18 - Problems and extensions

Summary

Some basic aspects of many-body theory remain to be developed in future sections where they will be connected more closely to their astronomical applications. Here I give several extensions which can be worked out as problems by the reader, or can be used to enter the literature. Other suggestions for practice problems are sprinkled lightly throughout the text.

The point mass approximation

So far we have usually supposed that gravitating bodies, from dust grains to galaxies, can be treated as point masses. This is clearly an idealization. Geometric collisions make this approximation fail; so does tidal disruption. A third reason for failure is angular momentum transfer from the orbit to the spin of an object. Consider two solid ellipsoids, each of mass m and semimajor axes a, b where a > b. Show that, as they pass by one another at average distance r and velocity v, each acquires an angular momentum of order G(mae)²/r²v, where e² = 1 - (b/a)². How close do they have to pass to transfer an appreciable fraction of their orbital energy into rotational energy? Is this a plausible process for stars? For galaxies? What happens if the masses are not approximated as solid bodies? What residual internal circulation would result? Is there a net vorticity? How does the transferred energy change the size of the galaxy? (See Harrison, MNRAS, 154, 167, 1971.)