3.4 Magnetic stereoscopy

Stereoscopic triangulation of curvi-linear coronal structures (such as loops, fans, jets, filaments, plumes) can usually only be carried out for segments and a limited number of structures within relatively isolated zones that are not too much congested (subject to confusion and mis-identification of stereoscopic correspondence). One additional a priori information that can be used to enhance stereoscopic 3D reconstruction is the auxiliary use of magnetic field models. Of course, the fundamental limitation is that we still do not have perfect magnetic field models for the solar corona, but a lot of progress has been made to improve magnetic field models based on stereoscopic information, and to beat down the discrepancy between theoretical magnetic field models and observed stereoscopically triangulated loop 3D coordinates.
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Figure 10: A 3D magnetic field representation is rendered from photospheric magnetograms (optical image in orange), from extrapolated magnetic field lines (black lines), and the iso-Gauss contours for three gyroresonant layers that correspond to gyrofrequencies of 5 GHz (green), 8 GHz (blue), and 11 GHz (yellow). The outer contours of each iso-Gauss surface demarcate the extent of radio emission at each frequency (courtesy of Stephen White and Jeong Woo Lee).

An early experiment with combined stereoscopy and magnetic field modeling has been conducted with multi-frequency radio maps observed with the Owens Valley Radio Observatory (Aschwanden et al., 1995). For a given 3D magnetic field, the gyroresonance layers at each frequency ν and harmonic s = 2,3,4 form a curved 2D surface that are layered like onion shells in the lower corona above sunspots, as shown for three frequencies (ν = 5,8,11 GHz) in Figure 10View Image. Some features (e.g., the brightest locations or centroids) of these gyroresonance layers can be either stereoscopically triangulated (using the solar rotation) or be modeled by matching the contours of their brightness distribution for a given projection. Such a method successfully demonstrated that in a strongly sheared region only a nonlinear force-free field could explain the observed radio brightness distribution, while a potential-field model failed (Lee et al., 1999). Also the polarization of gyroresonance emission in the theory of wave mode coupling in quasi-transverse regions was tested with combined radio mapping and magnetic field modeling (Ryabov et al., 2005).

The method of using the observed projections of coronal EUV loops to constrain a magnetic field model was proposed by Wiegelmann and Neukirch (2002Jump To The Next Citation Point) in the pre-STEREO area, who varied the α-parameter in a force-free magnetic field extrapolation to optimize the match to observed loop locations. Magnetic modeling was then also used to constrain tomographic reconstruction of the coronal density distribution (Wiegelmann and Inhester, 2003; Ruan et al., 2008). A variety of coronal magnetic field models, such as potential, linear, and nonlinear force-free field (NLFFF) models, were used to resolve ambiguities in stereoscopic triangulation of loops (Wiegelmann and Inhester, 2006Jump To The Next Citation Point; Feng et al., 2007bJump To The Next Citation Point; Conlon and Gallagher, 2010Jump To The Next Citation Point), reviewed in Wiegelmann et al. (2009Jump To The Next Citation Point). Comparisons of potential-field and NLFFF models with STEREO and Hinode images were also calculated for the global corona (Petrie et al., 2011; Schrijver and Title, 2011).

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Figure 11: SOHO/MDI magnetogram observed on 2007 Apr 30, 22:42 UT, with overlayed stereoscopically triangulated loop tracings (purple) and calculated NLFFF magnetic field model (colors in central box), where the colors indicate an increasing degree of misalignment from αmis < 5° (yellow) to αmis > 45° (red) (from DeRosa et al., 2009Jump To The Next Citation Point).
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Figure 12: Left: A potential dipole field is calculated with two unipolar magnetic charges buried in equal depth and equal magnetic field strength, but opposite magnetic polarity (B1 = − 0.1, B2 = +0.1). An asymmetric EUV loop is observed at the same location (grey torus) with a misalignment of αmis = 20° at the top of the loop. Right: The magnetic field strength of the right-hand side unipolar charge is adjusted (to B = +0.08 2) so that the misalignment of the loop reaches a minimum (αmis = 0°) (from Aschwanden and Sandman, 2010Jump To The Next Citation Point).

The success of magnetic stereoscopy depends on the quality of theoretical magnetic field models. A critical assessment of NLFFF models identified a substantial mismatch between theoretical magnetic field models extrapolated from photospheric magnetograms and stereoscopically triangulated loops, in the order of a 3D misalignment angle of αmis ≈ 20∘ –40∘ (DeRosa et al., 2009Jump To The Next Citation Point; Sandman et al., 2009Jump To The Next Citation Point). A comparison of a force-free field model with STEREO-triangulated loops is shown in Figure 11View Image. Some parameterization of the theoretical field models is required to minimize the mismatch. Such a parameterization was implemented in potential-field models in terms of unipolar magnetic charges (Aschwanden and Sandman, 2010Jump To The Next Citation Point) or magnetic dipoles (Sandman and Aschwanden, 2011Jump To The Next Citation Point), which enabled to improve the misalignment angle to α ≈ 10∘– 20∘ mis. The principle is illustrated in Figure 12View Image. A caveat of these first attempts to adjust theoretical magnetic field models to observed 3D loop coordinates (obtained from stereoscopic triangulation) is that the optimization algorithm takes only the coronal misalignment into account, but neglects the constraints of the observed photospheric magnetograms. The justification for this neglect is the fact that the magnetic field in the photosphere and lower chromosphere is not force-free (Metcalf et al., 1995), but ultimately we would like to have magnetic field models that can handle both the non force-free extrapolation in the chromosphere plus fitting of the force-free field in the corona to observed 2D projections or 3D triangulations of loops. Once we have an optimized (matching) coronal 3D magnetic field model, we can populate each field line with a magnetic flux tube and perform hydrodynamic modeling of entire coronal sections, constrained by the observed EUV and soft X-ray images in different temperature filters, a technique called instant stereoscopic tomography of active region (ISTAR) (Aschwanden et al., 2009cJump To The Next Citation Point).

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Figure 13: The basic tomography concept of solar tomography is visualized for multiple spacecraft located at different aspect angles from the Sun, which can reconstruct the 2D density distribution in the solar corona (for instance, in the ecliptic plane) by synthesizing the 1D brightness distributions observed from each spacecraft (from Davila, 1994Jump To The Next Citation Point).

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