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{\Large ASTR450 Homework \# 5 -- Two Body Problem\\
Due Tuesday, March 11}
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Reading: Read Chapter 9 \& start Chapter 8. Also, test your knowledge
of orbital elements with the 2D and 3D orbit viewers now available
from the class web page (click on: ``Celestial Mechanics Toolbox'').

1. Danby: Page 136, Problem 1. Given that the Moon's eccentricity is
0.05, what total fraction of the Moon's surface is visible to
observers on Earth over a month due to this effect?

2. Danby: Page 138, Problem 15. Give the initial conditions as six
orbital elements ($a$,$e$,$i$,$\Omega$,$\omega$, and $\nu$), taking
the equator plane to be the reference plane (the xy plane), the x-axis
to be the inertial reference direction, and the launch point to be in
the xz plane at the time of launch. You will probably need to use a
calculator to get the semimajor-axis and eccentricity. The six orbital
elements could be translated into three initial positions and
velocities with a little extra effort.

3. Danby: Page 143, Problem 21. Find the semimajor axis and
eccentricity of the orbit too.

4. Derive Equations 6.7.9 on page 163.  

5. In your favorite computer language, write six subroutines to
translate between the mean anomaly M, the eccentric anomaly $E$, and
the true anomaly $\nu$ for elliptic orbits. Find $E$ and $\nu$ given
$M=\pi/2$, $e=0.8$. Write down an inequality relating the three
anomalies over the pericenter to apocenter half of the orbit. How does
this change for the apocenter to pericenter half? Please turn in a
copy of your code.

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