One can imagine the fun Mark Twain would have had with the concept of black holes and their influence. Unfortunately, he died in 1910, six years before K. Schwarzschild discovered these singular solutions of general relativity. Although this book is concerned with Newtonian systems, in which the ratio GM/Rc2 of gravitational energy to rest mass energy is very small, the presence of a central singularity which destroys or absorbs stars can have a significant effect on the surrounding Newtonian dynamics. Actually a black hole is just one example of such a singularity. Others are a supermassive star or spinar, and even just a region of such high density that stars disrupt inside it by physically colliding (Section 52).

We have already seen (Section 40.2) how a massive gravitating point modifies the distribution of surrounding stars when this distribution obeys a simple polytropic equation of state. The large point mass induces a central density cusp, in contrast to the normal flat distribution. We did not approach the central mass very closely in Section 40.2 because it destroys the polytropic behavior. Here we venture further in to see how the stellar orbits are distorted.

Let us suppose that the mass of the hole Mh is much greater than the mass of any individual star m*, but much less than the total mass of all stars in the cluster m*N.