The coalescence of invariant manifolds and the spiral structure of barred galaxies

In a previous paper (Voglis et al., Paper I), we demonstrated that, in a rotating galaxy with a strong bar, the unstable asymptotic manifolds of the short-period family of unstable periodic orbits around the Lagrangian points L ₁ or L ₂ create correlations among the apocentric positions of many chaotic orbits, thus supporting a spiral structure beyond the bar. In this paper, we present evidence that the unstable manifolds of all the families of unstable periodic orbits near and beyond corotation contribute to the same phenomenon. Our results refer to a N -body simulation, a number of drawbacks of which, as well as the reasons why these do not significantly affect the main results, are discussed. We explain the dynamical importance of the invariant manifolds as due to the fact that they produce a phenomenon of 'stickiness' slowing down the rate of chaotic escape in an otherwise non-compact region of the phase space. We find a stickiness time of the order of 100 dynamical periods, which is sufficient to support a long-living spiral structure. Manifolds of different families become important at different ranges of values of the Jacobi constant. The projections of the manifolds of all the different families in the configuration space produce a pattern due to the 'coalescence' of the invariant manifolds. This follows closely the maxima of the observed   m = 2  component near and beyond corotation. Thus, the manifolds support both the outer edge of the bar and the spiral arms.     

Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies

In the presence of a strong   m = 2  component in a rotating galaxy, the phase-space structure near corotation is shaped to a large extent by the invariant manifolds of the short-period family of unstable periodic orbits terminating at L ₁ or L ₂. The main effect of these manifolds is to create robust phase correlations among a number of chaotic orbits large enough to support a spiral density wave outside corotation. The phenomenon is described theoretically by soliton-like solutions of a Sine–Gordon equation. Numerical examples are given in an N -body simulation of a barred spiral galaxy. In these examples, we demonstrate how the projection of unstable manifolds in configuration space reproduces essentially the entire observed bar–spiral pattern.     

Dynamics of the boxy elliptical galaxy NGC 1600

We use three-integral models to infer the distribution function (DF) of the boxy E3–E4 galaxy NGC 1600 from surface brightness and line-profile data on the minor and major axes. We assume axisymmetry and that the mass-to-light ratio is constant in the central ∼1 R _e. Stars in the resulting gravitational potential move mainly on regular orbits. We use an approximate third integral K from perturbation theory and write the DF as a sum of basis functions in the three integrals E , L _z and K . We then fit the projected moments of these basis functions to the kinematic observables and deprojected density, using a non-parametric algorithm. The deduced dynamical structure is radially anisotropic, with σ _θ σ _r ≈ σ _φ σ _r ≈0.7 on the major axis. Both on the minor axis and near the centre the velocity distribution is more isotropic; thus the model is flattened by equatorial radial orbits. The kinematic data are fitted without the need for a central black hole; the central mass determined previously from ground-based data therefore overestimates the actual black-hole mass. The mass-to-light ratio of the stars is M L _V =6  h ₅₀. The anisotropy structure of NGC 1600 with a radially anisotropic main body and more nearly isotropic centre is similar to that found recently in NGC 1399, 2434, 3379 and 6703, suggesting that this pattern may be common amongst massive elliptical galaxies. We discuss a possible merger origin of NGC 1600 in the light of these results.     

Contribution to the search for binaries among Am stars – VIII. New spectroscopic orbits of eight systems and statistical study of a sample of 91 Am stars

Red clump giant (RCG) stars can be used as distance indicators to trace the mass distribution of the Galactic bar. We use RCG stars from 44 bulge fields from the OGLE-II microlensing collaboration data base to constrain analytic triaxial models for the Galactic bar. We find the bar major-axis is oriented at an angle of 24°–27° to the Sun–Galactic Centre line-of-sight. The ratio of semimajor and semiminor bar axis scalelengths in the Galactic plane   x ₀, y ₀  , and vertical bar scalelength z ₀, is   x ₀ :  y ₀ :  z ₀= 10 : 3.5 : 2.6  , suggesting a slightly more prolate bar structure than the working model of Gerhard which gives the scalelength ratios as   x ₀ :  y ₀ :  z ₀= 10 : 4 : 3  .     

Survival and disruption of subsystems during a cold collapse

Cold collapse of a cluster composed of small identical clumps, each of which is in virial equilibrium, is considered. Since the clumps have no relative motion with respect to each other initially, the cluster collapses under its own gravity. At the first collapse of the cluster, most of the clumps are destroyed, but some survive. In order to find the condition for the clumps to survive, we made a systematic study in two-parameter space: the number of the clumps N _c and the size of the clump r _v . We obtained the condition N _c ≫ 1 and n _k  ≥ 1, where n _k is related to r _v and the initial radius of the cluster R _ini through the relation R _ini/ r _v  = 2 N ^{( n k +5)/6}_c. A simple analytical argument supports the numerical result. This n _k corresponds to the index of the power spectrum of the density fluctuation in the cosmological hierarchical clustering, and thus our result may suggest that in the systems smaller than 2/Ω h ²)Mpc, the first violent collapse is strong enough to sweep away all the substructures that exist before the collapse.     

Constraints on dark matter physics from dwarf galaxies through galaxy cluster haloes

One of the predictions of the standard cold dark matter model is that dark haloes have centrally divergent density profiles. An extensive body of rotation curve observations of dwarf and low surface brightness galaxies shows the dark haloes of those systems to be characterized by soft constant-density central cores. Several physical processes have been proposed to produce soft cores in dark haloes, each one with different scaling properties. With the aim of discriminating among them we have examined the rotation curves of dark-matter-dominated dwarf and low surface brightness galaxies and the inner mass profiles of two clusters of galaxies lacking a central cD galaxy and with evidence of soft cores in the centre. The core radii and central densities of these haloes scale in a well-defined manner with the depth of their potential wells, as measured through the maximum circular velocity. As a result of our analysis we identify self-interacting cold dark matter as a viable solution to the core problem, where a non-singular isothermal core is formed in the halo centre surrounded by a Navarro, Frenk & White profile in the outer parts. We show that this particular physical situation predicts core radii in agreement with observations. Furthermore, using the observed scalings, we derive an expression for the minimum cross-section ( σ ) which has an explicit dependence with the halo dispersion velocity ( v ). If m _x is the mass of the dark matter particle: σ m _x ≈4×10⁻²⁵ (100 km s⁻¹  v ⁻¹) cm² GeV⁻¹.     

Measuring Ω0 from the entropy evolution of clusters

In this paper we have extended the entropy-driven model of cluster evolution developed by Bower in order to be able to predict the evolution of galaxy clusters for a range of cosmological scenarios. We have applied this model to recent measurements of the evolution of the L _x− T normalization and X-ray luminosity function in order to place constraints on cosmological parameters. We find that these measurements alone do not select a particular cosmological framework. An additional constraint is required on the effective slope of the power spectrum to break the degeneracy that exists between this and the background cosmology. We therefore include a theoretical calculation of the Ω₀ dependence on the power spectrum, based on the cold dark matter paradigm, which infers Ω₀<0.55 (0.1<Ω₀<0.7 for Ω₀+Λ₀=1), at the 95 per cent confidence level. Alternatively, an independent measurement of the slope of the power spectrum from galaxy clustering requires Ω₀<0.6 (Ω₀<0.65 for Ω₀+Λ₀=1), again to 95 per cent confidence. The rate of entropy evolution is insensitive to the values of Ω₀ considered, although it is sensitive to changes in the distribution of the intracluster medium.     

The pattern speed of the OH/IR stars in the Milky Way

We show how the continuity equation can be used to determine pattern speeds in the Milky Way Galaxy (MWG). This method, first discussed by Tremaine & Weinberg in the context of external galaxies, requires projected positions, ( l , b ), and line-of-sight velocities for a spatially complete sample of relaxed tracers. If the local standard of rest (LSR) has a zero velocity in the radial direction ( u _LSR), then the quantity that is measured is  Δ V ≡Ω_p R ₀- V _LSR  , where Ω_p is the pattern speed of the non-axisymmetric feature, R ₀ is the distance of the Sun from the Galactic centre and V _LSR is the tangential motion of the LSR, including the circular velocity. We use simple models to assess the reliability of the method for measuring a single, constant pattern speed of either a bar or spiral in the inner MWG. We then apply the method to the OH/IR stars in the ATCA/VLA OH 1612-MHz survey of Sevenster et al., finding  Δ V =252±41 km s^-1,  if   u _LSR=0  . Assuming further that   R ₀=8 kpc  and   V _LSR=220 km s^-1,  this gives  Ω_p=59±5 km s^-1 kpc^-1  with a possible systematic error of perhaps 10 km s⁻¹ kpc⁻¹. The non-axisymmetric feature for which we measure this pattern speed must be in the disc of the MWG.     

Unravelling the mystery of the M31 bar

The inclination of M31 is too close to edge-on for a bar component to be easily recognized and is not sufficiently edge-on for a boxy/peanut bulge to protrude clearly out of the equatorial plane. Nevertheless, a sufficient number of clues allow us to argue that this galaxy is barred. We use fully self-consistent N -body simulations of barred galaxies and compare them with both photometric and kinematic observational data for M31. In particular, we rely on the near-infrared photometry presented in a companion paper. We compare isodensity contours to isophotal contours and the light profile along cuts parallel to the galaxy major axis and offset towards the north, or the south, to mass profiles along similar cuts on the model. All these comparisons, as well as position–velocity diagrams for the gaseous component, give us strong arguments that M31 is barred. We compare four fiducial N -body models to the data and thus set constraints on the parameters of the M31 bar, as its strength, length and orientation. Our 'best' models, although not meant to be exact models of M31, reproduce in a very satisfactory way the main relevant observations. We present arguments that M31 has both a classical and a boxy/peanut bulge. Its pseudo-ring-like structure at roughly 50 arcmin is near the outer Lindblad resonance of the bar and could thus be an outer ring, as often observed in barred galaxies. The shape of the isophotes also argues that the vertically thin part of the M31 bar extends considerably further out than its boxy bulge, that is, that the boxy bulge is only part of the bar, thus confirming predictions from orbital structure studies and from previous N -body simulations. It seems very likely that the backbone of M31's boxy bulge is families of periodic orbits, members of the x₁-tree and bifurcating from the x₁ family at its higher order vertical resonances, such as the x1v3 or x1v4 families.     

A numerical comparison of theories of violent relaxation

Using N -body simulations with a large set of massless test particles, we compare the predictions of two theories of violent relaxation, the well-known Lynden-Bell theory and the more recent theory by Nakamura. We derive 'weakened' versions of both the theories in which we use the whole equilibrium coarse-grained distribution function     as a constraint instead of the total energy constraint. We use these weakened theories to construct expressions for the conditional probability   K_i (τ)  that a test particle initially at the phase-space coordinate τ would end-up in the i th macro-cell at equilibrium. We show that the logarithm of the ratio   R_ij (τ) ≡ K_i (τ)/ K_j (τ)  is directly proportional to the initial phase-space density   f ₀(τ)  for the Lynden-Bell theory and inversely proportional to   f ₀(τ)  for the Nakamura theory. We then measure   R_ij (τ)  using a set of N -body simulations of a system undergoing a gravitational collapse to check the validity of the two theories of violent relaxation. We find that both the theories are at odds with the numerical results, both qualitatively and quantitatively.     

Stellar velocity dispersion in narrow-line Seyfert 1 galaxies

Several authors have recently explored, for narrow-line Seyfert 1 galaxies (NLS1s), the relationship between black hole mass ( M _BH) and stellar velocity dispersion (σ_*). Their results are more or less in agreement and seem to indicate that NLS1s fill the region below the fit obtained by Tremaine et al., showing a range of σ_* similar to that of Seyfert 1 galaxies, and a lower M _BH. Until now, the [O  iii ] width has been used in place of the stellar velocity dispersion, but some indications have begun to arise against the effectiveness of the gaseous kinematics in representing the bulge potential, at least in NLS1s. Bian & Zhao have stressed the urgency of producing true σ_* measurements. Here, we present new stellar velocity dispersions obtained through direct measurements of the Ca  ii absorption triplet (∼8550 Å) in the nuclei of eight NLS1 galaxies. The resulting σ_* values and a comparison with σ_[O III] confirm our suspicion that [O  iii ] typically overestimates the stellar velocity dispersion. We demonstrate that NLS1s follow the   M _BH–σ_*  relation as Seyfert 1, quasars and non-active galaxies.     

H i and radio continuum study of the isolated SBa Seyfert galaxy NGC 3783

NGC 3783 is a nearby SBa, type 1 Seyfert galaxy. We present H  i and radio continuum images of the galaxy made with the Australia Telescope Compact Array (ATCA). We find that NGC 3783 has an H  i mass of 8.4×10⁹ M_⊙, an H  i diameter of 1.9 D ₀ ( D ₀=37 kpc for h =0.5), and a nuclear depression in the H  i surface density. The H  i rotation curve is dominated by differential rotation, with little evidence of warping. The rotation curve suggests a mass-to-light ratio M L _B =7.2 and a bar-pattern speed of 19±7 km s⁻¹ kpc⁻¹. The total mass of gas in the inner 50 arcsec is ≳10 per cent of the dynamical mass, and consistent with models that require significant gas content to fuel the Seyfert nucleus. There is no evidence that the nuclear activity in NGC 3783 is being stimulated by an interaction or merger: it may be a self-generated, perhaps bar-driven, process.     

Variance and skewness in the FIRST survey

We investigate the large-scale clustering of radio sources in the FIRST 1.4-GHz survey by analysing the distribution function ( counts in cells ). We select a reliable sample from the the FIRST catalogue, paying particular attention to the problem of how to define single radio sources from the multiple components listed. We also consider the incompleteness of the catalogue. We estimate the angular two-point correlation function w (θ), the variance Ψ₂ and skewness Ψ₃ of the distribution for the various subsamples chosen on different criteria. Both w (θ) and Ψ₂ show power-law behaviour with an amplitude corresponding to a spatial correlation length of r ₀ ∼ 10  h ⁻¹Mpc. We detect significant skewness in the distribution, the first such detection in radio surveys. This skewness is found to be related to the variance through Ψ₃ =  S ₃(Ψ₂)^α, with α = 1.9 ± 0.1, consistent with the non-linear gravitational growth of perturbations from primordial Gaussian initial conditions. We show that the amplitude of variance and the skewness are consistent with realistic models of galaxy clustering.     

The quasar fraction in low-frequency-selected complete samples and implications for unified schemes

Low-frequency radio surveys are ideal for selecting orientation-independent samples of extragalactic sources because the sample members are selected by virtue of their isotropic steep-spectrum extended emission. We use the new 7C Redshift Survey along with the brighter 3CRR and 6C samples to investigate the fraction of objects with observed broad emission lines – the 'quasar fraction'– as a function of redshift and of radio and narrow-emission-line luminosity. We find that the quasar fraction is more strongly dependent upon luminosity (both narrow-line and radio) than it is on redshift. Above a narrow [O  ii ] emission-line luminosity of log₁₀( L _{[O  ii ]}/W)≳35 [or radio luminosity log₁₀( L ₁₅₁/W Hz⁻¹ sr⁻¹)≳ 26.5], the quasar fraction is virtually independent of redshift and luminosity; this is consistent with a simple unified scheme with an obscuring torus with a half-opening angle θ _trans≈53°. For objects with less luminous narrow lines, the quasar fraction is lower. We show that this is not due to the difficulty of detecting lower luminosity broad emission lines in a less luminous, but otherwise similar, quasar population. We discuss evidence which supports at least two probable physical causes for the drop in quasar fraction at low luminosity: (i) a gradual decrease in θ _trans and/or a gradual increase in the fraction of lightly reddened (0≲ A _V ≲5) lines of sight with decreasing quasar luminosity; and (ii) the emergence of a distinct second population of low-luminosity radio sources which, like M87, lack a well-fed quasar nucleus and may well lack a thick obscuring torus.     

The redshift evolution of clustering in the Hubble Deep Field

We present a correlation function analysis for the catalogue of photometric redshifts obtained from the Hubble Deep Field image by Fernandez-Soto, Lanzetta & Yahil. By dividing the catalogue into redshift bins of width Δ z =0.4 we measured the angular correlation function w ( θ ) as a function of redshift up to z ∼4.8. From these measurements we derive the trend of the correlation length r ₀. We find that r ₀( z ) is roughly constant with look-back time up to z ≃2, and then increases to higher values at z ≳2.4. We estimate the values of r ₀, assuming ξ ( r , z )=[ r r ₀( z )]^{− γ} , γ =1.8 and various geometries. For Ω₀=1 we find r ₀( z =3)≃7.00±4.87  h ⁻¹ Mpc, in good agreement with the values obtained from analysis of the Lyman break galaxies.     

Power-spectrum normalization from the local abundance of rich clusters of galaxies

The number density of rich galaxy clusters still provides the most robust way of normalizing the power spectrum of dark matter perturbations on scales relevant to large-scale structure. We revisit this constraint in the light of several recent developments: (1) the availability of well-defined samples of local clusters with relatively accurate X-ray temperatures; (2) new theoretical mass functions for dark matter haloes, which provide a good fit to large numerical simulations; (3) more accurate mass–temperature relations from larger catalogues of hydrodynamical simulations; (4) the requirement to consider closed as well as open and flat cosmologies to obtain full multiparameter likelihood constraints for CMB and SNe studies. We present a new sample of clusters drawn from the literature and use this sample to obtain improved results on σ ₈, the normalization of the matter power spectrum on scales of 8  h ⁻¹ Mpc, as a function of the matter density and cosmological constant in a universe with general curvature. We discuss our differences with previous work, and the remaining major sources of uncertainty. Final results on the normalization, approximately independent of power spectrum shape, can be expressed as constraints on σ at an appropriate cluster normalization scale R _Cl. We provide fitting formulas for R _Cl and σ ( R _Cl) for general cosmologies, as well as for σ ₈ as a function of cosmology and shape parameter Γ. For flat models we find approximately σ ₈≃(0.495_−0.037^+0.034)Ω_M^−0.60 for Γ=0.23, where the error bar is dominated by uncertainty in the mass–temperature relation.     

Peculiar velocities of galaxies and clusters

We present a simple model for the shape of the distribution function of galaxy peculiar velocities. We show how both non-linear and linear theory terms combine to produce a distribution which has an approximately Gaussian core with exponential wings. The model is easily extended to study how the statistic depends on the type of particle used to trace the velocity field (dark matter particles, dark matter haloes, galaxies), and on the density of the environment in which the test particles are located. Comparisons with simulations suggest that our model is accurate. We also show that the evolution of the peculiar velocities depends on the local, rather than the global, density. Since clusters populate denser regions on average, using cluster velocities with the linear theory scaling may lead to an overestimate of the global value of Ω₀. Conversely, using linear theory with the global value of Ω₀ to scale cluster velocities from the initial to the present time results in an underestimate of their true velocities. In general, however, the directions of motions of haloes are rather well described by linear theory. Our results help to simplify models of redshift-space distortions considerably.     

The distance–redshift equation in quintessence cosmology and the estimation of H0 through time delays

We calculate analytically and numerically the distance–redshift equation in perfect fluid quintessence models and give an accurate fit to the numerical solutions for all the values of the density parameter and the quintessence equation of state. Then we apply our solutions to the estimation of H ₀ from multiple image time delays and find that the inclusion of quintessence modifies significantly the likelihood distribution of H ₀, generally reducing the best estimate with respect to a pure cosmological constant. Marginalizing over the other parameters (Ω _m and the quintessence equation of state) we obtain H ₀=71±6 km s⁻¹ Mpc⁻¹ for an empty beam and H ₀=64±4 km s⁻¹ Mpc⁻¹ for a filled beam. These errors, however, do not take into account the uncertainty on the modelling of the lens. We also discuss the future prospects for distinguishing quintessence from a cosmological constant with time delays.     

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.