The SPSU Mathematics Colloquium

Presents

Dr.Jack Pace

Associate Professor of Mathematics

SPSU

 COMPUTER DISPLAY OF REMARKABLE ORBITS

 IN THE THREE-BODY PROBLEM

        Some interesting stable or unstable orbits are known in the general three-body case, and several new ones have been discovered just recently, including a remarkable figure-8 orbit. I will show some of these using computer programs.

          Even when several simplifying restrictions are imposed, the problem has no formula solutions. However, the system of differential equations can be solved by modern numerical techniques using a computer, and the orbital motion of masses 1, 2, and 3 can be shown on the computer screen. Also, there are five special equilibrium solutions, called L1 - L5, and a conserved quantity called the Jacobi constant, which can help us understand the behavior of the orbits. Furthermore, by using a numerical technique called the shooting method, special periodic orbits can be found interactively using the computer as a tool. Some of these orbits are remarkable, such as a horseshoe orbit, which is actually followed by an asteroid (Cruithne), occupying almost the same orbit as the Earth, and which can be called a second moon of the Earth.  Orbits around the Earth-Sun L1 - L5 points are just now being used as parking orbits for special-purpose satellites such as the solar observer SOHO and the cosmic microwave background mapper WMAP. Points L4 and L5 have been proposed as good locations for human colonies in space. Use of the L1 - L5 points of various planets is also being considered to allow missions to other planets with a minimum requirement of fuel. 

          After a little theory, the talk will consist of using a computer to show many examples of unusual orbits, both in a rotating coordinate system and in a non-rotating (inertial) system. The process of finding periodic orbits computationally, using the shooting method, will also be shown.  

Thursday  October 28

2:15 PM in D-219

 REFRESHMENTS AT 2:00 PM