Newsgroups: sci.space
Path: utzoo!henry
From: henry@utzoo.uucp (Henry Spencer)
Subject: Re: Finding Lagrange's Libration Points
Message-ID: <1989Jan22.235911.23395@utzoo.uucp>
Organization: U of Toronto Zoology
References: <1989Jan18.044744.18328@sq.uucp>
Date: Sun, 22 Jan 89 23:59:11 GMT

In article <1989Jan18.044744.18328@sq.uucp> msb@sq.com (Mark Brader) writes:
>... I don't even see why L4 and L5 exist.
>I hope someone can explain it to me so that I do....
>... now can someone explain to me why the Trojan points are stable
>equilibria, or even why they are equilibria at all?  The same condition
>of sideways force would seem to apply.  Clearly (I think) the equilateral-
>triangle position cannot be exact; the tertiary would then be in an orbit
>exactly like the secondary's despite the secondary's perturbing gravity.
>So where are the true L4 and L5 positions?  And why?

Odd though it sounds, the equilateral triangles are indeed exact.  The
key observation is that the tertiary is not in an orbit around the primary,
it is in an orbit around the center of mass of the primary-secondary system.
The secondary's "perturbing" gravity alters the net force vector on the
tertiary just enough to point it at the center of mass.  That's not a
sufficient condition, but it's necessary.  Beyond that, one pretty much
has to resort to math.  In particular, I know of no good intuitive way
of explaining why L4 and L5 are stable and L1-3 aren't; the key question
is not whether a nearby body feels a side force, but whether a perturbation
(of velocity or position) remains bounded (body remains near the point) or
grows (more or less) unboundedly.  The L1-3 points themselves are unstable;
carefully-chosen "orbits" around them are theoretically stable but in
practice the conditions are too fussy for true stability; L4 and L5 are
honestly stable, with slight gravity wells around them (subject to some
conditions, see below).

>Also, has anyone investigated situations where the masses are not so
>unequal?  I remember reading that the Trojan positions are stable if the
>some ratio exceeds 27; I think it was the secondary/tertiary mass ratio.

Not correct; the key requirement is that the primary/secondary ratio
exceed approximately 25.  (My favorite astrodynamics books, Archie Roy's
"Foundations of Astrodynamics", derives it as a requirement that the
ratio of secondary mass to total mass be under 1/2 - sqrt(23/108).)
Again, as far as I know, the question "why?" cannot be answered without
mathematics.
-- 
Allegedly heard aboard Mir: "A |     Henry Spencer at U of Toronto Zoology
toast to comrade Van Allen!!"  | uunet!attcan!utzoo!henry henry@zoo.toronto.edu