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10.1119/10.0000905.1A biased ball rolled and spun on a horizontal surface exhibits interesting dynamics. We investigate the motion of a truncated billiard ball, via experiments, analytical methods, and numerical solutions of the equations of motion for a biased sphere rolling without slipping. Solutions are identified where the center of mass moves in a circular or a square path, and we investigate other quasi-periodic motions of the ball.

Flare occurrence on the Sun is highly variable, exhibiting both short term variation due to the emergence and evolution of active regions, and long-term variation from the solar cycle. On solar-like stars, much larger stellar flares (superflares) have been observed, and it is of interest to determine whether observed rates of superflare occurrence exhibit similar variability to solar flares. We analyse 274 G-type stars using data from the Transiting Exoplanet Survey Satellite (TESS) and identify seven stars which exhibit statistically significant changes in the rate of superflare occurrence by fitting a piecewise constant-rate model with the Bayesian Blocks algorithm (Scargle et al 2012; arXiv:1207.5578). We investigate the properties of these stars and their flaring rates, and discuss the possible reasons for the low number of stars with detectable rate variation., 18 pages, 13 figures, ApJ Accepted 2022-11-21

The Variables and Slow Transients Survey (VAST) on the Australian Square Kilometre Array Pathfinder (ASKAP) is designed to detect highly variable and transient radio sources on timescales from 5 s to$\sim\!5$yr. In this paper, we present the survey description, observation strategy and initial results from the VAST Phase I Pilot Survey. This pilot survey consists of$\sim\!162$h of observations conducted at a central frequency of 888 MHz between 2019 August and 2020 August, with a typical rms sensitivity of$0.24\ \mathrm{mJy\ beam}^{-1}$and angular resolution of$12-20$arcseconds. There are 113 fields, each of which was observed for 12 min integration time, with between 5 and 13 repeats, with cadences between 1 day and 8 months. The total area of the pilot survey footprint is 5 131 square degrees, covering six distinct regions of the sky. An initial search of two of these regions, totalling 1 646 square degrees, revealed 28 highly variable and/or transient sources. Seven of these are known pulsars, including the millisecond pulsar J2039–5617. Another seven are stars, four of which have no previously reported radio detection (SCR J0533–4257, LEHPM 2-783, UCAC3 89–412162 and 2MASS J22414436–6119311). Of the remaining 14 sources, two are active galactic nuclei, six are associated with galaxies and the other six have no multi-wavelength counterparts and are yet to be identified.

We consider the motion of a light sail that is accelerated by a powerful laser beam. We derive the equations of motion for two proof-of-concept sail designs with damped internal degrees of freedom. Using linear stability analysis we show that perturbations of the sail movement in all lateral degrees of freedom can be damped passively. This analysis also shows complicated behaviour akin to that associated with exceptional points in PT-symmetric systems in optics and quantum mechanics. The excess heat that is produced by the damping mechanism is likely to be substantially smaller than the expected heating due to the partial absorption of the incident laser beam by the sail., 7 figures

A physical magnetic field has a divergence of zero. Numerical error in constructing a model field and computing the divergence, however, introduces a finite divergence into these calculations. A popular metric for measuring divergence is the average fractional flux $\langle |f_{i}| \rangle$. We show that $\langle |f_{i}| \rangle$ scales with the size of the computational mesh, and may be a poor measure of divergence because it becomes arbitrarily small for increasing mesh resolution, without the divergence actually decreasing. We define a modified version of this metric that does not scale with mesh size. We apply the new metric to the results of DeRosa et al. (2015), who measured $\langle |f_{i}| \rangle$ for a series of Nonlinear Force-Free Field (NLFFF) models of the coronal magnetic field based on solar boundary data binned at different spatial resolutions. We compute a number of divergence metrics for the DeRosa et al. (2015) data and analyze the effect of spatial resolution on these metrics using a non-parametric method. We find that some of the trends reported by DeRosa et al. (2015) are due to the intrinsic scaling of $\langle |f_{i}| \rangle$. We also find that different metrics give different results for the same data set and therefore there is value in measuring divergence via several metrics., Accepted for publication in ApJ

We investigate the ability of a nonlinear force-free code to calculate highly-twisted magnetic field configurations using the Titov and D\'{e}moulin (1999) equilibrium field as a test case. The code calculates a force-free field using boundary conditions on the normal component of the field in the lower boundary, and the normal component of the current density over one polarity of the field in the lower boundary. The code can also use the current density over both polarities of the field in the lower boundary as a boundary condition. We investigate the accuracy of the reconstructions with increasing flux-rope surface twist number $N_{\textrm{t}}$, achieved by decreasing the sub-surface line current in the model. We find that the code can approximately reconstruct the Titov-D\'{e}moulin field for surface twist numbers up to $N_{\textrm{t}} \approx 8.8$. This includes configurations with bald patches. We investigate the ability to recover bald patches, and more generally identify the limitations of our method for highly-twisted fields. The results have implications for our ability to reconstruct coronal magnetic fields from observational data., Comment: 23 pages, 6 figures, accepted by Solar Physics

Vector magnetogram data are often used as photospheric boundary conditions for force-free coronal magnetic field extrapolations. In general, however, vector magnetogram data are not consistent with the force-free assumption. In this article, we demonstrate a way to deal with inconsistent boundary data, by generalizing the "self-consistency procedure" of Wheatland & Regnier (2009). In that procedure, the inconsistency is resolved by an iterative process of constructing two solutions based on the values of the force-free parameter alpha on the two polarities of the field in the boundary (the P and N polarities), and taking uncertainty-weighted averages of the boundary alpha values in the P and N solutions. When the alpha values in the P and N regions are very different, the self-consistent solution may lose high alpha values from the boundary conditions. We show how, by altering the weighting of the uncertainties in the P or N boundary conditions, we can preserve high alpha values in the self-consistent solution. The weighted self-consistent extrapolation method is demonstrated on an analytic bipole field and applied to vector magnetogram data taken by the Helioseismic and Magnetic Imager (HMI) instrument for NOAA active region AR 12017 on 2014 March 29., 12 pages, 5 figures, submitted to Solar Physics

Studies of solar radio bursts play an important role in understanding the dynamics and acceleration processes behind solar space weather events, and the influence of solar magnetic activity on solar system planets. Similar low-frequency bursts detected from active M-dwarfs are expected to probe their space weather environments and therefore the habitability of their planetary companions. Active M-dwarfs produce frequent, powerful flares which, along with radio emission, reveal conditions within their atmospheres. However, to date, only one candidate solar-like coherent radio burst has been identified from these stars, preventing robust observational constraints on their space weather environment. During simultaneous optical and radio monitoring of the nearby dM5.5e star Proxima Centauri, we detected a bright, long-duration optical flare, accompanied by a series of intense, coherent radio bursts. These detections include the first example of an interferometrically detected coherent stellar radio burst temporally coincident with a flare, strongly indicating a causal relationship between these transient events. The polarization and temporal structure of the trailing long-duration burst enable us to identify it as a type IV burst. This represents the most compelling detection of a solar-like radio burst from another star to date. Solar type IV bursts are strongly associated with space weather events such as coronal mass ejections and solar energetic particle events, suggesting that stellar type IV bursts may be used as a tracer of stellar coronal mass ejections. We discuss the implications of this event for the occurrence of coronal mass ejections from Proxima Cen and other active M-dwarfs., Comment: 18 pages, 7 figures, 1 table. Published in the Astrophysical Journal

Modern mobile phones contain a three-axis microelectromechanical system (MEMS) gyroscope, capable of taking accurate measurements of the angular velocity along the three principal axes of the phone with a sampling rate of 100 Hz or better. If the phone is tossed in the air, then, neglecting air resistance, it is in free rotation (rotation in the absence of a torque) with respect to its centre of mass, and the phone's gyroscope can be used to record the rotational dynamics. This enables experimental investigation of free rotation. In this paper, we use a mobile phone to demonstrate the steady states for rotation of the phone about two of its principal axes, and the instability in rotation about the third corresponding to the intermediate moment of inertia. We also show the approximate conservation of angular momentum and rotational kinetic energy during motion in the air, and compare the data with numerical solution of Euler's equations for free rotation. Our results demonstrate the capability of smartphones for investigating free rotation, and should be of interest to college and university teachers developing "at home" physics labs for remote learning., Comment: 18 pages, 7 figures. Accepted for publication in the American Journal of Physics

Magnetic helicity is conserved under ideal magnetohydrodynamics (MHD) and quasi-conserved even under a resistive process. The standard definition for magnetic helicity cannot be applied directly to an open magnetic field in a volume, because it is gauge-dependent. Instead, the relative magnetic helicity is widely used. We find that the energy of a potential magnetic field in a rectangular domain with periodic lateral boundary conditions is less than that of the field with a fixed normal component on all six boundaries. To make use of this lower energy potential field in the analysis of relative magnetic helicity, we introducing a new definition for magnetic helicity for the magnetic field, which involves the periodic potential field. We apply this definition to a sequence of analytic solutions and a numerical simulation. The results show that our new gauge-invariant helicity is very close to the current-carrying part of the relative magnetic helicity of the original magnetic field. We find also that the ratio between the current-carrying helicity and the relative magnetic helicity for the original and our defined relative helicity show different behavior. It seems that the new helicity is more sensitive to the component of the field due to the electric current in the volume, which is the source for instabilities and solar eruptive phenomena., Comment: 33 pages, 7 figures, accepted by ApJ

The distributions of solar flare energies and waiting times have not been described simultaneously by a single physical model, yet. In this research, we investigate whether recent avalanche models can describe the distributions for both the released energies and waiting times of flares in an active region. Flaring events are simulated using the modified Lu and Hamilton model (Lu and Hamilton (1991), ApJ, 380, 89) and also the optimized model (Farhang et al. (2018), ApJ, 859, 41). Applying a probability balance equation approach, we study the statistics of the simulated flaring events and investigate the origin of the observed power law in the flare frequency-size distribution. The results indicate that the power law originates in the distribution of transition rates (the distribution of the probabilities of transitions between different energies) rather than the distribution of the energy of the active region. It is also observed that the waiting-time distribution of simulated flaring events follows a q-exponential function which approximates a simple Poisson distribution.

Since the early 1980s, decimetric spike bursts have been attributed to electron cyclotron maser emission (ECME) by the electrons that produce hard X-ray bursts as they precipitate into the chromosphere in the impulsive phase of a solar flare. Spike bursts are regarded as analogous to the auroral kilometric radiation (AKR), which is associated with the precipitation of auroral electrons in a geomagnetic substorm. Originally, a loss-cone-driven version of ECME, developed for AKR, was applied to spike bursts, but it is now widely accepted that a different, horseshoe-driven, version of EMCE applies to AKR. We explore the implications of the assumption that horseshoe-driven ECME also applies to spike bursts. We develop a 1D model for the acceleration of the electrons by a parallel electric field, and show that under plausible assumptions it leads to a horseshoe distribution of electrons in a solar flare. A second requirement for horseshoe-driven ECME is an extremely low plasma density, referred to as a density cavity. We argue that a coronal density cavity should develop in association with a hard X-ray burst, and that such a density cavity can overcome a long-standing problem with the escape of ECME through the second-harmonic absorption layer. Both the horseshoe distribution and the associated coronal density cavity are highly localized, and could not be resolved in the statistically large number of local precipitation regions needed to explain a hard X-ray burst. The model highlights the "number problem" in the supply of the electrons needed to explain a hard X-ray burst., Accepted for publication in Solar Physics

During eruptive flares, vector magnetograms show increasing horizontal magnetic field and downward Lorentz force in the Sun's photosphere around the polarity-inversion line. Such behavior has often been associated with the implosion conjecture and interpreted as the result of either momentum conservation while the eruption moves upward, or of the contraction of flare loops. We characterize the physical origin of these observed behaviors by analyzing a generic 3D MHD simulation of an eruptive flare. Even though the simulation was undesigned to recover the magnetic field and Lorentz force properties, it is fully consistent with them, and it provides key additional informations to understand them. The area where the magnetic field increases gradually develops between current ribbons, which spread away from each other and are connected to the coronal region. This area is merely the footprint of the coronal post-flare loops, whose contraction increases their shear field component and the magnetic energy density in line with the ideal induction equation. For simulated data, we computed the Lorentz force density map by applying the method used in observations. We obtained increase of the downward component of the Lorentz force density around the PIL -consistent with observations. However, this significantly differs from the Lorentz force density maps obtained directly from the 3D magnetic field and current. These results altogether question previous interpretations based on the implosion conjecture and momentum conservation with the CME, and rather imply that the observed increases in photospheric horizontal magnetic fields result from the reconnection-driven contraction of sheared flare-loops., Comment: 20 pages, 11 figures, accepted for publication in ApJ (April 19, 2019)

Mapping of magnetic field lines is important for studies of the solar wind and the sources and propagation of energetic particles between the Sun and observers. A recently developed mapping approach is generalised to use a more advanced solar wind model that includes the effects of solar wind acceleration, non-radial intrinsic magnetic fields and flows at the source surface/inner boundary, and conservation of angular momentum. The field lines are mapped by stepping along local magnetic field $\bm{B}$ and via a Runge-Kutta algorithm, leading to essentially identical maps. The new model's maps for Carrington rotation CR 1895 near solar minimum (19 April to 15 May 1995) and a solar rotation between CR 2145 and CR 2146 near solar maximum (14 January to 9 February 2014) are compared with the published maps for a constant solar wind model. The two maps are very similar on a large scale near both solar minimum and solar maximum, meaning that the field line orientations, winding angles, and connectivity generally agree very well. However, close inspection shows that the field lines have notable small-scale structural differences. An interpretation is that inclusion of the acceleration and intrinsic azimuthal velocity has significant effects on the local structure of the magnetic field lines. Interestingly, the field lines are more azimuthal for the accelerating solar wind model for both intervals. In addition, predictions for the pitch angle distributions (PADs) for suprathermal electrons agree at the $90$ -- $95\%$ level with observations for both solar wind models for both intervals.

We investigate a method to test whether a numerically computed model coronal magnetic field ${\boldsymbol {B}}$ departs from the divergence-free condition (also known as the solenoidality condition). The test requires a potential field ${\boldsymbol {B}}_{0}$ to be calculated, subject to Neumann boundary conditions, given by the normal components of the model field ${\boldsymbol {B}}$ at the boundaries. The free energy of the model field may be calculated using $\frac{1}{2\mu _{0}}\int ({\boldsymbol {B}}-{\boldsymbol {B}}_{0})^{2}\mathrm{d}V$ , where the integral is over the computational volume of the model field. A second estimate of the free energy is provided by calculating $\frac{1}{2\mu _{0}}\int {\boldsymbol {B}}^{2}\,\mathrm{d}V-\frac{1}{2\mu _{0}}\int {\boldsymbol {B}}_{0}^{2}\,\mathrm{d}V$ . If ${\boldsymbol {B}}$ is divergence free, the two estimates of the free energy should be the same. A difference between the two estimates indicates a departure from $\nabla \cdot {\boldsymbol {B}}=0$ in the volume. The test is an implementation of a procedure proposed by Moraitis et al. (Solar Phys. 289, 4453, 2014) and is a simpler version of the Helmholtz decomposition procedure presented by Valori et al. (Astron. Astrophys. 553, A38, 2013). We demonstrate the test in application to previously published nonlinear force-free model fields, and also investigate the influence on the results of the test of a departure from flux balance over the boundaries of the model field. Our results underline the fact that, to make meaningful statements about magnetic free energy in the corona, it is necessary to have model magnetic fields that satisfy the divergence-free condition to a good approximation.

A new book explores our understanding of electric currents, which are fundamental to the structure and dynamics of space plasmas.

We review nonlinear force-free field (NLFFF) modeling of magnetic fields in active regions. The NLFFF model (in which the electric current density is parallel to the magnetic field) is often adopted to describe the coronal magnetic field, and numerical solutions to the model are constructed based on photospheric vector magnetogram boundary data. Comparative tests of NLFFF codes on sets of boundary data have revealed significant problems, in particular associated with the inconsistency of the model and the data. Nevertheless NLFFF modeling is often applied, in particular to flare-productive active regions. We examine the results, and discuss their reliability.

Electric currents in solar active regions are thought to provide the energy released via magnetic reconnection in solar flares. Vertical electric current densities $J_z$ at the photosphere may be estimated from vector magnetogram data, subject to substantial uncertainties. The values provide boundary conditions for nonlinear force- free modelling of active region magnetic fields. A method is presented for estimating values of $J_z$ taking into account uncertainties in vector magnetogram field values, and minimizing $J_z^2$ across the active region. The method is demonstrated using the boundary values of the field for a force-free twisted bipole, with the addition of noise at randomly chosen locations., 11 pages, 2 figures

Rapid and stepwise changes of the magnetic field are often observed during flares but cannot be explained by models yet. Using a 45 min sequence of SDO/HMI 135 s fast-cadence vector magnetograms of the X1 flare on 2014-03-29 we construct, at each timestep, nonlinear force-free models for the coronal magnetic field. Observed flare-related changes in the line-of-sight magnetic field $B_{\rm LOS}$ at the photosphere and chromosphere are compared with changes in the magnetic fields in the models. We find a moderate agreement at the photospheric layer (the basis for the models), but no agreement at chromospheric layers. The observed changes at the photosphere and chromosphere are surprisingly different, and are unlikely to be reproduced by a force-free model. The observed changes are likely to require a change in the magnitude of the field, not just in its direction., Comment: accepted for publication in ApJ

Flares produce sudden and permanent changes in the horizontal photospheric magnetic field. In particular flares generally produce increased magnetic shear in the photospheric field along the neutral line. Recent observations show also that flares can produce sudden photospheric motion. We present a model for the observed changes as the response of the photosphere to a large-amplitude shear Alfv\'{e}n wave propagating down from the corona on either side of the neutral line. The Alfv\'{e}nic front is assumed to impact the photosphere close to the neutral line first, and then successively further away with time, such that the line of impact coincides with the flare ribbon. The wave introduces magnetic shear and velocity shear. The magnetic shear introduced at the photosphere has the same sign on either side of the neutral line, while the velocity shear has the opposite sign. We discuss the possibility that this process is responsible for particle acceleration in flares., Comment: 10 pages, 6 figures, accepted for publication in ApJ

Solar flares are an abrupt release of magnetic energy in the Sun's atmosphere due to reconnection of the coronal magnetic field. This occurs in response to turbulent flows at the photosphere which twist the coronal field. Similar to earthquakes, solar flares represent the behavior of a complex system, and expectedly their energy distribution follows a power law. We present a statistical model based on the principle of minimum energy in a coronal loop undergoing magnetic reconnection, which is described as an avalanche process. We show the distribution of peaks for the flaring events in this self-organized critical system is scale-free. The obtained power law index 1.84 $ \pm $ 0.02 for the peaks is in good agreement with satellite observations of soft X-ray flares. The principle of minimum energy can be applied for general avalanche models to describe many other phenomena.

Aims. Our main aim is to study the relationship between low-energy solar particles (energies below 1 MeV) and very narrow coronal mass ejections (“jets” with angular width ≤ 20°). Methods. For this purpose, we considered 125 very narrow coronal mass ejections (CMEs) from 1999 to 2003 that are potentially associated with low-energy solar particles (LESPs). These events were chosen on the basis of their source location. We studied only very narrow CMEs at the western limb, which are expected to have good magnetic connectivity with Earth. Results. We found 24 very narrow CMEs associated with energetic particles such as ions (protons and 3He), electrons, or both. We show that arrival times at Earth of energetic particles are consistent with onset times of the respective CMEs, and that in the same time intervals, there are no other potential sources of energetic particles. We also demonstrate statistical differences for the angular width distributions using the Kolmogorov–Smirnov test for angular widths for these 24 events. We consider a coherent sample of jets (mostly originating from boundaries of coronal holes) to identify properties of events that produce solar energetic particles (velocities, widths, and position angles). Our study presents a new approach and result: very narrow CMEs can generate low-energy particles in the vicinity of Earth without other activity on the Sun. The results could be very useful for space weather forecasting.

Magnetic field line mapping between the Sun and the Earth is important to trace the nonthermal particles. We generalize a recently developed mapping approach (B stepping) where this approach allows us to map the magnetic field lines by stepping along the local magnetic field direction. We employ an advanced solar wind model which includes acceleration, angular momentum conservation, and intrinsic non-radial velocities and magnetic fields at the inner boundary / source surface. We map the field lines using Wind spacecraft data for two solar rotation periods: one near a solar minimum between CR2118 and CR2119 and other CR1992 near a solar maximum. Maps using the accelerating solar wind model are compared with the maps using a constant solar wind model. On a broad scale, maps using two solar wind models for the same solar rotation periods are very similar. However, in a small scale, there are significant differences, e.g. differences are evident in connectivities, paths, and winding angles. In addition, field lines using the accelerating solar wind model are more azimuthally oriented for during the solar maximum. These differences demonstrate the significance of inclusion of the accelerating radial speed profile, intrinsic azimuthal velocity and magnetic field components.

Fil: Charbonneau, Paul. University Of Montreal; Canadá Fil: Leibacher, John. National Solar Observatory; Estados Unidos. Universite Paris-sud Xi; Francia Fil: Mandrini, Cristina Hemilse. Consejo Nacional de Investigaciónes Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Astronomía y Física del Espacio. - Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Astronomía y Física del Espacio; Argentina Fil: van Driel Gesztelyi, Lidia. Colegio Universitario de Londres; Reino Unido. Université Paris Diderot - Paris 7; Francia. Konkoly Observatory Hungarian Academy Of Sciences; Hungría Fil: Wheatland, Michael S.. The University Of Sydney; Australia

A Bayesian method for forecasting solar cycles is presented. The approach combines a Fokker--Planck description of short--timescale (daily) fluctuations in sunspot number (\citeauthor{NobleEtAl2011}, 2011, \apj{} \textbf{732}, 5) with information from other sources, such as precursor and/or dynamo models. The forecasting is illustrated in application to two historical cycles (cycles 19 and 20), and then to the current solar cycle (cycle 24). The new method allows the prediction of quantiles, i.e. the probability that the sunspot number falls outside large or small bounds at a given future time. It also permits Monte Carlo simulations to identify the expected size and timing of the peak daily sunspot number, as well as the smoothed sunspot number for a cycle. These simulations show how the large variance in daily sunspot number determines the actual reliability of any forecast of the smoothed maximum of a cycle. For cycle 24 we forecast a maximum daily sunspot number of $166\pm 24$, to occur in March 2013, and a maximum value of the smoothed sunspot number of $66\pm5$, indicating a very small solar cycle., Comment: 30 pages, 12 figures. Accepted for publication in Sol. Phys. (24/10/2011)

The NOAA active region AR 11029 was a small but highly active sunspot region which produced 73 GOES soft X-ray flares. The flares appear to show a departure from the well known power-law frequency-size distribution. Specifically, too few GOES C-class and no M-class flares were observed by comparison with a power-law distribution (Wheatland in Astrophys. J. 710, 1324, 2010). This was conjectured to be due to the region having insufficient magnetic energy to power large events. We construct nonlinear force-free extrapolations of the coronal magnetic field of active region AR 11029 using data taken on 24 October by the SOLIS Vector-SpectroMagnetograph (SOLIS/VSM), and data taken on 27 October by the Hinode Solar Optical Telescope SpectroPolarimeter (Hinode/SP). Force-free modeling with photospheric magnetogram data encounters problems because the magnetogram data are inconsistent with a force-free model, and we employ a recently developed `self-consistency' procedure which addresses this and accommodates uncertainties in the boundary data (Wheatland and Regnier in Astrophys. J. 700, L88, 2009). We calculate the total energy and free energy of the self-consistent solution and find that the free energy was 4x10^29 erg on 24 October, and 7x10^31 erg on 27 October. An order of magnitude scaling between RHESSI non-thermal energy and GOES peak X-ray flux is established from a sample of flares from the literature and is used to estimate flare energies from observed GOES peak X-ray flux. Based on the scaling, we conclude that the estimated free energy of AR 11029 on 27 October when the flaring rate peaked is sufficient to power M-class or X-class flares, and hence the modeling does not appear to support the hypothesis that the absence of large flares is due to the region having limited energy.

Type III solar radio storms, observed at frequencies below approximately 16 MHz by space borne radio experiments, correspond to the quasi-continuous, bursty emission of electron beams onto open field lines above active regions. The mechanisms by which a storm can persist in some cases for more than a solar rotation whilst exhibiting considerable radio activity are poorly understood. To address this issue, the statistical properties of a type III storm observed by the STEREO/WAVES radio experiment are presented, examining both the brightness distribution and (for the first time) the waiting-time distribution. Single power law behavior is observed in the number distribution as a function of brightness; the power law index is approximately 2.1 and is largely independent of frequency. The waiting-time distribution is found to be consistent with a piecewise-constant Poisson process. This indicates that during the storm individual type III bursts occur independently and suggests that the storm dynamics are consistent with avalanche type behavior in the underlying active region., 14 pages, 4 figures, 1 table. Accepted for publication in Astrophysical Journal Letters

A Monte-Carlo approach to solving a stochastic jump transition model for active-region energy (Wheatland and Glukhov, Astrophys. J. 494, 1998; Wheatland, Astrophys. J. 679, 2008) is described. The new method numerically solves the stochastic differential equation describing the model, rather than the equivalent master equation. This has the advantages of allowing more efficient numerical solution, the modelling of time-dependent situations, and investigation of details of event statistics. The Monte-Carlo approach is illustrated by application to a Gaussian test case, and to the class of flare-like models presented in Wheatland (2008), which are steady-state models with constant rates of energy supply, and power-law distributed jump transition rates. These models have two free parameters: an index ($\delta $), which defines the dependence of the jump transition rates on active-region energy, and a non-dimensional ratio ($\overline{r})$ of total flaring rate to rate of energy supply. For $\overline{r}\ll 1$ the non-dimensional mean energy $$ of the active-region satisfies $ \gg 1$, resulting in a power-law distribution of flare events over many decades in energy. The Monte-Carlo method is used to explore the behavior of the waiting-time distributions for the flare-like models. The models with $\delta\neq 0$ are found to have waiting times which depart significantly from simple Poisson behavior when $ \gg 1$. The original model from Wheatland and Glukhov (1998), with $\delta=0$ (no dependence of transition rates on active-region energy), is identified as being most consistent with observed flare statistics., Comment: 20 pages, 8 figures

A stochastic model for the energy of a flaring solar active region is presented, generalising and extending the approach of Wheatland & Glukhov (1998). The probability distribution for the free energy of an active region is described by the solution to a master equation involving deterministic energy input and random jump transitions downwards in energy (solar flares). It is shown how two observable distributions, the flare frequency-energy distribution and the flare waiting-time distribution, may be derived from the steady-state solution to the master equation, for given choices for the energy input and for the rates of flare transitions. An efficient method of numerical solution of the steady-state master equation is presented. Solutions appropriate for flaring, involving a constant rate of energy input and power-law distributed jump transition rates, are numerically investigated. The flare-like solutions exhibit power-law flare frequency-energy distributions below a high energy rollover, set by the largest energy the active region is likely to have. The solutions also exhibit approximately exponential (i.e. Poisson) waiting-time distributions, despite the rate of flaring depending on the free energy of the system., 24 pages, 6 figures. Accepted for publication in the Astrophysical Journal

We compare a variety of nonlinear force-free field (NLFFF) extrapolation algorithms, including optimization, magneto-frictional, and Grad – Rubin-like codes, applied to a solar-like reference model. The model used to test the algorithms includes realistic photospheric Lorentz forces and a complex field including a weakly twisted, right helical flux bundle. The codes were applied to both forced “photospheric” and more force-free “chromospheric” vector magnetic field boundary data derived from the model. When applied to the chromospheric boundary data, the codes are able to recover the presence of the flux bundle and the field’s free energy, though some details of the field connectivity are lost. When the codes are applied to the forced photospheric boundary data, the reference model field is not well recovered, indicating that the combination of Lorentz forces and small spatial scale structure at the photosphere severely impact the extrapolation of the field. Preprocessing of the forced photospheric boundary does improve the extrapolations considerably for the layers above the chromosphere, but the extrapolations are sensitive to the details of the numerical codes and neither the field connectivity nor the free magnetic energy in the full volume are well recovered. The magnetic virial theorem gives a rapid measure of the total magnetic energy without extrapolation though, like the NLFFF codes, it is sensitive to the Lorentz forces in the coronal volume. Both the magnetic virial theorem and the Wiegelmann extrapolation, when applied to the preprocessed photospheric boundary, give a magnetic energy which is nearly equivalent to the value derived from the chromospheric boundary, but both underestimate the free energy above the photosphere by at least a factor of two. We discuss the interpretation of the preprocessed field in this context. When applying the NLFFF codes to solar data, the problems associated with Lorentz forces present in the low solar atmosphere must be recognized: the various codes will not necessarily converge to the correct, or even the same, solution.

According to current practice, the results of each run of a radiochemical solar neutrino experiment comprise an estimate of the flux and upper and lower error estimates. These estimates are derived by a maximum-likelihood procedure from the times of decay events in the analysis chamber. This procedure has the following shortcomings: (a) published results sometimes include negative flux estimates; (b) even if the flux estimate is non-negative, the probability distribution function implied by the flux and error estimates will extend into negative territory; and (c) the overall flux estimate derived from the results of a sequence of runs may differ substantially from an estimate made by a “global” analysis of all of the timing data taken together. These defects indicate that the usual “packaging” of data in radiochemical solar neutrino experiments provides an inadequate summary of the data, which implies a loss of information. This article reviews this problem from a Bayesian perspective, and we suggest an alternative scheme for the packaging of radiochemical solar neutrino data, which we believe is free from these shortcomings.

Quantitative models of sunspot and starspot decay predict the timescale of magnetic diffusion and may yield important constraints in stellar dynamo models. Motivated by recent measurements of starspot lifetimes, we investigate the disintegration of a magnetic flux tube by nonlinear diffusion. Previous theoretical studies are extended by considering two physically motivated functional forms for the nonlinear diffusion coefficient $D$: an inverse power-law dependence $D \propto B^{-\nu}$ and a step-function dependence of $D$ on the magnetic field magnitude $B$. Analytical self-similar solutions are presented for the power-law case, including solutions exhibiting "superfast" diffusion. For the step-function case, the heat-balance integral method yields approximate solutions, valid for moderately suppressed diffusion in the spot. The accuracy of the resulting solutions is confirmed numerically, using a method which provides an accurate description of long-time evolution by imposing boundary conditions at infinite distance from the spot. The new models may allow insight into differences and similarities between sunspots and starspots., Comment: 12 pages, 5 figures, to appear in the Astrophysical Journal (accepted 11 November 2016)

Prediction of solar flares is an important task in solar physics. The occurrence of solar flares is highly dependent on the structure and the topology of solar magnetic fields. A new method for predicting large (M and X class) flares is presented, which uses machine learning methods applied to the Zernike moments of magnetograms observed by the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO) for a period of six years from 2 June 2010 to 1 August 2016. Magnetic field images consisting of the radial component of the magnetic field are converted to finite sets of Zernike moments and fed to the Support Vector Machine (SVM) classifier. Zernike moments have the capability to elicit unique features from any 2-D image, which may allow more accurate classification. The results indicate whether an arbitrary active region has the potential to produce at least one large flare. We show that the majority of large flares can be predicted within 48 hours before their occurrence, with only 10 false negatives out of 385 flaring active region magnetograms, and 21 false positives out of 179 non-flaring active region magnetograms. Our method may provide a useful tool for prediction of solar flares which can be employed alongside other forecasting methods., Comment: This paper has been accepted for publication in The Astrophysical Journal

Improvements to an existing method for calculating nonlinear force-free magnetic fields (Wheatland, Solar Phys. 238, 29, 2006) are described. In particular a solution of the 3-D Poisson equation using 2-D Fourier transforms is presented. The improved nonlinear force-free method is demonstrated in application to linear force-free test cases with localized nonzero values of the normal component of the field in the boundary. These fields provide suitable test cases for nonlinear force-free calculations because the boundary conditions involve localized nonzero values of the normal components of the field and of the current density, and because (being linear force-free fields) they have more direct numerical solutions. Despite their simplicity, fields of this kind have not been recognized as test cases for nonlinear methods before. The examples illustrate the treatment of the boundary conditions on current in the nonlinear force-free method, and in particular the limitations imposed by field lines that connect outside of the boundary region.

Existing methods for calculating nonlinear force-free magnetic fields are slow, and are likely to be inadequate for reconstructing coronal magnetic fields based on high-resolution vector magnetic field data from a new generation of spectro-polarimetric instruments. In this paper a new implementation of the current-field iteration method is presented, which is simple, fast, and accurate. The time taken by the method scales as N4, for a three-dimensional grid with N3 points. The method solves the field-updating part of the iteration by exploiting a three-dimensional Fast Fourier Transform solution of Ampere’s law with a current density field constructed to satisfy the required boundary conditions, and uses field line tracing to solve the current-updating part of the iteration. The method is demonstrated in application to a known nonlinear force-free field and to a bipolar test case.

There is a variety of observational evidence for solar flare sympathy, i.e., the triggering of a flare in one active region by a flare in another region. Models for solar flare statistics, however, usually ignore sympathy by assuming that flares occur as independent events. In this paper, we argue that statistical flare models should be robust to the effects of sympathetic flaring. Further, we investigate the consequences of flare sympathy for a specific model of flare statistics, the M. S. Wheatland and I. J. D. Craig (Astrophys. J.595, 458, 2003) model. The original treatment describes an assembly of reconnecting structures (labelled separators) that flare independently according to simple rules consistent with magnetohydrodynamic flare models. This description is modified by allowing a flare at one separator to increase the probability of flaring at all other separators for a period of time following the flare, by an amount proportional to the energy of the flare. Simulations illustrate the transition of the model from weak to strong sympathy. In the limit of weak sympathy, the model reproduces observed flare frequency–energy and waiting-time distributions.

Solar flare sympathy is the triggering of a flare in one active region by a flare in another. Statistical tests for flare sympathy have returned varying results. However, existing tests have relied on flaring rates in active regions being constant in time, or else have attempted to model the rate variation, which is a difficult task. A simple test is described which is independent of flaring rates. The test generalizes the approach of L. Fritzova-Svestkova, R.C. Chase, and Z. Svestka [Solar Phys. 48, 275, 1976], and examines the distribution of flare coincidences in pairs of active regions as a function of coincidence interval τ. The test is applied to available soft X-ray and Hα flare event listings. The soft X-ray events exhibit a deficit of flare coincidences for τ≤;20 min, which is most likely due to an event-selection effect whereby the increased soft X-ray emission due to one flare prevents a second flare being identified. The Hα events show an excess of flare coincidences for τ≤; 10 min, suggesting flare sympathy. The number of Hα event pairs occurring within 10 min of one another is higher than that expected on the basis of random coincidence by a fraction 0.12± 0.02. Nearby active regions (spatial separation

We compare six algorithms for the computation of nonlinear force-free (NLFF) magnetic fields (including optimization, magnetofrictional, Grad-Rubin based, and Green's function-based methods) by evaluating their performance in blind tests on analytical force-free-field models for which boundary conditions are specified either for the entire surface area of a cubic volume or for an extended lower boundary only. Figures of merit are used to compare the input vector field to the resulting model fields. Based on these merit functions, we argue that all algorithms yield NLFF fields that agree best with the input field in the lower central region of the volume, where the field and electrical currents are strongest and the effects of boundary conditions weakest. The NLFF vector fields in the outer domains of the volume depend sensitively on the details of the specified boundary conditions; best agreement is found if the field outside of the model volume is incorporated as part of the model boundary, either as potential field boundaries on the side and top surfaces, or as a potential field in a skirt around the main volume of interest. For input field (B) and modeled field (b), the best method included in our study yields an average relative vector error En =� |B− b|� /�| B|� of only 0.02 when all sides are specified and 0.14 for the case where only the lower boundary is specified, while

We outline a method for quantifying the performance of extrapolation methods for magnetic fields. We extrapolate the field for two model cases, using a linear force-free approach and a nonlinear approach. Each case contains a different topological feature of the field that may be of interest in solar energetic events. We are able to determine quantitatively whether either method is capable of reproducing the topology of the field. In one of our examples, a subjective evaluation of the performance of the extrapolation suggests that it has performed quite well, while our quantitative score shows that this is not the case, indicating the importance of being able to quantify the performance. Our method may be useful in determining which extrapolation techniques are best able to reproduce a force-free field and which topological features can be recovered.

The MHD virial theorem may be used to estimate the magnetic energy of active regions based on vector magnetic fields measured at the photosphere or chromosphere. However, the virial estimate depends on the measured vector magnetic field being force-free. Departure from force-freeness leads to an unknown systematic error in the virial energy estimate, and an origin dependence of the result. We present a method for estimating the systematic error by assuming that magnetic forces are confined to a thin layer near the photosphere. If vector magnetic field measurements are available at two levels in the low atmosphere (e.g. the photosphere and the chromosphere), the systematic error may be directly calculated using the observed horizontal and vertical field gradients, resulting in an energy estimate which is independent of the choice of origin. If (as is generally the case) measurements are available at only one level, the systematic error may be approximated using the observed horizontal field gradients together with a simple linear force-free model for the vertical field gradients. The resulting `improved' virial energy estimate is independent of the choice of origin, but depends on the choice of the model for the vertical field gradients, i.e. the value of the linear force-free parameter $\alpha$. This procedure is demonstrated for five vector magnetograms, including a chromospheric magnetogram., Comment: 17 pages, 1 figure

In solar physics, especially in exploratory stages of research, it is often necessary to compare the power spectra of two or more time series. One may, for instance, wish to estimate what the power spectrum of the combined data sets might have been, or one may wish to estimate the significance of a particular peak that shows up in two or more power spectra. One may also on occasion need to search for a complex of peaks in a single power spectrum, such as a fundamental and one or more harmonics, or a fundamental plus sidebands, etc. Visual inspection can be revealing, but it can also be misleading. This leads one to look for one or more ways of forming statistics, which readily lend themselves to significance estimation, from two or more power spectra. We derive formulas for statistics formed from the sum, the minimum, and the product of two or more power spectra. A distinguishing feature of our formulae is that, if each power spectrum has an exponential distribution, each statistic also has an exponential distribution., 26 pages, 4 figures

A numerical approach to calculating nonlinear force-free fields is presented. The approach is similar to Sakurai (1981) being a current-field iteration scheme using the integral solution to Ampere's law (the Biot–Savart law). However, the method of solution presented here is simpler than Sakurai's approach, in that the field is directly constructed on a grid without the intermediate solution of a large system of nonlinear equations. The method also permits straightforward implementation on parallel computers. Results of applying the method to a number test cases, including boundary conditions with substantial currents, are presented.

Wheatland and Litvinenko (2001) presented a model for dynamical energy balance in the flaring solar corona which predicts a time lag between flare occurrence and the supply of energy to the corona (`driving'). They also suggested that an observed net lag between flare numbers and sunspot numbers over cycles 21 and 22 might provide support for the model. Temmer, Veronig, and Hanslmeier (2003) examined data for five individual solar cycles (19–23) and confirmed a lag between flare and sunspot numbers for odd solar cycles, but found no lag for even cycles. Following the suggestion of Temmer, Veronig, and Hanslmeier, the energy balance model is here extended to incorporate 22-year driving consistent with the phenomenological Gnevyshev—Ohl rule. The model is found to exhibit a greater lag for the smaller (even) cycles, in contradiction with the findings of Temmer, Veronig, and Hanslmeier. A modification to the model is investigated in which the flaring rate is proportional to the free energy and to the driving rate for small driving rates, but is proportional only to the free energy for large driving rates. The modified model can in principle account for the observations.

Circuit models involving bulk currents and inductances are often used to estimate the energies of coronal magnetic field configurations, in particular configurations associated with solar flares. The accuracy of circuit models is tested by comparing calculated energies of linear force-free fields with specified boundary conditions with corresponding circuit estimates. The circuit models are found to provide reasonable (order of magnitude) estimates for the energies of the non-potential components of the fields, and to reproduce observed functional dependences of the energies. However, substantial departure from the circuit estimates is observed for large values of the force-free parameter, and this is attributed to the influence of the non-potential component of the field on the path taken by the current., Comment: 16 pages, 8 figures, to appear in Solar Physics

A model to account for observed solar flare statistics in terms of a superposition of independent random flaring elements (assumed to be sites of magnetic reconnection in the coronal magnetic field and hence termed '' separators '') is described. A separator of length l is assumed to flare as a Poisson process in time, with a rate � ðlÞ inversely proportional to the Alfven transit time for the structure. It is shown that a relationship E / l � between the mean energy of events E at a separator and the separator length implies a relationship E / � � between individual waiting timesand energies E of events at the separator. The most plausible � ¼ 2 model is found to be compatible with simple pictures for magnetohydrodynamic energy storage prior to magnetic reconnection in a current sheet with anomalous (turbulent) resistivity. Formal inversion of the observed flare frequency-energy distribution is shown to imply a distribution Pðl Þ/ l � 1 of the separator lengths in active regions. A simulation confirms the basic results of the model. It is also demonstrated that a model comprising time-dependent separator numbers N ¼ NðtÞ can reproduce an observed power-law tail in the flare waiting- time distribution, for large waiting times.

The distribution of times Δt between coronal mass ejections (CMEs) in the Large Angle and Spectrometric Coronagraph (LASCO) CME catalog for the years 1996–2001 is examined. The distribution exhibits a power-law tail ∼(Δt)γ with an index γ≈−2.36±0.11 for large waiting times (Δt>10 hours). The power-law index of the waiting-time distribution varies with the solar cycle: for the years 1996–1998 (a period of low activity), the power-law index is γ≈−1.86±0.14, and for the years 1999–2001 (a period of higher activity), the index is γ≈−2.98±0.20. The observed CME waiting-time distribution, and its variation with the cycle, may be understood in terms of CMEs occurring as a time-dependent Poisson process. The CME waiting-time distribution is compared with that for greater than C1 class solar flares in the Geostationary Operational Environmental Satellite (GOES) catalog for the same years. The flare and CME waiting-time distributions exhibit power-law tails with very similar indices and time variation.