A theorem on central velocity dispersions
  It is shown that, if the tracer population is supported by a spherical dark
halo with a core or a cusp diverging more slowly than that of a singular
isothermal sphere, the logarithmic cusp slope 'g' of the tracers must be given
exactly by g=2b where b is their velocity anisotropy parameter at the center
unless the same tracers are dynamically cold at the center. If the halo cusp
diverges faster than that of the singular isothermal sphere, the velocity
dispersion of the tracers must diverge at the center too. In particular, if the
logarithmic halo cusp slope is larger than two, the diverging velocity
dispersion also traces the behavior of the potential. The implication of our
theorem on projected quantities is also discussed. We argue that our theorem
should be understood as a warning against interpreting results based on
simplifying assumptions such as isotropy and spherical symmetry.
