method and results. the spirit sometimes observes a.m.uralov, g.v.rudenko institute of solar...

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Method and results. The SPIRIT sometimes observes A.M.Uralov, G.V.Rudenko Institute of Solar Terrestrial Physics, Irkutsk, Russia Comparison of 5.7 and 17 GHz solar images with extrapolated magnetograms of coronal magnetic field Active regions of the October/November 2003 period and forecast of powerful solar flares Abstract. A direct comparison of microwave images of a solar active region with MDI magnetograms of the longitudinal component of solar magnetic fields cannot correctly specify where a microwave source is located within an active region far from the solar disk center. We propose a way to do this by using coronal magnetic field magnetograms extrapolated from a single MDI magnetogram, which is the nearest to the time of observation. The method is illustrated for the October/November 2003 active regions period. Appearance and long-duration existence of Neutral Line associated Sources were detected. Such sources can be regarded as predictors of powerful flares. Application of the method of extrapolated B r magnetograms to the period of October/November 2003. During the period of October 19 to November 4, 2003, a large number of M and X class flares and CMEs occurred in AR 0484, 0486, and 0488 (Fig. 2). NLS dominated in NoRH maps of these ARs. AR 0484. Images in Fig. 3 represent this AR before the following flares: M1.4, M1.2 on October 22; M1.5 on October 25; X1.2, M7.6 on October 26. The upper row shows SSRT (contours) and NoRH (colors) intensity maps obtained simultaneously. The brightness temperatures are: T b_5.7 = (3–3.5)·10 6 K, T b_17max = (360, 460, 500)·10 3 K respectively. Middle row shows 5.7 GHz intensity maps and extrapolated low resolution B r magnetograms (colors). Red/yellow shows N-polarity (positive) of the radial component B r ; green/blue shows S-polarity (negative). Bottom: 17 GHz intensity maps (colors) overlaid with contours of B r magnetograms (solid – positive, dotted – negative). Thick yellow line marks the magnetic inversion line. One can see two NLS persisting in this AR (arrows). Microwave emission of AR 0484 is dominated by those NLS. The rightmost source is a sunspot-associated one. AR 0486. A comparison of microwave maps of this AR with calculated B r magnetograms for the period of October 28 – November 4 is shown in Fig. 4. Upper row represents 5.7 GHz I maps (contours) and B r magnetograms (colors), bottom row: 17 GHz I maps (colors) and B r magnetograms (contours). With the exception of the bursts, the brightness temperature range of a huge, weakly polarized 5.7 GHz source was T b_5.7 =(2–5)·10 6 K during its passage from the eastern to the western limb. The 17 GHz & B r row clearly shows the appearance and persistence of bright, compact Neutral Line associated Sources along with the development of the AR. Their brightness temperatures were T b_17 = (200–900)·10 3 K (except for the bursts). The birth of a NLS was observed several hours before X17.2 flare on October 28 (solid red arrows). The B r magnetograms presented clearly show intrusion of a new negative magnetic flux into the positive one (solid and dotted red arrows). Next day, on October 29, two NLS already exist before M3.5 and X10 (start time 20:37 UT) flares. An inspection of Huairou magnetogram and magnetograms of modulus of the magnetic field extrapolated from MDI magnetograms show that just in this case NLS emission can be, in principle, interpreted as the 4 th harmonic gyroresonance emission of rather hot and dense plasma. However, such interpretation does not seem to be appropriate for some other NLS observed. Between October 29 to October 30 Nobeyama daytime observations, the leftmost of two mentioned NLS shifted along the inversion line into the new position (blue arrow in Fig. 4). The displacement of the source likely occurred before the X10 flare on October 29, because an HXR (50-100 kev) flare source was observed by RHESSI in the same position. B r magnetograms for October 28 and 29 were extrapolated with the middle resolution. Three days, October 30, 31 and November 1, were rather calm: only one M1.5 flare occurred on October 30. A long-living, stationary NLS (leftmost in those frames) persisted in AR. This NLS remains in the same position at least 13 hours before an X8.3 flare on November 2. Low-resolution B r magnetograms were used for those days. By contrast, middle-resolution B r magnetograms of November 2, 3, and 4 correspond to the last three images in Fig.4. On November 3, a long-living NLS slightly displaced leftwards, most probably in response to the magnetic flux redistribution seen from the magnetograms. Possibly, it shifted to this new position before the X8.3 flare, during Nobeyama night. On November 3, only one M class flare (M3.9) occurred. On November 4, flux redistribution is seen again, and a new NLS appears in addition to the long-living one. This situation before M2.6, M1.1, and X28+ flares is shown in the last frame. AR 0487. No compact NLS was observed in this simple bipolar AR, which produced no M or X class flare. Its 5.7 GHz emission was located exactly above the leading sunspot. Its 17 GHz emission was dominated by a diffuse weak source (T b_17 ~30·10 3 K) located above the neutral line and the following diffuse part of AR. There was no appreciable 17 GHz emission above the leading sunspot. AR 0488. X2.7 and X3.9 flares were produced in this region. Its 17 GHz emission was dominated by one NLS and one sunspot-associated source. Introduction. Nobeyama Radio Heliograph (NoRH, 17 GHz) images of solar active regions (AR) reveal two types of quasi-stationary microwave sources. They are compact sunspot-associated and Neutral Line associated Sources (NLS), which are subdivided into weak, diffuse sources in simple bipolar regions and rather bright, compact sources in complex AR. We consider here bright, compact 17 GHz NLS only. Uralov, Nakajima Nakajima, Zandanov, and Grechnev (2000) first detected such quasi-stationary NLS in NoRH images and classified them as 1) slowly moving upward and 2) stationary, disposed low in the the AR magnetosphere, close to the photospheric magnetic field inversion line. Due to three times lower frequency, microwave sources in SSRT (Siberian Solar Radio Telescope, 5.7 GHz) images are larger and usually cover sources visible in NoRH images. SSRT observations of active regions containing NLS were reported by us earlier (e.g., Uralov et al. 1998), where these sources were regarded as predictors of powerful flares. At 5.7 GHz, NLS can be identified from I, V, and V/I maps, but under favorable conditions only. Several factors make identification of such sources from SSRT data difficult. These are: 1) very strong influence of the AR magnetosphere on the sign and degree of polarization of radio waves propagating from the source to observer; 2) commonly large sizes of 5.7 GHz sources; 3) positional uncertainties of SSRT images. NoRH 17 GHz data seem to be more preferable keeping in mind the factors mentioned. However, it is rather difficult or even impossible to recognize NLS in NoRH images, because inspection of 17 GHz I and V maps only cannot generally reveal considerable difference between compact NLS and sunspot-associated gyroresonance sources, the degree of polarization can vary for both type sources from small values to 100 %. A direct comparison of 17 GHz images with longitudinal magnetograms (B ) cannot give correct answer also. The main reason for this is well-known transformation of magnetograms obtained at oblique viewing angles. Due to this projection effect, B magnetograms observed do not represent the position of true magnetic field inversion line or true magnetic spot (Fig. 1). In a complex AR aside the solar disk center, it is practically impossible to determine the disposition of a microwave source: above magnetic spot (sunspot-associated source); in the vicinity of the magnetic inversion line (NLS); some other. The positions of true magnetic field inversion line or true magnetic spot can be determined, if the distribution of the radial magnetic component B r is known. This component is normal to the solar surface, and it is in close agreement with B for an AR near the solar disk center. In other cases, the B r component can be determined in calculations only. To calculate this and other magnetic components, we extrapolate the magnetic field in the potential-field approximation from full-disk MDI magnetograms into the corona (the method by Rudenko 2001). For this study, some new algorithms were developed. Extrapolation of B r magnetograms is possible in two ways: 1) for the whole solar disk (low resolution in this paper) and 2) for selected AR, where the magnetic field is calculated within a limited box (middle resolution). Extrapolated B r magnetograms in figures of this paper correspond to the spherical surface at a height of 2000 km above the photosphere, which is close to the lowest level of the solar emission at 17 GHz. Some deviation from that height does not appreciably change the results obtained. In all images, the Carrington coordinate system is used. Transformation of the observed B and extrapolated B r magnetograms at two different viewing angles is demonstrated by a cartoon in Fig.1. Here, orange vectors are true magnetic field vectors of some fragment of the solar surface. White-gray oval is the observed microwave source. When viewing from top, its arrangement with respect to the B and B r magnetograms is the same. At an oblique viewing angle, its arrangement in the B r magnetogram remains invariable, but it is drastically changes in the observed B magnetogram. B r = ( B n ) B = ( B l 2 ) B B = B r > 0 l 2 n l 1 B < 0 B > 0 fragmentofextrapolated B r -magnetogram ofsolar m icrow ave disk:R = R sun + h m icrow ave solar surface µ- source atangleview top view A ngle view transform ation of B - and extrapolated B r -m agnetogram s Fig. 1. Transformation of the longitudinal B and extrapolated B r magnetograms of an active region observed at different viewing angles. 22 October 25 October 26 October before flares М 1.4, М 1.2 М 1.5 X1.2, М 7.6 Fig.3. NOAA 0484 . Top: SSRT (5.7 GHz; contours) and NoRH (17 GHz; colors) intensity maps obtained at one time. Middle: 5.7 GHz intensity maps and extrapolated B r magnetograms (colors; low resolution). Red/yellow – positive (N), green/blue – S-polarity of the radial magnetic component B r . Bottom: 17 GHz intensity maps (colors) and B r magnetograms. Positive polarity – solid, negative – dotted. Inversion line is shown by thick yellow line. Two NLS persist in this AR (arrows). Rightmost source is sunspot-associated one. NLS L o w r e s o l u t i o n 5.7 GHz + B r magnetograms 5.7 GHz + 17 GHz 17 GHz +B r magnetograms Fig. 2. Left: SOHO/MDI 6767 Å continuum, 22:24 UT, October 27. Right: 5.7 GHz intensity map (contours) and MDI magnetogram (B ; background), 03:11 UT, October 28. 487 488 486 484 Fig.4. NOAA 0486 . Top: 5.7 GHz I maps (contours) and B r magnetograms (colors). Red/yellow – positive, green/blue – negative polarity of the radial magnetic component B r . Bottom: 17 GHz I maps (colors) and B r magnetograms (contours). Positive polarity – solid, negative – dotted. Inversion line is shown by thick yellow line. The 17 GHz & B r row clearly shows the appearance and persistence of bright, compact Neutral Line associated Sources along with the AR development. 28 October 28 October - two hours later two hours later 29 October 30 October 31October 1 November 2 November low res = 2 November middle res 3 November 4 November before flares: X 17.2 М 3.5, X 10 М 1.5 n o f l a r e s M 1.0 X 8.3 M 3.9 M 2.6, M1.1, X28+ L o w r e s o l u t i o n long-living NLS NLS birth NLS birth M i d d l e r e s o l u t i o n M i d d l e r e s o l u t i o n 5.7 GHz + B r magnetograms 17 GHz + B r magnetograms Conclusions. We have proposed a new method of extrapolated B r magnetograms to identify the type of a microwave source in NoRH maps. The method allows to avoid strong projection effects in longitudinal magnetograms observed. We demonstrate convincingly that quasi-stationary Neutral Line associated Sources are widespread at 17 GHz, especially in flare-productive regions. In contrast to sunspot-associated sources, the NLS are not cospatial with local maxima of the modulus of the magnetic field. The appearance of a NLS or their displacements precede powerful flares (CMEs). Their presence reflects the presence of a current sheets in AR magnetosphere and signifies the readiness of AR to flare production (Uralov, Nakajima Nakajima, Zandanov & Grechnev, 2000). These sources can be certainly regarded as predictors of powerful flares, and this is why NoRH microwave solar images should be included into the list of the major world data for Space Weather forecasts. More extensive material should be analyzed to study Neutral Line associated Sources in detail. References Uralov, A.M., H. H. Nakajima Nakajima, V.G. Zandanov, and V.V. Grechnev. Current-sheet-associated radio sources and development of the magnetosphere of an active region revealed from 17 GHz and Yohkoh data. Solar Physics, 2000, 197(2), 275. Uralov, A.M., R.A. Sych, V.L. Shchepkina, G.N. Zubkova, and G. Ya. Smolkov. Weakly polarized microwave sources in active regions prior to large X-ray flares. Solar Physics, 1998, 183(2), 359. Rudenko, G.V. Extrapolation of the solar magnetic field within the potential-field approximation from full-disk magnetograms. Solar Physics, 2001, 198, 5. Acknowledgments. We thank the instrumental teams operating SOHO/MDI, the Nobeyama Radioheliograph, and the Siberian Solar Radio Telescope for data used here. We thank Victor Grechnev for discussion and improvement of the text.

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Page 1: Method and results. The SPIRIT sometimes observes A.M.Uralov, G.V.Rudenko Institute of Solar Terrestrial Physics, Irkutsk, Russia Comparison of 5.7 and

Method and results. The SPIRIT sometimes observes

A.M.Uralov, G.V.RudenkoInstitute of Solar Terrestrial Physics, Irkutsk, Russia

Comparison of 5.7 and 17 GHz solar images with extrapolated magnetograms of coronal magnetic field Active regions of the October/November 2003 period and forecast of powerful solar flares

Abstract. A direct comparison of microwave images of a solar active region with MDI magnetograms of the longitudinal component of solar magnetic fields cannot correctly specify where a microwave source is located within an active region far from the solar disk center. We propose a way to do this by using coronal magnetic field magnetograms extrapolated from a single MDI magnetogram, which is the nearest to the time of observation. The method is illustrated for the October/November 2003 active regions period. Appearance and long-duration existence of Neutral Line associated Sources were detected. Such sources can be regarded as predictors of powerful flares.

Application of the method of extrapolated Br magnetograms to the period of October/November 2003. During the period

of October 19 to November 4, 2003, a large number of M and X class flares and CMEs occurred in AR 0484, 0486, and 0488 (Fig. 2). NLS dominated in NoRH maps of these ARs. 

AR 0484. Images in Fig. 3 represent this AR before the following flares: M1.4, M1.2 on October 22; M1.5 on October 25; X1.2, M7.6 on October 26. The upper row shows SSRT (contours) and NoRH (colors) intensity maps obtained simultaneously. The brightness temperatures are: Tb_5.7 = (3–3.5)·106K, Tb_17max = (360, 460, 500)·103K respectively. Middle row shows 5.7 GHz intensity maps and extrapolated low resolution Br

magnetograms (colors). Red/yellow shows N-polarity (positive) of the radial component Br; green/blue shows S-polarity (negative). Bottom: 17 GHz intensity maps (colors) overlaid with contours of Br magnetograms (solid – positive, dotted – negative). Thick yellow line marks the magnetic inversion line. One can see two NLS persisting in this AR (arrows). Microwave emission of AR 0484 is dominated by those NLS. The rightmost source is a sunspot-associated one.

AR 0486. A comparison of microwave maps of this AR with calculated Br magnetograms for the period of October 28 – November 4 is shown in Fig. 4. Upper row represents 5.7 GHz I maps (contours) and Br magnetograms (colors), bottom row: 17 GHz I maps (colors) and Br

magnetograms (contours). With the exception of the bursts, the brightness temperature range of a huge, weakly polarized 5.7 GHz source was Tb_5.7 =(2–5)·106 K during its passage from the eastern to the western limb. The 17 GHz & Br row clearly shows the appearance and persistence of bright, compact Neutral Line associated Sources along with the development of the AR. Their brightness temperatures were Tb_17 = (200–900)·103 K (except for the bursts).

The birth of a NLS was observed several hours before X17.2 flare on October 28 (solid red arrows). The Br magnetograms presented clearly show intrusion of a new negative magnetic flux into the positive one (solid and dotted red arrows). Next day, on October 29, two NLS already exist before M3.5 and X10 (start time 20:37 UT) flares. An inspection of Huairou magnetogram and magnetograms of modulus of the magnetic field extrapolated from MDI magnetograms show that just in this case NLS emission can be, in principle, interpreted as the 4th harmonic gyroresonance emission of rather hot and dense plasma. However, such interpretation does not seem to be appropriate for some other NLS observed. Between October 29 to October 30 Nobeyama daytime observations, the leftmost of two mentioned NLS shifted along the inversion line into the new position (blue arrow in Fig. 4). The displacement of the source likely occurred before the X10 flare on October 29, because an

HXR (50-100 kev) flare source was observed by RHESSI in the same position. Br magnetograms for October 28 and 29 were extrapolated with the middle resolution.

Three days, October 30, 31 and November 1, were rather calm: only one M1.5 flare occurred on October 30. A long-living, stationary NLS (leftmost in those frames) persisted in AR. This NLS remains in the same position at least 13 hours before an X8.3 flare on November 2. Low-resolution Br magnetograms were used for those days. 

By contrast, middle-resolution Br magnetograms of November 2, 3, and 4 correspond to the last three images in Fig.4. On November 3, a long-living NLS slightly displaced leftwards, most probably in response to the magnetic flux redistribution seen from the magnetograms. Possibly, it shifted to this new position before the X8.3 flare, during Nobeyama night. On November 3, only one M class flare (M3.9) occurred. On November 4, flux redistribution is seen again, and a new NLS appears in addition to the long-living one. This situation before M2.6, M1.1, and X28+ flares is shown in the last frame.  

AR 0487. No compact NLS was observed in this simple bipolar AR, which produced no M or X class flare. Its 5.7 GHz emission was located exactly above the leading sunspot. Its 17 GHz emission was dominated by a diffuse weak source (Tb_17 ~30·103K) located above the neutral line and the following diffuse part of AR. There was no appreciable 17 GHz emission above the leading sunspot.

AR 0488. X2.7 and X3.9 flares were produced in this region. Its 17 GHz emission was dominated by one NLS and one sunspot-associated source.

Introduction. Nobeyama Radio Heliograph (NoRH, 17 GHz) images of solar active regions (AR) reveal two types of quasi-stationary microwave sources. They are compact sunspot-associated and Neutral Line associated Sources (NLS), which are subdivided into weak, diffuse sources in simple bipolar regions and rather bright, compact sources in complex AR. We consider here bright, compact 17 GHz NLS only. Uralov, NakajimaNakajima, Zandanov, and Grechnev (2000) first detected such quasi-stationary NLS in NoRH images and classified them as 1) slowly moving upward and 2) stationary, disposed low in the the AR magnetosphere, close to the photospheric magnetic field inversion line.

Due to three times lower frequency, microwave sources in SSRT (Siberian Solar Radio Telescope, 5.7 GHz) images are larger and usually cover sources visible in NoRH images. SSRT observations of active regions containing NLS were reported by us earlier (e.g., Uralov et al. 1998), where these sources were regarded as predictors of powerful flares. At 5.7 GHz, NLS can be identified from I, V, and V/I maps, but under favorable conditions only. Several factors make identification of such sources from SSRT data difficult. These are: 1) very strong influence of the AR magnetosphere on the sign and degree of polarization of radio waves propagating from the source to observer; 2) commonly large sizes of 5.7 GHz sources; 3) positional uncertainties of SSRT images.

NoRH 17 GHz data seem to be more preferable keeping in mind the factors mentioned. However, it is rather difficult or even impossible to recognize NLS in NoRH images, because inspection of 17 GHz I and V maps only cannot generally reveal considerable difference between compact NLS and sunspot-associated gyroresonance sources, the degree of polarization can vary for both type sources from small values to 100 %. A direct comparison of 17 GHz images with longitudinal magnetograms (B) cannot give correct answer also. The main reason for this is well-known transformation of magnetograms obtained at oblique viewing angles. Due to this projection effect, B magnetograms observed do not represent the position of true magnetic field inversion line or true magnetic spot (Fig. 1). In a complex AR aside the solar disk center, it is practically impossible to determine the disposition of a microwave source: above magnetic spot (sunspot-associated source); in the vicinity of the magnetic inversion line (NLS); some other.

The positions of true magnetic field inversion line or true magnetic spot can be determined, if the distribution of the radial magnetic component Br is known. This component is normal to the solar surface, and it is in close agreement with B for an AR near the solar disk center. In other cases, the Br component can be determined in calculations only. To calculate this and other magnetic components, we extrapolate the magnetic field in the potential-field approximation from full-disk MDI magnetograms into the corona (the method by Rudenko 2001). For this study, some new algorithms were developed. Extrapolation of Br magnetograms is possible in two ways: 1) for the whole solar disk (low resolution in this paper) and 2) for selected AR, where the magnetic field is calculated within a limited box (middle resolution). Extrapolated Br magnetograms in figures of this paper correspond to the spherical surface at a height of 2000 km above the photosphere, which is close to the lowest level of the solar emission at 17 GHz. Some deviation from that height does not appreciably change the results obtained. In all images, the Carrington coordinate system is used.

Transformation of the observed B and extrapolated Br magnetograms at two different viewing angles is demonstrated by a cartoon in Fig.1. Here, orange vectors are true magnetic field vectors of some fragment of the solar surface. White-gray oval is the observed microwave source. When viewing from top, its arrangement with respect to the B and Br magnetograms is the same. At an oblique viewing angle, its arrangement in the Br magnetogram remains invariable, but it is drastically changes in the observed B magnetogram.

Br = (B • n)

B = (B• l2)

B

B = Br > 0

l2

n

l1

B < 0 B > 0

fragment of extrapolated

Br-magnetogram of solar microwave disk: R= Rsun+ h

microwave solar surface

µ-source

at angle viewtop view

Angle view transformation of B- and extrapolated Br-magnetograms

Fig. 1. Transformation of the longitudinal B and extrapolated Br

magnetograms of an active region observed at different viewing angles.

22 October 25 October 26 October

before flares М 1.4, М 1.2 М 1.5 X1.2, М 7.6

Fig.3. NOAA 0484. Top: SSRT (5.7 GHz; contours) and NoRH (17 GHz; colors) intensity maps obtained at one time. Middle: 5.7 GHz intensity maps

and extrapolated Br magnetograms (colors; low resolution). Red/yellow – positive (N), green/blue –

S-polarity of the radial magnetic component Br. Bottom: 17 GHz intensity maps (colors) and Br magnetograms. Positive polarity – solid, negative – dotted.

Inversion line is shown by thick yellow line. Two NLS persist in this AR (arrows). Rightmost source is sunspot-associated one.

NLS

L o w r e s o l u t i o n

5.7 GHz + Br magnetograms

5.7 GHz + 17 GHz

17 GHz +Br

magnetograms

Fig. 2. Left: SOHO/MDI 6767 Å continuum, 22:24 UT, October 27. Right: 5.7 GHz intensity map (contours) and MDI magnetogram (B; background), 03:11 UT,

October 28.

487488

486

484

Fig.4. NOAA 0486. Top: 5.7 GHz I maps (contours) and Br magnetograms (colors). Red/yellow – positive, green/blue – negative polarity of the radial magnetic component Br. Bottom: 17 GHz I maps (colors) and Br magnetograms (contours). Positive polarity – solid, negative – dotted. Inversion line is shown by thick yellow line. The 17 GHz & Br row clearly shows the appearance and persistence of bright, compact Neutral Line associated Sources along with the AR development.

28 October 28 October - two hours latertwo hours later 29 October 30 October 31October 1 November 2 November low res = 2 Novembermiddle res 3 November 4 November

before flares: X 17.2 М 3.5, X 10 М 1.5 n o f l a r e s M 1.0 X 8.3 M 3.9 M 2.6, M1.1, X28+

L o w r e s o l u t i o n

long-living NLS

NLS birth NLS birth

M i d d l e r e s o l u t i o n

M i d d l e r e s o l u t i o n

5.7 GHz + Br magnetograms

17 GHz + Br magnetograms

Conclusions. We have proposed a new method of extrapolated Br magnetograms to identify the type of a microwave source in NoRH maps. The method allows to avoid strong projection effects in longitudinal magnetograms observed. We demonstrate convincingly that quasi-stationary Neutral Line associated Sources are widespread at 17 GHz, especially in flare-productive regions. In contrast to sunspot-associated sources, the NLS are not cospatial with local maxima of the modulus of the magnetic field. The appearance of a NLS or their displacements precede powerful flares (CMEs). Their presence reflects the presence of a current sheets in AR magnetosphere and signifies the readiness of AR to flare production (Uralov, NakajimaNakajima, Zandanov & Grechnev, 2000). These sources can be certainly regarded as predictors of powerful flares, and this is why NoRH microwave solar images should be included into the list of the major world data for Space Weather forecasts. More extensive material should be analyzed to study Neutral Line associated Sources in detail.

ReferencesUralov, A.M., H.H.NakajimaNakajima, V.G. Zandanov, and V.V. Grechnev. Current-sheet-associated radio sources and development of the magnetosphere of an active region revealed from 17 GHz and Yohkoh data. Solar Physics, 2000, 197(2), 275.Uralov, A.M., R.A. Sych, V.L. Shchepkina, G.N. Zubkova, and G. Ya. Smolkov. Weakly polarized microwave sources in active regions prior to large X-ray flares. Solar Physics, 1998, 183(2), 359.Rudenko, G.V. Extrapolation of the solar magnetic field within the potential-field approximation from full-disk magnetograms. Solar Physics, 2001, 198, 5.

Acknowledgments. We thank the instrumental teams operating SOHO/MDI, the Nobeyama Radioheliograph, and the Siberian Solar Radio Telescope for data used here. We thank Victor Grechnev for discussion and improvement of the text.