Power-law Cusp Research Articles

Phase-space simulations of prompt cusps: simulating the formation of the first haloes without artificial fragmentation

ABSTRACT The first generation of haloes forms from the collapse of the smallest peaks in the initial density field. N-body simulations of this process suggest a prompt formation of a steep power-law cusp, but these calculations are plagued by numerical artefacts that cast some doubt on this result. Here, we develop new simulation methods based on the dark matter phase-space sheet approach and present results that are entirely free of artificial clumps. We find that a cusp with density ρ ∝ r−1.5 is indeed formed promptly, subsequently accreting a more extended halo and participating in the hierarchical growth of later halo generations. However, our simulations also suggest that the presence of artificial clumps just before peak collapse can significantly shallow the inner profiles of the cusps. We use N-body simulations with controlled amounts of small-scale power to place a conservative upper limit on the scales affected by artificial clumps. Finally, we used these results to simulate the collapse of the first generation of peaks of various types and in different cosmologies, finding prompt cusps to form in all cases. We conclude that prompt cusps are a generic feature of the collapse of peaks on the free-streaming scale of the initial density field, and their structure can safely be studied using N-body simulations provided care is taken to excise the region potentially affected by artificial clumps.

EFFECTS OF INTERMEDIATE MASS BLACK HOLES ON NUCLEAR STAR CLUSTERS

Nuclear star clusters (NSCs) are dense stellar clusters observed in galactic nuclei, typically hosting a central massive black hole. Here we study the possible formation and evolution of NSCs through the inspiral of multiple star clusters hosting intermediate mass black holes (IMBHs). Using an N-body code we examine the dynamics of the IMBHs and their effects on the NSC. We find that IMBHs inspiral to the core of the newly formed NSC and segregate there. Although the IMBHs scatter each other and the stars, none of them is ejected from the NSC. The IMBHs are excited to high eccentricities and their radial density profile develops a steep power-law cusp. The stars also develop a power-law cusp (instead of the central core that forms in their absence), but with a shallower slope. The relaxation rate of the NSC is accelerated due to the presence of IMBHs, which act as massive-perturbers. This in turn fills the loss-cone and boosts the tidal disruption rate of stars both by the MBH and the IMBHs to a value excluded by rate estimates based on current observations. Rate estimates of tidal disruptions can therefore provide a cumulative constraint on the existence of IMBHs in NSCs.

Self-interacting dark matter cusps around massive black holes

We adopt the conduction fluid approximation to model the steady-state distribution of matter around a massive black hole at the center of a weakly collisional cluster of particles. By "`weakly collisional" we mean a cluster in which the mean free time between particle collisions is much longer than the characteristic particle crossing (dynamical) time scale, but shorter than the cluster lifetime. When applied to a star cluster, we reproduce the familiar Bahcall-Wolf power-law cusp solution for the stars bound to the black hole. Here the star density scales with radius as $r^{-7/4}$ and the velocity dispersion as $r^{-1/2}$ throughout most of the gravitational well of the black hole. When applied to a relaxed, self-interacting dark matter (SIDM) halo with a velocity-dependent cross section $\sigma \sim v^{-a}$, the gas again forms a power-law cusp, but now the SIDM density scales as $r^{-\beta}$, where $\beta = (a+3)/4$, while its velocity dispersion again varies as $r^{-1/2}$. Results are obtained first in Newtonian theory and then in full general relativity. Although the conduction fluid model is a simplification, it provides a reasonable first approximation to the matter profiles and is much easier to implement than a full Fokker-Planck treatment or an $N$-body simulation of the Boltzmann equation with collisional perturbations.

Liquid interfaces in viscous straining flows: numerical studies of the selective withdrawal transition

This paper presents a numerical analysis of the transition from selective withdrawal to viscous entrainment. In our model problem, an interface between two immiscible layers of equal viscosity is deformed by an axisymmetric withdrawal flow, which is driven by a point sink located some distance above the interface in the upper layer. We find that steady-state hump solutions, corresponding to selective withdrawal of liquid from the upper layer, cease to exist above a threshold withdrawal flux, and that this transition corresponds to a saddle-node bifurcation for the hump solutions. Numerical results on the shape evolution of the steady-state interface are compared against previous experimental measurements. We find good agreement where the data overlap. However, the larger dynamic range of the numerical results allows us to show that the large increase in the curvature of the hump tip near transition is not consistent with an approach towards a power-law cusp shape, an interpretation previously suggested from inspection of the experimental measurements alone. Instead, the large increase in the curvature at the hump tip reflects a robust trend in the steady-state interface evolution. For large deflections, the hump height is proportional to the logarithm of the curvature at the hump tip; thus small changes in hump height correspond to large changes in the value of the hump curvature.

Gemini andHubble Space TelescopeEvidence for an Intermediate‐Mass Black Hole in ω Centauri

The globular cluster ω Centauri is one of the largest and most massive members of the galactic system. However, its classification as a globular cluster has been challenged making it a candidate for being the stripped core of an accreted dwarf galaxy; this together with the fact that it has one of the largest velocity dispersions for star clusters in our galaxy makes it an interesting candidate for harboring an intermediate-mass black hole. We measure the surface brightness profile from integrated light on an HST ACS image of the center, and find a central power-law cusp of logarithmic slope –0.08. We also analyze Gemini GMOS-IFU kinematic data for a 5'' × 5'' field centered on the nucleus of the cluster, as well as for a field 14'' away. We detect a clear rise in the velocity dispersion from 18.6 km s−1 at 14'' to 23 km s−1 in the center. A rise in the velocity dispersion could be due to a central black hole, a central concentration of stellar remnants, or a central orbital structure that is radially biased. We discuss each of these possibilities. An isotropic, spherical dynamical model implies a black hole mass of 4.0−1.0+0.75 × 104 M☉, and excludes the no black hole case at greater than 99% significance. We have also run flattened, orbit-based models and find similar results. While our preferred model is the existence of a central black hole, detailed numerical simulations are required to confidently rule out the other possibilities.

Central Dynamics of Globular Clusters: the Case for a Black Hole in ω Centauri

AbstractThe globular cluster ω Centauri is one of the largest and most massive members of the Galactic system. Its classification as a globular cluster has been challenged making it a candidate for being the stripped core of an accreted dwarf galaxy; this and the fact that it has one of the largest velocity dispersions for star clusters in our galaxy makes it an interesting candidate for harboring an intermediate mass black hole. We measure the surface brightness profile from integrated light on an HST/ACS image, and find a central power-law cusp of logarithmic slope -0.08. We also analyze Gemini GMOS-IFU kinematic data for a 5”x5” field centered on the nucleus of the cluster, as well as for a field 14″ away. We detect a clear rise in the velocity dispersion from 18.6 kms−1 at 14″ to 23 kms−1 in the center. Given the very large core in ω Cen (2.58'), an increase in the dispersion in the central 10″ is difficult to attribute to stellar remnants, since it requires too many dark remnants and the implied configuration would dissolve quickly given the relaxation time in the core. However, the increase could be consistent with the existence of a central black hole. Assuming a constant M/L for the stars within the core, the dispersion profile from these data and data at larger radii implies a black hole mass of 4.0+0.75−1.0×104M⊙. We have also run flattened, orbit-based models and find a similar mass. In addition, the no black hole case for the orbit model requires an extreme amount of radial anisotropy, which is difficult to preserve given the short relaxation time of the cluster.

The tidal disruption rate in dense galactic cusps containing a supermassive binary black hole

We consider the problem of tidal disruption of stars in the centre of a galaxy containing a supermassive binary black hole with unequal masses. We assume that over the separation distance between the black holes, the gravitational potential is dominated by the more massive primary black hole. Also, we assume that the number density of stars is concentric with the primary black hole and has a power-law cusp. We show that the bulk of stars with a small angular-momentum component normal to the black hole binary orbit can reach a small value of total angular momentum through secular evolution in the gravitational field of the binary, and hence they can be tidally disrupted by the larger black hole. This effect is analogous to the so-called Kozai effect well known in celestial mechanics. We develop an analytical theory for the secular evolution of the stellar orbits and calculate the rate of tidal disruption. We compare our analytical theory with a simple numerical model and find very good agreement. Our results show that for a primary black hole mass of ∼ 1.0 6 -10 7 M ○. , the black hole mass-ratio q > 10 -2 , cusp size ∼1 pc, the tidal disruption rate can be as large as ∼10 -2 -1 M ○. yr -1 . This is at least 10 2 -10 4 times larger than estimated for the case of a single supermassive black hole. The duration of the phase of enhanced tidal disruption is determined by the dynamical-friction time-scale, and it is rather short: ∼10 5 yr. The dependence of the tidal disruption rate on the mass ratio, and on the size of the cusp, is also discussed.

Evidence for a New Elliptical-Galaxy Paradigm: Srsic and Core Galaxies

We fitted the surface-brightness profiles of 21 elliptical galaxies using both the function and a new empirical model that combines an inner power with an outer function. The profiles are combinations of deconvolved Hubble Space Telescope (HST) profiles from the literature and ellipse fits to the full WFPC2 mosaic images and thus span a radial range from ~002 to about twice the half-light radius. We are able to accurately fit the entire profiles using either the function or our new model. In doing so, we demonstrate that most, if not all, so-called galaxies are better described as Sersic galaxies—they are well modeled by the three-parameter profile into the limits of HST resolution—and that galaxies are best understood as consisting of an outer profile with an inner power-law cusp, which is a downward deviation from the inward extrapolation of the profile. This definition of cores resolves ambiguities that result when the popular law is fitted to the profiles of ellipticals and bulges, particularly at lower luminosities. We also find that using the Nuker to model core-galaxy nuclear profiles systematically overestimates the core radii by factors of 1.5–4.5 and underestimates the inner power-law slope by ~20%–40% or more.

The Galactic Halo and CDM

This paper reviews the available information on the central density distribution and shape of the Milky Way's halo. At present, there is no strong evidence that the Milky Way's halo properties conflict with the predictions of cold dark matter (CDM): a primordial central power law cusp can be accommodated by the observations, and the current constraints on flattening are also consistent with the predictions of the theory. If you want to pick a fight with CDM, then the Milky Way is probably not the place to do it.

On the formation rate of close binaries consisting of a super-massive black hole and a white dwarf

The formation rate of a close binary consisting of a super-massive black hole and a compact object (presumably a white dwarf) in galactic cusps is calculated with the help of the so-called loss cone approximation. For a power-law cusp of radius ra, the black hole mass M ∼ 10 6 M� , and the fraction of the compact objects δ ∼ 0.1, this rate ˙ Nwd ∼ 4 · 10 −5 K ( p) [( GM )/(r 3 )] ≈ 3 × 10 −9 K ( p)(M/10 6 M� ) 1/2 (ra/1 pc) −3/2 yr −1 . The function K ( p) depends on parameter p determining the cusp profile, and for the standard cusp profiles with p = 1/4, K ( p) ∼ 2. We estimate the probability Pr of finding of a compact object orbiting around a black hole with the period P in one particular galaxy to be Pr ∼ 10 −7 [(P/10 3 s)/(M/10 6 M� )] 8/3 [(M/10 6 M� )/(ra/1 pc)] 3/2 . The object with the period P ∼ 10 3 s emits gravitational waves with amplitude sufficient to be detected by the LISAtype gravitational wave antenna from the distance ∼10 3 Mpc. Based on estimates of masses of super-massive black holes in nearby galaxies, we speculate that LISA would detect several such events during its mission.

Millisecond Pulsars as Probes of Mass Segregation in the Galactic Center

We propose a simple test for the existence of a cluster of black hole remnants around Sgr A* that is based on a small sample of any type of Galactic center objects, provided they are substantially less massive than the black holes and constitute part of an old (≳1 Gyr) population. The test relies on the fact that, in the presence of such a cluster of heavy remnants and because of energy equipartition, lower mass objects would be expelled from the central regions and settle into a distribution very different from the cusp expected to be induced by the supermassive black hole alone. We show that with a sample of just 50 objects and using only their angular positions on the sky relative to Sgr A*, it is possible to clearly differentiate between a distribution consistent with the presence of the cluster of black holes and a power-law cusp distribution. We argue that millisecond pulsars might currently be the best candidate to perform this test, because of the large uncertainties involved in the age determination of less exotic objects. In addition, by measuring their first and second period derivatives, millisecond pulsars offer the rare opportunity to determine the complete phase-space information of the objects. We show that this extra information improves the detection of mass segregation by about 30%.

Distribution Functions for Cuspy Dark Matter Density Profiles

Various analytic expressions have been proposed for the density profile of dark matter halos. We consider six of these expressions for which the density profile has a power-law falloff ρ ∝ r-3 at large radii and a power-law cusp ρ ∝ r-γ (γ = 0, , 1, ) at small radii. The phase-space distribution function for these models is calculated assuming spherical symmetry and either an isotropic velocity dispersion tensor or an anisotropic dispersion tensor of the type proposed by Osipkov and Merritt. The differential energy distribution for these models is also derived. Several applications are discussed, including the analysis of dark matter search experiments and the study of halo formation in a cosmological setting. Analytic fitting formulae for some of the models are provided.

Numerical Stability of a Family of Osipkov‐Merritt Models

We have investigated the stability of a set of nonrotating anisotropic spherical models with a phase-space distribution function of the Osipkov-Merritt type. The velocity distribution in these models is isotropic near the center and becomes radially anisotropic at large radii. The models are special members of the family studied by Dehnen and by Tremaine et al. in which the mass density has a power-law cusp ρ ∝ r-γ at small radii and decays as ρ ∝ r-4 at large radii. The radial-orbit instability of models with γ = 0, 1/2, 1, 3/2, and 2 was studied using an N-body code written by one of us and based on the self-consistent field method developed by Hernquist & Ostriker. These simulations have allowed us to delineate a boundary in the (γ, ra)-plane that separates the stable from the unstable models. This boundary is given by 2Tr/Tt = 2.31 ± 0.27 for the ratio of the total radial to tangential kinetic energy. We also found that the stability criterion df/dQ ≤ 0, recently raised by Hjorth, gives lower values compared with our numerical results. The stability to radial modes of some Osipkov-Merritt γ-models that fail to satisfy the Doremus-Feix criterion ∂f/∂E < 0 has been studied using the same N-body code but retaining only the l = 0 terms in the potential expansion. We have found no signs of radial instabilities for these models.

Chaos and the Shapes of Elliptical Galaxies

Hubble Space Telescope observations reveal that the density of stars in most elliptical galaxies rises toward the center in a power-law cusp. Many of these galaxies also contain central dark objects, possibly supermassive black holes. The gravitational force from a steep cusp or black hole will destroy most of the box orbits that constitute the “backbone” of a triaxial stellar system. Detailed modeling demonstrates that the resulting chaos can preclude a self-consistent, strongly triaxial equilibrium. Most elliptical galaxies may therefore be nearly axisymmetric, either oblate or prolate.

HST observations of the core of the globular cluster NGC 6624

We have used the Faint Object Camera (FOC) on the Hubble Space Telescope (HST) to observe three fields near the center of the post-collapse globular cluster NGC 6624 in B and V. We use individual stellar positions and a maximum-likelihood technique to measure a position for the cluster center. We then transfer this position and the position of the x-ray source 4U 1820-30 to right ascension and declination, and correct an error of 1.8 sec in an earlier paper. From star counts, we obtain a density profile, with a power-law cusp of slope -0.84 +/- 0.16. No flat core is evident. We construct color-magnitude diagrams down to the main-sequence turnoff near the cluster center, and more than three magnitudes below the turnoff in the outermost field at r = 28 sec. The former diagram reveals a central population of blue stragglers, which has not previously been observed in this cluster. These blue stragglers are quite similar in specific frequency to those recently observed by HST in the core of 47 Tuc, and their presence adds to the growing evidence of stellar interactions in dense globular-cluster cores.

Velocity profiles of Osipkov–Merritt models

A simple algorithm is presented for the calculation of the projected line-of-sight velocity profiles (VPs) of non-rotating anisotropic spherical stellar dynamical models with a phase-space distribution function of the Osipkov-Merritt type. The velocity distribution in these models is isotropic inside the anisotropy radius r_a and becomes increasingly radially biased at larger radii. VP shape parameters are presented for a family of models in which the luminous mass density has an r^{-gamma} power-law cusp at small radii and an r^{-4} power-law fall-off at large radii. Self-consistent models and models in which the luminous matter is embedded in a dark halo are discussed. The effects of changes in the cusp slope gamma and in the anisotropy radius r_a are documented, and the area in the (gamma,r_a)-plane that contains physical models is delineated. The shapes of the VPs of the models show a considerable (and observable) variation with projected galactocentric radius. These models will be useful for interpreting the data on the VP shapes of elliptical galaxies that are now becoming available.

Globular cluster photometry with the Hubble Space Telescope. 2: U, V, and I measurements of M15

The projected density distribution of resolved stars near the center of M15 is shown to be consistent with either a power-law cusp N(r) approximately r(exp alpha), with alpha approximately -0.85 +/- 0.2, or with a King model with a core of radius approximately less than 2 sec. The inferred slope is in agreement with the theoretical value, alpha = -0.75, calculated by Bahcall and Wolf for the distribution of equal-mass stars surrounding a massive black hole and is also consistent with the radial profile expected from core collapse without a central black hole. The object AC 214 is a candidate for the central density cusp. Analysis of Monte Carlo simulations of the diffuse light indicates that, using current analysis techniques and available data, the residual light is not a reliable indicator of the true density distribution. This is contrary to earlier work. Photometric measurements in V and I of more than 5 x 10(exp 3) stars (and in U, V, and I of approximately greater than 1500 stars) are used to construct color-magnitude diagrams in the central 1 min of M15. Fourteen blue straggler candidates are identified in the inner 20 sec. The central color gradient noticed by previous researchers is caused by a central depletion of bright red giant stars rather than an excess of blue stragglers or blue horizontal branch stars.

Core structure of M32 from seeing-deconvolved images

view Abstract Citations (11) References (42) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Core Structure of M32 From Seeing-Deconvolved Images Bendinelli, O. ; Zavatti, F. ; Parmeggiani, G. ; Djorgovski, S. Abstract We present results of two-color (Gunn-Thuan gr) CCD surface photometry and seeing deconvolution of the central region of M32 (NGC 221), using the regularized multi-Gaussian (RMG) method, a powerful nonlinear technique. The galaxy core remains essentially unresolved at the limit of our data, i.e., the core radius is r_c_<= 0.42 arcsec ~1.4 pc. We also present a deprojected luminosity density profile of the galaxy, from a multi-Gaussian fit to a combined profile from our own data, and the profile measured at large radii by Kent [AJ, 94, 306(1987)]. We obtain for the central luminosity density ~1.74x10^5^L_sun_v_/pc^3^, which should be regarded as a lower limit. Using the published measurements of the velocity dispersion, in the King-Minkowski approximation we obtain for the central (M/L_v_)~3, which is also probably a lower limit. Assuming this value of (M/L_v_), we derive the central relaxation time t_rc_~2 x 10^9^ yr, which should also probably be regarded as an upper limit. The surface brightness profile near the center is represented very well with a power-law cusp of the slope of - 1.2, which is characteristic of some theoretical models of post-core collapse stellar systems. It is thus plausible that normal dynamical evolution over the Hubble time, possibly accelerated by the tidal interaction with M31, has lead to the core collapse in M32, and even to the formation of a central black hole. A color gradient appears at radii <= l arcsec, with the center being redder, and with a roughly constant (g - r) color outside that radius; a similar effect has been seen independently by Michard & Nieto [A&AL, 243, L17 (1991)]. This may be caused by a compact dust feature near the center. Publication: The Astronomical Journal Pub Date: January 1992 DOI: 10.1086/116045 Bibcode: 1992AJ....103..110B Keywords: Elliptical Galaxies; Galactic Nuclei; Galactic Structure; Seeing (Astronomy); Brightness Distribution; Charge Coupled Devices; Mass To Light Ratios; Spatial Resolution; Astrophysics; GALAXIES: ELLIPTICAL; GALAXIES: INDIVIDUAL: M32; GALAXIES: NUCLEI full text sources ADS | data products SIMBAD (2) NED (1)

The postcollapse core of M15 imaged with the HST planetary camera

It is shown here that, despite the severe spherical aberration present in the HST, the Wide Field/Planetary Camera (WFPC) images still present useful high-resolution information on M15, the classic candidate for a cluster with a collapsed core. The stars in M15 have been resolved down to the main-sequence turnoff and have been subtracted from the images. The remaining faint, unresolved stars form a diffuse background with a surprisingly large core with r(c) = 0.13 pc. The existence of a large core interior to the power-law cusp may imply that M15 has evolved well past maximum core collapse and may rule out the presence of a massive central black hole as well.

The Structure of Collapsed Cluster Cores

In order to test the prediction that many Galactic globular clusters have undergone core collapse (Lightman 1982, Cohn and Hut 1984) and should therefore have central surface brightness cusps, we have obtained UBVR CCD frames of the cores of 72 clusters. We present and analyze U-band surface brightness profiles for three clusters: one “control cluster” with a normal flat core profile — NGC 6388 — and two with central power law cusps — NGC 6624 and M15 (NGC 7078).