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ISSN 0016 7932, Geomagnetism and Aeronomy, 2015, Vol. 55, No. 1, pp. 13­23. © Pleiades Publishing, Ltd., 2015. Original Russian Text © Yu.S. Zagainova, V.G. Fainshtein, V.N. Obridko, 2015, published in Geomagnetizm i Aeronomiya, 2015, Vol. 55, No. 1, pp. 15­25.

Comparison of the Properties of Leading and Trailing Sunspots
Yu. S. Zagainovaa, V. G. Fainshteinb, and V. N. Obridkoa
a

Pushkov Institute of Terrestrial Magnetism, Ionosphere, and Radiowave Propagation, Russian Academy of Sciences (IZMIRAN), Troitsk, Moscow oblast, 142190 Russia b Institute of Solar­Terrestrial Physics, Siberian Branch, Russian Academy of Sciences, ul. Lermontova 126A, Irkutsk, 664033 Russia e mails: yuliazag@izmiran.ru; vfain@iszf.irk.ru; obridko@mail.ru
Received July 8, 2014

Abstract--The magnetic properties of leading and trailing sunspots were compared based on SDO/HMI and SDO/AIA data with a high spatial resolution for the growth phase and maximum of cycle 24. The properties of the solar atmosphere above sunspots are also discussed independently for both of these sunspot types. It was shown that the contrast in the He II 304 (C304) line above the umbra of leading and single sunspots is on average smaller than such a contrast above the umbra of trailing sunspots and on average weakly depends on the umbra area for both C304 sunspot types. It was established that the minimal angle between the field direc tion and the normal to the solar surface at the field measurement site is smaller in leading sunspots than in trailing ones (min ­ ls < min ­ fs) in 84% of the considered magnetically connected "leading­trailing" sunspot pairs, and a positive correlation exists between angles min ­ ls and min ­ fs. It was found that the C304 contrast increases with decreasing min ­ ls, fs for leading and trailing sunspots, and the C304 ­ ls/C304 ­ fs ratio on average decreases with increasing min ­ ls/min ­ fs ratio. The dependences of the maximal and average magnetic induction values in an umbra on the umbra area were constructed for the first time and compared indepen dently for leading and trailing sunspots. It was concluded that the maximal and average magnetic field values do not vanish when the umbra area decreases to very small values. In all cases the magnetic field in leading and single sunspots is larger than in trailing ones. DOI: 10.1134/S001679321406022X

1. INTRODUCTION According to photospheric observations, sunspots are characterized by decreased values of the matter temperature and brightness and increased values of the magnetic field as compared to the remaining photo spheric areas (Bray and Loughed, 1964; Obridko, 1985; Maltby, 1992). The Wolf number, i.e., the index of the sunspot number and groups simultaneously observed on the visible solar surface, is one of the most popular solar activity measures (Vitinsky et al., 1986). Sunspots have been intensely studied from the beginning of their telescopic observations at the end of 1610. The first studies were mainly morphological. However, in the course of time, it became clear that sunspot origination and further evolution is a rather complex physical process, and the properties of single sunspots can substantially differ on the one hand and, on the other hand, are closely related to one another and to the surrounding solar areas in the subphoto spheric layers and at different altitudes in the solar atmosphere (see, e.g., (Pipin and Kosovichev, 2011) and references therein). Sunspots often form groups, in which spots with different properties can be distinguished; in this case a group itself has special characteristics governed by the set of all spots in a group. The westernmost sunspot in
13

a group, which has a larger area and is located closer to the equator as compared to the remaining sunspots in a group, is mostly called a leading or heading sunspot. Sunspots in a group of opposite field polarity are called tail or trailing sunspots. According to the Hale law on sunspot polarity in groups, "... in odd cycles the mag netic field of leading groups in the Northern Hemi sphere has north seeking polarity and that of trailing groups has south seeking polarity. This pattern reverses its sign in the Southern Hemisphere and in going to an even cycle" (Obridko, 1985). In the overwhelming majority of previous studies, sunspot properties were studied independently of their type: leading or trailing. The number of works com paring the properties of leading and trailing sunspots in one group and, on average, in many groups is rela tively small. It was shown that there is almost no differ ence between the dependences of the contrast in the sunspot emission (Sobotka, 1986) and the photo spheric magnetic field value in sunspots (Bray and Loughed, 1964) on the area of leading and trailing sunspots or the spot evolution stage. Gilman and Howard (1985) detected that the rotation velocities of leading and trailing sunspots slightly differ. Recent studies have shown that the properties of leading and trailing sunspots actually pronouncedly differ according to their observations in different spec


14 , arcsec 30 NOAA 11166

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30 NOAA 11130 NOAA 11092 20

20 10 10 0 0 ­20 ­10 0 ­5 5 15

10

0 ­25 ­15 ­5 L, arcsec

Fig. 1. Left: two large magnetically connected sunspots and new magnetically connected and independent smaller sunspots between them (March 8, 2011); center: a pair of small scale magnetically connected sunspots (November 29, 2010); right: a single sunspot with the magnetic properties of a leading sunspot (August 3, 2010).

tral ranges. Thus, it was shown (Zagainova, 2011) that the dependences of the contrast in the He II 304 line and He I 10830 parameters of the IR triplet on the umbra area of leading and trailing sunspots substan tially differ. The magnetic properties of leading and trailing sunspots were also different. Using magnetic field cal culations in a potential approximation based on Bd technology (Rudenko, 2001) and SDO data for 2010­2013, Zagainova et al. (2014) selected pairs of magnetically connected leading and trailing sunspots, the umbra of which are joined by magnetic field lines. It was found that the minimal angle between the field line and the normal to the solar surface (min) in a leading sunspot is smaller than in a trailing one in ~81% of the cases. A positive correlation between the values of this angle in leading and trailing sunspots was revealed for the spots that satisfy this condition. It was shown that the dependence of angle min on the umbra area differs in leading and trailing sunspots. A weak negative correlation was revealed between the min angle and the maximal magnetic induction ( Bmax ) value in an umbra. In other words, field lines are on average more radial in magnetic tubes that form umbrae of leading and trailing sunspots and have stronger fields at the level of the photosphere. Zagainova (2011) assumed that the difference in the magnetic field properties in an umbra of leading and trailing sunspots, which is characterized by the asymmetry of magnetic tubes connecting umbrae of two sunspot types, can result in increases in the He I atom layer optical depth in the 23S state and in the UV irradiance in 304 å above trailing sunspots as com pared to leading ones. In turn, the difference of the He I 10 830 IR triplet parameters from the leading and trailing sunspot umbra area can be explained by pre cisely this fact. This conclusion is based on the con

cept of the ionization­recombination formation of the chromospheric helium IR triplet. Such a forma tion includes helium atom ionization by UV irradi ance with the following transition of some atoms to the 23S metastable level after a certain delay, which is accompanied by the absorption of the photospheric continuum emission (Nikol'skaya, 1966; Livshits, 1975; Pozhalova, 1988). It was also indicated that the properties of single sunspots coincided with those of leading ones. The present work continues the studies that were started in (Zagainova, 2011; Zagainova et al., 2014). The dependences of the contrast values in He II 304 above a C304 umbra on the umbra area of leading and trailing sunspots at the cycle 24 growth phase and maximum were compared based on the solar observa tions in the SDO/AIA 304 å channel (Lemen et al., 2012). The magnetic properties of leading and trailing sunspots, determined using the SDO/HMI magnetic field vector measurements with a high spatial resolu tion, were also compared for the same period. 2. DATA AND METHODS OF STUDIES For analysis we selected the group of magnetically connected leading­trailing sunspot pairs that were observed in 2010­2013: the sunspots were connected by magnetic field lines calculated in the scope of the potential field­source surface model. Figure 1 from (Zagainova et al., 2014) illustrates magnetically con nected sunspot pairs and a single sunspot. It turned out that magnetically connected sunspots were character ized by a distinct regular umbra with a circular or ellip tical symmetry. In addition, we selected single sun spots with well shaped umbra and penumbra for anal ysis. The list of the magnetically connected sunspot pairs selected for analysis is presented in Table 1, where the active region numbers, umbra areas, and
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COMPARISON OF THE PROPERTIES OF LEADING AND TRAILING SUNSPOTS Table 1. List of magnetically connected sunspot pairs selected for analysis 1 Date Sept. 27, 2010 Oct. 25, 2010 Nov. Jan. Feb. Feb. 29, 2010 4, 2011 2, 2011 13, 2011 2 NOAA 11 109 11 117 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 130 142 150 158 158 166 169 183 184 190 193 195 203 234 260 263 263 3 4 5 N/S N N N N S S S S S N N N N N N S N S N N N 1 Date Aug. 7, 2011 Aug. 22, 2011 Oct. 15, 2011 Oct. 28, 2011 Nov. 1, 2011 Nov. 7, 2011 Nov. 30, 2011 Dec. 3, 2011 Dec. 5, 2011 Dec. 20, 2011 Dec. 25, 2011 Jan. 20, 2012 Feb. 1, 2012 Feb. 11, 2012 Feb. 20, 2012 2 NOAA 11 266 11 272 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 316 319 330 334 339 338 361 365 363 363 364 382 384 401 413 416 422 3 4 5

15

SL, MSH
51 15 21 9 4 7 12 6 21 30 22.5 19 7 10 21 30 21 3.3 22 54 63

SF , MSH
10 22 5 8 5 7 12 9 24 21 8 5 3 13 4 23 0.3 1 7.3 62 15.5

SL, MSH
3 4 9 20 8 68 15 37 24.5 18 7 27 9 61 23 71 50 15 33 45

SF , MSH
4 2 5 22 3 7 2 29 8 10 4 12 1 14.5 3 8 16 7 46 31

N/S N S S S N N N N S N N S S N S N N N S N

Feb. 14, 2011 Mar. 8, 2011 Mar. 11, 2011 Apr. 1, 2011 Apr. 3, 2011 Apr. 13, 2011 Apr. 18, 2011 Apr. 24, 2011 May 5, 2011 June 14, 2011 July 30, 2011 Aug. 1, 2011 Aug. 3, 2011

With the date (column 1); active region number (column 2); umbra area (the areas of leading (SL) and trailing (SF) sunspots are given in columns 3 and 4, respectively); and magnetic field sign (column 5, N/S).

magnetic field signs are indicated. The data on single sunspots are combined in Table 2. The SDO/AIA telescope has a spatial resolution of 0.5 (Lemen et al., 2011). The contrast in the 304 å line above an umbra was determined from the C304 = I S I 0 ratio, where I S and I 0 are the intensity readings in the umbra and in the quiet area, respec tively (for more detail, see (Zagainova, 2011)). The umbra area, expressed in millionths of the solar hemi sphere (MSH), was found based on the sunspot images in continuum according to SDO/HMI data. The present work analyzed the magnetic field char acteristics in an umbra, such as the minimal angle ( min ) between the field direction and the normal to the solar surface at the field measurement point and the maximal (Bmax ) and average ( B ) values of mag netic induction. To analyze the magnetic field proper ties in an umbra, we used the HMI vector magnetic field measurements (http://hmi.stanford.edu/), which make it possible to determine the magnetic induction (B), magnetic field vector inclination with respect to the angle of sight (), and azimuth (), which is counted off counterclockwise in the sky plane from the
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CCD matrix pixel column to the (transverse) magnetic field vector projection onto this plane. When the mag netic field is measured with HMI, the spatial resolu tion is 0.5. Solar images with the and distributions are obtained only several times a day. When the field vector characteristics are determined, the problem of uncertainty is solved when the transverse field direc tion is found. In our work we analyzed angle between the field direction and the normal to the solar surface at a point where the magnetic field is measured. To find this angle based on the measured and values, we obtained the relationships between angles , , and . The calculations were performed in the Cartesian coordinate system [X, Y, Z] with the origin at the solar center, where the 0X and 0Y axes are in the equatorial plane, the 0Y and 0Z axes are in the sky plane, and the 0Z axis crosses the North Pole (we neglect the ecliptic plane inclination with respect to the equatorial plane). The 0X axis is directed along the line of sight and is perpendicular to the sky plane. We considered that the line of sight is perpendicular to the sky plane at all points within the solar disk. We introduced unit vectors b and r, which are directed along the field magnetic
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Table 2. List of single sunspots selected for analysis 1 Date July 2, 2010 Aug. 3, 2010 Aug. 9, 2010 Sept. 21, 2010 Oct. 19, 2010 Oct. 19, 2010 Oct. 20, 2010 Oct. 20, 2010 Nov. 22, 2010 Dec. 8, 2010 Dec. 9, 2010 Jan. 5, 2011 Mar. 30, 2011 Apr. 10, 2011 2 NOAA 11 084 11 092 11 093 11 108 11 11 3 11 11 5 11 11 3 11 11 5 11 127 11 131 11 131 11 140 11 180 11 185 3 S, MSH 23 42 25 48 22 29 21 25 15 73 64 28 6 4 4 N/S S N N S N S N S N N N N N N Nov, 17, 2011 Nov. 24, 2011 Nov. 26, 2011 Nov. 12, 2011 1 Date May 22, 2011 June 7, 2011 July 17, 2011 Oct. 8, 2011 Oct. 10, 2011 2 NOAA 11 216 11 232 11 251 11 309 11 312 11 309 11 340 11 342 11 341 11 343 11 346 11 355 11 360 3 4 N/S S N N N N N S N N N S N N

SL, MSH
16 12 18 18 48 14 17 30 18 28 15 23 19

With the date (column 1); active region number (column 2); umbra area (column 3, S); and magnetic field sign (column 4, N/S)

induction and along the upward normal to the solar surface at a point where the field is measured. The required angle () is the angle between vectors b and r. We used the expression for the scalar product of these vectors. We also bore in mind that the b and r lengths are equal to unity. Angle between the magnetic field direction and the normal to the solar surface was found from the following relationship:

where A = cos(); B = -ta n() = c o s( ') s in(') sin( ') . Sign "+" corresponds to = [0 °, 9 0 °] or [270°, 360°]; sign "­", to = [9 0 °, 2 7 0 °] .
2 2 2 cos (') = ( A + B ) (1 + B ), ' = [- 90°, 90°] , (3) where ' 0,if = [0 °, 9 0 °] or [270°, 360°], and ' < 0 if = [9 0 °, 2 7 0 °] .
2 2 2 2 (4) sin(') = ± B (1 - A ) (B + A ), where sign is "+" if = [1 8 0 °, 3 6 0 °] and "­" if = [0 °,1 8 0 °] .

cos () = b X rX + bY rY + b Z rZ = co s(') cos (') co s() co s () + cos (')s i n (')co s ()s i n () + s i n (')s i n () .

(1)

Here is the vector r latitude (i.e., the inclination of this vector to the equatorial plane); ' is the vector b lat itude, , ' = [- 9 0 °, 9 0 °] ; is the vector r longitude (the angle counted off clockwise from the X axis in the equa torial plane to the vector r projection onto this plane); is the vector b longitude, = [0 °, 9 0 °]U [2 7 0 °, 3 6 0 °] , ' = [0 °, 3 6 0 °] west of the central meridian; bX, rX; bY, rY; bZ, rZ are the projections of vectors b and r onto the X, Y, and Z axes, respectively. Angle between b and the 0X axis, which is determined from the SDO data ( = [0 °,1 8 0 °]), and angle between the Z axis and the projection of b onto the YZ plane (the azimuth accord ing to the SDO data, = [0 °, 3 6 0 °] , counted off coun terclockwise from the 0Z axis) are specified.
sin(') = ± (1 - A ) (1 + B ) ,
2 2

2 2 2 2 (5) cos (') = ± A (1 + B ) ( A + B ), where sign is "+" if = [1 8 0 °, 3 6 0 °] , and "­" if = [0 °,1 8 0 °] . Note that angle was larger than 90° for magnetic field of negative polarity (the field vector is directed toward the Sun), and we subtracted the obtained val ues from 180° in order to compare this angle with the angles for the field of positive polarity.

(2)

3. RESULTS 3.1. Comparison of the Emission Contrast in the Helium 304 å Line in Leading and Trailing Sunspots at the Cycle 24 Growth Phase and Maximum It was shown (Zagainova, 2011) that the contrast in 304 å above an umbra of leading (C304 -ls ) and single
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COMPARISON OF THE PROPERTIES OF LEADING AND TRAILING SUNSPOTS 80 (a) 80 (b)

17

60 C304
fs ss

60

C304 20 40 S, MSH 60 80



304 ls,

40

40

20

20

0

0

20

40 S, MSH

60

80

Fig. 2. Contrast in the He II 304 line above an umbra depending on the sunspot area: (a) for leading and single sunspots (circles) and trailing sunspots (triangles); (b) for single sunspots. The regression lines are shown by straight lines.

(C304 - ss ), as well as trailing (C304 - fs ), sunspots on aver age slightly changes with increasing S for an area of S 5 MSH during the cycle 23 decline phase in 2002­ 2007. However, the contrast value in helium differs for leading and trailing sunspots. The average contrast value in the He II 304 line was C304 -ls , C304 - ss = 5 ± 0.67 according to the SOHO/EIT data ( C304 -ls , C304 - ss 7.5 according to the CORONAS F spacecraft data) for leading and single sunspots and C304 - fs = 12 ± 2.52 for trailing sunspots (the contrast value in 304 å on the regression line is presented here). For pores and small sunspots with a degenerate penumbra and an area smaller than 10 MSH, the contrast value in helium varied from 6 to 14.5. Figure 2 shows the C304 -ls , C 304 - ss , and C304 - fs values obtained according to the SDO/AIA data for the cycle 24 growth phase and maximum in 2010­2013. As in (Zagainova, 2011), the C304 -ls and C 304 - ss values are combined in one group. But small sunspots and pores (S < 10 MSH), which had a developed umbra with cir cular or close to circular symmetry, were included in this plot as against (Zagainova, 2011). It is clear that the contrast in 304 å above an umbra of leading/sin gle and trailing sunspots differs, and its value slightly varies depending on the area (S) in both cases. The con trast in 304 å varies from 8 to 50 above an umbra of leading sunspots/pores with a small area (S < 10 MSH) and from 12 to 71 for trailing sunspots. Only for single sunspots of leading polarity were contrasts C304 -ls and C 304 - ss in these sunspots comparable with contrast C304 - fs for trailing sunspots. Note that the average contrast value in 304 å for two sunspot groups was pronouncedly larger than in (Zagainova, 2011). According to the SDO data, C304-ls , C304 -ss 20, C304 - fs 32 for sunspots S =
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20 MSH. Such a difference is possibly related to the fact that the sunspot contrast was found using different instruments in two works. However, the contrast in helium above leading and single sunspots according to the CORONAS F data during the same period was close to the values determined based on the SOHO/EIT data and differs by not more than a factor of ~1.5 as compared to the EIT data. The CORONAS F spatial resolution is 1 pixel = 5.47 arcsec, which is twice as small as the SOHO/EIT spatial resolution (1 pixel = 2.63 arcsec). On the other hand, this work presents the results for the cycle 24 growth phase and maximum, which are char acterized by different anomalous characteristics: small cycle height, substantially smaller magnetic fields, etc. (Akhmetov et al., 2014), whereas the results presented in (Zagainova, 2011) correspond to the decline phase of cycle 23. The C 304 contrast values for sunspots in cycle 24 could also be anomalous. Figure 2b presents the contrast values in the helium line for single sunspots depending on their area. In Fig. 2 we omit C304 for sunspots with anomalously large (B > 3000 G) magnetic field values. In contrast to leading sunspots, relatively large contrast values in 304 å are absent for single sunspots. The C 304 - ss value was on average smaller by ~30% than the value for the combined data for leading and single sunspots. 3.2. Comparison of Magnetic Properties of Leading and Trailing Sunspots during the Cycle 24 Growth Phase Using the SDO/HMI vector measurements of the magnetic field, we determined and compared the magnetic field properties in an umbra of leading and trailing sunspots. These measurements have substan tially higher spatial resolution than the resolution pro vided by the magnetic field calculations in the solar atmosphere in a potential approximation, the data of
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trailing sunspots ( min - fs ), the regression line is almost parallel to the abscissa. These dependences were con structed for sunspots for which the min-ls < min- fs condition was satisfied. We also compared the magnetic field value ( Bmax ) in an umbra of leading and trailing sunspots with the umbra area (S) of these sunspots (see Fig. 4). The dependence of the magnetic induction value in an umbra on the umbra area (S) was discussed in several works (see the review in the monograph (Bray and Loughed, 1964)) and was summarized in (Ringnes and Jensen, 1960). Based on the results of several works, it was concluded that the relation between the magnetic induction (B) maximum and the area (S) agrees with the empirical relationship obtained in (Houtgast and van Sluiters, 1948) B = 3700S/(S + 66); here B and S are measured in Gauss and millionths of visible hemi sphere. The relation between the magnetic induction (B) maximum and the sunspot area (S) was also stud ied based on the magnetic field vector measurements in sunspots (Jin et al., 2006). It was shown that a loga rithmic dependence exists between the field maximum in the umbra upper layers and the umbra area. How ever, leading and trailing sunspots were not separated in all studies of the B­S dependence. The Houtgast and van Sluiters formula, derived in 1948, contains a very important disadvantage. Accord ing to this formula, the sunspot magnetic field van ishes when the sunspot area tends toward zero. This formula reflects an early period in sunspot studies, when it was assumed that the magnetic field is absent outside sunspots and varies from 4000 G in large sun spots to 100 G in the smallest ones (Ringnes and Jensen, 1960). However, as the observation quality increased, it became clear that magnetic fields are rather large even in small sunspots. Steshenko (1967), Bumba (1967), and Beckers and Schroter (1968) indi cated that the field in the smallest pores is not smaller than 1200 G and is even sometimes larger than 1800 G (Baranov, 1974). Antalova (1991) and Solov'ev and Kirichek (2014) also criticized this disadvantage of the Houtgast and van Sluiters formula. At approximately the same time (in the 1960s), there appeared the concept of "kilogaussian tubes" (Sheeley, 1966, 1967; Harvey, 1971; Harvey and Living ston, 1969; Livingston and Harvey, 1969; Stenflo, 1973), according to which very small formations with a strength reaching 2000 G can exist. At the same time, the Houtgast and van Sluiters formula expresses a very important property: satura tion is formed at a certain instant, and the dependence of the magnetic field on the area becomes strongly weaker. Figure 4 presents the magnetic field dependences on the area for leading (Bmax -L, Bmean-L ) and trailing (Bmax -F , Bmean-F ) sunspots and the formulas approxi
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0

20 10 min­ls, deg

30

Fig. 3. Relation between the minimal angles between the magnetic field direction and the normal to the solar surface in an umbra of leading (min ­ ls) and trailing (min ­ fs) sunspots. The correlation coefficient is k = 0.774.

which were used to perform a similar analysis of the field properties in leading and trailing sunspots in our previous work (Zagainova et al., 2014). We start analyzing magnetic field properties in an umbra with minimal values of the angle ( min ) between the magnetic field direction and the normal to the solar surface at a point where the magnetic field was measured. Recall that the measured angle (meas) between the field direction and the radial direction is more than 90° when field polarity is negative (the sunward magnetic field vector); in this case min = 180° ­ meas. It turned out that the minimal angle between the field direction and the normal to the solar surface in leading sunspots was smaller than the angle in trailing sunspots in 84% of the consid ered pairs. For sunspots satisfying this condition, the aver age min value was min -ls 6. 12°, in leading sunspots, min- fs 16°, and min-ls min- fs 0.422. All of these values are smaller than the corresponding values obtained when the magnetic field was calculated in a potential approximation (Zagainova et al., 2014). Figure 3 shows that a positive correlation with a corre lation coefficient of 0.774 exists between the min-ls and min- fs, values, for which min-ls min- fs . As in (Zagainova et al., 2014), a weak negative cor relation between min-ls and min- fs on the one hand and between the maximal magnetic induction value ( Bmax ) and the umbra area (S) on the other hand was established based on SDO/HMI data. The correlation is almost absent for the min- fs (S ) dependence. We omit the corresponding pots and only present the regression line equations, with indication of the correlation coefficient. For leading sunspots, the dependence of the minimal angle on the magnetic field maximal value is min-ls (Bmax ) = - 0. 00438B max + 16.223, and the correlation coefficient is k = ­0.309. For trailing sunspots, min- fs (Bmax-F ) = - 0. 0057B max + 27.437, and k = ­0.157. For leading sunspots, the angle value depending on area S is min-ls (S ) = ­0.376 ± 25.36 (the correlation coefficient k = ­0.119); for

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COMPARISON OF THE PROPERTIES OF LEADING AND TRAILING SUNSPOTS 3200 (a) 2500 2800 B ,G G
max,

19

(b)

2400

2000

B 2000

1500 0 20 40 Sl, MSH (d) 2000 60 80

1600 0 20 40 Sl, MSH (c) 2500 60 80

B ,G

G

B

max,

2000

1500

1500 1000 0 20 40 Sf, MSH 60 80 0 20 40 Sf, MSH 60 80

Fig. 4. The magnetic field value depending on the sunspot area: (a) the field maximal value for leading sunspots Bmax ­ L(SL) = 1700 + 1650S/(S + 25); (b) the average magnetic field value for leading sunspots B (SL) = 1400 + 1400S/(S + 25); (c) the max imal magnetic field value for trailing sunspots Bmax ­ F(SF) = 1650 + 1100S/(S + 10); (d) the average field value for trailing sun spots B (SF) = 1300 + 800S/(S + 10).

mating these dependences. The plots and approximat ing formulas shown in Fig. 4 indicate the following. (i) Neither maximal nor mean magnetic field vales vanish when the area decreases to very small values. (ii) In all cases the magnetic field in leading and single sunspots is smaller than in trailing ones. (iii) The threshold area value, when the curves start saturating, is of great importance. In the Houtgast and van Sluiters formula, this value was 66 MSH and cor responded to a sunspot with a radius of ~8000 km. Our approximation gives threshold area values of ~25 and ~10 MSH for leading and trailing sunspots, respec tively. This corresponds to radii of 5000 and 3100 km. We can conditionally consider that saturation is formed when the radius is comparable with the sun spot depth. The values obtained by us generally agree with the estimates made by Solov'ev and Kirichek (Solov'ev and Kirichek, 2014) and support the con cept of a "shallow sunspot" developed by these
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researchers. At such an interpretation, our data shows that trailing sunspots not only have a smaller magnetic field but are also potentially shallower formations. (iv) Asymptotic values for a very large area are 3550 G for leading sunspots and 2750 G for trailing ones, and the smallest values can be ~1000 G. We also analyzed the magnetic properties of well shaped single sunspots with clearly defined umbra and penumbra. For these sunspots, the mean value of the minimal angle between the magnetic field direction and the direction along the positive normal to the solar sur face is min - ss = 5. 79 °, Bmax 2736 G. In this case the average area of such sunspots is S 24. 64 MSH, and the average contrast in 304 å in an umbra of such sunspots was C304 14 . 7 . This means that single sun spots selected for analysis are characterized by smaller min values as compared to leading sunspots, which slightly differs from the contrast in helium, but by large
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30
ls

60 304­fs 10 20 deg 30

304­

20

40

10

20

0

0

10

min­ls,

20 30 min­fs, deg

40

Fig. 5. Contrast C304 depending on the minimal angle (min) between the magnetic field direction and the normal to the solar surface at the point where the field is measured for (a) leading and (b) trailing sunspots.

magnetic field values and a larger umbra area. This is confirmed by the conclusion that sunspots with stron ger magnetic fields in their umbrae and smaller areas are on average characterized by smaller minimal angles min. The relation of the minimal angle in single sunspots ( min-ss ) to the maximal magnetic induction ( Bmax ), as well to the area (S) and contrast in the helium line (C 304 ), is almost absent, or a weak negative correlation exists between these parameters. The absence of a relationship between min-ss and Bmax and S apparently results from the fact that the number of sin gle sunspots and sunspots with very small and very large Bmax and S values is very small in the sample. We also note one more result, which was achieved at the limit of statis tical significance and requires larger samples for its con firmation. It turned out that angle min - ss N = 6. 45 ° for sunspots of north seeking polarity is larger than
1.0
304­ls/C304­fs

the angle for sunspots of south seeking polarity ( min - ss S = 4. 35 °). 3.3. Relation between Contrast in the 304 å Line and Magnetic Properties of Leading and Trailing Sunspots Our analysis indicated that a negative correlation exists between min-ls and the contrast in 304 å above umbrae of leading sunspots (C304 -ls ,) as well as between min - fs and C304 - fs in trailing sunspots (Fig. 5). These dependences and the plot in Fig. 6 were constructed for sunspots for which min-ls min- fs and, simulta neously, C304-ls C304 - fs . The dependences shown in Fig. 5 show that the contrast in 304 å on average increases with a decreasing minimal angle between the magnetic field direction and the positive normal to the solar surface at a point where the field is measured for leading/single and trailing sunspots. At the same time, the character of the relation between min and C304 for leading and trailing sunspots differs (see Fig. 6): as the ratio of minimal angles in leading and trailing sunspots increases, the contrast ratio in an umbra of these sun spots increases. The dependences shown in Figs. 5 and 6 can be interpreted as follows: the magnetic field line configuration in sunspot groups is such that the amount of He II ions emitting in 304 å above an umbra of trailing sunspots is larger than such an amount above an umbra of leading sunspots. Figure 7 illustrates the dependence of the contrast in helium of leading and trailing sunspots on the max imal magnetic induction ( Bmax ) in an umbra. It is evi dent that C304 -ls (Bmax ) and C304 - fs (Bmax ) on average slightly vary when Bmax , increases, as does the depen dence on the sunspot area (see Fig. 1).
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0.8 0.6 0.4 0.2 0 0.2 0.6 0.4 min­ls/min­fs 0.8 1.0

C
Fig. 6. Ratio of the contrast in the 304 å line for leading and trailing sunspots (C304 ­ ls/C304 ­ fs) depending on the ratio of the minimal angles in an umbra of two type sun spots (min ­ ls/min ­ fs). The correlation coefficient is k = ­0.427.

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COMPARISON OF THE PROPERTIES OF LEADING AND TRAILING SUNSPOTS 80 304­ls, C304­
fs

21

60 40 20 0 2000 Bmax, G 3000

Fig. 7. Contrast in the 304 å line above an umbra of leading (C304 ­ ls, circles) and trailing (C304 ­ fs, triangles) sunspots depend ing on the maximal magnetic field value in an umbra of these sunspots (Bmax).

4. CONCLUSIONS A comparison of the contrast in an umbra of lead ing/single and trailing sunspots for the cycle 24 growth phase and maximum confirmed the main conclusions drawn in (Zagainova, 2011). Using the data from instruments with a high spatial resolution that were mounted on the SDO spacecraft, we showed that the contrast in the He II 304 line above an umbra of lead ing and single sunspots is smaller than that above an umbra of trailing sunspots. Both for leading/single and trailing sunspots, the umbra contrast in He II 304 on average weakly depends on umbra area. On the other hand, the results of this work concerning the sunspot contrast in the He II 304 line slightly differ from the results achieved previously in (Zagainova, 2011). Thus, according to the SDO data for cycle 24, the average contrast values in 304 å above an umbra of leading/single and trailing sunspots were larger than the values obtained based on an analysis of the SOHO/EIT data for cycle 23 by factors of ~3.6 and ~2.7, respectively. The possible cause of this difference for two sunspot samples is related to the fact that our work analyzed sunspots registered during the cycle 24 growth phase and maximum, whereas sunspots during the cycle 23 decline phase were analyzed in (Zagain ova, 2011). It is known that cycle 24 is characterized by several specific features and even anomalies as com pared to previous solar cycles. In addition to relatively small Wolf numbers, cycle 24 differs from other cycles by smaller values of the large scale solar magnetic field, pronounced differences in the polar magnetic field polarity reversal, etc. (Akhmetov et al., 2014). Finally, cycle 24 followed a prolonged solar activity minimum, which could affect the cycle characteris tics. Certainly, one more possible cause of the detected differences in the contrast values in the He II 304 line obtained in (Zagainova, 2001) and in the present work may be that the data used in these works was obtained using different instruments with different spatial reso lutions, etc. An analysis of the magnetic properties of magneti cally connected leading­trailing sunspot pairs and single sunspots, obtained using the SDO/HMI vector
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field measurements with a high spatial resolution, indicated that the magnetic properties agree with sun spot magnetic properties obtained based on the mag netic field calculations in a potential approximation. It was established that the minimal angle between the field direction and the positive (antisunward) normal to the solar surface at a point where the field was mea sured is smaller in leading sunspots than in trailing ones ( min-ls < min- fs ) in ~84% of the considered leading­trailing sunspot pairs. We also confirmed the conclusion made in our previous work (Zagainova et al., 2014) that a positive correlation exists between min-ls and min- fs. At the same time, according to the SDO data, the average value of the minimal angle ( min ) in leading and trailing sunspots is smaller than the values obtained based on the field calculations in a potential approximation by factors of ~2.4 and ~1.65, respectively. According to the SDO data, it turned out that the average min-ls value is smaller than the aver age min- fs value by a factor of ~2.6. According to (Zagainova et al., 2014), this difference is a factor of ~1.8. Our analysis of sunspot magnetic properties based on the SDO data also showed that min is smaller in single sunspots with well developed umbra and penumbra than in leading sunspots. We confirmed our previous conclusion that min slightly increases for sunspots of both types with increasing umbra magnetic field and area. Although min-ls is on average relatively small, it reached several tens of degrees in some sunspots. Large min - fs values were in many trailing sunspots. In our previous work (Zagainova et al., 2014), we explained that such large min-ls and min - fs angles originate because the inclination of the umbra mag netic tube axis with respect to the normal to the solar surface is considerable. In this case the inclination of magnetic field lines with respect to the magnetic tube axis can be insignificant. Such a sunspot model was considered, e.g., in (Kuklin, 1985). Nevertheless, the origin of large min values in sunspots of two types is still a problem to be solved.
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ZAGAINOVA et al. heights from the photosphere to the source surface, Bull. Crimean Astrophys. Observatory, 2014, vol. 110, pp. 108­118. Antalova, A., The relation of the sunspot magnetic field and penumbra­umbra radius ratio, Astron. Inst. Czechosl. Bull., 1991, vol. 42, pp. 316­320. Baranov, A.V., Magnetic fields of small sunspots, Astron. Tsirk., 1974, vol. 847, p. 5. Beckers, J.M. and Schroter, E.H., The intensity, velocity and magnetic structure of a sunspot region. I: Observa tional technique; properties of magnetic knots, Sol. Phys., 1968, vol. 4, no. 2, pp. 142­164. Bray, R. and Loughed, R., Sunspots, London: Chapman and Hall, 1964. Bumba, V., Magnetic fields in small and young sunspots, Sol. Phys., 1967, vol. 1, nos. 3­4, pp. 371­376. Harvey, J., Solar magnetic fields--small scale, Publ. Astron. Soc. Pac., 1971, vol. 83, no. 495, pp. 539­549. Harvey, J. and Livingston, W., Magnetograph measure ments with temperature sensitive lines, Sol. Phys., 1969, vol. 10, no. 2, pp. 283­293. Houtgast, J. and Sluiters, A.Van., Statistical investigations concerning the magnetic fields of sunspots. I, Bull. Aston. Inst. Netherlands, 1948, vol. 10, pp. 325­333. Jin, C.L., Qu, Z.Q., Xu, C.L., Jhang, X.Y., and Sun, M.G., The relationships of sunspot magnetic field strength with sunspot area, umbral area and penumbra­umbra radius ratio, Astrophys. Space Sci., 2006, vol. 306, nos. 1­2, pp. 23­27. Kuklin, G.V., East­west asymmetry of the Wilson effect, Issled. Geomagn. Aeron. Fiz. Solntsa, 1985, vol. 73, p. 52. Lemen, J.R., Title, A.M., Akin, D.J., et al., The atmo spheric imaging assembly (AIA) on the solar dynamics observatory (SDO), Sol. Phys., 2012, vol. 275, nos. 1­2, pp. 17­40. Livingston, W. and Harvey, J., Observational evidence for quantization in photospheric magnetic flux, Sol. Phys., 1969, vol. 10, pp. 294­296. Livshits, M.A., Constancy of tau /10 830/ in plages and helium emission in a shortwave radiation field, Astron. Rep., 1975, vol. 52, p. 970­974. Maltby, P., Continuum observations and empirical models of the thermal structure of sunspots, Proc. NATO Advanced Research Workshop on the Theory of Sunspots, Cambridge, 1992, pp. 103­120. Nikol'skaya, K.I., He I excitation in chromospheric spi cules, Astron. Rep., 1966, vol. 43, p. 936. Obridko, V.N., Solnechnye pyatna (Sunspots), Moscow: Nauka, 1985. Pipin, V.V. and Kosovichev, A.G., The subsurface shear shaped solar dynamo, Astrophys. J., 2011, vol. 727, no. 2, pp. 1­4. Pozhalova, Zh.A., The study of selected helium lines in the solar spectrum, Astron. Rep., 1988, vol. 65, pp. 1037­ 1046. Ringnes, T.B. and Jensen, E., On the relation between mag netic fields of sunspots in the interval 1917­56, Astro phys. Norvegica, 1960, vol. 7, no. 4, pp. 99­121.
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In this work we for the first time compared the dependences of the maximal and average values of magnetic induction on the umbra area separately for leading and trailing sunspots. The main results of our analysis of these dependences can be formulated as follows. 1. Neither maximal nor average magnetic field values vanish when the area decreases to very small values. 2. In all cases the magnetic field in leading and sin gle sunspots is larger than in trailing ones. 3. Our approximation gives the threshold area val ues, when the Bmax (S ) and B (S ) curves start saturat ing: 25 and 10 MSH for leading and trailing sun spots, respectively. This corresponds to umbra radii of 5000 and 3100 km. The values obtained by us generally agree with the estimates made by Solov'ev and Kirichek (2014) and confirm the concept of a "shallow" sun spot developed by these researchers. At such an inter pretation, our data indicate that trailing sunspots are possibly shallower formations. A new our achievement is that we compared min with the contrast in the 304 å line above an umbra (304). It turned out that the 304 value increases for leading and trailing sunspots with decreasing min (for single sunspots, this dependence is almost impercepti ble since the sample of such sunspots is specific). This result possibly reflects an increase in the height of the plasma column above an umbra, which contributes to the emission registered in 304 å when min decreases. It is interesting that a negative correlation was also obtained for the dependence of the C304 -ls C304 - fs ratio on min-ls min- fs : when the latter ratio increases, the former ratio decreases. This reflects a different charac ter of the C304-ls ( min-ls ) and C304 - fs ( min- fs ). depen dences. The dependence of the sunspot contrast in 304 å on the magnetic field maximum in an umbra (B max ) is one more new result achieved in our work. It turned out that C304 -ls and C304 - fs remain on average unchanged when B max increases. ACKNOWLEDGMENTS We are grateful to the SOLIS, SDO/HMI, and SDO/AIA teams for the opportunity to use the data of these instruments without restriction. We thank G.V. Rudenko for the presented program that was used to select magnetically connected sunspot pairs. This work was partially supported by the Russian Foundation for Basic Research, project no. 14 02 003 8. REFERENCES
Akhtemov, Z.S., Andreeva, O.A., Rudenko, G.V., Stepa nian, N.N., and Fainstein, V.G., Temporal variations in the large scale magnetic field of the solar atmosphere at

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COMPARISON OF THE PROPERTIES OF LEADING AND TRAILING SUNSPOTS Rudenko, G.V., Extrapolation of the solar magnetic field within the potential field approximation from full disk magnetograms, Sol. Phys., 2001, vol. 198, no. 1, pp. 5­30. Sheeley, N.R., Measurements of solar magnetic fields, Astrophys. J.,, 1966, vol. 144, pp. 723­732. Sheeley, N.R., Observations of small scale solar magnetic fields, Sol. Phys., 1967, vol. 1, pp. 171­179. Sobotka, M., Semi empirical models of sunspots in various phases of evolution, Contrib. Astron. Obs. Skalnate Pleso, 1986, vol. 15, pp. 315­318. Solov'ev, A. and Kirichek, E., Basic properties of sunspots: Equilibrium, stability and long term eigen oscillations, Astrophys. Space Sci., 2014, vol. 352, no. 1, pp. 23­42. Stenflo, J.O., Magnetic field structure of the photospheric network, Sol. Phys., 1973, vol. 32, no. 1, pp. 41­63.

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Steshenko, N.V., Magnetic field of small sunspots and pores, Bull. Crimean Astrophys. Observatory, 1967, vol. 37, pp. 21­26. Vitinsky, Yu.I., Kopetsky, M., and Kuklin, G.V., Statistika pyatnoobrazovatel'noi deyatel'nosti Solntsa (Sunspot Formation Activity Statistics), Moscow: Nauka, 1986. Zagainova, Yu.S., He II 304 emission above sunspot umbrae, Astron. Rep., 2011, vol. 55, no. 2, pp. 159­162. Zagainova, Yu.S., Fainshtein, V.G., Rudenko, G.V., and Obridko, V.N., Comparative analysis of magnetic field properties in leading and trailing sunspots, Astron. Rep., 2014, no. 9, pp, 19­27.

Translated by Yu. Safronov

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2015