Evolution of the phase-space density of dark matter haloes and mixing effects in merger events

Cosmological N -body simulations were performed to study the evolution of the phase-space density   Q =ρ/σ³  of dark matter haloes. No significant differences in the scale relations   Q ∝σ^−2.1  or   Q ∝ M ^−0.82  are seen for the 'cold' or 'warm' dark matter models. The follow-up of individual haloes from   z = 10  up to the present time indicate the existence of two main evolutionary phases: an early and fast one  (10 > z > 6.5)  , in which Q decreases on the average by a factor of 40 as a consequence of the randomization of bulk motions, and a late and long one  (6.5 > z ≥ 0)  , in which Q decreases by a factor of 20 because of mixing induced by merger events. The study of these haloes has also evidenced that rapid and positive variations of the velocity dispersion, induced by merger episodes, are related to a fast decrease of the phase-space density Q .     

Accretion in stellar clusters and the initial mass function

We present a simple physical mechanism that can account for the observed stellar mass spectrum for masses M ∗≳0.5 M_⊙ . The model depends solely on the competitive accretion that occurs in stellar clusters where each star's accretion rate depends on the local gas density and the square of the accretion radius. In a stellar cluster, there are two different regimes depending on whether the gas or the stars dominate the gravitational potential. When the cluster is dominated by cold gas, the accretion radius is given by a tidal-lobe radius. This occurs as the cluster collapses towards a ρ  ∝  R ⁻² distribution. Accretion in this regime results in a mass spectrum with an asymptotic limit of γ =−3/2 (where Salpeter is γ =−2.35) . Once the stars dominate the potential and are virialized, which occurs first in the cluster core, the accretion radius is the Bondi–Hoyle radius. The resultant mass spectrum has an asymptotic limit of γ =−2 with slightly steeper slopes ( γ ≈−2.5) if the stars are already mass-segregated. Simulations of accretion on to clusters containing 1000 stars show that, as expected, the low-mass stars accumulate the majority of their masses during the gas-dominated phase whereas the high-mass stars accumulate the majority of their masses during the stellar-dominated phase. This results in a mass spectrum with a relatively shallow γ ≈3/2 power law for low-mass stars and a steeper power law for high-mass stars −2.5≲ γ ≤−2 . This competitive accretion model also results in a mass-segregated cluster.     

Structures in a class of magnetized scale-free discs

We construct analytically stationary global configurations for both aligned and logarithmic spiral coplanar magnetohydrodynamics (MHD) perturbations in an axisymmetric background MHD disc with a power-law surface mass density  Σ₀∝ r ^−α  , a coplanar azimuthal magnetic field   B ₀∝ r ^−γ  , a consistent self-gravity and a power-law rotation curve   v ₀∝ r ^−β  , where v ₀ is the linear azimuthal gas rotation speed. The barotropic equation of state  Π∝Σ ⁿ   is adopted for both MHD background equilibrium and coplanar MHD perturbations where Π is the vertically integrated pressure and n is the barotropic index. For a scale-free background MHD equilibrium, a relation exists among  α, β, γ  and n such that only one parameter (e.g. β) is independent. For a linear axisymmetric stability analysis, we provide global criteria in various parameter regimes. For non-axisymmetric aligned and logarithmic spiral cases, two branches of perturbation modes (i.e. fast and slow MHD density waves) can be derived once β is specified. To complement the magnetized singular isothermal disc analysis of Lou, we extend the analysis to a wider range of  −1/4 < β < 1/2  . As an illustrative example, we discuss specifically the  β= 1/4  case when the background magnetic field is force-free. Angular momentum conservation for coplanar MHD perturbations and other relevant aspects of our approach are discussed.     

Re-identification of G35.6−0.4 as a supernova remnant

G35.6−0.4 is an extended radio source in the Galactic plane which has previously been identified as either a supernova remnant or an H  ii region. Observations from the Very Large Array Galactic Plane Survey at 1.4 GHz with a resolution of 1 arcmin allow the extent of G35.6−0.4 to be defined for the first time. Comparison with other radio survey observations show that this source has a non-thermal spectral index, with   S ∝ν^{−0.47±0.07}  . G35.6−0.4 does not have obvious associated infrared emission, so it is identified as a Galactic supernova remnant, not an H  ii region. It is  ≈15 × 11 arcmin²  in extent, showing partial limb brightening.     

The origin and formation of cuspy density profiles through violent relaxation of stellar systems

It is shown that the cuspy density distributions observed in the cores of elliptical galaxies can be realized by dissipationless gravitational collapse. The initial models consist of power-law density spheres such as ρ ∝ r ⁻¹ with anisotropic velocity dispersions. Collapse simulations are carried out by integrating the collisionless Boltzmann equation directly, on the assumption of spherical symmetry. From the results obtained, the extent of constant density cores, formed through violent relaxation, decreases as the velocity anisotropy increases radially, and practically disappears for extremely radially anisotropic models. As a result, the relaxed density distributions become more cuspy with increasing radial velocity anisotropy. It is thus concluded that the velocity anisotropy could be a key ingredient for the formation of density cusps in a dissipationless collapse picture. The velocity dispersions increase with radius in the cores according to the nearly power-law density distributions. The power-law index, n , of the density profiles, defined as ρ ∝ r ^{− n} , changes from n ≈2.1 at intermediate radii to a shallower power than n ≈2.1 toward the centre. This density bend can be explained from our postulated local phase-space constraint that the phase-space density accessible to the relaxed state is determined at each radius by the maximum phase-space density of the initial state.     

Application of a clumpy core model to NGC 2071

The multi-transitional observations of CS molecules towards the NGC 2071 core have been re-analysed by using a tri-dimensional Monte Carlo radiative transfer code. Better agreement with the observations is made by an introduction of clumpiness to this model than by smoothly varying density to the 1D microturbulent one. The best-fitting model shows that, when a unique density is assumed for clumps, the volume filling factor of the clumps varies as r ⁻² with an average of ∼5 per cent over the entire core, and that the H₂ number density and the CS abundance of the clump relative to H₂ are ∼ 2 × 10⁶ cm⁻³ and ∼ 6 × 10⁻¹⁰, respectively. The radial density gradient ∝ r ⁻² obtained from our clumpy core model is steeper than that (∝ r ^−1.3) obtained from the microturbulent model. Since all clumps are subject to random bulk motions in this 3D clumpy macroturbulent model, synthesized line profiles do not show self-absorption dips even for opaque transitions and the resulting linewidth is in good accordance with the observations.     

Self-similar solutions for the dynamical condensation of a radiative gas layer

A new self-similar solution describing the dynamical condensation of a radiative gas is investigated under a plane-parallel geometry. The dynamical condensation is caused by thermal instability. The solution is applicable to generic flow with a net cooling rate per unit volume and time  ∝ρ² T ^α  , where  ρ,  T   and α are the density, temperature and a free parameter, respectively. Given α, a family of self-similar solutions with one parameter η is found in which the central density and pressure evolve as follows:  ρ( x = 0, t ) ∝ ( t _c− t )^{−η/(2−α)}  and   P ( x = 0, t ) ∝ ( t _c− t )^{(1−η)/(1−α)}  , where t _c is the epoch at which the central density becomes infinite. For  η∼ 0  the solution describes the isochoric mode, whereas for  η∼ 1  the solution describes the isobaric mode. The self-similar solutions exist in the range between the two limits; that is, for  0 < η < 1  . No self-similar solution is found for  α > 1  . We compare the obtained self-similar solutions with the results of one-dimensional hydrodynamical simulations. In a converging flow, the results of the numerical simulations agree well with the self-similar solutions in the high-density limit. Our self-similar solutions are applicable to the formation of interstellar clouds (H  i clouds and molecular clouds) by thermal instability.     

Hot stars in old stellar populations: a continuing need for intermediate ages

We present numerical investigations into the formation of massive stars from turbulent cores of density structure  ρ∝ r ^−1.5  . The results of five hydrodynamical simulations are described, following the collapse of the core, fragmentation and the formation of small clusters of protostars. We generate two different initial turbulent velocity fields corresponding to power-law spectra   P ∝ k ⁻⁴  and   P ∝ k ^−3.5  , and we apply two different initial core radii. Calculations are included for both completely isothermal collapse, and a non-isothermal equation of state above a critical density  (10⁻¹⁴ g cm⁻³)  . Our calculations reveal the preference of fragmentation over monolithic star formation in turbulent cores. Fragmentation was prevalent in all the isothermal cases. Although disc fragmentation was largely suppressed in the non-isothermal runs due to the small dynamic range between the initial density and the critical density, our results show that some fragmentation still persisted. This is inconsistent with previous suggestions that turbulent cores result in the formation of a single massive star. We conclude that turbulence cannot be measured as an isotropic pressure term.     

A survey of hard spectrum ROSAT sources – I. X-ray source catalogue

We present a catalogue of 147 serendipitous X-ray sources selected to have hard spectra ( α <0.5) from a survey of 188 ROSAT fields. Such sources must be the dominant contributors to the X-ray background at faint fluxes. We have used Monte Carlo simulations to verify that our technique is very efficient at selecting hard sources: the survey has 10 times as much effective area for hard sources as it has for soft sources above a 0.5–2 keV flux level of 10⁻¹⁴ erg cm⁻² s⁻¹. The distribution of best-fitting spectral slopes of the hard sources suggests that a typical ROSAT hard source in our survey has a spectral slope α ∼0. The hard sources have a steep number flux relation (d N /d S ∝ S ^{− γ} with a best-fitting value of γ =2.72±0.12) and make up about 15 per cent of all 0.5–2 keV sources with S >10⁻¹⁴ erg cm⁻² s⁻¹. If their N ( S ) continues to fainter fluxes, the hard sources will comprise ∼40 per cent of sources with 5×10⁻¹⁵< S <10⁻¹⁴ erg cm⁻² s⁻¹. The population of hard sources can therefore account for the harder average spectra of ROSAT sources with S <10⁻¹⁴ erg cm⁻² s⁻¹. They probably make a strong contribution to the X-ray background at faint fluxes and could be the solution to the X-ray background spectral paradox.     

Spectral lags and the energy dependence of pulse width in gamma-ray bursts: contributions from the relativistic curvature effect

We compute the temporal profiles of the gamma-ray burst pulse in the four Burst and Transient Source Experiment (BATSE) Large Area Detector (LAD) discriminator energy channels, with the relativistic curvature effect of an expanding fireball being explicitly investigated. Assuming an intrinsic 'Band' shape spectrum and an intrinsic energy-independent emission profile, we show that merely the curvature effect can produce detectable spectral lags if the intrinsic pulse profile has a gradually decaying phase. We examine the spectral lag's dependences on some physical parameters, such as the Lorentz factor Γ, the low-energy spectral index, α, of the intrinsic spectrum, the duration of the intrinsic radiation   t '_d  and the fireball radius R . It is shown that approximately the lag  ∝Γ⁻¹  and  ∝ t '_d  , and a spectrum with a more extruded shape (a larger α) causes a larger lag. We find no dependence of the lag on R . Quantitatively, the lags produced from the curvature effect are marginally close to the observed ones, while larger lags require extreme physical parameter values, e.g.  Γ < 50  , or  α > −0.5  . The curvature effect causes an energy-dependent pulse width distribution but the energy dependence of the pulse width we obtained is much weaker than the observed   W ∝ E ^−0.4  one. This indicates that some intrinsic mechanism(s), other than the curvature effect, dominates the pulse narrowing of gamma-ray bursts.     

Interaction between the north-eastern boundary of Sgr A East and giant molecular clouds

We have detected the   v = 1 → 0 S(1) (λ= 2.1218 μm)  and   v = 2 → 1 S(1) (λ= 2.2477 μm)  lines of H₂ in the Galactic Centre, in a  90 × 27 arcsec²  region between the north-eastern boundary of the non-thermal source Sgr A East, and the giant molecular cloud (GMC)  M−0.02 − 0.07  . The detected  H₂ v = 1 → 0  S(1) emission has an intensity of  1.6–21 × 10⁻¹⁸ W m⁻² arcsec⁻²  and is present over most of the region. Along with the high intensity, the large linewidths  (FWHM = 40–70 km s⁻¹)  and the  H₂ v = 2 → 1 S(1)  to   v = 1 → 0 S(1)  line ratios (0.3–0.5) can be best explained by a combination of C-type shocks and fluorescence. The detection of shocked H₂ is clear evidence that Sgr A East is driving material into the surrounding adjacent cool molecular gas. The H₂ emission lines have two velocity components at ∼+50 and  ∼0 km s⁻¹  , which are also present in the NH₃(3, 3) emission mapped by McGary, Coil & Ho. This two-velocity structure can be explained if Sgr A East is driving C-type shocks into both the  GMC M−0.02 − 0.07  and the northern ridge of McGary et al.     

Density cusps: restrictions on non-axisymmetric models

Galactic nuclei are now generally thought to have density cusps in their centres, and the evidence is mounting that as a consequence they are unlikely to be triaxial. Self-consistent stellar dynamical models of non-axisymmetric cusps would be an interesting counter-argument to this conclusion. We consider 2D analogues of triaxial cusps: a sequence of non-axisymmetric, cuspy discs first described by Sridhar & Touma. Scale-free models with potential Φ ∝  r ^α are examined in detail. It is shown analytically for 0 < α ≲ 0.43 that self-consistent models with positive phase-space density do not exist. Numerical solutions of the combined Vlasov and Poisson equations suggest that the whole sequence of models with 0 < α < 1 are also unphysical. Together with existing work on cusps, we conclude on purely theoretical grounds that galactic nuclei are not expected to be triaxial.     

Long-term variability of the low-luminosity active galactic nucleus of M81

Long-term X-ray variability of the low-luminosity active galactic nucleus of M81 was studied, using 16 ASCA observations spanning 5.5 yr. The object exhibits a factor of 3 variation over the 5.5 yr. The source intensity was relatively constant within each observation which lasted typically for one day, but intra-day variability by 30 per cent was detected on the 15th observation. The power-spectral density (PSD) was estimated in a 'forward' manner, over a frequency range of 10^−8.2–10^−4.3 Hz (period range of 0.25 d–5.5 yr), by utilizing the structure function and extensive Monte Carlo simulations in order to overcome the very sparse and uneven data samplings. When the PSD is assumed to be white below a 'break frequency' f _b and falls off as ∝  f ^{− α} above f _b, where f is frequency and α is a positive parameter, the M81 light curve is well described with 1/ f _b≥800 d and α =1.4±0.2.     

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.     

Gravitational lensing of anisotropic sources

In strong gravitational lensing, the multiple images we see correspond to light rays that leave the source in slightly different directions. If the source emission is anisotropic, the images may differ from conventional lensing predictions (which assume isotropy). To identify scales on which source anisotropy may be important, we study the angle δ between the light rays emerging from the source, for different lensing configurations. If the lens has a power-law profile   M ∝ R ^γ  , the angle δ initially increases with lens redshift and then either diverges (for a steep profile  γ < 1  ), remains constant (for an isothermal profile  γ= 1  ), or vanishes (for a shallow profile  γ > 1  ) as   z _l→ z _s  . The scaling with lens mass is roughly  δ∝ M ^1/(2−γ)  . The results for an Navarro–Frenk–White (NFW) profile are qualitatively similar to those for a shallow power law, with δ peaking at about half the redshift of the source (not half the distance). In practice, beaming could modify the statistics of beamed sources lensed by massive clusters: for an opening angle  θ_jet  , there is a probability as high as   P ∼ 0.02–0.07(θ_jet/0.5°)⁻¹  that one of the lensed images may be missed (for  2 ≲ z _s≲ 6  ). Differential absorption within active galactic nuclei (AGNs) could modify the flux ratios of AGNs lensed by clusters; a sample of AGNs lensed by clusters could provide further constraints on the sizes of absorbing regions. Source anisotropy is not likely to be a significant effect in galaxy-scale strong lensing.     

Scaling up tides in numerical models of galaxy and halo formation

The purpose of this article is to show that when dynamically cold, dissipationless self-gravitating systems collapse, their evolution is a strong function of the symmetry in the initial distribution. We explore with a set of pressureless homogeneous fluids the time evolution of ellipsoidal distributions and map the depth of potential achieved during relaxation as function of initial ellipsoid axis ratios. We then perform a series of N -body numerical simulations and contrast their evolution with the fluid solutions. We verify an analytic relation between collapse factor and particle number N in spherical symmetry, such that  ∝ N ^1/3  . We sought a similar relation for axisymmetric configurations, and found an empirical scaling relation such that  ∝ N ^1/6  in these cases. We then show that when mass distributions do not respect spherical or axial symmetry, the ensuing gravitational collapse deepens with increasing particle number N but only slowly: 86 per cent of triaxial configurations may collapse by a factor of no more than 40 as   N →∞  . For   N ≈10⁵  and larger, violent relaxation develops fully under the Lin–Mestel–Shu instability such that numerical N -body solutions now resolve the different initial morphologies adequately.     

Parameters of core collapse

This paper considers the phenomenon of deep core collapse in collisional stellar systems, with stars of equal mass. The collapse takes place on some multiple,  ξ⁻¹  , of the central relaxation time, and produces a density profile in which  ρ∝ r ^−α  , where α is a constant. The parameters α and ξ have usually been determined from simplified models, such as gas and Fokker–Planck models, often with the simplification of isotropy. Here we determine the parameters directly from N -body simulations carried out using the newly completed GRAPE-6.     

The effects of anticorrelation on gravitational clustering

We use non-linear scaling relations (NSRs) to investigate the effects arising from the existence of negative correlations on the evolution of gravitational clustering in an expanding universe. It turns out that such anticorrelated regions have important dynamical effects on all scales. In particular, the mere existence of negative values for the linear two-point correlation function ξ¯ _L over some range of scales starting from l = L ₀ implies that the non-linear correlation function is bounded from above at all scales x < L ₀ . This also results in the relation ξ¯   ∝  x ⁻³ , at these scales, at late times, independent of the original form of the correlation function. Current observations do not rule out the existence of negative ξ¯ for 200  h ⁻¹ Mpc≲ ξ¯ ≲1000  h ⁻¹ Mpc; the present work may thus have relevance for the real Universe. The only assumption made in the analysis is the existence of NSR; the results are independent of the form of the NSR as well as of the stable clustering hypothesis.     

Amplification of primordial magnetic fields by anisotropic gravitational collapse

If a magnetic field is frozen into a plasma that undergoes spherical compression, then the magnetic field B varies with the plasma density ρ according to   B ∝ρ^2/3  . In the gravitational collapse of cosmological density perturbations, however, quasi-spherical evolution is very unlikely. In anisotropic collapses the magnetic field can be a much steeper function of gas density than in the isotropic case. We investigate the distribution of amplifications in realistic gravitational collapses from Gaussian initial fluctuations using the Zel'dovich approximation. Representing our results using a relation of the form   B ∝ρ^α  , we show that the median value of α can be much larger than the value  α= 2/3  resulting from spherical collapse, even if there is no initial correlation between magnetic field and principal collapse directions. These analytic arguments go some way towards understanding the results of numerical simulations.