Action-based dynamical models of dwarf spheroidal galaxies: application to Fornax

We present new dynamical models of dwarf spheroidal galaxies (dSphs) in which both the stellar component and the dark halo are described by analytic distribution functions that depend on the action integrals. In their most general form these distribution functions can represent axisymmetric and possibly rotating stellar systems. Here, as a first application, we model the Fornax dSph, limiting ourselves, for simplicity, to the non rotating, spherical case. The models are compared with state-of-the-art spectroscopic and photometric observations of Fornax, exploiting the knowledge of the line-of-sight velocity distribution of the models and accounting for the foreground contamination from the Milky Way. The model that best fits the structural and kinematic properties of Fornax has a cored dark halo, with core size $r_{\rm c}\simeq1.03$ kpc. The dark-to-luminous mass ratio is $(M_{\rm dm}/M_{\star})|_{R_{\rm eff}}\simeq9.6$ within the effective radius $R_{\rm eff} \simeq 0.62\,$kpc and $(M_{\rm dm}/M_{\star})|_{3 {\rm kpc}} \simeq 144$ within 3 kpc. The stellar velocity distribution is isotropic almost over the full radial range covered by the spectroscopic data and slightly radially anisotropic in the outskirts of the stellar distribution. The dark-matter annihilation $J$-factor and decay $D$-factor are, respectively, $\log_{10}(J$ $[$GeV$^2$ cm$^{-5}])\simeq18.34$ and $\log_{10}(D$ $[$GeV cm$^{-2}])\simeq18.55$, for integration angle $\theta = 0.5^{\circ}$. This cored halo model of Fornax is preferred, with high statistical significance, to both models with a Navarro, Frenk and White dark halo and simple mass-follows-light models.Comment: Accepted 2018 July 6; Received 2018 June 1; Submitted in original to MNRAS form 2018 February

The imprint of dark matter haloes on the size and velocity dispersion evolution of early-type galaxies

Early-type galaxies (ETGs) are observed to be more compact, on average, at $z \gtrsim 2$ than at $z\simeq 0$, at fixed stellar mass. Recent observational works suggest that such size evolution could reflect the similar evolution of the host dark matter halo density as a function of the time of galaxy quenching. We explore this hypothesis by studying the distribution of halo central velocity dispersion ($\sigma_0$) and half-mass radius ($r_{\rm h}$) as functions of halo mass $M$ and redshift $z$, in a cosmological $\Lambda$-CDM $N$-body simulation. In the range $0\lesssim z\lesssim 2.5$, we find $\sigma_0\propto M^{0.31-0.37}$ and $r_{\rm h}\propto M^{0.28-0.32}$, close to the values expected for homologous virialized systems. At fixed $M$ in the range $10^{11} M_\odot \lesssim M\lesssim 5.5 \times 10^{14} M_\odot$ we find $\sigma_0\propto(1+z)^{0.35}$ and $r_{\rm h}\propto(1+z)^{-0.7}$. We show that such evolution of the halo scaling laws is driven by individual haloes growing in mass following the evolutionary tracks $\sigma_0\propto M^{0.2}$ and $r_{\rm h}\propto M^{0.6}$, consistent with simple dissipationless merging models in which the encounter orbital energy is accounted for. We compare the $N$-body data with ETGs observed at $0\lesssim z\lesssim3$ by populating the haloes with a stellar component under simple but justified assumptions: the resulting galaxies evolve consistently with the observed ETGs up to $z \simeq 2$, but the model has difficulty reproducing the fast evolution observed at $z\gtrsim 2$. We conclude that a substantial fraction of the size evolution of ETGs can be ascribed to a systematic dependence on redshift of the dark matter haloes structural properties.Comment: 15 pages, 14 figures, 1 table. Matches the Accepted version from MNRA

The angular momentum-mass relation: a fundamental law from dwarf irregulars to massive spirals

In a $\Lambda$CDM Universe, the specific stellar angular momentum ($j_\ast$) and stellar mass ($M_\ast$) of a galaxy are correlated as a consequence of the scaling existing for dark matter haloes ($j_{\rm h}\propto M_{\rm h}^{2/3}$). The shape of this law is crucial to test galaxy formation models, which are currently discrepant especially at the lowest masses, allowing to constrain fundamental parameters, e.g. the retained fraction of angular momentum. In this study, we accurately determine the empirical $j_\ast-M_\ast$ relation (Fall relation) for 92 nearby spiral galaxies (from S0 to Irr) selected from the Spitzer Photometry and Accurate Rotation Curves (SPARC) sample in the unprecedented mass range $7 \lesssim \log M_\ast/M_\odot \lesssim 11.5$. We significantly improve all previous estimates of the Fall relation by determining $j_\ast$ profiles homogeneously for all galaxies, using extended HI rotation curves, and selecting only galaxies for which a robust $j_\ast$ could be measured (converged $j_\ast(<R)$ radial profile). We find the relation to be well described by a single, unbroken power-law $j_\ast\propto M_\ast^\alpha$ over the entire mass range, with $\alpha=0.55\pm 0.02$ and orthogonal intrinsic scatter of $0.17\pm 0.01$ dex. We finally discuss some implications for galaxy formation models of this fundamental scaling law and, in particular, the fact that it excludes models in which discs of all masses retain the same fraction of the halo angular momentum.Comment: A&A Letters, accepte

Mass and shape of the Milky Way's dark matter halo with globular clusters from Gaia and Hubble

We estimate the mass of the inner ($<20$ kpc) Milky Way and the axis ratio of its dark matter halo using globular clusters as tracers. At the same time, we constrain the phase-space distribution of the globular cluster system. We use the Gaia DR2 catalogue of 75 globular clusters' proper motions and recent measurements of the proper motions of another 20 distant clusters obtained with the Hubble Space Telescope. We describe the globular cluster system with a 2-component distribution function (DF), with a flat, rotating disc and a rounder, more extended halo. While fixing the Milky Way's disc and bulge, we let the mass and shape of the dark matter halo and we fit these two parameters, together with other six describing the DF, with a Bayesian method. We find the mass of the Galaxy within 20 kpc to be $M(<20{\,\rm kpc})=1.91^{+0.18}_{-0.17} \times 10^{11} M_\odot$, of which $M_{\rm DM}(<20{\,\rm kpc})=1.37^{+0.18}_{-0.17}\times 10^{11}M_\odot$ is in dark matter, and the density axis ratio of the dark matter halo to be $q=1.30 \pm 0.25$. This implies a virial mass $M_{\rm via} = 1.3 \pm 0.3 \times 10^{12} M_\odot$. Our analysis rules out oblate ($q1.9$) with 99\% probability. Our preferred model reproduces well the observed phase-space distribution of globular clusters and has a disc component that closely resembles that of the Galactic thick disc. The halo component follows a power-law density profile $\rho \propto r^{-3.3}$, has a mean rotational velocity of $V_{\rm rot}\simeq -14\,\rm km\,s^{-1}$ at 20 kpc, and has a mildly radially biased velocity distribution ($\beta\simeq 0.2 \pm 0.07$, fairly constant outside 15 kpc). We also find that our distinction between disc and halo clusters resembles, although not fully, the observed distinction in metal-rich ([Fe/H]$>-0.8$) and metal-poor ([Fe/H]$\leq-0.8$) cluster populations.Comment: Accepted for publication in A&

A discrete chemo-dynamical model of the dwarf spheroidal galaxy Sculptor: mass profile, velocity anisotropy and internal rotation

We present a new discrete chemo-dynamical axisymmetric modeling technique, which we apply to the dwarf spheroidal galaxy Sculptor. The major improvement over previous Jeans models is that realistic chemical distributions are included directly in the dynamical modelling of the discrete data. This avoids loss of information due to spatial binning and eliminates the need for hard cuts to remove contaminants and to separate stars based on their chemical properties. Using a combined likelihood in position, metallicity and kinematics, we find that our models naturally separate Sculptor stars into a metal-rich and a metal-poor population. Allowing for non-spherical symmetry, our approach provides a central slope of the dark matter density of $\gamma = 0.5 \pm 0.3$. The metal-rich population is nearly isotropic (with $\beta_r^{red} = 0.0\pm0.1$) while the metal-poor population is tangentially anisotropic (with $\beta_r^{blue} = -0.2\pm0.1$) around the half light radius of $0.26$ kpc. A weak internal rotation of the metal-rich population is revealed with $v_{max}/\sigma_0 = 0.15 \pm 0.15$. We run tests using mock data to show that a discrete dataset with $\sim 6000$ stars is required to distinguish between a core ($\gamma = 0$) and cusp ($\gamma = 1$), and to constrain the possible internal rotation to better than $1\,\sigma$ confidence with our model. We conclude that our discrete chemo-dynamical modelling technique provides a flexible and powerful tool to robustly constrain the internal dynamics of multiple populations, and the total mass distribution in a stellar system.Comment: Accepted by MNRA

On the Luminous and Dark Matter Distribution in Early-Type Galaxies

The way mass is distributed in galaxies plays a major role in shaping their evolution across cosmic time. The galaxy's total mass is usually determined by tracing the motion of stars in its potential, which can be probed observationally by measuring stellar spectra at different distances from the galactic centre, whose kinematics is used to constrain dynamical models. A class of such models, commonly used to accurately determine the distribution of luminous and dark matter in galaxies, is that of equilibrium models. In this Thesis, a novel approach to the design of equilibrium dynamical models, in which the distribution function is an analytic function of the action integrals, is presented. Axisymmetric and rotating models are used to explain observations of a sample of nearby early-type galaxies in the Calar Alto Legacy Integral Field Area survey. Photometric and spectroscopic data for round and flattened galaxies are well fitted by the models, which are then used to get the galaxies' total mass distribution and orbital anisotropy. The time evolution of massive early-type galaxies is also investigated with numerical models. Their structural properties (mass, size, velocity dispersion) are observed to evolve, on average, with redshift. In particular, they appear to be significantly more compact at higher redshift, at fixed stellar mass, so it is interesting to investigate what drives such evolution. This Thesis focuses on the role played by dark-matter haloes: their mass-size and mass-velocity dispersion correlations evolve similarly to the analogous correlations of ellipticals; at fixed halo mass, the haloes are more compact at higher redshift, similarly to massive galaxies; a simple model, in which all the galaxy's size and velocity-dispersion evolution is due to the cosmological evolution of the underlying halo population, reproduces the observed size and velocity-dispersion of massive compact early-type galaxies up to redshift of about 2

Leaves on trees: identifying halo stars with extreme gradient boosted trees

Extended stellar haloes are a natural by-product of the hierarchical formation of massive galaxies. If merging is a non-negligible factor in the growth of our Galaxy, evidence of such events should be encoded in its stellar halo. Reliable identification of genuine halo stars is a challenging task however. The 1st Gaia data release contains the positions, parallaxes and proper motions for over 2 million stars, mostly in the Solar neighbourhood. Gaia DR2 will enlarge this sample to over 1.5 billion stars, the brightest ~5 million of which will have a full phase-space information. Our aim is to develop a machine learning model to reliably identify halo stars, even when their full phase-space information is not available. We use the Gradient Boosted Trees algorithm to build a supervised halo star classifier. The classifier is trained on a sample extracted from the Gaia Universe Model Snapshot, convolved with the errors of TGAS, as well as with the expected uncertainties of the upcoming Gaia DR2. We also trained our classifier on the cross-match between the TGAS and RAVE catalogues, where the halo stars are labelled in an entirely model independent way. We then use this model to identify halo stars in TGAS. When full phase- space information is available and for Gaia DR2-like uncertainties, our classifier is able to recover 90% of the halo stars with at most 30% distance errors, in a completely unseen test set, and with negligible levels of contamination. When line-of-sight velocity is not available, we recover ~60% of such halo stars, with less than 10% contamination. When applied to the TGAS data, our classifier detects 337 high confidence RGB halo stars. Although small, this number is consistent with the expectation from models given the data uncertainties. The large parallax errors are the biggest limitation to identify a larger number of halo stars in all the cases studied.Comment: Accepted for publication in Astronomy & Astrophysics. 13 pages, 9 figure, 2 table

Galaxy spin as a formation probe:the stellar-to-halo specific angular momentum relation

We derive the stellar-to-halo specific angular momentum relation (SHSAMR) of galaxies at $z=0$ by combining i) the standard $\Lambda$CDM tidal torque theory ii) the observed relation between stellar mass and specific angular momentum (Fall relation) and iii) various determinations of the stellar-to-halo mass relation (SHMR). We find that the ratio $f_j = j_\ast/j_{\rm h}$ of the specific angular momentum of stars to that of the dark matter i) varies with mass as a double power-law, ii) it always has a peak in the mass range explored and iii) it is $3-5$ times larger for spirals than for ellipticals. The results have some dependence on the adopted SHMR and we provide fitting formulae in each case. For any choice of the SHMR, the peak of $f_j$ occurs at the same mass where the stellar-to-halo mass ratio $f_\ast = M_\ast/M_{\rm h}$ has a maximum. This is mostly driven by the straightness and tightness of the Fall relation, which requires $f_j$ and $f_\ast$ to be correlated with each other roughly as $f_j\propto f_\ast^{2/3}$, as expected if the outer and more angular momentum rich parts of a halo failed to accrete onto the central galaxy and form stars (biased collapse). We also confirm that the difference in the angular momentum of spirals and ellipticals at a given mass is too large to be ascribed only to different spins of the parent dark-matter haloes (spin bias).Comment: matches MNRAS published versio

The dynamically selected stellar halo of the Galaxy with Gaia and the tilt of the velocity ellipsoid

Aims. We study the dynamical properties of halo stars located in the solar neighbourhood. Our goal is to explore how the properties of the halo depend on the selection criteria used to define a sample of halo stars. Once this is understood, we proceed to measure the shape and orientation of the halo’s velocity ellipsoid and we use this information to put constraints on the gravitational potential of the Galaxy. Methods. We use the recently released Gaia DR1 catalogue cross-matched to the RAVE dataset for our analysis. We develop a dynamical criterion based on the distribution function of stars in various Galactic components, using action integrals to identify halo members, and we compare this to the metallicity and to kinematically selected samples. Results. With this new method, we find 1156 stars in the solar neighbourhood that are likely members of the stellar halo. Our dynamically selected sample consists mainly of distant giants on elongated orbits. Their metallicity distribution is rather broad, with roughly half of the stars having [M/H] ≥ −1 dex. The use of different selection criteria has an important impact on the characteristics of the velocity distributions obtained. Nonetheless, for our dynamically selected and for the metallicity selected samples, we find the local velocity ellipsoid to be aligned in spherical coordinates in a Galactocentric reference frame. This suggests that the total gravitational potential is rather spherical in the region spanned by the orbits of the halo stars in these samples