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.     

Triaxial stellar systems following the r1/n luminosity law: an analytical mass–density expression, gravitational torques and the bulge/disc interplay

We have investigated the structural and dynamical properties of triaxial stellar systems whose surface brightness profiles follow the   r ^{1/ n}   luminosity law – extending the analysis by Ciotti, who explored the properties of spherical   r ^{1/ n}   systems. A new analytical expression that accurately reproduces the spatial (i.e., deprojected) luminosity density profiles (error less than 0.1 per cent) is presented for detailed modelling of the Sérsic family of luminosity profiles. We evaluate both the symmetric and the non-axisymmetric components of the gravitational potential and force, and compute the torques as a function of position. For a given triaxiality, stellar systems with smaller values of n have a greater non-axisymmetric gravitational field component . We also explore the strength of the non-axisymmetric forces produced by bulges with differing n and triaxiality on systems having a range of bulge-to-disc ratios. The increasing disc-to-bulge ratio with increasing galaxy type (decreasing n ) is found to greatly reduce the amplitude of the non-axisymmetric terms, and therefore reduce the possibility that triaxial bulges in late-type systems may be the mechanism or perturbation for non-symmetric structures in the disc.
Using seeing-convolved   r ^{1/ n}   -bulge plus exponential-disc fits to the K -band data from a sample of 80 nearby disc galaxies, we probe the relations between galaxy type, Sérsic index n and the bulge-to-disc luminosity ratio. These relations are shown to be primarily a consequence of the relation between n and the total bulge luminosity. In the K band, the trend of decreasing bulge-to-disc luminosity ratio along the spiral Hubble sequence is predominantly, though not entirely, a consequence of the change in the total bulge luminosity; the trend between the total disc luminosity and Hubble type is much weaker.     

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 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.     

Lyman break galaxies at z= 4–6 in cosmological smoothed particle hydrodynamics simulations

We perform a spectrophotometric analysis of galaxies at redshifts z = 4–6 in cosmological smoothed particle hydrodynamics simulations of a Λ cold dark matter universe. Our models include radiative cooling and heating by a uniform ultraviolet (UV) background, star formation, supernova feedback, and a phenomenological model for galactic winds. Analysing a series of simulations of varying box size and particle number allows us to isolate the impact of numerical resolution on our results. Specifically, we determine the luminosity functions in B , V , R , i ' and z ' filters, and compare the results with observational surveys of Lyman break galaxies (LBGs) performed with the Subaru telescope and the Hubble Space Telescope . We find that the simulated galaxies have UV colours consistent with observations and fall in the expected region of the colour–colour diagrams used by the Subaru group. The stellar masses of the most massive galaxies in our largest simulation increase their stellar mass from   M _★∼ 10¹¹ M_⊙  at z = 6 to   M _★∼ 10^11.7 M_⊙  at z = 3. Assuming a uniform extinction of E ( B − V ) = 0.15, we also find reasonable agreement between simulations and observations in the space density of UV bright galaxies at z = 3–6, down to the magnitude limit of each survey. For the same moderate extinction level of E ( B − V ) ∼ 0.15, the simulated luminosity functions match observational data, but have a steep faint-end slope with α∼−2.0. We discuss the implications of the steep faint-end slope found in the simulations. Our results confirm the generic conclusion from earlier numerical studies that UV bright LBGs at z ≥ 3 are the most massive galaxies with E ( B − V ) ∼ 0.15 at each epoch.     

Populating a cluster of galaxies – I. Results at

We simulate the assembly of a massive rich cluster and the formation of its constituent galaxies in a flat, low-density universe. Our most accurate model follows the collapse, the star formation history and the orbital motion of all galaxies more luminous than the Fornax dwarf spheroidal, while dark halo structure is tracked consistently throughout the cluster for all galaxies more luminous than the SMC. Within its virial radius this model contains about     dark matter particles and almost 5000 distinct dynamically resolved galaxies. Simulations of this same cluster at a variety of resolutions allow us to check explicitly for numerical convergence both of the dark matter structures produced by our new parallel N -body and substructure identification codes, and of the galaxy populations produced by the phenomenological models we use to follow cooling, star formation, feedback and stellar aging. This baryonic modelling is tuned so that our simulations reproduce the observed properties of isolated spirals outside clusters. Without further parameter adjustment our simulations then produce a luminosity function, a mass-to-light ratio, luminosity, number and velocity dispersion profiles, and a morphology–radius relation which are similar to those observed in real clusters. In particular, since our simulations follow galaxy merging explicitly, we can demonstrate that it accounts quantitatively for the observed cluster population of bulges and elliptical galaxies.     

HST Planetary Camera images of quasar host galaxies

We present Hubble Space Telescope ( HST ) images of seven low-redshift quasars (six taken with the Planetary Camera, one with the Wide Field Camera). These complete the sample of 14 quasars observed by the Faint Object Camera Investigation Definition Team (FOC IDT). Following subtraction of the quasar nuclear light, host galaxies can be seen in all seven cases. A combination of the optical morphology and luminosity profiles of the residual host galaxies and the results of 2D cross-correlation model fitting implies that five of the objects have elliptical host galaxies and two have disc host galaxies. The luminosities vary from slightly fainter than L ^* to about 1.3 mag brighter than L ^*.   We discuss the properties of the complete sample of 14 quasars. Nine of the objects appear to have elliptical host galaxies (all six of the radio-loud quasars in the sample as well as three radio-quiet quasars). Two further radio-quiet quasars appear to lie in disc galaxies. The other three objects (radio-quiet, ultraluminous infrared quasars) all lie in violently interacting systems. The sample as a whole has an average luminosity about 0.8 mag brighter than L ^*, although the radio-loud objects have hosts on average 0.7 mag brighter than the radio-quiet objects.   We compare our results with those from HST imaging of quasars by other authors. Taken together, our observations are in broad agreement with those of Bahcall et al. Radio-loud quasars appear to lie in luminous elliptical galaxies whereas radio-quiet quasars are found to lie in either elliptical or spiral hosts. Host galaxy luminosities (of radio-quiet and radio-loud quasars) are much brighter than would be expected if they followed a Schechter luminosity function.     

Galaxy cores as relics of black hole mergers

We investigate the hypothesis that the cores of elliptical galaxies and bulges are created from the binding energy liberated by the coalescence of supermassive binary black holes during galaxy mergers. Assuming that the central density profiles of galaxies were initially steep power laws,   ρ ∼ r ^-2  , we define the 'mass deficit' as the mass in stars that had to be removed from the nucleus in order to produce the observed core. We use non-parametric deprojection to compute the mass deficit in a sample of 35 early-type galaxies with high-resolution imaging data. We find that the mass deficit correlates well with the mass of the nuclear black hole, consistent with the predictions of merger models. We argue that cores in haloes of non-interacting dark matter particles should be comparable in size to those observed in the stars.     

The black hole mass–stellar velocity dispersion correlation: bulges versus pseudo-bulges

We investigate the correlation between the supermassive black holes (SMBHs) mass ( M _bh) and the stellar velocity dispersion  (σ_*)  in two types of host galaxies: the early-type bulges (disc galaxies with classical bulges or elliptical galaxies) and pseudo-bulges. In the form  log ( M _bh/M_⊙) =α+β log (σ_*/200 km s⁻¹)  , the best-fitting results for the 39 early-type bulges are the slope  β= 4.06 ± 0.28  and the normalization  α= 8.28 ± 0.05  ; the best-fitting results for the nine pseudo-bulges are  β= 4.5 ± 1.3  and  α= 7.50 ± 0.18  . Both relations have intrinsic scatter in  log  M _bh  of ≲0.27 dex. The   M _bh–σ_*  relation for pseudo-bulges is different from the relation in the early-type bulges over the 3σ significance level. The contrasting relations indicate the formation and growth histories of SMBHs depend on their host type. The discrepancy between the slope of the   M _bh–σ_*  relations using different definition of velocity dispersion vanishes in our sample, a uniform slope will constrain the coevolution theories of the SMBHs and their host galaxies more effectively. We also find the slope for the 'core' elliptical galaxies at the high-mass range of the relation appears steeper  (β≃ 5–6)  , which may be the imprint of their origin of dissipationless mergers.     

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 alignment of clusters using large-scale simulations

The alignment of clusters of galaxies with their nearest neighbours and between clusters within a supercluster is investigated using simulations of 512³ dark matter particles for ΛCDM and τ CDM cosmological models. Strongly significant alignments are found for separations of up to 15  h ⁻¹ Mpc in both cosmologies, but for the ΛCDM model the alignments extend up to separations of 30  h ⁻¹ Mpc. The effect is strongest for nearest neighbours, but is not significant enough to be useful as an observational discriminant between cosmologies. As a check of whether this difference in alignments is present in other cosmologies, smaller simulations with 256³ particles are investigated for four different cosmological models. Because of poor number statistics, only the standard CDM model shows indications of having different alignments from the other models.     

Simulating galactic outflows with kinetic supernova feedback

Feedback from star formation is thought to play a key role in the formation and evolution of galaxies, but its implementation in cosmological simulations is currently hampered by a lack of numerical resolution. We present and test a subgrid recipe to model feedback from massive stars in cosmological smoothed particle hydrodynamics simulations. The energy is distributed in kinetic form among the gas particles surrounding recently formed stars. The impact of the feedback is studied using a suite of high-resolution simulations of isolated disc galaxies embedded in dark haloes with total mass 10¹⁰ and  10¹²  h ⁻¹ M_⊙  . We focus, in particular, on the effect of pressure forces on wind particles within the disc, which we turn off temporarily in some of our runs to mimic a recipe that has been widely used in the literature. We find that this popular recipe gives dramatically different results because (ram) pressure forces on expanding superbubbles determine both the structure of the disc and the development of large-scale outflows. Pressure forces exerted by expanding superbubbles puff up the disc, giving the dwarf galaxy an irregular morphology and creating a galactic fountain in the massive galaxy. Hydrodynamic drag within the disc results in a strong increase in the effective mass loading of the wind for the dwarf galaxy, but quenches much of the outflow in the case of the high-mass galaxy.     

The black hole mass – spheroid luminosity relation

The differing   M _bh– L   relations presented in McLure & Dunlop, Marconi & Hunt and Erwin et al. have been investigated. A number of issues have been identified and addressed in each of these studies, including but not limited to the removal of a dependency on the Hubble constant, a correction for dust attenuation in the bulges of disc galaxies, the identification of lenticular galaxies previously treated as elliptical galaxies and the application of the same ( Y ∣ X ) regression analysis. These adjustments result in relations which now predict similar black hole masses. The optimal K -band relation is  log( M _bh/M_⊙) =−0.37(±0.04)( M _K + 24) + 8.29(±0.08)  , with a total (not intrinsic) scatter in log M _bh equal to 0.33 dex. This level of scatter is similar to the value of 0.34 dex from the     relation of Tremaine et al. and compares favourably with the value of 0.31 dex from the   M _bh– n   relation of Graham & Driver. Using different photometric data, consistent relations in the B and R band are also provided, although we do note that the small  ( N = 13)  R -band sample used by Erwin et al. is found here to have a slope of −0.30 ± 0.06. Performing a symmetrical regression on the larger K -band sample gives a slope of ∼−0.40, implying M _bh∝ L ^1.00. Implications for galaxy–black hole co-evolution, in terms of dry mergers, are briefly discussed, as are the predictions for intermediate mass black holes. Finally, as noted by others, a potential bias in the galaxy sample used to define the   M _bh– L   relations is shown and a corrective formula provided.     

Collisional baryonic dark matter haloes

If dark haloes are composed of dense gas clouds, as has recently been inferred, then collisions between clouds lead to galaxy evolution. Collisions introduce a core in an initially singular dark matter distribution, and can thus help to reconcile scale-free initial conditions – such as are found in simulations – with observed haloes, which have cores. A pseudo-Tully–Fisher relation, between halo circular speed and visible mass (not luminosity), emerges naturally from the model: M _vis∝ V ^7/2.
Published data conform astonishingly well to this theoretical prediction. For our sample of galaxies, the mass–velocity relationship has much less scatter than the Tully–Fisher relation, and holds as well for dwarf galaxies (where diffuse gas makes a sizeable contribution to the total visible mass) as it does for giants. It seems very likely that this visible-mass/velocity relationship is the underlying physical basis for the Tully–Fisher relation, and this discovery in turn suggests that the dark matter is both baryonic and collisional.     

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  .     

Constraining the star formation histories of spiral bulges

Stellar populations in spiral bulges are investigated using the Lick system of spectral indices. Long-slit spectroscopic observations of line strengths and kinematics made along the minor axes of four spiral bulges are reported. Comparisons are made between central line strengths in spiral bulges and those in other morphological types [elliptical, spheroidal (Sph) and S0]. The bulges investigated are found to have central line strengths comparable to those of single stellar populations of approximately solar abundance or above. Negative radial gradients are observed in line strengths, similar to those exhibited by elliptical galaxies. The bulge data are also consistent with correlations between Mg₂, Mg₂ gradient and central velocity dispersion observed in elliptical galaxies. In contrast to elliptical galaxies, central line strengths lie within the loci defining the range of 〈Fe〉 and Mg₂ achieved by Worthey's solar abundance ratio, single stellar populations (SSPs). The implication of solar abundance ratios indicates significant differences in the star formation histories of spiral bulges and elliptical galaxies. A 'single zone with infall' model of galactic chemical evolution, using Worthey's SSPs, is used to constrain the possible star formation histories of our sample. We show that the 〈Fe〉, Mg₂ and H β line strengths observed in these bulges cannot be reproduced using primordial collapse models of formation but can be reproduced by models with extended infall of gas and star formation (2–17 Gyr) in the region modelled. One galaxy (NGC 5689) shows a central population with a luminosity-weighted average age of ∼5 Gyr, supporting the idea of extended star formation. Kinematic substructure, possibly associated with a central spike in metallicity, is observed at the centre of the Sa galaxy NGC 3623.     

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.     

The anomalous properties of Markarian 1460

We present and discuss optical, near-infrared and H  i measurements of the galaxy Markarian 1460 at a distance of 19 Mpc in the Ursa Major Cluster. This low-luminosity ( M _B =−14) galaxy is unusual because (i) it is blue ( B − R =0.8) and has the spectrum of an H  ii galaxy, (ii) it has a light profile that is smooth and well fitted by an r ^1/4 and not an exponential function at all radii larger than the seeing, and (iii) it has an observed central brightness of about μ _B =20 mag arcsec⁻² , intermediate between those of elliptical galaxies (on the bright μ _B side) and normal low-luminosity dwarf irregular (on the low μ _B side) galaxies. No other known galaxy exhibits all these properties in conjunction. On morphological grounds this galaxy looks like a normal distant luminous elliptical galaxy, since the Fundamental Plane tells us that higher luminosity normal elliptical galaxies tend to have lower surface-brightnesses. Markarian 1460 has 2×10⁷ M_⊙ of H  i and a ratio M (H  i )/ L _B of 0.2, which is low compared to the typical values for star-forming dwarf galaxies. From the high surface-brightness and r ^1/4 profile, we infer that the baryonic component of Markarian 1460 has become self-gravitating through dissipative processes. From the colours, radio continuum, H  i and optical emission line properties, and yet smooth texture, we infer that Markarian 1460 has had significant star formation as recently as ∼1 Gyr ago but not today.     

Evolution of the mass function of dark matter haloes

We use a high-resolution ΛCDM numerical simulation to calculate the mass function of dark matter haloes down to the scale of dwarf galaxies, back to a redshift of 15, in a  50 h ⁻¹ Mpc  volume containing 80 million particles. Our low-redshift results allow us to probe low-σ density fluctuations significantly beyond the range of previous cosmological simulations. The Sheth & Tormen mass function provides an excellent match to all of our data except for redshifts of 10 and higher, where it overpredicts halo numbers increasingly with redshift, reaching roughly 50 per cent for the  10¹⁰–10¹¹ M_⊙  haloes sampled at redshift 15. Our results confirm previous findings that the simulated halo mass function can be described solely by the variance of the mass distribution, and thus has no explicit redshift dependence. We provide an empirical fit to our data that corrects for the overprediction of extremely rare objects by the Sheth & Tormen mass function. This overprediction has implications for studies that use the number densities of similarly rare objects as cosmological probes. For example, the number density of high-redshift  ( z ≃ 6) QSOs  , which are thought to be hosted by haloes at 5σ peaks in the fluctuation field, are likely to be overpredicted by at least a factor of 50 per cent. We test the sensitivity of our results to force accuracy, starting redshift and halo-finding algorithm.