Restricted three-body problem

The three-body problem considers three mutually interacting masses , , and . In the restricted three-body problem, is taken to be small enough so that it does not influence the motion of and , which are assumed to be in circular orbits about their center of mass. The orbits of three masses are further assumed to all lie in a common plane. If and are in elliptical instead of circular orbits, the problem is variously known as the "elliptic restricted problem" or "pseudorestricted problem" (Szebehely 1967, pp. 30 and 39).

The efforts of many famous mathematicians have been devoted to this difficult problem, including Euler and Lagrange (1772), Jacobi (1836), Hill (1878), Poincaré (1899), Levi-Civita (1905), and Birkhoff (1915). In 1772, Euler first introduced a synodic (rotating) coordinate system. Jacobi (1836) subsequently discovered an integral of motion in this coordinate system (which he independently discovered) that is now known as the Jacobi integral. Hill (1878) used this integral to show that the Earth-Moon distance remains bounded from above for all time (assuming his model for the Sun-Earth-Moon system is valid), and Brown (1896) gave the most precise lunar theory of his time.

Poincaré published his monumental Méthodes Nouvelles in 1899, emphasizing qualitative aspects of celestial mechanics, including modern concepts such at phase space surfaces of section. Birkhoff (1915) further developed these qualitative methods. The important problem of regularization was considered by Thiele (1892), Painlevé (1897), Levi-Civita (1903), Burrau (1906), Sundman (1912), and Birkhoff (1915). Painlevé proved that all singularities are collisions for n = 3. Sundman found a uniformly convergent infinite series involving known functions that "solves" the restricted three-body problem in the whole plane (once singularities are removed through the process of regularization). Since such global regularizations are available for this problem, the restricted problem of three bodies can be considered to be complete "solved." However, this "solution" does not address issues of stability, allowed regions of motion, and so on, and so is of limited practical utility (Szebehely 1967, p. 42). Furthermore, an unreasonably large number of terms (of order ) of Sundman's series are required in to attain anything like the accuracy required for astronomical observations.

To set up the problem, let be the largest mass, be a mass in a circular orbit of semimajor axis a about the center of mass of and , and let be a massless test particle. Also pick dimensions so that the gravitational constant G = 1, then the orbital period is

and the mean motion is

so

Because of the definition of and , the radii of their orbits are and , respectively. Now enter a coordinate system which rotates with and . In this system, has fixed coordinates and has fixed coordinates . The equations of motion of are then

where

This is converted to a Hamiltonian system with two degrees of freedom with

There is one integral of motion called the Jacobi integral and defined as

No other integral is known. Making a mapping of vs. x (Hénon 1983) for , there are four elliptic fixed points corresponding to periodic orbits around and in either direction. At C = 4, the trajectory is chaotic.

Primary resonances occur when makes orbits in the same time that makes J , or

where J is an integer. The period between conjunctions is then . For

there is chaos