Disentangling <i>γ</i> − <i>β</i>: The Fourth-order Velocity Moments Based on Spherical Jeans Analysis

Abstract Distinguishing a core and a cusp within dark matter halos is complexified by the existence of mass-anisotropy degeneracy, where various combinations of velocity anisotropy ( β ) and inner density slope ( γ ) yield similar observational signatures. We construct a dynamical model that incorporates the fourth-order velocity moments to alleviate this challenge. The inclusion of the fourth-order velocity moments enables a star’s line-of-sight velocity distribution (LOSVD) to be flexible. This flexible LOSVD can range from a thin-tailed to a heavy-tailed distribution that is inaccessible if only the second-order moments are considered. We test the model on four mock galaxies having isotropic orbits, β = 0: two resembling dwarf spheroidal (dSph) galaxies and two resembling ultrafaint dwarfs (UFDs) in terms of velocity dispersion. Each category includes one galaxy with a cuspy Navarro–Frenk–White (NFW) profile and one with a cored density profile. The results show that a ratio of the global velocity dispersion to velocity error, σ los,global / δv los ≳ 4, is crucial to avoid systematic biases arising from the strong sensitivity of fourth-order moments to the LOSVD tails. In cases where this velocity ratio condition is met, our model reliably recovers γ in dSph mock galaxies, with the true value recovered within ∼1 σ , and strongly excludes a cuspy NFW profile for the cored dSph mock galaxy. However, recovering the density profiles of UFDs remains challenging due to their intrinsically low velocity dispersions.