The Astrophysical Journal, 695:12–29, 2009 April 10 C 2009.
doi:10.1088/0004-637X/695/1/12
The American Astronomical Society. All rights reserved. Printed in the U.S.A.
FIRST THREE-DIMENSIONAL RECONSTRUCTIONS OF CORONAL LOOPS WITH THE STEREO A+B SPACECRAFT. III. INSTANT STEREOSCOPIC TOMOGRAPHY OF ACTIVE REGIONS Markus J. Aschwanden, Jean-Pierre Wuelser, Nariaki V. Nitta, James R. Lemen, and Anne Sandman Lockheed Martin Advanced Technology Center, Solar & Astrophysics Laboratory, Org. ADBS, Bldg.252, 3251 Hanover Street, Palo Alto, CA 94304, USA;
[email protected] Received 2008 July 30; accepted 2009 January 2; published 2009 March 27
ABSTRACT Here we develop a novel three-dimensional (3D) reconstruction method of the coronal plasma of an active region by combining stereoscopic triangulation of loops with density and temperature modeling of coronal loops with a filling factor equivalent to tomographic volume rendering. Because this method requires only a stereoscopic image pair in multiple temperature filters, which are sampled within ≈ 1 minute with the recent STEREO/EUVI instrument, this method is about four orders of magnitude faster than conventional solar rotation-based tomography. We reconstruct the 3D density and temperature distribution of active region NOAA 10955 by stereoscopic triangulation of 70 loops, which are used as a skeleton for a 3D field interpolation of some 7000 loop components, leading to a 3D model that reproduces the observed fluxes in each stereoscopic image pair with an accuracy of a few percents (of the average flux) in each pixel. With the stereoscopic tomography we infer also a differential emission measure distribution over the entire temperature range of T ≈ 104 –107 , with predictions for the transition region and hotter corona in soft X-rays. The tomographic 3D model provides also large statistics of physical parameters. We find that the extreme-ultraviolet loops with apex temperatures of Tm 3.0 MK tend to be super-hydrostatic, while hotter loops with Tm ≈ 4–7 MK are near-hydrostatic. The new 3D reconstruction model is fully independent of any magnetic field data and is promising for future tests of theoretical magnetic field models and coronal heating models. Key words: Sun: corona – Sun: UV radiation Online-only material: color figures
in nonequilibrium, or flare loop systems, a major restriction than cannot be overcome with solar-rotation-based tomography methods. The coronal plasma is thought to be structured by the architecture of open and closed magnetic field lines, due to the low value of the coronal plasma-β parameter. A direct consequence of this concept is that an arbitrary 3D volume of the corona, say an active region, can be modeled as a composite of isolated onedimensional flux tubes, each one characterized by its own hydrodynamic density n(s) and temperature profile T (s), as a function of its 1D loop length coordinate s, whose geometry is defined by loop field line coordinates s(x, y, z). Such physical modeling has recently been carried out for the full corona (Schrijver et al. 2004), or for particular active regions (Warren & Winebarger 2006, 2007; Lundquist et al. 2008a, 2008b). The 3D geometry of loops has been modeled in all previous studies with magnetic field extrapolations (mostly using a potential field model), which often substantially deviate from the observed loop geometries (e.g., Sakurai et al. 1992; Jiao et al. 1997; Schmieder et al. 1996). Moreover, the hydrodynamic structure of individual loops has been assumed to correspond to a stationary equilibrium solution that can be described with a scaling law (where the heating and cooling rates are balanced in the energy equation), with the heating rate prescribed by some power-law scaling of the magnetic field B and loop length L, i.e., EH ∝ B α /Lβ . However, other studies have demonstrated that most of the coronal loops observed in extreme-ultraviolet (EUV) wavelengths are not in hydrostatic equilibrium (e.g., Lenz et al. 1999; Aschwanden et al. 2000, 2001). Therefore, since none of these three major assumptions used in previous modeling is expected to hold in detail, only a crude agreement with observations can be expected, possibly constraining the heating scaling law in the statistical average.
1. INTRODUCTION There are different methods of three-dimensional (3D) reconstruction of the coronal plasma. Coronal “tomography” reconstructs the 3D density distribution ne (x, y, z) with backprojection methods from many different aspect angles, usually obtained from synthesizing images during a half-solar rotation (e.g., Hurlburt et al. 1994; Frazin & Janzen 2002; Frazin & Kamalabadi 2005; Frazin et al. 2005a, 2005b, 2007; Zidowitz 1997, 1999), or from three different aspect angles (Kankelborg 2008). Solar “stereoscopy” reconstructs the 3D coordinates s(x, y, z) of individual curvi-linear features (e.g., loops, filaments, or fluxropes) by trianglation from two different aspect angles (Davila 1994; Gary et al. 1998; Aschwanden et al. 1999, 2008a, 2008b [Papers I and II]; Wiegelmann & Inhester 2006; Feng et al. 2007; Patsourakos et al. 2008). Here we explore a new method of 3D reconstruction that combines (1) the stereoscopic triangulation s(x, y, z) of coronal loops and (2) the tomographic reconstruction of the volumetric 3D density distribution ne (x, y, z) of an active region, which we call the Instant Stereosocpic Tomography of Active Regions (ISTAR)1 method. The essential difference to standard tomography, where the topology of the reconstructed 3D volume is amorphous, is the introduction of a substructuring by one-dimensional (1D) loop geometries s(x, y, z), which can accommodate for either sparse or dense space filling of the reconstructed volume, depending on the threshold density value. Our new ISTAR method makes full use of stereoscopic 3D information and can be applied to arbitrary short-time durations, which allows us instantaneously to reconstruct dynamic phenomena, such as coronal loops 1
Disclaimer: Our acronym has no relationship to the Intelligence, Surveillance, Target Acquisition, and Reconnaissance software also known as ISTAR.
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The new STEREO mission (Kaiser et al. 2007), launched in 2006, however, opens up a new avenue for 3D geometric modeling of coronal loops that is independent of any assumption on the coronal magnetic field. Moreover, the physical density and temperature structure of individual loops can be constrained by fitting the emission measure (EM) of an ensemble of model loops pixelwise to the multiwavelength images observed with both STEREO spacecraft, which allows us to model their hydrodynamic structure without any assumption on energy balance or equilibrium state, and thus provides much more accurate constraints on the spatio-temporal heating function of individual loops. In Paper I (Aschwanden et al. 2008a) of this series we determined for the first time the 3D geometry of 30 active region loops using stereoscopic triangulation from two spacecraft, while the electron densities and electron temperatures of these loops were determined in Paper II (Aschwanden et al. 2008b), agreeing within an accuracy of a few percents between the two spacecraft. In this paper we develop the method of stereoscopic loop reconstruction further by including new refinements in the data analysis method: (1) multiscale filtering to enhance loop structures, (2) stacking of a time series of multiple images in order to improve the signal-to-noise ratio and to obtain a more complete number of stereoscopically triangulated loops, (3) combining multiple temperature filters to obtain a more complete set of loops with different temperatures, (4) 3D interpolation of field lines within the skeleton of stereoscopically triangulated loops, and (5) plasma filling of all field lines s(x, y, z) to obtain a tomographic 3D volume rendering of the density ne (x, y, z) and temperature Te (x, y, z). This novel stereoscopic tomography method allows us to model the hydrodynamic structure of active regions with an unprecedented degree of detail. Timedependent hydrodynamic modeling will be pursued in Paper IV, while magnetic field modeling will be explored in Paper V. The plan of this paper is as follows: the data analysis method is described in Section 2; the results and discussion of new aspects are described in Section 3; and conclusions are summarized in Section 4. 2. DATA ANALYSIS AND MODELING METHOD Descriptions of the STEREO/EUVI instrument and various steps in the data analysis have been described in Papers I and II, such as image coalignment (Paper I: Section 2.5), stereoscopic loop triangulation (Paper I: Section 3.1 and 3.5), stereoscopic projection (Paper I: Section 3.2), loop tracing (Paper I: Section 3.4), EUVI temperature response functions (Paper II: Section 2.2), loop background subtraction methods (Paper II: Section 2.3), DEM modeling for density and temperature determination (Paper II: Section 2.4), which are not repeated here. In the following, we describe new features used in the data analysis here, such as multiscale filtering (Section 2.1), image stacking (Section 2.2), multifilter tracing (Section 2.3), 3D field line interpolation (Section 2.4), and 3D volume rendering with the Instant Stereoscopy Tomography of Active Region (ISTAR) method (Section 2.5). 2.1. Multiscale Filtering In the initial algorithm for loop tracing (Paper I) we used a high-pass filter with a single spatial scale, a method that is also called unsharp masking, where a smoothed image is subtracted from the original image to enhance the fine structures. In Paper I, we smoothed the unsharp image with a boxcar of nsm = 5.
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Two examples of single-scale filtering are shown in Figure 1, for a spatial scale (in the boxcar smoothing) of nsm = 3 to enhance the finest loop threads (Figure 1, bottom left), and for a coarser scale of nsm = 13 to enhance wider loop bundles (Figure 1, top right). The tradeoff between these two single filters is that the finer scale reveals the sharpest loop structures with highest positional accuracy, but has the lowest signal-tonoise ratio, while it is vice versa for the coarser scale. Since we aim for both, a high positional accuracy (important for the stereoscopic triangulation), as well as for a high signal-to-noise ratio in the tracing of loops (important for a high detection rate), we combine multiple scales in the filtering process with suitable proportions, which yields a good compromise for both quality criteria. Thus we superimpose multiple n high-pass-filtered images I sm,i (x, y), filtered with boxcars width of nsm,i = 2i + 1 for i = 1, . . . , n, n wi [I (x, y) − I sm,i (x, y)] n I HP (x, y) = i=1 (1) i=1 wi where the relative weighting factor wi is quadratically increasing toward smaller filter scales i, wi = [1 + (n − i)]2 .
(2)
Say, for a number of n = 6 multiple scales we have boxcars with widths of nsm = 2i + 1 = 3, 5, . . ., 13 pixels, and the relative weighting, i.e., wi = 36, 25, 16, 9, 4, 1, is quadratically decreasing with larger scales. With this definition, all larger filters combined have no more weight than a factor of 3 compared with the finest filter (for n 10 filter scales). This relative weighting maintains both the positional accuracy defined by the finest filter and a good signal-to-noise ratio boosted by the larger filter scales. An example of a multiscale filter with n = 6 filter scales is shown in Figure 1 (bottom right). 2.2. Image Stacking The signal-to-noise ratio can also be increased by stacking of a temporal sequence of images, if the evolution of features is minor during the accumulated time interval, provided that image coalignment is achieved with subpixel accuracy. STEREO/ EUVI has typical cadences of 150 s at 171 Åin the synoptic observing mode, or 75 s during campaigns. The spacecraft pointing does not track the solar rotation, but can be compensated in coaligned images by heliographic coordinate transformation or by cross-correlation. The solar rotation introduces a one-pixel shift (of 1. 6 for EUVI) in about 10 minutes. The angular resolution is about 2 EUVI pixels, corresponding to a solar rotation time interval of ≈20 minutes, an interval we choose to apply image stacking. This interval is also sufficiently short regarding observed temporal changes of loop brightnesses, using coaligned images (Paper I; Section 2.5) corrected for solar rotation by cross-correlation. An example of image stacking is shown in Figure 2, where we stacked multiple EUVI images observed during the time interval of 2007 April 30, 23:00–23:20 UT. Since the EUVI cadence was Δt = 150 during this time interval of Δt = 1200 s, we have n = 1200/150 = 8 images available for stacking. The stacked image has in principle a signal-to-noise √ ratio that is a factor of 8 ≈ 2.8 times better, which is clearly evident in the faint features of the high-pass-filtered images (Figure 2, bottom panels). However, at this lowest brightness level of ≈1 DN (data number) there are also some artifacts
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Figure 1. Part of a 171 Å image (pixel range x = 700–900, y = 800–1000) observed with STEREO/EUVI on 2007 May 9, 20:40:45 UT (see Paper I), is shown as original image with a linear grayscale (top left), after high-pass filtering with a boxcar of nsm = 3 pixels (bottom left), with a boxcar of nsm = 13 pixels (top right), and after multiscale filtering with boxcars of nsm = 3, 5, ..., 13 (bottom right). Note that the multiscale filter enhances both finer and wider structures with a balanced weighting and allows a more reliable loop tracing.
visible in the EUVI images, resulting from the entrance filter mesh of the EUVI telescope (visible as diagonal pattern in Figure 2, see also Figure 4 in Wuelser et al. 2004), that become even sharper after image stacking and have to be avoided when tracing loops. 2.3. Multifilter Loop Tracing The EUVI telescope has three coronal temperature filters, at the wavelengths of 171, 195, and 284 Å, with peak responses near the temperatures of T ≈ 1.0, 1.5, and 2.0 MK (Paper II, Figure 2). Therefore, coronal loops can be traced in all three filters, which approximately triples the number of traceable loops. Coronal loops have usually a narrow temperature range in their coronal part, so that they are only visible in one or two filters, but almost never in three filters, if strict cospatiality between the three filters is reassured (Aschwanden & Nightingale 2005). If a loop has a mean temperature that is in the middle range between the peak responses of two filters, the same loop could be traced in two filters, but most loops that are traced in multiple filters, even at near-cospatial locations, are not identical. Regardless whether an identical loop is traced in two filters or not, independent tracing in different filters improves the reconstruction of the 3D geometry in a coronal volume, increasing the accuracy for statistical models of the 3D geometry. An example of multifilter loop tracing is given in Figure 3 for AR 10955,
observed in 2007 May 9 (Paper I and II), where the three colors mark tracings in different temperature filters. Note that hottest loops (T ≈ 2.0 MK, yellow curves) mostly consist of complete closed field lines found in the core of the active region, while the cooler loops (T ≈ 1.0 MK, blue curves) mostly consist of incomplete segments of larger loops rooted in the periphery of the active region. The 3D geometry of these loops is shown in orthogonal projections to the image plane in Figure 4, calculated by stereoscopic triangulation as described in Paper I. 2.4. 3D Field Interpolation Stereoscopic triangulation of coronal loops can be performed only for the brightest loops that stand out of the background as curvi-linear features, while the bulk of fainter loops in the background cannot be triangulated directly. Quantitative modeling of the density structure of loops has shown that the combined flux associated with traceable loops amounts only to about 5%–10% of the total EUV flux in an image, and thus they represent only the tip of the iceberg. However, we can use these stereoscopically triangulated loops as a “skeleton” of the 3D geometry of an active region and fill in the geometry of undetected loops by a 3D-field interpolation scheme. In principle, if a skeleton of the 3D field is obtained from stereoscopy with a sufficiently high space filling factor, the field of the intervening space can be constrained with an
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arbitrary high degree of accuracy. In practice, in a 3D box that encompasses an active region, there will always be voids without stereoscopically detectable loops, where the 3D-field interpolation is less constrained and is biased by field lines that are not at close range. Such underconstrained regions bear larger uncertainties in delineating the magnetic field from stereoscopy, but they matter less for hydrodynamic 3D modeling, because only those spaces need to be filled with plasma for a 3D hydrodynamic model where bright loops exist, which are exactly the regions where we can detect and triangulate loops. In the following we describe our 3D field interpolation algorithm. The output of stereoscopic triangulation yields the 3D coordinates [x(si ), y(si ), z(si )] as a function of the loop length j coordinate si , i = 1, . . . , ns for a set of j = 1, . . . , nj loops. We define the normalized field direction vectors v(si ) at every point si along a loop, vx (si ) = [x(si+1 ) − x(si )]/|v| vy (si ) = [y(si+1 ) − y(si )]/|v| vz (si ) = [z(si+1 ) − z(si )]/|v|,
(3)
which are normalized to a length of unity by dividing with the length |v(si+1 ) − v(si )| of the directional vector vi , |v| = [x(si+1 ) − x(si )]2 + [y(si+1 ) − y(si )]2 + [z(si+1 ) − z(si )]2 . (4)
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These vectors vi defined for each loop length coordinate j si , i = 1, . . . , ns and loop j = 1, . . . , nj form the skeleton field of our field line interpolation now compute We can scheme. a new set of field lines x sik,l , y sik,l , z sik,l starting in a two-dimensional (2D)-Cartesian grid of footpoints k, l that covers the solar surface at the lower boundary of the 3D interpolation box with height range h = [0, hmax]. A new field line starts at a given footpoint position x s1k,l , y s1k,l , z s1k,l and is iteratively computed along the loop length coordinate sik,l by interpolating the field direction v sik,l at position k,l k,l k,l (as illustrated in Figure 5), x si , y s i , z s i vx sik,l = [vx (sm )w(sm )p(sm )] w(sm ) m
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[vy (sm )w(sm )p(sm )] w(sm ) vy sik,l = [vz (sm )w(sm )p(sm )] w(sm ), vz sik,l = m
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m j
where the index m runs over all ns old loop length coordinates j si , i = 0, . . . , ns of all j = 1, . . . , nj loops. The factor w(sm ) is a weighting factor of the skeleton vector v(sm ) that decreases
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Figure 3. Multifilter tracing of stereoscopically triangulated coronal loops for active region observed on 2007 May 9, 20:30–20:50 UT. The background image at 171 Å is multiscale high-pass filtered and is displayed with a gamma factor of 0.5. The superimposed curves correspond to 70 loops traced from spacecraft STEREO/A images at 171 Å (blue curves labeled 1–30), at 195 Å (red curves labeled 31–50), and at 284 Å (yellow curves labeled 51–70). The colors can considered to be an approximate temperature map, from T ≈ 1.0 MK (blue) over T ≈ 1.5 MK (red) to T ≈ 2.0 MK (red). Note that the hottest loops are found in the core of the active region, while the coolest are preferentially found in the peripheral plage region.
quadratically with distance, w(sm )
2 2 2 −2 = x(sm )−x sik,l + y sm −y sik,l + z sm −z sik,l , (6) which ensures that those skeleton field lines have the highest weight that are closest to the interpolation location. The coordik,l nate of the next position si+1 of the new loop is then defined by k,l = x sik,l + vx (xik,l ) |v| x si+1 k,l y si+1 = y sik,l + vy yik,l |v| k,l = z sik,l + vz zik,l |v|, z si+1
(7)
where |v| is the length of the field direction vector given in Equation (5).
The quantity p(sm ) in Equation (5) is a polarization sign of the skeleton field line, which is p = +1 if the magnetic field is parallel to the arbitrarily measured field line direction of the skeleton field line, i.e. v = +B/(|B|), and is vice versa p = −1 if the magnetic field is anti-parallel to the measured field line direction of the skeleton field line, i.e. v = −B/(|B|). This polarity sign corresponds to the 180◦ ambiguity in longitudinal magnetograms and has to be resolved before field lines are interpolated. So, the magnetic polarity has to be determined for every skeleton field line, which can be done either by using an observed magnetogram, or often from the relative topology of the stereoscopically triangulated magnetic skeleton field lines. In Figure 6, we show the polarity map of the skeleton field lines for AR 10953, which corresponds to a single dipole if we neglect some small-scale loops with parasitic magnetic polarity. In order to demonstrate how far we get from stereoscopic information alone, we do not make use of any magnetic information in this study (but see Paper V for magnetic modeling).
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Figure 4. 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 is shown in the top panel left, and the projection into the y–z plane is shown 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).
After we resolved the 180◦ -ambiguity of the skeleton field lines, we can now interpolate field lines at every point (x, y, z) in space (within some proximity to the skeleton field lines). We consider a cubic box that encompasses the volume of the skeleton field, with a square area in heliographic longitude and latitude, aligned with the solar surface, and covering a height range of h = [0, 0.1] solar radii. We interpolate field lines with footpoints distributed in a Cartesian (80 × 80) grid on the solar surface, over an area that corresponds to a field of view of 160 × 160 EUVI pixels (see Figures 1, 3, 4, and 6–8). The result of the interpolated field lines is shown in projection to the line of sight from spacecraft A in Figure 7, together with the skeleton field lines. 2.5. Tomographic Volume Rendering After having derived a 3D vector field v(x, y, z) at every point in space (x, y, z) from stereoscopic triangulation and interpolation, we can calculate a set of field lines with footpoints
starting in a Cartesian grid nk × nl with length coordinates s k,l (x, y, z). Since the set of field lines s k,l (x, y, z) fills a large fraction of the 3D space, we can now populate a 3D model of the density ne (x, y, z) and temperature distribution Te (x, y, z) by filling each field line with a 1D model of a fitting density ne (s) or temperature profile Te (s). For the physical parameterization of a 1D loop density model we define pressure profiles p(s) that decrease exponentially with height. Since many loops are asymmetric, we use two different base pressures (p1 , p2 ), and two pressure scale heights (λ1 , λ2 ), but the pressures have to match at the looptop, which constrains the pressure p2 at the second footpoint as a function of the other parameters. The height h(s) (vertical to the solar surface) as a function of the loop coordinate s is defined by the stereoscopically constrained 3D coordinates s(x, y, z). So, the two half sides of a loop have the following pressure functions:
h(s) − hchr (8) p(s < stop , h h1 ) = p1 exp − λ1
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vx(sik,l) x(sik,l)
x(s0j=2) x(s0j=1) x(s0k,l) x(s0j=3) Figure 5. 3D interpolation of a new vector vx (sik,l ) at position x(sik,l ) computed by interpolating the vector directions from all positions of the stereoscopically measured skeleton field lines s j =1 , s j =2 , s j =3 , weighted by a quadratically decreasing function to the distance between the new position and the old positions, see description in Section 2.4.
h(s) − hchr , p(s > stop , h h2 ) = p2 exp − λ2
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where the pressure p2 is constrained by the matching condition at the looptop,
htop − hchr htop − hchr p(s = stop ) = p1 exp − = p2 exp − λ1 λ2 (10) with an assumed chromospheric height of hchr = 2 Mm. The coronal height range of both footpoints is located at the top of the chromosphere, so h1 hchr and h2 hchr , but it can be somewhat higher in the transition region in the case of dynamic chromospheric processes. For the parameterization of the temperature profile Te (s) we use the function
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s − hchr s − hchr 2− Te (s) = Tchr + (Tm − Tchr ) , L − hchr L − hchr (11) which is a good approximation to hydrostatic temperature profiles for most uniform and nonuniform heating models (Aschwanden & Tsiklauri 2008). The temperature at the looptop (apex) is Te (s = stop ) = Tm , if we define the loop half-length by L = stop , while the temperature at the footpoints drops to Te (s = hchr ) = Te (s = 2L − hchr ) = Tchr ≈ 104 K. This functional form of the loop temperature profile seems generally closely to fit the observed EUV loops, regardless of whether they are in a state of hydrostatic equilibrium or nonequilibrium, as model fits to single loop threads at highest spatial resolution demonstrate (e.g., Figure 5 in Aschwanden et al. 2000). The electron density profile ne (s) is then simply defined by the relation for ideal gas as a function of the pressure p(s) and temperature Te (s), ne (s) =
p(s) 2kB Te (s)
with kB being the Boltzmann constant.
(12)
2 The EM profile per voxel (with constant cross-section wloop and incremental length ds) is then 2 ds. d EM(s) = n2e (s)dV ≈ n2e (s)wloop
(13)
The flux profile Ff (s) in a given filter wavelength f (171, 195, or 284 Å) is then defined (for a fully resolving instrument) by convolving the EM profile EM(s) with the instrumental response function Rf (T ), which is a function of the loop temperature profile Te (s), (14) Ff (s) = EM(s)Rf (T [s])ds. For a realistic instrument with a point spread function width of wres (i.e., 2.2 pixels for EUVI) and resolved loop width wloop , the image of an observed loop has to be simulated by a convolution with a Gaussian kernel with the widths added in quadrature, 2 + w2 . wobs = wres (15) loop We choose a nominal model loop width of wloop = 1500 km, which matches also the mean loop widths of elementary loops established from TRACE and EUVI (Paper II). This model loop width is not resolved with EUVI, and thus the chosen value does not matter in the fitting procedure, except that we have to keep in mind that the absolute values of the inferred electron −2 densities scale with ne ∝ wloop . In addition, we also add an empirical scattering function with a width of wscatter ≈ 4wobs at a level that corresponds to ≈ 10% of the loop flux, which represents a conservative estimate of the instrumental and/or coronal scattering seen in EUV images. Also we subtract an unmodeled background given by the median flux value at the four borders of each image. The physical loop model requires seven physical parameters for each loop: the loop top temperature Tm , the base pressures p1 , p2 (or base densities n1 , n2 ), the footpoint heights h1 , h2 , and pressure scale heights λ1 , λ2 . The two density parameters are constrained by (1) the observed flux in each field line (because
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Figure 6. Magnetic polarity map of stereoscopically triangulated field lines for AR 10955 on 2007 May 9. The magnetic configuration can be characterized by a singular dipolar domain, requiring two opposite magnetic poles (circles), where the leading spot (west) is arbitrarily assigned to be positive. The magnetic polarities of the stereoscopically triangulated field lines are defined by the sign of the pole that is closest to the footpoint of the field line. The height dependence of the stereoscopically triangulated field lines is indicated with variable thickness in the linestyle, with the broadest width near the footpoint.
we require a positive flux residual after subtraction of a model loop component) and (2) by the looptop matching condition. Using these two constraints we are left with five free parameters for each model loop component. Our best-fit procedure consists in an iterative subtraction of model loop fluxes Ff (s) in each of the six observed images (i.e., the dual stereoscopic image pairs of each of the three different temperature filters). In the observed images we have a set of nkl = nk × nl loop field lines (typically nk ≈ nl ≈ 80), which obs yield nkl ≈ 6400 flux profiles Ff,kl (x, y), that are sorted by their loop length. We fit then the model flux profiles Ff (s) to obs each of the six observed flux profiles Ff,kl (x, y) (in the six filter images, f = 1, . . . , 6), subtracting them iteratively in order of decreasing the total loop flux. The model flux profile fitting loops is performed by optimizing the five free parameters with the Powell algorithm (Press et al. 1986), in such a way that the subtracted loop fluxes remove a maximum fraction of the observed EUV flux in the six images, but not exceeding the observed flux (to warrant positive flux residuals). Some
additional constraints of the iterative subtraction algorithm are minimum EM per subtracted loop (EMf,kl 1042 cm−3 , corresponding to a base density range of ne 107.8 cm−3 , and the number of loop component fittings at the same field line position (nmulti = 3). Thus, we perform a number of nkl × nmulti ≈ 6400 × 3 = 19,200 iterative fits to the six observed images, resulting into typically Nloop ≈ 7000 fitted model loop components. An example of a fitted model Ifmodel (x, y) is shown for the viewing angle of spacecraft STEREO/A in Figure 8 (three filter images on the right-hand side), and is compared with the three corresponding observed images Ifobs (x, y) in Figure 8 (three filter images on the left-hand side). The model images show about the same level of details as the observed images. We would like to stress that the model images are not unique, but they should be considered as a representative statistical 3D distribution of loop densities and temperatures that are consistent with the stereoscopically measured 3D coordinates and the stereoscopic three filter image pairs. The observational
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constraints are given by the nsc × nf × nx × ny = 2 × 3 × 160 × 160 = 153,600 pixel fluxes (for the 160×160 pixel subimages recorded in nf = 3 wavelengths with nsc = 2 spacecraft; only those from spacecraft STEREO/A are shown in Figure 8 left), while the model images are produced here by npar × nloop ≈ 5 × 7000 = 35,000 free parameters. Although the model images are overconstrained, the model solution depends somewhat on the algorithm of the tomographic decomposition code. The iterative subtraction of loop fluxes is subject to uncertainties of the field line 3D coordinates, to loop-unrelated EUV fluxes (e.g., from moss structures), and to confusion problems resulting from the numerous loop superpositions. Nevertheless, the obtained model reproduces closely matching images and yields statistically representative samples of physical loop parameters than can be used to study the 1D hydrodynamic structure, the spatio-temporal heating functions, and cooling rates. In order to quantify the goodness
of fit of the model we show scatter plots of the modeled versus the observed pixel fluxes (of the six images) in Figure 9. The retrieved fluxes are ≈ 80% of the observed fluxes in the model shown in Figure 8, with a number of nloop ≈ 7000 model loop components. The missing flux of ≈ 20% comes mostly from faint image regions where no 3D field line is available (with our chosen field line resolution of ≈ 2 pixels) or from structures where no physical loop model could be fitted. Nevertheless, the tight correlation between the modeled and observed fluxes per pixel (≈ 0.1 dex in Figure 9) is about an order of magnitude better than in past modeling efforts (e.g., see Figure 6 in Schrijver et al. 2004; or Figure 9 in Lundquist et al. 2008a). 3. RESULTS AND DISCUSSION We modeled active region NOAA 10955, observed on 2007 May 9, the same case for which we conducted the first stereo-
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Figure 8. STEREO/A subimages of 2007 May 9, 20:30–20:50 UT recorded in three wavelength filters (171, 195, 284 Å) are shown on the left-hand side, and the model images based on 3D density and temperature distributions computed with the stereoscopic multiloop tomography method are shown on the right-hand side on the same color scale. The model images are reconstructed from 6991 model loop components and retrieve ≈ 81% of the total observed EUV flux. (A color version of this figure is available in the online journal.)
scopic 3D reconstruction of loops (Papers I and II). The two STEREO spacecraft A(head) and B(ehind) had a separation angle of ≈ 7◦ at this time. In this study we modeled the 3D density and temperature distribution using the new method of ISTAR. Different viewing projections of the 3D model of the active region for all three filters are shown in Figure 10. A visualization of different viewing angles of the 3D model active region seen through the three different EUVI filters are also shown in the Movies 1, 2, and 3 (viewable with QuickTime movie player),
which are included in the on-line electronic supplementary material of this paper. In the following we discuss the new aspects of this method and quantitative results of the obtained physical parameters. 3.1. Merits and Caveats of Coronal Tomography Methods The main advantage of our instant stereoscopic tomography method compared with standard tomography methods is the
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substantially higher spatial resolution and the much shorter time interval required for tomographic data acquisition. Earlier tomographic 3D reconstructions of the corona have been conducted with a spatial pixel size of ≈ 5 (Yohkoh pixel size; Hurlburt et al. 1994) or ≈ 10◦ heliographic degrees (using LASCO data; Frazin & Janzen 2002; Frazin et al. 2007), while we used a pixel size of ≈ 1. 6 (1 EUVI pixel) in our 3D reconstruction here, with an effective resolution of ≈ 3. 2 (two EUVI pixels). In principle, our method can be applied down to arbitrary small pixel sizes, e.g., to ≈ 0. 5, if one combines the stereoscopic STEREO/EUVI images with simultaneous TRACE data. The key advantage of our new ISTAR method is the instantaneity of the 3D reconstruction method. STEREO/A and B are operated in a simultaneous time schedule mode, corrected for the light travel time difference between the Sun and the arrival at the two spacecraft. Besides the light travel time correction, the 195 Å images were recorded 35 s after the 171 Å images, and the 284 Å images 70 s later (see Table 1 in Paper I). Therefore, our time resolution of ≈70 s is more than four orders of magnitude better in our instant stereoscopic tomography method than in previous standard tomography methods, which require a halfsolar rotation (≈ 14 days) to accumulate sufficient data for a 3D reconstruction, and thus are only applicable to long-lasting features that are quasistationary during a half-solar rotation. Another method of instant tomography using three spacecraft with a separation angle of 90◦ has recently been presented for a small number of coronal loops (Kankelborg 2008), but its performance has not been demonstrated for a realistic active region with a large number of loops. What are the caveats of our new tomography method? Our stereoscopic tomography method requires two stereoscopic spacecraft that operate with a suitable spacecraft angle. The
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limits of this spacecraft angle suitable for stereoscopy have not been explored yet. Conservative estimates restrict stereoscopy to small angles because of confusion limits, say 45◦ , which would yield only 1 year worth of STEREO data (2007 April– December), but stereoscopic information can be exploited in principle up to 180◦ , which would extend our database to ≈ 4 years (2007–2010). Another specialty of our tomography code is the topology of multiple loops, requiring 1D structures with some parameterization of the density and temperature profile. However, besides active region loops, any structure associated with a closed or open (magnetic) field line can be reconstructed, such as quiet-Sun loops, plumes in coronal holes, flare loops, and arcades, or even quiescent and eruptive filaments, as long as they can be stereoscopically triangulated as curvi-linear 1D features. A 3D model can be reconstructed on arbitrary short timescales (only limited by the exposure time and the time difference between multiple wavelength filters). Our new method cannot easily be applied to optically thick or diffuse structures, such as coronal mass ejections (CMEs), though a modified code is conceivable. Our stereoscopic tomography code works the more accurately, the more structures can be triangulated stereoscopically, but bears larger uncertainties in voids of sparsely filled volumes. Nevertheless, the saving grace of density voids is that the high-density regions of the 3Dreconstructed volumes are correlated with regions of feasible stereoscopic triangulation. 3.2. Statistics of Model Parameters Our best-fit tomographic model provides statistics on physical parameters of some nloop ≈ 7000 individual loop components. The statistical distributions of our tomographic model parameters are shown in form of histograms in Figure 11. It is important to understand that our tomographic method is based on a segmentation of loop structures into elemental loop components that have a fixed width (chosen to be wloop = 1500 km here), which represents an arbitrary substructuring of real loops. This elemental loop structures are building blocks of our 3D volume rendering in our model, optimized in such a way that the density or temperature averaged over any subvolume larger than the instrumental resolution should be consistent with the data in each wavelength filter and spacecraft. The loop half-length L corresponds to the length from the loop footpoint to the lop top of the field lines that have been stereoscopically reconstructed in a volume box with a height range of hmax /R 0.1. The 3D reconstruction box restricts the detected loop half-length of closed loops and open field lines. The vertical height of the 3D box is hmax = 0.1 R or 70 Mm. This restricts the detected half-length L of a circular loop with an inclination angle ϑ of the loop plane to L
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which is L 109 Mm for a vertical loop plane, or L 154 pixels for a loop with an inclination angle of ϑ = 45◦ . The distribution N (L) of detected loop half-lengths L is found to show an exponential distribution (Figure 11, top right),
L N (L) ∝ exp − (17) 57 Mm with a clear cutoff in the range of 110 L 150 Mm, as expected from the truncation bias of the 3D box. Interestingly,
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Figure 10. Three different viewing projections of the 3D model: from vertical (left panels), from east (middle panels), and from south (right panels) of the active region, for all three filters: 171 (top panels), 195 (middle panels), and 184 Å panels. See also movie for different viewing angles of the model. (A color version and an mpeg animation of this figure are available in the online journal.)
the e-folding constant of the exponential distribution is similar to the size scale of the dipole in this active region (see Figure 1). For the loop top temperatures Tm we find a distribution with a peak around T ≈ 2.0 MK, falling off exponentially toward higher temperatures in the range of 2.0 T 7 MK (Figure 11, top left),
Tm N (Tm ) ∝ exp − . (18) 0.6 MK Although the primary sensitivity range of the three EUVI filters is limited to temperatures of T ≈ 0.7–2.7 MK (Figure 2 in Paper II), our model also requires loops with apex temperatures up to Tm 7.0 MK, because the lower parts of the loops (above the footpoints) have temperatures in the range of T ≈ 1–2 MK where our tomographic code fits the observed data. Thus, hotter loops with soft X-ray temperatures can clearly be predicted and
constrained from tomographic EUV data alone. In fact, 52% of the 7000 fitted loop components in our model have a loop top temperature of Tm 2.0 MK, and 13% have Tm 3.0 MK, respectively. For the distribution of (base) electron densities n1 , n2 we find a distribution that is monotonically decreasing at higher densities, approximately like a power-law function (Figure 11, second row), N (n1 ) ≈ N (n2 ) ∝ n−1.4 . (19) 1 The fact that the footpoint densities n1 and n2 at both footpoints show near-identical distributions indicates that asymmetries of loop densities cancel out in large statistical samples. The cutoff at densities of n1 108.7 cm−3 for the first footpoint (see Figure 11, second row left) is caused by the lower limit for the emission measure EM 1042 cm−3 , which was applied to
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Figure 11. Statistical distributions of loop top temperatures Tm (top left), loop half-lengths L (top right), loop base densities n1 , n2 (second row), pressure scale heights λ1 , λ2 (third right), and loop footpoint heights h1 , h2 (bottom row) of the nloop = 6991 loop components of our tomographic 3D model. The primary temperature sensitivity range of the three coronal EUVI filters (T ≈ 0.7–2.7 MK) is indicated with a gray box (top left).
limit the number of faint loop components in our tomographic reconstruction. For the fitted scale heights λ1 and λ2 we find the host of values in the range of 200 Mm (Figure 11, third row), as expected for near-hydrostatic loops with apex temperatures of Tm 4.0 MK. Finally, for the footpoint heights we imposed a lower limit of h1 , h2 hchr = 2.0 Mm. The footpoint heights h1 , h2 are free parameters in our tomographic fitting, for which we found a strong clustering near the lower limit, approximately falling off with an e-folding constant of ≈ 350 km (Figure 11, bottom row),
h1 . (20) N(h1 ) ∝ N (h2 ) ≈ exp − 0.35 Mm A small fraction of fitted loop components (≈ 30%) has footpoint heights of larger than h1 5 Mm, which mostly results from confusion problems of closely spaced and overlapping loops in the tomographic fitting procedure, which is a numerical artifact rather than a physical reality, but has no effect in
our 3D model on spatial scales larger than the instrumental resolution. 3.3. Tomographic Differential Emission Measure Distribution Because our tomographic 3D reconstruction yields the density ne (s kl ) and temperature Te (s kl ) as function of the loop length coordinates skl , for a set of nk ×nl loops (with k = 1, . . . , nk ; l = 1, . . . , nl ), we can easily sample the distribution of dn2e (Te )/dT by summing the squared densities n2e (s) in given temperature bins dT along each loop to obtain a complete differential EM (DEM) distribution of our tomographic 3D model. Since temperatures are modeled over the entire loop length with a parameterized temperature model (Equation (11)), which has lower limits of Tchr 0.01 MK and yielded values up to Tm 7.0 MK, we can reconstruct the DEM over this entire temperature range, which is much wider than the temperature sensitivity range of Te ≈ 0.7–2.7 MK of the three used EUVI filters. Thus, our tomographic reconstruction also constrains a substantial temperature range below and above the EUV temperature range.
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Figure 12. DEM distribution dEM(T)/dT computed from our tomographic model, and compared with two active regions and two quiet Sun regions from Brosius et al. (1996). Note that the primary temperature sensitivity range of EUVI is log(T ) ≈ 5.8–6.3 (gray range), but the DEM could be constrained in the range of log(T ) ≈ 4–7, based on the parameterized temperature profiles used in our stereoscopic tomography code. The area of the active region is A = 1.7 × 1020 cm2 (about half of the (160 × 160 pixel image size), and a canopy correction was applied in the temperature range of log(T ) = 5.7–6.0, with a quadratic area expansion from 10% to 100% of the coronal flux tube area (histogram with thick line style; the uncorrected DEM is also shown as histogram with gray color).
The obtained DEM distribution is shown in Figure 12, which shows a maximum at log(T ) ≈ 4.8 and a broad secondary peak at a coronal temperature of log(T ) ≈ 6.0–6.6 (T ≈ 1.0– 4.0 MK). Our DEM of AR 10955 is comparable in magnitude and functional shape with DEMs calculated for other active regions (e.g., Brosius et al. 1996; shown in Figure 12 here), which have been derived from data over a much larger wavelength and temperature range. This quantitative agreement corroborates that the temperature model used in our tomographic reconstruction is reasonably representative for coronal loops. It demonstrates moreover that the stereoscopic tomography code (ISTAR) can retrieve the DEM distribution over a much larger temperature range than that of the used EUV data. In fact, the temperature resolution of the obtained DEM can be made arbitrarily fine, independent of the filter width and separation of the used EUV filters, and thus there are no numerical inversion problems of the DEM. Besides the DEM of an active region, our stereoscopic tomography method allows us also to investigate the hydrostatic equilibrium of coronal loops (Figure 13 and Section 3.4), the density map of an active region (Figure 14), and the temperature map of an active region (Figure 15). In Figure 14, we show a density map ne (x, y) with a spatial resolution of 0. 8, where the color scale represents the accumulated density in each pixel. Clearly the density distribution is extremely inhomogeneous and shows a high filling factor, at a density threshold of ne 107.6 . The density map includes all loops in the range of T = 0.01– 7.0 MK. In Figure 15, we visualize a temperature map of AR 10955, where the color in each pixel marks the highest temperature that is found along the line of sight associated with each pixel. We can clearly see the cooler loop footpoint segments above the leading and following sunspot, which may contain many open diverging (fanning-out) field lines, while the loop apexes all show higher temperatures in the range of T ≈ 2–4 MK. Our stereoscopic tomography method displays
all temperatures in the range of T = 0.01–7.0 MK, regardless of the used EUV filter sensitivity range of T ≈ 0.7–2.7 MK. Soft X-ray images can easily be predicted based on our stereoscopic tomography method and compared with actual observed ones, e.g., from the X-Ray Telescope (XRT) of Hinode (a future study is planned). 3.4. Hydrostaticity of Coronal Loops Our multiloop model contains some nloop ≈ 7000 coronal loops, for which both the pressure scale height λ and the loop top temperature Tmax were independently fitted. We can thus calculate the hydrostatic temperature scale height λT (T = Tmax ), which is defined as λT (Te ) =
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with kB being the Boltzmann constant, μ ≈ 1.27 (for a hydrogen to helium abundance of H:He = 10:1 in the solar corona) the molecular (mean atomic) weight, mH the hydrogen mass, and g the solar gravitation. In Figure 13, we show the average pressure scale height values λ(T ) for temperature bins with widths of dT = 0.5 MK from T = 0.5 MK to T = 6.0 MK, and compare them with the expected hydrostatic temperature scale heights λT (Equation (21)), which is drawn as a thick solid curve in Figure 13. Interestingly we find that fitted pressure scale heights λ(T ) exceed the expected hydrostatic temperature scale heights λT (T ) for EUV temperatures of T 3.0 MK, while they match (within 25%) for the hotter soft X-ray temperatures (T ≈ 3–6 MK). Therefore, we conclude that EUV loops are super-hydrostatic, while soft X-ray loops appear to be near-hydrostatic. This result is consistent with earlier findings (e.g., Figure 7 in Aschwanden et al. 2000), where super-hydrostatic pressure scale
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heights with ratios of λ/λT ≈ 1–4 were found for 29 loops analyzed from TRACE data with 1 spatial resolution. The effect of super-hydrostaticity is more pronounced for longer loops than for shorter loops. The deviation from hydrostatic equilibrium is generally interpreted in terms of dynamic processes, which could include (1) accelerated upflows, (2) impulsive heating of multiple loop strands with subsequent cooling, or (3) wave pressures that exceed the hydrodynamic pressure (see discussion in Section 4.6.2 of Aschwanden 2004). Numerical hydrodynamic simulations of impulsively heated multistrand loops indeed are able to reproduce such super-hydrostatic loop densities (Warren et al. 2002; Mendoza-Briceno et al. 2002). Thus, our stereoscopic tomography method confirms that most coronal loops are not in hydrostatic equilibrium. The nonequilibrium in coronal loops is likely to be associated with impulsive heating. In future studies we plan to derive the spatial heating rates with our stereoscopic tomography method, which possibly could produce a 2D map of the heating rate, EH (x, y), which can then be compared with photospheric magnetic field maps, B(x, y), in order to test scaling relationships of coronal heating models, such as EH (x, y) ∝ B(x, y)/L(x, y) (Schrijver et al. 2004), where our stereoscopic measurements already provide accurate measurements of loop lengths L(x,y) at a given spatial location (x, y).
Figure 14. Density map of AR 10955 reconstructed from our tomography code with a resolution of a half EUVI pixel size (0. 8), representing individual loop components with a finite width of w = 1500 km (2 ). The density distribution is independent of the temperature, the color scale is linear to the total density along the line of sight of each pixel. (A color version of this figure is available in the online journal.) 7.0
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Figure 15. Maximum temperature map of AR 10955 reconstructed from our tomography code with a resolution of a half EUVI pixel size (0. 8), representing individual loop components with a finite width of w = 1500 km (2 ). The color in each pixel marks the highest temperature that is encountered along the line of sight of each pixel. A logarithmic temperature scale is indicated on the right side.
Standard tomography, such as computer-aided tomography (CAT) scanners require many aspect angles to obtain a 3D reconstruction of an object. For solar tomography, many aspect angles can be obtained by using the solar rotation, but this requires a sampling time (about two weeks for full angle coverage) that is way too long for the dynamically evolving processes that are observed in the solar corona. With the new STEREO mission, however, at least two simultaneous aspect
angles are available, which offers full 3D information for curvilinear features, such as coronal loops. We find that we can triangulate about 102 coronal loops per active region with STEREO, after optimizing our loop detection algorithm with multiscale filtering, image stacking, and multifilter tracing. The 3D coordinates of some 102 coronal loops provides the essential skeleton structure of an active region. We develop a
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new method to infer the full 3D vector field of loop-aligned magnetic field lines in an active region by 3D interpolation of the directional vectors of the skeleton of stereoscopically triangulated loops. This quantification of the 3D vector field is fully independent of any magnetic field measurement and can be used to test theoretical magnetic field models. However, our 3D field line reconstruction method yields only the directionality v = B/|B| of the magnetic field (supposed that the coronal loops observed in EUV are aligned with the local magnetic field, v B), but it does not provide the magnetic field strength B itself. Nevertheless, the main consequence of our 3D field reconstruction method is a quantification of a coronal 3D volume into 1D field lines, which reduces the standard 3D tomography approach into an array of 1D substructures. In our application we find that an active region can be modeled as space-filling volume by some 104 1D structures. Using the 3D vector field we developed a novel method that we call Instant Stereoscopic Tomography of Active Regions (ISTAR), which combines the geometric 3D information of stereoscopic triangulation with density and temperature modeling of coronal loops with a filling factor equivalent to tomographic 3D rendering. The underlying density and temperature modeling of 1D loop components uses a parameterization with five free parameters per loop structure, which are iteratively fitted to a stereoscopic image pair with three temperature filters each. This parameterization yields loop apex temperatures, footpoint densities, footpoint heights, and loop pressure scale heights. No assumption on any physical model is made, such as hydrostatic equilibrium or a heating scaling law. The main results of this study are 1. The 3D density and temperature model obtained with the ISTAR method reproduces the images of an active region (AR 10955 here) observed in three different EUV temperature filters from two spacecraft with a stereoscopic separation of ≈ 7◦ with an unprecedented accuracy of |Fmodel (x, y) − Fobs (x, y)|/max(Fobs ) ≈ 0.02 ± 0.05 in every image pixel, at the instrumental resolution of ≈ 3. 2. 2. The stereoscopic tomography method using STEREO data is about four orders of magnitude faster than conventional solar-rotation-based tomography, and thus allows 3D reconstruction of dynamic phenomena within ≈ 1 minute. 3. The stereoscopic tomography method provides an accurate determination of the DEM distribution in any location of a coronal active region over the entire temperature range of T ≈ 104 –107 K with arbitrarily high temperature resolution, using EUV filters with a limited temperature sensitivity range of T ≈ 0.7–2.7 MK. Our method of DEM determination does not suffer from ill-posedness or mathematical inversion uncertainties from a small number of temperature filters. Slight corrections of the DEM at temperatures below T 1.0 MK may apply from unknown loop width variations in the canopy below the transition region. 4. The statistical pressure scale heights λ(T ) of coronal loops are found to exceed the hydrostatic temperature scale heights λT (T ) for EUV temperatures of T 3.0 MK, while they approximately match for the hotter soft X-ray temperatures (T ≈ 3–6 MK). The super-hydrostaticity of EUV loops is interpreted as a consequence of impulsive heating, which applies to all loops in an active region. 5. The ISTAR method provides a 3D vector field of coronal field lines v = B/|B| that can be used to test theoretical
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magnetic field models, such as potential fields, or linear and nonlinear force-free field extrapolation models. 6. The ISTAR method can provide 2D maps of the heating rate EH (x, y) and loop lengths L(x, y), which can be used to test coronal heating models that depend on the magnetic field B(x, y) and loop lengths L(x,y). In future studies we will explore the stereoscopic tomography method to quantify the time evolution of coronal heating and cooling processes (Paper IV), to test theoretical magnetic field models (Paper V), to test the consistency of high-temperature images predicted from EUV data with observed soft X-ray images (e.g., from Hinode), and to test various coronal heating models by quantitative means. APPENDIX VALIDATION OF THE ISTAR METHOD The ISTAR method provides a 3D model of an active region that fits the observed EUV emission in images of different wavelengths and temperatures as close a possible, with spatial substructures that are consistent with the stereoscopically triangulated loops, but has some freedom in the number of unresolved loop strands and cross-sectional widths of the loop strands. Nevertheless, a comparison of the electron densities of background-subtracted loops obtained from the images of the two STEREO spacecraft with the densities of co-spatial ISTAR model components may be instructive to obtain a feeling for the accuracy in the determination of absolute values of 3D electron densities in general. We show the electron densities ne (s) measured along stereoscopically triangulated 30 loops in Figure 16 (identical with those shown in Figure 10 of Paper II), which were obtained by co-spatial cross-sectional background modeling with the triplefilter (171, 195, 284 Å) method described in Paper II. The density measurements from both spacecraft STEREO/A and B (cross and diamond symbols in Figure 16) are shown, along with a hydrostatic model fit (thick black curve in Figure 16). For the absolute values of electron densities determined with the triplefilter method one has to bear in mind that the observed loop width w(s) was corrected for the finite point-spread function 2 , which leads with width wres = 1. 59 by wc (s) = w(s)2 − wres √ to a corrected electron density of ne,c (s) = ne (s) w(s)/wc (s). Also we have to be aware that the background-subtracted loop fluxes typically contain about 10% of the total EUV flux at the same location, while the background subtraction has an accuracy of similar magnitude (≈ 10%), and thus the absolute value of the obtained electron densities may be uncertain by a factor of ≈ 2, besides other uncertainties resulting from unknown atomic abundances, ionization equilibrium, etc. Moreover, the densities evaluated from the triple-filter method assume a filling factor of unity, and thus represent only a lower limit in the case of unresolved substrands with a filling factor less than unity. Now we compare with cospatial electron densities obtained from the ISTAR model. Although the ISTAR model fits the same data (six EUV images taken with three wavelength filters from two STEREO spacecraft), and uses even the same 3D geometry of stereoscopically triangulated loops, the loop substructure is different. In our ISTAR model presented in this paper we used some 6400 field lines and populated each field line with up to three loop strands with a fixed diameter of w = 1500 km, which is the average diameter of the finest loop threads measured with TRACE, corresponding to the average diameter wEUVI =
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Figure 16. Comparison of electron densities obtained from triple-filter modeling (Paper II) with those of cospatial loop strands modeled with the ISTAR method. The profiles show the electron density ne (s) in units of 109 cm−3 (y-axis) vs. the normalized loop length s/L (x-axis). The densities inferred from the triple-filter method are shown for both STEREO/A and B spacecraft (cross nad diamond symbols), along with a hydrostatic model fit (thick curve), identical to Figure 10 in Paper II. The densities of three cospatial loop strand components of the ISTAR model are shown with thin curves, with different linestyles indicating the temperature difference to the triple-filter method (solid if dT 0.5 MK, dashed if 0.5 MK dT 1.0 MK, and dotted if dT 1.0 MK). The median density ratios are indicated for those loop strands with similar temperatures (dT 0.5 MK).
No. 1, 2009
STEREOSCOPY OF CORONAL LOOPS. III.
2700 km observed with EUVI, after correction for the pointspread function (Figure 11 in Paper II). Since we populated up to three loop strands in every field line, which have different temperatures, there is some ambiguity in comparing the densities of those ISTAR loop strands with those from the triple-filter background subtraction method. Thus we show the density profiles of all three cospatial loop strands in Figure 16 (thin curves), where we mark their difference in loop top temperature (with respect to the triple-filter temperature) with different linestyles (solid linestyle for dT 0.5 MK, dashed linestyle for 0.5 MK dT 1.0 MK, and dotted linestyle for dT 1.0 MK). In order to quantify the agreement in electron densities between the two methods we calculate the median densities nstr of those cospatial ISTAR loop strands with similar temperatures (dT 0.5 MK) and compare them with the median densities nobs of the triple-filter method (indicated for each of the 30 loops in Figure 16). This density ratio could be determined for 50 (out of the 60) loop strands (with matching temperatures dT 0.5 MK), for which we find a mean and standard deviation of nstr = 1.4 ± 0.7. nobs
(A1)
So, this means that we obtain a agreement within a factor of 1.4 in the densities derived by both methods in the statistical average, or about a factor of 2 for individual loops. This agreement in the absolute electron density is quite remarkable, given the uncertainties of unknown widths of individual loop strands, since a factor of 1.4 change in loop diameter causes a factor 2 change in electron densities. As a disclaimer, the ISTAR method is designed to model the 3D density distribution of an active region on spatial scales down to the instrumental resolution (which is 2500 km or 3. 4 for EUVI), but cannot exactly determine the absolute densities of unresolved smaller loop widths. Our corrections for unresolved loop strands are based on measurements with the highest spatial resolution available to date (i.e., 900 km or 1. 2 from TRACE). Nonetheless, the best validation of the ISTAR model is the temperature-convolved EM maps, which show a remarkable detailed agreement with the data (Figures 8 and 9). This work is supported by the NASA STEREO under NRL contract N00173-02-C-2035. The STEREO/SECCHI data used here are produced by an international consortium of the Naval Research Laboratory (USA), Lockheed Martin Solar and Astrophysics Lab (USA), NASA Goddard Space Flight Center (USA), Rutherford Appleton Laboratory (UK), University of Birmingham (UK), Max-Planck-Institut f¨ur Sonnensystemforschung (Germany), Centre Spatiale de Li`ege (Belgium), Institut d’Optique Th´eorique et Applique (France), Institute d’Astrophysique Spatiale (France). The USA institutions were funded by NASA; the UK institutions by the Science & Technology Facility Council (which used to be the Particle Physics and Astronomy Research Council, PPARC); the German institutions by Deutsches Zentrum f¨ur Luft- und Raumfahrt e.V. (DLR); the Belgian institutions by Belgian Science Policy Of-
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fice; the French institutions by Centre National d’Etudes Spatiales (CNES), and the Centre National de la Recherche Scientifique (CNRS). The NRL effort was also supported by the USAF Space Test Program and the Office of Naval Research. REFERENCES Aschwanden, M. J. 2004, Physics of the Solar Corona: An Introduction (Chichester, UK: PRAXIS Publishing Ltd./Springer: New York, Berlin) Aschwanden, M. J., Newmark, J. S., Delaboudiniere, J. P., Neupert, W. M., Klimchuk, J. A., Gary, G. A., Portier-Fornazzi, F., & Zucker, A. 1999, ApJ, 515, 842 Aschwanden, M. J., & Nightingale, R. W. 2005, ApJ, 633, 499 Aschwanden, M. J., Nightingale, R. W., & Alexander, D. 2000, ApJ, 541, 1059 Aschwanden, M. J., Nitta, N. V., Wuelser, J. P., & Lemen, J. R. 2008b, ApJ, 680, 1477 (Paper II) Aschwanden, M. J., Schrijver, C. J., & Alexander, D. 2001, ApJ, 550, 1036 Aschwanden, M. J., & Tsiklauri, D. 2008, ApJ, submitted Aschwanden, M. J., Wuelser, J. P., Nitta, N. V., & Lemen, J. R. 2008a, ApJ, 679, 827 (Paper I) Brosius, J. W., Davila, J. M., Thomas, R. J., & Monsignori-Fossi, B. C. 1996, ApJS, 106, 143 Davila, J. M. 1994, ApJ, 423, 871 Feng, L., Inhester, B., Solanki, S., Wiegelmann, T., Podlipnik, B., Howard, R. S., & Wuelser, J. P. 2007, ApJ, 671, L205 Frazin, R. A., Butala, M. D., Kemball, A., & Kamalabadi, F. 2005a, ApJ, 635, L197 Frazin, R. A., & Janzen, P. 2002, ApJ, 570, 408 Frazin, R. A., & Kamalabadi, F. 2005, ApJ, 628, 1061 Frazin, R. A., Kamalabadi, F., & Weber, M. A. 2005b, ApJ, 628, 1070 Frazin, R. A., Vasquez, A. M., Kamalabadi, F., & Park, H. 2007, ApJ, 671, L201 Gary, A., Davis, J. M., & Moore, R. 1998, Solar Phys., 183, 45 Hurlburt, N. E., Martens, P. C. H., Slater, G. L., & Jaffey, S. M. 1994, in ASP Conf. Ser. 68, Proc. Solar Active Region Evolution: Comparing Models with Observations, ed. K. S. Balasubramaniam & G. W. Simon (San Francisco, CA: ASP), 30 Jiao, L., McClymont, A. N., & Mikic, Z. 1997, Solar Phys., 174, 311 Kaiser, M. L., Kucera, T. A., Davila, J. M., St. Cyr, O. C., Guhathakurta, M., & Christian, E. 2007, Space Sci. Rev., 198, 1 Kankelborg, C. C. 2008, ApJ, in press Lenz, D. D., DeLuca, E. E., Golub, L., Rosner, R., & Bookbinder, J. A. 1999, ApJ, 517, L155 Lundquist, L. L., Fisher, G. H., & McTiernan, J. M. 2008a, ApJS, 179, 509 Lundquist, L. L., Fisher, G. H., & McTiernan, J. M. 2008b, ApJ, 689, 1388 Mendoza-Briceno, C. A., Erdelyi, R., & Sigalotti, L. D. G. 2002, ApJ, 579, L49 Patsourakos, S., Pariat, E., Vourlidas, A., Antiochos, S. K., & Wuelser, J. P. 2008, ApJ, 680, L73 Press, W. H., Flannery, B. P., Teukolsky, S. A., & Vetterling, W. T. 1986, Numerical Recipes: The Art of Scientific Computing (Cambridge: Cambridge Univ. Press) Sakurai, T., Shibata, K., Ichimoto, K., Tsuneta, S., & Acton, L. W. 1992, PASJ, 44, L123 Schmieder, B., Demoulin, P., Aulanier, G., & Golub, L. 1996, ApJ, 467, 881 Schrijver, C. J., Sandman, A. W., Aschwanden, M. J., & DeRosa, M. L. 2004, ApJ, 615, 512 Warren, H. P., & Winebarger, A. R. 2006, ApJ, 645, 711 Warren, H. P., & Winebarger, A. R. 2007, ApJ, 666, 1245 Warren, H. P., Winebarger, A. R., & Hamilton, P. S. 2002, ApJ, 579, L41 Wiegelmann, T., & Inhester, B. 2006, Solar Phys., 236, 25 Wuelser, et al. 2004, SPIE Proc., 5171, 111 Zidowitz, S. 1997, in Proc. 5th SoHO Workshop, ESA, Vol. 404, The Corona and Solar Wind Near Minimum Activity, ed. A. Wilson (Noordwijk: ESTEC), 757 Zidowitz, S. 1999, AGR, 104, 9727
