Dark matter subhaloes in numerical simulations

We use cosmological Λ cold dark matter (CDM) numerical simulations to model the evolution of the substructure population in 16 dark matter haloes with resolutions of up to seven million particles within the virial radius. The combined substructure circular velocity distribution function (VDF) for hosts of 10¹¹ to  10¹⁴ M_⊙  at redshifts from zero to two or higher has a self-similar shape, is independent of host halo mass and redshift, and follows the relation  d n /d v = (1/8)( v _cmax/ v _cmax,host)⁻⁴  . Halo to halo variance in the VDF is a factor of roughly 2 to 4. At high redshifts, we find preliminary evidence for fewer large substructure haloes (subhaloes). Specific angular momenta are significantly lower for subhaloes nearer the host halo centre where tidal stripping is more effective. The radial distribution of subhaloes is marginally consistent with the mass profile for   r ≳ 0.3 r _vir  , where the possibility of artificial numerical disruption of subhaloes can be most reliably excluded by our convergence study, although a subhalo distribution that is shallower than the mass profile is favoured. Subhalo masses but not circular velocities decrease towards the host centre. Subhalo velocity dispersions hint at a positive velocity bias at small radii. There is a weak bias towards more circular orbits at lower redshift, especially at small radii. We additionally model a cluster in several power-law cosmologies of   P ∝ kⁿ   , and demonstrate that a steeper spectral index, n , results in significantly less substructure.     

Constraints on dark matter physics from dwarf galaxies through galaxy cluster haloes

One of the predictions of the standard cold dark matter model is that dark haloes have centrally divergent density profiles. An extensive body of rotation curve observations of dwarf and low surface brightness galaxies shows the dark haloes of those systems to be characterized by soft constant-density central cores. Several physical processes have been proposed to produce soft cores in dark haloes, each one with different scaling properties. With the aim of discriminating among them we have examined the rotation curves of dark-matter-dominated dwarf and low surface brightness galaxies and the inner mass profiles of two clusters of galaxies lacking a central cD galaxy and with evidence of soft cores in the centre. The core radii and central densities of these haloes scale in a well-defined manner with the depth of their potential wells, as measured through the maximum circular velocity. As a result of our analysis we identify self-interacting cold dark matter as a viable solution to the core problem, where a non-singular isothermal core is formed in the halo centre surrounded by a Navarro, Frenk & White profile in the outer parts. We show that this particular physical situation predicts core radii in agreement with observations. Furthermore, using the observed scalings, we derive an expression for the minimum cross-section ( σ ) which has an explicit dependence with the halo dispersion velocity ( v ). If m _x is the mass of the dark matter particle: σ m _x ≈4×10⁻²⁵ (100 km s⁻¹  v ⁻¹) cm² GeV⁻¹.     

The Norma cluster (ACO 3627) – I. A dynamical analysis of the most massive cluster in the Great Attractor

A detailed dynamical analysis of the nearby rich Norma cluster (ACO 3627) is presented. From radial velocities of 296 cluster members, we find a mean velocity of 4871 ± 54 km s⁻¹ and a velocity dispersion of 925 km s⁻¹. The mean velocity of the E/S0 population (4979 ± 85 km s⁻¹) is offset with respect to that of the S/Irr population (4812 ± 70 km s⁻¹) by  Δ v = 164 km s⁻¹  in the cluster rest frame. This offset increases towards the core of the cluster. The E/S0 population is free of any detectable substructure and appears relaxed. Its shape is clearly elongated with a position angle that is aligned along the dominant large-scale structures in this region, the so-called Norma wall. The central cD galaxy has a very large peculiar velocity of 561 km s⁻¹ which is most probably related to an ongoing merger at the core of the cluster. The spiral/irregular galaxies reveal a large amount of substructure; two dynamically distinct subgroups within the overall spiral population have been identified, located along the Norma wall elongation. The dynamical mass of the Norma cluster within its Abell radius is  1–1.1 × 10¹⁵  h ⁻¹₇₃ M_⊙  . One of the cluster members, the spiral galaxy WKK 6176 which recently was observed to have a 70 kpc X-ray tail, reveals numerous striking low-brightness filaments pointing away from the cluster centre suggesting strong interaction with the intracluster medium.     

Dark matter halo structure in CDM hydrodynamical simulations

We have carried out a comparative analysis of the properties of dark matter haloes in N -body and hydrodynamical simulations. We analyse their density profiles, shapes and kinematical properties with the aim of assessing the effects that hydrodynamical processes might produce on the evolution of the dark matter component. The simulations performed allow us to reproduce dark matter haloes with high resolution, although the range of circular velocities is limited. We find that for haloes with circular velocities of [150–200] km s⁻¹ at the virial radius, the presence of baryons affects the evolution of the dark matter component in the central region, modifying the density profiles, shapes and velocity dispersions. We also analyse the rotation velocity curves of disc-like structures and compare them with observational results.     

Dynamical friction and galaxy merging time-scales

The time-scale for galaxies within merging dark matter haloes to merge with each other is an important ingredient in galaxy formation models. Accurate estimates of merging time-scales are required for predictions of astrophysical quantities such as black hole binary merger rates, the build-up of stellar mass in central galaxies and the statistical properties of satellite galaxies within dark matter haloes. In this paper, we study the merging time-scales of extended dark matter haloes using N -body simulations. We compare these results to standard estimates based on the Chandrasekhar theory of dynamical friction. We find that these standard predictions for merging time-scales, which are often used in semi-analytic galaxy formation models, are systematically shorter than those found in simulations. The discrepancy is approximately a factor of 1.7 for M _sat/ M _host≈ 0.1 and becomes larger for more disparate satellite-to-host mass ratios, reaching a factor of ∼3.3 for M _sat/ M _host≈ 0.01. Based on our simulations, we propose a new, easily implementable fitting formula that accurately predicts the time-scale for an extended satellite to sink from the virial radius of a host halo down to the halo's centre for a wide range of M _sat/ M _host and orbits. Including a central bulge in each galaxy changes the merging time-scale by ≲10 per cent. To highlight one concrete application of our results, we show that merging time-scales often used in the literature overestimate the growth of stellar mass by satellite accretion by ≈40 per cent, with the extra mass gained in low mass ratio mergers.     

The Planetary Nebula Spectrograph elliptical galaxy survey: the dark matter in NGC 4494

We present new Planetary Nebula Spectrograph observations of the ordinary elliptical galaxy NGC 4494, resulting in positions and velocities of 255 planetary nebulae out to seven effective radii (25 kpc). We also present new wide-field surface photometry from MMT/Megacam, and long-slit stellar kinematics from VLT/FORS2. The spatial and kinematical distributions of the planetary nebulae agree with the field stars in the region of overlap. The mean rotation is relatively low, with a possible kinematic axis twist outside  1 R _e  . The velocity dispersion profile declines with radius, though not very steeply, down to  ∼70 km s⁻¹  at the last data point.
We have constructed spherical dynamical models of the system, including Jeans analyses with multi-component Λ cold dark matter (CDM) motivated galaxies as well as logarithmic potentials. These models include special attention to orbital anisotropy, which we constrain using fourth-order velocity moments. Given several different sets of modelling methods and assumptions, we find consistent results for the mass profile within the radial range constrained by the data. Some dark matter (DM) is required by the data; our best-fitting solution has a radially anisotropic stellar halo, a plausible stellar mass-to-light ratio and a DM halo with an unexpectedly low central density. We find that this result does not substantially change with a flattened axisymmetric model.
Taken together with other results for galaxy halo masses, we find suggestions for a puzzling pattern wherein most intermediate-luminosity galaxies have very low concentration haloes, while some high-mass ellipticals have very high concentrations. We discuss some possible implications of these results for DM and galaxy formation.     

The concentration–velocity dispersion relation in galaxy groups

Based on results from cold dark matter N -body simulations, we develop a dynamical model for the evolution of subhaloes within group-sized host haloes. Only subhaloes more massive than 5 × 10⁸ M_⊙ are considered, because they are massive enough to possibly host luminous galaxies. On their orbits within a growing host potential the subhaloes are subject to tidal stripping and dynamical friction. At the present time  ( z = 0)  , all model hosts have equal mass  ( M _vir= 3.9 × 10¹³ M_⊙)  but different concentrations associated with different formation times. We investigate the variation of subhalo (or satellite galaxy) velocity dispersion with host concentration and/or formation time. In agreement with the Jeans equation, the velocity dispersion of subhaloes increases with the host concentration. Between concentrations of ∼5 and ∼20, the subhalo velocity dispersions increase by a factor of ∼1.25. By applying a simplified tidal disruption criterion, that is, rejection of all subhaloes with a tidal truncation radius below 3  kpc at   z = 0  , the central velocity dispersion of the 'surviving' subhalo sample increases substantially for all concentrations. The enhanced central velocity dispersions in the surviving subhalo samples are caused by a lack of slow tangential motions. Additionally, we present a fitting formula for the anisotropy parameter which does not depend on concentration if the group-centric distances are scaled by r _s, the characteristic radius of the Navarro, Frenk & White profile. Since the expected loss of subhaloes and galaxies due to tidal disruption increases the velocity dispersion of surviving galaxies, the observed galaxy velocity dispersion can substantially overestimate the virial mass.     

The sizes of minivoids in the local Universe: an argument in favour of a warm dark matter model?

Using high-resolution simulations within the cold dark matter (CDM) and warm dark matter (WDM) models, we study the evolution of small-scale structure in the local volume, a sphere of 8-Mpc radius around the Local Group. We compare the observed spectrum of minivoids in the local volume with the spectrum of minivoids determined from the simulations. We show that the ΛWDM model can easily explain both the observed spectrum of minivoids and the presence of low-mass galaxies observed in the local volume, provided that all haloes with circular velocities greater than 20 km s⁻¹ host galaxies. On the contrary, within the ΛCDM model the distribution of the simulated minivoids reflects the observed one if haloes with maximal circular velocities larger than  35 km s⁻¹  host galaxies. This assumption is in contradiction with observations of galaxies with circular velocities as low as 20 km s⁻¹ in our local Universe. A potential problem of the ΛWDM model could be the late formation of the haloes in which the gas can be efficiently photoevaporated. Thus, star formation is suppressed and low-mass haloes might not host any galaxy at all.     

The dark matter distribution in disc galaxies

We use high-quality optical rotation curves of nine low-luminosity disc galaxies to obtain the velocity profiles of the surrounding dark matter haloes. We find that they increase linearly with radius at least out to the edge of the stellar disc, implying that, over the entire stellar region, the density of the dark halo is about constant.
The properties of the mass structure of these haloes are similar to those found for a number of dwarf and low surface brightness galaxies, but provide a more substantial evidence of the discrepancy between the halo mass distribution predicted in the cold dark matter scenario and those actually detected around galaxies. We find that the density law proposed by Burkert reproduces the halo rotation curves, with halo central densities ( ρ ₀∼1–4×10⁻²⁴ g cm⁻³) and core radii ( r ₀∼5–15 kpc) scaling as ρ ₀∝ r ₀^−2/3.     

Numerical action reconstruction of the dynamical history of dark matter haloes in N-body simulations

We test the ability of the numerical action method (NAM) to recover the individual orbit histories of mass tracers in an expanding universe, given the masses and redshift-space coordinates at the present epoch. The mass tracers are represented by dark matter (DM) haloes identified in a region of radius  26  h ⁻¹ Mpc  of a high-resolution N -body simulation of the standard Λ cold dark matter (CDM) cosmology. Since previous tests of NAM at this scale have traced the underlying distribution of DM particles rather than extended haloes, our study offers an assessment of the accuracy of NAM in a scenario which more closely approximates the complex dynamics of actual galaxy haloes. We show that NAM can recover the present-day halo distances with typical distance errors of less than 3 per cent and radial peculiar velocities with a dispersion of  ∼130 km s⁻¹  . The accuracy of individual orbit reconstructions was limited by the inability of NAM, in some instances, to correctly model the positions of haloes at early times solely on the basis of the redshifts, angular positions and masses of the haloes at the present epoch. Improvements in the quality of NAM reconstructions may be possible using the present-day three-dimensional halo velocities and distances to further constrain the dynamics. This velocity data is expected to become available for nearby galaxies in the coming generations of observations by Space Interferometry Mission ( SIM ) and Global Astrometric Interferometer for Astrophysics ( GAIA ).     

Large-scale galactic turbulence: can self-gravity drive the observed H i velocity dispersions?

Observations of turbulent velocity dispersions in the H  i component of galactic discs show a characteristic floor in galaxies with low star formation rates and within individual galaxies the dispersion profiles decline with radius. We carry out several high-resolution adaptive mesh simulations of gaseous discs embedded within dark matter haloes to explore the roles of cooling, star formation, feedback, shearing motions and baryon fraction in driving turbulent motions. In all simulations the disc slowly cools until gravitational and thermal instabilities give rise to a multiphase medium in which a large population of dense self-gravitating cold clouds are embedded within a warm gaseous phase that forms through shock heating. The diffuse gas is highly turbulent and is an outcome of large-scale driving of global non-axisymmetric modes as well as cloud–cloud tidal interactions and merging. At low star formation rates these processes alone can explain the observed H  i velocity dispersion profiles and the characteristic value of  ∼10 km s⁻¹  observed within a wide range of disc galaxies. Supernovae feedback creates a significant hot gaseous phase and is an important driver of turbulence in galaxies with a star formation rate per unit area  ≳10⁻³ M_⊙ yr⁻¹ kpc⁻²  .     

The orbit and mass of the Sagittarius dwarf galaxy

Possible orbital histories of the Sgr dwarf galaxy are explored. A special-purpose N -body code is used to construct the first models of the Milky Way–Sgr dwarf system in which both the Milky Way and the Sgr dwarf are represented by full N -body systems and followed for a Hubble time. These models are used to calibrate a semi-analytic model of the Sgr dwarf's orbit that enables us to explore a wider parameter space than is accessible to the N -body models. We conclude that the extant data on the Sgr dwarf are compatible with a wide range of orbital histories. At one extreme the Sgr dwarf initially possesses ∼10¹¹ M_⊙ and starts from a Galactocentric distance R _D(0)≳200 kpc. At the other extreme the Sgr dwarf starts with ∼10⁹ M_⊙ and R _D(0)∼60 kpc, similar to its present apocentric distance. In all cases the Sgr dwarf is initially dark matter dominated and the current velocity dispersion of the Sgr dwarf's dark matter is tightly constrained to be 21±2 km s⁻¹. This number is probably compatible with the smaller measured dispersion of the Sgr dwarf's stars because of (i) the dynamical difference between dark and luminous matter, and (ii) velocity anisotropy.     

NGC 3741: the dark halo profile from the most extended rotation curve

We present new H  i observations of the nearby dwarf galaxy NGC 3741. This galaxy has an extremely extended H  i disc, which allows us to trace the rotation curve out to unprecedented distances in terms of the optical disc: we reach 42 B -band exponential scalelengths or about 7 kpc. The H  i disc is strongly warped, but the warp is very symmetric. The distribution and kinematics are accurately derived by building model data cubes, which closely reproduce the observations. In order to account for the observed features in the data cube, radial motions of the order of 5–13 km s⁻¹ are needed. They are consistent with an inner bar of several hundreds of pc and accretion of material in the outer regions.
The observed rotation curve was decomposed into its stellar, gaseous and dark components. The Burkert dark halo (with a central constant density core) provides very good fits. The dark halo density distribution predicted by the Λ cold dark matter (CDM) theory fails to fit the data, unless NGC 3741 is a 2.5σ exception to the predicted relation between concentration parameter and virial mass and at the same time a high value of the virial mass (though poorly constrained) of  10¹¹ M_⊙  . Noticeably, modified Newtonian dynamics (MOND) seems to be consistent with the observed rotation curve. Scaling up the contribution of the gaseous disc also gives a good fit.     

Global velocity field and bubbles in the blue compact dwarf galaxy Mrk 86

We have studied the velocity field of the blue compact dwarf galaxy Mrk 86 (NGC 2537) using data provided by 14 long-slit optical spectra obtained in 10 different orientations and positions. This kinematical information is complemented with narrow-band ([O  iii ]5007 Å and H α ) and broad-band ( B , V , Gunn r and K ) imaging. The analysis of the galaxy global velocity field suggests that the ionized gas could be distributed in a rotating inclined disc, with projected central angular velocity of Ω=34 km s⁻¹ kpc⁻¹. The comparison between the stellar, H  i and modelled dark matter density profile indicates that the total mass within its optical radius is dominated by the stellar component. Peculiarities observed in its velocity field can be explained by irregularities in the ionized gas distribution or local motions induced by star formation.
Kinematical evidences for two expanding bubbles, Mrk 86–B and Mrk 86–C, are given. They show expanding velocities of 34 and 17 km s⁻¹, H α luminosities of 3×10³⁸ and 1.7×10³⁹ erg s⁻¹, and physical radii of 374 and 120 pc, respectively. The change in the [S  ii ]/H α , [N  ii ]/H α , [O  ii ]/[O  iii ] and [O  iii ]/H β line ratios with the distance to the bubble precursor suggests a diminution in the ionization parameter and, in the case of Mrk 86–B, an enhancement of the shock-excited gas emission. The optical–near-infrared colours of the bubble precursors are characteristic of low‐metallicity star‐forming regions (∼0.2 Z_⊙) with burst strengths of about 1 per cent in mass.     

Galactic halo cusp–core: tidal compression in mergers

We explain in simple terms how the build-up of dark haloes by merging compact satellites, as in the cold dark matter (CDM) cosmology, inevitably leads to an inner cusp of density profile  ρ∝ r ^−α  with  α≳ 1  , as seen in cosmological N -body simulations. A flatter halo core with  α < 1  exerts on the satellites tidal compression in all directions, which prevents the deposit of stripped satellite material in the core region. This makes the satellite orbits decay from the radius where  α∼ 1  to the halo centre with no local tidal mass transfer, and thus causes a rapid steepening of the inner profile to  α > 1  . These tidal effects, the resultant steepening of the profile to a cusp, and the stability of this cusp to tandem mergers with compact satellites are demonstrated using N -body simulations. The transition at  α∼ 1  is then addressed using toy models in the limiting cases of impulse and adiabatic approximations and using tidal radii for satellites on radial and circular orbits. In an associated paper, we address the subsequent slow convergence from either side to an asymptotic stable cusp with  α≳ 1  . Our analysis thus implies that an inner cusp is enforced when small haloes are typically more compact than larger haloes, as in the CDM scenario, such that enough satellite material makes it intact into the inner halo and is deposited there. We conclude that a necessary condition for maintaining a flat core, as indicated by observations, is that the inner regions of the CDM satellite haloes be puffed up by about 50 per cent such that when they merge into a larger halo they would be disrupted outside the halo core. This puffing up could be due to baryonic feedback processes in small haloes, which may be stimulated by the tidal compression in the halo cores.     

The distribution of ejected subhaloes and its implication for halo assembly bias

Using a high-resolution cosmological N -body simulation, we identify the ejected population of subhaloes, which are haloes at redshift   z = 0  but were once contained in more massive 'host' haloes at high redshifts. The fraction of the ejected subhaloes in the total halo population of the same mass ranges from 9 to 4 per cent for halo masses from  ∼10¹¹  to  ∼10¹²  h ⁻¹ M_⊙  . Most of the ejected subhaloes are distributed within four times the virial radius of their hosts. These ejected subhaloes have distinct velocity distribution around their hosts in comparison to normal haloes. The number of subhaloes ejected from a host of given mass increases with the assembly redshift of the host. Ejected subhaloes in general reside in high-density regions, and have a much higher bias parameter than normal haloes of the same mass. They also have earlier assembly times, so that they contribute to the assembly bias of dark matter haloes seen in cosmological simulations. However, the assembly bias is not dominated by the ejected population, indicating that large-scale environmental effects on normal haloes are the main source for the assembly bias.     

Dark matter in dwarf spheroidals – II. Observations and modelling of Draco

We present stellar radial velocity data for the Draco dwarf spheroidal (dSph) galaxy obtained using the AF2/WYFFOS instrument combination on the William Herschel Telescope. Our data set consists of 186 member stars, 159 of which have good quality velocities, extending to a magnitude   V ≈19.5  with a mean velocity precision of ≈2 km s⁻¹. As this survey is based on a high-precision photometric target list, it contains many more Draco members at large radii. For the first time, this allows a robust determination of the radial behaviour of the velocity dispersion in a dSph.
We find statistically strong evidence of a rising velocity dispersion consistent with a dark matter halo that has a gently rising rotation curve. There is a <2 σ signature of rotation about the long axis, inconsistent with tidal disruption as the source of the rising dispersion. By comparing our data set with earlier velocities, we find that Draco probably has a binary distribution and fraction comparable to those in the solar neighbourhood.
We apply a novel maximum likelihood algorithm and fit the velocity data to a two parameter spherical model with an adjustable dark matter content and velocity anisotropy. Draco is best fit by a weakly tangentially anisotropic distribution of stellar orbits in a dark matter halo with a very slowly rising rotation law  ( v _circ∝ r ^0.17)  . We are able to rule out both a mass-follows-light distribution and an extended halo with a harmonic core at the 2.5 to 3 σ significance level, depending on the details of our assumptions about Draco's stellar binary population. Our modelling lends support to the idea that the dark matter in dwarf spheroidals is distributed in the form of massive, nearly isothermal haloes.     

Numerical simulations of hot halo gas in galaxy mergers

Galaxy merger simulations have explored the behaviour of gas within the galactic disc, yet the dynamics of hot gas within the galaxy halo have been neglected. We report on the results of high-resolution hydrodynamic simulations of colliding galaxies with metal-free hot halo gas. To isolate the effect of the halo gas, we simulate only the dark matter halo and the hot halo gas over a range of mass ratios, gas fractions and orbital configurations to constrain the shocks and gas dynamics within the progenitor haloes. We find that (i) a strong shock is produced in the galaxy haloes before the first passage, increasing the temperature of the gas by almost an order of magnitude to   T ∼ 10^6.3 K  . (ii) The X-ray luminosity of the shock is strongly dependent on the gas fraction; it is  ≳10³⁹ erg s⁻¹  for halo gas fractions larger than 10 per cent. (iii) The hot diffuse gas in the simulation produces X-ray luminosities as large as  10⁴² erg s⁻¹  . This contributes to the total X-ray background in the Universe. (iv) We find an analytic fit to the maximum X-ray luminosity of the shock as a function of merger parameters. This fit can be used in semi-analytic recipes of galaxy formation to estimate the total X-ray emission from shocks in merging galaxies. (v) ∼10–20 per cent of the initial gas mass is unbound from the galaxies for equal-mass mergers, while 3–5 per cent of the gas mass is released for the 3:1 and 10:1 mergers. This unbound gas ends up far from the galaxy and can be a feasible mechanism to enrich the intergalactic medium with metals.     

Star formation and figure rotation in the early-type galaxy NGC 2974

We present Galaxy Evolution Explorer ( GALEX ) far-ultraviolet (FUV) and near-ultraviolet (NUV) imaging of the nearby early-type galaxy NGC 2974, along with complementary ground-based optical imaging. In the ultraviolet, the galaxy reveals a central spheroid-like component and a newly discovered complete outer ring of radius 6.2 kpc, with suggestions of another partial ring at an even larger radius. Blue FUV–NUV and UV-optical colours are observed in the centre of the galaxy and from the outer ring outwards, suggesting young stellar populations (≲1 Gyr) and recent star formation in both locations. This is supported by a simple stellar population model which assumes two bursts of star formation, allowing us to constrain the age, mass fraction and surface mass density of the young component pixel by pixel. Overall, the mass fraction of the young component appears to be just under 1 per cent (lower limit, uncorrected for dust extinction). The additional presence of a nuclear and an inner ring (radii 1.4 and 2.9 kpc, respectively), as traced by [O  iii ] emission, suggests ring formation through resonances. All three rings are consistent with a single pattern speed of  78 ± 6  km s⁻¹ kpc⁻¹, typical of S0 galaxies and only marginally slower than expected for a fast bar if traced by a small observed surface brightness plateau. This thus suggests that star formation and morphological evolution in NGC 2974 at the present epoch are primarily driven by a rotating asymmetry (probably a large-scale bar), despite the standard classification of NGC 2974 as an E4 elliptical.     

Dark matter in elliptical galaxies – I. Is the total mass density profile of the NFW form or even steeper?

Elliptical galaxies are modelled as Sérsic luminosity distributions with density profiles (DPs) for the total mass adopted from the DPs of haloes within dissipationless ΛCDM (cold dark matter) N -body simulations. Ellipticals turn out to be inconsistent with cuspy low-concentration NFW models representing the total mass distribution, neither are they consistent with a steeper −1.5 inner slope, nor with the shallower models proposed by Navarro et al., nor with NFW models 10 times more concentrated than predicted, as deduced from several X-ray observations – the mass models, extrapolated inwards, lead to local mass-to-light ratios that are smaller than the stellar value inside an effective radius ( R _e), and to central aperture velocity dispersions that are much smaller than observed. This conclusion remains true as long as there is no sharp steepening (slope < −2) of the dark matter DPs just inside 0.01 virial radii.
The very low total mass and velocity dispersion produced within R _e by an NFW-like total mass profile suggests that the stellar component should dominate the dark matter component out to at least R _e. It should then be difficult to kinematically constrain the inner slope of the DP of ellipticals. The high-concentration parameters deduced from X-ray observations appear to be a consequence of fitting an NFW model to the total mass DP made up of a stellar component that dominates inside and a dark matter component that dominates outwards.
An appendix gives the virial mass dependence of the concentration parameter, central density and total mass of the Navarro et al. model. In a second appendix are given single integral expressions for the velocity dispersions averaged along the line of sight, in circular apertures and in thin slits, for general luminosity density and mass distributions, with isotropic orbits.