# Zero-velocity surfaces

Consider the surface

 (9.60)

where

 (9.61)

Note that . It follows, from Equation (9.35), that if the mass has the Jacobi integral and lies on the surface specified in Equation (9.60), 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; that is, regions where . 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

 (9.62) (9.63) (9.64) (9.65) (9.66)

Note 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.