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{\Large ASTR450 Homework \# 8 -- The 3-Body Problem\\
Due Tuesday, April 18}
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1. Use the ``General Three-Body Integrator'' on the class webpage to
investigate orbits near the L4 and L5 Lagrange points. Try different
initial conditions to see how Tadpole (orbits around one of L4 or L5)
and Horseshoe orbit (one that surround both L4 and L5) look. For a
Jupiter-mass planet, roughly how many orbital periods does it take to
go once around the equilibrium point? How about for an Earth mass
planet?  For small Tadpole orbits, try increasing the secondary mass -
at what mass ratio is stability lost? Compare this with the value
discussed in class.  Write up a page discussing your findings and
attach some relevant orbits. Explore and have fun!

2. Do you taxes!

3. 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}
   Amir Caspi: Initially Circular Orbits with $i=0\dg$ \\
   Albert Davison: Initially Circular Orbits with $i=15\dg$ \\
   Ross Henry: Initially Circular Orbits with $i=30\dg$ \\
   Jacob Pimentel: Initially Circular Orbits with $i=45\dg$ \\
   Amanda Proctor: Initially Circular Orbits with $i=60\dg$ \\
   Roman Pyatkov: Initially Circular Orbits with $i=75\dg$ \\
   Kelly Shockey: Initially Circular Orbits with $i=90\dg$ \\
   Stacy Teng: Initially Circular Orbits with $i=105\dg$ \\
   Jian Chen: Initially Circular Orbits with $i=120\dg$ \\
   Kenn Flynn: Initially Circular Orbits with $i=135\dg$ \\
   Jianyang Li: Initially Circular Orbits with $i=150\dg$ \\
   James Marshall: Initially Circular Orbits with $i=165\dg$ \\
   Donna Pierce: Initially Circular Orbits with $i=180\dg$ \\
\end{center}

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, April
18. 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)!
 
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