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What is the Length of a Year on Planet Sitnikov?The natural question is: How can one describe the motion? The answer is surprisingly complicated,...Bill Casselman
In an impossibly distant galaxy far, far away lies the Sitnikov system, in which a single bizarre planet orbits a pair of stars. Well ... "orbit" isn't quite the word, as we shall see, although the planet certainly moves under the influence of the gravity due to the massive stars. In our solar system, the planets are far smaller than the sun, and their motion is not affected in a major way by each other  each of them moves in an orbit very close to the ellipse that it would describe if there were in fact no other planets in the system. Planets in a binary star system, where there are two suns of large mass instead of one, behave very differently. We'll take a look at one of the simplest possible examples, that first examined by the Soviet mathematician K. Sitnikov. Binary star motionThe stars are assumed to have equal mass, and we assume their center of gravity to be at rest  in fact I'll take it to be the origin of our coordinate system. The stars therefore move in congruent ellipses with the origin as common focus.
I'll allow the eccentricity
In this figure, The position of the stars is shown at equal time intervals. Introducing the planetWe now place a very, very small planet into the system. It will 'orbit' in a rather peculiar manner  we assume that it simply moves along the line perpendicular to the plane of the stars, passing through the common focus of the stellar orbits. We shall look at what happens in the limiting case where the mass of the planet vanishes. The gravitational force on the planet varies for two reasons  for one thing, the planet moves back and forth away from the stellar plane, and on the other the stars themselves are moving. The force of gravity is greatest when the planet crosses the stellar plane and the stars are at periastron (nearest to the focus).
A very small planet moves perpendicularly to the motion of the stars. The natural question is: How can one describe the motion? The answer is surprisingly complicated, and much of what happens is a good model for what happens in much more complicated situations. The starting point is the differential equation expressing Newton's Law of Gravity. Choose units of mass, length, and time so that each star has mass where The case of circular motionThere is one situation which is far, far simpler than all others  when the eccentricity of the motion of the stars vanishes, or in other words when the stars move in a circle.
The only simple case is when the stars move in a circle. In this case, the function in which there is no remains constant. This is the limit of the expression for energy per unit mass as the mass of the planet tends to The constant must lie at
The curves of constant energy. Here The blue area marks the bounded trajectories. When the motion in this simple case is bounded, the planet just keeps cycling through a given family of states. This motion amounts to an oscillation back and forth across the stellar plane. Since the restoring force of gravity weakens with distance from equilibrium, the period of oscillation is larger for larger swings. This is opposed to what happens for a harmonic oscillator, where the period is independent of the amplitude. The following figure exhibits the graphs of
Here Plots of In summary, the case where When eccentricity does not vanishWhen
Here, with
The same impression of chaotic behavior remains even more valid for
It looks very much as if the planet were about to escape to infinity, or at the very least oscillate with increasing and unbounded magnitude as time goes on. This kind of behavior can be intuitively explained in terms of a 'sling shot' effect, similar to what happens when a comet's orbit in our solar system is seriously modified when it passes close to Jupiter. In fact, just about anything is possible. The basic theorem in the subject is essentially due to Sitnikov and Alekseev. It is proven carefully and with reasonable clarity in Moser's book. Suppose we associate to every planet trajectory a sequence For any given value of This theorem allows coming from or escaping to infinity, in which case the sequence stops at one end or the other. Moser's proof introduces structures much like Smale's famous horseshoe, where a similar theorem is proven about sequences of bits. Sections of motionOne way to visualize what is going on is to take snapshots of the planet in various situations. In Moser's book, he does this by recording, as we have already seen, when the planet crosses the stellar plane. Another way to do this is by recording position and velocity at regular time intervals. The natural choice is the period of the stars. In the first figure below, this is done for the case
Plots of In the next two pictures we see the plots for
Plots of
Plots of Explaining exactly what's going on is rather complicated. One thing that is not so complicated, however, is explaining the 'islands' one sees, say in the second plot. One of the islands is centered at (z, z') = (1.847, 0). If I plot the solution of the differential equation with those initial conditions, I get:
Graph of This is very, very close to a periodic solution that is in a How the plotting was carried outSolutions to the differential equations describing the planet's motion were computed by a variant of the extrapolation method of BulirschStoer described in the wellknown book Numerical Recipes. To find out moreThe literature on Sitnikov's system is very large, but in spite of that I am not aware of any single account of Sitnikov's system that is at once easily readable, thorough, and reliable. Accounts that attempt to be rigorous are not always clear, and those that attempt to give an intuitive explanation of what's going on rarely make any attempt at all to prove claims.
Bill Casselman Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal , which also provides bibliographic services. 
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