Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 9/18/84; site utastro.UUCP
Path: utzoo!linus!philabs!cmcl2!seismo!ut-sally!utastro!bill
From: bill@utastro.UUCP (William H. Jefferys)
Newsgroups: net.physics
Subject: Re: slingshot effect
Message-ID: <88@utastro.UUCP>
Date: Thu, 21-Nov-85 10:42:05 EST
Article-I.D.: utastro.88
Posted: Thu Nov 21 10:42:05 1985
Date-Received: Sat, 23-Nov-85 10:02:16 EST
References: <395@wuphys.UUCP> <693@lasspvax.UUCP>
Organization: U. Texas, Astronomy, Austin, TX
Lines: 41

> 
> At least two explanations have appeared for the slingshot effect that do NOT
> involve a burn at the bottom of the planet's gravity well. It might at first
> appear that this violates conservation of energy. In fact what happens is
> that both energy and momentum are transferred from teh planet to the
> spacecraft -- the planet *slows down* in response to the flyby.
> 
> I would be interested in seeing this worked out in some detail. Might even
> do it myself if I have time.

This whole thing has been blown out of porportion.  Ethan Vishniac had it
right.  No burn is required at the bottom of the potential well.  The 
easiest way to think about it is as follows:

In the center-of-mass frame of reference of the planet and the probe,
the two bodies approach and recede on symmetrically placed hyperbolic
orbits.  The two bodies approach and recede with the same speed.
(The massive planet hardly moves at all in this encounter and to
a first approximation we can regard it as fixed, i.e., on a degenerate
hyperbolic orbit).

But we observe the event in a "laboratory" frame of reference fixed with
respect to the Sun.  In that frame, the probe enters the encounter with
one speed and leaves with another.  As pointed out above, however, no
violation of energy/momentum conservation is involved, since the orbit
of the massive planet is also affected, albeit imperceptibly.  The net
result is that energy has been transferred from the planet to the probe.
This is Freshman Physics stuff.  To solve it, transform to the C.O.M.
frame, do the calculation, then transform back to the lab frame.

If you ignore the mass of the probe and the eccentricity of the planet's
orbit, there is a conserved quantity in the problem.  In this simplified
case (the Restricted Problem of Three Bodies) the Jacobi Constant is
conserved.  The conservation of this constant is the basis of
Tisserand's criterion for determining whether or not two comets that
appeared at different times are in fact the same comet at two apparitions,
the comet's orbit having been strongly perturbed by Jupiter in the interim.
(This is a not-uncommon occurrence).

You can read all about this in Szebehely's "Theory of Orbits" (Academic
Press 1967), which is the standard work.