Let be the value of at the Lagrange point, for . When , it is easily demonstrated that
Figures 9.7 through 9.11 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 that lies beyond a large circle centered on the origin. All three allowed regions are separated from one another by an excluded region. (See Figure 9.7.) When , the two inner allowed regions merge at the point. (See Figure 9.8.) When , the inner and outer allowed regions merge at the point, forming a horseshoe-like excluded region. (See Figure 9.9.) When , the excluded region splits in two at the point. (See Figure 9.10.) For , the two excluded regions are localized about the and points. (See Figure 9.11.) Finally, for , there is no excluded region.
Figure 9.12 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.