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{\Large ASTR450 Homework \# 3 -- Central Force Motion\\
Due Tuesday, February 28}
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Reading: Start Danby's Chapter 6.

1. Danby: Page 82, Problem 8. To start, draw a picture of the
elliptical orbit and label the semimajor axis, semiminor axis and
latus rectum. The fact that a particle sweeps out equal areas in equal
times for all central forces will be useful for this problem. You will
need to look up some hairy integrals!

2. Danby: Page 84, Problem 10. You can assume that the force is
gravity. Draw a picture to start and use the equation that describes
the velocity of a particle in terms of $a$ and $r$. Under what
conditions will the future orbit be a circle? An ellipse? A parabola?
A hyperbola?

3. Danby: Page 84, Problem 14. The ``apsidal angle'' is the angle
between two successive turning points (apses) - it is 180 degrees for
a Keplerian ellipse. You'll need a good understanding of the
derivation on page 60 to do this problem.

4. Danby: Page 85, Problem 25. The six cases refer to how r varies
with time; r increasing monotonically with time from 0 to infinity is
one case. Use the Central Force Integrator to determine numerically
what happens to each of the six cases when the force law is changed to
$r^{-2.9}$.

5. Danby: Page 86, Problem 31. Find $h$ and $C$ and use Eq. 4.3.1.
Sketch the orbit for several different values of a.

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