4.4 Coronal loops

Coronal loops essentially represent curvi-linear 1D structures with a small transverse cross-section and a noticable density contrast to the background corona, so that their 3D geometry can be triangulated with stereoscopy (Section 3.3). Stereoscopy of coronal loops is pursued for at least three important reasons in coronal physics: (i) The 3D geometry measures the inclination angle of the average loop plane, which is a necessary parameter to derive the vertical pressure scale height; (ii) The 3D geometry measures the loop length and the local line-of-sight angles, which are key parameters for hydrodynamic flux tube modeling and related scaling laws; (iii) The 3D geometry is the most important observational constraint to test theoretical magnetic field models of the solar corona, and it also reveals whether loops follow a magnetic closed-field or open-field line. We will discuss these physical aspects in the context of stereoscopic observations in turn.

4.4.1 Hydrostatic scale height of loops

The 3D loop geometry can be parameterized by a set of 3D coordinates [xi,yi,zi], where the projected positions [xi,yi] are directly measured in the image plane, and the distance zi 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 (xc, yc,zc), the loop curvature radius r c, the azimuth angle of the footpoint baseline α, and the inclination angle πœƒ of the loop plane to the vertical, where the latter two angles are defined in a heliographic coordinate system (Figure 23View Image).

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Figure 23: Definition of loop parameters: loop point positions (xi,yi),i = 1,...,n starting at the primary footpoint at height h = h 1 foot, the azimuth angle α between the loop footpoint baseline and heliographic east-west direction, and the inclination angle πœƒ between the loop plane and the vertical to the solar surface.

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

2L (πs ) h(s) = r(s) − R βŠ™ = ---sin --- , (25 ) π 2L
where L is the loop half length. The hydrodynamic pressure p(s) in a vertical semi-circular loop is then, based on the pressure balance or momentum equation, taking only gravity into account (Aschwanden, 2005Jump To The Next Citation Point),
⌊ ⌋ [ ] ⌈ ---(h-[s]-−-h0)--⌉ (h[s] −-h0) p(s) = p0exp − h[s] ≈ p0exp − λ (T ) , (26 ) λp (Te )(1 + RβŠ™) p e
with λp (Te ) being the vertical pressure scale height,
-2kBTe-- 9( -Te--) λp(Te) = μm g ≈ 4.7 × 10 1 MK (cm) . (27 ) H βŠ™
Note that we use the so-called coronal approximation Te = Ti 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 πœƒ with respect to the local vertical direction on the solar surface, the observed scale height obs λp as derived from the exponential density drop along the loop, has to be corrected by the cosine of the inclination angle,

λobs= -λp--. (28 ) p cosπœƒ
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 24View Image). 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 rc ≈ 100 Mm, which approximately corresponds to two density or pressure scale heights at a temperature of T = 1.0 MK (Eq. (27View Equation)), the electron density at the loop top is a factor of e−2 = 0.14 lower than at the footpoint, which makes the EUV brightness (2 I ∝ ne) drop by a factor of −4 e = 0.02, while the loop apex can be almost equally bright as the loop footpoints for a highly inclined loop (πœƒ >∼ 0). 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 25View Image, which illustrates that the vertical range of detected loop segments does not exceed about one pressure scale height (see height projection above limb, Figure 25View Image top left), and that complete loops are all highly inclined (see horizontal projection and side view in Figure 25View Image bottom left and right).
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Figure 24: Vertical and apparent density scale heights in coronal loops (right) and analogy with communicating water tubes (left) (from Aschwanden, 2005Jump To The Next Citation Point).
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Figure 25: Orthogonal projections of the stereoscopically triangulated 70 coronal loops in AR 10955 observed on 2007 May 9 in three filters (171 Å = blue; 195 Å = red; 284 Å = yellow). The observed projection in the x-y image plane seen from spacecraft A is shown in the bottom panel left, the projection into the x-z plane in the top panel left, and the projection into the y-z plane in the bottom panel right. The three orthogonal projections correspond to rotations by 90° to the north or west (to positions indicated on the solar sphere in the top right panel) (from Aschwanden et al., 2009c).

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 πœƒ ≈ 7∘–25 ∘. 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 2L ≈ 300 –800 Mm, loop heights of h ≈ 70– 330 Mm, and inclination angles in a range of |πœƒ| <∼ 40 ∘ (Aschwanden et al., 1999Jump To The Next Citation Point, 2000Jump To The Next Citation Point). 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., 2007aJump To The Next Citation Point; Aschwanden et al., 2008cJump To The Next Citation Point). Feng et al. (2007aJump To The Next Citation Point) 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 α-parameter. Aschwanden et al. (2008c) was able to determine the non-planarity (|Δy |βˆ•rc ≈ 3% –11%), the non-circularity (|Δr |βˆ•rc ≈ 11% – 30%), and inclination (πœƒ ≈ 35∘– 73∘) 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).

4.4.2 Hydrodynamics of loops

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

∫ ∫ F ∝ n2(z,T )R (T) dT dz , (29 ) λ e λ
which in the case of a 1D flux tube can be reduced to constant values of the local loop density n (s) e and temperature Te(s), integrated over a column depth wz, which corresponds to the loop width w divided by the cosine of the local line-of-sight angle ψ (s),
w wz[xi,yi,zi] = ---------------, (30 ) cos(ψ[xi,yi,zi])
and can be calculated from the stereoscopically triangulated loop coordinates [x(s),y(s),z(s)],
∘ -------------------------- (xi+1 − xi)2 + (yi+1 − yi)2 cos(ψ[xi,yi,zi]) = ∘----------------------------------------. (31 ) (xi+1 − xi)2 + (yi+1 − yi)2 + (zi+1 − zi)2
Thus, stereoscopic measurements of the 3D loop coordinates and background-subtracted EUV and soft X-ray fluxes Fλ(s) along the loop coordinate s allows us to fully constrain the physical parameters ne(s) and Te(s) of the loop, as well as to measure the loop half length L in 3D space. Since multiple temperature filters are required, forward-fitting of a hydrodynamic loop model to observed fluxes Fλ(s) 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 26View Image.
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Figure 26: Left panel: the effect of the variable column depth w (s) z measured parallel to the line-of-sight z is illustrated as a function of the loop length parameter s, for a loop with a constant diameter w. Right panel: the effect of the inclination angle πœƒ of the loop plane on the inferred density scale height λ(πœƒ) is shown. Both effects have to be accounted for when determining the electron density ne(s) along the loop (from Aschwanden, 2005).

Simulations of multiple coronal loops that form an active region based on 1D hydrostatic models (Figure 14View Image) 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 ne(s) and temperature profiles Te(s) has been carried out with STEREO/A and B triple-filter data (Aschwanden et al., 2008bJump To The Next Citation Point). Although the same parameters can also approximately be inferred from a single spacecraft, as demonstrated with EIT data (Aschwanden et al., 1999, 2000Jump To The Next Citation Point), 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 (TB βˆ•TA = 1.05 ± 0.09), densities (nB βˆ•nA = 0.94 ± 0.12), and loop widths (wB βˆ•wA = 0.96 ± 0.05), as shown in Figure 27View Image (Aschwanden et al., 2008bJump To The Next Citation Point).

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Figure 27: Self-consistency of mean loop temperatures TA, TB (top left), base electron densities n ,n A B (middle left), and mean loop widths w ,w A B (bottom left) measured with spacecraft STEREO/A vs. STEREO/B. These loop parameters are inferred from the background-subtracted loop-associated flux (top right), based on independent background subtractions for the different line-of-sights of both spacecraf A and B (bottom right) (from Aschwanden et al., 2008bJump To The Next Citation Point).

The stereoscopic measurement of the loop half length L, the electron density ne, and loop apex temperature Te enables us to test 1D loop scaling laws, such as the RTV scaling law, pRTV = (Teβˆ•1400 )3L −1, by comparing the theoretically predicted pressure pRTV with the observed pressure pobs = 2nekBTe. 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 q of (in cgs-units),

nobs pobs 2nekBTe neL q = ------= ----- = ---------------3 = 7.57 × 10 −7 --2-. (32 ) nRTV pRTV (1βˆ•L )(Teβˆ•1400 ) T e
This over-pressure is shown for stereoscopically triangulated loops in Figure 28View Image. The overpressure amounts to q ≈ 3 –15, 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, 2008bJump To The Next Citation Point), once the heating rate fades, in particular in the late cooling phase when the radiative loss rate dominates.
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Figure 28: The loop overpressure factor qp = p βˆ•pRTV (normalized by Rosner-Tucker-Vaiana scaling law with uniform heating) is shown versus the loop half length L. Datapoints are given for the complete 7 loops detected along their full length, measured with STEREO-A (large diamonds) and with STEREO-B (small diamonds) (from Aschwanden et al., 2008b).

Loops with known density ne(s), temperature Te (s), and width profiles w (s) 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).

4.4.3 Magnetic fields of loops

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 Φ (r),

B (r) = ∇ Φ (r) , (33 )
which fulfills Maxwell’s equation of divergence-freeness,
∇ ⋅ B = ∇2 Φ = 0. (34 )
In principle, for a given boundary condition, e.g., from vector magnetograph data B (l,b,h = 0), and ignoring data noise, the solution of a potential magnetic field B(r) 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 j(r), are the force-free models, characterized by an α-parameter that is a constant for linear force-free (LFF) models,

(∇ × B ) = 4πj = αB , (35 )
or a spatially varying function α(r) for nonlinear force-free (NLFF) models,
(∇ × B ) = 4πj = α (r)B . (36 )
Using stereoscopic triangulation, the 3D coordinates robs(s ) = [x(s),y(s),z(s)] of coronal loops can be determined, which can be compared with the 3D coordinates rff (s,α ) = [x (s),y (s ),z (s)] of the nearest field line of a force-free model, e.g., by varying the α-parameter and minimizing the average least-square deviation between the loop coordinates,
∫ s ∘ -------------------- min [Δ (α)] = --1-- max [r (s) − r (s,α )]2 ds. (37 ) 2 s2max 0 obs ff
This method was used by Wiegelmann and Neukirch (2002) and a best-fit value α was determined for the linear force-free model. Thus, the stereoscopic information provides a measurement of the average current (Eq. (35View Equation)) for this model,
j = αB--. (38 ) 4π
Instead of fitting a constant α 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 α (r) that converges to a global solution of Eq. (36View Equation).

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. (37View Equation)). This method has then been applied to real STEREO data for the first time by Feng et al. (2007a), who found values of −1 |α | ≈ (2 –8) × 10−3 Mm for 5 loops, which corresponds to a helical twist angle of Φ = 2πn < 0.5, well below the kink instability threshold (Φ ≤ 3.5π). 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 11View Image) was carried out by DeRosa et al. (2009Jump To The Next Citation Point), who find average 3D misalignment angles in the range of αmis = 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. (2009Jump To The Next Citation Point), who finds 3D misalignment angles,

( ) --robsrpot-- αmis = arccos |robs| ⋅ |rpot| , (39 )
of αmis = 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 N multiple unipolar charges Bj (Aschwanden and Sandman, 2010Jump To The Next Citation Point),
( ) N∑ ∑N zj 2 rj B (r) = Bj (r ) = Bj -- -- , (40 ) j=1 j=1 rj rj
or in terms of a N submerged magnetic dipoles defined by the magnetic moments mj (Sandman and Aschwanden, 2011Jump To The Next Citation Point),
∑N μ ∑N [ 3ˆr (ˆr ⋅ m ) − m ] B (r) = Bj (r) = -0- --j-j----j------j- , (41 ) j=1 4π j=1 |r − rj|3
where rj is the position of the j-th dipole, with r being the position vector, and ˆrj = (r − rj)βˆ•|r − rj| the normal vector.
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Figure 29: The mean misalignment angle for four active regions as a function of the GOES soft X-ray flux: for the potential field source surface model (PFSS: diamonds), for the unipolar potential field model bootstrapped with observed STEREO loops (PFU: triangles), and contributions from stereoscopic measurement errors (SE; crosses). The difference between the best-fit potential field model (triangles) and stereoscopic errors (crosses) can be considered as a measure of the non-potentiality of the active region (hatched area) (from Aschwanden and Sandman, 2010Jump To The Next Citation Point).
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Figure 30: Best-fit potential field model of AR observed on 2007 Apr 30. The stereoscopically triangulated loops are shown in blue color, while field lines starting at identical footpoints as the STEREO loop extrapolated with the best-fit potential field (composed of nc = 200 unipolar magnetic charges) are shown in red. Side views are shown in the top and right panels. A histogram of misalignment angles measured between the two sets of field lines is displayed in the bottom panel. The distribution is fitted with a Gaussian, where the vertical solid line indicates the peak of the Gaussian (or most probable value), while the vertical dashed line indicates the median value (from Aschwanden and Sandman, 2010Jump To The Next Citation Point).

Both potential-field parameterizations achieved a smaller misalignment, which reduced to of αmis = 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, 2010Jump To The Next Citation Point). An example is shown in Figure 30View Image, which represents the first of the compared active regions, observed on 2007 Apr 30 (which is identical to the NLFF modeling shown in Figure 11View Image). Since stereoscopic errors are estimated to have uncertainties of similar magnitude (Δ αSE ≈ 10∘), 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 29View Image 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 (αSE), the residual misalignment that could be attributed to the non-potentiality of the magnetic field are in the order of ∘-------------- αNP = α2 − Δ α2 ≈ 11∘– 17∘ PFU SE. Interestingly, the average misalignment angles are correlated with the soft X-ray flux of the active region (Figure 29View Image), which implies a relationship between electric currents j and plasma heating (Aschwanden and Sandman, 2010Jump To The Next Citation Point).

Table 1: Misalignment statistics of stereoscopically triangulated loops in four active regions (Aschwanden and Sandman, 2010Jump To The Next Citation Point).
Parameter 2007-Apr-30 2007-May-9 2007-May-19 2007-Dec-11
Misalignment NLFFF1 24 – 44      
Misalignment PFSS2 25 ± 8 19 ± 6 36 ± 13 32 ± 10
Misalignment PFU3 14.3 ± 11.5 13.3 ± 9.3 20.3 ± 16.5 15.2 ± 12.3
Median PFU3 20.0 16.2 25.8 15.7
Stereoscopy error4 9.4 7.6 11.5 8.9
Non-potentiality5 11 ± 9 11 ± 8 17 ± 14 12 ± 10
GOES soft X-ray flux6 10–7.3 10–7.6 10–6.0 10–6.9
GOES class A7 A4 C0 B1
1) Measured with nonlinear force-free field code (DeRosa et al., 2009).
2) Measured with potential field source surface code (Sandman et al., 2009).
3) Measured with unipolar potential field model (Aschwanden and Sandman, 2010).
4) Measured from inconsistency between adjacent loops.
5) Residual misalignment of unipolar best-fit model with stereoscopic error subtracted in quadrature, ∘ ------------- αNP = α2 − Δ α2 PFU SE.
6) GOES flux in units of [W m–2].

Linear force-free (LFF) models of coronal loops in active regions have one free parameter, the twist α, 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 α(t) and non-potential magnetic energy ELFF (t), which was found to change significantly during a magnetic flux emergence event (Conlon and Gallagher, 2010). The free (non-potential) energy ENLFF 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).

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