Magnetic Virial Theorem Research Articles

Identification of Pre-flare Processes and Their Possible Role in Driving a Large-scale Flux Rope Eruption with Complex M-class Flare in the Active Region NOAA 12371

In this article, we study the origin of precursor flare activity and investigate its role towards triggering the eruption of a flux rope which resulted into a dual-peak M-class flare (SOL2015-06-21T02:36) in the active region NOAA 12371. The flare evolved in two distinct phases with peak flux levels of M2.1 and M2.6 at an interval of $\approx$54 min. The active region exhibited striking moving magnetic features (MMFs) along with sunspot rotation. Non-linear force free field (NLFFF) modelling of the active region corona reveals a magnetic flux rope along the polarity inversion line in the trailing sunspot group which is observationally manifested by the co-spatial structures of an active region filament and a hot channel identified in the 304 and 94 \AA\ images, respectively, from the Atmospheric Imaging Assembly (AIA). The active region underwent a prolonged phase of flux enhancement followed by a relatively shorter period of flux cancellation prior to the onset of the flare which led to the build up and activation of the flux rope. Extreme ultra-violet (EUV) images reveal localised and structured pre-flare emission, from the region of MMFs, adjacent to the location of the main flare. Our analysis reveals strong, localised regions of photospheric currents of opposite polarities at the precursor location, thereby making the region susceptible to small-scale magnetic reconnection. Precursor reconnection activity from this location most likely induced a slipping reconnetion towards the northern leg of the hot channel which led to the destabilization of the flux rope. The application of magnetic virial theorem suggests that there was an overall growth of magnetic free energy in the active region during the prolonged pre-flare phase which decayed rapidly after the hot channel eruption and its successful transformation into a halo coronal mass ejection (CME).

The magnetic virial theorem and the nature of flares on the Sun and other G stars

Modern observations of magnetic fields confirm basic ideas obtained earlier by A.B. Severny about field changes that occur over the course of a flare. For interpretation of new data of the magnetic field vector, the methods of the non-linear force-free field (NLFFF) extrapolation were developed. As a test of these methods, we carried out the NLFFF extrapolation for the active region AR 11158 (February 2011) using the algorithm by Rudenko and Myshyakov (2009). This NLFFF approximation gives information on the magnetic field structure in the corona of active regions (AR). When considering the only total energy of non-stationary processes, it is possible to shift from the equations describing the structural features to considering the magnetic virial theorem. We obtain new analytic expression for the free energy of the solar corona as a whole and discuss circumstances, which prevent using this analytic expression as a measure of the free energy of a size-limited AR, when magnetic fields of the photosphere of AR are not force-free. Observations and NLFFF extrapolation give the upper estimate of the energy of flares, which are able to occur in a given large active region. For the Sun, this value does not exceed 3×1032ergs. Because the average longitudinal magnetic fields of G stars (at an age of 1–2Gyr) are ten-times stronger than the maximum magnetic field of the Sun as a star, the energy of solar-type stellar flares can only slightly exceed 1034ergs. These flares are associated with the energy deposit above the chromosphere and its subsequent release. Phenomena with the total energy up to 1036ergs which were registered also on the younger G stars have the nature which differs substantially from those non-stationary processes occurring on the Sun.

MAGNETIC ENERGY BUILDUP FOR RELATIVISTIC MAGNETAR GIANT FLARES

Motivated by coronal mass ejection studies, we construct general relativistic models of a magnetar magnetosphere endowed with strong magnetic fields. The equilibrium states of the stationary, axisymmetric magnetic fields in the magnetar magnetosphere are obtained as solutions of the Grad-Shafranov equation in a Schwarzschild spacetime. To understand the magnetic energy buildup in the magnetar magnetosphere, a generalized magnetic virial theorem in the Schwarzschild metric is newly derived. We carefully address the question whether the magnetar magnetospheric magnetic field can build up sufficient magnetic energy to account for the work required to open up the magnetic field during magnetar giant flares. We point out the importance of the Aly-Sturrock constraint, which has been widely studied in solar corona mass ejections, as a reference state in understanding magnetar energy storage processes. We examine how the magnetic field can possess enough energy to overcome the Aly-Sturrock energy constraint and open up. In particular, general relativistic (GR) effects on the Aly-Sturrock energy constraint in the Schwarzschild spacetime are carefully investigated. It is found that, for magnetar outbursts, the Aly-Sturrock constraint is more stringent, i.e., the Aly-Sturrock energy threshold is enhanced due to the GR effects. In addition, neutron stars with greater mass have a higher Aly-Sturrock energy threshold and are more difficult to erupt. This indicates that magnetars are probably not neutron stars with extreme mass. For a typical neutron star with mass of $1-2 M_{\odot}$, we further explore the effects of cross-field current effects, caused by the mass loading, on the possibility of stored magnetic field energy exceeding the Aly-Sturrock threshold.

Nonlinear Force-Free Modeling of Coronal Magnetic Fields. II. Modeling a Filament Arcade and Simulated Chromospheric and Photospheric Vector Fields

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.

Free Magnetic Energy in Solar Active Regions above the Minimum-Energy Relaxed State

To understand the physics of solar flares, including the local reorganization of the magnetic field and the acceleration of energetic particles, we have first to estimate the free magnetic energy available for such phenomena, which can be converted into kinetic and thermal energy. The free magnetic energy is the excess energy of a magnetic configuration compared to the minimum-energy state, which is a linear force-free field if the magnetic helicity of the configuration is conserved. We investigate the values of the free magnetic energy estimated from either the excess energy in extrapolated fields or the magnetic virial theorem. For four different active regions, we have reconstructed the nonlinear force-free field and the linear force-free field corresponding to the minimum-energy state. The free magnetic energies are then computed. From the energy budget and the observed magnetic activity in the active region, we conclude that the free energy above the minimum-energy state gives a better estimate and more insights into the flare process than the free energy above the potential field state.

Structure and evolution of magnetized clusters: entropy profiles,S–TandLX–Trelations

We study the impact of an intracluster magnetic field on the main structural properties of clusters and groups of galaxies: the radial density and entropy profiles, the entropy – temperature relation and the X-ray luminosity – temperature relation for groups and clusters of galaxies. To this aim, we develop a description of the intra-cluster gas based on the Hydrostatic Equilibrium condition and on the Magnetic Virial Theorem in the presence of a radial distribution of the magnetic field $B(r) = B_* (\rho_{\rm g}(r))^{\alpha}$, with $\alpha \approx 0.9$, as the one indicated by observations and numerical simulation. Our analysis shows that such a description is able to provide, at once, a possible explanation of three problematic aspects of the cluster structure: i) the flattening of the entropy profile in the cluster center; ii) the flatness of the $S - T$ relation; iii) the increasing steepening of the $L_{\rm X}-T$ relation from the cluster scale towards the group scales. The available entropy and X-ray luminosity data indicate that an increase of the magnetic field $B_* \sim T^{0.5 \pm 0.1}$ is required to reproduce at the same time both the $S - T$ and the $L_{\rm X}-T$ relations. It follows that a consistent description of the magnetized ICM can provide a simple explanation of several (or of all) of these still open problems, and thus weakens the need for the inclusion of other non-gravitational effects which have been proposed so far for the explanation of some of these features. This (initial, but not conclusive) analysis can be regarded as a starting point for a more refined analytical exploration of the physics of the magnetized intra-cluster medium, and it provides testable predictions that can be proven or disproven with the next coming sensitive observations of groups and clusters in the X-ray band and in the radio frequency band.

A Virial Theorem Investigation of Magnetic Variations in the Sun

The magnetic virial theorem is applied here to a long-standing astrophysical problem, namely, the sign of the Sun's radius change during the solar activity cycle. The solar radius is theoretically found to decrease around the time of maximum magnetic field strength, in agreement with the best available observational evidence. This theoretical prediction, although simply based, instills some confidence by explicitly satisfying the conservation of total energy.

The impact of magnetic field on the cluster M – T relation

We discuss the impact of magnetic field on the mass–temperature relation for groups and clusters of galaxies based on the derivation of the general Magnetic Virial Theorem. The presence of a magnetic field B yields a decrease of the virial temperature T for a fixed mass M: such a decrease in T is stronger for low-mass systems than for high-mass systems. We outline several implications of the presence of B-field and of its mass scaling for the structure and evolution of groups and clusters.

Magnetic Free Energy in NOAA Active Region 10486 on 2003 October 29

We calculate the total and the free magnetic energy for solar NOAA active region 10486 on 2003 October 29 using chromospheric vector magnetograms observed with the Imaging Vector Magnetograph at Mees Solar Observatory in the Na I λ5896 spectral line. The magnetic energy is derived from the magnetic virial theorem using observations spanning the X10 flare that occurred at 20:39 UT. Although poor atmospheric seeing prevented us from discerning changes in the free energy associated with the flare, there was an unusually large amount of free magnetic energy in NOAA AR 10486: (5.7 ± 1.9) × 1033 ergs, which is consistent with the very high level of activity observed in this active region. It is thus plausible that the extreme activity was powered by the magnetic free energy.

Aspects of the Global MHD Equilibria and Filament Eruptions in the Solar Corona

This is a review of several promising approaches for analyzing the accumulation and release of magnetic energy in filament eruptions and coronal mass ejections in the solar corona. The importance of the magnetic virial theorem for understanding the role of slowly changing boundary conditions in the photosphere is stressed. A possible role of the magnetic expulsion force in the solar filament dynamics is also discussed.

The practical application of the magnetic virial theorem

The magnetic virial theorem states that the magnetic energy contained in a coronal force-free magnetic field is given by a surface integral at the photospheric boundary involving the three vector magnetic field components. It is possible to use this theorem together with vector magnetograph data to compute the magnetic energy of solar active regions. In particular, one might compute the energy of an active region both before and after a flare occurs to determine whether the energy observed to be released during the flare is actually extracted from the magnetic field, as is commonly believed