The double-power approach to spherically symmetric astrophysical systems

Abstract

In this paper, we present two simple approaches for deriving anisotropic distribution functions for a wide range of spherical models. The first method involves multiplying and dividing a basic augmented density with polynomials in r and constructing more complex augmented densities in the process, from which we obtain the double-power distribution functions. This procedure is applied to a specific case of the Veltmann models that is known to closely approximate the Navarro-Frenk-White (NFW) profile, and also to the Plummer and Hernquist profiles (in the appendix). The second part of the paper is concerned with obtaining hypervirial distribution functions, i.e. distribution functions that satisfy the local virial theorem, for several well-known models. In order to construct the hypervirial augmented densities and the corresponding distribution functions, we start with an appropriate ansatz for the former and proceed to determine the coefficients appearing in that ansatz by expanding the potential-density pair as a series, around r = 0 and r = ∞. By doing so, we obtain hypervirial distribution functions, valid in these two limits, that can generate the potential-density pairs of these models to an arbitrarily high degree of accuracy. This procedure is explicitly carried out for the Hénon isochrone, Jaffe, Dehnen and NFW models and the accuracy of this procedure is established. Finally, we derive some universal properties for these hypervirial distribution functions, involving the asymptotic behaviour of the anisotropy parameter and its relation to the density slope in this regime. In particular, we show that the cusp slope-central anisotropy inequality is saturated.

Analytical solutions for weak black hole kicks

“…where we also used M m and the explicit expression for the Plummer distribution function (Binney and Tremaine 1987;Lingam and Nguyen 2014). Since β is independent of v, we find that the dynamical friction is linearly proportional to the velocity.…”

New Strong Constraints on the Central Behaviour of Spherical Galactic Models

“…The Dehnen models-and in particular the Jaffe and Hernquist models-have routinely been used to represent the stellar distribution within spheroidal elliptical galaxies and bulges (just to quote a few papers out of so many, see Refs. [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]).…”

Exploring the consideration of university teachers’ basic psychological needs in the design of professional development initiatives

“…This concludes our construction and analysis of our DF components. These provide a significant extension of previously implemented DFs, and in principle they can be extended even more (Lingam & Nguyen 2014). We now have all the necessary tools to use our modelling technique in several applications, building general DFs as linear combinations of our components, fitting given data sets.…”