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Unformatted text preview: These lecture notes were prepared for Rutgers Physics 341/342: Principles of Astrophysics by Prof. Chuck Keeton, and modified by Profs. Saurabh Jha and Eric Gawiser. All rights reserved. c 2011 Lecture 12: Three-Body Problem We have now completed the one-body and two-body problems. The extension to the three- body problem should be easy, right? No! In fact, it was the gravitational three-body problem that led Henri Poincar´ e to discover ... I. General Three-Body Problem: Chaos Or, “sensitive dependence on initial conditions.” A classical chaotic system is still deterministic , meaning that its behavior over time is gov- erned by specific laws. If you know the laws, together with the state of the system at any given time, then you can predict the behavior at any future time. There is nothing random about the system (unlike quantum mechanics). However, it may be the case that very slight changes in the initial conditions can lead to very different results. If we don’t know the initial conditions perfectly, then we cannot predict the long-term behavior in practice. This is nicely illustrated with a Flash applet at: http://faraday.physics.utoronto.ca/ PVB/Harrison/Flash/Chaos/ThreeBody/ThreeBody.html One immediate implication is that we don’t actually know what will happen to the Solar System millions/billions of years into the future! So we cannot actually solve the three-body problem in general. But we can make progress, and uncover some interesting applications, by starting with ... II. The Restricted Three-Body Problem II.1. Theory: Lagrange Points (See § 18.1 of Carroll & Ostlie.) Specifically, consider the problem depicted below, with the following simplifying assumptions: • take m M 1 ,M 2 so the “planet” does not affect the motion of the two “stars” 1 Figure 1: Set-up for the restricted three-body problem. (From Carroll & Ostlie Figure 18.1.) • take the two stars to have circular orbits, and keep the planet in the orbital plane of the two stars So the two stars just go around in their orbits. What does the planet do? In this problem it is convenient to work in a rotating reference frame, in which the stars are fixed. But we must be careful, because Newton’s laws in their usual form hold only in an inertial (non-rotating) reference frame. However, we can modify Newton’s laws to account for the rotating axes. Consider Figure 1, where the stars are executing circular orbits around the center of mass in the x- y plane. Now consider rotating the x and y axes (around the z-axis coming out of and going into the page) at the same rate as the orbiting stars, an angular frequency of ω = 2 π/P . Then the stars will remain fixed in the rotating coordinates (with x 1 =- r 1 , x 2 = + r 2 , and y 1 = y 2 = 0 at all times). Because their positions are fixed, the stars’ velocities and accelerations in the rotating frame are zero....
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This note was uploaded on 10/20/2011 for the course PH 341 taught by Professor Gawiser during the Fall '11 term at Rutgers.
- Fall '11