Analytical potential–density pairs from complex-shifted Kuzmin–Toomre discs

The complex-shift method is applied to the Kuzmin–Toomre family of discs to generate a family of non-axisymmetric flat distributions of matter. These are then superposed to construct non-axisymmetric flat rings. We also consider triaxial potential–density pairs obtained from these non-axisymmetric flat systems by means of suitable transformations. The use of the imaginary part of complex-shifted potential–density pairs is also discussed.     

Simple potential–density pairs for flat rings

Potential–density pairs representing flat-ring structures in terms of elementary functions are presented. Structures representing one or several concentric flat rings, and discs surrounded by concentric flat rings are examined. The stability of concentric circular orbits of particles moving on a flat-ring structure is analyzed for radial perturbations.     

Exact potential–density pairs for flattened dark haloes

Cosmological simulations suggest that dark matter haloes are not spherical, but typically moderately to strongly triaxial systems. We investigate methods to convert spherical potential–density pairs into axisymmetric ones, in which the basic characteristics of the density profile (such as the slope at small and large radii) are retained. We achieve this goal by replacing the spherical radius r by an oblate radius m in the expression of the gravitational potential  Φ( r )  .
We extend and formalize the approach pioneered by Miyamoto & Nagai to be applicable to arbitrary potential–density pairs. Unfortunately, an asymptotic study demonstrates that, at large radii, such models always show a   R ⁻³  disc superposed on a smooth roughly spherical density distribution. As a result, this recipe cannot be used to construct simple flattened potential–density pairs for dynamical systems such as dark matter haloes. Therefore, we apply a modification of our original recipe that cures the problem of the discy behaviour. An asymptotic analysis now shows that the density distribution has the desired asymptotic behaviour at large radii (if the density falls less rapidly than   r ⁻⁴  ). We also show that the flattening procedure does not alter the shape of the density distribution at small radii: while the inner density contours are flattened, the slope of the density profile is unaltered.
We apply this recipe to construct a set of flattened dark matter haloes based on the realistic spherical halo models by Dehnen & McLaughlin. This example illustrates that the method works fine for modest flattening values, whereas stronger flattening values lead to peanut-shaped density distributions.     

Barred galaxy resonance rings: analytically explaining morphology and predicting dissipative misalignment

Many barred disc galaxies show rings of gas clouds and young stars thought to be in periodic orbits near the two-fold inner and outer Lindblad resonances (ILR and OLR) plus a four-fold ultraharmonic resonance (UHR) of the turning bar with oscillations about the disc orbital motion. To confirm and extend simulations by Schwarz and by Byrd et al. of resonance ring formation, we present an analytical formulation of the clouds' orbital motion which includes dissipative damping of oscillations relative to the local interstellar medium plus the rotation curve, bar pattern speed, and strength. Observed ring morphology matches our plots of periodic orbits where the density is enhanced but clouds do not collide violently. Pairs of 'outer rings' bracket the OLR. Dimpled outer rings like that of ESO 507-16 can be matched by plots with strong bars. Slightly dimpled outer rings like that of ESO 509-98 can be matched by weak bar plots. For flat rotation curves, a pair of two-fold rings bracket the ILR; the smaller can be identified with the tiny 'nuclear rings'. We find narrow UHR rings just outside this pair as well as just inside the OLR pair. We confirm the identification of the larger ILR ring and the inner UHR ring with 'inner rings'. Disagreeing with the common identification, we associate the dimpled outer rings with the UHR just inside the OLR. See ESO 507-16 as an example. We predict that damping can misalign the ILR and OLR rings relative to the bar as seen in our match to ESO 507-16. We find that for weak bars, if the linearly rising portion of the rotation curve is a significant fraction of the corotation radius, nuclear and inner rings are absent with outer rings still present. We show this in a match to ESO 509-98. Success of the matches to ESO 507-16 and 509-98 shows how the analytic formulation can be used to estimate disc orientation and pattern speed if rotation curve observations are available.     

New biorthogonal potential–density basis functions

We use the weighted integral form of spherical Bessel functions and introduce a new analytical set of complete and biorthogonal potential–density basis functions. The potential and density functions of the new set have finite central values and they fall off, respectively, similar to   r ^{−(1+ l )}  and   r ^{−(4+ l )}  at large radii, where l is the latitudinal quantum number of spherical harmonics. The lowest order term associated with   l = 0  is the perfect sphere of de Zeeuw. Our basis functions are intrinsically suitable for the modelling of three-dimensional, soft-centred stellar systems and they complement the basis sets of Clutton-Brock, Hernquist & Ostriker and Zhao. We test the performance of our functions by expanding the density and potential profiles of some spherical and oblate galaxy models.     

N-body simulations using customized potential–density pair basis sets

Potential–density pair basis sets can be used for highly efficient N -body simulation codes, but they suffer from a lack of versatility, i.e. a basis set has to be constructed for each different class of stellar system. We present numerical techniques for generating a biorthonormal potential–density pair basis set that has a general specified pair as its lowest-order member. We go on to demonstrate how the set can be used to construct N -body equilibria, which we then evolve using an N -body code that calculates forces using the basis set.     

Smoke rings around supernova 1987a

Detailed observations of the [Oiii]5007 Å emission from the elliptical rings around SN 1987A suggest a model wherein the two faint, outer rings are due to emission from two circular toroids moving outwards (at 25 km s^–1) along a bipolar cone centred on the site of the supernova. The brighter, central ring is expanding radially outward at 8.3 km s^–1. The rings must have been created 2-3 × 10⁴ years before the supernova explosion and are thought to be a consequence of the interaction of stellar winds emanating from the progenitor system during the final stages of its evolution to a supernova.     

A Roche model for uniformly rotating rings

A Roche model for describing uniformly rotating rings is presented, and the results are compared with the numerical solutions to the full problem for polytropic rings. In the thin ring limit, the surfaces of constant pressure including the surface of the ring itself are given in analytical terms, even in the mass-shedding case.     

The color of the Uranian rings

Observations of the Uranian rings were made in several color filters by the Voyager Imaging Science experiment in January 1986 for the purpose of determining the color of the rings. Selected images were taken through the Violet (λ = 0.41 μm), Clear (λ = 0.48 μm), and Green (λ = 0.55 μm) filters of the Voyager 2 narrow angle camera. The results of the analysis are consistent with the α, β, η, γ, δ, and ϵ rings being very dark, with flat spectra throughout the visible, and are comparable to the latest Voyager results showing a lack of color for the Uranian satellites. The general lack of color in the ring/satellite system of Uranus is remarkably different than the case of the distinctly reddish systems of Jupiter and Saturn. The unique combination of low absolute reflectivity and flat spectrum which characterizes the Uranian rings supports the concept that the Uranian ring material is compositionally distinct from either the Si- and S-rich Jovian ring and inner satellites, or the water-ice-rich rings and inner satellites of Saturn. Of all cosmically abundant materials, the candidate which best matches the low brightness and flat spectrum of the Uranian rings is carbon.     

A granular flow model for dense planetary rings

We study the viscosity of a differentially rotating particle disk in the limiting case where the particles are densely packed and their collective behavior resembles that of a liquid. The pressure tensor is derived from the equations of hydrodynamics and from a simple kinetic model of collisions described by Haff (1983). We find that density waves and narrow circular rings are unstable if the liquid approximation applies. The resulting development of nonlinear perturbations may give rise to “splashing” of the ring material in the vertical direction. These results may help in understanding the origin of the ellipticities of ringlets, the nonaxisymmetric features near the outer edge of the Saturnian B ring, and the unexplained residuals in kinematic models of the Saturnian and Uranian rings.     

The morphology of galactic rings exterior to evolving bars: test-particle simulations

The morphology of the outer rings of early-type spiral galaxies is compared to integrations of massless collisionless particles initially in nearly circular orbits. Particles are perturbed by a quadrupolar gravitational potential corresponding to a growing and secularly evolving bar. We find that outer rings with R1R2 morphology and pseudo-rings are exhibited by the simulations even though they lack gaseous dissipation. Simulations with stronger bars form pseudo-rings earlier and more quickly than those with weaker bars. We find that the R1 ring, perpendicular to the bar, is fragile and dissolves after a few bar rotation periods if the bar pattern speed increases by more than ∼8 per cent, bar strength increases (by ≳140 per cent) after bar growth or the bar is too strong  ( Q_T > 0.3)  . If the bar slows down after formation, pseudo-ring morphology persists and the R2 ring perpendicular to the bar is populated due to resonance capture. The R2 ring remains misaligned with the bar and increases in ellipticity as the bar slows down. The R2 ring becomes scalloped and does not resemble any ringed galaxies if the bar slows down more than 3.5 per cent suggesting that bars decrease in strength before they slow down this much. We compare the morphology of our simulations to B -band images of nine ringed galaxies from the Ohio State University Bright Spiral Galaxy Survey, and we find a reasonable match in morphologies to R1R2' pseudo-rings seen within a few bar rotation periods of bar formation. Some of the features previously interpreted in terms of dissipative models may be due to transient structure associated with recent bar growth and evolution.     

Stability of general relativistic Miyamoto–Nagai galaxies

The stability of a recently proposed general relativistic model of galaxies is studied in some detail. This model is a general relativistic version of the well-known Miyamoto–Nagai model that represents well a thick galactic disc. The stability of the disc is investigated under a general first-order perturbation keeping the space–time metric frozen (no gravitational radiation is taken into account). We find that the stability is associated with the thickness of the disc. We find that flat galaxies have more non-stable modes than the thick ones, i.e. flat galaxies have a tendency to form more complex structures like rings, bars and spiral arms.     

An Electron Density Model above the Sunspot from a Mapping of NOAA 7260 at 17 GHz

The brightness temperature distribution of microwave emission in a solar active region generally shows a ring structure, with a dip at the centre. However, no dip was found in the Nobeyama Radioheliograph left handed circular polarization (LCP) image on 1992 August 18; instead, there was a peak. This is a completely LCP source with zero right-handed circular polarization (RCP). We examine this structure in terms of the joint effect of gyroresonance and bremsstrahlung mechanism with a raised electron density above the central part of the sunspot, and the commonly assumed temperature and vertical dipole magnetic field models. The raised electron density is found to be 1.4 × 1011 cm-3 at the chromosphere base.     

Inelastic collisions in narrow planetary rings

It is shown that a particle ring with energy dissipation has an extremum in its energy when all the particles are in the same circular orbit. This extremum is a relative maximum in radial directions, indicating possible radial expansion; but it is a relative minimum in the particle velocity components, indicating a tendency for the velocity distribution to collapse. An N-body model of ring evolution incorporating two-body dynamics, oblateness perturbations, inelastic collisions, and phase averaging is described. By local analysis of impact statistics, it is shown that the velocity distribution of the ring will collapse if the coefficient of restitution ∈ ≲ 0.7. The collapse of the velocity distribution stabilizes a ring of point particles against radial dispersion. Furthermore, for ∈ ≲ 0.25, the semimajor axis distribution tends to collapse toward its local mean value, leading to radial collapse. A survey of ring evolution is presented for different values of coefficient of restitution and initial velocity dispersion. As has been predicted uy Goldreich and Tremaine, rings with equilibrium velocity distributions are unstable and expand, while rings where the velocity distribution collapses are shown to undergo massive, pervasive fragmentation into a myriad of ringlets. It is proposed that such fragmented rings are stable in their own right, and that the observational test to discriminate between the two cases is simply the presence of a smooth, featureless surface or the presence of intricate radial structure in the ring.     

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.     

Bending waves and the structure of Saturn's rings

The surface mass density profiles at four locations within Saturn's rings are calculated using Voyager spacecraft images of spiral bending waves. Bending waves are vertical corrugations in Saturn's rings which are excited at vertical resonances of a moon, e.g., Mimas, whose orbit is inclined with respect to the mean plane of the rings. Bending waves propagate toward Saturn by virtue of the rings' self-gravity; their wavelength depends on the local surface mass density of the rings. Observations of bending waves can thus be used to determine the surface density in regions of Saturn's rings near vertical resonances. The average surface density of the outer B ring near Mimas' 4:2 inner vertical resonance is 54 ± 10 g cm^?2. Surface density in this region probably varies by ～ 30% over radial length scales of tens of kilometers; and irregular radial structure is present on similar length scales in this region. Surface densities ranging from 24 g cm^?2 to 45 g cm^?2 are found in the A ring. Small scale variations in surface density are not seen in the A ring, consistent with its more uniform optical appearance.     

Projection effects in X-ray cores of cooling flow galaxy clusters

Recent analyses of Newton-XMM and Chandra data of the cores of X-ray bright clusters of galaxies show that modelling with a multi-phase gas in which several temperatures and densities are in equilibrium might not be appropriate. Instead, a single-phase model seems able to reproduce properly the spectra collected in annuli from the central region. The measured single-phase temperature profiles indicate a steep positive gradient in the central  100–200 kpc  and the gas density shows a flat profile in the central few 10s of kpc. Given this observational evidence, we estimate the contribution to the projected-on-the-sky rings from the cluster emissivity as function of the shell volume fraction sampled. We show that the observed projected X-ray emission mimics the multi-phase status of the plasma even though the input distribution is single-phase. This geometrical projection affects (i) analyses of data where insufficient spatial resolution is accessible, (ii) the central bin when its dimension is comparable to the extension of any flatness in the central gas density profile.     

Observation and photometry of an outer ring of saturn

A faint outer ring (E ring), which lies outside the classical rings A, B, C, and F, has been detected out to eight Saturn radii. We first observed it on November 1, 1979, and thereby confirmed the 1966 observation by Feibelman. Our plates were taken with a coronographic design and are specially intended for photometry. They are directly scaled in reflectance by reference to the Saturn disk which is properly attenuated. Photometry of the edge-on ring E lineament shows a strong brightness increase at small phase angles, which is compatible with scattering by particles of several microns in radius. The excess reflectivity in blue compared to the B ring implies a significant contribution of small particles in the scattering process. The E ring shows brightness and radial gradient changes, with condensations, which differ between east and west limbs and are not always the same from night to night. The E ring is probably a flat structure with a condensation centered at a distance of 4 R_s, but without a simple axial symmetry. It is probably shaped by segments or lumps and may have streamerlike structures.     

On the 3D dynamics and morphology of inner rings

We argue that inner rings in barred spiral galaxies are associated with specific 2D and 3D families of periodic orbits located just beyond the end of the bar. These are families located between the inner radial ultraharmonic 4 : 1 resonance and corotation. They are found in the upper part of a type-2 gap of the x1 characteristic, and can account for the observed ring morphologies without any help from families of the x1-tree. Due to the evolution of the stability of all these families, the ring shapes that are favoured are mainly ovals, as well as polygons with 'corners' on the minor axis, on the sides of the bar. On the other hand, pentagonal rings, or rings of the NGC 7020-type hexagon, should be less probable. The orbits that make the rings belong in their vast majority to 3D families of periodic orbits and orbits trapped around them.     

Uniformly rotating homogeneous and polytropic rings in Newtonian gravity

An analytical method is presented for treating the problem of a uniformly rotating, self-gravitating ring without a central body in Newtonian gravity. The method is based on an expansion about the thin ring limit, where the cross-section of the ring tends to a circle. The iterative scheme developed here is applied to homogeneous rings up to the 20th order and to polytropes with the index   n = 1  up to the third order. For other polytropic indices no analytic solutions are obtainable, but one can apply the method numerically. However, it is possible to derive a simple formula relating mass to the integrated pressure to leading order without specifying the equation of state. Our results are compared with those generated by highly accurate numerical methods to test their accuracy.