Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!linus!decvax!harpo!floyd!clyde!ihnp4!zehntel!hplabs!sri-unix!RHB@MIT-MC From: RHB@MIT-MC@sri-unix.UUCP Newsgroups: net.physics Subject: Stable Grav. Points Message-ID: <12255@sri-arpa.UUCP> Date: Mon, 3-Oct-83 11:14:00 EDT Article-I.D.: sri-arpa.12255 Posted: Mon Oct 3 11:14:00 1983 Date-Received: Thu, 6-Oct-83 02:28:07 EDT Lines: 40 From: Robert H. Berman A "Modified Three Body problem" consistes of two finite mass bodies and a third infinitesimal test body. The test body is used to find gravitational equilibirum points. There are actually 5 equilibirum points of which 2 are stable. They may be located by the following intuitive arguments. First, the two finite mass bodies (M_1 and M_2) revolve around their center of mass with a definite angular velocity Om. Consider what happens in a frame of reference rotating with that velocity. Pick the origin of the frame to be the center of mass. Then the two bodies can be placed at coordinates (-x_1,0) and (x_2,0) in the rotating frame. Next, the acceleration of the test particle is the combined gravitational force from the two bodies ** and ** the centripetal acceleration in the rotating frame. Equilirbirum points are found when there is no acceleration on the test body. Stability is determined by what happens when small displacements from equilbirum occur, namely whether they stay small or grow larger. This can be determined by checking the "tidal" forces on the test particle in equilbirum. The first Lagranian equilibrium point is the orgin (0,0), denoted as L1. The acceleration on the test particle is 0* Om**2 - GM_1/x_1**2 + GM_2/x_2**2 = 0 (because it is the CM) The next two Lagrangian points are somewhere to the right of (x_2,0) and to the left of (x_1,0). The diminshed gravitational pull from the further body is compensated for by the increased centripetal acceleration. Together, these points L1, L2 and L3 are known as the "straight-line" solutions. They are all unstable because changes in the centripetal force don't match changes in the gravitational forces. The stable equilbirum points L4 and L5 are located symmetrically at (0,y_L) and (0,-y_L). They are both stable, as small displacements from the points continue to orbit around the points in epicyclic orbits.