Distribution Functions

Logarithm of the distribution function, f(E) for spherical isotropic systems obeying the deVaucauleurs profile; the Jaffe profile; the King profile. These have been derived from the density profiles, (r), using the Eddington formula. Recall, E (r, v) = ½v2 + (r) is the net energy (per unit mass), and is more negative for more bound stars. Clearly, places in the 6-D phase space which are tightly bound (e.g. near the center with low velocity) have more stars, while places in the 6-D phase space which are nearly unbound (e.g. at larger radii with higher velocities) have few stars. Don't forget, however, that there is a much larger phase space for the nearly unbound stars (4 r2 dr & 4 v2 dv) so when the integral over phase space volume is done we find that most stars are nearly unbound and only a few are tightly bound (see next section and figure).