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Three-body problem and variants

The three-body problem is the oldest unsolved .   .   .   . but you already knew that, else why would you be reading this page?

What this page presents is an applet for simulating and visualizing (i) the classical problem of three gravitating point masses, and (ii) a variant problem with two point masses and one smooth cluster.

     

You should see a button to the left saying Show Jacobi window.   (If not, see here.)   To download, see below.

To get started, click in the timestep field and then press return. You should see two bodies moving along Keplerian ellipses which are themselves slowly varying. What are these two bodies? Well, let's go back a little .   .   .   .

Let's call the three bodies in the problem Uno, Dos, Tres. We now introduce two fictitious bodies: Tweedledum represent Tres relative to the barycenter of Uno and Dos; Tweedledee represents Uno relative to Dos.

Here are two important special cases:

Uno    Dos    Tres    Tweedledum    Tweedledee
Sun Earth Jupiter Jupiter - SunEarth Earth - Sun
Earth Moon Sun Sun - EarthMoon Moon - Earth

or more generally, a planetary case and a satellite case.

The applet shows the motion of Tweedledum (left panel) and Tweedledee (right panel), each in its instantaneous orbital plane. The inclinations of the two orbital planes are not shown, but (amazingly) they don't need to be; provided we know the initial mutual inclination, the inclination variation with time is implicit in the eccentricity variation. In other words, the two planar Keplerian ellipses encode the full three-dimensional motion.

The above is Jacobi's formulation of the three-body problem as two coupled planar Keplerian orbits. Traditionally the stuff of theoretical celestial mechanics, it emerges now as an aid to visualization.

Try pausing the run with the pause button. You can now resume the run with a new timestep, or you can change the initial conditions and start a new run. Which brings us to the orbital parameters. The first two lines refer to Tweedledum and Tweedledee respectively. Here a and e are obvious, while the other two parameters are mean anomaly and argument of pericenter (both in degrees). In the last line we have the initial mutual inclination (also in degrees); more than 90 degrees puts Tweedledee on a retrograde orbit. The input timestep refers only to the display, the integration timestep is different and chosen internally. That leaves the masses. Now the input masses are not the effective masses of Tweedledum and Tweedledee. Instead, the first line gives the mass of Dos and second line gives the mass of Tres; meanwhile the sum of Uno and Dos is taken to be unity.

A word about the user interface. To start a new run you have to press enter twice: first in any text field except the timestep field to set new initial conditions, and then in the timestep field to start the clock. If you miss the first `enter' the applet will just resume the last run.


The Jacobi program is written in literate Java and is free software. For source and Java byte code, download the Jacobi.jar file.