Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83; site ho95b.UUCP
Path: utzoo!watmath!clyde!bonnie!akgua!whuxlm!whuxl!houxm!ho95b!ran
From: ran@ho95b.UUCP (RANeinast)
Newsgroups: net.space
Subject: Lagrange Points
Message-ID: <298@ho95b.UUCP>
Date: Mon, 28-Jan-85 16:17:43 EST
Article-I.D.: ho95b.298
Posted: Mon Jan 28 16:17:43 1985
Date-Received: Wed, 30-Jan-85 04:14:12 EST
Organization: AT&T-Bell Labs, Holmdel, NJ
Lines: 63




Here are the Lagrange points:


                (4)
                 x


                          ^
                          |
  (1)       (2)       CoM      (3)
   x     O   x         .  O     x

         |
         V

                (5)
                 x


O-locations of large masses.
x-locations of Lagrange points.
CoM-Center of Mass of the two large masses (in this picture the right one
    is more massive than the left one)
Arrows try to show direction of travel around the CoM.
Everything in the picture rotates about the CoM at the same rate.

The Lagrange points are those where the gravitational forces from the
masses and the centripetal force from the rotation of the point
cancel (for the nitpickers: where the grav forces equal the mass of an
object located at the point times the centripetal acceleration -- F=ma).
These are the only ones where this is true.  1, 2, & 3 are fairly easy
to see.  4 & 5 occur because the centripetal force is aimed at the
CoM so that the components of gravitational force perpendicular to
that line cancel (this is easiest to see if you consider the big
masses to be equal; the CoM is then exactly between them, one mass
pulls the point forward, the other back;  the net effect is for the
point to just rotate around the center of mass at the same speed as
the big masses).

Regarding stability:
Stability is much more difficult to show (at least in a posting).
Essentially, the technique is,
for a particle located at one of the points,
to expand the potential (gravitational and centripetal)
in a small perturbation around that point.
This expended potential can look like either a hill or a valley.
If a valley, then the point is stable (that is, a particle at this point,
when perturbed, oscillates about that point, but stays near it, like
a ball at the bottom of a bowl).
If a hill, then the point is unstable (when perturbed, the particle
leaves the point, like a ball on top of an inverted bowl).
1, 2, & 3 are unstable points.
4 & 5 are stable.


-- 

". . . and shun the frumious Bandersnatch."
       Robert Neinast (ihnp4!ho95b!ran)
       AT&T-Bell Labs