Circular restricted three-body problem

Let us define a Cartesian coordinate system in an inertial reference frame whose origin coincides with the center of mass, , of the two orbiting masses, and . Furthermore, let the orbital plane of these masses coincide with the - plane, and let them both lie on the -axis at time . (See Figure 9.1.) Suppose that is the constant distance between the two orbiting masses, the constant distance between mass and the origin, and the constant distance between mass and the origin. Moreover, let be the constant orbital angular velocity. It follows, from Section 4.16, that

(9.1) | ||||||

and | (9.2) |

where .

It is convenient to choose our unit of length such that , and our unit of mass such that . It follows, from Equation (9.1), that . However, we shall continue to retain in our equations, for the sake of clarity. Let and . It is easily demonstrated that and . Hence, the two orbiting masses, and , have position vectors

respectively. (See Figure 9.1.) Let the third mass have position vector . The Cartesian components of the equation of motion of this mass are thus

where