Semianalytical Dark Matter Halos and the Jeans Equation

Abstract

Although N-body studies of dark matter halos show that the density profiles, ρ(r), are not simple power-laws, the quantity ρ/σ 3 , where σ(r) is the velocity dispersion, is in fact a featureless power-law over ∼ 3 decades in radius. In the first part of the paper we demonstrate, using the semi-analytic Extended Secondary Infall Model (ESIM), that the nearly scale-free nature of ρ/σ 3 is a robust feature of virialized halos in equilibrium. By examining the processes in common between numerical N-body and semi-analytic approaches, we argue that the scale-free nature of ρ/σ 3 cannot be the result of hierarchical merging, rather it must be an outcome of violent relaxation. The empirical results of the first part of the paper motivate the analytical work of the second part of the paper, where we use ρ/σ 3 ∝ r −α as an additional constraint in the isotropic Jeans equation of hydrostatic equilibrium. Our analysis shows that the constrained Jeans equation has different types of solutions, and in particular, it admits a unique "periodic" solution with α = 1.9444. We derive the analytic expression for this density profile, which asymptotes to inner and outer profiles of ρ ∼ r −0.78 , and ρ ∼ r −3.44 , respectively.

On Universal Halos and the Radial Orbit Instability

“…Moreover, Taylor & Navarro (2001) found that / 3 , which they argue is a proxy for the ''phase-space density,'' is well approximated by a pure power law in radius. This observation prompted a number of analytic studies in which the Jeans equation, under the assumption that / 3 is a power law, is solved for the radial density profile ( Hansen 2004;Dehnen & McLaughlin 2005;Austin et al 2005; Barnes et al 2006). Finally, and identified what appears to be a universal relation between the velocity anisotropy parameter profile (r) (see Binney & Tremaine 1987 and eq.…”

A Universal Power-Law Profile of Pseudo-Phase-Space Density-Like Quantities in Elliptical Galaxies

“…As first noted by Taylor & Navarro (2001), N-body simulations of hierarchical cold dark matter (DM) halo formation show that the combination ρ pDM (r)/σ 3 pDM (r) of mass density ρ pDM (r) and velocity dispersion (VD) σ pDM (r) of the pristine DM halo, which is called pseudo-phase-space (PPS) density because it has the dimension of the phase-space density (or distribution function) but is not a true measure of it (e.g., Ascasibar & Binney 2005;Sharma & Steinmetz 2005;Vass et al 2009), is closer to universal than ρ pDM (r) and can be well described by a scale-free power-law profile with slope χ pDM ≡ −d ln[ρ pDM (r)/σ 3 pDM (r)]/d ln r ≈ 1.9 over three orders of magnitude in radius (e.g., Ascasibar et al 2005;Austin et al 2005;Wang & White 2009;Vass et al 2009;Navarro et al 2010;Ludlow et al 2011). It appears that ρ pDM (r)/σ 3 pDM (r) rather than ρ pDM (r) offers a more powerful route to the universal nature of pristine halos.…”

“…It appears that ρ pDM (r)/σ 3 pDM (r) rather than ρ pDM (r) offers a more powerful route to the universal nature of pristine halos. In this context, significant efforts have been made to investigate the physical origin of the universality, scale-free nature and slope value of the ρ pDM (r)/σ 3 pDM (r)-profile (e.g., Taylor & Navarro 2001;Austin et al 2005;Barnes et al 2006;Wang & White 2009;Vass et al 2009;Ludlow et al 2011;Lapi & Cavaliere 2011) and their implications for the structure of pristine DM halos (e.g., Austin et al 2005;Dehnen & McLaughlin 2005).…”

Gamma rays from annihilations at the galactic center in a physical dark matter distribution

“…On the other hand, tangential components (corresponding to β 0) must develop toward the center, as expected from the increasing strength of angular momentum effects. This view is supported by numerical simulations (see Austin et al 2005;Hansen & Moore 2006;Dehnen & McLaughlin 2005), which in detail suggest the effective linear approximation…”

“…in terms of the logarithmic density slope γ ≡ −d log ρ/d log r. As first shown by Austin et al (2005) and Dehnen & McLaughlin (2005), Jeans supplemented with the mass definition M(<r) ≡ 4π r 0 dr r 2 ρ(r ) entering v 2 c ≡ GM(<r)/r provides an integrodifferential equation for ρ(r), which by double differentiation reduces to a handy 2nd-order differential equation for γ.…”