Variations on the theme of Michel Henon's Isochrone
  A variation of Newton's method of mapping Kepler's orbits into orbits in the
simple harmonic oscillator is shown to give Henon's Isochrone. The statistical
mechanics of a micro-canonical ensemble of isochrone oscillators shows that the
temperature reaches a maximum as a function of the energy and then declines to
zero at the escape energy. In that declining region adding heat (energy)
decreases the temperature, as occurs in star clusters. We then define the
internal temperature of an ensemble of binary stars all at the same (negative)
energy and show that they too get cooler when heated and hotter when cooled.
When the internal temperature of a binary is less that the temperature of the
stars it interacts with, then on average heat will flow into it, making it less
bound and of still lower temperature. Conversely hard binaries have higher
internal temperatures than the local stars, so they lose energy and become
hotter and yet more strongly bound, a process invoked by Michel Henon in his
explanation of star-cluster evolution. Finally we give an isochronal variation
of Newton's exactly soluble N-body problem.
