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{\Large ASTR450 Homework \# 5 -- Two Body Problem\\
Due Tuesday, March 14}
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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 available from
the class web page.

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 numerically. The
six orbital elements could be translated into three initial positions
and velocities with a little extra effort.

3. {\bf ASTR688O Problem} Danby: Page 143, Problem 21. Find the
semimajor axis and eccentricity of the orbit too.

4. Derive Equations 6.7.9 on page 163.  Start with the definitions
6.7.3.

5. (Due in two weeks). 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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