A dynamical attractor in the evolution of dwarf spheroidal galaxies

Abstract

We use controlled $N$-body experiments to study the dynamical evolution of dwarf spheroidal galaxies (dSphs) embedded in dark-matter (DM) haloes containing a large population of dark subhaloes. We show that stellar orbits subject to stochastic force fluctuations irreversibly gain energy and expand toward a dynamical attractor characterized by a stellar half-light radius $r_{\rm half} \approx r_{\rm max}$ and a velocity dispersion $σ\approx 0.5\,v_{\rm max}$, where $v_{\rm max}$ is the peak circular velocity of the host halo at radius $r_{\rm max}$. This state is reached both in isolation and under tidal stripping, although tidal mass loss significantly accelerates the evolution. Assuming that the Milky Way (MW) dSphs have reached this state, we find that the inferred halo masses collapse onto narrow sequences as a function of $r_{\rm half}$. Under this assumption, MW satellites with $r_{\rm half} \lesssim 1\,\mathrm{kpc}$ follow the tidal tracks of cuspy haloes, while larger systems deviate in a manner consistent with cored DM profiles. Moreover, the mass--luminosity relation follows the slope expected from abundance matching, but with halo masses systematically lowered from their peak values at fixed luminosity. These results suggest that the structural diversity of dSphs is largely an evolutionary outcome driven by internal heating and tides, rather than by the conditions of star formation. This framework predicts that isolated, early-quenched dSphs should have systematically larger sizes than satellites, a prediction testable with upcoming surveys.

Figures (3)

• Figure 1: Left panel: Stellar velocity dispersion $\sigma$ as a function of half-light radius $\,r_{\rm half}$, derived from the virial theorem for stellar tracers with Plummer (solid) or Hernquist (dashed) profiles embedded in Dehnen (colored) and NFW (black) haloes with $M_h=10^9\,\, M_{\odot}$ and $c_h=2.26\,\,{\rm kpc}$. The dependence on the stellar profile is weak in cuspy haloes and increases as the inner density slope approaches $\gamma\to0$. Middle panel: Same relations normalized to the halo peak velocity $\,v_{\rm max}$ and radius $\,r_{\rm max}$. As $\,r_{\rm half} \rightarrow r_{\rm max}$ (vertical dotted line), the velocity dispersion peaks at $\eta_\sigma=\sigma_{\rm max}/\,v_{\rm max}\simeq0.54$ for a Plummer profile and $0.48$ for a Hernquist profile. Right panel: Comparison with $N$-body models (see text). Stochastic heating drives progressive expansion of the tracers, proceeding more rapidly in haloes with shallower density profiles. At early times, the evolution follows the virial-theorem prediction (dash–dotted lines). In all cases, the stellar velocity dispersion peaks at $\eta_\sigma\simeq0.63$, independent of the halo profile, exceeding the virial prediction by a factor $\simeq1.16$.
• Figure 2: Top panel: Time evolution of the bound subhalo mass fraction (black solid line) for a dSph with $M_h=10^9\,\, M_{\odot}$ and $c_h=2.26\,\,{\rm kpc}$ orbiting a MW-like halo with $M_G=10^{12}\,\, M_{\odot}$ and $c_G=21.5\,\,{\rm kpc}$. The orbit has apocentre $r_{\rm apo}=140\,\,{\rm kpc}$ and pericentre $r_{\rm peri}=25\,\,{\rm kpc}$ (shown with a black dotted line in arbitrary units). Smooth and clumpy halo models are shown in orange and blue, respectively. While the extended DM halo initially shields the stars from stripping, subhaloes enhance stellar mass loss at late times. Second panel: Evolution of the DM peak velocity radius (black solid line), which follows the tidal track $r_{\rm max}/r_{\rm max,0}=(M/M_0)^\kappa$ with $\kappa=0.44$ (Errani & Peñarrubia 2020). Stellar half-light radii for the smooth and clumpy models are shown in orange and blue; the red line shows a clumpy halo evolving in isolation. Third panel: Stellar segregation within the halo, $\eta_r=\,r_{\rm half}/r_{\rm max}$, as a function of time. All models evolve toward $\eta_r\to1$ (black dashed line), with the fastest convergence occurring in tidally stripped, clumpy haloes. Bottom panel: Stellar-to-DM velocity dispersion ratio, $\eta_\sigma=\sigma/v_{\rm max}$; the asymptotic limit $\eta_\sigma\to0.5$ is indicated by the black dashed line.
• Figure 3: Left panel: Dwarf galaxy masses inferred from Equation (\ref{['eq:totalm']}), with $(\eta_r,\eta_\sigma)=(1,0.5)$, as a function of half-light radius. Here, we adopt $\gamma=1$, although $M_{\rm HA}$ values are largely insensitive to this choice. Orange circles indicate galaxies with associated stellar streams, while green circles mark systems exhibiting signatures of tidal stripping (Pace et al. 2022). Magenta squares correspond to galaxies that may reside in cored DM haloes. Red and blue curves show the tidal tracks expected for galaxies embedded in cuspy ($\gamma=1)$ and cored ($\gamma=0$) haloes, respectively (see text). Note that dSphs with $\,r_{\rm half}\lesssim 1\,{\rm kpc}$ lie close to the cuspy tidal track, as well as to the subhalo mass--size relation in the Aquarius CDM simulation (green dotted line), $r_{\rm max}\sim M^{0.43}$ (Springel et al. 2008). Right panel: HA masses as a function of dSph luminosity. The green-solid line shows the best-fit luminosity--peak halo mass relation derived from the MW satellite luminosity function (Nadler et al. 2020). The green-dotted line shows the same relation with halo masses lowered by 1.5 dex at a fixed luminosity.