Fractal-diffusive Self-organized Criticality Research Articles

Scale-invariant Features of X-Ray Bursts from SGR J1935+2154 Detected by Insight-HXMT

In this work, we restudy the scale-invariant features of X-ray bursts from the soft gamma repeater (SGR) J1935+2154. To compare with previous studies, we choose 75 bursts from a dedicated 33 days-long observation carried out by Insight-HXMT. We investigate the size difference distributions of net counts, duration, and waiting time. It is found that the cumulative difference distributions of net counts and duration follow the q-Gaussian models with approximately steady q-values, confirming that the scale-invariant features exist in X-ray bursts of SGR J1935+2154. Regarding the varying results of waiting time reported by Sang & Lin and Wei et al, we find that the distributions of waiting time can be well described by the q-Gaussian model. Furthermore, the q-values of waiting time remain relatively stable at the 3σ confidence level, corroborating the scale invariance in the X-ray bursts. Additionally, we note that there is no significant q-value evolution across three Insight-HXMT telescopes. These findings statistically affirm that the X-ray bursts from SGR J1935+2154 can be attributed to an fractal-diffusive self-organized criticality system with a plausible Euclidean spatial dimension S = 3, implying that X-ray bursts from SGR J1935+2154 and associated astrophysical phenomena may share a similar magnetically dominated stochastic process.

Automated detection and analysis of coronal active region structures across solar cycle 24

ABSTRACT Observations from NASA’s Solar Dynamic Observatory Atmospheric Imaging Assembly were employed to investigate targeted physical properties of coronal active region structures across the majority of solar cycle 24 (From 2010 May to end of 2020 December). This is the largest consistent study to date which analyses emergent trends in structural width, location, and occurrence rate by performing an automatic and long-term examination of observable coronal and chromospheric limb features within equatorial active region belts across four extreme ultraviolet wavelengths (171, 193, 211, and 304 Å). This has resulted in over 30 000 observed coronal structures and hence allows for the production of spatial and temporal distributions focused upon the rise, peak, and decay activity phases of solar cycle 24. Employing a self-organized-criticality approach as a descriptor of coronal structure formation, power-law slopes of structural widths versus frequency are determined, ranging from -1.6 to -3.3 with variations of up to 0.7 found between differing periods of the solar cycle, compared to a predicted Fractal Diffusive Self-Organized Criticality (FD-SOC) value of -1.5. The North–South hemispheric asymmetry of these structures was also examined with the Northern hemisphere exhibiting activity that is peaking earlier and decaying slower than the Southern hemisphere, with a characteristic ‘butterfly’ pattern of coronal structures detected. This represents the first survey of coronal structures performed across an entire solar cycle, demonstrating new techniques available to examine the composition of the corona by latitude in varying wavelengths.

Evidence for the Self-organized Criticality Phenomenon in the Prompt Phase of Short Gamma-Ray Bursts

The prompt phase of gamma-ray bursts (GRBs) contains essential information regarding their physical nature and central engine, which are as yet unknown. In this paper, we investigate the self-organized criticality phenomenon in GRB prompt phases as done in X-ray flares of GRBs. We obtain the differential and cumulative distributions of 243 short GRB pulses, such as peak flux, FWHM, rise time, decay time, and peak time in the fourth BATSE Time-Tagged Event Catalog with the Markov Chain Monte Carlo technique. It is found that these distributions can be well described by power-law models. In particular, comparisons are made with 182 short GRB pulses in the third Swift GRB Catalog from 2004 December to 2019 July. The results are essentially consistent with the BATSE ones. We notice that there is no obvious power-law index evolution across different energy bands for either BATSE or Swift short GRBs. The joint analysis suggests that the GRB prompt phase can be explained by a fractal-diffusive self-organized criticality system with the spatial dimension S = 3 and the classical diffusion β = 1. Our findings show that GRB prompt phases and X-ray flares possess the very same magnetically dominated stochastic process and mechanism.

Self-organized criticality in multi-pulse gamma-ray bursts

The variability in multi-pulse gamma-ray bursts (GRBs) may help to reveal the mechanism of underlying processes from the central engine. To investigate whether the self-organized criticality (SOC) phenomena exist in the prompt phase of GRBs, we statistically study the properties of GRBs with more than 3 pulses in each burst by fitting the distributions of several observed physical variables with a Markov Chain Monte Carlo approach, including the isotropic energy Eiso, the duration time T, and the peak count rate P of each pulse. Our sample consists of 454 pulses in 93 GRBs observed by the CGRO/BATSE satellite. The best-fitting values and uncertainties for these power-law indices of the differential frequency distributions are: $$\alpha_E^d=1.54\pm0.09, \alpha_T^d=1.82\pm_{0.15}^{+0.14}\;and\;\alpha_P^d=2.09_{-019}^{+0.18}$$ , while the power-law indices in the cumulative frequency distributions are: $$\alpha_E^c=1.44_{-0.10}^{+0.08}, \alpha_T^c=1.75_-{0.13}^{+0.141}\;and\;\alpha_P^c=1.99_{-019}^{+0.16}$$ . We find that these distributions are roughly consistent with the physical framework of a Fractal-Diffusive, Self-Organized Criticality (FD-SOC) system with the spatial dimension S = 3 and the classical diffusion I²=1. Our results support that the jet responsible for the GRBs should be magnetically dominated and magnetic instabilities (e.g., kink model, or tearing-model instability) lead the GRB emission region into the SOC state.

GLOBAL ENERGETICS OF SOLAR FLARES. II. THERMAL ENERGIES

We present the second part of a project on the global energetics of solar flares and CMEs that includes about 400 M- and X-class flares observed with AIA/SDO during the first 3.5 years of its mission. In this Paper II we compute the differential emission measure (DEM) distribution functions and associated multi-thermal energies, using a spatially-synthesized Gaussian DEM forward-fitting method. The multi-thermal DEM function yields a significantly higher (by an average factor of $\approx 14$), but more comprehensive (multi-)thermal energy than an isothermal energy estimate from the same AIA data. We find a statistical energy ratio of $E_{th}/E_{diss} \approx 2\%-40\%$ between the multi-thermal energy $E_{th}$ and the magnetically dissipated energy $E_{diss}$, which is an order of magnitude higher than the estimates of Emslie et al.~2012. For the analyzed set of M and X-class flares we find the following physical parameter ranges: $L=10^{8.2}-10^{9.7}$ cm for the length scale of the flare areas, $T_p=10^{5.7}-10^{7.4}$ K for the DEM peak temperature, $T_w=10^{6.8}-10^{7.6}$ K for the emission measure-weighted temperature, $n_p=10^{10.3}-10^{11.8}$ cm$^{-3}$ for the average electron density, $EM_p=10^{47.3}-10^{50.3}$ cm$^{-3}$ for the DEM peak emission measure, and $E_{th}=10^{26.8}-10^{32.0}$ erg for the multi-thermal energies. The deduced multi-thermal energies are consistent with the RTV scaling law $E_{th,RTV} = 7.3 \times 10^{-10} \ T_p^3 L_p^2$, which predicts extremal values of $E_{th,max} \approx 1.5 \times 10^{33}$ erg for the largest flare and $E_{th,min} \approx 1 \times 10^{24}$ erg for the smallest coronal nanoflare. The size distributions of the spatial parameters exhibit powerlaw tails that are consistent with the predictions of the fractal-diffusive self-organized criticality model combined with the RTV scaling law.

UNIVERSAL BEHAVIOR OF X-RAY FLARES FROM BLACK HOLE SYSTEMS

X-ray flares have been discovered in black hole systems, such as gamma-ray bursts, the tidal disruption event Swift J1644+57, the supermassive black hole Sagittarius A$^*$ at the center of our Galaxy, and some active galactic nuclei. Their occurrences are always companied by relativistic jets. However, it is still unknown whether there is a physical analogy among such X-ray flares produced in black hole systems spanning nine orders of magnitude in mass. Here we report the observed data of X-ray flares, and show that they have three statistical properties similar to solar flares, including power-law distributions of energies, durations, and waiting times, which both can be explained by a fractal-diffusive self-organized criticality model. These statistical similarities, together with the fact that solar flares are triggered by a magnetic reconnection process, suggest that all of the X-ray flares are consistent with magnetic reconnection events, implying that their concomitant relativistic jets may be magnetically dominated.

GLOBAL ENERGETICS OF SOLAR FLARES. I. MAGNETIC ENERGIES

We present the first part of a project on the global energetics of solar flares and coronal mass ejections (CMEs) that includes about 400 M- and X-class flares observed with AIA and HMI onboard SDO. We calculate the potential energy, free energy, and the flare-dissipated magnetic energy. We calculate these magnetic parameters using two different NLFFF codes: The COR-NLFFF code uses the line-of-sight magnetic field component $B_z$ from HMI to define the potential field, and the 2D coordinates of automatically detected coronal loops in 6 coronal wavelengths from AIA to measure the helical twist of coronal loops caused by vertical currents, while the PHOT-NLFFF code extrapolates the photospheric 3D vector fields. We find agreement between the two codes in the measurement of free energies and dissipated energies within a factor of $ \approx 3$. The size distributions of magnetic parameters exhibit powerlaw slopes that are approximately consistent with the fractal-diffusive self-organized criticality model. The magnetic parameters exhibit scaling laws for the nonpotential energy, $E_{np} \propto E_p^{1.02}$, for the free energy, $E_{free} \propto E_p^{1.7}$ and $E_{free} \propto B_{\varphi}^{1.0} L^{1.5}$, for the dissipated energy, $E_{diss} \propto E_p^{1.6}$ and $E_{diss} \propto E_{free}^{0.9}$, and the energy dissipation volume, $V \propto E_{diss}^{1.2}$. The potential energies vary in the range of $E_p = 1 \times 10^{31} - 4 \times 10^{33}$ erg, while the free energy has a ratio of $E_{free}/E_p \approx 1%-25%$. The Poynting flux amounts to $F_{flare} \approx 5 \times 10^{8} - 10^{10}$ erg cm$^{-2}$ s$^{-1}$ during flares, which averages to $F_{AR} \approx 6 \times 10^6$ erg cm$^{-2}$ s$^{-1}$ during the entire observation period and is comparable with the coronal heating rate requirement in active regions.

MULTI-WAVELENGTH OBSERVATIONS OF THE SPATIO-TEMPORAL EVOLUTION OF SOLAR FLARES WITH AIA/SDO. II. HYDRODYNAMIC SCALING LAWS AND THERMAL ENERGIES

In this study we measure physical parameters of the same set of 155 M and X-class solar flares observed with AIA/SDO as analyzed in Paper I, by performing a {\sl differential emission measure (DEM)} analysis to determine the flare peak emission measure $EM_p$, peak temperature $T_p$, electron density $n_p$, and thermal energy $E_{th}$, in addition to the spatial scales $L$, areas $A$, and volumes $V$ measured in Paper I. The parameter ranges for M and X-class flares are: $\log(EM_p)=47.0-50.5$, $T_p=5.0-17.8$ MK, $n_p=4 \times 10^9-9 \times 10^{11}$ cm$^{-3}$, and thermal energies of $E_{th}=1.6 \times 10^{28}-1.1 \times 10^{32}$ erg. We find that these parameters obey the Rosner-Tucker-Vaiana (RTV) scaling law $T_p^2 \propto n_p L$ and $H \propto T^{7/2} L^{-2}$ during the peak time $t_p$ of the flare density $n_p$, when energy balance between the heating rate $H$ and the conductive and radiative loss rates is achieved for a short instant, and thus enables the applicability of the RTV scaling law. The application of the RTV scaling law predicts powerlaw distributions for all physical parameters, which we demonstrate with numerical Monte-Carlo simulations as well as with analytical calculations. A consequence of the RTV law is also that we can retrieve the size distribution of heating rates, for which we find $N(H) \propto H^{-1.8}$, which is consistent with the magnetic flux distribution $N(\Phi) \propto \Phi^{-1.85}$ observed by Parnell et al.(2009) and the heating flux scaling law $F_H \propto H L \propto B/L$ of Schrijver et al.(2004). The fractal-diffusive self-organized criticality model in conjunction with the RTV scaling law reproduces the observed powerlaw distributions and their slopes for all geometrical and physical parameters and can be used to predict the size distributions for other flare datasets, instruments, and detection algorithms.

STEREO/Extreme Ultraviolet Imager (EUVI) Event Catalog 2006 – 2012

We generated an event catalog with an automated detection algorithm based on the entire EUVI image database observed with the two Solar Terrestrial Relations Observatory (STEREO)-A and -B spacecraft over the first six years of the mission (2006 – 2012). The event catalog includes the heliographic positions of some 20 000 EUV events, transformed from spacecraft coordinates to Earth-based coordinates, and information on associated GOES flare events (down to the level of GOES A5-class flares). The 304 Å wavelength turns out to be the most efficient channel for flare detection (79 % of all EUVI event detections), while the 171 Å (4 %), 195 Å (10 %), and the 284 Å channel (7 %) retrieve substantially fewer flare events, partially due to the suppressing effect of EUV dimming, and partially due to the lower cadence in the later years of the mission. Due to the Sun-circling orbits of STEREO-A and -B, a large number of flares have been detected on the farside of the Sun, invisible from Earth, or seen as partially occulted events. The statistical size distributions of EUV peak fluxes (with a power-law slope of α P =2.5±0.2) and event durations (with a power-law slope of α T =2.4±0.3) are found to be consistent with the fractal-diffusive self-organized criticality model. The EUVI event catalog is available on-line at secchi.lmsal.com/EUVI/euvi_autodetection/euvi_events.txt and may serve as a comprehensive tool to identify stereoscopically observed flare events for 3D reconstruction and to study occulted flare events.

DOES A SCALING LAW EXIST BETWEEN SOLAR ENERGETIC PARTICLE EVENTS AND SOLAR FLARES?

Among many other natural processes, the size distributions of solar X-ray flares and solar energetic particle (SEP) events are scale-invariant power laws. The measured distributions of SEP events prove to be distinctly flatter, i.e., have smaller power-law slopes, than those of the flares. This has led to speculation that the two distributions are related through a scaling law, first suggested by Hudson, which implies a direct nonlinear physical connection between the processes producing the flares and those producing the SEP events. We present four arguments against this interpretation. First, a true scaling must relate SEP events to all flare X-ray events, and not to a small subset of the X-ray event population. We also show that the assumed scaling law is not mathematically valid and that although the flare X-ray and SEP event data are correlated, they are highly scattered and not necessarily related through an assumed scaling of the two phenomena. An interpretation of SEP events within the context of a recent model of fractal-diffusive self-organized criticality by Aschwanden provides a physical basis for why the SEP distributions should be flatter than those of solar flares. These arguments provide evidence against a close physical connection of flares with SEP production.

THE SPATIO-TEMPORAL EVOLUTION OF SOLAR FLARES OBSERVED WITH AIA/SDO: FRACTAL DIFFUSION, SUB-DIFFUSION, OR LOGISTIC GROWTH?

We explore the spatio-temporal evolution of solar flares by fitting a radial expansion model $r(t)$ that consists of an exponentially growing acceleration phase, followed by a deceleration phase that is parameterized by the generalized diffusion function $r(t) \propto \kappa (t-t_1)^{\beta/2}$, which includes the logistic growth limit ($\beta=0$), sub-diffusion ($\beta = 0-1$), classical diffusion ($\beta=1$), super-diffusion ($\beta = 1-2$), and the linear expansion limit ($\beta=2$). We analyze all M and X-class flares observed with GOES and AIA/SDO during the first two years of the SDO mission, amounting to 155 events. We find that most flares operate in the sub-diffusive regime ($\beta=0.53\pm0.27$), which we interpret in terms of anisotropic chain reactions of intermittent magnetic reconnection episodes in a low plasma-$\beta$ corona. We find a mean propagation speed of $v=15\pm12$ km s$^{-1}$, with maximum speeds of $v_{max}=80 \pm 85$ km s$^{-1}$ per flare, which is substantially slower than the sonic speeds expected for thermal diffusion of flare plasmas. The diffusive characteristics established here (for the first time for solar flares) is consistent with the fractal-diffusive self-organized criticality (FD-SOC) model, which predicted diffusive transport merely based on cellular automaton simulations.

AUTOMATED SOLAR FLARE STATISTICS IN SOFT X-RAYS OVER 37 YEARS OFGOESOBSERVATIONS: THE INVARIANCE OF SELF-ORGANIZED CRITICALITY DURING THREE SOLAR CYCLES

We analyzed the soft X-ray light curves from the {\sl Geostationary Operational Environmental Satellites (GOES)} over the last 37 years (1975-2011) and measured with an automated flare detection algorithm over 300,000 solar flare events (amounting to $\approx 5$ times higher sensitivity than the NOAA flare catalog). We find a powerlaw slope of $\alpha_F=1.98\pm0.11$ for the (background-subtracted) soft X-ray peak fluxes that is invariant through three solar cycles and agrees with the theoretical prediction $\alpha_F=2.0$ of the {\sl fractal-diffusive self-organized criticality (FD-SOC)} model. For the soft X-ray flare rise times we find a powerlaw slope of $\alpha_T =2.02\pm0.04$ during solar cycle minima years, which is also consistent with the prediction $\alpha_T=2.0$ of the FD-SOC model. During solar cycle maxima years, the powerlaw slope is steeper in the range of $\alpha_T \approx 2.0-5.0$, which can be modeled by a solar cycle-dependent flare pile-up bias effect. These results corroborate the FD-SOC model, which predicts a powerlaw slope of $\alpha_E=1.5$ for flare energies and thus rules out significant nanoflare heating. While the FD-SOC model predicts the probability distribution functions of spatio-temporal scaling laws of nonlinear energy dissipation processes, additional physical models are needed to derive the scaling laws between the geometric SOC parameters and the observed emissivity in different wavelength regimes, as we derive here for soft X-ray emission. The FD-SOC model yields also statistical probabilities for solar flare forecasting.