Properties of a Spherical Galaxy with Exponential Energy Distribution

Some analytical relations for the phase space functions of a self-consistent spherical stellar system are derived. The integral constraints on the distribution function by imposing a given ϱ(r) density distribution andN(E) fractional energy distribution are determined. For the case of radially-anisotropic velocity distribution in theE→0 limit the constraint by an exponentialN(E) implies thatf(E, J ²) tends to zero in the order (-E)^3/2. This lends analytical support to the use of the Stiavelli and Bertin (1985) distribution function for modeling elliptical galaxies. Maximum phase space density constraint confirms the necessity of high collapse factors to produce such a distribution function. Limits on the steepness of an exponentialN(E) for the case when ϱ(r) resembles the emissivity law of ellipticals are also derived.