Argument to be remarkably precise for systems similar to the Local Group.

The broadening of the distribution as the selection requirements are relaxed is easy to understand. Removing the isolation requirements allows third bodies to play a significant role in the dynamics. This can extend the upper tail of the $A_{200}$ distribution if mass from the third body falls inside $R_{200}$ for one of the pair galaxies or if the gravity of the third galaxy produces a tidal field which opposes the attraction between the pair members. It can extend the lower tail if the mass of the third body lies between the pair members but outside their $R_{200}$ spheres, thus enhancing their mutual attraction without adding to their mass. Removing the morphology constraint moves the whole distribution towards larger values and this effect is most pronounced in the large $A_{200}$ tail. This is because objects with more dominant bulges have more complex merger histories. They typically form in denser regions and their halos tend to be more massive and to have more complex structure.

Loosening the requirements on $V_{max}$ affects the distribution in a complex way. There is a tight relation between $V_{max}$ and $M_{200}$ (also $M_{halo}$) in the 