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 ($\alpha_{sep}\approx 6^\circ, 43^\circ, 89^\circ, 127^\circ$, 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 $\alpha_{sep} \approx 22^\circ-125^\circ$, 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 $\alpha_{sep}\approx 6^\circ-127^{\circ}$ we find a mean 3D misalignment angle of $\mu_{PF} \approx 21^\circ-39^\circ$ of stereoscopically triangulated loops with magnetic potential field models, and $\mu_{FFF} \approx 15^\circ-21^\circ$ for a force-free field model, which is partly caused by stereoscopic uncertainties $\mu_{SE} \approx 9^\circ$. We predict optimum conditions for solar stereoscopy during the time intervals of 2012--2014, 2016--2017, and 2021--2023.Comment: Solar Physics, (in press), 22 pages, 9 figure

Transverse oscillations of systems of coronal loops

We study the collective kinklike normal modes of a system of several cylindrical loops using the T-matrix theory. Loops that have similar kink frequencies oscillate collectively with a frequency which is slightly different from that of the individual kink mode. On the other hand, if the kink frequency of a loop is different from that of the others, it oscillates individually with its own frequency. Since the individual kink frequency depends on the loop density but not on its radius for typical 1 MK coronal loops, a coupling between kink oscillations of neighboring loops take place when they have similar densities. The relevance of these results in the interpretation of the oscillations studied by \citet{schrijver2000} and \citet{verwichte2004}, in which transverse collective loop oscillations seem to be detected, is discussed. In the first case, two loops oscillating in antiphase are observed; interpreting this motion as a collective kink mode suggests that their densities are roughly equal. In the second case, there are almost three groups of tubes that oscillate with similar periods and therefore their dynamics can be collective, which again seems to indicate that the loops of each group share a similar density. All the other loops seem to oscillate individually and their densities can be different from the rest

Transverse oscillations of a multi-stranded loop

We investigate the transverse oscillations of a line-tied multi-stranded coronal loop composed of several parallel cylindrical strands. First, the collective fast normal modes of the loop are found with the T-matrix theory. There is a huge quantity of normal modes with very different frequencies and a complex structure of the associated magnetic pressure perturbation and velocity field. The modes can be classified as bottom, middle, and top according to their frequencies and spatial structure. Second, the temporal evolution of the velocity and magnetic pressure perturbation after an initial disturbance are analyzed. We find complex motions of the strands. The frequency analysis reveals that these motions are a combination of low and high frequency modes. The complexity of the strand motions produces a strong modulation of the whole tube movement. We conclude that the presumed internal fine structure of a loop influences its transverse oscillations and so its transverse dynamics cannot be properly described by those of an equivalent monolithic loop.Comment: Accepted in Ap

Deconvolution of directly precipitating and trap-precipitating electrons in solar flare hard x-rays. III.Yohkoh hard x-ray telescope data analysis

We analyze the footpoint separation d and flux asymmetry A of magnetically conjugate double footpoint sources in hard X-ray images from the Yohkoh Hard X-Ray Telescope (HXT). The data set of 54 solar flares includes all events simultaneously observed with the Compton Gamma Ray Observatory (CGRO) in high time resolution mode. From the CGRO data we deconvolved the direct-precipitation and trap-precipitation components previously (in Paper II). Using the combined measurements from CGRO and HXT, we develop an asymmetric trap model that allows us to quantify the relative fractions of four different electron components, i.e., the ratios of direct-precipitating (q_P1, q_P2) and trap-precipitating electrons (q_T1, q_T2) at both magnetically conjugate footpoints. We find mean ratios of q_P1=0.14+/-0.06, q_P2=0.26+/-0.10, and q_T=q_T1+q_T2=0.60+/-0.13. We assume an isotropic pitch-angle distribution at the acceleration site and double-sided trap precipitation (q_T2/q_T1=q_P2/q_P1) to determine the conjugate loss-cone angles (alpha_1=42^deg+/-11^deg and alpha_2=52^deg+/-10^deg) and magnetic mirror ratiosat both footpoints (R_1=1.6,...,4.0 and R_2=1.3,...,2.5). From the relative displacement of footpoint sources we also measure altitude differences of hard X-ray emission at different energies, which are found to decrease systematically with higher energies, with a statistical height difference of h_Lo-h_M1=980+/-250 km and h_M1-h_M2=310+/-300 km between the three lower HXT energy channels (Lo, M1, M2

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 $\mu=19^\circ-46^\circ$ for a potential field model, which is reduced to $\mu=14^\circ-19^\circ$ for a non-potential field model parameterized by twist parameters. The residual error is commensurable with stereoscopic measurement errors ($\mu_{SE} \approx 8^\circ-12^\circ$). 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 $\alpha$-parameter (median $\alpha \approx 4 \times 10^{-11}$ 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 $\alpha$), represents the first code that enables fast forward-fitting to photospheric magnetograms and stereoscopically triangulated loops in the solar corona.Comment: The Astrophysical Journal (in press), 37 pages, 14 Figure

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 $\alpha({\bf x})$, the electric current density ${\bf j}({\bf x})$, and are accurate to second-order (of the nonlinear force-free $\alpha$-parameter). The explicit expressions of a force-free field can easily be applied to modeling or forward-fitting of many coronal phenomena.Comment: Solar Physics (in press), 26 pages, 11 figure

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 $\mu \approx 1^\circ-10^\circ$), (ii) relatively fast computation times (from seconds to a few minutes), and (iii) the high fidelity of retrieved force-free $\alpha$-parameters ($\alpha_{\rm fit}/\alpha_{\rm model} \approx 0.9-1.0$ for simulations and $\alpha_{\rm fit}/\alpha_{\rm model} \approx 0.7\pm0.3$ 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.Comment: Solar PHysics, (in press), 25 pages, 11 figure

The Role of fast magnetosonic waves in the release and conversion via reconnection of energy stored by a current sheet

Using a simple two-dimensional, zero-beta model, we explore the manner by which reconnection at a current sheet releases and dissipates free magnetic energy. We find that only a small fraction (3%-11% depending on current sheet size) of the energy is stored close enough to the current sheet to be dissipated abruptly by the reconnection process. The remaining energy, stored in the larger-scale field, is converted to kinetic energy in a fast magnetosonic disturbance propagating away from the reconnection site, carrying the initial current and generating reconnection-associated flows (inflow and outflow). Some of this reflects from the lower boundary (the photosphere) and refracts back to the X-point reconnection site. Most of this inward wave energy is reflected back again, and continues to bounce between X-point and photosphere until it is gradually dissipated, over many transits. This phase of the energy dissipation process is thus global and lasts far longer than the initial purely local phase. In the process a significant fraction of the energy (25%-60%) remains as undissipated fast magnetosonic waves propagating away from the reconnection site, primarily upward. This flare-generated wave is initiated by unbalanced Lorentz forces in the reconnection-disrupted current sheet, rather than by dissipation-generated pressure, as some previous models have assumed. Depending on the orientation of the initial current sheet the wave front is either a rarefaction, with backward directed flow, or a compression, with forward directed flow

Deterministically Driven Avalanche Models of Solar Flares

We develop and discuss the properties of a new class of lattice-based avalanche models of solar flares. These models are readily amenable to a relatively unambiguous physical interpretation in terms of slow twisting of a coronal loop. They share similarities with other avalanche models, such as the classical stick--slip self-organized critical model of earthquakes, in that they are driven globally by a fully deterministic energy loading process. The model design leads to a systematic deficit of small scale avalanches. In some portions of model space, mid-size and large avalanching behavior is scale-free, being characterized by event size distributions that have the form of power-laws with index values, which, in some parameter regimes, compare favorably to those inferred from solar EUV and X-ray flare data. For models using conservative or near-conservative redistribution rules, a population of large, quasiperiodic avalanches can also appear. Although without direct counterparts in the observational global statistics of flare energy release, this latter behavior may be relevant to recurrent flaring in individual coronal loops. This class of models could provide a basis for the prediction of large solar flares.Comment: 24 pages, 11 figures, 2 tables, accepted for publication in Solar Physic