Density–potential pairs for spherical stellar systems with Sérsic light profiles and (optional) power-law cores

Popular models for describing the luminosity-density profiles of dynamically hot stellar systems (e.g. Jaffe, Hernquist, Dehnen) were constructed with the desire to match the deprojected form of an   R ^1/4  light profile. Real galaxies, however, are now known to have a range of different light-profile shapes that scale with mass. Consequently, although highly useful, the above models have implicit limitations, and this is illustrated here through their application to a number of real galaxy density profiles. On the other hand, the analytical density profile given by Prugniel & Simien closely matches the deprojected form of Sérsic   R ^{1/ n}   light profiles – including deprojected exponential light profiles. It is thus applicable for describing bulges in spiral galaxies, dwarf elliptical galaxies, and both ordinary and giant elliptical galaxies. Moreover, the observed Sérsic quantities define the parameters of the density model. Here we provide simple equations, in terms of elementary and special functions, for the gravitational potential and force associated with this density profile. Furthermore, to match galaxies with partially depleted cores, and better explore the supermassive black hole/galaxy connection, we have added a power-law core to this density profile and derived similar expressions for the potential and force of this hybrid profile. Expressions for the mass and velocity dispersion, assuming isotropy, are also given. These spherical models may also prove appropriate for describing the dark matter distribution in haloes formed from ΛCDM cosmological simulations.     

Inclination- and dust-corrected galaxy parameters: bulge-to-disc ratios and size–luminosity relations

While galactic bulges may contain no significant dust of their own, the dust within galaxy discs can strongly attenuate the light from their embedded bulges. Furthermore, such dust inhibits the ability of observationally determined inclination corrections to recover intrinsic (i.e. dust-free) galaxy parameters. Using the sophisticated 3D radiative transfer model of Popescu et al. and Tuffs et al., together with the recent determination of the average face-on opacity by Driver et al. in nearby disc galaxies, we provide simple equations to correct (observed) disc central surface brightness and scalelengths for the effects of both inclination and dust in the B , V , I , J and K passbands. We then collate and homogenize various literature data sets and determine the typical intrinsic scalelengths, central surface brightness and magnitudes of galaxy discs as a function of morphological type. All galaxies have been carefully modelled in their respective papers with a Sérsic   R ^{1/ n}   bulge plus an exponential disc. Using the bulge magnitude corrections from Driver et al., we additionally derive the average, dust-corrected, bulge-to-disc flux ratio as a function of galaxy type. With values typically less than 1/3, this places somewhat uncomfortable constraints on some current semi-analytic simulations. Typical bulge sizes, profile shapes, surface brightness and deprojected densities are provided. Finally, given the two-component nature of disc galaxies, we present luminosity–size and (surface brightness)–size diagrams for discs and bulges. We also show that the distribution of elliptical galaxies in the luminosity–size diagram is not linear but strongly curved.     

Elliptical galaxies from mergers of discs

We analyse N -body galaxy merger experiments involving disc galaxies. Mergers of disc–bulge–halo models are compared to those of bulgeless, disc–halo models to quantify the effects of the central bulge on merger dynamics and the structure of the remnant. Our models explore galaxy mass ratios 1:1 through 3:1, and use higher bulge mass fractions than previous studies. A full comparison of the structural and dynamical properties with our observations is carried out. The presence of central bulges results in longer tidal tails, oblate final intrinsic shapes, surface brightness profiles with a higher Sérsic index, steeper rotation curves and oblate-rotator internal dynamics. Mergers of bulgeless galaxies do not generate long-lasting tidal tails, and their strong triaxiality seems inconsistent with observations; these remnants show shells, which we do not find in models including central bulges. Giant ellipticals with boxy isophotes and anisotropic dynamics cannot be produced by the mergers modelled here; they could be the result of mergers between lower luminosity ellipticals, themselves plausibly formed in disc-disc mergers.     

The rotation curves of flattened Sérsic bulges

I present a method to deproject the observed intensity profile of an axisymmetric bulge with arbitrary flattening to derive the three-dimensional luminosity density profile, and to calculate the contribution of the bulge to the rotation curve. I show the rotation curves for a family of fiducial bulges with Sérsic surface brightness profiles and with various concentrations and intrinsic axis ratios. Both parameters have a profound impact on the shape of the rotation curve. In particular, I show how the peak rotation velocity, as well as the radius where it is reached, depends on both parameters.
I also discuss the implications of the flattening of a bulge for the decomposition of a rotation curve and use the case of NGC 5533 to show the errors that result from neglecting it. For NGC 5533, neglecting the flattening of the bulge leads to an overestimate of its mass-to-light ratio by approximately 30 per cent and an underestimate of the contributions from the stellar disc and dark matter halo in the regions outside the bulge-dominated area.     

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.     

Rates of tidal disruption of stars by massive central black holes

There is strong evidence for some kind of massive dark object in the centres of many galaxy bulges. The detection of flares from tidally disrupted stars could confirm that these objects are black holes (BHs). Here we present calculations of the stellar disruption rates in detailed dynamical models of real galaxies, taking into account the refilling of the loss cone of stars on disruptable orbits by two-body relaxation and tidal forces in non-spherical galaxies. The highest disruption rates (one star per 10⁴ yr) occur in faint ( L ≲10¹⁰ L_⊙) galaxies, which have steep central density cusps. More luminous galaxies are less dense and have much longer relaxation times and more massive BHs. Dwarf stars in such galaxies are swallowed whole by the BH and hence do not emit flares; giant stars could produce flares as often as every 10⁵ yr, although the rate depends sensitively on the shape of the stellar distribution function. We discuss the possibility of detecting disruption flares in current supernova searches. The total mass of stars consumed over the lifetime of the galaxy is of the order of 10⁶ M_⊙, independent of galaxy luminosity; thus, disrupted stars may contribute significantly to the present BH mass in galaxies fainter than ∼10⁹ L_⊙.     

The effects of a disc field on bulge surface brightness

Collisionless N -body simulations are used in an effort to reproduce the observed tendency of the surface brightness profile of bulges to change progressively from an R ^1/4 law to an exponential, going from early- to late-type spirals. A possible cause for this is the formation of the disc, later in the history of the galaxy, and this is simulated by applying on the N -body bulge the force field of an exponential disc the surface density of which increases with time. It is shown that n , the index of the Sersic law Σ _n  ( r ) ∝ exp [−( r / r ₀)^{1/ n} ] that best describes the surface brightness profile, does indeed decrease from 4 (de Vaucouleurs law) to smaller values; this decrease is larger for more massive and more compact discs. A large part of the observed trend of n with B/D ratio is explained, and many of the actual profiles can be matched exactly by the simulations. The correlation between the disc scalelength and bulge effective radius, used recently to support the 'secular evolution' origin for bulges, is also shown to arise naturally in a scenario like this. This mechanism, however, saturates at around n  = 2 and exponential bulges cannot be produced; as n gets closer to 1, the profile becomes increasingly robust against a disc field. These results provide strong support to the old-bulge hypothesis for the early-type bulges. The exponential bulges, however, remain essentially unexplained; the results here suggest that they did not begin their lives as R ^1/4 spheroids, and hence were probably formed, at least in part, by different processes from those of early-type spirals.     

Structure and dynamics of galaxies with a low surface-brightness disc – I. The stellar and ionized-gas kinematics

Photometry and long-slit spectroscopy are presented for a sample of six galaxies with a low surface-brightness stellar disc and a bulge. The characterizing parameters of the bulge and disc components were derived by means of a two-dimensional photometric decomposition of the images of the sample galaxies. Their surface-brightness distribution was assumed to be the sum of the contribution of a Sérsic bulge and an exponential disc, with each component being described by elliptical and concentric isophotes of constant ellipticity and position angle. The stellar and ionized-gas kinematics were measured along the major and minor axes in half of the sample galaxies, whereas the other half was observed only along two diagonal axes. Spectra along two diagonal axes were obtained also for one of the objects with major and minor axis spectra. The kinematic measurements extend in the disc region out to a surface-brightness level  μ _R ≈ 24  mag arcsec⁻², reaching in all cases the flat part of the rotation curve. The stellar kinematics turns out to be more regular and symmetric than the ionized-gas kinematics, which often shows the presence of non-circular, off-plane and non-ordered motions. This raises the question about the reliability of the use of the ionized gas as the tracer of the circular velocity in the modelling of the mass distribution, in particular in the central regions of low surface-brightness galaxies.     

The Hubble constant from galaxy lenses: impacts of triaxiality and model degeneracies

The Hubble constant can be constrained using the time delays between multiple images of gravitationally lensed sources. In some notable cases, typical lensing analyses assuming isothermal galaxy density profiles produce low values for the Hubble constant, inconsistent with the result of the HST Key Project  (72 ± 8 km s⁻¹ Mpc⁻¹)  . Possible systematics in the values of the Hubble constant derived from galaxy lensing systems can result from a number of factors, for example, neglect of environmental effects, assumption of isothermality, or contamination by line-of-sight structures. One additional potentially important factor is the triaxial structure of the lensing galaxy halo; most lens models account for halo shape simply by perturbing the projected spherical lensing potential, an approximation that is often necessary but that is inadequate at the levels of triaxiality predicted in the cold dark matter paradigm. To quantify the potential error introduced by this assumption in estimates of the Hubble parameter, we strongly lens a distant galaxy through a sample of triaxial softened isothermal haloes and use an Markov Chain Monte Carlo method to constrain the lensing halo profile and the Hubble parameter from the resulting multiple image systems. We explore the major degeneracies between the Hubble parameter and several parameters of the lensing model, finding that without a way to accurately break these degeneracies accurate estimates of the Hubble parameter are not possible. Crucially, we find that triaxiality does not significantly bias estimates of the Hubble constant, and offer an analytic explanation for this behaviour in the case of isothermal profiles. Neglected triaxial halo shape cannot contribute to the low Hubble constant values derived in a number of galaxy lens systems.     

The dynamical evolution of stellar superclusters

Recent images taken with the Hubble Space Telescope ( HST ) of the interacting disc galaxies NGC 4038/4039 (the Antennae) reveal clusters of many dozens and possibly hundreds of young compact massive star clusters within projected regions spanning about 100 to 500 pc. It is shown here that a large fraction of the individual star clusters merge within a few tens to a hundred Myr. Bound stellar systems with radii of a few hundred parsecs, masses ≲ 10⁹ M⊙ and relaxation times of 10¹¹ − 10¹² yr may form from these. These spheroidal dwarf galaxies contain old stars from the pre-merger galaxy and much younger stars formed in the massive star clusters, and possibly from later gas accretion events. The possibility that star formation in the outer regions of gas-rich tidal tails may also lead to superclusters is raised. The mass-to-light ratio of these objects is small, because they contain an insignificant amount of dark matter. After many hundred Myr such systems may resemble dwarf spheroidal satellite galaxies with large apparent mass-to-light ratios, if tidal shaping is important.     

Strong size evolution of the most massive galaxies since z∼ 2

Using the combined capabilities of the large near-infrared Palomar/DEEP-2 survey, and the superb resolution of the Advanced Camera for Surveys HST camera, we explore the size evolution of 831 very massive galaxies  ( M _⋆≥ 10¹¹ h ⁻²₇₀ M_⊙)  since   z ∼ 2  . We split our sample according to their light concentration using the Sérsic index n . At a given stellar mass, both low  ( n < 2.5)  and high  ( n > 2.5)  concentrated objects were much smaller in the past than their local massive counterparts. This evolution is particularly strong for the highly concentrated (spheroid like) objects. At   z ∼ 1.5  , massive spheroid-like objects were a factor of 4 (±0.4) smaller (i.e. almost two orders of magnitudes denser) than those we see today. These small sized, high-mass galaxies do not exist in the nearby Universe, suggesting that this population merged with other galaxies over several billion years to form the largest galaxies we see today.     

The host galaxy of a narrow-line Seyfert 1 galaxy, RE J1034+396, with X-ray quasi-periodic oscillations

Using simple stellar population synthesis, we model the bulge stellar contribution in the optical spectrum of a narrow-line Seyfert 1 galaxy, RE J1034+396. We find that its bulge stellar velocity dispersion is  67.7 ± 8 km s⁻¹  . The supermassive black hole (SMBH) mass is about  (1–4) × 10⁶ M_⊙  if it follows the well-known   M _BH–σ_*  relation found in quiescent galaxies. We also derive the SMBH mass from the Hβ second moment, which is consistent with that from its bulge stellar velocity dispersion. The SMBH mass of (1–4)  × 10⁶ M_⊙  implies that the X-ray quasi-periodic oscillation (QPO) of RE J1034+396 can be scaled to a high-frequency QPO at 27–108 Hz found in Galactic black hole binaries with a  10-M_⊙  black hole. With the mass distribution in different age stellar populations, we find that the mean specific star formation rate (SSFR) over the past 0.1 Gyr is  0.0163 ± 0.0011  Gyr⁻¹, the stellar mass in the logarithm is  10.155 ± 0.06  in units of solar mass and the current star formation rate is  0.23 ± 0.016 M_⊙ yr⁻¹  . For RE J1034+396, there is no relation between the Eddington ratio and the SSFR as suggested by Chen et al., despite a larger scatter in their relation. We also suggest that about 7.0 per cent of the total Hα luminosity and 50 per cent of the total [O  ii ] luminosity come from the star formation process.     

The growth of supermassive black holes in pseudo-bulges, classical bulges and elliptical galaxies

Using results from structural analysis of a sample of nearly 1000 local galaxies from the Sloan Digital Sky Survey, we estimate how the mass in central black holes is distributed amongst elliptical galaxies, classical bulges and pseudo-bulges, and investigate the relation between their stellar masses and central stellar velocity dispersion σ. Assuming a single relation between elliptical galaxy/bulge mass, M _Bulge, and central black hole mass, M _BH, we find that  55⁺⁸₋₄  per cent of the mass in black holes in the local universe is in the centres of elliptical galaxies,  41⁺⁴₋₂  per cent in classical bulges and  4^+0.9_−0.4  per cent in pseudo-bulges. We find that ellipticals, classical bulges and pseudo-bulges follow different relations between their stellar masses and σ, and the most significant offset occurs for pseudo-bulges in barred galaxies. This structural dissimilarity leads to discrepant black hole masses if single   M _BH– M _Bulge  and   M _BH–σ  relations are used. Adopting relations from the literature, we find that the   M _BH–σ  relation yields an estimate of the total mass density in black holes that is roughly 55 per cent larger than if the   M _BH– M _Bulge  relation is used.     

The structure of spiral galaxies — I. Near-infrared properties of bulges, discs and bars

We present data for a sample of 45 spiral galaxies over a range of Hubble types, imaged in the near-IR J K bands. Parameters are calculated describing the bulge, disc and bar K -band light distributions, and we look for correlations showing the interrelation between these components. We find that bulge profiles are not well-fitted by the classic de Vaucouleurs profile, and that exponential or R ^1/2 fits are preferred. The bulge-to-disc ratio correlates only weakly with Hubble type. Many of the galaxies show central reddening of their J  −  K colours, which we interpret as due to nuclear starbursts or dusty AGN. We define a new method for measuring the strength of bars, which we call 'equivalent angle'. We stress that this is better than the traditional bar–interbar contrast, as it is not subject to seeing and resolution effects. Bars are found in 40 of the 45 galaxies, nine of which had been previously classified as unbarred. Bar strengths are found not to correlate with disc surface brightness or the presence of near neighbours, but a tendency is found for the most strongly barred galaxies to lie within a restricted, intermediate range of bulge-to-disc ratio. Bar light profiles are found to be either flat or exponentially decreasing along their long axes, with profile type not correlating strongly with Hubble type. Bar short axis profiles are significantly asymmetric, with the steeper profile being generally on the leading edge, assuming trailing arms. In the K band we find bars with higher axial ratios than have been found previously in optical studies.     

The radial Tully–Fisher relation for spiral galaxies – I

We find a new Tully–Fisher-like relation for spiral galaxies holding at different galactocentric radii. This radial Tully–Fisher relation allows us to investigate the distribution of matter in the optical regions of spiral galaxies. This relation, applied to three different samples of rotation curves of spiral galaxies, directly proves that: (i) the rotation velocity of spirals is a good measure of their gravitational potential and both the rotation curve's amplitudes and profiles are well predicted by galaxy luminosity, (ii) the existence of a dark component, less concentrated than the luminous one, and (iii) a scaling law, according to which, inside the disc optical size:   M _dark/ M _lum= 0.5( L _B /10¹¹ L _{B ⊙})^−0.7  .     

The census of nuclear activity of late-type galaxies in the Virgo cluster

The first spectroscopic census of active galactic nuclei (AGNs) associated with late-type galaxies in the Virgo cluster was carried out by observing 213 out of a complete set of 237 galaxies more massive than   M _dyn > 10^8.5 M_⊙  . Among them, 77 are classified as AGNs [including 21 transition objects, 47 low-ionization nuclear emission regions (LINERs) and nine Seyferts] and comprise 32 per cent of the late-type galaxies in Virgo. Due to spectroscopic incompleteness, at most 21 AGNs are missed in the survey, so that the fraction would increase up to 41 per cent. Using corollary near-infrared observations that enable us to estimate galaxy dynamical masses, it is found that AGNs are hosted exclusively in massive galaxies, i.e.   M _dyn≳ 10¹⁰ M_⊙  . Their frequency increases steeply with the dynamical mass from zero at   M _dyn≈ 10^9.5 M_⊙  to virtually 1 at   M _dyn > 10^11.5 M_⊙  . These frequencies are consistent with those of low-luminosity AGNs found in the general field by the Sloan Digital Sky Survey. Massive galaxies that harbour AGNs commonly show conspicuous r -band star-like nuclear enhancements. Conversely, they often, but not necessarily, contain massive bulges. A few well-known AGNs (e.g. M61, M100, NGC 4535) are found in massive Sc galaxies with little or no bulge. The AGN fraction seems to be only marginally sensitive to galaxy environment. We infer the black hole masses using the known scaling relations of quiescent black holes. No black holes lighter than  ∼10⁶ M_⊙  are found active in our sample.     

Photometric properties of the Arp 220 super star clusters

We investigate the photometric properties of six super stellar clusters (SSCs) seen in both the optical and near-infrared Hubble Space Telescope images of the local ultraluminous starburst galaxy Arp 220. Three of the SSCs are located in the central 0.5-kpc region. The remaining three are in the circumnuclear region between 0.5 and 2.5 kpc from the centre. Comparing the observed spectral energy distributions (SEDs) of the SSCs with the Starburst99 models of Leitherer et al., we confirm that all three nuclear SSCs are heavily obscured     Considering the results from this comparison in conjunction with measurements of the near-infrared CO absorption index and of millimetre CO linewidths and luminosities, we estimate the ages of the nuclear SSCs to be 10⁷–10⁸ yr. The bolometric luminosity of the three nuclear SSCs is at most one-fifth of the total bolometric luminosity of Arp 220. On the other hand, the circumnuclear SSCs have little internal extinction     These contribute negligibly to the total bolometric luminosity.     

Quantifying the coexistence of massive black holes and dense nuclear star clusters

In large spheroidal stellar systems, such as elliptical galaxies, one invariably finds a  10⁶–10⁹ M_⊙  supermassive black hole at their centre. In contrast, within dwarf elliptical galaxies one predominantly observes a  10⁵–10⁷ M_⊙  nuclear star cluster. To date, few galaxies have been found with both types of nuclei coexisting and even less have had the masses determined for both central components. Here, we identify one dozen galaxies housing nuclear star clusters and supermassive black holes whose masses have been measured. This doubles the known number of such hermaphrodite nuclei – which are expected to be fruitful sources of gravitational radiation. Over the host spheroid (stellar) mass range  10⁸–10¹¹ M_⊙  , we find that a galaxy's nucleus-to-spheroid (baryon) mass ratio is not a constant value but decreases from a few per cent to ∼0.3 per cent such that  log[( M _BH+ M _NC)/ M _sph]=−(0.39 ± 0.07) log[ M _sph/10¹⁰ M_⊙]− (2.18 ± 0.07)  . Once dry merging commences and the nuclear star clusters disappear, this ratio is expected to become a constant value.
As a byproduct of our investigation, we have found that the projected flux from resolved nuclear star clusters is well approximated with Sérsic functions having a range of indices from ∼0.5 to ∼3, the latter index describing the Milky Way's nuclear star cluster.     

Evolution of massive binary black holes

Since many or most galaxies have central massive black holes (BHs), mergers of galaxies can form massive binary black holes (BBHs). In this paper we study the evolution of massive BBHs in realistic galaxy models, using a generalization of techniques used to study tidal disruption rates around massive BHs. The evolution of BBHs depends on BH mass ratio and host galaxy type. BBHs with very low mass ratios (say, ≲0.001) are hardly ever formed by mergers of galaxies, because the dynamical friction time-scale is too long for the smaller BH to sink into the galactic centre within a Hubble time. BBHs with moderate mass ratios are most likely to form and survive in spherical or nearly spherical galaxies and in high-luminosity or high-dispersion galaxies; they are most likely to have merged in low-dispersion galaxies (line-of-sight velocity dispersion ≲90 km s⁻¹) or in highly flattened or triaxial galaxies.
The semimajor axes and orbital periods of surviving BBHs are generally in the range  10^-3–10 pc  and  10–10⁵ yr;  they are also larger in high-dispersion galaxies than in low-dispersion galaxies, larger in nearly spherical galaxies than in highly flattened or triaxial galaxies, and larger for BBHs with equal masses than for BBHs with unequal masses. The orbital velocities of surviving BBHs are generally in the range  10²–10⁴ km s^-1  . The methods of detecting surviving BBHs are also discussed.
If no evidence of BBHs is found in AGNs, this may be either because gas plays a major role in BBH orbital decay or because nuclear activity switches on soon after a galaxy merger, and ends before the smaller BH has had time to spiral to the centre of the galaxy.