Spherical Jeans Equation Research Articles

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.

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.

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.

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.

Milky Way mass with K giants and BHB stars using LAMOST, SDSS/SEGUE, and Gaia: 3D spherical Jeans equation and tracer mass estimator

ABSTRACT We measure the enclosed Milky Way mass profile to Galactocentric distances of ∼70 and ∼50 kpc using the smooth, diffuse stellar halo samples of Bird et al. The samples are Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) and Sloan Digital Sky Survey/Sloan Extension for Galactic Understanding and Exploration (SDSS/SEGUE) K giants (KG) and SDSS/SEGUE blue horizontal branch (BHB) stars with accurate metallicities. The 3D kinematics are available through LAMOST and SDSS/SEGUE distances and radial velocities and Gaia DR2 proper motions. Two methods are used to estimate the enclosed mass: 3D spherical Jeans equation and Evans et al. tracer mass estimator (TME). We remove substructure via the Xue et al. method based on integrals of motion. We evaluate the uncertainties on our estimates due to random sampling noise, systematic distance errors, the adopted density profile, and non-virialization and non-spherical effects of the halo. The tracer density profile remains a limiting systematic in our mass estimates, although within these limits we find reasonable agreement across the different samples and the methods applied. Out to ∼70 and ∼50 kpc, the Jeans method yields total enclosed masses of 4.3 ± 0.95 (random) ±0.6 (systematic) × 1011 M⊙ and 4.1 ± 1.2 (random) ±0.6 (systematic) × 1011 M⊙ for the KG and BHB stars, respectively. For the KG and BHB samples, we find a dark matter virial mass of $M_{200}=0.55^{+0.15}_{-0.11}$ (random) ±0.083 (systematic) × 1012 M⊙ and $M_{200}=1.00^{+0.67}_{-0.33}$ (random) ±0.15 (systematic) × 1012 M⊙, respectively.

Non-parametric spherical Jeans mass estimation with B-splines

ABSTRACTSpherical Jeans modelling is widely used to estimate mass profiles of systems from star clusters to galactic stellar haloes to clusters of galaxies. It derives the cumulative mass profile, M(<r), from kinematics of tracers of the potential under the assumptions of spherical symmetry and dynamical equilibrium. We consider the application of Jeans modelling to mapping the dark matter distribution in the outer reaches of the Milky Way using field halo stars. We present a novel non-parametric routine for solving the spherical Jeans equation by fitting B-splines to the velocity and density profiles of halo stars. While most implementations assume parametric forms for these profiles, B-splines provide non-parametric fitting curves with analytical derivatives. Our routine recovers the mass profiles of equilibrium systems with flattened haloes or a stellar disc and bulge excellently (${\lesssim} 10{{\ \rm per\ cent}}$ error at most radii). Tests with non-equilibrium, Milky Way-like galaxies from the Latte suite of the Feedback In Realistic Environments model 2 (FIRE-2) simulations perform quite well (${\lesssim} 15{{\ \rm per\ cent}}$ error for r$\lesssim$ 100 kpc). We also create observationally motivated data sets for the Latte suite by imposing selection functions and errors on phase-space coordinates characteristic of Gaia and the Dark Energy Spectroscopic Instrument (DESI) Milky Way Survey. The resulting imprecise and incomplete data require us to introduce a Markov chain Monte Carlo (MCMC)-based subroutine to obtain deconvolved density and velocity dispersion profiles from the tracer population. With these observational effects taken into account, the accuracy of the Jeans mass estimate remains at the level 20 per cent or better.

Dynamical analysis of clusters of galaxies from cosmological simulations

Context. Studies of cluster mass and velocity anisotropy profiles are useful tests of dark matter models and of the assembly history of clusters of galaxies. These studies might be affected by unknown systematics caused by projection effects. Aims. We aim to test observational methods for the determination of mass and velocity anisotropy profiles of clusters of galaxies. Particularly, we focus on the MAMPOSSt technique Methods. We used results from two semi-analytic models of galaxy formation, coupled with high-resolution N-body cosmological simulations, the DLB07 catalog, and the FIRE catalog based on the new GAlaxy Evolution and Assembly model. We tested the reliability of the Jeans equation in recovering the true mass profile when full projected phase-space information is available. We examined the reliability of the MAMPOSSt method in estimating the true mass and velocity anisotropy profiles of the simulated halos when only projected phase-space information is available, as in observations. Results. The spherical Jeans equation provides a reliable tool to determine cluster mass profiles, both when considering the whole population of cluster galaxies, and when considering subsamples of tracers separated by galaxy color; the exception to this is for the central region, where deviations may be attributed to dynamical friction effects or galaxy mergers. The results are equally good for prolate and oblate clusters. Using only projected phase-space information, MAMPOSSt provides estimates of the mass profile with a standard deviation of 35–69% and a negative bias of 7–17%, nearly independent of radius, which we attribute to the presence of interlopers in the projected samples. The bias changes sign; that is, the mass is over-estimated, for prolate clusters with their major axis aligned along the line of sight. The MAMPOSSt method measures the velocity anisotropy profiles accurately in the inner cluster regions and there is a slight overestimate in the outer regions for the whole sample of observationally identified cluster members, and, separately, for red and blue galaxies.

What to expect from dynamical modelling of cluster haloes – I. The information content of different dynamical tracers

ABSTRACT Using hydrodynamical simulations, we study how well the underlying gravitational potential of a galaxy cluster can be modelled dynamically with different types of tracers. In order to segregate different systematics and the effects of varying estimator performances, we first focus on applying a generic minimal assumption method (oPDF) to model the simulated haloes using the full 6D phase-space information. We show that the halo mass and concentration can be recovered in an ensemble unbiased way, with a stochastic bias that varies from halo to halo, mostly reflecting deviations from steady state in the tracer distribution. The typical systematic uncertainty is ∼0.17 dex in the virial mass and ∼0.17 dex in the concentration as well when dark matter (DM) particles are used as tracers. The dynamical state of satellite galaxies are close to that of DM particles, while intracluster stars are less in a steady state, resulting in an ∼0.26-dex systematic uncertainty in mass. Compared with galactic haloes hosting Milky-Way-like galaxies, cluster haloes show a larger stochastic bias in the recovered mass profiles. We also test the accuracy of using intracluster gas as a dynamical tracer modelled through a generalized hydrostatic equilibrium equation, and find a comparable systematic uncertainty in the estimated mass to that using DM. Lastly, we demonstrate that our conclusions are largely applicable to other steady-state dynamical models including the spherical Jeans equation, by quantitatively segregating their statistical efficiencies and robustness to systematics. We also estimate the limiting number of tracers that leads to the systematics-dominated regime in each case.

Stellar profile independent determination of the dark matter distribution of the Fornax Local Group dwarf spheroidal galaxy

ABSTRACT We detail a method to measure the correspondence between dark matter (DM) models and observations of stellar populations within Local Group dwarf spheroidal galaxies (LG dSphs) that assumes no parametric stellar distribution. Solving the spherical or cylindrical Jeans equations, we calculate the consistency of DM and stellar kinematic models with stellar positions and line-of-sight velocities. Our method can be used to search for signals of standard and exotic DM distributions. Applying our methodology to the Fornax LG dSph and using statistical bootstrapping, we find: (i) that oblate or prolate cored DM haloes match the stellar data, respectively, ≃60 or ≃370 times better than oblate or prolate cusped DM haloes for isotropic and isothermal stellar velocity dispersions, (ii) that cusped spherical DM haloes and cored spherical DM haloes match the Fornax data similarly well for isotropic stellar velocity dispersions, (iii) that the semiminor to semimajor axial ratio of spheroidal DM haloes are more extreme than 80 per cent of those predicted by Lambda cold dark matter with baryon simulations, (iv) that oblate cored or cusped DM haloes are, respectively, ≃5 or ≃30 times better matches to Fornax than prolate cored or cusped DM haloes, and (v) that Fornax shows no evidence of a disc-like structure with more than two per cent of the total DM mass. We further note that the best-fitting cusped haloes universally favour the largest mass and size fit parameters. If these extreme limits are decreased, the cusped halo likelihoods decrease relative to those of cored haloes.

The relaxation of galaxy clusters at redshift z = 0 in IllustrisTNG simulation

We study the dynamical states of the 30 most massive galaxy clusters in the TNG100 simulation at redshift z = 0 using three types of tracers: stars, dark matter particles and satellite galaxies. If the massive galaxy cluster is spherically symmetric and relaxed, we can obtain the underlying total mass distribution accurately from its dynamical tracers using the spherical Jeans equations. Although the three tracers of clusters have very different number densities, velocity dispersions and anisotropies, they still trace the same total mass profile. We obtain the total mass profiles of clusters using these tracers separately and compare them with the true mass distributions. We find that: (1) the kinematics of dark matter trace the total mass of all clusters well and the mass inferred from dark matter are generally consistent with the true mass profiles with relative deviations smaller than ∼ 25% at all radii; (2) stars in ∼ 60% massive clusters are approaching equilibrium and the total mass of these clusters inferred from stars have relative deviations smaller than ∼ 50% at all radii. Stellar substructures are rich and the mass inferred from stars tend to be over-estimated in the inner region; and (3) satellite galaxies are unrelaxed in the inner region and become more relaxed as the radius increases. The total mass inferred from satellites are under-estimated in all regions.

On the Presence of a Universal Acceleration Scale in Elliptical Galaxies

Abstract Dark matter phenomena in rotationally supported galaxies exhibit a characteristic acceleration scale of g † ≈ 1.2 × 10−10 m s−2. Whether this acceleration is a manifestation of a universal scale, or merely an emergent property with an intrinsic scatter, has been debated in the literature. Here we investigate whether a universal acceleration scale exists in dispersion-supported galaxies using two uniform sets of integral field spectroscopy (IFS) data from SDSS-IV MaNGA and ATLAS3D. We apply the spherical Jeans equation to 15 MaNGA and 4 ATLAS3D slow-rotator E0 (i.e., nearly spherical) galaxies. Velocity dispersion profiles for these galaxies are well determined with observational errors under control. Bayesian inference indicates that all 19 galaxies are consistent with a universal acceleration of m s−2. Moreover, all 387 data points from the radial bins of the velocity dispersion profiles are consistent with a universal relation between the radial acceleration traced by dynamics and that predicted by the observed distribution of baryons. This universality remains if we include 12 additional non-E0 slow-rotator elliptical galaxies from ATLAS . Finally, the universal acceleration from MaNGA and ATLAS3D is consistent with that for rotationally supported galaxies, so our results support the view that dark matter phenomenology in galaxies involves a universal acceleration scale.

The mass of our Milky Way

We perform an extensive review of the numerous studies and methods used to determine the total mass of the Milky Way. We group the various methods into seven broad classes, including: i) estimating Galactic escape velocity using high velocity objects; ii) measuring the rotation curve through terminal and circular velocities; iii) modeling halo stars, globular clusters and satellite galaxies with the Spherical Jeans equation and iv) with phase-space distribution functions; v) simulating and modeling the dynamics of stellar streams and their progenitors; vi) modeling the motion of the Milky Way, M31 and other distant satellites under the framework of Local Group timing argument; and vii) measurements made by linking the brightest Galactic satellites to their counterparts in simulations. For each class of methods, we introduce their theoretical and observational background, the method itself, the sample of available tracer objects, model assumptions, uncertainties, limits and the corresponding measurements that have been achieved in the past. Both the measured total masses within the radial range probed by tracer objects and the extrapolated virial masses are discussed and quoted. We also discuss the role of modern numerical simulations in terms of helping to validate model assumptions, understanding systematic uncertainties and calibrating the measurements. While measurements in the last two decades show a factor of two scatters, recent measurements using \textit{Gaia} DR2 data are approaching a higher precision. We end with a detailed discussion of future developments, especially as the size and quality of the observational data will increase tremendously with current and future surveys. In such cases, the systematic uncertainties will be dominant and thus will necessitate a much more rigorous testing and characterization of the various mass determination methods.

Accurate mass estimates from the proper motions of dispersion-supported galaxies

ABSTRACT We derive a new mass estimator that relies on internal proper motion measurements of dispersion-supported stellar systems, one that is distinct and complementary to existing estimators for line-of-sight velocities. Starting with the spherical Jeans equation, we show that there exists a radius where the mass enclosed depends only on the projected tangential velocity dispersion, assuming that the anisotropy profile slowly varies. This is well-approximated at the radius where the log-slope of the stellar tracer profile is −2: r−2. The associated mass is $M(r_{-2}) = 2 G^{-1} \langle \sigma _{\mathcal {T}}^{2}\rangle ^{*} r_{-2}$ and the circular velocity is $V^{2}({r_{-2}}) = 2\langle \sigma _{\mathcal {T}}^{2}\rangle ^{*}$. For a Plummer profile r−2 ≃ 4Re/5. Importantly, r−2 is smaller than the characteristic radius for line-of-sight velocities derived by Wolf et al. Together, the two estimators can constrain the mass profiles of dispersion-supported galaxies. We illustrate its applicability using published proper motion measurements of dwarf galaxies Draco and Sculptor, and find that they are consistent with inhabiting cuspy NFW subhaloes of the kind predicted in CDM but we cannot rule out a core. We test our combined mass estimators against previously published, non-spherical cosmological dwarf galaxy simulations done in both cold dark matter (CDM; naturally cuspy profile) and self-interacting dark matter (SIDM; cored profile). For CDM, the estimates for the dynamic rotation curves are found to be accurate to $10\rm { per\, cent}$ while SIDM are accurate to $15\rm { per\, cent}$. Unfortunately, this level of accuracy is not good enough to measure slopes at the level required to distinguish between cusps and cores of the type predicted in viable SIDM models without stronger priors. However, we find that this provides good enough accuracy to distinguish between the normalization differences predicted at small radii (r ≃ r−2 < rcore) for interesting SIDM models. As the number of galaxies with internal proper motions increases, mass estimators of this kind will enable valuable constraints on SIDM and CDM models.

Galaxy Cluster Mass Estimates in the Presence of Substructure

Abstract We develop and implement a model to analyze the internal kinematics of galaxy clusters that may contain subpopulations of galaxies that do not independently trace the cluster potential. The model allows for substructures within the cluster environment and disentangles cluster members from contaminating foreground and background galaxies. We estimate the cluster velocity dispersion and/or mass while marginalizing over uncertainties in all of the above complexities. Using mock observations from the MultiDark simulation, we compare the true substructures from the simulation with the substructures identified by our model, showing that 50% of the identified substructures have at least 79% of its members are also members of the same true substructure, which is on par with other substructure identification algorithms. Furthermore, we show a ∼35% decrease in scatter in the inferred velocity dispersion versus true cluster mass relationship when comparing a model that allows three substructures to a model that assumes no substructure. In a first application to our published data for A267, we identify up to four distinct galaxy subpopulations. We use these results to explore the sensitivity of inferred cluster properties to the treatment of substructure. Compared to a model that assumes no substructure, our substructure model reduces the dynamical mass of A267 by ∼22% and shifts the cluster mean velocity by ∼100 km s−1, approximately doubling the offset with respect to the velocity of A267's brightest cluster galaxy. Embedding the spherical Jeans equation within this framework, we infer for A267 a halo mass M 200 = (7.0 ± 1.3) × 1014 M ⊙ h −1 and concentration , consistent with the mass–concentration relation found in cosmological simulations.

Stellar 3D kinematics in the Draco dwarf spheroidal galaxy

Aims. We present the first three-dimensional internal motions for individual stars in the Draco dwarf spheroidal galaxy. Methods. By combining first-epoch Hubble Space Telescope observations and second-epoch Gaia Data Release 2 positions, we measured the proper motions of 149 sources in the direction of Draco. We determined the line-of-sight velocities for a sub-sample of 81 red giant branch stars using medium resolution spectra acquired with the DEIMOS spectrograph at the Keck II telescope. Altogether, this resulted in a final sample of 45 Draco members with high-precision and accurate 3D motions, which we present as a table in this paper. Results. Based on this high-quality dataset, we determined the velocity dispersions at a projected distance of ∼120 pc from the centre of Draco to be σR = 11.0−1.5+2.1 km s−1, σT = 9.9−3.1+2.3 km s−1 and σLOS = 9.0−1.1+1.1 km s−1 in the projected radial, tangential, and line-of-sight directions. This results in a velocity anisotropy β = 0.25−1.38+0.47 at r ≳ 120 pc. Tighter constraints may be obtained using the spherical Jeans equations and assuming constant anisotropy and Navarro-Frenk-White (NFW) mass profiles, also based on the assumption that the 3D velocity dispersion should be lower than ≈1/3 of the escape velocity of the system. In this case, we constrain the maximum circular velocity Vmax of Draco to be in the range of 10.2−17.0 km s−1. The corresponding mass range is in good agreement with previous estimates based on line-of-sight velocities only. Conclusions. Our Jeans modelling supports the case for a cuspy dark matter profile in this galaxy. Firmer conclusions may be drawn by applying more sophisticated models to this dataset and with new datasets from upcoming Gaia releases.

Basilisk: Bayesian hierarchical inference of the galaxy–halo connection using satellite kinematics – I. Method and validation

ABSTRACT We present a Bayesian hierarchical inference formalism (Basilisk) to constrain the galaxy–halo connection using satellite kinematics. Unlike traditional methods, Basilisk does not resort to stacking the kinematics of satellite galaxies in bins of central luminosity, and does not make use of summary statistics, such as satellite velocity dispersion. Rather, Basilisk leaves the data in its raw form and computes the corresponding likelihood. In addition, Basilisk can be applied to flux-limited, rather than volume-limited samples, greatly enhancing the quantity and dynamic range of the data. And finally, Basilisk is the only available method that simultaneously solves for halo mass and orbital anisotropy of the satellite galaxies, while properly accounting for scatter in the galaxy–halo connection. Basilisk uses the conditional luminosity function to model halo occupation statistics, and assumes that satellite galaxies are a relaxed tracer population of the host halo’s potential with kinematics that obey the spherical Jeans equation. We test and validate Basilisk using mocks of varying complexity, and demonstrate that it yields unbiased constraints on the galaxy–halo connection and at a precision that rivals galaxy–galaxy lensing. In particular, Basilisk accurately recovers the full PDF of the relation between halo mass and central galaxy luminosity, and simultaneously constrains the orbital anisotropy of the satellite galaxies. Basilisk ’s inference is not affected by potential velocity bias of the central galaxies, or by slight errors in the inferred, radial profile of satellite galaxies that arise as a consequence of interlopers and sample impurity.

Deriving Galaxy Cluster Velocity Anisotropy Profiles from a Joint Analysis of Dynamical and Weak Lensing Data

Abstract We present an analytic approach to lift the mass-anisotropy degeneracy in clusters of galaxies by utilizing the line-of-sight velocity dispersion of clustered galaxies jointly with weak lensing inferred masses. More specifically, we solve the spherical Jeans equation by assuming a simple relation between the line-of-sight velocity dispersion and the radial velocity dispersion and recast the Jeans equation as a Bernoulli differential equation that has a well-known analytic solution. We first test our method in cosmological N-body simulations and then derive the anisotropy profiles for 35 archival data galaxy clusters with an average redshift of . The resulting profiles yield a weighted average global value of (stat) ±0.15 (sys). This indicates that clustered galaxies tend to globally fall on radially anisotropic orbits. We note that this is the first attempt to derive velocity anisotropy profiles for a cluster sample of this size utilizing joint dynamical and weak lensing data.

Effects of mass models on dynamical mass estimate: the case of ultradiffuse galaxy NGC 1052-DF2

ABSTRACT NGC 1052-DF2 was recently discovered as the dark-matter-deficient galaxy claimed by van Dokkum et al. However, large uncertainties on its dynamical mass estimate have been pointed out, concerning the paucity of sample, statistical methods, and distance measurements. In this work, we discuss the effects of the difference in modelling of the tracer profile of this galaxy on the dynamical mass estimate. To do this, we assume that the tracer densities are modelled with power-law and Sérsic profiles, and then we solve the spherical Jeans equation to estimate the dynamical mass. Applying these models to kinematic data of globular clusters in NGC 1052-DF2, we compare 90 per cent upper limits of dynamical mass-to-light ratios estimated from this analysis and from van Dokkum et al. We find that the upper limit obtained by the power-law is virtually the same as the result from van Dokkum et al, whilst this limit estimated by the Sérsic is significantly greater than that from van Dokkum et al, thereby suggesting that NGC 1052-DF2 can still be a dark-matter-dominated system. Consequently, we propose that dynamical mass estimate of a galaxy is largely affected by not only the small kinematic sample but the choice of tracer distributions, and thus the estimated mass still remains quite uncertain.

The mass of the Galactic dark matter halo from ∼9000 LAMOST DR5 K giants

We constrain the mass of the Milky Wayʼs dark matter halo, based on the kinematics of 9627 K giants at Galactocentric distances ranging over 5 kpc drawn from LAMOST DR5. The substructure in this sample has been identified and removed carefully to enable construction of the underlying line-of-sight velocity dispersion at different radii from the Galactic center. We interpret the radial profile of the line-of-sight velocity dispersion using a spherical Jeans equation under the assumptions of anisotropy/isotropy and that radial velocity dispersion is approximately equal to line-of-sight velocity dispersion . If we assume that the dark matter halo follows an NFW profile and the stellar halo is isotropic (β = 0), then can be directly used to estimate the virial mass of the Galactic dark matter halo, , and concentration parameter . In case that the stellar halo is anisotropic, we cannot avoid differentiation of sparse velocity dispersions according to the Jeans equation, which may cause overestimation of the mass. We use an isotropic case to test and find that overestimates the virial mass by 15% but within 1-σ error. We use to fit the NFW profile and get and in case of β = 0.3.

What to expect from dynamical modelling of galactic haloes – II. The spherical Jeans equation

The spherical Jeans equation (SJE) is widely used in dynamical modelling of the Milky Way (MW) halo potential. We use haloes and galaxies from the cosmological Millennium-II simulation and hydrodynamical APOSTLE simulations to investigate the performance of the SJE in recovering the underlying mass profiles of MW mass haloes. The best-fitting halo mass and concentration parameters scatter by 25% and 40% around their input values, respectively, when dark matter particles are used as tracers. This scatter becomes as large as a factor of 3 when using star particles instead. This is significantly larger than the estimated statistical uncertainty associated with the use of the SJE. The existence of correlated phase-space structures that violate the steady state assumption of the SJE as well as non-spherical geometries are the principal sources of the scatter. Binary haloes show larger scatter because they are more aspherical in shape and have a more perturbed dynamical state. Our results confirm the previous study of Wang et al. (2017) that the number of independent phase-space structures sets an intrinsic limiting precision on dynamical inferences based on the steady state assumption. Modelling with a radius-independent velocity anisotropy, or using tracers within a limited outer radius, result in significantly larger scatter, but the ensemble-averaged measurement over the whole halo sample is approximately unbiased.