Coronal loops are magnetic structures and might therefore be mapped easily and safely by mapping reliably the coronal magnetic field. Unfortunately, it is well-known that it is very difficult to measure the magnetic field in the corona, and it can only be done in very special conditions, e.g., very strong local field (White et al., 1991). In some cases it is possible to use coronal seismology (first proposed by Uchida, 1970) to determine the average magnetic field strength in an oscillating loop first used by Nakariakov et al. (1999) and Nakariakov and Ofman (2001) on TRACE loops, and recently investigated in a number of studies. The accuracy of this method depends on the correct detection of the temporally and spatially resolved mode of oscillation, and on the details of the loop geometry.
Since we cannot well determine the coronal magnetic field, coronal loop geometry deserves specific analysis. As a good approximation, loops generally have a semicircular shape (Figure 3). The loop aspect, of course, depends on the loop orientation with respect to the line of sight: loops with the footpoints on the limb more easily appear as semicircular, as well as loops very inclined on the surface near the center of the disk. The assumption of semicircular shape can be useful to measure the loop length even in the presence of important deformations due to projection effects: the de-projected distance of the loop footpoints is the diameter of the arc. However, deviations from circularity are rather common and, in general, the detailed analysis of the loop geometry is not a trivial task. The accurate determination of the loop geometry is rather important for the implications on the magnetic field topology and reconstruction. It is less important for the structure and evolution of the confined plasma, which follow the field lines whatever shape they have and change little also with moderate changes of the gravity component along the field lines. First works on the accurate determination of the loop geometry date back to the sixties (Saito and Billings, 1964). More specific ones take advantage of stereoscopic views allowed by huge loops during solar rotation, with the aid of magnetic field reconstruction methods. These studies find deviations from ideal circularity and symmetry, not surprising for such large structures (Berton and Sakurai, 1985). The geometry of a specific loop observed with TRACE was measured in the framework of a complete study including time-dependent hydrodynamic modeling (Reale et al., 2000b,a). In that case, the discrepancy between the length derived from the distance of the footpoints taken as loop diameter and the length measured along the loop itself allowed to assess the loop as elongated. Later, a reconstruction of loop geometry was applied to TRACE observation of medium-sized oscillating loops, to derive the properties of the oscillations. In this case, a semicircular pattern was applied (Aschwanden et al., 2002). The importance of the deviations from circularity on constraining loop oscillations was remarked later (Dymova and Ruderman, 2006).
The STEREO mission is actually contributing much to the analysis of loop morphology and geometry, thanks to its unique capability to observe the Sun simultaneously from different positions. Feng et al. (2007) presented a first stereoscopic reconstruction of the three-dimensional shape of magnetic loops in an active region from two different vantage points based on simultaneously recorded STEREO/SECCHI images. They derived parameters of five relatively long loops and constraints on the local magnetic field, and found reconstructed loops to be non-planar and more curved than field lines extrapolated from SoHO/MDI measurements, probably due to the inadequacy of the linear force-free field model used for the extrapolation. A misalignment of 20 – 40 deg between theoretical model and observed loops has been quantified from STEREO results and discussed (Sandman et al., 2009; DeRosa et al., 2009). Aschwanden et al. (2008b) presented triangulations and 3D reconstructions of 30 coronal loops, using the Extreme UltraViolet Imager (EUVI) telescopes of both STEREO spacecrafts and deriving a series of loop characteristics, such as the loop plane inclination angles, and the coplanarity and circularity. They derived the parameters of seven complete and quite inclined loops and found deviations from circularity within 30%, and less significant from coplanarity. Aschwanden et al. (2009) applied a reconstruction method to an active region observed with STEREO by combining stereoscopic triangulation of 70 loops and addressing mainly density and temperature modeling with a filling factor equivalent to tomographic volume rendering.
Another interesting issue regarding coronal loop geometry is the analysis of the loop cross-section, which also provides information about the structure of the coronal magnetic field. Yohkoh / SXT allowed for systematic and quantitative studies of loop morphology and showed that the cross-section of coronal loops is approximately constant along their length and do not increase significantly. More in detail, a systematic analysis of a sample of ten loops showed that the loops tend to be only slightly ( 30%) wider at their midpoints than at their footpoints, while for a bipolar field configuration we would expect expansions by factors. One possible explanation of this effect is the presence of significant twisting of the magnetic field lines, and therefore the development of electric currents and strong deviations from a potential field. The effect might be seen either as a twisting of a single loop or as a “braiding” of a bundle of unresolved thin loops. At the same time it was found that the variation of width along each loop tends to be modest, implying that the cross section has an approximately circular shape (Klimchuk et al., 1992). Implications of these results on the theory of coronal heating are discussed in Klimchuk (2000), but the conclusion is that none of the current models alone is able to explain all observed properties.
Important information about the internal structuring of coronal loops comes from the joint analysis of the photospheric and coronal magnetic field (Figure 4). An analysis of the magnetic field at the footpoints of hot and cool loops showed, among other results, that the magnetic filling factor is lower in hot loops (0.05 – 0.3 out of sunspots) than in warm loops (0.2 – 0.6) (Katsukawa and Tsuneta, 2005). López Fuentes et al. (2006) investigated the magnetic structure of loops observed with TRACE, applying a linear force-free extrapolation model to SoHO/MDI data and comparing the resulting configuration of the magnetic field directly with TRACE images of the same region. They confirmed that, whereas the model predicts a significant expansion of the magnetic field structures, at least a factor two, from the footpoints to the corona, and a significant asymmetry of the structures, because the magnetic field lines starting from the same footpoint can diverge to different other end footpoints, these features are not observed: also TRACE loops are quite symmetric and their cross-section is constant to a good degree of approximation, as it had been already found for Yohkoh loops (Klimchuk et al., 1992). The results in López Fuentes et al. (2006) suggest that the tangling of the magnetic flux strands driven by the photospheric convection might be very strong, and confirm, therefore, that the magnetic field structure is far more complicated than it can be modelled even with linear force-free extrapolation. Schrijver (2007) studied braiding-induced interchange reconnection of the magnetic field and the width of solar coronal loops and showed that loop width observations support the hypothesis that granular braiding is countered statistically by frequent coronal reconnections, which in turn explain the general absence of entangled coronal field structures in high-resolution observations of the quiescent solar corona. DeForest (2007) addressed the apparent uniform cross section of bright threadlike structures in the corona, long apparent scale height, and the inconsistency between loop densities derived by spectral and photometric means. They found that, if coronal loops are interpreted as a mixture of diffuse background and very dense, unresolved stranded structures, this requires a combination of high plasma density within the structures, which greatly increases the emissivity of the structures, and geometric effects that attenuate the apparent brightness of the feature at low altitudes.
New methods to improve the morphological analysis of loops are being developed. For instance, Dudok de Wit and Auchère (2007) made multispectral analysis of solar EUV images and explored the possibility of separating the different solar structures from a linear combination of images. They found source images with more contrast than the original ones.
It has been long claimed (e.g., Gomez et al., 1993) that coronal loops consist of bundles of thin strands, to scales below the current instrumental resolution. The task to investigate this substructuring is not easy because the thickness of the elementary components may be as small as a few km, according to some nanoflare models (e.g., Vekstein, 2009), and the measured one goes down to the resolution limit of the most powerful imaging instruments (e.g., Gomez et al., 1993). First limited evidence of fine structuring was the low filling factor inferred for loops observed with NIXT (Di Matteo et al., 1999) (see Section 3.3.2). The high spatial resolution achieved by the TRACE normal incidence telescope allowed to address the transverse structure of the imaged coronal loops. TRACE images visibly show that coronal loops are substructured (Figure 5).
There were some early attempts to study the structure along the single strands in TRACE observations (Testa et al., 2002). Later, it was shown that in many cases, hot loop structures observed with the Yohkoh/SXT are not exactly co-spatial with warm structures observed with the SoHO/EIT, which is sensitive in the same bands as TRACE, nor they cool down to become visible to the EIT (Nagata et al., 2003; Schmieder et al., 2004).
The detailed morphological comparison of an active region showed that hot loops seen in SXT () and warm loops seen in the SoHO/EIT 195 Å band () are located in almost alternating manner (Nagata et al., 2003). The anti-coincidence of the hot and the warm loops is conserved for a duration longer than the estimated cooling timescale. These results suggest that loops are not isothermal in the transverse direction, rather they have a differential emission measure distribution of modest but finite width that peaks at different temperatures for different loops (see Section 3.3).
In apparent alternative, Winebarger and Warren (2005) showed that hot monolithic loops visible with the Yohkoh/SXT are later resolved as stranded cooler structures with TRACE. However, this occurs with a time delay of 1 to 3 hours, much longer than the plasma cooling times, and, therefore, correlation can be hardly established between the plasma detected by Yohkoh and that detected by TRACE. Direct density measurements of loop plasma made from multi-line observations at the solar limb indicate not very dense plasma and relatively high plasma filling factor (0.2 – 0.9, Ugarte-Urra et al., 2005). This is in the direction of a moderate structuring of the loops.
Aschwanden and Nightingale (2005) analyzed specifically and systematically TRACE images to search for the thinnest coherent structures that can be resolved with TRACE. They found that about 10% of the positions can be fitted with an isothermal model and proposed that, since the corresponding structures have a uniform thermal distribution, they should be elementary loop components, with an average width of about 2000 km. Aschwanden et al. (2007) studied statistically a large set of coronal loops and found further evidence of elementary loop strands resolved by TRACE.
Other studies based both on models and on analysis of observations independently suggest that elementary loop components should be even finer, with typical cross-sections of the strands to be of the order of 10 – 100 km (Beveridge et al., 2003; Cargill and Klimchuk, 2004; Vekstein, 2009).
Living Rev. Solar Phys. 7, (2010), 5
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