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{\Large ASTR450 Homework \# 6 -- The N-Body and 3-Body Problems\\
Due Tuesday, March 18}
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Reading: Read Chapter 8. 

1. Danby: Page 280, Problem 5.

2. Danby: Page 281, Problem 10.

3. Chapter 8. Rewrite Tisserand's Criterion so that it applies to i)
parabolic-parabolic transitions and ii) parabolic-elliptical and
parabolic-hyperbolic transitions. Answer the following questions using
the approximations that i) the Tisserand Constant is exactly conserved
and ii) Jupiter affects the comet only when it is very close to
Jupiter. Can Jupiter put parabolic $i=0$ comets onto circular
uninclined orbits? Can a prograde parabolic ($i<90^\circ$) comet be
perturbed onto a retrograde ($i>90^\circ$) parabolic orbit? Onto a
retrograde elliptical or hyperbolic orbit?

4. Finish the 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 subroutine, with
several example runs showing how you tested them.

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