Power-law Density Distribution Research Articles

The Correlations of Stellar Tidal Disruption Rates with Properties of Massive Black Holes and Their Host Galaxies

Stars can be either disrupted as tidal disruption events (TDEs) or swallowed whole by massive black holes (MBHs) at galactic centers when they approach sufficiently close to these MBHs. In this work, we investigate the correlations of such stellar consumption rates with both the MBH mass M BH and the inner slope of the host-galaxy mass density distribution α. We introduce a simplified analytical power-law model with a power-law stellar-mass density distribution surrounding MBHs and separate the contributions of two-body relaxation and stellar orbital precession for the stellar orbital angular momentum evolution in nonspherical galaxy potentials. The stellar consumption rates derived from this simplified model can be well consistent with the numerical results obtained with a more realistic treatment of stellar distributions and dynamics around MBHs, providing an efficient way to estimate TDE rates. The origin of the correlations of stellar consumption rates with M BH and α is explained by the dependence of this analytical model on those MBH/host-galaxy properties and by the separation of the stellar angular momentum evolution mechanisms. We propose that the strong positive correlation between the rates of stellar consumption due to two-body relaxation and α provides one interpretation for the overrepresentation of TDEs found in some rare E+A/poststarburst galaxies. We find high TDE rates for giant stars, up to those for solar-type stars. Understanding the origin of the correlations of the stellar consumption rates will be necessary for obtaining the demographics of MBHs and their host galaxies via TDEs.

Modeling Two First Hydrostatic Core Candidates Barnard 1b-N and 1b-S

A first hydrostatic core (FHC) is proposed to form after the initial collapse of a prestellar core, as a seed of a Class 0 protostar. FHCs are difficult to observe because they are small, compact, embedded, and short lived. In this work, we explored the physical properties of two well-known FHC candidates, B1-bN and B1-bS, by comparing interferometric data from Submillimeter Array (SMA) 1.1 and 1.3 mm and Atacama Large Millimeter/submillimeter Array (ALMA) 870 μm observations with simulated synthesis images of the two sources. The simulated images are based on a simple model containing a single, hot compact first-core-like component at the center surrounded by a large-scale, cold and dusty envelope described by a broken power-law density distribution with an index, α. Our results show that the hot compact components of B1-bN and B1-bS can be described by temperatures of ∼500 K with a size of ∼4 au, which are in agreement with theoretical predictions of an FHC. If the α inside the broken radii is fixed to −1.5, we find α ∼−2.9 and ∼−3.3 outside the broken radii for B1-bN and B1-bS, respectively, consistent with theoretical calculations of a collapsing, bounded envelope and previous observations. Comparing the density and temperature profiles of the two sources with radiation-hydrodynamic simulations of an FHC, we find both sources lie close to, but before, the second collapse stage. We suggest that B1-bS may have started the collapsing process earlier compared to B1-bN, since a larger discontinuity point is found in its density profile.

Environmental effects on the dynamical evolution of star clusters in turbulent molecular clouds

Context. Star clusters form within giant molecular clouds that are strongly altered by the feedback action of the massive stars, but the cluster still remains embedded in a dense, highly turbulent medium and interactions with ambient structures may modify its dynamical evolution from that expected if it were isolated. Aims. We aim to study coupling mechanisms between the dynamical evolution of the cluster, accelerated by the mass segregation process, with harassment effects caused by the gaseous environment. Methods. We simulated the cluster dynamical evolution combining N-body and hydrodynamic codes within the Astronomical Multipurpose Software Environment (AMUSE). Results. Tidal harassment produces a sparser configuration more rapidly than the isolated reference simulations. The evolution of the asymptotic power-law density distribution exponent also shows substantially different behaviour in the two cases. The background is more effective on clusters in advanced stages of dynamical development.

The Viscosity Parameter for Late-type Stable Be Stars

Abstract Using hydrodynamic principles we investigate the nature of the disk viscosity following the parameterization by Shakura & Sunyaev adopted for the viscous decretion model in classical Be stars. We consider a radial viscosity distribution including a constant value, a radially variable α assuming a power-law density distribution, and isothermal disks, for a late-B central star. We also extend our analysis by determining a self-consistent temperature disk distribution to model the late-type Be star 1 Delphini, which is thought to have a nonvariable, stable disk as evidenced by Hα emission profiles that have remained relatively unchanged for decades. Using standard angular momentum loss rates given by Granada et al., we find values of α of approximately 0.3. Adopting lower values of angular momentum loss rates, i.e., smaller mass loss rates, leads to smaller values of α. The values for α vary smoothly over the Hα emitting region and exhibit the biggest variations nearest the central star within about five stellar radii for the late-type, stable Be stars.

The possible equation of state of dark matter in low surface brightness galaxies * *Supported by the National Natural Science Foundation (NSF) of China (11973081, 11573062, 11403092, 11390374, 11521303), the YIPACAS Foundation (2012048), the Chinese Academy of Sciences (CAS, KJZD-EW-M06-01), the NSF of Yunnan Province (2019FB006) and the Youth Project of Western Light of CAS

The observed rotation curves of low surface brightness (LSB) galaxies play an essential role in studying dark matter, and indicate the existence of a central constant density dark matter core. However, the cosmological N-body simulations of cold dark matter predict an inner cusped halo with a power-law mass density distribution, and cannot reproduce a central constant-density core. This phenomenon is called cusp-core problem. When dark matter is quiescent and satisfies the condition for hydrostatic equilibrium, the equation of state can be adopted to obtain the density profile in the static and spherically symmetric space-time. To address the cusp-core problem, we assume that the equation of state is independent of the scaling transformation. Its lower order approximation for this type of equation of state can naturally lead to a special case, i.e., , where p and represent the pressure and density, respectively, depicts the rotation velocity of galaxy, and and are positive constants. It can obtain a density profile that is similar to the pseudo-isothermal halo model when is approximately 0.15. To obtain a more universally used model, let the equation of state include the polytropic model, i.e. , from which we can obtain other types of density profiles, such as the profile that is nearly same as the Burkert profile, where s and are positive constants.

Cosmic structures from a mathematical perspective 1: dark matter halo mass density profiles

The shapes of individual self-gravitating structures of an ensemble of identical, collisionless particles have remained elusive for decades. In particular, a reason why mass density profiles like the Navarro–Frenk–White or the Einasto profile are good fits to simulation- and observation-based dark matter halos has not been found. Given the class of three dimensional, spherically symmetric power-law probability density distributions to locate individual particles in the ensemble mentioned above, we derive the constraining equation for the power-law index for the most and least likely joint ensemble configurations. We find that any dark matter halo can be partitioned into three regions: a core, an intermediate part, and an outskirts part up to boundary radius r_mathrm {max}. The power-law index of the core is determined by the mean radius of the particle distribution within the core. The intermediate region becomes isothermal in the limit of infinitely many particles. The slope of the mass density profile far from the centre is determined by the choice of r_mathrm {max} with respect to the outmost halo particle, such that two typical limiting cases arise that explain the r^{-3}-slope for galaxy-cluster outskirts and the r^{-4}-slope for galactic outskirts. Hence, we succeed in deriving the mass density profiles of common fitting functions from a general viewpoint. These results also allow to find a simple explanation for the cusp-core-problem and to separate the halo description from its dynamics.

Systematic errors in strong gravitational lensing reconstructions, a numerical simulation perspective

ABSTRACT We present the analysis of a sample of 24 SLACS-like galaxy–galaxy strong gravitational lens systems with a background source and deflectors from the Illustris-1 simulation. We study the degeneracy between the complex mass distribution of the lenses, substructures, the surface brightness distribution of the sources, and the time delays. Using a novel inference framework based on Approximate Bayesian Computation, we find that for all the considered lens systems, an elliptical and cored power-law mass density distribution provides a good fit to the data. However, the presence of cores in the simulated lenses affects most reconstructions in the form of a Source Position Transformation. The latter leads to a systematic underestimation of the source sizes by 50 per cent on average, and a fractional error in H0 of around $25_{-19}^{+37}$ per cent. The analysis of a control sample of 24 lens systems, for which we have perfect knowledge about the shape of the lensing potential, leads to a fractional error on H0 of $12_{-3}^{+6}$ per cent. We find no degeneracy between complexity in the lensing potential and the inferred amount of substructures. We recover an average total projected mass fraction in substructures of fsub < 1.7–2.0 × 10−3 at the 68 per cent confidence level in agreement with zero and the fact that all substructures had been removed from the simulation. Our work highlights the need for higher resolution simulations to quantify the lensing effect of more realistic galactic potentials better, and that additional observational constraint may be required to break existing degeneracies.

Hills and holes in the microlensing light curve due to plasma environment around gravitational lens

ABSTRACT In this paper, we investigate the influence of the plasma surrounding the gravitational lens on the effect of microlensing. In presence of plasma around the lens, the deflection angle is determined by both the gravitational field of the lens and the chromatic refraction in the inhomogeneous plasma. We calculate microlensing light curves numerically for point-mass lens surrounded by power-law density distribution of plasma. A variety of possible curves is revealed, depending on the plasma density and frequency of observations. In the case of significant influence of plasma, the shape of microlensing light curve is strongly deformed in comparison with vacuum case. If the refractive deflection is large enough to compensate or to overcome the gravitational deflection, microlensing images can completely disappear for the observer. In this case, the remarkable effect occurs: formation of a ‘hole’ instead of a ‘hill’ in the center of microlensing light curve. Observational prospects of ‘hill-hole’ effect in different microlensing scenarios are discussed.

Dispersion of sausage waves in coronal waveguides with transverse density structuring

ABSTRACT We study dispersion properties of fast-sausage waves in a radially structured coronal magnetic tube with continuous radial density distribution. The models, containing either a non-uniform core or inhomogeneous external medium are considered. The dispersion relations are obtained for a power law density distribution in the corresponding non-uniform region, where the power-law index controls the steepness of the tube boundary. The governing wave equations with varying coefficients were solved with the Wentzel–Kramers–Brillouin (WKB) approximation. The model with the non-uniform core supports the existence of trapped and leaky sausage modes. The density non-uniformity in the core modifies the values of cut-off wave numbers kc. The smaller values of cut-offs, normalized to the effective tube radius r0, correspond to the smaller power index p. The wave dispersion (i.e. dVph/dk) decreases for smaller p. This occurs in the range of not too small longitudinal wave numbers k > kc. For the model, containing inhomogeneous environment the basic dispersion properties are generally identical to that for the monolithic tube model, studied in Lopin & Nagorny (2015b). The waves are trapped for all wave numbers, if the power-law index 0 < n < 2. There are both trapped and leaky regimes for n ≥ 2. The wave dispersion decreases for smaller n, in the range of the intermediate values of the longitudinal wave numbers k > kc. The seismological application of the obtained results is discussed.

Gravitational lensing around Kehagias\u2013Sfetsos compact objects surrounded by plasma

We study the optical properties of the Kehagias–Sfetsos (KS) compact objects, characterized by the “Hořava” parameter omega _{_{KS}}, in the presence of plasma, considering its homogeneous or power-law density distribution. The strong effects of both “Hořava” parameter omega _{_{KS}} and plasma on the shadow cast by the KS compact objects are demonstrated. Using the weak field approximation, we investigate the gravitational lensing effect. Strong dependence of the deflection angle of the light on both the “Hořava” and plasma parameter is explicitly shown. The magnification of image source due to the weak gravitational lensing is given for both the homogeneous and inhomogeneous plasma.

X-ray and SZ constraints on the properties of hot CGM

We use observations of stacked X-ray luminosity and Sunyaev-Zel'dovich (SZ) signal from a cosmological sample of $\sim 80,000$ and $104,000$ massive galaxies, respectively, with $ 10^{12.6}\lesssim M_{500} \lesssim 10^{13} M_{\odot}$ and mean redshift, \={z} $\sim$ 0.1 - 0.14 to constrain the hot Circumgalactic Medium (CGM) density and temperature. The X-ray luminosities constrain the density and hot CGM mass, while the SZ signal helps in breaking the density-temperature degeneracy. We consider a simple power-law density distribution ($n_e \propto r^{-3\beta}$) as well as a hydrostatic hot halo model, with the gas assumed to be isothermal in both cases. The datasets are best described by the mean hot CGM profile $\propto r^{-1.2}$, which is shallower than an NFW profile. For halo virial mass $\sim 10^{12}$ - $10^{13} M_{\odot}$, the hot CGM contains $\sim$ 20 - 30\% of galactic baryonic mass for the power-law model and 4 - 11\% for the hydrostatic halo model, within the virial radii. For the power-law model, the hot CGM profile broadly agrees with observations of the Milky Way. The mean hot CGM mass is comparable to or larger than the mass contained in other phases of the CGM for $L^*$ galaxies.

Constraining the H2 column density distribution at z ∼ 3 from composite DLA spectra

Abstract We present the detection of the average H2 absorption signal in the overall population of neutral gas absorption systems at z∼ 3 using composite absorption spectra built from the Sloan Digital Sky Survey-III damped Lyman α catalogue. We present a new technique to directly measure the H2 column density distribution function $f_{\rm H_2}(N)$ from the average H2 absorption signal. Assuming a power-law column density distribution, we obtain a slope $\beta = -1.29 \pm 0.06(\rm stat) \pm 0.10 (\rm sys)$ and an incidence rate of strong H2 absorptions [with N(H2) ≳ 1018 cm−2] to be $4.0 \pm 0.5(\rm stat) \pm 1.0 (\rm sys)\, \hbox{ per cent}$ in H i absorption systems with N(H i) ≥1020 cm−2. Assuming the same inflexion point where $f_{\rm H_2}(N)$ steepens as at z = 0, we estimate that the cosmological density of H2 in the column density range $\log N(\rm H_2) ({\rm cm}^{-2})= 18{\text{--}}22$ is ${\sim } 15\hbox{ per cent}$ of the total. We find one order of magnitude higher H2 incident rate in a sub-sample of extremely strong damped Lyman α absorption systems (DLAs) [$\log N(\rm{H\,\small {I}}) ({\rm cm}^{-2}) \ge 21.7$], which, together with the derived shape of $f_{\rm H_2}(N)$, suggests that the typical H i–H2 transition column density in DLAs is log N(H)(cm−2) ≳ 22.3 in agreement with theoretical expectations for the average (low) metallicity of DLAs at high-z.

Utsu aftershock productivity law explained from geometric operations on the permanent static stress field of mainshocks

Abstract. The aftershock productivity law is an exponential function of the form K∝exp(αM), with K being the number of aftershocks triggered by a given mainshock of magnitude M and α≈ln (10) being the productivity parameter. This law remains empirical in nature although it has also been retrieved in static stress simulations. Here, we parameterize this law using the solid seismicity postulate (SSP), the basis of a geometrical theory of seismicity where seismicity patterns are described by mathematical expressions obtained from geometric operations on a permanent static stress field. We first test the SSP that relates seismicity density to a static stress step function. We show that it yields a power exponent q = 1.96 ± 0.01 for the power-law spatial linear density distribution of aftershocks, once uniform noise is added to the static stress field, in agreement with observations. We then recover the exponential function of the productivity law with a break in scaling obtained between small and large M, with α=1.5ln (10) and ln (10), respectively, in agreement with results from previous static stress simulations. Possible biases of aftershock selection, proven to exist in epidemic-type aftershock sequence (ETAS) simulations, may explain the lack of break in scaling observed in seismicity catalogues. The existence of the theoretical kink, however, remains to be proven. Finally, we describe how to estimate the solid seismicity parameters (activation density δ+, aftershock solid envelope r∗ and background stress amplitude range Δo∗) for large M values.

Energy balance in a Z pinch with suppressed Rayleigh–Taylor instability

At present Z-pinch has evolved into a powerful plasma source of soft x-ray. This paper considers the energy balance in a radiating metallic gas-puff Z pinch. In this type of Z pinch, a power-law density distribution is realized, promoting suppression of Rayleigh–Taylor (RT) instabilities that occur in the pinch plasma during compression. The energy coupled into the pinch plasma, is determined as the difference between the total energy delivered to the load from the generator and the magnetic energy of the load inductance. A calibrated voltage divider and a Rogowski coil were used to determine the coupled energy and the load inductance. Time-gated optical imaging of the pinch plasma showed its stable compression up to the stagnation phase. The pinch implosion was simulated using a 1D two-temperature radiative magnetohydrodynamic code. Comparison of the experimental and simulation results has shown that the simulation adequately describes the pinch dynamics for conditions in which RT instability is suppressed. It has been found that the proportion of the Ohmic heating in the energy balance of a Z pinch with suppressed RT instability is determined by Spitzer resistance and makes no more than ten percent.

Self-Similar Solutions to Isothermal Shock Problems

We investigate exact solutions for isothermal shock problems in different one-dimensional geometries. These solutions are given as analytical expressions if possible, or are computed using standard numerical methods for solving ordinary differential equations. We test the numerical solutions against the analytical expressions to verify the correctness of all numerical algorithms. We use similarity methods to derive a system of ordinary differential equations (ODE) yielding exact solutions for power law density distributions as initial conditions. Further, the system of ODEs accounts for implosion problems (IP) as well as explosion problems (EP) by changing the initial or boundary conditions, respectively. Taking genuinely isothermal approximations into account leads to additional insights of EPs in contrast to earlier models. We neglect a constant initial energy contribution but introduce a parameter to adjust the initial mass distribution of the system. Moreover, we show that due to this parameter a constant initial density is not allowed for isothermal EPs. Reasonable restrictions for this parameter are given. Both, the (genuinely) isothermal implosion as well as the explosion problem are solved for the first time.

Millimeter-wave Line Ratios and Sub-beam Volume Density Distributions

Abstract We explore the use of mm-wave emission line ratios to trace molecular gas density when observations integrate over a wide range of volume densities within a single telescope beam. For observations targeting external galaxies, this case is unavoidable. Using a framework similar to that of Krumholz & Thompson, we model emission for a set of common extragalactic lines from lognormal and power law density distributions. We consider the median density of gas that produces emission and the ability to predict density variations from observed line ratios. We emphasize line ratio variations because these do not require us to know the absolute abundance of our tracers. Patterns of line ratio variations have the potential to illuminate the high-end shape of the density distribution, and to capture changes in the dense gas fraction and median volume density. Our results with and without a high-density power law tail differ appreciably; we highlight better knowledge of the probability density function (PDF) shape as an important area. We also show the implications of sub-beam density distributions for isotopologue studies targeting dense gas tracers. Differential excitation often implies a significant correction to the naive case. We provide tabulated versions of many of our results, which can be used to interpret changes in mm-wave line ratios in terms of adjustments to the underlying density distributions.

Income distribution in the Colombian economy from an econophysics perspective

Recently, in econophysics, it has been shown that it is possible to analyze economic systems as equilibrium thermodynamic models. We apply statistical thermodynamics methods to analyze income distribution in the Colombian economic system. Using the data obtained in random polls, we show that income distribution in the Colombian economic system is characterized by two specific phases. The first includes about 90% of the interviewed individuals, and is characterized by an exponential Boltzmann-Gibbs distribution. The second phase, which contains the individuals with the highest incomes, can be described by means of one or two power-law density distributions that are known as Pareto distributions.

Supernova blast waves in wind-blown bubbles, turbulent, and power-law ambient media

Supernova (SN) blast waves inject energy and momentum into the interstellar medium (ISM), control its turbulent multiphase structure and the launching of galactic outflows. Accurate modelling of the blast wave evolution is therefore essential for ISM and galaxy formation simulations. We present an efficient method to compute the input of momentum, thermal energy, and the velocity distribution of the shock-accelerated gas for ambient media with uniform (and with stellar wind blown bubbles), power-law, and turbulent density distributions. Assuming solar metallicity cooling, the blast wave evolution is followed to the beginning of the momentum conserving snowplough phase. The model recovers previous results for uniform ambient media. The momentum injection in wind-blown bubbles depend on the swept-up mass and the efficiency of cooling, when the blast wave hits the wind shell. For power-law density distributions with $n(r) \sim$ $r^{-2}$ (for $n(r) > n_{_{\rm floor}}$) the amount of momentum injection is solely regulated by the background density $n_{_{\rm floor}}$ and compares to $n_{_{\rm uni}}$ = $n_{_{\rm floor}}$. However, in turbulent ambient media with log-normal density distributions the momentum input can increase by a factor of 2 (compared to the homogeneous case) for high Mach numbers. The average momentum boost can be approximated as $p_{_{\rm turb}}/\mathrm{p_{_{0}}}\ =23.07\, \left(\frac{n_{_{0,\rm turb}}}{1\,{\rm cm}^{-3}}\right)^{-0.12} + 0.82 (\ln(1+b^{2}\mathcal{M}^{2}))^{1.49}\left(\frac{n_{_{0,\rm turb}}}{1\,{\rm cm}^{-3}}\right)^{-1.6}$. The velocity distributions are broad as gas can be accelerated to high velocities in low-density channels. The model values agree with results from recent, computationally expensive, three-dimensional simulations of SN explosions in turbulent media.

Liouville’s theorem and the canonical measure for nonconservative systems from contact geometry

Standard statistical mechanics of conservative systems relies on the symplectic geometry of the phase space. This is exploited to derive Hamilton’s equations, Liouville’s theorem and to find the canonical invariant measure. In this work we analyze the statistical mechanics of a class of nonconservative systems stemming from contact geometry. In particular, we find out the generalized Hamilton’s equations, Liouville’s theorem and the microcanonical and canonical measures invariant under the contact flow. Remarkably, the latter measure has a power law density distribution with respect to the standard contact volume form. Finally, we argue on the several possible applications of our results.

Herschel Hi-GAL imaging of massive young stellar objects

We used Herschel Hi-GAL survey data to determine whether massive young stellar objects (MYSOs) are resolved at 70$\mu$m and to study their envelope density distribution. Our analysis of three relatively isolated sources in the l=30{\deg} and l=59{\deg} Galactic fields show that the objects are partially resolved at 70$\mu$m. The Herschel Hi-GAL survey data have a high scan velocity which makes unresolved and partially resolved sources appear elongated in the 70$\mu$m images. We analysed the two scan directions separately and examine the intensity profile perpendicular to the scan direction. Spherically symmetric radiative transfer models with a power law density distribution were used to study the circumstellar matter distribution. Single dish sub-mm data were also included to study how different spatial information affects the fitted density distribution. The density distribution which best fits both the 70$\mu$m intensity profile and SED has an average index of ~0.5. This index is shallower than expected and is probably due to the dust emission from bipolar outflow cavity walls not accounted for in the spherical models. We conclude that 2D axisymmetric models and Herschel images at low scan speeds are needed to better constrain the matter distribution around MYSOs.