Analytical potential–density pairs for flat rings and toroidal structures

The Kuzmin–Toomre family of discs is used to construct potential–density pairs that represent flat ring structures in terms of elementary functions. Systems composed of two concentric flat rings, a central disc surrounded by one ring and a ring with a centre of attraction are also presented. The circular velocity of test particles and the epicyclic frequency of small oscillations about circular orbits are calculated for these structures. A few examples of three-dimensional potential–density pairs of 'inflated' flat rings (toroidal mass distributions) are presented.     

Global lopsided instability in a purely stellar galactic disc

It is shown that pure exponential discs in spiral galaxies are capable of supporting slowly varying discrete global lopsided modes, which can explain the observed features of lopsidedness in the stellar discs. Using linearized fluid dynamical equations with the softened self-gravity and pressure of the perturbation as the collective effect, we derive self-consistently a quadratic eigenvalue equation for the lopsided perturbation in the galactic disc. On solving this, we find that the ground-state mode shows the observed characteristics of the lopsidedness in a galactic disc, namely the fractional Fourier amplitude A ₁, increases smoothly with the radius. These lopsided patterns precess in the disc with a very slow pattern speed with no preferred sense of precession. We show that the lopsided modes in the stellar disc are long-lived because of a substantial reduction (approximately a factor of 10 compared to the local free precession rate) in the differential precession. The numerical solution of the equations shows that the ground-state lopsided modes are either very slowly precessing stationary normal mode oscillations of the disc or growing modes with a slow growth rate depending on the relative importance of the collective effect of the self-gravity. N -body simulations are performed to test the spontaneous growth of lopsidedness in a pure stellar disc. Both approaches are then compared and interpreted in terms of long-lived global   m = 1  instabilities, with almost zero pattern speed.     

Vertical dynamics of disc galaxies in modified Newtonian dynamics

We investigate the possibility of discriminating between modified Newtonian dynamics (MOND) and Newtonian gravity with dark matter, by studying the vertical dynamics of disc galaxies. We consider models with the same circular velocity in the equatorial plane (purely baryonic discs in MOND and the same discs in Newtonian gravity embedded in spherical dark matter haloes), and we construct their intrinsic and projected kinematical fields by solving the Jeans equations under the assumption of a two-integral distribution function. We find that the vertical velocity dispersion of deep MOND discs can be much larger than in the equivalent spherical Newtonian models. However, in the more realistic case of high surface density discs, this effect is significantly reduced, casting doubt on the possibility of discriminating between MOND and Newtonian gravity with dark matter by using current observations.     

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.     

The shape of 'dark matter' haloes of disc galaxies according to MOND

Analyses of halo shapes for disc galaxies are said to give conflicting results. I point out that the modified dynamics (MOND) predicts for disc galaxies a distribution of fictitious dark matter that comprises two components: a pure disc and a rounder halo. The former dominates the true disc in regions of small accelerations, where it controls the z -dynamics in the disc (disc flaring etc.); it has a finite total mass. It also dominates the round component near the centre where the geometry is nearly planar. The second component controls motions far from the plane, has a total enclosed mass that diverges linearly with radius, and determines the rotation curve at large radii. Its ellipticity may be appreciable at small radii but vanishes asymptotically. This prediction of MOND differs from what one expects from galaxy formation scenarios with dark matter. Analyses to date, which, as they do, assume one component – usually with a constant ellipticity – perforce give conflicting results for the best value of ellipticity, depending on whether they probe the disc or the sphere, small radii or large ones.     

The paradox of the scale-free discs

Scale-free discs have no preferred length or time-scale. The question has been raised whether such discs have a continuum of unstable linear modes or perhaps no unstable modes at all. We resolve this paradox by analysing the particular case of a gaseous, isentropic disc with a completely flat rotation curve (the Mestel disc) exactly . The heart of the matter is this: what are the correct boundary conditions to impose at the origin or central cusp? We argue that the linear stability problem is ill-posed and that similar ambiguities may afflict general disc models with power-law central cusps. From any finite radius, waves reach the origin after finite time but with logarithmically divergent phase. Instabilities exist, but their pattern speeds depend upon an undetermined phase with which waves are reflected from the origin. For any definite choice of this phase, there is an infinite but discrete set of growing modes. The ratio of growth rate to pattern speed is independent of the central phase. This ratio is derived in closed form for non-self-gravitating normal modes and is shown to agree with approximate results obtained from the shearing sheet in the short-wavelength limit. This provides the first exact, analytically solved stability analysis for a differentially rotating disc. For self-gravitating normal modes, the ratio of growth rate to pattern is found numerically by solving recurrence relations in Mellin-transform space.     

Radial heating of a galactic disc by multiple spiral density waves

We consider a differentially rotating, 2D stellar disc perturbed by two steady-state spiral density waves moving at different pattern speeds. Our investigation is based on direct numerical integration of initially circular test-particle orbits. We examine a range of spiral strengths and spiral speeds and show that stars in this time-dependent gravitational field can be heated (their random motions increased). This is particularly noticeable in the simultaneous propagation of a two-armed spiral density wave near the corotation resonance (CR), and a weak four-armed one near the inner and outer 4:1 Lindblad resonances. In simulations with two spiral waves moving at different pattern speeds, we find: (i) the variance of the radial velocity,  σ² _R   , exceeds the sum of the variances measured from simulations with each individual pattern; (ii)  σ² _R   can grow with time throughout the entire simulation; (iii)  σ² _R   is increased over a wider range of radii compared to that seen with one spiral pattern; and (iv) particles diffuse radially in real space, whereas they do not when only one spiral density wave is present. Near the CR with the stronger, two-armed pattern, test-particles are observed to migrate radially. These effects take place at or near resonances of both spirals, so we interpret them as the result of stochastic motions. This provides a possible new mechanism for increasing the stellar velocity dispersion in galactic discs. If multiple spiral patterns are present in the Galaxy, we predict that there should be large variations in the stellar velocity dispersion as a function of radius.     

Discs and haloes in pre-main-sequence stars

We study the infrared (IR) emission from flared discs with and without additional optically thin haloes. Flux calculations of a flared disc in vacuum can be considered a special case of the more general family of models in which the disc is imbedded in an optically thin halo. In the absence of such a halo, flux measurements can never rule out its existence because the disc flaring surface defines a mathematically equivalent halo that produces the exact same flux at all IR wavelengths. When a flared disc with height H at its outer radius R is imbedded in a halo whose optical depth at visual wavelengths is  τ_halo  , the system IR flux is dominated by the halo whenever  τ_halo > (1/4) H / R   . Even when its optical depth is much smaller, the halo can still have a significant effect on the disc temperature profile. Imaging is the only way to rule out the existence of a potential halo, and we identify a decisive test that extracts a signature unique to flared discs from imaging observations.     

Kinematics from spectroscopy with a wide slit: detecting black holes in galaxy centres

We consider long-slit emission-line spectra of galactic nuclei when the slit is wider than the instrumental point spread function, and the target has large velocity gradients. The finite width of the slit generates complex distributions of brightness at a given spatial point in the measured spectrum, which can be misinterpreted as coming from additional physically distinct nuclear components. We illustrate this phenomenon for the case of a thin disc in circular motion around a nuclear black hole (BH). We develop a new method for estimating the mass of the BH that exploits a feature in the spectrum at the outer edge of the BH's sphere of influence, and therefore gives higher sensitivity to BH detection than traditional methods. Moreover, with this method we can determine the BH mass and the inclination of the surrounding disc separately, whereas the traditional approach to BH estimation requires two long-slit spectra to be taken. We show that, with a given spectrograph, the detectability of a BH depends on the sense of rotation of the nuclear disc. We apply our method to estimate the BH mass in M84 from a publicly available spectrum, and recover a value four times lower than that published previously from the same data.     

On the number of young stellar discs in the Galactic Centre

Observations of the Galactic Centre show evidence of disc-like structures of very young stars orbiting the central supermassive black hole within a distance of a few 0.1 pc. While it is widely accepted that about half of the stars form a relatively flat disc rotating clockwise on the sky, there is a substantial ongoing debate on whether there is a second, counter-clockwise disc of stars.
By means of N -body simulations using our bhint code, we show that two highly inclined stellar discs with the observed properties cannot be recognized as two flat circular discs after 5 Myr of mutual interaction. Instead, our calculations predict a significant warping of the two discs, which we show to be apparent among the structures observed in the Galactic Centre. While the high eccentricities of the observed counter-clockwise orbits suggest an eccentric origin of this system, we show the eccentricity distribution in the inner part of the more massive clockwise disc to be perfectly consistent with an initially circular disc in which stellar eccentricities increase due to both non-resonant and resonant relaxation.
We conclude that the relevant question to ask is therefore not whether there are two discs of young stars, but whether there were two such discs to begin with.     

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.     

The peculiar rotation curve of NGC 157

We present the results of a new H i , optical, and Hα interferometric study of the nearby spiral galaxy NGC 157. Our combined C- and D-array observations with the VLA show a large-scale, ring-like structure in the neutral hydrogen underlying the optical disc, together with an extended, low surface density component going out to nearly twice the Holmberg radius. Beginning just inside the edge of the star-forming disc, the line of nodes in the gas disc commences a 60° warp, while at the same time, the rotation velocity drops by almost half its peak value of 200 km s⁻¹, before levelling off again in the outer parts. While a flat rotation curve in NGC 157 cannot be ruled out, supportive evidence for an abrupt decline comes from the ionized gas kinematics, the optical surface photometry, and the global H i profile. A standard 'maximum-disc' mass model predicts comparable amounts of dark and luminous matter within NGC 157. Alternatively, a model employing a disc truncated at 2 disc scalelengths could equally well account for the unusual form of the rotation curve in NGC 157.     

A bar in the inner halo of barred galaxies – I. Structure and kinematics of a representative model

N -body simulations argue that the inner haloes of barred galaxies should not be spherical, nor even axisymmetric, but triaxial. The departure from sphericity is the strongest near the centre and decreases outwards; typical axial ratios for the innermost parts are of the order of 0.8. The halo shape is prolate-like in the inner parts up to a certain radius and then turns to oblate-like. I call this inner halo structure the 'halo bar' and analyse here in depth its structure and kinematics in a representative model. It is always considerably shorter than the disc bar. It lags the disc bar by only a few degrees at all radii and the difference between the two bar phases increases with distance from the centre. The two bars turn with roughly the same pattern speed. This means that the halo bar is a slow bar, since its corotation radius is much larger than its length. The bisymmetric component in the halo continues well outside the halo bar in the form of an open spiral, trailing behind the disc bar. The inner parts of the halo display some mean rotation in the same sense as the disc rotation. This is more important for particles nearer to the equatorial plane and decreases with increasing distance from it, but is always much smaller than the disc rotation.     

Intensity and polarization of radiation reflected from accretion disc

We consider the reflection of non-polarized radiation from the point-like sources above the accretion discs both the optically thick and optically thin. We investigate the dependence of the polarization of reflected radiation on the aperture angle of incident radiation. The aperture angle is determined by the radius of accretion disc and the height of the source above the disc. For optically thick accretion discs we show that, if the aperture angle is smaller 70 grad, then the wave electric field oscillations of reflected radiation are parallel to the accretion disc plane. For aperture angle greater than 70 grad the electric field oscillations are parallel to the plane “normal to accretion disc—the line of sight”. The latter also holds for reflection from the optically thin accretion disc independent of the aperture angle value.     

Three-integral models for axisymmetric galactic discs

We present new equilibrium component distribution functions that depend on three analytic integrals in a Stäckel potential, and that can be used to model stellar discs of galaxies. These components are generalizations of two-integral ones and can thus provide thin discs in the two-integral approximation. Their most important properties are the partly analytical expression for their moments, the disc-like features of their configuration space densities (exponential decline in the galactic plane and finite extent in the vertical direction) and the anisotropy of their velocity dispersions. We further show that a linear combination of such components can fit a van der Kruit disc.     

Damped Lyα systems at high redshift and models of protogalactic discs

We employ observationally determined intrinsic velocity widths and column densities of damped Lyman alpha (Lyα) systems at high redshift to investigate the distribution of baryons in protogalaxies within the context of a standard cold dark matter (CDM) model. We proceed under the assumption that damped Lyα systems represent a population of cold, rotationally supported, protogalactic discs, and that the abundance of dark matter haloes is well approximated by a CDM model with critical density and vanishing cosmological constant. Using conditional cross-sections to observe a damped system with a given velocity width and column density, we compare observationally inferred velocity width and column density distributions to the corresponding theoretically determined distributions for a variety of disc parameters and CDM normalizations. In general, we find that the observations cannot be reproduced by the models for most disc parameters and CDM normalizations. Whereas the column density distribution favours small discs with large neutral gas fraction, the velocity width distribution favours large and thick discs with small neutral gas fraction. The possible resolutions of this problem in the context of this CDM model may be (1) an increased contribution of rapidly rotating discs within massive dark matter haloes to damped Lyα absorption, or (2) the abandoning of simple disc models within this CDM model for damped Lyα systems at high redshift. Here the first possibility may be achieved by supposing that damped Lyα system formation occurs only in haloes with fairly large circular velocities, and the second possibility may result from a large contribution of mergers and double discs to damped Lyα absorption at high redshift.     

The origin of the light distribution in spiral galaxies

We analyse a high-resolution, fully cosmological, hydrodynamical disc galaxy simulation, to study the source of the double-exponential light profiles seen in many stellar discs, and the effects of stellar radial migration upon the spatiotemporal evolution of both the disc age and metallicity distributions. We find a 'break' in the pure exponential stellar surface brightness profile, and trace its origin to a sharp decrease in the star formation per unit surface area, itself produced by a decrease in the gas volume density due to a warping of the gas disc. Star formation in the disc continues well beyond the break. We find that the break is more pronounced in bluer wavebands. By contrast, we find little or no break in the mass density profile. This is, in part, due to the net radial migration of stars towards the external parts of the disc. Beyond the break radius, we find that ∼60 per cent of the resident stars migrated from the inner disc, while ∼25 per cent formed in situ . Our simulated galaxy also has a minimum in the age profile at the break radius but, in disagreement with some previous studies, migration is not the main mechanism producing this shape. In our simulation, the disc metallicity gradient flattens with time, consistent with an 'inside-out' formation scenario. We do not find any difference in the intensity or the position of the break with inclination, suggesting that perhaps the differences found in empirical studies are driven by dust extinction.     

Simulations of minor mergers – II. The phase-space structure of thick discs

We analyse the phase-space structure of simulated thick discs that are the result of a 5:1 mass-ratio merger between a disc galaxy and a satellite. Our main goal is to establish what would be the imprints of a merger origin for the Galactic thick disc. We find that the spatial distribution predicted for thick-disc stars is asymmetric, seemingly in agreement with recent observations of the Milky Way thick disc. Near the Sun, the accreted stars are expected to rotate more slowly, to have broad velocity distributions and to occupy preferentially the wings of the line-of-sight velocity distributions. The majority of the stars in our model thick discs have low eccentricity orbits (in clear reference to the pre-existing heated disc) which give rise to a characteristic (sinusoidal) pattern for their line-of-sight velocities as a function of galactic longitude. The z -component of the angular momentum of thick-disc stars provides a clear discriminant between stars from the pre-existing disc and those from the satellite, particularly at large radii. These results are robust against the particular choices of initial conditions made in our simulations.     

Stability of a finite disc under the influence of a spherical halo

We have studied the stability of finite gaseous discs, against large-scale perturbations, under the influence of spherical, massive haloes. A surface-density distribution consistent with the observed spiral-tracer profiles in disc galaxies is considered for the disc. We find that growing eigenmodes with both ‘trailing’ and ‘leading’ spirals exist in ‘cold’ discs for a wide range of values of the halo mass and its radius. The amplification rates of the unstable modes reduce as the ratio of the mass of the halo to the mass of the disc is increased. A uniform halo is not very effective towards stabilizing the disc against these modes. The results from the present study are consideredvis-a-vis previous studies on the global modes of self-gravitating discs.