Gravothermal expansion of dwarf spheroidal galaxies heated by dark subhaloes

TL;DR

The paper shows that dwarf spheroidal galaxies embedded in dark matter halos hosting substantial subhalo populations experience gradual, self-similar gravothermal-like expansion driven by stochastic heating from subhaloes. Stars within the dSphs gain energy, increasing the inner velocity dispersion while the outer regions cool, with a maximum dispersion reached when the half-light radius approaches the halo's scale radius $c_h$ and a corresponding phase-transition-like divergence in heat capacity. The heating is analyzed via virial equations and a Chandrasekhar-type kinetic theory, revealing that the relaxation time scales as $t_{\rm rel} \propto r_{\rm half}^{3/2}$ and is dominated by the most massive subhaloes, causing expansion to slow as $r_{\rm half}$ grows. The study predicts that some ultra-faint dSphs could evolve into stealth or ultra-diffuse galaxy–like states with large sizes but very low luminosities, providing a potential observational window into subhalo demographics and the interplay between baryons and dark matter. The results bear on the interpretation of dSphs in a cosmological context and suggest broader implications for the nature of dark matter and the detectability of faint satellites.

Abstract

We use analytical and $N$-body methods to study the evolution of dwarf spheroidal galaxies (dSphs) embedded in dark matter (DM) haloes that host a sizeable subhalo population. Dark subhaloes generate a fluctuating gravitational field that injects energy into stellar orbits, driving a gradual expansion of dSphs. Despite the overall expansion, the stellar density profile preserves its initial shape, suggesting that the evolution proceeds in a self-similar manner. Meanwhile, the velocity dispersion profile, initially flat, evolves as the galaxy expands: the inner regions heat up, while the outer regions cool down. Kinematically, this resembles gravothermal collapse but with an inverted evolution, instead of collapsing the stellar system expands within a fluctuating halo potential. As the half-light radius $r_{\rm half}$ approaches the halo peak velocity radius $r_{\rm max}$, the expansion slows, and the velocity dispersion peaks at $σ_{\rm max} \simeq 0.54 v_{\rm max}$. The stellar heat capacity remains positive for deeply embedded stars but diverges near $r_{\rm max}$, turning negative beyond this threshold, which indicates a phase transition in the dynamical response to energy injection. The relaxation time scales as $t_{\rm rel} \sim r_{\rm half}^{3/2}$, showing that orbital diffusion slows as the galaxy expands. Ultra-faint dSphs, having the smallest sizes and shortest relaxation times, are particularly sensitive to the presence of dark subhaloes. Some of our dSph models expand beyond the detection of current photometric surveys, becoming `stealth' galaxies with half-light radii and velocity dispersions similar to ultra-diffuse galaxies (UDGs), but orders of magnitude lower luminosities, representing a distinct, yet currently undetected, population of DM-dominated satellites.

Figures (12)

• Figure 1: Projected half-light radius ($R_{\rm half}$) versus mean densities ($\langle \rho\rangle$) of Milky Way dSphs with available velocity dispersion. To compute the mean densities, $\langle \rho\rangle = 3\,M_h(<r')/(4\pi r'^3)$, we use the enclosed mass estimated as $M_h(<r')=3.5\times r' \,\sigma^2/G$ (Errani et al. 2018), where $r'=1.8\,R_{\rm half}$ and $\sigma$ are the half-light radius luminosity-averaged velocity dispersion, respectively. Coloured lines show the mean densities of Hernquist haloes with masses $M_h/M_\odot=3\times 10^8,10^9$ and $10^{10}$, with scale radii $c_h/\,{\rm kpc}=0.75, 2.26$ and 9.95, respectively. In red we highlight 4 dSphs that will be used in the remainder of this paper for reference.
• Figure 2: Cumulative subhalo mass function sampled from 16 realizations of the Aquarius model (red line) within a mass range $\xi\equiv M/M_{200}>10^{-4}$. The total number of subhaloes derived from (\ref{['eq:N']}) in each realization is $N\approx 152$.
• Figure 3: Distribution of pericentric radii (upper panel) and orbital periods (lower panel) of subhaloes moving the in dSph haloes plotted in Fig. \ref{['fig:rhdens']}. Solid and dotted lines show models with a low-mass truncation at $\xi_1=10^{-4}$ and $\xi_1=10^{-5}$, respectively. Black-dashed lines mark the half-light radii ($\, r_{\rm half}$) and characteristic orbital times ($T_\star=2\pi \, r_{\rm half}/\sigma$) of 4 dSphs used for reference in this paper. Notice that stars typically have shorter orbital periods than subhaloes because they are more spatially segregated in the halo potential.
• Figure 4: Peak velocity radius of individual subhaloes with masses $10^{-4}<M/M_h<10^{-1}$ orbiting in the halo models plotted in Fig. \ref{['fig:rhdens']}. Exponentially-truncated NFW profiles (\ref{['eq:rhoexp']}) have a circular velocity profile that peaks at $r_{\rm max}\approx 1.8\,c$ (Errani & Navarro 2021). Dotted lines show power-law fits $c=c_0(M/M_\odot)^\eta$ with $\eta=0.38$ and $c_0/\,{\rm pc}=0.25, 0.49$ and $0.93$ for dSph halo masses $M_h/M_\odot=3\times 10^8,10^9$ and $10^{10}$, respectively. For comparison, we overplot an extrapolation of the relation found in the Aquarius subhaloes by Springel et al. (2008), which exhibits a slightly steeper index $\eta=0.43$ with $c_0=0.21\,{\rm pc}$ (red line).
• Figure 5: Snap-shots of the evolution of a Sculptor-like dSph in the three halo models presented in §\ref{['sec:haloes']} with $\xi_1=10^{-4} (N=151)$. Blue and green circles mark the size of the initial half-light radius, $\, r_{\rm half}(t=0)=276\,{\rm pc}$, and the scale radius $c_h$ of the host DM halo, respectively. Subhaloes are shown with red open dots with a size that scales according to their mass, while black dots show stellar particles. Notice that as time goes by, dSphs tend to expand within the DM haloes as a result of repeated encounters with subhaloes, with signatures of disequilibrium that are more clearly visible in dSph halo models.