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# Co-Rotating Frame

Let us transform to a non-inertial frame of reference rotating with angular velocity about an axis normal to the orbital plane of masses and , and passing through their center of mass. It follows that masses and appear stationary in this new reference frame. Let us define a Cartesian coordinate system in the rotating frame of reference which is such that masses and always lie on the -axis, and the -axis is parallel to the previously defined -axis. It follows that masses and have the fixed position vectors and in our new coordinate system. Finally, let the position vector of mass be --see Figure 48. According to Chapter 7, the equation of motion of mass in the rotating reference frame takes the form (1050)

where , and   (1051)   (1052)

Here, the second term on the left-hand side of Equation (1050) is the Coriolis acceleration, whereas the final term on the right-hand side is the centrifugal acceleration. The components of Equation (1050) reduce to   (1053)   (1054)   (1055)

which yield   (1056)   (1057)   (1058)

where (1059)

is the sum of the gravitational and centrifugal potentials.

Now, it follows from Equations (1056)-(1058) that   (1060)   (1061)   (1062)

Summing the above three equations, we obtain (1063)

In other words, (1064)

is a constant of the motion, where . In fact, is the Jacobi integral introduced in Section 13.3 [it is easily demonstrated that Equations (1039) and (1064) are identical]. Note, finally, that the mass is restricted to regions in which (1065)

since is a positive definite quantity.   Next: Lagrange Points Up: The Three-Body Problem Previous: Tisserand Criterion
Richard Fitzpatrick 2011-03-31