The 3D loop geometry can be parameterized by a set of 3D coordinates , where the projected positions are directly measured in the image plane, and the distance is calculated via stereoscopy (Section 3.3). If we approximate the 3D loop geometry with a semi-circular shape, we can characterize it with 6 parameters: the loop curvature center position , the loop curvature radius , the azimuth angle of the footpoint baseline , and the inclination angle of the loop plane to the vertical, where the latter two angles are defined in a heliographic coordinate system (Figure 23).

The pressure scale height in a hydrostatic (gravitationally stratified) atmosphere depends on the height above the surface, which can be specified as a function of the loop length coordinate , e.g., for a vertical semi-circular loop as

where is the loop half length. The hydrodynamic pressure in a vertical semi-circular loop is then, based on the pressure balance or momentum equation, taking only gravity into account (Aschwanden, 2005), with being the vertical pressure scale height, Note that we use the so-called coronal approximation for the coronal temperature throughout this paper, although differences may exist in non-collisional regimes in the upper corona.However, if the loop plane is inclined by an inclination angle with respect to the local vertical direction on the solar surface, the observed scale height as derived from the exponential density drop along the loop, has to be corrected by the cosine of the inclination angle,

This can be understood also in terms of communicating water tubes where the water level depends only on the height, but not on the actual length of an inclined water tube segment (Figure 24). This is the reason why 3D stereoscopy is important for the understanding of the hydrostatics of a coronal plasma loop. If a semi-circular loop has a curvature radius of , which approximately corresponds to two density or pressure scale heights at a temperature of (Eq. (27)), the electron density at the loop top is a factor of lower than at the footpoint, which makes the EUV brightness () drop by a factor of , while the loop apex can be almost equally bright as the loop footpoints for a highly inclined loop . As a consequence, vertical loops are typically only visible in the lowest scale height, while highly inclined loops are visible over their entire length. An example of a stereoscopic 3D reconstruction of 70 loops in an active region using STEREO/A and B is shown in Figure 25, which illustrates that the vertical range of detected loop segments does not exceed about one pressure scale height (see height projection above limb, Figure 25 top left), and that complete loops are all highly inclined (see horizontal projection and side view in Figure 25 bottom left and right).First determinations of the 3D geometry of individual coronal loops have been carried out using Skylab data and the solar rotation (Berton and Sakurai, 1985), finding loop asymmetry, non-planar geometry, and mean inclination angles of . This study demonstrated the feasibility of 3D reconstruction, the ability to measure deviations from simple circular and planar loop geometries, but worked only for very large (inter-active region) loops due to the static restriction for solar rotation-based methods. Similarly, the 3D geometry of 65 coronal loops has been triangulated using EIT images and the solar rotation, yielding loop lengths of , loop heights of , and inclination angles in a range of (Aschwanden et al., 1999, 2000). Using parameterized 3D geometry models, non-planar loop geometries can even be determined from forward-fitting to observed loop projections in a single-spacecraft image without stereoscopy, as demonstrated for twisted helical geometries using EIT images (Portier-Fozzani et al., 2001; Portier-Fozzani and Inhester, 2001).

The first true stereoscopic triangulations of loops have been performed with the dual STEREO/A and B spacecraft (Feng et al., 2007a; Aschwanden et al., 2008c). Feng et al. (2007a) fitted the 3D shape of 5 loops with a linear force-free magnetic field model and could constrain the twist of the loops in term of the nonlinear force-free -parameter. Aschwanden et al. (2008c) was able to determine the non-planarity (), the non-circularity (), and inclination () in 7 complete loops. One problem of solar loop triangulation is the incompleteness of tracable loop segments due to confusion in crowded locations. This problem is particularly present in automated loop detection algorithms, for instance using the loop segmentation method (Inhester et al., 2008) or the Oriented Coronal CUrved Loop Tracing (OCCULT) code (Aschwanden, 2010).

1D hydrodynamic loop models can be parameterized by the electron density and temperature evolution as a function of the loop length coordinate and time . Observational measurements of these parameters can be inferred from the soft X-ray and EUV intensities, which are proportional to the squared density (Eqs. (4) and (5)) and the line-of-sight column depth ,

which in the case of a 1D flux tube can be reduced to constant values of the local loop density and temperature , integrated over a column depth , which corresponds to the loop width divided by the cosine of the local line-of-sight angle , and can be calculated from the stereoscopically triangulated loop coordinates , Thus, stereoscopic measurements of the 3D loop coordinates and background-subtracted EUV and soft X-ray fluxes along the loop coordinate allows us to fully constrain the physical parameters and of the loop, as well as to measure the loop half length in 3D space. Since multiple temperature filters are required, forward-fitting of a hydrodynamic loop model to observed fluxes is generally a more viable way than direct inversion. The geometric effect of the column depth integration and scale height correction in inclined loops is illustrated in Figure 26.Simulations of multiple coronal loops that form an active region based on 1D hydrostatic models (Figure 14) and their 3D reconstruction from stereoscopic images has been discussed and tested in the pre-STEREO era (Gary et al., 1998), including magnetic modeling (Gary and Alexander, 1999). The first stereoscopically constrained inversion of loop density and temperature profiles has been carried out with STEREO/A and B triple-filter data (Aschwanden et al., 2008b). Although the same parameters can also approximately be inferred from a single spacecraft, as demonstrated with EIT data (Aschwanden et al., 1999, 2000), dual spacecraft stereoscopy yields more accurate values because of a true measurement of the line-of-sight angle, inclination of loop plane, and independent background subtraction from two different aspect angles. The self-consistency between the two independent spacecraft measurements from STEREO/A and B is evident from the obtained ratios of loop temperatures (), densities (), and loop widths (), as shown in Figure 27 (Aschwanden et al., 2008b).

The stereoscopic measurement of the loop half length , the electron density , and loop apex temperature enables us to test 1D loop scaling laws, such as the RTV scaling law, , by comparing the theoretically predicted pressure with the observed pressure . Interestingly, while the RTV law approximately holds for hot soft X-ray emitting loops, it completely fails for EUV loops at cooler temperatures, giving rise to an over-density or over-pressure ratio of (in cgs-units),

This over-pressure is shown for stereoscopically triangulated loops in Figure 28. The overpressure amounts to , which indicates a strong deviation from the RTV energy balance equilibrium between heating and cooling rates. In the impulsive heating scenario (e.g., Aschwanden and Tsiklauri, 2009) EUV loops exhibit a higher density than predicted by the RTV steady-state equilibrium (Lenz et al., 1999; Aschwanden et al., 2000, 2008b), once the heating rate fades, in particular in the late cooling phase when the radiative loss rate dominates.Loops with known density , temperature , and width profiles represent the building blocks of active regions and the solar corona at large and, thus, can be used for tomographic modeling of active regions (Aschwanden, 2009a). Physical properties of cooling plasma in quiescent active region loops, such as its emission measure and filling factor can also be determined using the two line-of-sights of EUVI from STEREO/A and B, in combination with SOHO/SUMER, UVCS, EIT, LASCO, and Hinode (Landi et al., 2009). Direct detections of small-scale siphon flows in funnel-like legs of coronal loops by both STEREO spacecraft were also reported (Tian et al., 2009), as well as cool flows along coronal loops (Zhang and Li, 2009). Quantitative modeling of the hydrodynamic evolution of coronal loops, based on STEREO/EUVI, Hinode/EIS, XRT, and TRACE data, however, still represents a major challenge that cannot easily be reconciled in all observed wavelengths (Warren et al., 2010).

While we outlined several methods of “magnetic stereoscopy” in Section 3.4, we review here some observational results of stereoscopic magnetic field modeling for coronal loops in more detail.

The simplest magnetic field model is a potential field, which can be characterized by a magnetic scalar potential function ,

which fulfills Maxwell’s equation of divergence-freeness, In principle, for a given boundary condition, e.g., from vector magnetograph data , and ignoring data noise, the solution of a potential magnetic field is unique and has no free parameters that can be adjusted to match observed loop geometries.Widely used non-potential magnetic field models that include currents , are the force-free models, characterized by an -parameter that is a constant for linear force-free (LFF) models,

or a spatially varying function for nonlinear force-free (NLFF) models, Using stereoscopic triangulation, the 3D coordinates of coronal loops can be determined, which can be compared with the 3D coordinates of the nearest field line of a force-free model, e.g., by varying the -parameter and minimizing the average least-square deviation between the loop coordinates, This method was used by Wiegelmann and Neukirch (2002) and a best-fit value was determined for the linear force-free model. Thus, the stereoscopic information provides a measurement of the average current (Eq. (35)) for this model, Instead of fitting a constant for all loops (in the LFF model), each loop could be fitted separately, which may yield an approximative force-free model (NLFF) model, but a self-consistent NLFF solution would require an iteration of a space-filling distribution that converges to a global solution of Eq. (36).If a pair of corresponding loops can be unambiguously triangulated with stereoscopy, the obtained 3D coordinates can serve in assessing the accuracy of a theoretical magnetic field model. However, since loop segmentation and identification of corresponding loop pairs are often ambiguous, theoretical magnetic field models can be used to improve stereoscopic triangulation. This method of “magnetic stereoscopy” was proposed by Wiegelmann and Inhester (2006) and Wiegelmann et al. (2009), who applied multiple (potential, linear, and nonlinear force-free) magnetic field models to solve the correspondence problem of stereoscopic loop pairs (by optimization of the criterion Eq. (37)). This method has then been applied to real STEREO data for the first time by Feng et al. (2007a), who found values of for 5 loops, which corresponds to a helical twist angle of , well below the kink instability threshold (). The same method was also applied to SOHO/EIT and TRACE data using solar rotation stereoscopy (Feng et al., 2007b).

A comprehensive comparison of 11 NLFF and one potential-field model with a set of stereoscopically triangulated coronal loops (Figure 11) was carried out by DeRosa et al. (2009), who find average 3D misalignment angles in the range of = 24° – 44°, which was attributed to three problems: (i) limited area of vector magnetograph data (from Hinode); (ii) uncertainties of boundary data; and (iii) non-force-freeness at photosphere-corona interface. In fact, the NLFF models did not fare better than the potential-field model in this case. More cross-comparisons have been performed by Sandman et al. (2009), who finds 3D misalignment angles,

of = 25° ± 8°, 19° ± 6°, and 36° ± 13° for 370 stereoscopically triangulated loops in three active regions observed with STEREO. In a next step it was attempted to forward-fit potential field models with adjustable parameters to bootstrap a best-fit solution, either in terms of multiple unipolar charges (Aschwanden and Sandman, 2010), or in terms of a submerged magnetic dipoles defined by the magnetic moments (Sandman and Aschwanden, 2011), where is the position of the -th dipole, with being the position vector, and the normal vector.Both potential-field parameterizations achieved a smaller misalignment, which reduced to of = 16° ± 14°, 11° ± 7°, and 18° ± 11° for the same three active regions in the case of the dipolar model (Sandman and Aschwanden, 2011), and similar values for the case of the unipolar model (Aschwanden and Sandman, 2010). An example is shown in Figure 30, which represents the first of the compared active regions, observed on 2007 Apr 30 (which is identical to the NLFF modeling shown in Figure 11). Since stereoscopic errors are estimated to have uncertainties of similar magnitude (), future attempts may envision bootstrapping of stereoscopy with iterative magnetic field variations in order to obtain a best match. A summary of the observed misalignment angles measured in four active regions is shown as a function of the soft X-ray flux in Figure 29 and tabulated in Table 1: The optimized unipolar potential field model (PFU) has about half the misalignment than a standard potential source surface (PFSS) model, and taking the stereoscopic errors into account (), the residual misalignment that could be attributed to the non-potentiality of the magnetic field are in the order of . Interestingly, the average misalignment angles are correlated with the soft X-ray flux of the active region (Figure 29), which implies a relationship between electric currents and plasma heating (Aschwanden and Sandman, 2010).

Parameter | 2007-Apr-30 | 2007-May-9 | 2007-May-19 | 2007-Dec-11 |

Misalignment NLFFF^{1} |
24 – 44 | |||

Misalignment PFSS^{2} |
25 ± 8 | 19 ± 6 | 36 ± 13 | 32 ± 10 |

Misalignment PFU^{3} |
14.3 ± 11.5 | 13.3 ± 9.3 | 20.3 ± 16.5 | 15.2 ± 12.3 |

Median PFU^{3} |
20.0 | 16.2 | 25.8 | 15.7 |

Stereoscopy error^{4} |
9.4 | 7.6 | 11.5 | 8.9 |

Non-potentiality^{5} |
11 ± 9 | 11 ± 8 | 17 ± 14 | 12 ± 10 |

GOES soft X-ray flux^{6} |
10^{–7.3} |
10^{–7.6} |
10^{–6.0} |
10^{–6.9} |

GOES class | A7 | A4 | C0 | B1 |

Linear force-free (LFF) models of coronal loops in active regions have one free parameter, the twist , which can be constrained by stereoscopic triangulation. Iterating this stereoscopic fitting procedure as a function of time yields then the time evolution of the twist parameter and non-potential magnetic energy , which was found to change significantly during a magnetic flux emergence event (Conlon and Gallagher, 2010). The free (non-potential) energy calculated with a NLFF model with Hinode data was also found to be correlated with the soft X-ray flare index in a sample of 75 active regions (Jing et al., 2010).

Living Rev. Solar Phys. 8, (2011), 5
http://www.livingreviews.org/lrsp-2011-5 |
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