot.

\begin{center}
   Anthony Alarcon: Initially Circular Orbits with $i=0\dg$ \\
   Mia Bovill: Initially Circular Orbits with $i=15\dg$ \\
   Damon Ellingston: Initially Circular Orbits with $i=30\dg$ \\
   Carmella House: Initially Circular Orbits with $i=45\dg$ \\
   Leila Malayeri: Initially Circular Orbits with $i=60\dg$ \\
   Jeremy Miller: Initially Circular Orbits with $i=75\dg$ \\
   Kaveh Pahlevan: Initially Circular Orbits with $i=90\dg$ \\
   Kajal Pancholi: Initially Circular Orbits with $i=105\dg$ \\
   Robyn Sanderson: Initially Circular Orbits with $i=120\dg$ \\
   Joseph Simons: Initially Circular Orbits with $i=135\dg$ \\
   David SSimpson: Initially Circular Orbits with $i=150\dg$ \\
   Patrick Taylor: Initially Circular Orbits with $i=165\dg$ \\
   Andrew Weber: Initially Circular Orbits with $i=180\dg$ \\
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Run simulations for different initial distances larger than the Moon's
current distance (60 Earth Radii), and determine the fate of orbits:
do they remain bound, escape the Earth, or crash into our planet?
Where do transitions between the various regimes take place? Note: to
determine if an orbit crashes into Earth or not, check to see if i)
the integration ends early and ii) the final distance of the satellite
is near 1 Earth radius. What is the escape signature in orbital
elements? Look at orbits in inertial X,Y,Z space, in coordinates that
rotate with the Earth's mean motion (Xsun, Ysun, Z), and in orbital
elements to get an understanding of what is happening to each orbit.
Do your orbits ever cross the Zero Velocity Curves (ZVCs)?  Are the
ZVCs a good indicator of the bound-escape transition? Print out
examples of interesting orbits to discuss in class on Tuesday,
November 19. Those of you doing the homeworks regularly, please write
up the results of your investigations and attach plots of interesting
orbits (with the initial conditions labeled!).  If you'd like to
explore further, try some eccentric orbits (be sure to note all of the
initial orbital elements)!

2. Using the paper by Burns et al. (1979), fill in the missing steps
in the derivations of Eqs 24 and 28 for $da/dt$ and $de/dt$. Show
enough work that I can see that you understand the derivations!

 
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