The previous two chapters were intended to develop a qualitative understanding of the nature of the interactions between binary and single stars, with no more than order-of-magnitude estimates. The present chapter attempts to sharpen these ideas with some approximate quantitative results. We imagine that binaries and single stars are distributed throughout some region of space, such as a part of a star cluster, and we want to know how frequently three-body interactions are taking place.

Cross sections

What is important in applications (Chapters 23f) is the energetics of these interactions, and that is why such stress was laid on the distinction between soft and hard pairs in Chapter 19. In the present chapter this consideration implies that we may be interested in interactions with binaries of a given energy. Encounters with such binaries, however, are taking place all the time with stars which approach from random directions and random distances. Therefore, besides the energy of the binary, we usually do not know or care about the other properties of the participants, except for their statistical distribution. This is true of the approach path of the third body, and also usually it is true of the other five parameters (besides the energy) which determine the relative motion of the binary components.