Evolution of Massive Black Hole Binaries

We present the result of large-scale N-body simulations of the stellar-dynamical evolution of a massive black-hole binary at the center of a spherical galaxy. We focus on the dependence of the hardening rate on the relaxation timescale of the parent galaxy. A simple theoretical argument predicts that a binary black hole creates the ``loss cone'' around it. Once the loss cone is formed, the hardening rate is determined by the rate at which field stars diffuse into the loss cone. Therefore the hardening timescale becomes proportional to the relaxation timescale. Recent N-body simulations, however, have failed to confirm this theory and various explanations have been proposed. By performing simulations with sufficiently large N (up to $10^6$) for sufficiently long time, we found that the hardening rate does depend on N. Our result is consistent with the simple theoretical prediction that the hardening timescale is proportional to the relaxation timescale. This dependence implies that most massive black hole binaries are unlikely to merge within the Hubble time through interaction with field stars and gravitational wave radiation alone.Comment: Reviced version accepted for publication in ApJ. Scheduled to appear in the February 10, 2004 issu

Non-Poissonian level spacing statistics of classically integrable quantum systems based on the Berry-Robnik approach

Along the line of thoughts of Berry and Robnik\cite{[1]}, we investigated the gap distribution function of systems with infinitely many independent components, and discussed the level-spacing distribution of classically integrable quantum systems. The level spacing distribution is classified into three cases: Case 1: Poissonian if $\bar{\mu}(+\infty)=0$, Case 2: Poissonian for large $S$, but possibly not for small $S$ if $0<\bar{\mu}(+\infty)< 1$, and Case 3: sub-Poissonian if $\bar{\mu}(+\infty)=1$. Thus, even when the energy levels of individual components are statistically independent, non-Poisson level spacing distributions are possible.Comment: 5 pages, 0 figur

Long-Range Spectral Statistics of Classically Integrable Systems --Investigation along the Line of the Berry-Robnik Approach--

Extending the argument of Ref.\citen{[4]} to the long-range spectral statistics of classically integrable quantum systems, we examine the level number variance, spectral rigidity and two-level cluster function. These observables are obtained by applying the approach of Berry and Robnik\cite{[0]} and the mathematical framework of Pandey \cite{[2]} to systems with infinitely many components, and they are parameterized by a single function $\bar{c}$, where $\bar{c}=0$ corresponds to Poisson statistics, and $\bar{c}\not=0$ indicates deviations from Poisson statistics. This implies that even when the spectral components are statistically independent, non-Poissonian spectral statistics are possible.Comment: 13 pages, 4 figure

Long-Term Evolution of Massive Black Hole Binaries. II. Binary Evolution in Low-Density Galaxies

We use direct-summation N-body integrations to follow the evolution of binary black holes at the centers of galaxy models with large, constant-density cores. Particle numbers as large as 400K are considered. The results are compared with the predictions of loss-cone theory, under the assumption that the supply of stars to the binary is limited by the rate at which they can be scattered into the binary's influence sphere by gravitational encounters. The agreement between theory and simulation is quite good; in particular, we are able to quantitatively explain the observed dependence of binary hardening rate on N. We do not verify the recent claim of Chatterjee, Hernquist & Loeb (2003) that the hardening rate of the binary stabilizes when N exceeds a particular value, or that Brownian wandering of the binary has a significant effect on its evolution. When scaled to real galaxies, our results suggest that massive black hole binaries in gas-poor nuclei would be unlikely to reach gravitational-wave coalescence in a Hubble time.Comment: 13 pages, 8 figure

Time-Symmetrized Kustaanheimo-Stiefel Regularization

In this paper we describe a new algorithm for the long-term numerical integration of the two-body problem, in which two particles interact under a Newtonian gravitational potential. Although analytical solutions exist in the unperturbed and weakly perturbed cases, numerical integration is necessary in situations where the perturbation is relatively strong. Kustaanheimo--Stiefel (KS) regularization is widely used to remove the singularity in the equations of motion, making it possible to integrate orbits having very high eccentricity. However, even with KS regularization, long-term integration is difficult, simply because the required accuracy is usually very high. We present a new time-integration algorithm which has no secular error in either the binding energy or the eccentricity, while allowing variable stepsize. The basic approach is to take a time-symmetric algorithm, then apply an implicit criterion for the stepsize to ensure strict time reversibility. We describe the algorithm in detail and present the results of numerical tests involving long-term integration of binaries and hierarchical triples. In all cases studied, we found no systematic error in either the energy or the angular momentum. We also found that its calculation cost does not become higher than those of existing algorithms. By contrast, the stabilization technique, which has been widely used in the field of collisional stellar dynamics, conserves energy very well but does not conserve angular momentum.Comment: figures are available at http://grape.c.u-tokyo.ac.jp/~funato/; To appear in Astronomical Journal (July, 1996

Cluster Mass Estimate and a Cusp of the Mass Density Distribution in Clusters of Galaxies

We study density cusps in the center of clusters of galaxies to reconcile X-ray mass estimates with gravitational lensing masses. For various mass density models with cusps we compute X-ray surface brightness distribution, and fit them to observations to measure the range of parameters in the density models. The Einstein radii estimated from these density models are compared with Einstein radii derived from the observed arcs for Abell 2163, Abell 2218, and RX J1347.5-1145. The X-ray masses and lensing masses corresponding to these Einstein radii are also compared. While steeper cusps give smaller ratios of lensing mass to X-ray mass, the X-ray surface brightnesses estimated from flatter cusps are better fits to the observations. For Abell 2163 and Abell 2218, although the isothermal sphere with a finite core cannot produce giant arc images, a density model with a central cusp can produce a finite Einstein radius, which is smaller than the observed radii. We find that a total mass density profile which declines as $\sim r^{-1.4}$ produces the largest radius in models which are consistent with the X-ray surface brightness profile. As the result, the extremely large ratio of the lensing mass to the X-ray mass is improved from 2.2 to 1.4 for Abell 2163, and from 3 to 2.4 for Abell 2218. For RX J1347.5-1145, which is a cooling flow cluster, we cannot reduce the mass discrepancy.Comment: 23 pages, 10 figures, Latex, uses aasms4.sty, accepted for publication in ApJ, Part

A general framework for online audio source separation

We consider the problem of online audio source separation. Existing algorithms adopt either a sliding block approach or a stochastic gradient approach, which is faster but less accurate. Also, they rely either on spatial cues or on spectral cues and cannot separate certain mixtures. In this paper, we design a general online audio source separation framework that combines both approaches and both types of cues. The model parameters are estimated in the Maximum Likelihood (ML) sense using a Generalised Expectation Maximisation (GEM) algorithm with multiplicative updates. The separation performance is evaluated as a function of the block size and the step size and compared to that of an offline algorithm.Comment: International conference on Latente Variable Analysis and Signal Separation (2012

Reactive oxygen species induced by water containing nano-bubbles and its role in the improvement of barley seed germination

This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.The study of reactive oxygen species (ROS) generation caused by nano-bubbles (NBs) is of great importance for its application in both physiological activity promotion aspect and sterilization aspect. In this paper, Aminophenyl Fluorescein (APF) was used as a fluorescent reagent for the detection of ROS generation by NBs. The NBs scattering could cause the decrease of fluorescence intensity. Meanwhile, the quenching effect of oxygen could also cause the decrease of fluorescence intensity. Although the above two factors masked the fluorescence intensity generation by ROS, the fluorescence intensity of the water containing NBs still increased with NBs generation time, which proved that oxygen NBs could generate ROS. Moreover, the endogenous ROS in the barley seed cells were measured in the seed that germinated in the water containing NBs and the distilled water respectively. According to the results of seed staining experiments, both the microscope images and the absorbance at 560nm proved that the seeds dipped in the water containing NBs could generate more ROS compared to those in the distilled water. These findings greatly increase the NBs potential use both in agricultural field and environmental field

The dynamics of spiral arms in pure stellar disks

It has been believed that spirals in pure stellar disks, especially the ones spontaneously formed, decay in several galactic rotations due to the increase of stellar velocity dispersions. Therefore, some cooling mechanism, for example dissipational effects of the interstellar medium, was assumed to be necessary to keep the spiral arms. Here we show that stellar disks can maintain spiral features for several tens of rotations without the help of cooling, using a series of high-resolution three-dimensional $N$-body simulations of pure stellar disks. We found that if the number of particles is sufficiently large, e.g., $3\times 10^6$, multi-arm spirals developed in an isolated disk can survive for more than 10 Gyrs. We confirmed that there is a self-regulating mechanism that maintains the amplitude of the spiral arms. Spiral arms increase Toomre's $Q$ of the disk, and the heating rate correlates with the squared amplitude of the spirals. Since the amplitude itself is limited by the value of $Q$, this makes the dynamical heating less effective in the later phase of evolution. A simple analytical argument suggests that the heating is caused by gravitational scattering of stars by spiral arms, and that the self-regulating mechanism in pure-stellar disks can effectively maintain spiral arms on a cosmological timescale. In the case of a smaller number of particles, e.g., $3\times 10^5$, spiral arms grow faster in the beginning of the simulation (while $Q$ is small) and they cause a rapid increase of $Q$. As a result, the spiral arms become faint in several Gyrs.Comment: 18 pages, 19 figures, accepted for Ap