TAB 33

The Three Body Problem

The most famous unsolved problem in mathematics is the Three Body Problem. All the famous mathematicians have tried to solve it - from Euler, Laplace, Lagrange, Jacobi, Poincare, Lemaitre, and many more. Solutions are known to special cases - e.g. two large bodies in circular orbits and a third small body, all in the same plane - but the general problem of three bodies in random motion has no known solution.

This website considers one special case, called the circular coplanar problem. Consider one example, the Sun-Jupiter-Asteroids system (assume Jupiter is in a circular orbit, which is not too far off because its orbit is of very small eccentricity). (intro) also covers this basic material, with an emphasis on "anti-gravity."

The Three Body Problem

If you draw a line from the sun to Jupiter, the center of mass or balancing point between them is near the sun. This is called the barycenter. Now make a coordinate system at this center, and let it rotate so the x-axis follows Jupiter around the sun. This makes the sun orbit the barycenter in a small circle and Jupiter in a very large circle. The sun and Jupiter are the two primary bodies. The primaries are always very large masses and the third body is much, much smaller by comparison - e.g. a satellite orbiting Earth in the Earth-moon system.

The 3BP is in Rotating Coordinates

There are five important points in the Three Body Problem (3BP) called Lagrange Points, numbered L1 through L5. The L4 and L5 points are stable equilibrium points and are situated at the apex of an equilateral triangle with one side formed by the primary masses. A small body at L4 or L5 will stay there, indefinitely. The small mass can also be in a small orbit around the L4 or L5 point, and will remain in that orbit even if there are small forces acting upon it. In the sun-Jupiter system there are small clusters of asteroids orbiting L4 and L5 and they are called the Trojan and Apollo asteroid groups.

The Sun-Jupiter-Asteroids 3BP

The Trojan and Apollo Asteroids Rotate with Jupiter

Keep in mind always with the 3BP that the coordinate system is rotating along with the primary bodies, so that the two big masses always remain on the x-axis. That means the Trojan and Apollo asteroids always keep the same triangular orientation to the sun and Jupiter, and in fact have the same period of revolution around the sun as Jupiter.

The Colinear Lagrange Points

There are three more Lagrange Points: L1, L2, and L3. The figure above shows an approximate location for the Earth-moon system. There is always a figure-8 shaped closed orbit (also called a periodic orbit) around both primary masses and through the L2 point. This is called a "free return" orbit ( 9 ) and early Apollo missions to the moon followed exactly this trajectory because if anything went wrong (as it did on Apollo 13; on the far side of the moon, where their engines lost all power) then the spacecraft will stay right on the loop and return directly to Earth without the need for any thrusts or expenditure of fuel.

These three Lagrange points are unstable equilibrium points. An object can orbit one of them but if there is the slightest perturbation, or disturbance, to the orbit then the small object will draft out of the orbit and never return. Satellites are in these "halo orbits" at L2 and L3 to study the moon from a fixed altitude - and occasionally in an L1 orbit as well.

Stable Orbits around the Colinear Lagrange Points

These five Lagrange points (named after the famous French mathematician who first solved the problem) exist for every 3BP, even if the two primary bodies are stars. The L1, L2, and L3 points are sometimes called the colinear Lagrange points and the L4 and L5 the equilateral Lagrange points.

It may help to understand these points a little better if you look at a plot of what are called the zero velocity curves.

 

Zero Velocity Curves

It may help to understand these points a little better if you look at a plot of what are called the zero velocity curves.

You can see the two primary masses, and all the Lagrange Points. Notice also the free return loop that intersects at L2. There are several other periodic, or closed orbits, if you look carefully. These curves are just like contour lines for a hilly area on a land map. A spacecraft with a specific velocity, or amount of total energy, cannot cross a line - just like a person hiking the hills follows a line of constant height. The L4 and L5 points are in small valleys - if you move away from either of them it is uphill in every direction, so if a ball is located there it will always roll right back again. That's why they are called stable equilibrium points. The other three Lagrange points are much more precarious because those points are at saddles - the slope is both up and down around them, so if you move away in any direction, you keep right on going and do not return.

 

The Twin Towers

The above illustration is a three dimensional plot of the gravity around two large masses. The previous figure is just a cross section of this three dimensional image in the plane of the two masses.

The Subatomic 3BP

There is also a Three Body Problem on the atomic level because the equation for the force of gravity and the equation for the force between charged particles like electrons is exactly the same except for the constant. ( 19 ) The original atomic model, the Bohr atom - one electron orbiting one positron in the nucleus - came from Celestial Mechanics, and is what we call the Two Body Problem.

Not much will be said about the subatomic 3BP here but it is explored at length in the Theory. ( 22 ) There the free return loop becomes a molecular bonding orbital, and Lagrange points are bonding points between and within molecules. Everything that we know about the 3BP in astronomy has an equivalent in subatomic physics. This is important because it suggests profound new ways of looking at matter - perspectives that could conceivably lead to such science fiction devices as the medical tricorder, the material transporter, or food synthesizers.

About the 3BP

That's all the theory you need to follow this section, and in fact not much more is needed to follow the Theory because everything is based on the 3BP. Just remember that the three bodies are in the same plane, two large bodies and one very small third body; the coordinate system is rotating at a constant rate, and there are periodic orbits that follow a closed, repeating path.

The Twin Towers

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© 2004 WH Clark