arXiv subjects

Markus J. Aschwanden

Publications and source records attributed to Markus J. Aschwanden.

At least 55 records · Page 3Linked to original sources

arXiv2015

Magnetic Energy Dissipation during the 2014 March 29 Solar Flares

We calculated the time evolution of the free magnetic energy during the 2014-Mar-29 flare (SOL2014-03-29T17:48), the first X-class flare detected by IRIS. The free energy was calculated from the difference between the nonpotential field, constrained by the geometry of observed loop structures, and the potential field. We use AIA/SDO and IRIS images to delineate the geometry of coronal loops in EUV wavelengths, as well as to trace magnetic field directions in UV wavelengths in the chromosphere and transition region. We find an identical evolution of the free energy for both the coronal and chromospheric tracers, as well as agreement between AIA and IRIS results, with a peak free energy of $E_{free}(t_{peak}) \approx (45 \pm 2) \times 10^{30}$ erg, which decreases by an amount of $ΔE_{free} \approx (29 \pm 3) \times 10^{30}$ erg during the flare decay phase. The consistency of free energies measured from different EUV and UV wavelengths for the first time here, demonstrates that vertical electric currents (manifested in form of helically twisted loops) can be detected and measured from both chromospheric and coronal tracers.

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arXiv2014

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 $E_{free} \propto B_φ^{1.0} L^{1.5}$, 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.

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arXiv2014

25 Years of Self-Organized Criticality: Solar and Astrophysics

Shortly after the seminal paper {\sl "Self-Organized Criticality: An explanation of 1/f noise"} by Bak, Tang, and Wiesenfeld (1987), the idea has been applied to solar physics, in {\sl "Avalanches and the Distribution of Solar Flares"} by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme.

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arXiv2014

The Magnetic Field of Active Region 11158 During the 2011 February 12-17 Flares : Differences between Photospheric Extrapolation and Coronal Forward-Fitting Methods

We developed a {\sl coronal non-linear force-free field (COR-NLFFF)} forward-fitting code that fits an approximate {\sl non-linear force-free field (NLFFF)} solution to the observed geometry of automatically traced coronal loops. In contrast to photospheric NLFFF codes, which calculate a magnetic field solution from the constraints of the transverse photospheric field, this new code uses coronal constraints instead, and this way provides important information on systematic errors of each magnetic field calculation method, as well as on the non-forcefreeness in the lower chromosphere. In this study we applied the COR-NLFFF code to active region NOAA 11158, during the time interval of 2011 Feb 12 to 17, which includes an X2.2 GOES-class flare plus 35 M and C-class flares. We calcuated the free magnetic energy with a 6-minute cadence over 5 days. We find good agreement between the two types of codes for the total nonpotential $E_N$ and potential energy $E_P$, but find up to a factor of 4 discrepancy in the free energy $E_{free}=E_N-E_P$, and up to a factor of 10 discrepancy in the decrease of the free energy $ΔE_{free}$ during flares. The coronal NLFFF code exhibits a larger time variability, and yields a decrease of free energy during the flare that is sufficient to satisfy the flare energy budget, while the photospheric NLFFF code shows much less time variability and an order of magnitude less free energy decrease during flares. The discrepancy may partly be due to the pre-processing of photospheric vector data, but more likely due to the non-forcefreeness in the lower chromosphere. We conclude that the coronal field cannot be correctly calculated based on photospheric data alone, but requires additional information on coronal loop geometries.

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arXiv2014

The Compatibility of Flare Temperatures Observed with AIA, GOES, and RHESSI

We test the compatibility and biases of multi-thermal flare DEM (differential emission measure) peak temperatures determined with AIA with those determined by GOES and RHESSI using the isothermal assumption. In a set of 149 M- and X-class flares observed during the first two years of the SDO mission, AIA finds DEM peak temperatures at the time of the peak GOES 1-8 A flux to have an average of Tp = 12.0+/-2.9 MK and Gaussian DEM widths of log10(sigma_T) = 0.50+/-0.13. From GOES observations of the same 149 events, a mean temperature of Tp = 15.6+/-2.4 MK is inferred, which is systematically higher by a factor of TGOES/TAIA = 1.4+/-0.4. We demonstrate that this discrepancy results from the isothermal assumption in the inversion of the GOES filter ratio. From isothermal fits to photon spectra at energies of E ~ 6-12 keV of 61 of these events, RHESSI finds the temperature to be higher still by a factor of TRHESSI/TAIA = 1.9+/-1.0. We find that this is partly a consequence of the isothermal assumption. However, RHESSI is not sensitive to the low-temperature range of the DEM peak, and thus RHESSI samples only the high-temperature tail of the DEM function. This can also contribute to the discrepancy between AIA and RHESSI temperatures. The higher flare temperatures found by GOES and RHESSI imply correspondingly lower emission measures. We conclude that self-consistent flare DEM temperatures and emission measures require simultaneous fitting of EUV (AIA) and soft X-ray (GOES and RHESSI) fluxes.

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arXiv2013

A Macroscopic Description of a Generalized Self-Organized Criticality System: Astrophysical Applications

We suggest a generalized definition of self-organized criticality (SOC) systems: SOC is a critical state of a nonlinear energy dissipation system that is slowly and continuously driven towards a critical value of a system-wide instability threshold, producing scale-free, fractal-diffusive, and intermittent avalanches with powerlaw-like size distributions. We develop here a macroscopic description of SOC systems that provides an equivalent description of the complex microscopic fine structure, in terms of fractal-diffusive transport (FD-SOC). Quantitative values for the size distributions of SOC parameters (length scales $L$, time scales $T$, waiting times $Δt$, fluxes $F$, and energies $E$) are derived from first principles, using the scale-free probability conjecture, $N(L) dL \propto L^{-d}$, for Euclidean space dimension $d$. We apply this model to astrophysical SOC systems, such as lunar craters, the asteroid belt, Saturn ring particles, magnetospheric substorms, radiation belt electrons, solar flares, stellar flares, pulsar glitches, soft gamma-ray repeaters, black-hole objects, blazars, and cosmic rays. The FD-SOC model predicts correctly the size distributions of 8 out of these 12 astrophysical phenomena, and indicates non-standard scaling laws and measurement biases for the others.

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arXiv2013

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 $N(Φ) \propto Φ^{-1.85}$ 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.

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arXiv2013

Multi-Wavelength Observations of the Spatio-Temporal Evolution of Solar Flares with AIA/SDO: I. Universal Scaling Laws of Space and Time Parameters

We extend a previous statistical solar flare study of 155 GOES M- and X-class flares observed with AIA/SDO (Aschwanden 2012) to all 7 coronal wavelengths (94, 131, 171, 193, 211, 304, 335 \ang) to test the wavelength-dependence of scaling laws and statistical distributions. Except for the 171 and 193 \ang\ wavelengths, which are affected by EUV dimming caused by coronal mass ejections (CMEs), we find near-identical size distributions of geometric (lengths $L$, flare areas $A$, volumes $V$, fractal dimension $D_2$), temporal (flare durations $T$), and spatio-temporal parameters (diffusion coefficient $κ$, spreading exponent $β$, and maximum expansion velocities $v_{max}$) in different wavelengths, which are consistent with the universal predictions of the fractal-diffusive avalanche model of a slowly-driven self-organized criticality (FD-SOC) system, i.e., $N(L) \propto L^{-3}$, $N(A) \propto A^{-2}$, $N(V) \propto V^{-5/3}$, $N(T) \propto T^{-2}$, $D_2=3/2$, for a Euclidean dimension $d=3$. Empirically we find also a new strong correlation $κ\propto L^{0.94\pm0.01}$ and the 3-parameter scaling law $L \propto κ T^{0.1}$, which is more consistent with the logistic-growth model than with classical diffusion. The findings suggest long-range correlation lengths in the FD-SOC system that operate in the vicinity of a critical state, which could be used for predictions of individual extreme events. We find also that eruptive flares (with accompanying CMEs), have larger volumes $V$, longer flare durations $T$, higher EUV and soft X-ray fluxes, and somewhat larger diffusion coefficients $κ$ than confined flares (without CMEs).

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arXiv2013

Optimization of Curvi-Linear Tracing Applied to Solar Physics and Biophysics

We developed an automated pattern recognition code that is particularly well suited to extract one-dimensional curvi-linear features from two-dimensional digital images. A former version of this {\sl Oriented Coronal CUrved Loop Tracing (OCCULT)} code was applied to spacecraft images of magnetic loops in the solar corona, recorded with the NASA spacecraft {\sl Transition Region And Coronal Explorer (TRACE)} in extreme ultra-violet wavelengths. Here we apply an advanced version of this code ({\sl OCCULT-2}) also to similar images from the {\sl Solar Dynamics Observatory (SDO)}, to chromospheric H-$α$ images obtained with the {\sl Swedish Solar Telescope (SST)}, and to microscopy images of microtubule filaments in live cells in biophysics. We provide a full analytical description of the code, optimize the control parameters, and compare the automated tracing with visual/manual methods. The traced structures differ by up to 16 orders of magnitude in size, which demonstrates the universality of the tracing algorithm.

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arXiv2012

Nonlinear Force-Free Magnetic Field Fitting to Coronal Loops with and without Stereoscopy

We developed a new nonlinear force-free magnetic field (NLFFF) forward-fitting algorithm based on an analytical approximation of force-free and divergence-free NLFFF solutions, which requires as input a line-of-sight magnetogram and traced 2D loop coordinates of coronal loops only, in contrast to stereoscopically triangulated 3D loop coordinates used in previous studies. Test results of simulated magnetic configurations and from four active regions observed with STEREO demonstrate that NLFFF solutions can be fitted with equal accuracy with or without stereoscopy, which relinquishes the necessity of STEREO data for magnetic modeling of active regions (on the solar disk). The 2D loop tracing method achieves a 2D misalignment of $μ_2=2.7^\circ\pm 1.3^\circ$ between the model field lines and observed loops, and an accuracy of $\approx 1.0%$ for the magnetic energy or free magnetic energy ratio. The three times higher spatial resolution of TRACE or SDO/AIA (compared with STEREO) yields also a proportionally smaller misalignment angle between model fit and observations. Visual/manual loop tracings are found to produce more accurate magnetic model fits than automated tracing algorithms. The computation time of the new forward-fitting code amounts to a few minutes per active region.

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arXiv2012

A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. III. The Free Energy

An analytical approximation of a nonlinear force-free magnetic field (NLFFF) solution was developed in Paper I, while a numerical code that performs fast forward-fitting of this NLFFF approximation to a line-of-sight magnetogram and coronal 3D loops has been described and tested in Paper II. Here we calculate the free magnetic energy $E_{\rm free}=E_{\rm N}-E_{\rm P}$, i.e. the difference of the magnetic energies between the nonpotential field and the potential field. A second method to estimate the free energy is obtained from the mean misalignment angle change $Δμ=μ_{\rm P}-μ_{\rm N}$ between the potential and nonpotential field, which scales as $E_{\rm free}/E_{\rm P} \approx \tan^2{(Δμ)}$. For four active regions observed with STEREO in 2007 we find free energies in the range of $q_{\rm free}=(E_{\rm free}/E_{\rm P}) \approx 1%-10%$, with an uncertainty of less than $\pm 2%$ between the two methods, while the free energies obtained from 11 other NLFFF codes exhibit a larger scatter of order $\approx\pm10%$. We find also a correlation between the free magnetic energy and the GOES flux of the largest flare that occurred during the observing period, which can be quantified by an exponential relationship, $F_{\rm GOES} \propto \exp{(q_{\rm free}/0.015)}$, implying an exponentiation of the dissipated currents.

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arXiv2012

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 $r(t) \propto κ(t-t_1)^{β/2}$, which includes the logistic growth limit ($β=0$), sub-diffusion ($β= 0-1$), classical diffusion ($β=1$), super-diffusion ($β= 1-2$), and the linear expansion limit ($β=2$). 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 ($β=0.53\pm0.27$), which we interpret in terms of anisotropic chain reactions of intermittent magnetic reconnection episodes in a low plasma-$β$ 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.

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arXiv2012

Self-Organized Criticality Systems in Astrophysics (Chapter 13)

Chapter 13: SOC Systems in Astrophysics --- Content list: 13.1 Theory -- 13.1.1 The Sacle-Free Probability Theorem - 13.1.2 The Fractal-Diffusive Spatio-Temporal Relationship - 13.1.3 Size Distributions of Astrophysical Observables - 13.1.4 Scaling Laws for Thermal Emission of Astrophysical Plasmas - 13.1.5 Scaling Laws for Astrophysical Acceleration Mechanisms - 13.2 Observations -- 13.2.1 Lunar Craters - 13.2.2 Asteroid Belt - 13.2.3 Saturn Ring - 13.2.4 Magnetospheric Substorms and Auroras - 13.2.5 Solar Flares - 13.2.6 Stellar Flares - 13.2.7 Pulsars - 13.2.8 Soft Gamma-Ray Repeaters - 13.2.9 Black-Hole Objects - 13.2.10 Blazars - 13.2.11 Cosmic Rays - 13.3 Conclusions

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arXiv2012

A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. I. Theory

We derive an analytical approximation of nonlinear force-free magnetic field solutions (NLFFF) that can efficiently be used for fast forward-fitting to solar magnetic data, constrained either by observed line-of-sight magnetograms and stereoscopically triangulated coronal loops, or by 3D vector-magnetograph data. The derived NLFFF solutions provide the magnetic field components $B_x({\bf x})$, $B_y({\bf x})$, $B_z({\bf x})$, the force-free parameter $α({\bf x})$, the electric current density ${\bf j}({\bf x})$, and are accurate to second-order (of the nonlinear force-free $α$-parameter). The explicit expressions of a force-free field can easily be applied to modeling or forward-fitting of many coronal phenomena.

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arXiv2012

A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. II. Numeric Code and Tests

Based on a second-order approximation of nonlinear force-free magnetic field solutions in terms of uniformly twisted field lines derived in Paper I, we develop here a numeric code that is capable to forward-fit such analytical solutions to arbitrary magnetogram (or vector magnetograph) data combined with (stereoscopically triangulated) coronal loop 3D coordinates. We test the code here by forward-fitting to six potential field and six nonpotential field cases simulated with our analytical model, as well as by forward-fitting to an exactly force-free solution of the Low and Lou (1990) model. The forward-fitting tests demonstrate: (i) a satisfactory convergence behavior (with typical misalignment angles of $μ\approx 1^\circ-10^\circ$), (ii) relatively fast computation times (from seconds to a few minutes), and (iii) the high fidelity of retrieved force-free $α$-parameters ($α_{\rm fit}/α_{\rm model} \approx 0.9-1.0$ for simulations and $α_{\rm fit}/α_{\rm model} \approx 0.7\pm0.3$ for the Low and Lou model). The salient feature of this numeric code is the relatively fast computation of a quasi-forcefree magnetic field, which closely matches the geometry of coronal loops in active regions, and complements the existing {\sl nonlinear force-free field (NLFFF)} codes based on photospheric magnetograms without coronal constraints.

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arXiv2012

Solar Stereoscopy with STEREO/EUVI A and B spacecraft from small (6 deg) to large (170 deg) spacecraft separation angles

We performed for the first time stereoscopic triangulation of coronal loops in active regions over the entire range of spacecraft separation angles ($α_{sep}\approx 6^\circ, 43^\circ, 89^\circ, 127^\circ$, and $170^\circ$). The accuracy of stereoscopic correlation depends mostly on the viewing angle with respect to the solar surface for each spacecraft, which affects the stereoscopic correspondence identification of loops in image pairs. From a simple theoretical model we predict an optimum range of $α_{sep} \approx 22^\circ-125^\circ$, which is also experimentally confirmed. The best accuracy is generally obtained when an active region passes the central meridian (viewed from Earth), which yields a symmetric view for both STEREO spacecraft and causes minimum horizontal foreshortening. For the extended angular range of $α_{sep}\approx 6^\circ-127^{\circ}$ we find a mean 3D misalignment angle of $μ_{PF} \approx 21^\circ-39^\circ$ of stereoscopically triangulated loops with magnetic potential field models, and $μ_{FFF} \approx 15^\circ-21^\circ$ for a force-free field model, which is partly caused by stereoscopic uncertainties $μ_{SE} \approx 9^\circ$. We predict optimum conditions for solar stereoscopy during the time intervals of 2012--2014, 2016--2017, and 2021--2023.

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arXiv2012

First 3D Reconstructions of Coronal Loops with the STEREO A+B Spacecraft: IV. Magnetic Modeling with Twisted Force-Free Fields

The three-dimensional (3D) coordinates of stereoscopically triangulated loops provide strong constraints for magnetic field models of active regions in the solar corona. Here we use STEREO/A and B data from some 500 stereoscopically triangulated loops observed in four active regions (2007 Apr 30, May 9, May 19, Dec 11), together with SOHO/MDI line-of-sight magnetograms. We measure the average misalignment angle between the stereoscopic loops and theoretical magnetic field models, finding a mismatch of $μ=19^\circ-46^\circ$ for a potential field model, which is reduced to $μ=14^\circ-19^\circ$ for a non-potential field model parameterized by twist parameters. The residual error is commensurable with stereoscopic measurement errors ($μ_{SE} \approx 8^\circ-12^\circ$). We developed a potential field code that deconvolves a line-of-sight magnetogram into three magnetic field components $(B_x, B_y, B_z)$, as well as a non-potential field forward-fitting code that determines the full length of twisted loops ($L \approx 50-300$ Mm), the number of twist turns (median $N_{twist}=0.06$), the nonlinear force-free $α$-parameter (median $α\approx 4 \times 10^{-11}$ cm$^{-1}$), and the current density (median $j_z \approx 1500$ Mx cm$^{-2}$ s$^{-1}$). All twisted loops are found to be far below the critical value for kink instability, and Joule dissipation of their currents is found be be far below the coronal heating requirement. The algorithm developed here, based on an analytical solution of nonlinear force-free fields that is accurate to second order (in the force-free parameter $α$), represents the first code that enables fast forward-fitting to photospheric magnetograms and stereoscopically triangulated loops in the solar corona.

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arXiv2012

Automated Solar Flare Statistics in Soft X-rays over 37 Years of GOES Observations - 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 $α_F=1.98\pm0.11$ for the (background-subtracted) soft X-ray peak fluxes that is invariant through three solar cycles and agrees with the theoretical prediction $α_F=2.0$ 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 $α_T =2.02\pm0.04$ during solar cycle minima years, which is also consistent with the prediction $α_T=2.0$ of the FD-SOC model. During solar cycle maxima years, the powerlaw slope is steeper in the range of $α_T \approx 2.0-5.0$, 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 $α_E=1.5$ 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.

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