A Quantitative Method to Optimise Magnetic Field Line Fitting of Observed Coronal Loops

Many authors use magnetic-field models to extrapolate the field in the solar corona from magnetic data in the photosphere. The accuracy of such extrapolations is usually judged qualitatively by eye, where a less judgemental quantitative approach would be more desirable. In this paper, a robust method for obtaining the best fit between a theoretical magnetic field and intensity observations of coronal loops on the solar disk will be presented. The method will be applied to Yohkoh data using a linear force-free field as an illustration. Any other theoretical model for the magnetic field can be used, provided there is enough freedom in the model to optimize the fit.     

Optimization code with weighting function for the reconstruction of coronal magnetic fields

We developed a code for the reconstruction of nonlinear force-free and non-force-free coronal magnetic fields. The 3D magnetic field is computed numerically with the help of an optimization principle. The force-free and non-force-free codes are compiled in one program. The force-free approach needs photospheric vector magnetograms as input. The non-force-free code additionally requires the line-of-sight integrated coronal density distribution in combination with a tomographic inversion code. Previously the optimization approach has been used to compute magnetic fields using all six boundaries of a computational box. Here we extend this method and show how the coronal magnetic field can be reconstructed only from the bottom boundary, where the boundary conditions are measured with vector magnetographs. The program is planed for use within the Stereo mission.     

Coronal Magnetic Field Structure and Evolution for Flaring AR 11117 and Its Surroundings

In this study, photospheric vector magnetograms obtained with the Synoptic Optical Long-term Investigations of the Sun (SOLIS) survey are used as boundary conditions to model three-dimensional nonlinear force-free (NLFF) coronal magnetic fields as a sequence of NLFF equilibria in spherical geometry. We study the coronal magnetic field structure inside an active region and its temporal evolution. We compare the magnetic field configuration obtained from NLFF extrapolation before and after the flaring event in active region (AR) 11117 and its surroundings observed on 27 October 2010, and we also compare the magnetic field topologies and the magnetic energy densities and study the connectivities between AR 11117 and its surroundings. During the investigated time period, we estimate the change in free magnetic energy from before to after the flare to be 1.74×10³²?erg, which represents about 13.5?% of the NLFF magnetic energy before the flare. In this study, we find that electric currents from AR 11117 to its surroundings were disrupted after the flare.     

First 4D Reconstruction of an Eruptive Prominence Using Three Simultaneous View Directions

Data from the STEREO (Solar Terrestrial Relations Observatory) mission are intensively used for 3D reconstruction of solar coronal structures. After the launch of the SDO (Solar Dynamic Observatory) satellite, its additional observations give the possibility to have a third eye for more accurate 3D reconstruction in the very low corona (<?1.5?R _??). With our reconstruction code MBSR (Multi-view B-spline Stereoscopic Reconstruction), we use three view directions (STEREO A, B, and SDO) to perform the 3D reconstruction and evolution of a prominence which triggered a CME on 1 August 2010. In the paper we present the reconstruction of this prominence from the moment it starts to erupt until it leaves the field of view of the coronagraph. We also determine the evolution of the leading edge of the CME. Based on the temporal evolution, we analyze some of its properties, such as velocity, acceleration, opening and rotation angles and evolution of the cavity.     

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. On 07/07/2007, the NLFFF team was saddened by the news that Tom Metcalf had died as the result of an accident. We remain grateful for having had the opportunity to benefit from his unwavering dedication to the problems encountered in attempting to understand the Sun’s magnetic field; Tom had completed this paper several months before his death, leading the team through the many steps described above.     

Evolution of the Fine Structure of Magnetic Fields in the Quiet Sun: Observations from Sunrise/IMaX and Extrapolations

Observations with the balloon-borne Sunrise/Imaging Magnetograph eXperiment (IMaX) provide high spatial resolution (roughly 100 km at disk center) measurements of the magnetic field in the photosphere of the quiet Sun. To investigate the magnetic structure of the chromosphere and corona, we extrapolate these photospheric measurements into the upper solar atmosphere and analyze a 22-minute long time series with a cadence of 33 seconds. Using the extrapolated magnetic-field lines as tracer, we investigate temporal evolution of the magnetic connectivity in the quiet Sun’s atmosphere. The majority of magnetic loops are asymmetric in the sense that the photospheric field strength at the loop foot points is very different. We find that the magnetic connectivity of the loops changes rapidly with a typical connection recycling time of about 3±1 minutes in the upper solar atmosphere and 12±4 minutes in the photosphere. This is considerably shorter than previously found. Nonetheless, our estimate of the energy released by the associated magnetic-reconnection processes is not likely to be the sole source for heating the chromosphere and corona in the quiet Sun.     

Magnetic Stereoscopy of Coronal Loops in NOAA 8891

The Solar TErrestrial RElations Observatory (STEREO) requires powerful tools for the three-dimensional (3D) reconstruction of the solar corona. Here we test such a program with data from SOHO and TRACE. By taking advantage of solar rotation, a newly developed stereoscopy tool for the reconstruction of coronal loops is applied to the solar active region NOAA 8891 observed from 1 March to 2 March 2000. The stereoscopic reconstruction is composed of three steps. First, we identify loop structures in two TRACE images observed from two vantage viewpoints approximately 17 degrees apart, which corresponds to observations made about 30 hours apart. In the second step, we extrapolate the magnetic field in the corona with the linear force-free field model from the photospheric line-of-sight SOHO/MDI data. Finally, combining the extrapolated field lines and one-dimensional loop curves from two different viewpoints, we obtain the 3D loop structures with the magnetic stereoscopy tool. We demonstrate that by including the magnetic modeling this tool is more powerful than pure geometrical stereoscopy, especially in resolving the ambiguities generated by classical stereoscopy. This work will be applied to the STEREO mission in the near future.     

Computing Nonlinear Force-Free Coronal Magnetic Fields in Spherical Geometry

We describe a newly developed code for the extrapolation of nonlinear force-free coronal magnetic fields in spherical coordinates. The program uses measured vector magnetograms on the solar photosphere as input and solves the force-free equations in the solar corona. The method is based on an optimization principle and the heritage of the newly developed code is a corresponding method in Cartesian geometry. We test the newly developed code with the help of a semi-analytic solution and rate the quality of our reconstruction qualitatively by magnetic field line plots and quantitatively with a number of comparison metrics. We find that we can reconstruct the original test field with high accuracy. The method is fast if the computation is limited to low co-latitudes (say 30°≤θ≤150°), but it becomes significantly slower if the polar regions are included.     

Including stereoscopic information in the reconstruction of coronal magnetic fields

We present a method to include stereoscopic information about the three-dimensional structure of flux tubes into the reconstruction of the coronal magnetic field. Due to the low plasma beta in the corona we can assume a force-free magnetic field, with the current density parallel to the magnetic field lines. Here we use linear force-free fields for simplicity. The method uses the line-of-sight magnetic field on the photosphere as observational input. The value of is determined iteratively by comparing the reconstructed magnetic field with the observed structures. The final configuration is the optimal linear force-free solution constrained by both the photospheric magnetogram and the observed plasma structures. As an example we apply our method to SOHO MDI/EIT data of an active region. In the future it is planned to apply the method to analyse data from the SECCHI instrument aboard the STEREO mission.     

Preprocessing of Vector Magnetograph Data for a Nonlinear Force-Free Magnetic Field Reconstruction

Knowledge regarding the coronal magnetic field is important for the understanding of many phenomena, like flares and coronal mass ejections. Because of the low plasma beta in the solar corona, the coronal magnetic field is often assumed to be force-free and we use photospheric vector magnetograph data to extrapolate the magnetic field into the corona with the help of a nonlinear force-free optimization code. Unfortunately, the measurements of the photospheric magnetic field contain inconsistencies and noise. In particular, the transversal components (say B _x and B _y) of current vector magnetographs have their uncertainties. Furthermore, the magnetic field in the photosphere is not necessarily force free and often not consistent with the assumption of a force-free field above the magnetogram. We develop a preprocessing procedure to drive the observed non–force-free data towards suitable boundary conditions for a force-free extrapolation. As a result, we get a data set which is as close as possible to the measured data and consistent with the force-free assumption.