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# Zero-Velocity Surfaces

Consider the surface (1089)

where (1090)

Note that . It follows, from Equation (1064), that if the mass has the Jacobi integral , and lies on the surface specified in Equation (1089), then it must have zero velocity. Hence, such a surface is termed a zero-velocity surface. The zero-velocity surfaces are important because they form the boundary of regions from which the mass is dynamically excluded: i.e., regions in which . Generally speaking, the regions from which is excluded grow in area as increases, and vice versa.

Let be the value of at the Lagrange point, for . When , it is easily demonstrated that   (1091)   (1092)   (1093)   (1094)   (1095)

Note that .     Figures 50-54 show the intersection of the zero-velocity surface with the - plane for various different values of , and illustrate how the region from which is dynamically excluded--which we shall term the excluded region--evolves as the value of is varied. Of course, any point not in the excluded region is in the so-called allowed region. For , the allowed region consists of two separate oval regions centered on and , respectively, plus an outer region which lies beyond a large circle centered on the origin. All three allowed regions are separated from one another by an excluded region--see Figure 50. When , the two inner allowed regions merge at the point--see Figure 51. When , the inner and outer allowed regions merge at the point, forming a horseshoe-like excluded region--see Figure 52. When , the excluded region splits in two at the point--see Figure 53. For , the two excluded regions are localized about the and points--see Figure 54. Finally, for , there is no excluded region. Figure 55 shows the zero-velocity surfaces and Lagrange points calculated for the case . It can be seen that, at very small values of , the and Lagrange points are almost equidistant from mass . Furthermore, mass , and the , , and Lagrange points all lie approximately on a unit circle, centered on mass . It follows that, when is small, the Lagrange points , and all share the orbit of mass about (in the inertial frame) with being directly opposite , (by convention) ahead of , and  behind.   Next: Stability of Lagrange Points Up: The Three-Body Problem Previous: Lagrange Points
Richard Fitzpatrick 2011-03-31