A Method for Measuring (Slopes of) the Mass Profiles of Dwarf Spheroidal Galaxies

TL;DR

This paper introduces a novel method to measure the slope of the mass profile in dwarf spheroidal galaxies directly from stellar kinematics without imposing a dark-matter halo model. By combining Jeans-based mass estimates at halflight radii with the existence of two chemo-dynamically distinct subcomponents that trace the same potential, it infers two mass points and computes the slope $\Gamma$ as a direct indicator of the inner mass distribution. Applying the method to Fornax and Sculptor using Magellan/MMFS data, the authors find $\Gamma=2.61_{-0.37}^{+0.43}$ and $\Gamma=2.95_{-0.39}^{+0.51}$, respectively, which strongly favor constant-density cores over cuspy NFW halos with significances $s(\gamma_{DM}\geq 1) \gtrsim 95.9\%$ (Fornax) and $\gtrsim 99.8\%$ (Sculptor). Synthetic-data tests show the slope estimates are systematically biased low due to inner-component mass overestimation in embedded configurations, implying the NFW exclusions are conservative. Overall, the work provides direct, model-independent evidence against cuspy DM halos in these classical dSphs and demonstrates a pathway for testing small-scale predictions of the CDM paradigm with stellar chemo-dynamics.

Abstract

We introduce a method for measuring the slopes of mass profiles within dwarf spheroidal (dSph) galaxies directly from stellar spectroscopic data and without adopting a dark matter halo model. Our method combines two recent results: 1) spherically symmetric, equilibrium Jeans models imply that the product of halflight radius and (squared) stellar velocity dispersion provides an estimate of the mass enclosed within the halflight radius of a dSph stellar component, and 2) some dSphs have chemo-dynamically distinct stellar \textit{sub}components that independently trace the same gravitational potential. We devise a statistical method that uses measurements of stellar positions, velocities and spectral indices to distinguish two dSph stellar subcomponents and to estimate their individual halflight radii and velocity dispersions. For a dSph with two detected stellar subcomponents, we obtain estimates of masses enclosed at two discrete points in the same mass profile, immediately defining a slope. Applied to published spectroscopic data, our method distinguishes stellar subcomponents in the Fornax and Sculptor dSphs, for which we measure slopes $Γ\equiv Δ\log M / Δ\log r=2.61_{-0.37}^{+0.43}$ and $Γ=2.95_{-0.39}^{+0.51}$, respectively. These values are consistent with 'cores' of constant density within the central few-hundred parsecs of each galaxy and rule out `cuspy' Navarro-Frenk-White (NFW) profiles ($d\log M/d\log r \leq 2$ at all radii) with significance $\ga 96%$ and $\ga 99%$, respectively. Tests with synthetic data indicate that our method tends systematically to overestimate the mass of the inner stellar subcomponent to a greater degree than that of the outer stellar subcomponent, and therefore to underestimate the slope $Γ$ (implying that the stated NFW exclusion levels are conservative).

Figures (10)

• Figure 1: Top: Projected stellar velocity dispersion profile for the Fornax dSph adopted from walker09d. Overlaid are spherical Jeans models that assume either a cored dark matter halo, an NFW dark matter halo, or if one lets the shape of the dark matter halo vary freely, velocity distributions that are either isotropic, radially anisotropic, or tangentially anisotropic. Bottom: Enclosed-mass profiles corresponding to the same models. The vertical dotted line indicates Fornax's projected halflight radius ih95, where the simple estimator specified by Equation \ref{['eq:walker']} gives $M(r_h)=[5.3\pm 0.9] \times 10^7M_{\odot}$, in agreement with the value common to the various successful Jeans models.
• Figure 2: Spatial sampling bias in the Magellan dSph spectroscopic samples walker09a adopted here. Panels plot the observational selection probability, $w\equiv dN_{\mathrm{obs}}/dN_{\mathrm{cand}}$, as a function of projected radius, estimated via kernel smoothing (Equation \ref{['eq:weight']}). Different linestyles correspond to different smoothing bandwidths. For this study we adopt the $w(R)$ estimates that correspond to bandwidth $k_1=2$ arcmin (dotted red); our results and conclusions are not sensitive to this choice.
• Figure 3: Top: Differential distribution of stars as a function of (projected) radius for the stellar subcomponents in our tests. Bottom five panels: Projected velocity dispersion profiles for the physical dynamical models used to construct the synthetic data sets on which we test our method. For clarity we plot profiles corresponding only to models with outer stellar density profiles specified by $\beta_*=5$ (velocity dispersion profiles for models with $\beta_*=4$ and $\beta_*=6$ behave similarly). Notice that for a given halo potential, projected velocity dispersion profiles corresponding to isotropic ($r_a=\infty$) and anisotropic ($r_a=r_*$) velocity distributions cross at $r\sim r_*$, the radius where the number of stars reaches a maximum. This phenomenon helps to explain the insensitivity of our mass estimates to velocity anisotropy (Section \ref{['subsubsec:masses']}).
• Figure 4: Enclosed mass profiles (top) and slopes of logarithmic mass profiles (bottom), for the cored and cusped dark matter halos considered in tests of our method. Discrete points identify the luminous scale radii of various dSph-like stellar subcomponents that we embed in these halos for our tests.
• Figure 5: Recovery of the free parameters in our likelihood function (Equations \ref{['eq:likelihood']} and \ref{['eq:likelihood2']}), from tests with synthetic data. Panels display composite error distributions (red/blue for the distributions obtained by stacking error distributions obtained in individual realizations corresponding to cored/cusped input halos) evaluated by subtracting the known input value of each parameter from the MCMC-sampled values. Peaks at $E(\log_{10}[r_{h,2}/\mathrm{pc}])\sim -0.2, 0.0, +0.3$ correspond to input models with outer stellar density profiles specified by $\beta_{*,2}=6,5,4$, respectively. Since the test models generally do not have constant velocity dispersion, errors associated with velocity dispersion estimates are poorly defined and therefore not shown (but see Figure \ref{['fig:jorge2compnojeansflatpmpmem_masses']} for errors associated with derived masses).