Probabilistic Inference of the Structure and Orbit of Milky Way Satellites with Semi-Analytic Modeling

TL;DR

This paper develops a novel semi-analytic approach to infer the dark matter halo structure of Milky Way satellites by constructing a large, variance-rich population of MW-like systems with SatGen and then weight-select satellites to match observed properties. The core method couples a flexible SMHM relation and baryonic feedback models with a statistical weighting framework to infer halo parameters such as $r_{ ext{max}}$ and $v_{ ext{max}}$, accounting for correlated and non-Gaussian uncertainties. Applied to the MW classical dwarfs, the results broadly agree with Jeans analyses and simple scaling relations, while highlighting substantial systematic uncertainties stemming from feedback physics and halo–galaxy connections. The framework is versatile, allowing arbitrary observational constraints and enabling inferences about orbital histories and accretion modes, with broad implications for constraining baryonic feedback and the galaxy–halo connection in CDM.

Abstract

Semi-analytic modeling furnishes an efficient avenue for characterizing the properties of dark matter halos associated with satellites of Milky Way-like systems, as it easily accounts for uncertainties arising from halo-to-halo variance, the orbital disruption of satellites, baryonic feedback, and the stellar-to-halo mass (SMHM) relation. We use the SatGen semi-analytic satellite generator -- which incorporates both empirical models of the galaxy-halo connection in the field as well as analytic prescriptions for the orbital evolution of these satellites after they enter a host galaxy -- to create large samples of Milky Way-like systems and their satellites. By selecting satellites in the sample that match the observed properties of a particular dwarf galaxy, we can then infer arbitrary properties of the satellite galaxy within the Cold Dark Matter paradigm. For the Milky Way's classical dwarfs, we provide inferred values (with associated uncertainties) for the maximum circular velocity $v_{max}$ and the radius $r_{max}$ at which it occurs, varying over two choices of feedback model and two prescriptions for the SMHM relation that populate dark matter halos with physically distinct galaxies. While simple empirical scaling relations can recover the median inferred value for $v_{max}$ and $r_{max}$, this approach provides realistic correlated uncertainties and aids interpretability through variation of the model. For these different models, we also demonstrate how the internal properties of a satellite's dark matter profile correlate with its orbit, and we show that it is difficult to reproduce observations of the Fornax dwarf without strong baryonic feedback. The technique developed in this work is flexible in its application of observational data and can leverage arbitrary information about the satellite galaxies to make inferences about their dark matter halos and population statistics.

Figures (15)

• Figure 1: The satellites of all SatGen realizations form a correlated probability distribution in a high-dimensional space, shown with contours enclosing 68 and 95 per cent of satellites, while those outside the contours are shown as individual points. This figure shows the joint distribution of profile parameters $(r_\text{max},\,v_\text{max})$ against (i) the stellar mass $M_\star$, (ii) the mass $M_{1/2}$ contained within Fornax's half-light radius, (iii) the pericentre $r_\mathrm{peri}$, and (iv) the infall time $t_\mathrm{infall}$. Because many of these parameters correlate with $r_\text{max}$ and $v_\text{max}$, they can be used to infer these halo properties. As an example, the blue bands highlight the 68 per cent confidence regions for the Fornax dwarf, collected in \ref{['tab:1']}. SatGen satellites that lie in these bands contribute most strongly to the inference of Fornax's halo profile parameters, following the procedure detailed in \ref{['sec:2.2']}. These satellites are generated with the SMHM relation of Rodriguez-Puebla17 and use the NIHAO-calibrated feedback emulator. The strong core formation of this model can be seen in the second column, where the large ratio of stellar mass to halo mass puffs up the satellites, leading to a population with large values for $r_\text{max}$ and $v_\text{max}$ with a relatively small $M_{1/2}$. The turnover in these panels is not present in the APOSTLE feedback emulator. Note that the sub-unity values in the $M_\star$ space reflect extrapolations of the semi-analytic tidal tracks of Errani18, which allow for tidal stripping to be tracked even below what is physically meaningful. The region of interest is well within the reasonable parts of this space, but the full distribution is shown for clarity.
• Figure 2: (Left) The inferred profile parameters $r_\text{max}$ and $v_\text{max}$ for the Fornax dwarf according to the procedure described in \ref{['sec:2.2']}. The contours enclose 68 and 95 per cent confidence regions based on the mass within the observed half-light radius ($M_{1/2}$, in green), the stellar mass ($M_\star$, in gold), and the combination of both parameters simultaneously (in blue). Observationally, Fornax has a fairly low $M_{1/2}$, leading this inference to be consistent both with the massive cored satellites and with smaller uncored satellites. Fornax's stellar mass, on the other hand, is quite large, and selects only this former population. This selection is further refined by requiring that satellites simultaneously satisfy the observed $M_{1/2}$ and $M_{\star}$ values. The pericentric distance $r_\mathrm{peri}$ and infall time $t_\mathrm{infall}$ do not correlate strongly with $v_\text{max}$ or $r_\text{max}$ and are thus unable to constrain the space from the overall distribution of satellites; as such, they are not shown here. (Right) The combined inference from the left plot is shown under various models for baryonic feedback and the SMHM relation. The blue and magenta 'NIHAO' contours assume strong feedback, while the purple and orange 'APOSTLE' contours use a weaker model of baryonic feedback. The blue and purple contours assume the Rodriguez-Puebla17 SMHM relation, which has a steeper faint-end slope than Behroozi13, taken for the magenta and orange contours. Notably, without the strong core formation of the NIHAO feedback emulator, the satellites consistent with both $M_\star$ and $M_{1/2}$ are much smaller, with lower $v_\text{max}$ and $r_\text{max}$. In both panels, the probability density is defined as a function of the base-ten logarithm of $r_\text{max}$ and $v_\text{max}$, not the quantities themselves, though $r_\text{max}$ and $v_\text{max}$ are provided on the top and right axes, respectively, for convenience. This convention is followed throughout this work. Furthermore, in this plot and throughout this work, we plot contours that are statistics-limited as dashed lines rather than complete contours. See \ref{['tab:2']} for more information.
• Figure 3: This figure shows the results of the SatGen analysis for the Fornax (left) and Draco (right) dwarfs, assuming the Rodriguez-Puebla17 SMHM relation and varying over feedback emulation, as compared with other results from the literature. Data points are kinematic measurements, including Errani18, shown in leftward- and rightward-pointing triangles; Kaplinghat19, shown in upward- and downward-pointing triangles; and Andrade23, shown as a square. For the first two models, the direction of the triangle indicates the assumption of a cored or cuspy DM profile, respectively, while Andrade23 models the DM halo with a profile where the inner slope is allowed to vary; the other two analyses model the DM halo with both a cuspy profile and a cored profile. Additionally, we compare these results to the prediction obtained by the simple scaling relations of the Rodriguez-Puebla17 SMHM relation and the concentration--mass relation from Moline23. The SatGen inference is generally in agreement with these analyses, particularly due to the highly correlated and non-Gaussian errors in the observational analyses. For results on the other dwarfs, see \ref{['fig:A1']}.
• Figure 4: 68 and 95 per cent confidence intervals for the pericentre $r_\mathrm{peri}$ and central density $\rho_\mathrm{150}$ of Fornax (left) and Draco (right), based on the combined $M_\star$ and $M_{1/2}$-based inference. The bands represent the maximum and minimum one-sigma observational results for Fritz18, Battaglia22, and Pace22 for $r_\mathrm{peri}$; and Read19, Kaplinghat19, and Hayashi20 for $\rho_{150}$, marginalizing over all MW potentials in the pericentre modelling and all assumed profiles (including both cusps and cores) in the density modelling. The SatGen results assume the Rodriguez-Puebla17 model and are shown for both the NIHAO and APOSTLE feedback emulators (in blue and purple, respectively). In the case of Fornax, the satellites able to match both masses in the NIHAO feedback emulator also match the observed low central density and fairly large pericentre -- Fornax-like satellites do not remain Fornax-like with low pericentres. Conversely, the APOSTLE feedback emulator requires the tidal stripping enhancement of a low pericentre to match both mass constraints, which brings it in slight tension with observations. Unlike Fornax, the high central density of Draco can be achieved in either feedback emulator, and satellites that match the observed masses also satisfy observational limits on the central density and pericentre.
• Figure 5: The probability that any given classical satellite is accreted as part of a group rather than directly from the field (as inferred though the $\boldsymbol{\theta} = (M_\star,\,M_{1/2})$ selection). The vertical axis shows the probability that, given a group-accretion scenario, the original group host persists as a satellite of the MW. While the dwarfs presented here are unlikely to have been substructure at infall, in the case that they were, it is unlikely that the group host survived to the present day.