Pure Power Law Research Articles

On the Early‐Time X‐Ray Spectra ofSwiftAfterglows. I. Evidence for Anomalous Soft X‐Ray Emission

We have conducted a thorough and blind search for emission lines in >70 Swift X-ray afterglows of total exposure ~10^7s. We find that most afterglows are consistent with pure power-laws plus extinction. Significant outliers to the population exist at the 5-10% level and have anomalously soft, possibly thermal spectra. Four bursts are singled out via possible detections of 2-5 lines: GRBs 060218, 060202, 050822, and 050714B. Alternatively, a blackbody model with kT~0.1-0.5 keV can describe the soft emission in each afterglow. The most significant soft component detections in the full data set of ~2000 spectra correspond to GRB060218/SN2006aj, with line significances ranging up to \~20-sigma. A thermal plasma model fit to the data indicates that the flux is primarily due to L-shell transitions of Fe at ~ solar abundance. We associate (>4-sigma significant) line triggers in the 3 other events with K-shell transitions in light metals. We favor a model where the possible line emission in these afterglows arises from the mildly relativistic cocoon of matter surrounding the GRB jet as it penetrates and exits the surface of the progenitor star. The emitting material in each burst is at a similar distance \~10^12--10^13 cm, a similar density ~10^17 cm^-3, and subject to a similar flux of ionizing radiation. The lines may correlate with the X-ray flaring. For the blackbody interpretation, the soft flux may arise from break out of the GRB shock or plasma cocoon from the progenitor stellar wind, as recently suggested for GRB060218 (Campana et al. 2006). Due to the low z of GRB060218, bursts faint in Gamma-rays with fluxes dominated by this soft X-ray component could outnumber classical GRBs 100-1.

The Global Kinematics of the Globular Cluster M92

We report the determination of high-accuracy radial velocities for 306 members of the globular cluster M92 using the Hydra multiobject spectrograph on the WIYN telescope. We have concentrated on stars outside of the central region of the cluster, located up to 14.4' from the cluster center. Candidate members were selected for spectroscopy based on a photometric metallicity index determined from three-band Washington photometry, also obtained with the WIYN telescope. The median error in the velocities is 0.35 km s-1. We find the heliocentric radial velocity of the cluster to be -121.2 ± 0.3 km s-1. We have used an improved Bayesian analysis to determine the velocity dispersion profile of M92. The most probable profile is a cored power law with a scale radius of 2', a velocity dispersion at 1' of 6.3 km s-1, and an outer power law with a slope of -0.6. We have also reanalyzed the M15 radial velocities of Drukier et al. and find that a pure power law with a 1' velocity dispersion of 8 km s-1 and a slope of -0.5 and the combination of a power law with a slope of -0.4 and a scale of 7.5 km s-1 inside 9' and a dispersion of 4 km s-1 outside are equally likely. In both clusters there is evidence that the samples include escaping stars. We present results from a GRAPE-based N-body simulation of an isolated cluster that demonstrates this effect. We suggest additional tests to determine the relative importance of tidal heating and stellar ejection for establishing the velocity field in globular cluster halos.

Observations of TeV γ-rays from Mrk 421 during December 2005 to April 2006 with the TACTIC telescope

The TACTIC γ-ray telescope has observed Mrk 421 on 66 clear nights from December 07, 2005 to April 30, 2006, totalling ∼202 h of on-source observations. Here, we report the detection of flaring activity from the source at ⩾1 TeV energy and the time-averaged differential γ-ray spectrum in the energy range 1–11 TeV for the data taken between December 27, 2005 and February 07, 2006 when the source was in a relatively higher state as compared to the rest of the observation period. Analysis of this data spell, comprising ∼97 h reveals the presence of a ∼12.0 σ γ-ray signal with daily flux of >1 Crab unit on several days. A pure power law spectrum with exponent −3.11 ± 0.11 as well as a power law spectrum with an exponential cutoff ( Γ = −2.51 ± 0.26 and E 0 = (4.7 ± 2.1) TeV) are found to provide reasonable fits to the inferred differential spectrum within statistical uncertainties. We believe that the TeV light curve presented here, for nearly 5 months of extensive coverage, as well as the spectral information at γ-ray energies of >5 TeV provide a useful input for other groups working in the field of γ-ray astronomy.

Energy dependent γ-ray morphology in the pulsar wind nebula HESS J1825–137

Aims. We present results from deep $\gamma$-ray observations of the Galactic pulsar wind nebula HESS J1825-137 performed with the HESS array. Methods. Detailed morphological and spatially resolved spectral studies reveal the very high-energy (VHE) $\gamma$-ray aspects of this object with unprecedented precision. Results. We confirm previous results obtained in a survey of the Galactic Plane in 2004. The $\gamma$-ray emission extends asymmetrically to the south and south-west of the energetic pulsar PSR J1826-1334 , that is thought to power the pulsar wind nebula. The differential $\gamma$-ray spectrum of the whole emission region is measured over more than two orders of magnitude, from 270 GeV to 35 TeV, and shows indications for a deviation from a pure power law. Spectra have also been determined for spatially separated regions of HESS J1825-137. The photon indices from a power-law fit in the different regions show a softening of the spectrum with increasing distance from the pulsar and therefore an energy dependent morphology. Conclusions. This is the first time that an energy dependent morphology has been detected in the VHE $\gamma$-ray regime. The VHE $\gamma$-ray emission of HESS J1825-137 is phenomenologically discussed in the scenario where the $\gamma$-rays are produced by VHE electrons via Inverse Compton scattering. The high $\gamma$-ray luminosity of the source cannot be explained on the basis of constant spin-down power of the pulsar and requires higher injection power in past.

Infrared gluon propagator from lattice QCD: Results from large asymmetric lattices

The infrared limit of the lattice Landau gauge gluon propagator is studied. We show that the lattice data are compatible with the pure power law $({q}^{2}{)}^{2\ensuremath{\kappa}}$ solution of the Dyson-Schwinger equations. Using various lattice volumes, the infinite volume limit for the exponent $\ensuremath{\kappa}$ is measured. Although, the results allow $\ensuremath{\kappa}=0.498--0.525$, the lattice data favor $\ensuremath{\kappa}\ensuremath{\sim}0.52$, which would imply a vanishing zero momentum gluon propagator.

A detailed spectral and morphological study of the gamma-ray supernova remnant RX J1713.7–3946 with HESS

We present results from deep observations of the Galactic shell-type supernova remnant (SNR) RX J1713.7-3946 (also known as G347.3-0.5) conducted with the complete H.E.S.S. array in 2004. Detailed morphological and spatially resolved spectral studies reveal the very-high-energy (VHE -- Energies E > 100 GeV) gamma-ray aspects of this object with unprecedented precision. Since this is the first in-depth analysis of an extended VHE gamma-ray source, we present a thorough discussion of our methodology and investigations of possible sources of systematic errors. Gamma rays are detected throughout the whole SNR. The emission is found to resemble a shell structure with increased fluxes from the western and northwestern parts. The differential gamma-ray spectrum of the whole SNR is measured over more than two orders of magnitude, from 190 GeV to 40 TeV, and is rather hard with indications for a deviation from a pure power law at high energies. Spectra have also been determined for spatially separated regions of RX J1713.7-3946. The flux values vary by more than a factor of two, but no significant change in spectral shape is found. There is a striking correlation between the X-ray and the gamma-ray image. Radial profiles in both wavelength regimes reveal the same shape almost everywhere in the region of the SNR. The VHE gamma-ray emission of RX J1713.7-3946 is phenomenologically discussed for two scenarios, one where the gamma rays are produced by VHE electrons via Inverse Compton scattering and one where the gamma rays are due to neutral pion decay from proton-proton interactions. In conjunction with multi-wavelength considerations, the latter case is favoured. However, no decisive conclusions can yet be drawn regarding the parent particle population dominantly responsible for the gamma-ray emission from RX J1713.7-3946.

Exact distribution of Poisson shot noise with constant marks under power-law attenuation

Considered is a wireless network where nodes transmitting simultaneously at some common power are located at the points of a Poisson process in the infinite plane. In the case of a pure power-law path loss model, the probability distribution of the interference power sum at an arbitrary reference point is evaluated exactly. In particular, when the attenuation exponent equals 4, the distribution in closed form is obtained.

From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion

Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit anomalous diffusion. We consider here the case of subdiffusive processes, which correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to known fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.

Analysis of fragment size distributions in collisions of monocharged ions with the C60 molecule

Fragmentation of the C60 molecule is investigated using a multicorrelation technique. We first focus on the transition from asymmetrical dissociation (AD) to multifragmentation (MF). These processes are studied in collisions between H+x(x = 1–3) hydrogenic projectiles and C60 fullerene in the gas phase, in the 2–130 keV collisional energy range. A rather sharp transition from pure AD to predominant MF is observed when plotting the AD/(AD + MF) ratio against the average deposited energy Edep; it occurs in the 80–240 eV Edep range; this ratio is also found to be independent of the projectile species (scaling law). The evolution of the size distribution shape is also discussed and compared with other data available in the literature. A pure power law is never reached in the present experimental conditions. Finally, an event-by-event analysis of the fragmentation data is developed for the first time in the study of the C60 molecule fragmentation and discussed in terms of the predictions of the percolation model near a critical behaviour. Moments of order 2, 3 and 5 are determined for each correlation event. Moments of order 3 and 5 follow a linear behaviour when plotted against the moment of order 2, as predicted, and the exponent τ that is extracted takes a value near 2. The Campi scatter plot is also determined and discussed for total and multiplicity-selected events. Both slopes of the two branches in the Campi plots and τ value are near those that are expected in the percolation of a 2D lattice.

Scaling, cluster dynamics and complex oscillations in a multispecies Lattice Lotka–Volterra Model

The cluster formation in the cyclic (4+1)-Lattice Lotka–Volterra Model is studied by Kinetic Monte Carlo simulations on a square lattice support. At the Mean Field level this model demonstrates conservative four-dimensional oscillations which, depending on the parameters, can be chaotic or quasi-periodic. When the system is realized on a square lattice substrate the various species organize in domains (clusters) with fractal boundaries and this is consistent with dissipative dynamics. For small lattice sizes, the entire lattice oscillates in phase and the size distribution of the clusters follows a pure power law distribution. When the system size is large many independently oscillating regions are formed and as a result the cluster size distribution in addition to the power law, acquires a exponential decay dependence. This combination of power law and exponential decay of distributions and correlations is indicative, in this case, of mixing and superposition of regions oscillating asynchronously.

Detection of TeV gamma-rays from the BL Lac 1ES 1959+650 in its low states and during a major outburst in 2002

TeV γ-rays from the BL Lac object 1ES 1959+650 have been measured during the years 2000 and 2001 with a significance of 5.2 σ at a value of 5.3% of the Crab flux and in May 2002 during strong outbursts with >23 σ at a flux level of up to 2.2 Crab, making 1ES 1959+650 the TeV Blazar with the third best event statistics. The deep observation of 197.4 h has been performed with the HEGRA stereoscopic system of 5 imaging atmospheric Cherenkov telescopes (IACT system). 1ES 1959+650 is located at a redshift of z = 0.047, providing an intermediate distance between the nearby Blazars Mkn 421 and Mkn 501, and the much more distant object H1426+428. This makes 1ES 1959+650 an important member of the class of TeV Blazars in view of the absorption of TeV photons by the diffuse extragalactic background radiation (DEBRA). The differential energy spectrum of 1ES 1959+650 during the flares can be fitted by a power law with a spectral index of 2.83 ± 0.14stat ± 0.08sys or by a power law with an exponential cut-off at (4.2 +0.8 −0.6 stat ± 0.9sys) TeV and a spectral index of 1.83 ± 0.15stat ± 0.08sys. The low state differential energy spectrum obtained with lower statistics can be described by a pure power law with a spectral index of 3.18 ± 0.17stat ± 0.08sys.

DOES THE "KNEE" IN PRIMARY COSMIC RAY SPECTRUM EXIST?

The problem of the knee in primary cosmic ray at energy about 3–5 PeV is the most exciting problem in cosmic ray physics. Since 1958, physicists have been trying to solve this problem. In our opinion, the problem could be solved from the experimental point of view, whereas the primary spectrum would follow a pure power law. A key to the "knee" problem lies in the hadronic structure of EAS and its propagation in the Earth's atmosphere. Neither exotic processes nor new physics are used. An explanation of the approach and some results of Monte Carlo simulations are given below.

Stochastic models and statistical analysis for clock noise

In this primer we first give an overview of stochastic models that can be used to interpret clock noise. Because of their statistical tractability, we concentrate on fractionally differenced (FD) processes, which we relate to fractional Brownian motion, discrete fractional Brownian motion, fractional Gaussian noise and discrete pure power law processes. We discuss several useful extensions to FD processes, namely, composite FD processes, autoregressive fractionally integrated moving average processes and time-varying FD processes. We then consider the statistical analysis of clock noise in terms of how these models are manifested via the spectral density function (SDF) and the wavelet variance (WV), the latter being a generalization of the well-known Allan variance. Both the SDF and the WV decompose the process variance with respect to an independent variable (Fourier frequency for the SDF, and scale for the WV); similarly, judiciously chosen estimators of the SDF and WV decompose the sample variance of clock noise. We give an overview of the statistical properties of estimators of the SDF and the WV and show how these estimators can be used to deduce the parameters of stochastic models for clock noise.

Singular crack-tip fields for pressure sensitive plastic solids

Plain strain mode-I singular plastic fields are examined for cracks embedded in pressure sensitive solids. Material response is described by a small strain deformation theory in conjunction with elliptic yield criterion and plastic potential. Non-associativity is accounted for and a pure power law is assumed to characterize strain hardening. The material does not admit a strain energy function hence it is not possible to deducea-priori theJ-integral motivated stress singularities. A standard separation of variables representation of near-tip eigenfunctions has been evaluated numerically, over a range of material parameters. It has been found that stress singularities may deviate fromJ-integral predictions, with increasing non-associativity, by up to nearly 20%. Sample illustrations are provided for singular field profiles and some aspects of pressure sensitive non- associated plasticity are discussed.

Gamma‐Ray Burst Spectra and Light Curves as Signatures of a Relativistically Expanding Plasma

Temporal and spectral characteristics of prompt emission of gamma-ray burst (GRB) pulses are the primary observations for constraining the energizing and emission mechanisms. In spite of very complex temporal behavior of the GRBs, several patterns have been discovered in how some spectral characteristics change during the decaying phase of individual, well-defined long (greater than a few seconds) pulses. In this paper we compare these observed signatures with those expected from a relativistically expanding, shock-heated, and radiation-emitting plasma shell. Within the internal shock model and assuming a short cooling time, we show that the angular dependence in arrival time from a spherical expanding shell can explain the general characteristics of some well-defined long GRB pulses. This includes the pulse shape, with a fast rise and a slower decay, ∝(1 + t/τ)2, where τ is a time constant, and the spectral evolution, which can be described by the hardness-intensity correlation (HIC), with the intensity being proportional to the square of the hardness measured by the value of the peak, e.g., Ep of the νFν spectrum. A variation of the relevant timescales involved (the angular spreading and the dynamic) can explain the broad, observed dispersion of the HIC index. Reasonable estimates of physical parameters lead to situations where the HIC relation deviates from a pure power law, features that are indeed present in the observations. Depending on the relative values of the rise and decay times of the intrinsic light curve, the spectral/temporal behavior, as seen by an observer, will produce the hard-to-soft evolution and the so-called tracking pulses. In our model the observed spectrum is a superposition of many intrinsic spectra arriving from different parts of the fireball shell with varying spectral shifts. Therefore, it will be broader than the emitted spectrum and its spectral parameters could have complex relations with the intrinsic ones. Furthermore, we show that the softening of the low-energy power-law index, that has been observed in some pulses, can be explained by geometric effects and does not need to be an intrinsic behavior.

Generalized comparison theorems in quantum mechanics

This paper is concerned with the discrete spectra of Schrödinger operators H = −Δ + V, where V(r) is an attractive potential in N spatial dimensions. Two principal results are reported for the bottom of the spectrum of H in each angular-momentum subspace ℓ: (i) an optimized lower bound when the potential is a sum of terms V(r) = V(1)(r) + V(2)(r), and the bottoms of the spectra of −Δ + V(1)(r) and −Δ + V(2)(r) in ℓ are known, and (ii) a generalized comparison theorem which predicts spectral ordering when the graphs of the comparison potentials V(1)(r) and V(2)(r) intersect in a controlled way. Pure power-law potentials are studied and an application of the results to the Coulomb-plus-linear potential V(r) = −a/r + br is presented in detail: for this problem an earlier formula for energy bounds is sharpened and generalized to N dimensions.

Winners don't take all: Characterizing the competition for links on the web.

As a whole, the World Wide Web displays a striking "rich get richer" behavior, with a relatively small number of sites receiving a disproportionately large share of hyperlink references and traffic. However, hidden in this skewed global distribution, we discover a qualitatively different and considerably less biased link distribution among subcategories of pages-for example, among all university homepages or all newspaper homepages. Although the connectivity distribution over the entire web is close to a pure power law, we find that the distribution within specific categories is typically unimodal on a log scale, with the location of the mode, and thus the extent of the rich get richer phenomenon, varying across different categories. Similar distributions occur in many other naturally occurring networks, including research paper citations, movie actor collaborations, and United States power grid connections. A simple generative model, incorporating a mixture of preferential and uniform attachment, quantifies the degree to which the rich nodes grow richer, and how new (and poorly connected) nodes can compete. The model accurately accounts for the true connectivity distributions of category-specific web pages, the web as a whole, and other social networks.

X‐Ray Spectral Variability and Rapid Variability of the Soft X‐Ray Spectrum Seyfert 1 Galaxies Arakelian 564 and Ton S180

The bright, soft X-ray spectrum Seyfert 1 galaxies Ark 564 and Ton S180 were monitored for 35 days and 12 days, respectively, with ASCA and RXTE (and EUVE for Ton S180). These represent the most intensive X-ray monitoring of any such soft-spectrum Seyfert 1 to date. Light curves were constructed for Ton S180 in six bands spanning 0.1-10 keV and for Ark 564 in five bands spanning 0.7-10 keV. The short-timescale (hours-days) variability patterns were very similar across energy bands, with no evidence of lags between any of the energy bands studied. The fractional variability amplitude was almost independent of energy band, unlike hard-spectrum Seyfert 1 galaxies, which show stronger variations in the softer bands. It is difficult to simultaneously explain soft Seyfert galaxies stronger variability, softer spectra, and weaker energy dependence of the variability relative to hard Seyfert galaxies. There was a trend for soft- and hard-band light curves of both objects to diverge on the longest timescales probed (~weeks), with the hardness ratio showing a secular change throughout the observations. This is consistent with the fluctuation power density spectra that showed relatively greater power on long timescales in the softest bands. The simplest explanation of all of these is that two continuum emission components are visible in the X-rays: a relatively hard, rapidly variable component that dominates the total spectrum and a slowly variable soft excess that only shows up in the lowest energy channels of ASCA. Although it would be natural to identify the latter component with an accretion disk and the former with a corona surrounding it, a standard thin disk could not get hot enough to radiate significantly in the ASCA band, and the observed variability timescales are much too short. It also appears that the hard component may have a more complex shape than a pure power law. The most rapid factor of 2 flares and dips occurred within ~1000 s, in Ark 564 and a bit more slowly in Ton S180. The speed of the luminosity changes rules out viscous or thermal processes and limits the size of the individual emission regions to 15 Schwarzschild radii (and probably much less), that is, to either the inner disk or small regions in a corona.

Spatial variations of a passive tracer in a random wave field

The effect of random wave fields on passive tracer spatial variations is studied. We derive a closed-form expression for the spatial autocorrelation function (or power spectrum) of the tracer fluctuations that is quantitatively accurate so long as wave field nonlinearities are small. The theory is illustrated for the case of long internal gravity waves in the ocean. We find that even if the (rear face of the) spectrum of the advecting velocity field is a pure power law, the tracer spectrum has two separate power law subranges. Most important to oceanographic applications, in the larger scale subrange, the effective horizontal compressibility of the wave velocity field becomes a dominant factor of the tracer variations. In such cases, the concentration spectrum becomes approximately proportional to the spectrum of the wave potential energy. The latter, which decays with increasing wavenumber much more rapidly than that known for two-dimensional eddy turbulence, is confirmed by satellite observations in wave-dominated ocean regions. As an additional confirmation of the theory, we demonstrate the occurrence of spectral peaks at wavenumbers corresponding to the semi-diurnal tide frequency.

Non-Fermi-liquid behavior ind- andf-electron metals

A relatively new class of materials has been found in which the basic assumption of Landau Fermi-liquid theory---that at low energies the electrons in a metal should behave essentially as a collection of weakly interacting particles---is violated. These ``non-Fermi-liquid'' systems exhibit unusual temperature dependences in their low-temperature properties, including several examples in which the specific heat divided by temperature shows a singular $\mathrm{log}T$ temperature dependence over more than two orders of magnitude, from the lowest measured temperatures in the milliKelvin regime to temperatures over 10 K. These anomalous properties, with their often pure power-law or logarithmic temperature dependences over broad temperature ranges and inherent low characteristic energies, have attracted active theoretical interest from the first experimental report in 1991. This article first describes the various theoretical approaches to trying to understand the source of strong temperature- and frequency-dependent electron-electron interactions in non-Fermi-liquid systems. It then discusses the current experimental body of knowledge, including a compilation of data on non-Fermi-liquid behavior in over 50 systems. The disparate data reveal some interesting correlations and trends and serve to point up a number of areas where further theoretical and experimental work is needed.