Jeans Equation Research Articles

Equilibrium dynamical models in the inner region of the Large Magellanic Cloud based on Gaia DR3 kinematics

Context. The Large Magellanic Cloud (LMC) contains complex dynamics driven by both internal and external processes. The external forces are due to tidal interactions with the Small Magellanic Cloud and the Milky Way, while internally its dynamics mainly depend on the stellar, gas, and dark matter mass distributions. Despite this complexity, simple physical models often provide valuable insights into the primary driving factors. Aims. We used Gaia Data Release 3 (DR3) to explore how well equilibrium dynamical models based on the Jeans equations and the Schwarzschild orbit superposition method are able to describe the LMC’s five-dimensional phase-space distribution and line-of-sight (LOS) velocity distribution, respectively. In the Schwarzschild model, we incorporated a triaxial bar component for the first time and derived the LMC’s bar pattern speed. Methods. We fit comprehensive Jeans dynamical models to all Gaia DR3 stars with proper motion and LOS velocity measurements found in the footprint of the VISTA near-infrared survey of the Magellanic System using a discrete maximum likelihood approach. These models are very efficient at discriminating genuine LMC member stars from Milky Way foreground stars and background galaxies. They constrain the shape, orientation, and enclosed mass of the galaxy under the assumption of axisymmetry. We used the Jeans model results as a stepping stone to more complex two-component Schwarzschild models, which include an axisymmetric disc and a co-centric triaxial bar, which we fit to the LMC Gaia DR3 LOS velocity field using a χ2 minimisation approach. Results. The Jeans models describe the rotation and velocity dispersion of the LMC disc well, and we find an inclination angle of θ = 25.5° ±0.2°, line of nodes orientation of ψ = 124° ±0.4°, and an intrinsic thickness of the disc of q0d = b/a = 0.23 ± 0.01 (minor to major axis ratio). However, bound to axisymmetry, these models fail to properly describe the kinematics in the central region of the galaxy dominated by the LMC bar. We used the derived disc orientation and the Gaia DR3 density image of the LMC to obtain the intrinsic shape of the bar. Using these two components as input to our Schwarzschild models, we performed orbit integration and weighting in a rotating reference frame fixed to the bar, deriving an independent measurement of the LMC bar pattern speed of Ω = 11 ± 4 km s−1 kpc−1. Both the Jeans and Schwarzschild models predict the same enclosed mass distribution within a radius of 6.2 kpc of ∼ 1.4 × 1010 M⊙.

Gas thermodynamics meets galaxy kinematics: Joint mass measurements for eROSITA galaxy clusters

The mass of galaxy clusters is a critical quantity for probing cluster cosmology and testing theories of gravity, but its measurement could be biased, given that assumptions are inevitable in order to make use of any approach. In this paper, we employ and compare two mass proxies for galaxy clusters: thermodynamics of the intracluster medium and kinematics of member galaxies. We selected 22 galaxy clusters from the cluster catalog in the first SRG/eROSITA All-Sky Survey (eRASS1) that have sufficient optical and near-infrared observations. We generated multiband images in the energy range of (0.3, 7) keV for each cluster, and derived their temperature profiles, gas mass profiles, and hydrostatic mass profiles using a parametric approach that does not assume dark matter halo models. With spectroscopically confirmed member galaxies collected from multiple surveys, we numerically solved the spherical Jeans equation for their dynamical mass profiles. Our results quantify the correlation between dynamical mass and the line-of-sight velocity dispersion, $ dyn los ^2r_ proj /G) - (3.87 with a root mean square (rms) scatter of 0.14 dex. We find that the two mass proxies lead to roughly the same total mass, with no observed systematic bias. As such, the $ tension is not specific to hydrostatic mass or weak lensing shears, but also appears with galaxy kinematics. Interestingly, the hydrostatic-to-dynamical mass ratios decrease slightly toward large radii, which could possibly be evidence for accreting galaxies in the outskirts. We also compared our hydrostatic masses with the latest weak lensing masses inferred with scaling relations. The comparison shows that the weak lensing mass is significantly higher than our hydrostatic mass by sim 110<!PCT!>. This might explain the significantly larger value of $ from the latest measurement using eRASS1 clusters than almost all previous estimates in the literature. Finally, we tested the radial acceleration relation established in disk galaxies. We confirm the missing baryon problem in the inner region of galaxy clusters using three independent mass proxies for the first time. As ongoing and planned surveys are providing deeper X-ray observations and more galaxy spectra for cluster members, we expect to extend the study to cluster outskirts in the near future.

HSTPROMO Internal Proper-motion Kinematics of Dwarf Spheroidal Galaxies. I. Velocity Anisotropy and Dark Matter Cusp Slope of Draco

We analyze four epochs of Hubble Space Telescope imaging over 18 yr for the Draco dwarf spheroidal galaxy. We measure precise proper motions for hundreds of stars and combine these with existing line-of-sight (LOS) velocities. This provides the first radially resolved 3D velocity dispersion profiles for any dwarf galaxy. These constrain the intrinsic velocity anisotropy and resolve the mass–anisotropy degeneracy. We solve the Jeans equations in oblate axisymmetric geometry to infer the mass profile. We find the velocity dispersion to be radially anisotropic along the symmetry axis and tangentially anisotropic in the equatorial plane, with a globally averaged value βB¯=−0.20−0.53+0.28 , (where 1 – βB≡〈vtan2〉/〈vrad2〉 in 3D). The logarithmic dark matter (DM) density slope over the observed radial range, Γdark, is −0.83−0.37+0.32 , consistent with the inner cusp predicted in ΛCDM cosmology. As expected given Draco’s low mass and ancient star formation history, it does not appear to have been dissolved by baryonic processes. We rule out cores larger than 487, 717, and 942 pc at 1σ, 2σ, and 3σ confidence, respectively, thus imposing important constraints on the self-interacting DM cross section. Spherical models yield biased estimates for both the velocity anisotropy and the inferred slope. The circular velocity at our outermost data point (900 pc) is 24.19−2.97+6.31kms−1 . We infer a dynamical distance of 75.37−4.00+4.73 kpc and show that Draco has a modest LOS rotation, with v/σ=0.22±0.09 . Our results provide a new stringent test of the so-called “cusp–core” problem that can be readily extended to other dwarfs.

Total and dark mass from observations of galaxy centers with machine learning

Context.The galaxy total mass inside the effective radius is a proxy of the galaxy dark matter content and the star formation efficiency. As such, it encodes important information on the dark matter and baryonic physics.Aims.Total central masses can be inferred via galaxy dynamics or gravitational lensing, but these methods have limitations. We propose a novel approach based on machine learning to make predictions on total and dark matter content using simple observables from imaging and spectroscopic surveys.Methods.We used catalogs of multiband photometry, sizes, stellar mass, kinematic measurements (features), and dark matter (targets) of simulated galaxies from the Illustris-TNG100 hydrodynamical simulation to train a Mass Estimate machine Learning Algorithm (MELA) based on random forests.Results.We separated the simulated sample into passive early-type galaxies (ETGs), both normal and dwarf, and active late-type galaxies (LTGs) and showed that the mass estimator can accurately predict the galaxy dark masses inside the effective radius in all samples. We finally tested the mass estimator against the central mass estimates of a series of low-redshift (z ≲ 0.1) datasets, including SPIDER, MaNGA/DynPop, and SAMI dwarf galaxies, derived with standard dynamical methods based on the Jeans equations. We find that MELApredictions are fully consistent with the total dynamical mass of the real samples of ETGs, LTGs, and dwarf galaxies.Conclusions.MELAlearns from hydro-simulations how to predict the dark and total mass content of galaxies, provided that the real galaxy samples overlap with the training sample or show similar scaling relations in the feature and target parameter space. In this case, dynamical masses are reproduced within 0.30 dex (∼2σ), with a limited fraction of outliers and almost no bias. This is independent of the sophistication of the kinematical data collected (fiber vs. 3D spectroscopy) and the dynamical analysis adopted (radial vs. axisymmetric Jeans equations, virial theorem). This makes MELAa powerful alternative to predict the mass of galaxies of massive stage IV survey datasets using basic data, such as aperture photometry, stellar masses, fiber spectroscopy, and sizes. We finally discuss how to generalize these results to account for the variance of cosmological parameters and baryon physics using a more extensive variety of simulations and the further option of reverse engineering this approach and using model-free dark matter measurements (e.g., via strong lensing), plus visual observables, to predict the cosmology and the galaxy formation model.

First spiral arm detection using dynamical mass measurements of the Milky Way disk

We applied the vertical Jeans equation to the Milky Way disk in order to study non-axisymmetric variations in the thin disk surface density. We divided the disk plane into area cells with a 100 pc grid spacing and used four separate subsets of the Gaia DR3 stars, defined by cuts in absolute magnitude, that reach distances up to 3 kpc. The vertical Jeans equation is informed by the stellar number density field and the vertical velocity field; for the former, we used maps produced via Gaussian process regression; for the latter, we used Bayesian neural network radial velocity predictions, which allowed us to utilise the full power of the Gaia DR3 proper motion sample. For the first time, we find evidence of a spiral arm in the form of an over-density in the dynamically measured disk surface density, detected in all four data samples, which agrees very well with the spiral arm as traced by stellar age and chemistry. We fitted a simple spiral arm model to this feature and infer a relative over-density of roughly 20% and a width of roughly 400 pc. We also infer a thin disk surface density scale length of 3.3–4.2 kpc when restricting the analysis to stars within a distance of 2 kpc.

Anisotropy ansatz for the axisymmetric Jeans equations

Abstract The Jeans equations do not form a closed system, and to solve them a parametrization relating the velocity moments is often adopted. For axisymmetric models, a phenomenological choice (the “b-ansatz”) is widely used for the relation between the vertical ($\sigma _z^2$) and radial ($\sigma _R^2$) components of the velocity dispersion tensor, thus breaking their identity present in two-integral systems. However, the way in which the ansatz affects the resulting kinematical fields can be quite complicated, so that the analysis of these fields is usually performed only after numerically computing them. We present here a general procedure to study the properties of the ansatz-dependent fields $\overline{v_{\varphi }^2}$, $\Delta =\overline{v_{\varphi }^2}- \sigma _z^2$ and $\Delta _R= \overline{v_{\varphi }^2}- \sigma _R^2$. Specifically, the effects of the b-ansatz can be determined before solving the Jeans equations once the behaviour over the (R, z)-plane of three easy-to-build ansatz-independent functions is known. The procedure also constrains the ansatz to exclude unphysical results (as a negative $\overline{v_{\varphi }^2}$). The method is illustrated by discussing the cases of three well-known galaxy models: the Miyamoto & Nagai and Satoh disks, and the Binney logarithmic halo, for which the regions and the constraints on the ansatz values can be determined analytically; a two-component (Miyamoto & Nagai plus logarithmic halo) model is also discussed.

The Radial Orbits of Ram-pressure-stripped Galaxies in Clusters from the GASP Survey

We analyze a sample of 244 ram-pressure-stripped candidate galaxy members within the virial radius of 62 nearby clusters to determine their velocity anisotropy profile β(r). We use previously determined mass profiles for the 62 clusters to build an ensemble cluster by stacking the 62 cluster samples in projected phase space. We solve the Jeans equation for dynamical equilibrium by two methods, MAMPOSSt and the Jeans inversion technique, and determine β(r) both in parametric form and nonparametrically. The two methods consistently indicate that the orbits of the ram-pressure-stripped candidates are increasingly radial with distance from the cluster center, from almost isotropic (β ≃ 0) at the center, to very radial at the virial radius (β ≃ 0.7). The orbits of cluster galaxies undergoing ram pressure stripping are similar to those of spiral cluster galaxies but more radially elongated at large radii.

A distinct radial acceleration relation across the brightest cluster galaxies and galaxy clusters

Recent studies reveal a radial acceleration relation (RAR) in galaxies, which illustrates a tight empirical correlation connecting the observational acceleration and the baryonic acceleration with a characteristic acceleration scale. However, a distinct RAR has been revealed on brightest cluster galaxy (BCG) cluster scales with a seventeen-times-larger acceleration scale due to the gravitational lensing effect. In this work, we systematically explore the acceleration and mass correlations between dynamical and baryonic components in 50 BCGs. To investigate the dynamical RAR in BCGs, we derived their dynamical accelerations from the stellar kinematics using the Jeans equation through Abel inversion and adopted the baryonic mass from Sloan Digital Sky Survey photometry. We explored the spatially resolved kinematic profiles with the largest integral field spectroscopy (IFS) data collected by the Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) survey. Our results demonstrate that the dynamical RAR in BCGs is consistent with the lensing RAR on BCG-cluster scales as well as a larger acceleration scale. This finding may imply that BCGs and galaxy clusters have fundamental differences from field galaxies. We also find a mass correlation, but it is less tight than the acceleration correlation.

The rotation curve and mass distribution of M31

ABSTRACT To gain a better understanding of the Andromeda galaxy M31 and its role in the Local Group, measuring its mass precisely is essential. In this work, we have constructed the rotation curve of M31 out to ∼125 kpc using 13 679 M31 objects obtained from various sources, including the LAMOST data release 9 (LAMOST DR9), the DESI survey, and relevant literature. We divide all objects in our sample into bulge, disc, and halo components. For the sources in the M31 disc, we have measured their circular velocities by a kinematic model with asymmetric drift corrections. For the bulge and halo objects, we calculate their velocity dispersions and use the spherical and projected Jeans equation to obtain the circular velocities. Our findings indicate a nearly isotropic nature for the M31 bulge, while the halo exhibits tangential anisotropy. The results show that the rotation curve remains constant at ∼220 km s−1 up to radius ∼25 kpc and gradually decreases to ∼170 km s−1 further out. Based on the newly determined rotation curve, we have constructed a mass distribution model for M31. Our measurement of the M31 virial mass is $M_{\rm vir} = 1.14^{+0.51}_{-0.35} \times 10^{12} M_\odot$ within rvir = 220 ± 25 kpc.

Predicting the scaling relations between the dark matter halo mass and observables from generalised profiles I: Kinematic tracers

Abstract We investigate the relationship between a dark matter halo’s mass profile and measures of the velocity dispersion of kinematic tracers within its gravitational potential. By predicting the scaling relation of the halo mass with the aperture velocity dispersion, $M_\mathrm{vir} - \unicode{x03C3}_\mathrm{ap}$ , we present the expected form and dependence of this halo mass tracer on physical parameters within our analytic halo model: parameterised by the halo’s negative inner logarithmic density slope, $\unicode{x03B1}$ , its concentration parameter, c, and its velocity anisotropy parameter, $\unicode{x03B2}$ . For these idealised halos, we obtain a general solution to the Jeans equation, which is projected over the line of sight and averaged within an aperture to form the corresponding aperture velocity dispersion profile. Through dimensional analysis, the $M_\mathrm{vir} - \unicode{x03C3}_\mathrm{ap}$ scaling relation is devised explicitly in terms of analytical bounds for these aperture velocity dispersion profiles: allowing constraints to be placed on this relation for motivated parameter choices. We predict the $M_{200} - \unicode{x03C3}_\mathrm{ap}$ and $M_{500} - \unicode{x03C3}_\mathrm{ap}$ scaling relations, each with an uncertainty of $60.5\%$ and $56.2\%$ , respectively. These halo mass estimates are found to be weakly sensitive to the halo’s concentration and mass scale, and most sensitive to the size of the aperture radius in which the aperture velocity dispersion is measured, the maximum value for the halo’s inner slope, and the minimum and maximum values of the velocity anisotropy. Our results show that a halo’s structural and kinematic profiles impose only a minor uncertainty in estimating its mass. Consequently, spectroscopic surveys aimed at constraining the halo mass using kinematic tracers can focus on characterising other, more complex sources of uncertainty and observational systematics.

Jeans modelling of weakly flattened ellipsoidal systems

ABSTRACT In the homoeoidal expansion, a given ellipsoidally stratified density distribution and its associated potential are expanded in the (small) density flattening parameter η, and usually truncated at the linear order. The truncated density–potential pair obeys exactly the Poisson equation, and it can be interpreted as the first-order expansion of the original ellipsoidal density–potential pair, or as a new autonomous system. In the first interpretation, in the solutions of the Jeans equations the quadratic terms in η must be discarded (‘η-linear’ solutions), while in the second (‘η-quadratic’) all terms are retained. In this work, we study the importance of the quadratic terms by using the ellipsoidal Plummer model and the Perfect Ellipsoid, which allow for fully analytical η-quadratic solutions. These solutions are then compared with those obtained numerically for the original ellipsoidal models, finding that the η-linear models already provide an excellent approximation of the numerical solutions. As an application, the η-linear Plummer model (with a central black hole) is used for the phenomenological interpretation of the dynamics of the weakly flattened and rotating globular cluster NGC 4372, confirming that this system cannot be interpreted as an isotropic rotator, a conclusion reached previously with more sophisticated studies.

CLASH-VLT: The Inner Slope of the MACS J1206.2-0847 Dark Matter Density Profile

The inner slope (γ DM) of the dark matter (DM) density profile of cosmological halos carries information about the properties of DM and/or baryonic processes affecting the halo gravitational potential. Cold DM cosmological simulations predict steep inner slopes, γ DM ≃ 1. We test this prediction on the MACS J1206.2-0847 cluster at redshift z = 0.44, whose DM density profile has been claimed to be cored at the center. We determine the cluster DM density profile from 2 kpc from the cluster center to the virial radius (∼2 Mpc), using the velocity distribution of ≃500 cluster galaxies and the internal velocity dispersion profile of the Brightest Cluster Galaxy (BCG), obtained from VIMOS@VLT and MUSE@VLT data. We solve the Jeans equation of dynamical equilibrium using an upgraded version of the MAMPOSSt method. The total mass profile is modeled as a sum of a generalized Navarro–Frenk–White profile that describes the DM component, allowing for a free inner slope of the density profile, a Jaffe profile that describes the BCG stellar mass component, and a nonparametric baryonic profile that describes the sum of the remaining galaxy stellar mass and of the hot intra-cluster gas mass. Our total mass profile is in remarkable agreement with independent determinations based on X-ray observations and strong lensing. We find (68% confidence levels), consistent with predictions from recent Lambda cold dark matter cosmological numerical simulations.

The surface mass density of the Milky Way: does the traditional KZ approach work in the context of new surveys?

ABSTRACT We revisit the classical KZ problem – determination of the vertical force and implied total mass density distribution of the Milky Way disc – for a wide range of Galactocentric radius and vertical height using chemically selected thin and thick disc samples based on Apache Point Observatory Galactic Evolution Experiment spectroscopy combined with the Gaia astrometry. We derived the velocity dispersion profiles in Galactic cylindrical coordinates, and solved the Jeans equation for the two samples separately. The result is surprising that the total surface mass density as a function of vertical height as derived for these two chemically distinguished populations is different. The discrepancies are larger in the inner compared to the outer Galaxy, with the density calculated from thick disc being larger, independent of the Galactic radius. Furthermore, while there is an overall good agreement between the total mass density derived for the thick disc population and the standard halo model for vertical heights larger than 1 kpc, close to the mid-plane the mass density observed using the thick disc population is larger than that predicted from the standard halo model. We explore various implications of these discrepancies, and speculate their sources, including problems associated with the assumed density laws, velocity dispersion profiles, and the Galactic rotation curve, potential non-equilibrium of the Galactic disc, or a failure of the Navarro-Frenk-White (NFW) dark matter halo profile for the Milky Way. We conclude that the growing detail in hand on the chemodynamical distributions of Milky Way stars challenges traditional analytical treatments of the KZ problem.

Measuring galaxy cluster mass profiles into the low-acceleration regime with galaxy kinematics

We probed the dynamical mass profiles of ten galaxy clusters from the HIghest X-ray FLUx Galaxy Cluster Sample (HIFLUGCS) using galaxy kinematics. We numerically solved the spherical Jeans equation and parameterize the dynamical mass profile and the galaxy velocity anisotropy profile using two general functions to ensure that our results are not biased toward any specific model. The mass-velocity anisotropy degeneracy is ameliorated by using two “virial shape parameters” that depend on the fourth moment of velocity distribution. The resulting velocity anisotropy estimates consistently show a nearly isotropic distribution in the inner regions, with an increasing radial anisotropy toward large radii. We compared our derived dynamical masses with those calculated from X-ray gas data assuming hydrostatic equilibrium, finding that massive and rich relaxed clusters generally present consistent mass measurements, while unrelaxed or low-richness clusters have systematically larger total masses than hydrostatic masses by, on average, 50%. This might help alleviate current tensions in the measurement of σ8, but it also leads to cluster baryon fractions below the cosmic value. Finally, our approach probes accelerations as low as 10−11 m s−2, comparable to the outskirts of individual late-type galaxies. We confirm that galaxy clusters deviate from the radial acceleration relation defined by galaxies.

Dynamics of the galactic component of Abell S1063 and MACS J1206.2−0847

Context.The galactic component in clusters is commonly thought to be generally nonrotating and in a dynamical state different from that of a collisionally relaxed system. In practice, a test for such conditions is often not available.Aims.We consider the member galaxies of the two clusters Abell S1063 and MACS J1206.2−0847 and study the possible presence of mean rotation and some properties of their distribution in phase space. We look for empirical evidence of factors normally found in collisionally relaxed systems and other characteristics of violently relaxed collisionless systems.Methods.Starting from the CLASH-VLT data, we obtained positions, stellar masses, and individual line-of-sight velocities for a large number of galaxies (NAS1063 ≈ 1200 andNM1206 ≈ 650) extending out to ≈1.6 (Abell) and ≈2.5 (MACS) times the radiusr200. We studied the spatial distribution of the galaxy velocities and the properties of the available galaxy sets when divided into stellar mass bins. To test the presence of velocity dispersion anisotropy, we compared the results based on the Jeans equations with those obtained by assuming a specific form of the galaxy distribution function incorporating the picture of violent relaxation, where the total gravitational potential is imposed as set by the available gravitational lensing observations.Results.We find evidence of systematic rotation in both clusters, with significant rotation in each core (within 0.5′ from the center) and no signatures of rotation at large radii. While no signs of energy equipartition were found, there is a clear indication of (stellar) mass segregation. Velocity dispersion anisotropy is present and qualitatively similar to what has been found in violently relaxed collisionless systems. This last conclusion is strengthened by the overall success in matching the observations with the predictions of the physically justified distribution function.

A Bayesian estimation of the Milky Way’s circular velocity curve using Gaia DR3

Aims. Our goal is to calculate the circular velocity curve of the Milky Way, along with corresponding uncertainties that quantify various sources of systematic uncertainty in a self-consistent manner. Methods. The observed rotational velocities are described as circular velocities minus the asymmetric drift. The latter is described by the radial axisymmetric Jeans equation. We thus reconstruct the circular velocity curve between Galactocentric distances from 5 kpc to 14 kpc using a Bayesian inference approach. The estimated error bars quantify uncertainties in the Sun’s Galactocentric distance and the spatial-kinematic morphology of the tracer stars. As tracers, we used a sample of roughly 0.6 million stars on the red giant branch stars with six-dimensional phase-space coordinates from Gaia Data Release 3 (DR3). More than 99% of the sample is confined to a quarter of the stellar disc with mean radial, rotational, and vertical velocity dispersions of (35 ± 18) km s−1, (25 ± 13) km s−1, and (19 ± 9) km s−1, respectively. Results. We find a circular velocity curve with a slope of 0.4 ± 0.6 km s−1 kpc−1, which is consistent with a flat curve within the uncertainties. We further estimate a circular velocity at the Sun’s position of vc(R0) = 233 ± 7 km s−1 and that a region in the Sun’s vicinity, characterised by a physical length scale of ∼1 kpc, moves with a bulk motion of VLSR = 7 ± 7 km s−1. Finally, we estimate that the dark matter (DM) mass within 14 kpc is log10 MDM(R < 14kpc)/ M⊙ =(11.2+2.0-2.3) and the local spherically averaged DM density is ρDM(RO)=(0.41+0.10-0.09) GeV cm-3 = (0.011+0.003-0.002) M⊙pc-3. In addition, the effect of biased distance estimates on our results is assessed.

Virial theorem for a cloud of stars obtained from the Jeans equations with second correlation moments

A hydrodynamic model for small acoustic oscillations in a cloud of stars is built, taking into account the self-consistent gravitational field in equilibrium with a non-zero second correlation moment. It is assumed that the momentum flux density tensor should include the analog of the anisotropic pressure tensor and the second correlation moment of both longitudinal and transverse gravitational field strength. The non-relativistic temporal equation for the second correlation moment of the gravitational field strength is derived from the Einstein equations using the first-order post-Newtonian approximation. One longitudinal and two transverse branches of acoustic oscillations are found in a homogeneous and isotropic star cloud. The requirement for the velocity of transverse oscillations to be zero provides the boundary condition for the stability of the cloud. The critical radius of the spherical cloud of stars is obtained, which is precisely consistent with the virial theorem.

Forecasts on the Dark Matter Density Profiles of Dwarf Spheroidal Galaxies with Current and Future Kinematic Observations

We forecast parameter uncertainties on the mass profile of a typical Milky Way dwarf spheroidal galaxy (dSph) using the spherical Jeans equation and Fisher matrix formalism. For a Draco-like system we show that radial velocity measurements for 1000 individual stars can constrain the mass contained within the effective radius of a dSph to within 5%. This is consistent with constraints extracted from current observational data. We compare two systems, a cusp and core, and demonstrate that a minimum sample of 100,000 (10,000) stars with both radial and proper motions measurements is required to disentangle their inner slopes at the 2σ (1σ) level. If using the log-slope measured at the half-light radius as a proxy for differentiating between a core or cusp slope, only 1000 line-of-sight and proper motions measurements are required; however, we show this choice of radius does not always unambiguously differentiate between core and cusped profiles. Once observational errors are below half the value of the intrinsic dispersion, improving the observational precision yields little change in the density profile uncertainties. The choice of priors in our profile shape analysis plays a crucial role when the number of stars in a system is less than 100 but does not affect the resulting uncertainties for larger kinematic samples. Our predicted 2D confidence regions agree well with those from a full likelihood analysis run on a mock kinematic data set taken from the Gaia Challenge, validating our Fisher predictions. Our methodology is flexible, allowing us to predict density profile uncertainties for a wide range of current and future kinematic data sets.

Structural and dynamical modeling of WINGS clusters

Numerical simulations indicate that cosmological halos display power-law radial profiles of pseudo phase-space density (PPSD), Q ≡ ρ/σ3, where ρ is the mass density and σ is the velocity dispersion. We tested these predictions for Q(r) using the parameters derived from the Markov chain Monte Carlo (MCMC) analysis performed with the MAMPOSSt mass-orbit modeling code on the observed kinematics of a velocity dispersion based stack (σv) of 54 nearby regular clusters of galaxies from the WINGS data set. In the definition of PPSD, the density is either in total mass ρ (Qρ) or in galaxy number density ν (Qν) of three morphological classes of galaxies (ellipticals, lenticulars, and spirals), while the velocity dispersion (obtained by inversion of the Jeans equation using the MCMC parameters) is either the total (Qρ and Qν) or its radial component (Qr, ρ and Qr, ν). We find that the PPSD profiles are indeed power-law relations for nearly all MCMC parameters. The logarithmic slopes of our observed Qρ(r) and Qr, ρ(r) for ellipticals and spirals are in excellent agreement with the predictions for particles in simulations, but slightly shallower for S0s. For Qν(r) and Qr, ν(r), only the ellipticals have a PPSD slope matching that of particles in simulations, while the slope for spirals is much shallower, similar to that of subhalos. However, for cluster stacks based on the richness or gas temperature, the fraction of power-law PPSDs is lower (esp. Qν) and the Qρ slopes are shallower, except for S0s. The observed PPSD profiles, defined using ρ rather than ν, appear to be a fundamental property of galaxy clusters. They would be imprinted during an early phase of violent relaxation for dark matter and ellipticals, and later for spirals as they move toward dynamical equilibrium in the cluster gravitational potential, while S0s are either intermediate (richness and temperature-based stacks) or a mixed class (σv stack).

Dynamics of dwarf galaxies in f(R) gravity

ABSTRACT We use the kinematic data of the stars in eight dwarf spheroidal galaxies to assess whether f(R) gravity can fit the observed profiles of the line-of-sight velocity dispersion of these systems without resorting to dark matter. Our model assumes that each galaxy is spherically symmetric and has a constant velocity anisotropy parameter β and constant mass-to-light ratio consistent with stellar population synthesis models. We solve the spherical Jeans equation that includes the Yukawa-like gravitational potential appearing in the weak field limit of f(R) gravity, and a Plummer density profile for the stellar distribution. The f(R) velocity dispersion profiles depends on two parameters: the scale length ξ−1, below which the Yukawa term is negligible, and the boost of the gravitational field δ > −1. δ and ξ are not universal parameters, but their variation within the same class of objects is expected to be limited. The f(R) velocity dispersion profiles fit the data with a value $\xi ^{-1}= 1.2^{+18.6}_{-0.9}$ Mpc for the entire galaxy sample. On the contrary, the values of δ show a bimodal distribution that picks at $\overline{\delta }=-0.986\pm 0.002$ and $\overline{\delta }=-0.92\pm 0.01$. These two values disagree at 6σ and suggest a severe tension for f(R) gravity. It remains to be seen whether an improved model of the dwarf galaxies or additional constraints provided by the proper motions of stars measured by future astrometric space missions can return consistent δ’s for the entire sample and remove this tension.