arXiv subjects

M. S. Wheatland

Publications and source records attributed to M. S. Wheatland.

At least 37 records · Page 2Linked to original sources

arXiv2010

Evidence for departure from a power-law flare size distribution for a small solar active region

Active region 11029 was a small, highly flare-productive solar active region observed at a time of extremely low solar activity. The region produced only small flares: the largest of the $>70$ Geostationary Observational Environmental Satellite (GOES) events for the region has a peak 1--$8Å$ flux of $2.2\times 10^{-6} {\rm W} {\rm m}^{-2}$ (GOES C2.2). The background-subtracted GOES peak-flux distribution suggests departure from power-law behavior above $10^{-6} {\rm W} {\rm m}^{-2}$, and a Bayesian model comparison strongly favors a power-law plus rollover model for the distribution over a simple power-law model. The departure from the power law is attributed to this small active region having a finite amount of energy. The rate of flaring in the region varies with time, becoming very high for two days coinciding with the onset of an increase in complexity of the photospheric magnetic field. The observed waiting-time distribution for events is consistent with a piecewise-constant Poisson model. These results present challenges for models of flare statistics and of energy balance in solar active regions.

astro-ph.SR↗

arXiv2009

On the Brightness and Waiting-time Distributions of a Type III Radio Storm observed by STEREO/WAVES

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.

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arXiv2009

A self-consistent nonlinear force-free solution for a solar active region magnetic field

Nonlinear force-free solutions for the magnetic field in the solar corona constructed using photospheric vector magnetic field boundary data suffer from a basic problem: the observed boundary data are inconsistent with the nonlinear force-free model. Specifically, there are two possible choices of boundary conditions on vertical current provided by the data, and the two choices lead to different force-free solutions. A novel solution to this problem is described. Bayesian probability is used to modify the boundary values on current density, using field-line connectivity information from the two force-free solutions and taking into account uncertainties, so that the boundary data are more consistent with the two nonlinear force-free solutions. This procedure may be iterated until a set of self-consistent boundary data (the solutions for the two choices of boundary conditions are the same) is achieved. The approach is demonstrated to work in application to Hinode/SOT observations of NOAA active region 10953.

astro-ph.SR↗

arXiv2009

Monte-Carlo Simulation of Solar Active-Region Energy

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 ($δ$), 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 $<\overline{E}>$ of the active-region satisfies $<\overline{E}> \gg 1$, 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 $δ\neq 0$ are found to have waiting times which depart significantly from simple Poisson behavior when $<\overline{E}> \gg 1$. The original model from Wheatland and Glukhov (1998), with $δ=0$ (no dependence of transition rates on active-region energy), is identified as being most consistent with observed flare statistics.

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arXiv2008

Dynamics of a double pendulum with distributed mass

We investigate a variation of the simple double pendulum in which the two point masses are replaced by square plates. The double square pendulum exhibits richer behavior than the simple double pendulum and provides a convenient demonstration of nonlinear dynamics and chaos. It is also an example of an asymmetric compound double pendulum, which has not been studied in detail. We obtain the equilibrium configurations and normal modes of oscillation and derive the equations of motion, which are solved numerically to produce Poincaré sections. We show how the behavior varies from regular motion at low energies, to chaos at intermediate energies, and back to regular motion at high energies. We also show that the onset of chaos occurs at a significantly lower energy than for the simple double pendulum.

physics.comp-ph↗

arXiv2008

The Energetics of a Flaring Solar Active Region, and Observed Flare Statistics

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.

astro-ph↗

arXiv2007

Non-linear force-free field modeling of a solar active region around the time of a major flare and coronal mass ejection

Solar flares and coronal mass ejections are associated with rapid changes in field connectivity and powered by the partial dissipation of electrical currents in the solar atmosphere. A critical unanswered question is whether the currents involved are induced by the motion of pre-existing atmospheric magnetic flux subject to surface plasma flows, or whether these currents are associated with the emergence of flux from within the solar convective zone. We address this problem by applying state-of-the-art nonlinear force-free field (NLFFF) modeling to the highest resolution and quality vector-magnetographic data observed by the recently launched Hinode satellite on NOAA Active Region 10930 around the time of a powerful X3.4 flare. We compute 14 NLFFF models with 4 different codes and a variety of boundary conditions. We find that the model fields differ markedly in geometry, energy content, and force-freeness. We discuss the relative merits of these models in a general critique of present abilities to model the coronal magnetic field based on surface vector field measurements. For our application in particular, we find a fair agreement of the best-fit model field with the observed coronal configuration, and argue (1) that strong electrical currents emerge together with magnetic flux preceding the flare, (2) that these currents are carried in an ensemble of thin strands, (3) that the global pattern of these currents and of field lines are compatible with a large-scale twisted flux rope topology, and (4) that the ~10^32 erg change in energy associated with the coronal electrical currents suffices to power the flare and its associated coronal mass ejection.

astro-ph↗

arXiv2007

Analysis and packaging of radiochemical solar neutrino data. 1. Bayesian approach

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 suggests an alternative scheme for the packaging of radiochemical solar neutrino data, which is we believe free from the above objections.

hep-ph↗

arXiv2005

An Improved Virial Estimate of Solar Active Region Energy

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 $α$. This procedure is demonstrated for five vector magnetograms, including a chromospheric magnetogram.

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arXiv2005

Power-Spectrum Analyses of Super-Kamiokande Solar Neutrino Data: Variability and its Implications for Solar Physics and Neutrino Physics

There have been conflicting claims as to whether or not power-spectrum analysis of the Super-Kamiokande solar neutrino data yields evidence of variability. Comparison of these claims is complicated by the fact that the relevant articles may use different datasets, different methods of analysis, and different procedures for significance estimation. The purpose of this article is to clarify the role of power spectrum analysis. To this end, we analyze only the Super-Kamiokande 5-day dataset, and we use a standard procedure for significance estimation proposed by the Super-Kamiokande collaboration. We then analyze this dataset, with this method of significance estimation, using six methods of power spectrum analysis. We find that the significance of the principal peak in the power spectrum (that at 9.43 yr-1with a depth of modulation of 7%) shows a clear correlation with the amount and relevance of the information being processed, as would be expected if there were a real signal in the data. The significance level reaches 99.3% for one method of analysis. We discuss, in terms of sub-dominant processes, possible neutrino-physics interpretations of the apparent variability of the Super-Kamiokande measurements, and we suggest steps that could be taken to resolve the question of variability of the solar neutrino flux.

hep-ph↗

arXiv2005

A Statistical Solar Flare Forecast Method

A Bayesian approach to solar flare prediction has been developed, which uses only the event statistics of flares already observed. The method is simple, objective, and makes few ad hoc assumptions. It is argued that this approach should be used to provide a baseline prediction for certain space weather purposes, upon which other methods, incorporating additional information, can improve. A practical implementation of the method for whole-Sun prediction of Geostationary Observational Environment Satellite (GOES) events is described in detail, and is demonstrated for 4 November 2003, the day of the largest recorded GOES flare. A test of the method is described based on the historical record of GOES events (1975-2003), and a detailed comparison is made with US National Oceanic and Atmospheric Administration (NOAA) predictions for 1987-2003. Although the NOAA forecasts incorporate a variety of other information, the present method out-performs the NOAA method in predicting mean numbers of event days, for both M-X and X events. Skill scores and other measures show that the present method is slightly less accurate at predicting M-X events than the NOAA method, but substantially more accurate at predicting X events, which are important contributors to space weather.

astro-ph↗

arXiv2005

Combined and Comparative Analysis of Power Spectra

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.

astro-ph↗

arXiv2004

Initial test of a Bayesian approach to solar flare prediction

A test of a new Bayesian approach to solar flare prediction (Wheatland 2004a) is presented. The approach uses the past history of flaring together with phenomenological rules of flare statistics to make a prediction for the probability of occurrence of a large flare within an interval of time, or to refine an initial prediction (which may incorporate other information). The test of the method is based on data from the Geostationary Observational Environmental Satellites (GOES), and involves whole-Sun prediction of soft X-ray flares for 1976-2003. The results show that the method somewhat over-predicts the probability of all events above a moderate size, but performs well in predicting large events.

astro-ph↗

arXiv2004

Comment on "Search for periodic modulations of the solar neutrino flux in Super-Kamiokande" by J. Yoo et al

We comment on a recent article by Yoo et al. that presents an analysis of Super-Kamiokande 10-day and 5-day data, correcting certain errors in that article. We also point out that, in using the Lomb-Scargle method of power spectrum analysis, Yoo et al. ignore much of the relevant data. A likelihood analysis, that can take account of all of the relevant data, yields evidence indicative of modulation by solar processes.

hep-ph↗

arXiv2004

A Bayesian Approach to Solar Flare Prediction

A number of methods of flare prediction rely on classification of physical characteristics of an active region, in particular optical classification of sunspots, and historical rates of flaring for a given classification. However these methods largely ignore the number of flares the active region has already produced, in particular the number of small events. The past history of occurrence of flares (of all sizes) is an important indicator to future flare production. We present a Bayesian approach to flare prediction, which uses the flaring record of an active region together with phenomenological rules of flare statistics to refine an initial prediction for the occurrence of a big flare during a subsequent period of time. The initial prediction is assumed to come from one of the extant methods of flare prediction. The theory of the method is outlined, and simulations are presented to show how the refinement step of the method works in practice.

astro-ph↗

arXiv2003

Testing Circuit Models for the Energies of Coronal Magnetic Field Configurations

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.

astro-ph↗

arXiv2003

Statistics of the Chi-Square Type, with Application to the Analysis of Multiple Time-Series Power Spectra

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. Also, one may occasionally 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. The familiar chi-square statistic provides a convenient mechanism for combining variables drawn from normal distributions, and one may generalize the chi-square statistic to be any function of any number of variables with arbitrary distributions. In dealing with power spectra, we are interested mainly in exponential distributions. One well-known statistic, formed from the sum of two or more variables with exponential distributions, satisfies the gamma distribution. We show that a transformation of this statistic has the convenient property that it has an exponential distribution. We introduce two additional statistics formed from two or more variables with exponential distributions. For certain investigations, we may wish to study the minimum power (as a function of frequency) drawn from two or more power spectra. In other investigations, it may be helpful to study the product of the powers. We give numerical examples and an example drawn from our solar-neutrino research.

astro-ph↗

arXiv2003

The coronal mass ejection waiting-time distribution

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 $\propto (Δt)^γ$ with an index $γ\approx -2.36\pm 0.11$ for large waiting times ($Δt>10 {\rm hours}$). 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 $γ\approx-1.86\pm 0.14$, and for the years 1999-2001 (a period of higher activity), the index is $γ\approx-2.98\pm 0.20$. 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.

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