Single Power-law Distribution Research Articles

Analysis of Acoustic Emission Energy Distribution and Avalanche Dynamics of Sandstone with Different Particle Sizes.

The mechanical properties of sandstone have an important impact on the stability of the coal mine roof and floor, sandstone gas mining, and underground engineering safety. In order to study the critical characteristics on the failure process of sandstone with different particle sizes under uniaxial compression conditions, avalanche dynamics theory and a critical model are used to analyze the distribution of acoustic emission (AE) parameters, and the maximum likelihood estimation is used to accurately estimate the critical parameters. The results showed that the AE phenomenon of sandstone can be divided into four stages: initial compaction period, quiet period, crack stable growth period and outbreak period. During the process of compression failure, the larger the particle size is, the more seriously the sandstone is damaged. The AE energy probability density distribution follows single power-law distribution, and the AE energy critical exponent is 1.20 and follows the characteristics of scale-free regarding the power-law distribution on the particle sizes. When the stress runs up to 90% of peak stress, the bifurcation ratio increases sharply and shows the characteristics of the critical state. The waiting time and the avalanche size distribution follow double power-law distribution, and the inflection points are 0.03 and 37. Before and after the inflection point, the waiting time critical exponent and the avalanche size critical exponent are 1.90, 0.40 and 2.40, 1.60. This shows that the dynamic evolution process of sandstone under uniaxial compression condition can be characterized well by the fiber bundle model.

The Radio to GeV Afterglow of GRB 221009A

GRB 221009A (z = 0.151) is one of the closest known long γ-ray bursts (GRBs). Its extreme brightness across all electromagnetic wavelengths provides an unprecedented opportunity to study a member of this still-mysterious class of transients in exquisite detail. We present multiwavelength observations of this extraordinary event, spanning 15 orders of magnitude in photon energy from radio to γ-rays. We find that the data can be partially explained by a forward shock (FS) from a highly collimated relativistic jet interacting with a low-density, wind-like medium. Under this model, the jet’s beaming-corrected kinetic energy (E K ∼ 4 × 1050 erg) is typical for the GRB population. The radio and millimeter data provide strong limiting constraints on the FS model, but require the presence of an additional emission component. From equipartition arguments, we find that the radio emission is likely produced by a small amount of mass (≲6 × 10−7 M ⊙) moving relativistically (Γ ≳ 9) with a large kinetic energy (≳1049 erg). However, the temporal evolution of this component does not follow prescriptions for synchrotron radiation from a single power-law distribution of electrons (e.g., in a reverse shock or two-component jet), or a thermal-electron population, perhaps suggesting that one of the standard assumptions of afterglow theory is violated. GRB 221009A will likely remain detectable with radio telescopes for years to come, providing a valuable opportunity to track the full lifecycle of a powerful relativistic jet.

Great balls of FIRE – I. The formation of star clusters across cosmic time in a Milky Way-mass galaxy

ABSTRACT The properties of young star clusters formed within a galaxy are thought to vary in different interstellar medium conditions, but the details of this mapping from galactic to cluster scales are poorly understood due to the large dynamic range involved in galaxy and star cluster formation. We introduce a new method for modelling cluster formation in galaxy simulations: mapping giant molecular clouds (GMCs) formed self-consistently in a FIRE-2 magnetohydrodynamic galaxy simulation on to a cluster population according to a GMC-scale cluster formation model calibrated to higher resolution simulations, obtaining detailed properties of the galaxy’s star clusters in mass, metallicity, space, and time. We find $\sim 10{{\ \rm per\ cent}}$ of all stars formed in the galaxy originate in gravitationally bound clusters overall, and this fraction increases in regions with elevated Σgas and ΣSFR, because such regions host denser GMCs with higher star formation efficiency. These quantities vary systematically over the history of the galaxy, driving variations in cluster formation. The mass function of bound clusters varies – no single Schechter-like or power-law distribution applies at all times. In the most extreme episodes, clusters as massive as 7 × 106 M⊙ form in massive, dense clouds with high star formation efficiency. The initial mass–radius relation of young star clusters is consistent with an environmentally dependent 3D density that increases with Σgas and ΣSFR. The model does not reproduce the age and metallicity statistics of old ($\gt 11\rm Gyr$) globular clusters found in the Milky Way, possibly because it forms stars more slowly at z > 3.

Enhancing fracture network characterization: A data-driven, outcrop-based analysis

We utilize a pixel-based fracture detection algorithm to digitize 80 published outcrop maps of different scales at different locations. The key fracture properties, including fracture lengths, orientations, intensities, topological structures, clusters, and flow, are analyzed. Our findings provide significant justifications for statistical distributions used in SDFN modelings. We find that fracture lengths follow multiple (instead of single) power-law distributions with varying exponents. Large fractures tend to have large exponents, possibly because of a small coalescence probability. Most small-scale natural fracture networks have scattered orientations, corresponding to a small κ value (κ<3) in a von Mises–Fisher distribution. Large fracture systems analyzed in this research usually have more concentrated orientations with large κ values. Fracture intensities are spatially clustered at all scales. A fractal spatial density distribution, which introduces clustered fracture positions, can better capture the spatial clustering than a uniform distribution. Natural fracture networks usually have a significant proportion of T-type nodes, which is unavailable in conventional SDFN models. Thus, a rule-based algorithm is necessary to mimic the fracture growth and form T-type nodes. Most outcrop maps show good topological connectivity. However, sealing patterns and stress impact must be considered to evaluate the hydraulic conductivity of fracture networks.

Evolution of the Complex Supply Chain Network Based on Deviation from the Power-Law Distribution

The power-law distribution is an important descriptive characteristic of scale-free complex supply chain networks (SCN). The power-law distribution and deviation phenomena of SCN nodes are explored in combination with complex network theory, so it is imporant to accurately characterize the dynamic characteristics of network evolution on a time scale. Based on the analysis of the topological structure and evolutionary characteristics of the small-world network and scale-free SCN, the single and double power-law distribution and evolutionary dynamic characteristics of the complex SCN are further analyzed, and the deviation phenomenon of the power-law distribution is analyzed. On the premise of setting three parameters of new nodes, new edges, and node reconnection in the process of network evolution, the power-law distribution deviation evolution model under a complex network environment is constructed, and then the parameters of the SCN evolution model are analyzed. Combined with numerical simulation and model simulation, the evolution of SCN with two kinds of power-law deviation is analyzed. The results show that the deviation of the two-stage power-law distribution is not caused by the process of adding nodes or connecting edges, while it has a certain influence on the change of the power index, and the deviation of the power-law distribution in SCN increases with the extension of the evolution time. When p1=0, the single power-law distribution of SCN tends to change to a δ distribution when the time step is large enough.

Polydisperse streaming instability – III. Dust evolution encourages fast instability

ABSTRACT Planet formation via core accretion requires the production of kilometre-sized planetesimals from cosmic dust. This process must overcome barriers to simple collisional growth, for which the streaming instability (SI) is often invoked. Dust evolution is still required to create particles large enough to undergo vigorous instability. The SI has been studied primarily with single-size dust, and the role of the full evolved dust distribution is largely unexplored. We survey the polydisperse streaming instability (PSI) with physical parameters corresponding to plausible conditions in protoplanetary discs. We consider a full range of particle stopping times, generalized dust size distributions, and the effect of turbulence. We find that while the PSI grows in many cases more slowly with an interstellar power-law dust distribution than with a single size, reasonable collisional dust evolution, producing an enhancement of the largest dust sizes, produces instability behaviour similar to the monodisperse case. Considering turbulent diffusion, the trend is similar. We conclude that if fast linear growth of PSI is required for planet formation, then dust evolution producing a distribution with peak stopping times on the order of 0.1 orbits and an enhancement of the largest dust significantly above the single power-law distribution produced by a fragmentation cascade is sufficient, along with local enhancement of the dust to gas volume mass density ratio to order unity.

Double Power-Law in Leucosome Width Distribution: Implications for Recognizing Melt Movement in Migmatites

Leucosome sizes in migmatites has been shown to follow power-law distribution, which is indicative of self-organized criticality governing accumulation and transport of partial melts in anatexis. In our measurements of leucosome widths in the Olkiluoto migmatite in western Finland, we found double power-law distributions in addition to single power-law distributions. We divided the double power-law behaviors into those with positive and negative kinks separating the two power-law regimes. We propose that double power-law distributions in migmatites result from impediments on the bottom-up self-organized criticality that melt transport processes commonly display. These impediments can be caused by some leucosomes solidifying before others, in which case the leucosomes were effectively formed in two distinct events or phases. We also consider the emergence of double power-laws as a result of melt accumulation in and sudden melt removal from conduits of a specific size.

The flux distribution of Sgr A*

The Galactic center black hole Sagittarius A* is a variable near-infrared (NIR) source that exhibits bright flux excursions called flares. When flux from Sgr A* is detected, the light curve has been shown to exhibit red noise characteristics and the distribution of flux densities is non-linear, non-Gaussian, and skewed to higher flux densities. However, the low-flux density turnover of the flux distribution is below the sensitivity of current single-aperture telescopes. For this reason, the median NIR flux has only been inferred indirectly from model fitting, but it has not been directly measured. In order to explore the lowest flux ranges, to measure the median flux density, and to test if the previously proposed flux distributions fit the data, we use the unprecedented resolution of the GRAVITY instrument at the VLTI. We obtain light curves using interferometric model fitting and coherent flux measurements. Our light curves are unconfused, overcoming the confusion limit of previous photometric studies. We analyze the light curves using standard statistical methods and obtain the flux distribution. We find that the flux distribution of Sgr A* turns over at a median flux density of (1.1 ± 0.3) mJy. We measure the percentiles of the flux distribution and use them to constrain the NIR K-band spectral energy distribution. Furthermore, we find that the flux distribution is intrinsically right-skewed to higher flux density in log space. Flux densities below 0.1 mJy are hardly ever observed. In consequence, a single powerlaw or lognormal distribution does not suffice to describe the observed flux distribution in its entirety. However, if one takes into account a power law component at high flux densities, a lognormal distribution can describe the lower end of the observed flux distribution. We confirm the rms–flux relation for Sgr A* and find it to be linear for all flux densities in our observation. We conclude that Sgr A* has two states: the bulk of the emission is generated in a lognormal process with a well-defined median flux density and this quiescent emission is supplemented by sporadic flares that create the observed power law extension of the flux distribution.

An Argument for a Kilometer-scale Nucleus of C/2019 Q4

C/2019 Q4 (Borisov), discovered by Gennady Borisov on August 30, 2019, is tentatively an interstellar comet. We show that a kilometer-scale nucleus would provide consistency with both `Oumuamua and CNEOS-2014-01-08, resulting in a single power-law distribution with an equal amount of mass per logarithmic bin of interstellar objects.

Finding patterns in the degree distribution of real-world complex networks: going beyond power law

The most important structural characteristics in the study of large-scale real-world complex networks in pattern analysis are degree distribution. Empirical observations on the pattern of the real-world networks have led to the claim that their degree distributions follow, in general, a single power law. However, a closer observation, while fitting, shows that the single power-law distribution is often inadequate to meet the data characteristics properly. Since the degree distribution in the log–log scale actually displays, under inspection, two different slopes unlike what happens while fitting with the single power law. These two slopes with a transition in between closely resemble the pattern of the sigmoid function. This motivates us to derive a novel double power-law distribution for accurately modeling the real-world networks based on the sigmoid function. The proposed modeling approach further leads to the identification of a transition degree which, it has been demonstrated, may have a significant implication in analyzing the complex networks. The applicability, as well as effectiveness of the proposed methodology, is shown using rigorous experiments and also validated using statistical tests.

Olivine Grain Size Distributions in Faults and Shear Zones: Evidence for Nonsteady State Deformation

AbstractThe grain size distribution of deformed rocks may provide valuable information about their deformation history and the associated mechanisms. Here we present a unique set of olivine grain size distributions from ultramafic rocks deformed under a wide range of stress and strain rate conditions. Both experimentally deformed and naturally deformed samples are included. We observe a surprisingly uniform behavior, and most samples show power law grain size distributions. Convincing lognormal distributions across all scales were only observed for samples experimentally deformed at high temperature (1200 °C) and for some mantle‐deformed natural samples. Single power law distributions were observed for natural samples deformed by brittle mechanisms and by samples deformed experimentally in the regime of low‐temperature plasticity. Most natural samples show a crossover in power law scaling behavior near the median grain size from a steep slope for the larger grain fraction to a more gentle slope for the smaller grains. The small grain fraction shows a good data collapse when normalized to the crossover length scale. The associated power law slope indicates a common grain size controlling process. We propose a model that explains how such a scaling behavior may arise in the dislocation creep regime from the competition between the rate involved in the dislocation dynamics and the imposed strain rate. The common departure from lognormal distributions suggests that naturally deformed samples often have a deformation history that is far from a steady state scenario and probably reflects deformation under highly variable stress and strain rates.

Strength of the Solar Coronal Magnetic Field – A Comparison of Independent Estimates Using Contemporaneous Radio and White-Light Observations

We estimated the coronal magnetic field strength ($B$) during the 23 July 2016 coronal mass ejection (CME) event using i) the flux rope structure of the CME in the whitelight coronagraph images and ii) the band splitting in the associated type {\sc II} burst. No models were assumed for the coronal electron density ($N(r)$) used in the estimation.The results obtained using the above two independent methods correspond to different heliocentric distances ($r$) in the range $\approx$\,2.5\,--\,4.5$\rm R_{\odot}$, but they show excellent consistency and could be fitted with a single power-law distribution of the type $B(r)=5.7r^{-2.6}$ \textrm{G}, which is applicable in the aforementioned distance range. The power law index i.e. $-2.6$) is in good agreement with the results obtained in previous studies by different methods.

Anomalous Distributions of Primary Cosmic Rays as Evidence for Time-dependent Particle Acceleration in Supernova Remnants

Abstract Recent precise measurements of cosmic-ray (CR) spectra show that the energy distribution of protons is softer than those of heavier nuclei, and there are spectral hardenings for all nuclear compositions above ∼200 GV. Models proposed for these anomalies generally assume steady-state solutions of the particle acceleration process. We show that if the diffusion coefficient has a weak dependence on the particle rigidity near shock fronts of supernova remnants (SNRs), time-dependent solutions of the linear diffusive shock acceleration at two stages of SNR evolution can naturally account for these anomalies. The high-energy component of CRs is dominated by acceleration in the free expansion and adiabatic phases with enriched heavy elements and a high shock speed. The low-energy component may be attributed to acceleration by slow shocks propagating in dense molecular clouds with low metallicity in the radiative phase. Instead of a single power-law distribution, the spectra of time-dependent solutions soften gradually with the increase of energy, which may be responsible for the “knee” of CRs.

Statistical Analysis of Individual Solar Active Regions

AbstractIn the last decades, numerous observational and computational studies have shown that the global flare distribution is a power-law with a slope less than 2. In these studies, active regions are treated as statistically indistinguishable. To test this, we identify and separately analyze the flares produced by ten individual active regions (2006-2016). In five regions, we find a single power-law distribution, with a slope of a &lt; 2. In the other five, we find a broken double power-law distribution, with slopes a1 &lt; 2 and a2 &gt; 2.

Planetesimal Clearing and Size-dependent Asteroid Retention by Secular Resonance Sweeping during the Depletion of the Solar Nebula

Abstract The distribution of heavy elements is anomalously low in the asteroid main belt region compared with elsewhere in the solar system. Observational surveys also indicate a deficit in the number of small (≲50 km size) asteroids, which is two orders of magnitude lower than what is expected from the single power-law distribution that results from a collisional coagulation and fragmentation equilibrium. Here, we consider the possibility that a major fraction of the original asteroid population may have been cleared out by Jupiter’s secular resonance, as it swept through the main asteroid belt during the depletion of the solar nebula. This effect leads to the excitation of the asteroids’ orbital eccentricities. Concurrently, hydrodynamic drag and planet–disk tidal interaction effectively damp the eccentricities of sub-100 km-size and of super-lunar-size planetesimals, respectively. These combined effects lead to the asteroids’ orbital decay and clearing from the present-day main belt region (∼2.1–3.3 au). Eccentricity damping for the intermediate-size (50 to several hundreds of kilometers) planetesimals is less efficient than for small or large planetesimals. These objects therefore preferentially remain as main belt asteroids near their birthplaces, with modest asymptotic eccentricities. The smaller asteroids are the fragments of subsequent disruptive collisions at later times as suggested by the present-day asteroid families. This scenario provides a natural explanation for both the observed low surface density and the size distribution of asteroids in the main belt, without the need to invoke special planetesimal formation mechanisms. It also offers an explanation for the confined spatial extent of the terrestrial planet building blocks without the requirement of extensive migration of Jupiter, which is required in the grand-tack scenario.

The double power law in human collaboration behavior: The case of Wikipedia

Abstract We study human behavior in terms of the inter-event time distribution of revision behavior on Wikipedia, an online collaborative encyclopedia. We observe a double power law distribution for the inter-editing behavior at the population level and a single power law distribution at the individual level. Although interactions between users are indirect or moderate on Wikipedia, we determine that the synchronized editing behavior among users plays a key role in determining the slope of the tail of the double power law distribution.

HESS J1943+213: A NON-CLASSICAL HIGH-FREQUENCY-PEAKED BL LAC OBJECT

ABSTRACT HESS J1943+213 is an unidentified TeV source that is likely a high-frequency-peaked BL Lac (HBL) object, but that is also compatible with a pulsar wind nebula (PWN) nature. Each of these enormously different astronomical interpretations is supported by some of the observed unusual characteristics. In order to finally classify and understand this object, we took a three-pronged approach, through time-domain, high angular resolution, and multi-frequency radio studies. First, our deep time-domain observations with the Arecibo telescope failed to uncover the putative pulsar powering the proposed PWN. We conclude with ∼70% certainty that HESS J1943+213 does not host a pulsar. Second, long-baseline interferometry of the source with e-MERLIN at 1.5 and 5 GHz shows only a core, that is, a point source at –100 mas resolution. Its 2013 flux density is about one-third lower than that detected in the 2011 observations with similar resolution. This radio variability of the core strengthens the HBL object hypothesis. Third, additional evidence against the PWN scenario comes from the radio spectrum we compiled. The extended structure follows a power-law behavior with spectral index while the core component displays a flat spectrum ( ). In contrast, the radio synchrotron emission of PWNe predicts a single power-law distribution. Overall, we rule out the PWN hypothesis and conclude that the source is a BL Lac object. The consistently high fraction (70%) of the flux density from the extended structure then leads us to conclude that HESS J1943+213 must be a non-classical HBL object.

Proton activity of the Sun in current solar cycle 24**Supported by the National Natural Science Foundation of China.

We present a study of seven large solar proton events in the current solar cycle 24 (from 2009 January up to the current date). They were recorded by the GOES spacecraft with the highest proton fluxes being over 200 pfu for energies >10 MeV. In situ particle measurements show that: (1) The profiles of the proton fluxes are highly dependent on the locations of their solar sources, namely flares or coronal mass ejections (CMEs), which confirms the “heliolongitude rules” associated with solar energetic particle fluxes; (2) The solar particle release (SPR) times fall in the decay phase of the flare emission, and are in accordance with the times when the CMEs travel to an average height of 7.9 solar radii; and (3) The time differences between the SPR and the flare peak are also dependent on the locations of the solar active regions. The results tend to support the scenario of proton acceleration by the CME-driven shock, even though there exists a possibility of particle acceleration at the flare site, with subsequent perpendicular diffusion of accelerated particles in the interplanetary magnetic field. We derive the integral time-of-maximum spectra of solar protons in two forms: a single power-law distribution and a power law roll-over with an exponential tail. It is found that the unique ground level enhancement that occurred in the event on 2012 May 17 displays the hardest spectrum and the largest roll-over energy which may explain why this event could extend to relativistic energies.

Polarized light from Sagittarius A* in the near-infraredKs-band

We present a statistical analysis of polarized near-infrared (NIR) light from Sgr A*, the radio source associated with the supermassive black hole at the center of the Milky Way. The observations have been carried out using the adaptive optics instrument NACO at the VLT UT4 in the infrared $K_\mathrm{s}$-band from 2004 to 2012. Several polarized flux excursions were observed during these years. Linear polarization at 2.2 $\mu m$, its statistics and time variation, can be used constrain the physical conditions of the accretion process onto this supermassive black hole. With an exponent of about 4 for the number density histogram of fluxes above 5~mJy, the distribution of polarized flux density is closely linked to the single state power-law distribution of the total $K_\mathrm{s}$-band flux densities reported earlier. We find typical polarization degrees of the order of 20\%$\pm$10\% and a preferred polarization angle of 13$^o$$\pm$15$^o$. Simulations show the uncertainties under a total flux density of $\sim 2\,{\rm mJy}$ are probably dominated by observational effects. At higher flux densities there are intrinsic variations of polarization degree and angle within rather well constrained ranges. Since the emission is most likely due to optically thin synchrotron radiation, this preferred polarization angle we find is very likely coupled to the intrinsic orientation of the Sgr A* system i.e. a disk or jet/wind scenario associated with the super massive black hole. If they are indeed linked to structural features of the source the data imply a rather stable geometry and accretion process for the Sgr A* system.

A STATISTICAL ANALYSIS OF SEEDS AND OTHER HIGH-CONTRAST EXOPLANET SURVEYS: MASSIVE PLANETS OR LOW-MASS BROWN DWARFS?

We conduct a statistical analysis of a combined sample of direct imaging data, totalling nearly 250 stars. The stars cover a wide range of ages and spectral types, and include five detections ($\kappa$ And b, two $\sim$60 M$_{\rm J}$ brown dwarf companions in the Pleiades, PZ Tel B, and CD$-$35 2722B). For some analyses we add a currently unpublished set of SEEDS observations, including the detections GJ 504b and GJ 758B. We conduct a uniform, Bayesian analysis of all stellar ages using both membership in a kinematic moving group and activity/rotation age indicators. We then present a new statistical method for computing the likelihood of a substellar distribution function. By performing most of the integrals analytically, we achieve an enormous speedup over brute-force Monte Carlo. We use this method to place upper limits on the maximum semimajor axis of the distribution function derived from radial-velocity planets, finding model-dependent values of $\sim$30--100 AU. Finally, we model the entire substellar sample, from massive brown dwarfs to a theoretically motivated cutoff at $\sim$5 M$_{\rm Jup}$, with a single power law distribution. We find that $p(M, a) \propto M^{-0.65\pm0.60} a^{-0.85\pm0.39}$ (1$\sigma$ errors) provides an adequate fit to our data, with 1.0--3.1\% (68\% confidence) of stars hosting 5--70 $M_{\rm Jup}$ companions between 10 and 100 AU. This suggests that many of the directly imaged exoplanets known, including most (if not all) of the low-mass companions in our sample, formed by fragmentation in a cloud or disk, and represent the low-mass tail of the brown dwarfs.