Impulsive mixing of stellar populations in dwarf spheroidal galaxies

TL;DR

This work analyzes how dwarf spheroidal galaxies respond to perturbations of their dark matter potentials by modeling spherical, isotropic stellar tracers as sums of mono-energetic components within cuspy and cored halos. Using controlled N-body experiments, it shows that adiabatic changes preserve a single evolutionary track with $r_h \,\langle\sigma^2\rangle^{1/2}$ constant and centrally isotropic kinematics, while impulsive perturbations broaden energy distributions, enable energetic mixing, and produce long-lasting transients—especially in cored halos where phase mixing is inefficient. The results illuminate how tidal mass loss, outflows, and halo evolution can flatten metallicity gradients under slow changes but potentially steepen or erase chemo-kinematic distinctions under fast perturbations, potentially biasing dynamical mass estimates from Jeans modeling. Overall, the study provides a framework to interpret the diverse dynamical and chemical structures of dSphs and to quantify biases in mass inferences under different perturbation regimes.

Abstract

We study the response of mono-energetic stellar populations with initially isotropic kinematics to impulsive and adiabatic changes to an underlying dark matter potential. Half-light radii expand and velocity dispersions decrease as enclosed dark matter is removed. The details of this expansion and cooling depend on the time scale on which the underlying potential changes. In the adiabatic regime, the product of half-light radius and average velocity dispersion is conserved. We show that the stellar populations maintain centrally isotropic kinematics throughout their adiabatic evolution, and their densities can be approximated by a family of analytical radial profiles. Metallicity gradients within the galaxy flatten as dark matter is slowly removed. In the case of strong impulsive perturbations, stellar populations develop power-law-like density tails with radially biased kinematics. We show that the distribution of stellar binding energies within the dark matter halo substantially widens after an impulsive perturbation, no matter the sign of the perturbation. This allows initially energetically separated stellar populations to mix, to the extent that previously chemo-dynamically distinct populations may masquerade as a single population with large metallicity and energy spread. Finally, we show that in response to an impulsive perturbation, stellar populations that are deeply embedded in cored dark matter halos undergo a series of damped oscillations before reaching a virialised equilibrium state, driven by inefficient phase mixing in the harmonic potentials of cored halos. This slow return to equilibrium adds substantial systematic uncertainty to dynamical masses estimated from Jeans modeling or the virial theorem.

Figures (23)

• Figure 1: Top panel: 3D exponential number density $\nu_\star(r)$, Eq. \ref{['eq:3Dexp_rho']}, of a stellar tracer with $N_\star$ stars and scale radius $r_\star\approx r_\mathrm{h}/2.67$ (black solid curve), as well as an approximation to this profile consisting of the sum of five mono-energetic stellar distributions. Red curves show the individual mono-energetic profiles, Eq. \ref{['eq:monoenergetic_rho']}, and a dashed curve shows their sum, with each component weighted by $\left. \mathrm{d} N_\star / \mathrm{d} \mathcal{E} \right|_{\mathcal{E} = \mathcal{E}_\star}$. Bottom panel: Differential energy distribution $\mathrm{d} N_\star / \mathrm{d} \mathcal{E}$, Eq. \ref{['eq:energydist']}, of an exponential stellar profile embedded deeply in the density cusp of a Hernquist dark matter halo, Eq. \ref{['eq:Hernquist_pot']}. Energies are expressed in units of the energy $\hat{\mathcal{E}}_\star$ where $\mathrm{d} N_\star / \mathrm{d} \mathcal{E}$ peaks, and the energy distribution is normalized here so that $\int_0^1 \left(\mathrm{d} N_\star/\mathrm{d} \mathcal{E}\right) \mathrm{d} \mathcal{E} = 1$. The five $\delta$-functions corresponding to the mono-energetic profiles shown in the top panel are depicted as arrows.
• Figure 2: Top panel: cuspy (black, Eq.\ref{['eq:RhoDehnenGamma']} with $\gamma=1$) and cored (gray, $\gamma=0$) dark matter halo density profiles considered in this study. The half-light radii of four mono-energetic stellar tracers with $r_\mathrm{h} / r_\mathrm{mx} =1$, $1/4$, $1/16$ and $1/64$ are indicated by filled circles in dark blue, light blue, orange and red, respectively (see Tab. \ref{['tab:parameter_overview']} for the corresponding energies). Bottom panel: 3D logarithmic slope of the dark matter density profiles shown above.
• Figure 3: Mono-energetic stellar number density profiles $\nu_\star(r)$ (Eq. \ref{['eq:monoenergetic_rho']}) with energies as listed in Tab. \ref{['tab:parameter_overview']}, with half-light radii of $r_\mathrm{h} / r_\mathrm{mx} =1$, $1/4$, $1/16$ and $1/64$. Stellar profiles embedded in a cuspy (Eq. \ref{['eq:StellarCuspy']}) and a cored (Eq. \ref{['eq:StellarCored']}) dark matter halo are shown using solid and dashed lines, respectively. Filled circles indicate the locations of the respective half-light radii. The profiles are sharply truncated at a radius $r_\mathrm{t}$. Note that for these mono-energetic energy distributions, the relative density and velocity dispersion profiles coincide.
• Figure 4: Top panel: Relation between stellar binding energy $\mathcal{E}_\star$ and half-light radius $r_\mathrm{h}$ for mono-energetic stellar tracers embedded in a cuspy (black solid curve) and a cored (gray solid curve) potential. The values corresponding to the models of Fig. \ref{['fig:StellarDensitiesCuspCore']} are shown as filled circles. The dashed curves show the truncation radii $r_\mathrm{t}$ beyond which there is no stellar density (Eq. \ref{['eq:HernquistStellarTruncation']}, \ref{['eq:DehnenStellarTruncation']}). The energetic spacing of stellar populations with matching half-light radii differs between models embedded deeply in a cuspy and cored halo. Bottom panel: Crossing time $T_\mathrm{cross} = r_\mathrm{h} \langle \sigma^2 \rangle^{1/2}$ as a function of energy $\mathcal{E}_\star$. Stellar populations that are deeply embedded in the density core all have virtually identical crossing times, regardless of their binding energy.
• Figure 5: Initial ( ) and final ( ) equilibrium configurations of stellar tracer populations in the numerical experiments with an adiabatically evolving dark matter halo. Shown are half-light radii $r_\mathrm{h}$ and average velocity dispersions $\langle \sigma^2 \rangle^{1/2}$ of four stellar tracers with mono-energetic initial conditions, embedded in a cuspy ($\gamma=1$ in Eq. \ref{['eq:RhoDehnenGamma']}, left panel) and a cored ($\gamma=0$, right panel) dark matter halo. Stellar tracers with initial half-light radii $r_\mathrm{h0} / r_\mathrm{mx0} = 1$, $1/4$, $1/16$ and $1/64$ are shown in dark blue, light blue, orange and red, respectively. All initial conditions lie along the thick gray curve labeled $M/M_0=1$. In each simulation, the mass sourcing the underlying dark matter potential is lowered slowly by $-0.2\,\mathrm{dex}$, $-0.4\,\mathrm{dex}$, $\dots$, $-2\,\mathrm{dex}$. Solid gray curves are computed from Eq. \ref{['eq:LumAveragedDispersion']}, assuming mono-energetic distribution functions with isotropic kinematics. All equilibrium configurations fall along lines of $r_\mathrm{h} \langle\sigma^2\rangle^{1/2}=\text{const}$.