The differential energy distribution and the total integrated binding energy of dynamical models

Abstract

We revisit the differential energy distribution of steady-state dynamical models. It has been shown that the differential energy distribution of steady-state spherical models does not vary strongly with the anisotropy profile, and that it is hence mainly determined by the density distribution of the model. We explore this similarity in more detail. Through a worked example and a simple proof, we show that the mean binding energy per unit mass $\langle{\cal{E}}\rangle$, or equivalently the total integrated binding energy $B_{\text{tot}} = M\langle{\cal{E}}\rangle$, is independent of the orbital structure, not only for spherical models but for any steady-state dynamical model. Only the higher-order moments of the differential energy distribution depend on the details of the orbital structure. We show that the standard deviation of the differential energy distribution of spherical dynamical models varies systematically with the anisotropy profile: radially anisotropic models tend to prefer more average binding energies, whereas models with a more tangential orbital distribution slightly favour more extreme binding energies. Finally, we find that the total integrated binding energy supplements the well-known trio consisting of total kinetic energy, total potential energy, and total energy on an equal footing. Knowledge of any one out of these four energies suffices to calculate the other three.