Issue 
A&A
Volume 620, December 2018



Article Number  A104  
Number of page(s)  7  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201833441  
Published online  05 December 2018 
Magnetic properties of a longlived sunspot
Vertical magnetic field at the umbral boundary^{⋆}
Kiepenheuer Institut für Sonnenphysik (KIS), Schöneckstr. 6, 79104
Freiburg i.Br., Germany
email: schmassmann@leibnizkis.deschliche@leibnizkis.denbello@leibnizkis.de
Received:
16
May
2018
Accepted:
5
October
2018
Context. In a recent statistical study of sunspots in 79 active regions, the vertical magnetic field component B_{ver} averaged along the umbral boundary is found to be independent of sunspot size. The authors of that study conclude that the absolute value of B_{ver} at the umbral boundary is the same for all spots.
Aims. We investigate the temporal evolution of B_{ver} averaged along the umbral boundary of one longlived sunspot during its stable phase.
Methods. We analysed data from the HMI instrument onboard SDO. Contours of continuum intensity at I_{c} = 0.5I_{qs}, whereby I_{qs} refers to the average over the quiet sun areas, are used to extract the magnetic field along the umbral boundary. Projection effects due to different formation heights of the Fe I 617.3 nm line and continuum are taken into account. To avoid limb artefacts, the spot is only analysed for heliocentric angles smaller than 60°.
Results. During the first disc passage, NOAA AR 11591, B_{ver} remains constant at 1693 G with a rootmeansquare deviation of 15 G, whereas the magnetic field strength varies substantially (mean 2171 G, rms of 48 G) and shows a long term variation. Compensating for formation height has little influence on the mean value along each contour, but reduces the variations along the contour when away from disc centre, yielding a better match between the contours of B_{ver} = 1693 G and I_{c} = 0.5I_{qs}.
Conclusions. During the disc passage of a stable sunspot, its umbral boundary can equivalently be defined by using the continuum intensity I_{c} or the vertical magnetic field component B_{ver}. Contours of fixed magnetic field strength fail to outline the umbral boundary.
Key words: sunspots / Sun: magnetic fields / Sun: photosphere / Sun: activity
Movies associated with Figs. 3 and 5 are available at https://www.aanda.org
© ESO 2018
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The boundary between umbra and penumbra of sunspots has long been defined in terms of the continuum intensity I_{c}. This brightness difference is the consequence of the different magnetoconvective processes running in umbrae and penumbrae. We have evaluated magnetic quantities to identify which of them may cause the different behaviour on the two sides of the umbral boundary.
Jurčák (2011) investigated the properties of the magnetic field at umbral boundaries and noted that the vertical magnetic field component B_{ver} changes little along the boundaries of the ten sunspots he analysed and could neither verify nor falsify a dependence of the median value along the boundary on the area of the sunspot. The tenspot average of the median along the boundary was 1860 G, whereas the mean of the standard deviations along the boundary was given as 190 G for Hinode/SP data.
Jurčák et al. (2015) extended the analysis by investigating a 4.5h time series of a forming sunspot using GFPI/VTT data and noting an increase of B_{ver} at the migrating umbral boundary during penumbra formation and stabilization of this value after completion of the formation. Shortly thereafter, that part of the umbral boundary was observed with Hinode/SP and a B_{ver} value of 1810 G measured. They propose that the umbral mode of magnetoconvection prevails in areas with , whereas outside, the penumbral mode takes over.
Following this line of investigation, Jurčák et al. (2017) studied a pore whose B_{ver} remained below this critical value. They found that a developing penumbra completely cannibalized the pore, thus supporting the assertion that in umbral areas with , the penumbral mode of magnetoconvection takes over the umbral mode.
Jurčák et al. (2018) extends the analysis of Jurčák (2011) to 88 scans of 79 different active regions again using Hinode/SP and showed that the I_{c} = 0.5I_{qs} contours match mostly the B_{ver}=1867 G contours. A Bayesian linear regression showed that a model with constant B_{ver} is more likely to explain the data than a first or second order polynomial with logarea as independent variable. Furthermore the most likely B_{ver}=1867 G, with a 99% probability for 1849 G ≤ B_{ver}≤1885 G. A dependence on the solar cycle could not be verified.
These findings have led to the Jurčák criterion, an empirical law stating that the umbral boundary of stable sunspots can be equivalently defined by a continuum intensity I_{c} or a vertical magnetic field component B_{ver}. In other words, in areas with , only the umbral mode of convection exists, hindering other modes of magnetoconvection. A conjecture can also be stated from these findings: umbral areas with are unstable against more vigorous modes of convection, that is, they are prone to vanish.
In this work we have investigated the behaviour of the magnetic field along the umbral boundary in a time series of a single stable sunspot. We used the spot of NOAA AR 11591 during its first disc passage. This allows us to verify whether remains constant over ≈10 days, which would provide support to the Jurčák criterion. Hereby ⟨ ⋅ ⟩_{ψ} stands for average along the I_{c} contour.
2. Data and analysis
The used data are retrieved after processing by the Solar Dynamics Observatory’s (SDO) Helioseismic and Magnetic Imager (HMI) Vector Magnetic Field Pipeline (Hoeksema et al. 2014) cutout service for NOAA AR 11591. Using this NOAA AR number on the JSOC website^{1} in the im_patch processing option automatically gives the reference coordinates listed in the final three columns of Table 1. t_{< E} and t_{> W} are the first and last time steps processed, where t_{< E} is before the sunspot rotates over the east limb onto the sun and t_{> W} is after the sunspot rotates off the west limb. A cutout size of 500 × 500 pixel was chosen. The data series used are hmi.IcnoLimbDark720s and hmi.B720s. For the full disc passage, there are 1599 time steps.
Timestamps of our spot, year = 2012.
For the 180°disambiguation the potential acute solution provided by the pipeline was adopted. This can be done using hmi_disambig with method=0. We note that for all pixels 180° must be added because the azimuth is defined relative to the positive yaxis of the maps in CCDframe and exportdata’s im_patch option rotates the maps 180° so that solar north is up.
The heliographic Stonyhurst coordinates are calculated using procedures modified from and tested against sswidl’s wcs routines fitshead2wcs, wcs_get_coord, wcs_convert_from_coord and those they call (see Thompson 2006). The canonical value for HMI of R_{⊙} = 696 Mm is used. The transformation of the magnetic field vector into the local reference frame was performed with a code modified from and tested against Xudong Sun’s sswidl routine hmi_b2ptr (see Gary & Hagyard 1990; Thompson 2006; Sun 2013).
Quiet sun intensity. The limb darkening correction in the HMI pipeline was based on Pierce & Slaughter (1977, Eq. (9)), which does not consider all orbital artefacts introduced into the continuum intensity I_{c} of SDO/HMI data. Even after limb darkening removal and normalization there is a change over the day in I_{c} of the order of 1% towards the limb with opposite signs on the western and eastern hemisphere. To compensate for this, the quiet sun intensity I_{qs} for each time step was chosen such that I_{qs} is the mean of all the quiet sun pixels within the 500 × 500 cutout, where quiet sun is defined as having I_{c} > 0.9I_{qs}.
Contours were taken at I_{c} = 0.4 and 0.5I_{qs}, and the positions of the contours are used to interpolate the values of the vertical magnetic field component B_{ver}, the magnetic field strength B and the inclination to the surface normal γ_{LRF}. Vertical is to be understood in the local reference frame, in other words, it is the direction of the surface normal.
Due to different formation heights of I_{c} and the Fe I 617.3 nm line, as well as the Wilson depression (Wilson 1774) and differential lineofsight opacity effects (see e.g. Rimmele 1995; Westendorp Plaza et al. 2001a,b), the magnetic contours are projected towards the limb (i.e. outwards) relative to the intensity contours. To compensate for these shifts and get a better match between I_{c} and B_{ver} contours, we transformed the coordinates obtained from I_{c} contours, (x, y), using
before retrieving the magnetic field values at coordinates (x′,y′). (x, y),(x′,y′) are helioprojective coordinates in arcseconds from disc centre and Δh is the formation height difference. Later on, when the contours from magnetic field maps are plotted onto the I_{c} map (cf. Sect. 3.2 and Figs. 3 and 5), the inverse of Eq. (1) is applied, meaning that the magnetic contours are shifted inward. The value of Δh = 465 km results from a minimization procedure, which is explained on page 4. The effect of neglecting this compensation is discussed in Sect. 3.3.
The limits of the time series we analyse are given as t_{start} and t_{end} in Table 1. A total of 1063 time steps in this time range are available. This time range was chosen to select data sets, for which the heliocentric angle^{2} of the centroid of the umbra was smaller than 60°.
Time series fit. For every time step and magnetic quantity, an average was computed along the contours, thereby creating time series of the form , , . Similarly, standard deviations along the contours σ_{ψ}(t) were calculated. These time series (cf. Sect. 3 and Figs. 1 and 2) show a daily variation of an approximately sinusoidal shape. We believe them to be an artefact of SDO’s geosynchronous orbital motion. For the ranges from t_{start} to t_{end} given in Table 1, these time series are least square fitted against functions of the form
Fig. 1. Mean magnetic field strength in black (it’s vertical component in blue) along the I_{c} = 0.5I_{qs} contour, with Δh = 465 km accounted for, Sinusodial fits and the residuals for NOAA AR 11591. 

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whereby t is in days and t ∈ ℕ is at noon. X_{0} is the value we are interested in and will be henceforth called offset. It is used instead of a time average ⟨X(t)⟩_{t} because it correctly accounts for missing data (most importantly the gap in the afternoon of Oct 17) and that t_{start} and t_{end} have a different time of day. Here we have ⟨X(t)⟩_{t} − X_{0} < 0.5 G for all X(t) in G and < 0.01° for . While X_{1} and X_{2} are used internally during the fitting process to guarantee numeric stability, the results are presented with parameters X_{0}, X_{3}, and X_{4} in Table 2. X_{3} and X_{4} are the amplitude and phase of the orbital artefacts. Also listed are the standard deviations of the residuals σ_{t} = σ(X(t)−X_{fit}(t)) and the means of the standard deviations along the contours over the same range in time ⟨σ_{ψ}⟩_{t}.
Results: fit parameters and time averages.
Levels of magnetic contours. The offsets X_{0} from the fits to , and for the 0.5(0.4)I_{qs} contours are then used as contour level on the B_{ver}, B and γ_{LRF} maps, respectively. They are discussed in Sect. 3.2 and plotted in Figs. 3 and 5 and the videos.
Distance between contours. To quantify how well two contours match we calculated the average distance between them ⟨d⟩_{ψ}, which we define as the area of symmetric difference divided by the length of the intensity contour, ℓ(t). The area of symmetric difference, Δa(t), is the area surrounded by either of the contours but not both. When averaging in time we weighed by the contour length, giving
These average distances between contours are listed in Table 2 in pixel. For ⟨d⟩_{ψ} ≪ 1pixel only the total ordering should be relied upon due to griding and other computational effects.
Fit along each contour. For every point along a contour, a reference angle ψ = ∡(PCD) is calculated, whereby P is the point on the contour, C is the centroid of the I_{c} = 0.5 I_{qs} contour in the CCD frame and D is the centre of the solar disc as observed by SDO. The angles are calculated on the sphere. For every time step and every contour, is least square fitted against functions of the form
Those fits are plotted in the right panels of the videos (cf. Sect. 3.2 and bottom panels of Figs. 3 and 5). Furthermore the time averages of the fit amplitudes are listed in Table 2.
Optimal height difference.Δh = 465 km was chosen because it minimizes the average distance ⟨d⟩_{ψ, t} between the I_{c} = 0.5I_{qs} contours after transformation with Eq. (1) and the B_{ver} contours, whereby the contour level X_{0} on the B_{ver} map has been derived with the fit to ⟨B_{ver}⟩_{ψ}(t) as described above (Eq. (2)). An optimal height difference of Δh = 465 km means that the intensity contour at the limb is shifted outwards by 465 km ⋅ r_{⊙}/R_{⊙} ≈ 0.65″ ≈ 1.3 pixel. The difference of the formation heights for continuum and Fe I 617.3 nm line core amounts to ≈250 km for a typical umbral model atmosphere (see e.g. Norton et al. 2006, Table 1). The fact that the value for Δh is larger may be explained with the Wilson depression of the umbra, which typically amounts to 800 km. The latter causes the τ = 1 surface to be strongly inclined relative to horizontal. Minimizing the standard deviation of B_{ver} along the I_{c} = 0.5I_{qs} contour (⟨σ_{ψ}⟩_{t} column in Table 2) instead would give an optimal Δh = 520 km.
3. Results
Based on the time series of approximately ten days, in which the spot of NOAA AR 11591 has heliocentric angles smaller than 60°, we determine the magnetic properties for two distinct contour levels of the continuum intensity. As intensity levels we use I_{c} = 0.5 (0.4)I_{qs}. Along each contour, the azimuthal average of B_{ver}, B and γ_{LRF} are calculated. The respective values of those averages for B_{ver} (in blue) and B (in black) as well as sinusoidal fit of the orbital variation are displayed in the upper panels of Fig. 1 for I_{c} = 0.5I_{qs} and of Fig. 2 for I_{c} = 0.4I_{qs}. The lower panels show the residuals after subtracting the fit.
Fig. 2. As Fig. 1, but from contours at I_{c} = 0.4I_{qs}. 

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3.1. Temporal evolution
The parameters of the sinusoidal fits, offset X_{0}, amplitude X_{3}, and the rms of the corresponding residuals, σ_{t}, are given in Table 2 for all considered cases. In addition, they are printed into the plots of Figs. 1 and 2. For the contours at I_{c} = 0.5I_{qs}, we find for that σ_{t} = 15 G is smaller than the orbital amplitude X_{3} = 18 G, with an offset of X_{0} = 1693 G. For the contours of I_{c} = 0.4I_{qs}, σ_{t} = 19 G is also smaller than X_{3} = 20 G with an X_{0} = 1850 G. For the residuals of B_{ver} no longterm trend is noticeable.
In contrast, the residuals of amount to σ_{t} = 48 G which is larger than the amplitudes of the sinusoidal fit (16 G), and it shows a longterm variation. Since γ_{LRF} is dependent on B_{ver} and B, it has a longterm variation which compensates for that of B (not shown). The offsets X_{0} for and are 2171 G and 141.4° respectively at the contours with I_{c} = 0.5I_{qs}.
The fact that the residuals are smaller than is remarkable, but is even more remarkable if one considers that the gradient of B_{ver} perpendicular to the contour is larger than that of B. This can be inferred from Table 2: the difference of the offset, X_{0}, between the two different intensities amounts to 157 G while that of is only 94 G. Hence, a small shift of the contour implies a larger deviation in B_{ver} than in B. Therefore, our result of a smaller deviation in B_{ver} relative to B gives further evidence that can be considered constant in time.
3.2. Contours
Using the offsets X_{0} from the fits in Table 2 with I_{c} = 0.5I_{qs} and Δh = 465 km, the upper panels of Fig. 3 overplot the contours of intensity I_{c} = 0.5I_{qs} (red), B=2171 G (green), −B_{ver} = 1693 G (blue), and γ_{LRF} = 141.4° (yellow). The background images consist of 100 × 100 pixel cutouts of greyscale intensity maps with a minimum (maximum) of I_{c} = 0.1 (1.2)I_{qs}. A close inspection of the figure shows that the B_{ver} contour matches best with the intensity contour. The cyan arrow originates in the centroid of the umbra and points towards disc centre. The centroid is determined by the I_{c} = 0.5I_{qs} contour and is derived using CCD coordinates.
Fig. 3. NOAA AR 11591, top panels: continuum intensity maps for longitudes −60°, 0° and 30°. The legend in the lower right corner of the top left panel defines the contour levels. Different formation heights are accounted for (Eq. (1), Δh = 465 km). The cyan arrow originates in the centroid of the umbra and points towards disc centre. Bottom three rows: magnetic field parameters retrieved along the I_{c} = 0.5I_{qs} contour. The temporal evolution is available online. 

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The three bottom rows of panels of Fig. 3 show the magnetic field quantities along the I_{c} = 0.5I_{qs} contour as well as their sinusoidal fits in black. The azimuth is determined relative to the centroid and the direction towards disc centre, which corresponds to ψ = 0° and runs counterclockwise.
To quantify the azimuthal variation of the magnetic parameters, Table 2 gives the time average of the standard deviations along the contours, ⟨σ_{ψ}⟩_{t}. Again ⟨σ_{ψ}⟩_{t} is smaller for B_{ver} (81 G) than for B (111 G). As before, the small value for B_{ver} is remarkable, since its gradient perpendicular to the contour is larger than for B. The lower panels demonstrate that the azimuthal variations are smallest for B_{ver}. Again, we note that this is remarkable considering the fact that the gradient of B_{ver} perpendicular to the contour is larger than the gradient of B.
A video of the temporal evolution of those contours during the disc passage of the spot is available online. This animation demonstrates that an isocontour of B_{ver} = −1693 G coincides nicely with the intensity contour at 0.5I_{qs}. This animation also demonstrates that contours of B and γ_{LRF} do not coincide.
To quantify the match or mismatch of two contours, we have introduced the average distance between two sets of contours, (cf. Eq. (3)). It is given in the last column of Table 2. is smallest for the B_{ver} contours with Δh = 465 km (see the first and final row of Table 2).
In Fig. 4 the average distance is plotted for intensities changing from 0.30 to 0.65. The corresponding contour levels for B_{ver} are calculated as described in Sect. 2 (fit to Eq. (2)). The best match, , is found for I = 0.53I_{qs} with −B_{ver} = 1639 G (X_{3} = 17 G, σ_{t} = 15 G, and ⟨σ_{ψ}⟩_{t} = 82 G). Distances for B and γ_{LRF} are in all cases larger and not plotted. Hence, by minimizing the distance, −B_{ver} = 1639 G results as the value that defines the umbral boundary at I = 0.53I_{qs}. This is additional proof that our chosen value of I = 0.5I_{qs} is very close to the optimum value.
Fig. 4. Average distance, , between contours of varying intensity and B_{ver}. The contour level of B_{ver} is determined by fitting Eq. (2). has a minimum for I_{c} = 0.53I_{qs} corresponding to −B_{ver} = 1639 G. 

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3.3. Effect of neglecting formation heights compensations
For the results presented so far, we corrected for the projection effects due to different formation heights of continuum and line. As discussed in the end Sect. 2 we assume a height difference of Δh = 465 km. Table 2 also gives the results for the case in which these projection effects are not considered, i.e. Δh = 0 km. As a general trend, it is seen that the values for X_{0}, X_{3}, and σ_{t} change only marginally. A plot like in Fig. 1 with Δh = 0 km looks almost identical (not shown).
However, ⟨σ_{ψ}⟩_{t} and increase significantly. For example, for B_{ver} at I = 0.5I_{qs}, ⟨σ_{ψ}⟩_{t} and increase by more than 30% from 81 to 113 G, and from 0.45 to 0.59 pixel, respectively. This is illustrated in Fig. 5, which shows the same snapshot as in the left column of Fig. 3, with the only difference that Δh = 0 km. In this case, the heliocentric angle is 60°. It is seen that the magnetic contours are shifted relative to the intensity, which results in an increase of , and the variation of B_{ver} along the contour (bottom panels) are larger for Δh = 0 km. This can also be seen in the corresponding video of the disc passage of the spot, which is available online.
Fig. 5. As Fig. 3, left column, but without compensation for different formation heights (Δh = 0). The temporal evolution is available online. 

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4. Conclusion
Investigating the physical properties along the umbrapenumbral boundary of a stable sunspot for a time span of approximately ten days, we find three main results:

B_{ver} averaged along the I = 0.5I_{qs} contour is nearly constant in time.

Contours of intensity and of B_{ver} match at the umbral boundary. The best match is obtained for I = 0.53I_{qs} and B_{ver}=1639 G.

Projection effects due to different formations height of the spectral line and continuum need to be considered. If not, variation of B_{ver} along the contour increases significantly.
These results are obtained by analysing 1063 consecutive SDO/HMI data sets (with a time step of 12 min) of the first disc passage of NOAA AR 11591.
Using I_{c} = 0.5I_{qs} to define the umbral boundary, we obtain B_{ver} = 1693 G ± 15 (1σ_{t}error). Jurčák et al. (2018) used Hinode/SP data to find (99%error) at I_{c} = 0.5I_{qs}.
The values for B_{ver} differ by some 175 G. In general, a difference is expected due to differences in the experimental setup and analysis methods. Sainz Dalda (2017) investigates the differences between HMI and SP vector magnetograms and obtained comparable differences. He concludes that the filling factor followed by spatial and spectral resolution are the main source. At the umbral boundary the filling factor is 1, and causes therefore no differences. The other effects are particularly strong at the sharp boundary between umbra and penumbra, where the intensity gradient is large.
Hence, these investigations provide evidence that B_{ver} is constant for a statistical sample of sunspots as well as during the evolution of one stable spot, thereby supporting the Jurčák criterion.
We note the subtle difference between the heliocentric angle and the angle between the LOS and the local vertical. The heliocentric angle, θ, is the angle between the centre of the umbra and the observer as measured from the centre of the sun. The angle, α, between the LOS and the local vertical at the umbral centre is given by: α = θ + r, whereby For any position on the solar disc, r is smaller than r_{⊙} ≈ 0.27°. The angle, α, is used to transform between the LOS and LRF coordinate systems.
Acknowledgments
We wish to thank Jan Jurčák, Juan Manuel Borrero and the anonymous reviewer for valuable discussions, Xudong Sun for making available the vector transformation routine as well as Hanna Strecker and various members of the IDL user group (http://www.idlcoyote.com/comp.lang.idlpvwave/) for help with IDL and tex.sx (https://tex.stackexchange.com/) users for help with LaTeX. The data used is courtesy of NASA/SDO and the HMI science team (see e.g. Metcalf 1994; Leka et al. 2009; Borrero et al. 2011; Pesnell et al. 2012; Schou et al. 2012; Hoeksema et al. 2014). This research has made use of NASA’s Astrophysics Data System (http://adsabs.harvard.edu/abstract_service.html).
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Movies
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Movie of Fig. 3 (Original mov) (Access here)
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All Tables
All Figures
Fig. 1. Mean magnetic field strength in black (it’s vertical component in blue) along the I_{c} = 0.5I_{qs} contour, with Δh = 465 km accounted for, Sinusodial fits and the residuals for NOAA AR 11591. 

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In the text 
Fig. 2. As Fig. 1, but from contours at I_{c} = 0.4I_{qs}. 

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In the text 
Fig. 3. NOAA AR 11591, top panels: continuum intensity maps for longitudes −60°, 0° and 30°. The legend in the lower right corner of the top left panel defines the contour levels. Different formation heights are accounted for (Eq. (1), Δh = 465 km). The cyan arrow originates in the centroid of the umbra and points towards disc centre. Bottom three rows: magnetic field parameters retrieved along the I_{c} = 0.5I_{qs} contour. The temporal evolution is available online. 

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In the text 
Fig. 4. Average distance, , between contours of varying intensity and B_{ver}. The contour level of B_{ver} is determined by fitting Eq. (2). has a minimum for I_{c} = 0.53I_{qs} corresponding to −B_{ver} = 1639 G. 

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In the text 
Fig. 5. As Fig. 3, left column, but without compensation for different formation heights (Δh = 0). The temporal evolution is available online. 

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In the text 
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