Coronal Magnetid Field Bootstrapping. I.

Bootstrapping the Coronal Magnetic Field with STEREO: I. Unipolar Potential Field Modeling

Abstract

We investigate the recently quantified misalignment of between the 3-D geometry of stereoscopically triangulated coronal loops observed with STEREO/EUVI (in four active regions) and theoretical (potential or nonlinear force-free) magnetic field models extrapolated from photospheric magnetograms. We develop an efficient method of bootstrapping the coronal magnetic field by forward-fitting a parameterized potential field model to the STEREO-observed loops. The potential field model consists of a number of unipolar magnetic charges that are parameterized by decomposing a photospheric magnetogram from MDI. The forward-fitting method yields a best-fit magnetic field model with a reduced misalignment of . We evaluate also stereoscopic measurement errors and find a contribution of , which constrains the residual misalignment to , which is likely due to the nonpotentiality of the active regions. The residual misalignment angle of the potential field due to nonpotentiality is found to correlate with the soft X-ray flux of the active region, which implies a relationship between electric currents and plasma heating.

Sunb : EUV — Sun : Magnetic fields

1 Introduction

The STEREO mission provides us an unprecedented view of the solar corona, enabling us for the first time to fully constrain the three-dimensional (3-D) geometry of the coronal magnetic field. Stereoscopic triangulation of coronal loops has been conducted at small STEREO spacecraft separation angles (), for several active regions observed with STEREO A(head) and B(ehind) in April and May 2007 (Aschwanden et al. 2008a,b; 2009). The reconstructed 3-D geometry of STEREO-observed coronal loops has been compared with theoretical magnetic field models based on extrapolations from photospheric magnetograms, using nonlinear force-free field (NLFFF) models (DeRosa et al. 2009), as well as potential and stretched potential field models (Sandman et al. 2009), but surprisingly it turned out that the two types of magnetic field lines exhibited an average misalignment angle of , regardless of what type of theoretical magnetic field model was used. From this dilemma it was concluded that a more realistic physical model is needed to quantify the transition from the non-force-free photospheric boundary condition to the nearly force-free field at the base of the solar corona (DeRosa et al. 2009).

At this junction, it is not clear what a viable method is to obtain a force-free boundary of the magnetic field at the coronal base, or how to correct the non-force-free magnetograms. However, the stereoscopic triangulation supposedly provides the correct 3-D directions of the magnetic field , which together with Maxwell’s equation of divergence-freeness (, constrain also the absolute values of the field strengths. In this Paper I we choose a magnetic field model that is defined in terms of multiple unipolar charges. An approach in terms of multiple dipoles is employed in Paper II (Sandman and Aschwanden 2010). Since both unipolar or dipolar magnetic fields represent potential magnetic fields that fulfill the divergence-free condition, the superposition of multiple unipolar and dipolar magnetic field components fulfill the same condition. We develop a numerical code of such a parameterized divergence-free magnetic field that can be forward-fitted to the 3-D geometry of stereoscopically triangulated coronal loops. So, the simple goal of this study is to evaluate how closely the stereoscopically observed loops can be modeled in terms of potential fields, a goal that was already attempted with Skylab observations (Sakurai and Uchida 1977). Modeling with non-potential fields, such as nonlinear force-free field (NLFFF) models, will be considered in future studies.

This paper is organized as follows: The definition of a parameterized potential field is described in Section 2, the development and tests of a numeric magnetic field code and the results of forward-fitting to stereoscopically triangulated loops is presented in Section 3, and conclusions follow in Section 4. An alternative approach with dipolar magnetic fields is the subject of Paper II (Sandman and Aschwanden 2010).

2 Theory and Definitions

2.1 Divergence-Free Magnetic Field

Since the coronal plasma- parameter is generally less than unity (Gary 2001), the magnetic pressure exceeds the thermal pressure, and thus all soft X-ray or EUV-emitting plasma that fills or flows through coronal flux tubes traces out the coronal field. Consequently, stereoscopic triangulation of EUV loops provides the correct 3-D field directions along coronal loops. We denote the normalized 3-D field direction along a loop with the unity vector , which is parameterized as a function of a loop length coordinate , starting at the footpoint position at the base of the corona,

(1)

where the magnetic field is defined by the cartesian components . However, the absolute magnitude of the magnetic field strength, , is not known a priori. For a physical solution of the magnetic field, Maxwell’s equation of a divergence-free magnetic field has to be satisfied,

(2)

which (in its integral form) corresponds to the theorem of magnetic flux conservation along a fluxtube,

(3)

where is the integral over the cross-sectional area of a fluxtube defined in perpendicular direction to the magnetic field line (at loop position ). Therefore, since the stereoscopic triangulation defines the magnetic field directions in adjacent flux tubes, it defines also the divergence of the field and the relative change of the magnetic field strength along the flux tubes, and this way implicitly defines also the isogauss surfaces perpendicular to each fluxtube, and therefore the full 3-D vector field , except for a scaling constant. The derivation of a 3-D magnetic field in an active region,

(4)

requires also the knowledge of the scalar function in every 3-D location , which we constrain with a forward-fitting method of a divergene-free field model.

A divergence-free 3-D magnetic field model can be parameterized by a superposition of divergence-free fields, because the divergence-free condition is linear (or Abelian), i.e., if two components and fulfill and , then also their sum is divergence-free, . Divergence-free magnetic field components are, for instance, a parallel field, unipolar fields (a magnetic charge with spherical isogauss surfaces and a field that falls off with the square of the distance), dipole fields, quadrupolar fields, other multi-pole representations, or any potential field. The Abelian property warrants that any superposition of divergence-free fields is also divergence-free. Specifically, we will use divergence-free potential field models that consist of either i) multiple unipolar charges (in this Paper I), or ii) multiple dipoles (in Paper II).

Our philosophy is the following: We will employ magnetic field models of the category of potential-field models, which are divergence-free by definition. We use particular potential field models that can be quantified with a finite number of free parameters. Potential field models are not as general as non-potential and force-free field (NLFFF) models. However, since both potential and NLLLF models currently exhibit an equally poor misalignment with observed EUV loops, we need first to investigate whether the misalignement between observations and any theoretical model can be mimized for the simplest class of magnetic field models, such as potential field models. If our approach proves to be successfull, refinements with non-potential or NLLLF models can then be pursued in the future along the same avenue (e.g., see Conlon and Gallagher 2010).

2.2 Multiple Unipolar Magnetic Charges

Unipolar potential fields often provide a good approximation to the magnetic field of sunspots, and thus can be used also for an active region that is composed of a finite number of spot-like magnetic polarities. Conceptually, a unipolar field can be considered as an approximation to the upper half of a vertically positioned dipole. The simplest representation of a unipolar magnetic field that is a potential field, and hence fulfills Maxwell’s divergence-free condition, is a spherically symmetric field that drops off with an -dependence with distance, so it has a potential function that drops off with ,

(5)

where is the potential field value at the solar surface vertically above the buried magnetic charge in depth . Of course, the extrapolated magnetic field is only computed in the coronal domain , so that no “magnetic monopole” exists in the solar corona. The magnetic field model of a single magnetic charge requires 4 parameters: the maximum value of the potential field and the location of the buried charge, having a distance to any point in the solar corona,

(6)

The resulting magnetic field has then only a radial component in direction of ,

(7)

with the surface field strength . This unipolar potential field fulfills the divergence-free condition, as it can be calculated from the Laplacian operator of the potential function,

(8)

The fulfillment of the divergence-free condition can also be verified from the conservation of the magnetic flux theorem (Eq. 3), if the envelope of a fluxtube is defined by radial field lines, so that the cross-sectional area remains constant for . In our first model we employ a superposition of multiple unipolar charges,

(9)

in terms of the vector . For a single unipolar charge, the field lines will all be straight lines in radial direction away from the buried charge, which can approximate open-field regions. Burying multiple magnetic charges of opposite magnetic polarity, however, can mimic closed-field regions. An example is given in Fig. 1, where we compare the magnetic field of a dipole with that of a combination of two unipolar charges with opposite magnetic polarity. Actually, the two magnetic field models become identical when the two unipolar charges are moved close together at the location of the dipole moment, as it can be shown mathematically. Although the two models are equivalent in the far-field approximation, a combination of two unipolar charges (with free parameters) allows more general solutions than a single dipole (with 6 free parameters), especially in the case of strongly asymmetric fields (sunspots) or open-field regions, as they exist is most active regions.

2.3 Definition of Misalignment Angle

The forward-fitting of our analytical magnetic field model to a set of observed magnetic field vectors (e.g., using stereoscopically triangulated loop coordinates), is the task of optimizing the free parameters of the analytical model until the best match with the observed field lines is obtained. For the evaluation of the goodness or consistency of the analytical magnetic field models with the observed field line model , we define the 3-D misalignment angle , which is defined by the scalar product between the two field vectors and ,

(10)

or equivalently, between the unity field vectors and ,

(11)

This parameter is a single value at every spatial point , which can be averaged at the coronal base or over the lengths of the observed field lines, at positions,

(12)

which is similar to a -criterion. Since a unipolar magnetic charge can be parameterized with 4 free parameters ( (Eq. 9), a model with components has free parameters.

3 Numerical Code and Results

Our strategy to bootstrap the coronal magnetic field with stereoscopic data consists of the following steps: (1) we create a model of subphotospheric magnetic charges by deconvolving an observed photospheric magnetogram into point charges, which defines the unit vectors of our parameterized theoretical magnetic field model ; (2) we perform stereoscopic triangulation for a set of coronal loops observed with STEREO/EUVI, which are quantified in terms of directional field vectors ; (3) we forward-fit the theoretical magnetic field model with a number of free parameters in the setup of unipolar magnetic charges by minimizing the mean misalignment angle , and which yields a best-fit solution . We can then compare the minimized misalignment of the bootstrapped best-fit model with those of standard methods based on extrapolation of the the photospheric boundaries using a magnetogram, e.g., with the Potential Field Source Surface (PFSS) model. The procedure is illustrated in Fig. 2, where a dipolar EUV loop is observed and a magnetic field model is constructed from two unipolar charges. By adjusting the field strength of the second unipolar charge from to , the misalignment angle between the observed EUV loop and the model field can be reduced from (Fig. 2 top) to (Fig. 2, bottom). Note that the adjusted field is still a potential field and divergence-free, but represents a better match to the observed EUV loop.

3.1 Data Selection

We select four active regions observed with STEREO/EUVI and the Michelson Doppler Imager (MDI) (Scherrer et al. 1995) onboard the Solar and Heliospheric Observatory (SoHO): 2007 April 30, May 9, May 19, and Dec 11. The first AR 10953 (2007 Apr 30) is identical to the case previously analyzed with STEREO and Hinode (DeRosa et al. 2009; Sandman et al. 2009). The second AR 10955 (2007 May 9) was subject of the first stereoscopically triangulated coronal loops, temperature and density measurements, and stereoscopic tomographic reconstruction (Aschwanden et al. 2008a,b, 2009; Sandman et al. 2009). The third AR 10953 (2007 May 9) displayed a small flare (during 12:40-13:20 UT) as well as a partial filament eruption during the time of observations, and was featured in a few studies (Li et al. 2008; Liewer et al. 2009; Sandman et al. 2009). The fourth AR 10978 (2007 Dec 11) is also subject of recent magnetic field modeling (Aad Van Ballegooijen and Alec Engell, private communication 2010). Some details of these four active regions are given in Table 1, such as the heliographic position of the AR center, the magnetic area for fluxes of G, the minimum and maximum field strengths, and the total unsigned magnetic flux.

3.2 Parameterization of Magnetic Field Model

All four active regions were observed with the Michelson Doppler Imager (MDI) onboard the Solar and Heliospheric Observatory (SoHO), which provides full-disk MDI magnetograms with a pixel size of 2. Subimages encompassing the active region of interest are shown in Fig. 3 (left column), with quadratic field-of-view sizes ranging from 145 to 339 pixels, or 0.3 to 0.7 solar radii, respectively. In order to create a realistic 3D magnetic field model we decompose the partial magnetograms into a number of positive and negative Gaussian 2-D components, using an iterative 2-D Gaussian fitting scheme that determines local maxima in decreasing order of field strengths. The composite magnetogram of these 200 decomposed sources is shown in Fig. 2 (middle column), and the difference to the original magnetogram is also shown (right column). For each of the 200 magnetic source components we store the peak magnetic field value , the center position (), and the half widths at full maximum. For a parameterization in terms of unipolar charges, however, we need to convert the half width into the corresponding depth at which the unipolar charge is buried. From the definition of the full width at half maximum (FWHM) at of a unipolar field (Eq. 9), we have for the vertical magnetic field component (see geometry in Fig. 4),

(13)

with the angle between the vertical and a surface ring with radius (Fig. 4), from which we can calculate the dipole depth ,

(14)

which corresponds approximately to the half width of the fitted Gaussian component. Since we have now all input parameters , , for a definition of a magnetic field model with multiple unipolar magnetic charges (Eq. 9), we can calculate the full 3D magnetic field by superimposing the fields of all components. A particular field line is simply calculated by starting with the field at a footpoint position and by iterative stepping in the field direction, until the field line returns to the solar surface (in case of closed field lines) or to a selected boundary of the computation box (for open field lines).

3.3 Stereoscopic Triangulation of EUV Loops

For each of the four active regions we triangulate as many loops as can be discerned with a highpass filter in image pairs from STEREO/EUVI A and B. The method of stereoscopic triangulation is described in detail in Aschwanden (2008a), from which we use identical loop coordinates for AR 10955 observed on 2007 May 9. The stereoscopic triangulation requires accurate coalignment of both EUVI/A and B images in an epipolar coordinate system. Furthermore, we transform the 3-D loop coordinates measured in the epipolar coordinate system with the line-of-sight of the EUVI/A spacecraft into the coordinate system of MDI, which sees the Sun from the Langrangian point L1, almost in the same direction as seen from Earth. The advantage of transforming the EUVI loop coordinates into the MDI reference frame is the direct modeling of the longitudinal magnetic field component in the MDI reference frame, without requiring any knowledge of the absolute magnetic field strength , which is model-dependent, i.e., it depends on the choice of the extrapolation method (potential, force-free, or NLFFF) from photospheric magnetograms (which measures only the longitudinal component ). Some parameters of the stereoscopic triangulation procedure are listed in Table 1, such as the heliographic coordinates of the active region, the spacecraft separation angle, and the number of triangulated EUV loops (varying between 70 and 200 per active region, combined from all three coronal wavelengths, 171, 195, and 284 Å). 3-D representations of the stereoscopically triangualted EUV loops are shown in Figs. 5-8 (with blue color), seen along the line-of-sight of MDI (in greyscale maps of Figs. 5-8) and in the two orthogonal directions (sideview and topview in Figs. 5-8). The height range of stereosopically triangulated loops generally does not exceed 0.1 solar radii, due to the drop of dynamic range in flux for altitudes in excess of one hydrostatic scale height.

3.4 Forward-Fitting of Potential Field Models

After we have parameterized the magnetic field with unipolar charges, each one defined by 4 parameters , using the procedure of iterative decomposition of a photospheric magnetogram as described in Section 3.2, we can vary these free model parameters to adjust it to the 3D geometry of the stereoscopically triangulated loops. However, since we typically represent an MDI magnetogram with components, we have free parameters, which are too many to optimize independently. Standard optimization procedures, such as the Powell algorithm (Press et al. 1986), allow for optimization of free parameters, with good convergence behavior if the initial guesses are suitably chosen. Therefore we choose 10-20 small circular zones (with typical radii of solar radii) containing the strongest field regions in the magnetogram of equal magnetic polarity and optimize the magnetic field strengths with a common correction factor in each zone. This is an empirical optimization of the lower boundary of the coronal magnetic field, optimized by varying the (decomposed) photospheric MDI magnetogram in such a way that the resulting potential field more closely matches the stereoscopically triangulated loops. The results of the best fits, based on the mimization of the median misalignment angle (Eqs. 11-12), are shown in Figs. 5-9, in form of (red) model field lines that are extrapolated at the same locations as the footpoints of the stereoscopic loops (blue in Figs. 5-9). The distribution of all misalignment angles, evaluated at about 80 locations of every loop is shown in form of histograms at the bottom of Figs. 5-9. Each histogram is characterized with a gaussian peak, which are found to have a mean value and standard deviation of (Fig.5: 2007 Apr 30), (Fig.6: 2007 May 9), (Fig.7: 2007 May 19), and (Fig.8: 2007 Dec 11). If we compare these misalignment angles in the range of with the previously measured values using a potential field source surface (PFSS) model (; Sandman et al. 2009), or using a nonlinear force-free field (NLFFF) model (; DeRosa et al. 2009), we see an improvement of the misalignment angle by about a factor of two. This is a remarkable result that demonstrates that the magnetic field at the lower boundary of the corona can be bootstrapped with stereoscopic measurements and with a suitable parameterization of a potential field model. Although the extrapolation from a photospheric magnetogram should be unique, if there is no data noise present, an infinite number of potential field solutions can be obtained depending on how the boundary field is parameterized (e.g., by unipolar charges or dipolar magnetic moments). In our case, for every variation of the free parameters, a slightly different field with a different misalignment to the stereosocpic loops is obtained. The PFSS code is designed to compute the potential field of the entire (front and backside) Sun, and thus has a relatively coarse spatial resolution of typically (one heliographic degree, i.e., 12 Mm) for standard computations, but the small-scale magnetic field should not matter too much for our large-scale loops ( Mm). Hence, the best-fit potential-field bootstraps a force-free photospheric boundary field that is significantly different from the observed photospheric magnetograms which is observed in a non-force-free zone.

3.5 Convergence Behavior

The remaining misalignment between our best-fit potential field solution and the stereoscopic loops could be due to three sources of errors: (1) The bootstrapping method did not converge to the best solution; (2) The 3D coordinates of the stereoscopically triangulated loops have some error; or (3) The real coronal magnetic field that is represented by the stereoscopic loops could be non-potential. We consider the convergence of our code as satisfactory, because we ran many attempts for each case with different initial conditions and obtained about the same minimum misalignment. One possibility to vary the initial conditions is to vary the number of (gaussian) unipolar components. Fig. 9 shows the convergence behavior as a function of the number of (decomposed) unipolar components, where we computed a best-fit potential field solution for unipolar components for each of the 4 active regions. Convergence to the minimum misalignment value typically recquires components for the simplest dipolar active regions (2007 May 9 or Dec 11), and for more complex active regions (2007 Apr 30 or May 19).

3.6 Estimates of Stereoscopy Error

We investigate the second possible source of errors that could contribute to the measured misalignment, i.e., errors associated with the stereoscopic triangulation method. The errors of stereoscopic triangulation have been discussed in Aschwanden et al. (2008a), which depend on (1) the ratio of the stereoscopic parallax angle to the spatial resolution of the instrument, and thus on the spacecraft separation angle, (2) the angle between the loop segment and East-West direction in the epipolar plane, being largest for loop segments parallel to the epipolar plane, and (3) the proper identification of a corresponding loop in image B to a selected loop in image A. While the first two sources of errors can be formally calculated, the correspondence problem is difficult to quantify. Since stereoscopic triangulation cannot be accomplished in an automated way at present time, the error of identifying corresponding loops could be estimated from the scatter obtained with different observers, but this is time-consuming. Here we pursue another approach that is based on a self-consistency test. The procedure works as follows. If a sufficient large number of loops are triangulated in an active region, there should be for every triangulated loop (which we call primary loop) a neighbored (secondary) loop with an almost parallel direction. Whatever the direction of the true magnetic field in the same neighborhood is, both the primary and the secondary STEREO loop should have a similar misalignment angle with the local magnetic field, , or , in order to be self-consistent. Therefore we can define a stereoscopic error (SE) angle by averaging these differences in misalignements over all STEREO loop positions with suitable weighting factors ,

(15)

where the index runs over all loop positions for each primary loop and the index runs over all loop positions of secondary loops, excluding the primary loop. For the weighting factor we should give less weight to more distant neighbors, because the true magnetic field and thus the local misalignment angle is likely to increase with distance. Thus we should choose a negative power of the relative distance ,

(16)

where is a power index. If the index is small, say the weighting is reciprocal to the distance and the long-range neighbors are relatively strongly weighted. If the index is large, say , the short-range neighbors are relatively strongly weighted while the longe-range neighbors have almost no weight. So we expect that the so-defined stereoscopic error monotonously decreases with short-range weighting towards higher power indices . In Fig. 10 we show the stereoscopic error calculated with Eqs. (15-16) as a function of the power index from to , for all 4 active regions. Indeed, the stereoscopic error monotonously decreases from to , because we give progressively less weight to the long-range misalignments. However, from towards the error increases again, probably because of the inhomogeneity of the closest neighbors at the shortest range and the excessive weighting to the very nearest neighbors. However, the plateau in the range of is a good indication that we measure a stable value at the minumum of . From this we obtain a misalignment contribution of (2007 Apr 30), (2007 May 9), (2007 May 19), and (2007 Dec 11), as listed in Table 2.

3.7 Nonpotentiality of Magnetic Field

A diagram of the various misalignment angles is shown in Fig. 11, which we plot as a function of the maximum GOES soft X-ray flux during the observing interval. The GOES flux was dominanted by the emission from the active regions analyzed here, since there was no other comparable active region present on the solar disk during the times of the observations. The case of 2007 May 9 with the lowest GOES shows a single dipolar structure, while the case of 2007 May 19 with the highest GOES flux exhibits multiple dipolar groups. The misalignment angles indicate those obtained with the potential field source surface (PFSS) code, those with the nonlinear force-free field (NLFFF) code (only for 2007 Apr 30), those optimized with the potential field with unipolar charge (PFU) code, and indicates the stereosocpic errors. As a working hypothesis we might attribute the remaining residuals to the non-potentiality of the active regions,

(17)

for which we measure the values: (2007 Apr 30), (2007 May 9), (2007 May 19), and (2007 Dec 11), listed also in Table 2. We consider these values as a new method to quantify the nonpotentialiy of active regions. We find that quiescent active regions that contain simple dipoles (2007 Apr 30 and May 9) have small misalignments of with best-fit potential field models, while more complex active regions (2007 May 19 and Dec 11) have somewhat larger non-potential misalignments in the order of compared with best-fit potential field models.

While we express thge degree of nonpotentiality in terms of a mean misalignment angle here, other measures are the ratios of the total nonpotential to the potential energy in an active region, which amounts up to in one flaring active region (Schrijver et al. 2008).

3.8 Quiescent and Flaring Active Regions

Since the four investigated active regions have quite different misalignment angles, and also the inferred nonpotentiality of the magnetic field varies significantly, we quantify the activity level of the active regions from the soft X-ray flux measured by GOES. Fig. 12 shows the GOES light curves for the 4 active regions during the times of stereoscopic triangulation. The lowest GOES flux is measured for the AR of 2007 May 9, with a level of W m (GOES class A4), while the highest GOES flux is measured for the AR of 2007 May 19, which has a flare occurring during the observing period with a GOES flux of W m (GOES class C0). The three active regions with the low soft X-ray flux levels appear to be quiescent, judging from the GOES flux profile, or may be subject to micro-flaring at a low level.

There is a clear correlation between the soft X-ray level of the active region (when it was on disk) and the overall misalignment angle (Fig. 11), as well as with the misalignment angle attributed to the non-potentiality, varying from for the lowest GOES A-class levels to for an active region with a GOES C-class flare. A higher soft X-ray flux generally means a higher heating rate with stronger impulsive heating or flaring. A higher degree of non-potentiality, on the other hand, indicates the presence of a higher level of electric currents (which are non-potential). Therefore, the observed correlation suggests a physical relationship between the electric currents in an active region and the amount of heating input. This is not surprising, since evidence for current-carrying emerging flux was demonstrated previously for H and soft X-ray structures that are non-potential (e.g., Leka et al. 1996; Jiao et al. 1997; Schmieder et al. 1996). Our measurement of the degree of non-potentiality with the magnetic field misalignment averaged over the entire active region, is a very coarse technique, but a more detailed investigation of the misalignment in separate parts of the active region that are quiescent or flaring will be pursued in Paper II.

4 Conclusions

The agreement between theoretical magnetic field models of active regions in the solar corona with the true 3-D magnetic field as delineated from the stereoscopic triangulation of coronal loops in EUV wavelengths has never been quantified until the recent advent of the STEREO mission. To everybody’s surprise, the average misalignment between the theoretical and observed magnetic field was quite substantial, in the amount of for both potential and nonlinear force-free field models (DeRosa et al. 2009; Sandman et al. 2009). In this study we investigate the various contributions of this large misalignment for four different active regions observed with STEREO and arrive at the following conclusions:

  1. The amount of misalignment can be reduced to about half of the values for potential-field models optimized by a bootstrapping method that minimizes the field directions with the stereoscopically triangulated loops. Our potential-field model is parameterized with unipolar charges per active region, whose positions and field strengths are approximately derived from a gaussian decomposition of a photospheric magnetogram, and then varied until a best fit is obtained. The best-fit potential field model has an improved misalignment of . Because the best-fit potential field model defines an improved magnetic field boundary condition at the bottom of the corona, the difference to the observed photospheric magnetogram contains information on the currents between the photosphere and the base of the force-free corona.

  2. We estimate the misalignment contriubtion caused by stereoscopic correlation errors from self-consistency measurements between the magnetic field misalignments of adjacent loops. We find contributions in the order of .

  3. We estimate the contributions to the field misalignment due to non-potentiality caused by electric currents from the residuals between the best-fit potential field and the stereoscopic triangulation errors and find misalignment contributions in the order of .

  4. The overall average misalignemnt angle between potential field models and stereoscopic loop directions, as well as the contribution to the misalignment due to non-potentiality, are found to correlate with the soft X-ray flux of the active region, which suggests a correlation between the amount of electric currents and the amount of energy dissipation in form of plasma heating in an active region.

In this study we identify for the first time the contributions to the misalignment of the magnetic field, in terms of optimized potential field models, non-potentiality due to electric currents, and stereoscopic triangulation errors. These results open up a number of new avenues to improve theoretical modeling of the coronal magnetic field. First of all, optimized potential field models can be found that represent a suitable lower boundary condition at the base of the force-free corona, which provides a less computing-expensive method than nonlinear force-free codes. Second, methods can be developed that allow us to localize electric currents in the non-force-free photophere and chromosphere. Third, the misalignment angle can be used as a sensitive parameter to probe the evolution of current dissipation, energy build-up in form of non-potential magnetic energy in different quiescenct and flaring zones of active regions. The high-resolution magnetic field data from Hinode and Solar Dynamics Observatory provide excellent opportunities to obtain better theoretical models of the coronal magnetic field using our bootstrapping method, which is not restricted to stereoscopic data only, but can also be applied to single-spacecraft observations.

Acknowledgements: We are grateful to helpful discussions with Marc DeRosa and Allen Gary. This work was partially supported by the NASA contract NAS5-38099 of the TRACE mission and by NASA STEREO under NRL contract N00173-02-C-2035. The STEREO/SECCHI data used here are produced by an international consortium of NRL, LMSAL, RAL, MPI, ISAS, and NASA. The MDI/SoHO data were produced by the MDI Team at Stanford University and NASA.

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Schrijver,C.J., M.L. DeRosa, T. Metcalf, G. Barnes, B. Lites, T. Tarbell, J. McTiernan, G. Valori, T. Wiegelmann, M.S. Wheatland, T. Amari, G. Aulanier, P. Demoulin, M. Fuhrmann, K. Kusano, S. Regnier, and J.K. Thalmann 2008, Astrophys. J. 675, 1637.

Active Observing Observing Spacecraft Number Magnetic Magnetic Magnetic
Region date times separation of EUVI area (B100 G) field strength flux
Active (UT) angle (deg) (loops) ( cm) B(G) ( Mx)
10953 (S05E20) 2007-Apr-30 23:00-23:20 5.966 200 24.2 [-3134,+1425] 8.7
10955 (S09E24) 2007-May-9 20:30-20:50 7.129 70 4.4 [-2396,+1926] 1.6
10953 (N03W03) 2007-May-19 12:40-13:00 8.554 100 12.2 [-2056,+2307] 4.0
10978 (S09E06) 2007-Dec-11 16:30-16:50 42.698 87 8.2 [-2270,+2037] 4.8
Table 1: Data selection of four Active Regions observed with STEREO/EUVI and SOHO/MDI.
Parameter 2007-Apr-30 2007-May-9 2007-May-19 2007-Dec-11
Misalignment NLFFF 2444
Misalignment PFSS 25 8 19 6 36 13 32 10
Misalignment PFU 14.311.5 13.39.3 20.316.5 15.212.3
Median PFU 20.0 16.2 25.8 15.7
Stereoscopy error 9.4 7.6 11.5 8.9
Nonpotentiality 4.9 5.7 8.8 7.2
GOES soft X-ray flux
GOES class A7 A4 C0 B1

Measured with nonlinear force-free field code (DeRosa et al. 2009),

Measured with potential field source surface code (Sandman et al. 2009),

Measured with unpolar potential field model (this study),

Measured from inconsistency between adjacent loops,

Difference of mean misalignment in unipolar model, stereoscopy error,

Goes flux in units of [W m].

Table 2: Misalignment statistics of four analyzed active regions.
Figure 1: The magnetic field of a symmetric dipole (dashed lines) is shown, together with the field resulting from the superimposition of two unipolar magnetic charges (solid lines). The two field models become identical once the two unipolar charges are moved to the location of the dipole moment at position . The radial field of each unipolar positive charge is also shown for comparison (dotted lines).
Figure 2: Top: A potential dipole field is calculated with two unipolar magnetic charges buried in equal depth and equal magnetic field strength, but opposite polarization (, ). An asymmetric EUV loop is observed at the same location (grey torus) with a misalighment of at the top of the loop. Bottom: The magnetic field strength of the right-hand side unipolar charge is adjusted (to ) until the misalignment of the loop reaches a minimum of .
Figure 3: Left column: Photospheric magnetograms observed with SOHO/MDI on four different dates. Middle: The magnetic field is decomposed into typically unipolar charges and the model displays the longitudinal magnetic field strength on the same grey scale. Right columne: Difference between observed MDI magnetogram and model.
Figure 4: Cross-section of gaussian magnetic field component showing the geometric relation between the half width and depth of a unipolar charge. The radial magnetic field drops off quadratically with the distance , while the vertical component is forshortened by a factor .
Figure 5: 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 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.
Figure 6: Best-fit potential field model of AR observed on 2007 May 9. Otherwise similar representation as Fig. 5.
Figure 7: Best-fit potential field model of AR observed on 2007 May 19. Otherwise similar representation as Fig. 5.
Figure 8: Best-fit potential field model of AR observed on 2007 Dec 11. Otherwise similar representation as Fig. 5.
Figure 9: Mean misalignment angle of best-fit potential field solution as a function of the number of unipolar components (right side) and distribution of misalignment angles for one particular value of (left side). The solid curves denote the mean of the misalignment angles, and the dashed curves denote the median misalignment angles.
Figure 10: Evaluation of stereoscopic error from the consistency of misaligment angles as a function of the short- or long-range weighting.
Figure 11: 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 measuement 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).
Figure 12: GOES soft X-ray light curves of the 0.5-4 Å (upper curve) and 1-8 Å channel (lower curve) during the time of stereoscopic triangulation and magnetic modeling of the active region. The peak level of the GOES flux during the observing time is indicated with a thick bar.
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