How general is the global density slope-anisotropy inequality?

Abstract

Following the seminal result of An & Evans, known as the central density slope-anisotropy theorem, successive investigations unexpectedly revealed that the density slope-anisotropy inequality holds not only at the center, but at all radii in a very large class of spherical systems whenever the phase-space distribution function is positive. In this paper we derive a criterion that holds for all spherical systems in which the augmented density is a separable function of radius and potential: this new finding allows to unify all the previous results in a very elegant way, and opens the way for more general investigations. As a first application, we prove that the global density slope-anisotropy inequality is also satisfied by all the explored additional families of multi-component stellar systems. The present results, and the absence of known counter-examples, lead us to conjecture that the global density slope-anisotropy inequality could actually be a universal property of spherical systems with positive distribution function.Comment: 6 pages, MNRAS accepte

Mass, velocity anisotropy, and pseudo phase-space density profiles of Abell 2142

“…The β − γ relation found by Hansen & Moore (2006) for singlecomponent dissipationless simulations is shown as the dotted lines. The dashed line is the limit below which the relation by Ciotti & Morganti (2010) holds. The vertical dot-dashed line locates the value of γ at the virial radius.…”

“…are not too much radially anisotropic at the center), necessarily satisfy β(r) < −γ(r)/2, where the velocity anisotropy β and the logarithmic slope of density γ are for that component, as shown in Ciotti & Morganti (2010;see also Van Hese et al 2011). It is not clear Fig.…”

On the augmented density of a spherical anisotropic dynamic system

“…Here, γ is the (negative) logarithmic density slope, whereas β is the so‐called Binney anisotropy parameter. Extending the earlier finding by An & Evans (2006) that the inequality is necessary at the centre (given a finite potential well), Ciotti & Morganti (2010) have shown that, for wide varieties of anisotropic spherical systems built by flexible families of analytic two‐integral distributions of certain ansatz (e.g., Cuddeford 1991; Cuddeford & Louis 1995; Baes & van Hese 2007), the forementioned inequality holds everywhere in radial positions given that the DF is also non‐negative everywhere in the accessible phase space and that the central anisotropy parameter β 0 is restricted to be β 0 ≤ 1/2. They also presented the equivalent condition to the inequality for the separable augmented density, which has subsequently been proven by Van Hese, Baes & Dejonghe (2011) to be satisfied by such a system if β 0 ≤ 1/2.…”

“…Recently, Ciotti & Morganti (2010) raised a question whether the density slope–anisotropy inequality γ≥ 2β is the necessary condition for the consistency of the underlying two‐integral phase‐space distribution function (DF). Here, γ is the (negative) logarithmic density slope, whereas β is the so‐called Binney anisotropy parameter.…”

An analytic distribution function for a mass-less cored stellar system in a cuspy dark-matter halo

“…This implies that if the light profile is perfectly cored, as is often assumed, that is γ 0 = 0, the velocity ellipsoid must be isotropic, independently of the dark-matter halo profile (which should be shallower than the singular isothermal sphere). However, if the system is cold at the center, that is σ r,0 = 0, the only constraint is that γ 0 > 2β 0 , which for cored stellar profiles is satisfied by tangentially anisotropic ellipsoids (Ciotti & Morganti 2010). Since these conditions refer to the intrinsic velocity dispersion, they do not impose strong constraints on the line-of-sight velocity dispersion (σ los ), which is the observable, and one may obtain flat σ los profiles even if the system is intrinsically cold at the center.…”