When I wrote the program here, I tested it out with two stars, and they rotated nicely in ellipses about their common center of gravity. I tried the stable configurations of two stars with a planet close to one of them (worked fine), and with a planet at quite a distance from them both (also worked fine). I also tried 3 stars, 2 in a binary system and a third coming in from the outside. I had seen an article describing this motion: usually there is too much energy for the third to be captured, but instead, sometimes after an incredible amount of chaotic motion, one of the stars is ejected to infinity, with a binary system remaining. Any one of the three might be ejected. (See an example below under "Triple Star Systems".)

Finally, I tried out the system in Asimov's story. At first I used two stars with equal mass, and a planet with nearly zero mass. The stars rotated about each other in a perfectly circular orbit. (This is the red-green circle below.) I carefully started out the planet at the third vertex of an equilateral triangle, with the proper vector for its motion. (This is the blue orbit below. The initial position has the two stars at opposite ends of a horizontal diameter, with the blue planet at directly below them at the third vertex of an equilateral triangle.) It would stay at the vertex of the triangle for much of one revolution, and then drift into chaos. I thought that something was wrong with my approach or with my equations. See Figure 1 for a snapshop of this situation:

Figure 1.

• 2 stars: green and red, of equal mass, each rotating in the same exact circle about their common center of gravity,
• 1 planet: blue, of negligible mass, starting at the Lagrange Point, with the proper velocity vector.
• The initial position of the two stars is on the circle at the center, where the red changes to green. The line through them is horizontal.
• The initial position of the planet is directly below the stars at the third vertex of an equilateral angle.
• The planet stays in the L5 position for most of one rotation, but then it deviates more and more.
• After a chaotic orbit, the planet is eventually ejected to infinity.
• The applet Chaotic orbit gives this behavior. (Keep pushing the "Go" button.)

earth mass/moon mass  =  5.98 x 1024 kg / 7.35 x 1022 kg =   81.36
sun mass/jupiter mass = 1.989 x 1030 kg / 1.90 x 1027 kg = 1046

The number 24.96 is an interesting dimensionless constant, a product of basic Newtonian physics.

Going back to my simulation, I couldn't get a stable orbit with the ratio very close to 25 , but I was able to get it easily at ratios above 50 . The earth-moon L4 and L5 points are evidently stable even with perturbations from the sun and other planets. With the sun-jupiter system, there are two sets of asteroids in the asteroid belt (called the trojans) that move along in the L4 and L5 stable locations, ahead of and behind Jupiter and moving along with it.

Figure 2 shows a run with two stars having a mass ratio of 100 and a planet with negligible mass at the stable Lagrange Point:

Figure 2.

• 2 stars: green with mass 0.01 and red with mass 1.0. Red (almost exactly at the center of mass) is nearly stationary, wobbling and drifting downward very slowly, while green goes in a circular orbit about red.
• 1 planet: blue, of mass 0.000005, starting at the Lagrange Point, with the proper velocity vector.
• The arrows show the positions of all three bodies after about 10 revolutions. They are still in the Lagrange configuration.
• The applet Stable orbit gives this behavior. (Keep pushing the "Go" button.)

Figure 3 shows a run with a similar configuration, except that here there are 2 planets (each blue), one in each of the two stable Lagrange points. Again the 4 arrows show the postions of the two stars and the two planets after about 12 revolutions. The applet Stable orbit gives this behavior.

Figure 3.

Many double stars would not have as big a ratio, so their Lagrange points would not be stable. All these facts are explained in detail in the following articles:

References about Lagrange points:

• Wikipedia article on Lagrange Points
• Web Page About Lagrange Points
• Web Page About Lagrange Points
• Mathematical Stability Analysis by Neil J. Cornish

Other References:

• http://www.merlyn.demon.co.uk/gravity4.htm
• http://www.santafe.edu/~moore/gallery.html
• http://www.ams.org/featurecolumn/archive/orbits1.html
• http://www.orbitsimulator.com/gravity/articles/pluto.html

It's interesting that as of 2006-09-05, the Wikipedia article above refers to another article about Asimov's story but does not say that the configuration in the story is impossible.

Suppose one tries to rescue the story. One could imagine a binary system with one red star having 0.1 solar masses, and another bright star with 3 solar masses. The mass ratio is 30 , so this works. You still have to wonder how the planet got into position, but in Asimov's story, this was the only such planet known, so it was presented as very rare. The problem is that the planet had evolved life. Many people feel that even small red suns might have planets around them with life, and certainly the red sun would last long enough for this to happen (such a small red star lasts much longer than our sun). However the large star in the system would have a life span of well under half a billion years -- perhaps not enough time for life to develop. So in the end, even with the most optimistic assumptions, it's hard to make Asimov's story work.

Figure 4.

• 3 stars: Green, Red, and Blue, of equal mass.
• Red and Green form a binary system, while Blue is an interloper, coming in from the upper right.
• After a chaotic interaction, Green is ejected off to the left, while Blue and Red form a new binary system that leaves at the bottom.
• The applet Chaotic interaction of 3 stars gives this behavior.

Figure 5.
Very similar to Figure 4.

• 3 stars: Green, Red, and Blue, of equal mass.
• Red and Green form a binary system, originally coming from the middle, moving right and upward.
• Blue is an interloper, coming in from the upper right.
• First Red is thrown into a small orbit (toward the upper left), while Blue and Green rotate about one another (toward the lower right).
• Then Blue is thrown out into a larger orbit (straight down), while Red and Green rotate about one another (toward the top).
• Finally, Green is ejected to infinity off to the upper right, while Blue and Red form a new binary system that leaves toward the lower left.

Figure 6.

• Red and Green have mass 1 , while Blue has mass 0.1 .
• Blue starts at the unstable Lagrange L5 point, with the correct initial velocity.
• Blue stays at the Lagrange L5 point for most of one revolution, and then drifts off.
• Blue goes through a chaotic orbit, and affects the orbits of Red and Green somewhat, even with its small mass.
• Finally, Blue is ejected to infinity (upward), and Red and Green continue toward the left without Blue.