Realistic triaxial density–potential–force profiles for stellar systems and dark matter haloes

Popular models for describing the luminosity–density profiles of dynamically hot stellar systems (e.g. Jaffe, Hernquist, Dehnen) were constructed to match the deprojected form of de Vaucouleurs' R ^1/4 light-profile. However, we now know that elliptical galaxies and bulges display a mass-dependent range of structural profiles. To compensate this, the model in Terzić & Graham was designed to closely match the deprojected form of Sérsic R ^{1/ n} light-profiles, including deprojected exponential light-profiles and galaxies with partially depleted cores. It is thus applicable for describing bulges in spiral galaxies, dwarf elliptical galaxies, both 'power-law' and 'core' elliptical galaxies, also dark matter haloes formed from Λ cold dark matter cosmological simulations. In this paper, we present a new family of triaxial density–potential–force triplets, which generalizes the spherical model reported in Terzić & Graham to three dimensions. If the (optional) power-law core is present, it is a five-parameter family, while in the absence of the core it reduces to three parameters. The isodensity contours in the new family are stratified on confocal ellipsoids and the potential and forces are expressed in terms of integrals which are easy to evaluate numerically. We provide the community with a suite of numerical routines for orbit integration, which feature: optimized computations of potential and forces for this family; the ability to run simulations on parallel platforms; and modular and easily editable design.     

The shape of the dark matter halo in the early-type galaxy NGC 2974

We present H  i observations of the elliptical galaxy NGC 2974, obtained with the Very Large Array. These observations reveal that the previously detected H  i disc in this galaxy is in fact a ring. By studying the harmonic expansion of the velocity field along the ring, we constrain the elongation of the halo and find that the underlying gravitational potential is consistent with an axisymmetric shape.
We construct mass models of NGC 2974 by combining the H  i rotation curve with the central kinematics of the ionized gas, obtained with the integral-field spectrograph SAURON. We introduce a new way of correcting the observed velocities of the ionized gas for asymmetric drift, and hereby disentangle the random motions of the gas caused by gravitational interaction from those caused by turbulence. To reproduce the observed flat rotation curve of the H  i gas, we need to include a dark halo in our mass models. A pseudo-isothermal sphere provides the best model to fit our data, but we also tested an NFW halo and modified Newtonian dynamics, which fit the data marginally worse.
The mass-to-light ratio M / L _I increases in NGC 2974 from 4.3 M_⊙/L_⊙, _I at one effective radius to 8.5 M_⊙/L_{⊙, I} at 5  R _e. This increase of M / L already suggests the presence of dark matter: we find that within 5  R _e at least 55 per cent of the total mass is dark.     

Central mass-to-light ratios and dark matter fractions in early-type galaxies

Dynamical studies of local elliptical galaxies and the Fundamental Plane point to a strong dependence of the total mass-to-light ratio ( M / L ) on luminosity with a relation of the form   M / L ∝ L ^γ  . The 'tilt'γ may be caused by various factors, including stellar population properties (metallicity, age and star formation history), initial mass function, rotational support, luminosity profile non-homology and dark matter (DM) fraction. We evaluate the impact of all these factors using a large uniform data set of local early-type galaxies from Prugniel & Simien. We take particular care in estimating the stellar masses, using a general star formation history, and comparing different population synthesis models. We find that the stellar M / L contributes little to the tilt. We estimate the total M / L using simple Jeans dynamical models, and find that adopting accurate luminosity profiles is important but does not remove the need for an additional tilt component, which we ascribe to DM. We survey trends of the DM fraction within one effective radius, finding it to be roughly constant for galaxies fainter than   M _B∼−20.5  , and increasing with luminosity for the brighter galaxies; we detect no significant differences between S0s and fast- and slow-rotating ellipticals. We construct simplified cosmological mass models and find general consistency, where the DM transition point is caused by a change in the relation between luminosity and effective radius. A more refined model with varying galaxy star formation efficiency suggests a transition from total mass profiles (including DM) of faint galaxies distributed similarly to the light to near-isothermal profiles for the bright galaxies. These conclusions are sensitive to various systematic uncertainties which we investigate in detail, but are consistent with the results of dynamical studies at larger radii.     

The fundamental plane of elliptical galaxies with modified Newtonian dynamics

The modified Newtonian dynamics (MOND), suggested by Milgrom as an alternative to dark matter, implies that isothermal spheres with a fixed anisotropy parameter should exhibit a near-perfect relation between the mass and velocity dispersion of the form M ∝ σ  ⁴. This is consistent with the observed Faber–Jackson relation for elliptical galaxies: a luminosity–velocity dispersion relation with large scatter. However, the observable global properties of elliptical galaxies comprise a three-parameter family; they lie on a 'fundamental plane' in a logarithmic space consisting of central velocity dispersion, effective radius ( r _e) and luminosity. The scatter perpendicular to this plane is significantly less than that about the Faber–Jackson relation. I show here that, in order to match the observed properties of elliptical galaxies with MOND, models must deviate from being strictly isothermal and isotropic; such objects can be approximated by high-order polytropic spheres with a radial orbit anisotropy in the outer regions. MOND imposes boundary conditions on the inner Newtonian regions which restrict these models to a dynamical fundamental plane of the form where the exponents may differ from the Newtonian expectations ( α =2, γ =1). Scatter about this plane is relatively insensitive to the necessary deviations from homology.     

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.     

The X-ray emission in post-merger ellipticals

The evolution in X-ray properties of early-type galaxies is largely unconstrained. In particular, little is known about how, and if, remnants of mergers generate hot gas haloes. Here we examine the relationship between X-ray luminosity and galaxy age for a sample of early-type galaxies. Comparing normalized X-ray luminosity to three different age indicators, we find that L _X L _B increases with age, suggesting an increase in X-ray halo mass with time after the last major star formation episode of a galaxy. The long-term nature of this trend, which appears to continue across the full age range of our sample, poses a challenge for many models of hot halo formation. We conclude that models involving a declining rate of type Ia supernovae, and a transition from outflow to inflow of the gas originally lost by galactic stars, offer the most promising explanation for the observed evolution in X-ray luminosity.     

Using bright ellipticals as dark energy cosmic clocks

The age of the Universe has been increasingly constrained by different techniques, such as the observations of type Ia supernovae (SNIa) at high redshift or dating the stellar populations of globular clusters. In this paper, we present a complementary approach using the colours of the brightest elliptical galaxies in clusters over a wide redshift range  ( z ≲ 1)  . We put new and independent bounds on the dark energy equation of state parametrized by a constant pressure-to-density ratio   w _Q  and by a parameter (ξ) which determines the scaling between the matter and dark energy densities. We find that accurate estimates of the metallicities of the stellar populations in moderate and high-redshift cluster galaxies can pose stringent constraints on the parameters that describe dark energy. Our results are in good agreement with the analysis of dark energy models using SNIa data as a constraint. Accurate estimates of the metallicities of stellar populations in cluster galaxies at   z ≲ 2  will make this approach a powerful complement to studies of cosmological parameters using high-redshift SNIa.     

Encounters between spherical galaxies – I. Systems without a dark halo

We report here on a survey of N -body simulations of encounters between spherical galaxies. Initial systems are isotropic Jaffe models. Different sets of mass ratios, impact parameters and orbital energies are studied. Both merger remnants and systems perturbed after a non-merging encounter are analysed and compared to real-life elliptical galaxies. The properties of merger remnants show a large variety. Merger remnants resulting from head-on encounters are mainly non-rotating prolate spheroids. Merger remnants from models with   J _orb≠ 0  are tri-axial or mildly oblate spheroids, supported in part by rotation. The velocity distributions are biased towards the radial direction in the prolate case and the tangential direction in the oblate case. Non-mergers are affected in various ways, depending on the orbital characteristics. We conclude that many of the global properties of real-life ellipticals can, in principle, be attributed to a merger of spherical progenitors.     

The universal rotation curve of spiral galaxies – II. The dark matter distribution out to the virial radius

In the current ΛCDM cosmological scenario, N -body simulations provide us with a universal mass profile, and consequently a universal equilibrium circular velocity of the virialized objects, as galaxies. In this paper we obtain, by combining kinematical data of their inner regions with global observational properties, the universal rotation curve of disc galaxies and the corresponding mass distribution out to their virial radius. This curve extends the results of Paper I, concerning the inner luminous regions of Sb–Im spirals, out to the edge of the galaxy haloes.