The dynamical structure of broken power-law and double power-law models for dark matter haloes

Abstract

Galaxy kinematics and gravitational lensing are two complementary ways to constrain the distribution of dark matter on galaxy scales. The typical dark matter density profiles adopted in dynamical studies cannot easily be adopted in lensing studies. Ideally, a mass model should be used that has the global characteristics of realistic dark matter distributions, and that allows for an analytical calculation of the magnifications and deflection angles. A simple model with these properties, the broken-power-law (BPL) model, has very recently been introduced. We examine the dynamical structure of the family of BPL models. We derive simple closed expressions for basic dynamical properties, and study the distribution function under the assumption of velocity isotropy. We find that none of the BPL models with realistic parameters has an isotropic distribution function that is positive over the entire phase space, implying that the BPL models cannot be supported by an isotropic velocity distribution, or models with a more radially anisotropic orbital structure. This result limits the attractiveness of the BPL family as a tool for lensing studies to some degree. More generally, we find that not all members of the general family of double power-law or Zhao models, often used to model dark matter haloes, can be supported by an isotropic or radially anisotropic distribution function. In other words, the distribution function may become negative even for spherically symmetric models with a well-behaved density profile.

SpheCow: Flexible dynamical models for galaxies and dark matter haloes

“…This implies that these models are self-consistent and physical, and stable against radial and non-radial perturbations (Antonov 1962;Doremus et al 1971;Binney & Tremaine 2008). We do note, however, that this conclusion cannot be generalised for all members of the family of algebraic sigmoid models, in a similar way as for the family of Zhao models (Baes & Camps 2021). If the parameter α that determines the sharpness of the transition between the inner and outer profiles grows larger, the distribution function will first develop a kink around E = Ψ(r b ).…”

“…When α increases even more, the distribution function will become negative, making the models with an isotropic orbital structure unphysical. In the limit α → ∞, the models reduce to the broken power-law (BPL) models discussed by Du et al (2020) and Baes & Camps (2021). These models are characterised by the simple density profile…”

Forecasts on the Dark Matter Density Profiles of Dwarf Spheroidal Galaxies with Current and Future Kinematic Observations

“…This five parameter ({ρ 0 , r 0 , a, b, c,}) model is extremely flexibility and reduces to many well known cases such as the Navarro-Frenk-White (NFW) profile (a = 1, b = 1, c = 3), Plummer profile (a = 0, b = 2, c = 5), or the generalized NFW profile, gNFW (b = 1, c = 3). The only restriction is that one should be careful when choosing specific parameters since not every combination produces a positive definite distribution function (Baes & Camps 2021) A useful quantity derived from density profile is the so-called log-slope, γ(r) = ∂ ln ρ ∂ ln r . With our parametrization of the density function the log-slope behaves such that γ(r → 0) = a and γ(r → ∞) = c. The shape of the curve in between these limits is dictated by scale radius r 0 and slope parameter b.…”

SpheCow: flexible dynamical models for galaxies and dark matter haloes

“…This implies that these models are self-consistent and physical, and stable against radial and nonradial perturbations (Antonov 1962;Doremus et al 1971;Binney & Tremaine 2008). We do note, however, that this conclusion cannot be generalised for all members of the family of algebraic sigmoid models, in a similar way as for the family of Zhao models (Baes & Camps 2021). If the parameter α that determines the sharpness of the transition between the inner and outer profiles grows larger, the distribution function will first develop a kink around E = Ψ(r b ).…”