LATITUDE AND MAGNETIC FLUX DEPENDENCE OF THE TILT ANGLE OF BIPOLAR REGIONS LIRONG TIAN1 , YANG LIU2 and HUANING WANG1 1 National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, China (e-mail:
[email protected]) 2 W. W. Hansen Experimental Physics Lab., Stanford University, Stanford, CA 94305-4085, U.S.A.
(Received 4 February 2002; accepted 23 April 2003)
Abstract. Magnetogram data of 517 bipolar active regions are analyzed to study latitude, magnetic flux, polarity separation dependence of tilt angle of the active regions with well-defined bipolar magnetic configurations. The data were obtained at Huairou Solar Observing Station in Beijing during 1988 to October 2001. By statistical analysis, it is found: (1) The tilt angle (ψ) is a function of the latitude (θ). Our observed result, sin ψ = 0.5 sin θ, is in good agreement with that obtained by Wang and Sheeley (1991). (2) The tilt angle is a function of the magnetic flux. The tilt angle increases (decreases) with flux increasing when the flux is smaller (larger) than 5 × 1021 Mx. (3) The tilt angle is a function of the magnetic polarity separation. The tilt angle increases (decreases) with the separation increasing when the separation is smaller (larger) than 8 × 109 cm. (4) The magnetic flux (φ in 1020 Mx) is correlated to the magnetic polarity separation (d in Mm), following φ20 ∼ d 1.15 . The result is close to the observed result of Wang and Sheeley (1989), φ20 ∼ d 1.3 . (5) The tilt fluctuations are independent of the latitude, but depend slightly on the separation, which is similar to the result obtained by Fisher, Fan, and Howard (1995). (6) The distribution function of the ratio of net magnetic flux to total magnetic flux is almost centered around zero net flux. The imbalance of magnetic flux is lower than 10% for 47% of our samples; 31% of active regions are in imbalance of the magnetic flux between 10% and 20%.
1. Introduction Magnetic fields are now believed to be generated by a dynamo operating in the overshoot region beneath the convection zone. Strands of fields that enter the convection zone are magnetically buoyant (Parker, 1979) and rise to the surface in the form of -shaped flux loops, forming leading and following spots. Hale et al. (1919) showed that a line joining the leading and following spots of a bipolar active region makes an angle, on average, with the local parallel of latitude, which is called tilt. Tilts are observed to increase with increasing latitude, which is called Joy’s law (Zirin, 1988). Joy’s law plays a very important role in several classical solar cycle dynamo models (Babcock, 1961; Leighton, 1969; Durney, 1997). Magnetic flux and separation of the leading and following polarities are parameters of region importance. Several investigators (Schmidt, 1968; Wang and Sheeley, 1991) believed that the Coriolis force could be responsible for the tilt of bipolar active regions. The observed magnetic polarity separation dependence of the tilt Solar Physics 215: 281–293, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.
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angle suggests that the dominant tilting occurs from the Coriolis force (reviewed by Fisher et al., 2000). Fan, Fisher, and McClymont (1994) presented a numerical model in which the buoyancy-driven rise of a flux tube is balanced by an aerodynamic drag force, leading to a rising, diverging loop geometry. The Coriolis force then acts on the diverging velocity field to twist this loop into a backward ‘S’-shaped geometry (in the northern hemisphere) when viewed from above. If the Coriolis force is then balanced with a magnetic tension force opposite the twisting motion, one can show −5/4 that a scaling law of the tilt angle ψ ∼ B0 φ 1/4 sin θ, where B0 is initial field strength of rising loop, φ is magnetic flux in the rising loop, and θ is latitude. This simple analysis not only yields Joy’s law, but also predicts a new relationship between tilt and magnetic flux that can be tested with observations. Fisher, Fan, and Howard (1995) tested this prediction by examining the tilt angles of a large number (24701) of spot groups observed at Mount Wilson over many decades from 1917–1985. They used separation distance (d) of spot groups as a substitute for magnetic flux (φ) and found that tilt angles were indeed an increasing function of both θ and d, i.e., ψ ∼ d 1/4 sin θ. However, white-light spot group measurements contain no magnetic information at Mount Wilson during 1917–1985. What is the observed scaling law called by Fan, Fisher, and McClymont (1994) for magnetogram data of bipolar active regions? In this paper, we will focus on this question. In Section 2, data and parameters used in this paper are introduced. In Section 3, we will show the quantitative relationship between the tilt angle, the latitude, the magnetic flux and polarity separation. In Section 4, we give the conclusions and discussion.
2. Data The magnetic field data come from a data set obtained with Solar Magnetic Field Telescope at Huairou Solar Observatory Station (HSOS). The field of view is about 5.23 × 3.63 (512 × 512 pixels of the CCD). The noise level is lower than 10 G in the line-of-sight magnetograms and the spatial resolution is 2 × 2 . Three hundred fifteen bipolar active regions (BARs) are chosen in 22nd cycle (some of which have been used by Tian et al., 1999, 2001), and 202 BARs from 1997 to October 2001. These BARs were formed by two main bigger spots with simpler magnetic configuration (i.e., the magnetic fields can be simply divided into areas with N and S polarities with little, opposite weak fields). They are isolated from others and observed in good seeing and weather. When the active regions are almost mature at the center of the disk (from −15◦ to 15◦ ), they are selected as samples by studying magnetograms. The projection effects for high-latitude active regions were removed according to the formulae given by Gary and Hagyard (1990). In this study, four parameters are used: the magnetic flux, magnetic polarity separation, tilt angle and emerging latitude of BARs. The magnetic flux φ is defined
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as (|φp |+|φf |)/2, where φp and φf are the magnetic flux measured from proceeding and following regions, respectively, when the line-of-sight magnetic field is more than 20 G. In calculation of the tilt angle and the separation, N and S polarity position could be approximately indicated by the magnetic flux-weighted center of each magnetic polarity in the line-of-sight magnetogram. The tilt angle (ϕ) of an active region is defined as the angle between the line joining opposite polarities and the solar equator. The tangent of the tilt angle (tan ϕ) is given by δy/δx (Wang and Sheeley, 1989; Tian et al., 1999), where δy and δx are the distances between the leading polarity and the following polarity along the Y - and X-axis, respectively, in Cartesian coordinates in the heliographic plane. Here, the tilt angle in the northern/southern hemisphere is defined as positive/negative when the active region is tilted clockwise/counter-clockwise from the E–W direction. Here, we believe that the measurement error of the tilt angle mainly comes from the variable morphology of individual active region, rather than from errors in determining the position of each polarity in an active region.
3. Analysis The theoretical model (Fan, Fisher, and Howard, 1994) predicted that the tilt angle −5/4 (ψ) is increased with increasing flux (φ) and latitude (θ) as ψ ∼ B0 φ 1/4 sin θ. Fisher, Fan, and Howard (1995) tested this prediction by examining the tilt angles of a large number (24701) of spot groups. They used separation distance (d) of spot groups as a substitute for magnetic flux (φ) because white-light spot group measurements contain no magnetic information at Mount Wilson during 1917–1985. It was found that tilt angles are indeed an increasing function of both θ and d, i.e., ψ ∼ d 1/4 sin θ. How do the tilt angles depend on magnetic flux, the separation, and latitude in magnetic field observations? We will test this with data sets from HSOS in the 22nd and 23rd cycles. 3.1. T HE TILT- ANGLE DEPENDENCE ON THE LATITUDE In order to study the relationship of the tilt-angle (ψ) dependence on the latitude (θ), we fix the magnetic flux in the intervals of (a) 1 < φ21 ≤ 8, (b) 8 < φ21 ≤ 16, (c) 16 < φ21 ≤ 24, (d) φ21 > 24. Here, φ21 denotes magnetic flux as unit of 1021 Mx. In these intervals, there are 163, 136, 66, and 137 active regions, respectively. The distributions of tilt angles with latitude are shown in the left panel of Figure 1. From the left panel, it is found that those samples are distributed almost symmetrically in the two hemispheres. The correlation coefficients denoted by ‘C’ are generally larger than the values (0.208, 0.228, 0.325, and 0.218) expected with confidence level of 99%. In Figure 1(d) of the left panel, the correlation coefficient is larger than 0.164 with confidence level of 95%. Then the relationship between ‘sin θ’ and ‘sin ψ’ is studied and shown in the right panel of Figure 1. It can be
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Figure 1. The relationship between the tilt angles (ψ) and the latitude (θ) of bipolar active regions. Magnetic flux φ21 (1021 Mx) of active regions is fixed in the intervals of (a) 1 < φ21 ≤ 8, (b) 8 < φ21 ≤ 16, (c) 16 < φ21 ≤ 24, (d) φ21 > 24. There are 163, 136, 66, and 137 active regions, respectively. ‘C’ denotes the correlation coefficient and ‘K’ denotes the slope of least-squares linear fit. The solid lines are the results of the linear fit to the data.
found that the correlation coefficients are larger than that between ‘θ’ and ‘ψ’ and larger than the values expected with 99% confidence level. Therefore, we find that a linear relation roughly exists between the two parameters, especially between ‘sin θ’ and ‘sin ψ’. However, the relationships between ‘θ’ and ‘ψ’ or ‘sin θ’ and ‘sin ψ’ are different in the four flux bins that are indicated by different slopes ‘K’ of the linear fits. From this, we infer that the tilt angles are also related to the magnetic flux. The correlation coefficients in Figure 1 indicate that linear relations exist between the two parameters. Thus, we have a function by a least-squares linear fit as sin ψ ≈ K(φ) sin θ,
(1)
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where K(φ) is a quantity dependent on the magnetic flux. We combine these data from four bins to study comparatively the relation between the two parameters with others, which are shown in the left panel of Figure 2. For all active regions, K(φ) = 0.5 is obtained from Figure 2(b), which is in good agreement with the result obtained by Wang and Sheeley (1991). From this result, we can find the same sign between the tilt and latitude. Then, the tilt angles from both hemispheres can be combined: the tilt angles in the northern/southern hemisphere are positive when the active regions are tilted clockwise/counter-clockwise from the E–W direction; and the latitude is replaced by the absolute value of the latitude. 3.2. T HE TILT- ANGLE DEPENDENCE ON THE MAGNETIC FLUX Since the slopes ‘K’ of the linear-fit function are affected by fluxes, shown in Equation (1), we study the relationship between the slopes and mean fluxes in the bins in the right panel of Figure 2. From the two figures, we know that the different relationships K(φ) possibly exist between the tilt angles and the fluxes. When magnetic fluxes are roughly smaller (larger) than 2 × 1022 Mx, the tilt angles increase (decrease) with the flux increasing. As we have calculated magnetic fluxes of active regions, the tilt angle dependence on the magnetic fluxes can be studied statistically, as shown in Figure 3. After a least-squares polynomial fit in Figure 3(a), we find that the dependence on the magnetic flux seems different when the fluxes are larger or smaller than 5 × 1021 Mx. In order to minimize the latitude effect on the result, we show sin ψ/ sin θ dependence on the flux in Figure 3(b). Similar results are obtained when the flux is smaller and larger than 5 × 1021 Mx. There are 209 and 298 active regions in each case, respectively. After the magnetic flux values are averaged in 0.1 width bins of the flux as units of 1022 Mx, shown as ‘diamonds’ in Figures 3(a) and 3(b), we show logarithmic relationship between the average tilt angle and the mean flux in Figures 3(c) and 3(d). The solid lines denote least-squares polynomial fits to the logarithms of the mean values. It is obvious that the tilt angle is increased (decreased) with increasing flux when the log φ20 is roughly smaller (larger) than 1.7, corresponding to 5.0 × 1021 Mx flux. We obtained the following approximate functions that are φ20 < 50 ±K (2) sin ψ ∼ φ20 sin θ φ20 > 50. 3.3. T HE RELATIONSHIP BETWEEN THE MAGNETIC FLUX AND THE SEPARATION
We study statistically the relationship between the magnetic flux and the polarity separation using data of 517 BARs in 22nd and 23rd cycles. In Figure 4, it can be found that the correlation coefficients ‘Co’ are much larger than 0.118 with
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Figure 2. The relationship between the tilt angles and the latitude of bipolar active regions (left panel). The relationship between the mean fluxes in the four bins and slopes of linear-fit between the two parameters (right panels).
a confidence level of 99% for 500 samples. The correlation coefficients between ‘log φ20 ’ and ‘log d’ are much better than between ‘φ’ and ‘d’. Thus, we obtained from Figure 4(c) that the relationship between the magnetic flux and the separation approximately follows φ20 ∼ d 1.15 ,
(3)
where the unit of the magnetic flux is 1020 Mx, the unit of the separation is ‘Mm’. This relationship is close to the observed result φ = 4 × 1020 d 1.3 (‘d’ shown by degree), obtained by Wang and Sheeley (1989). We know that the result in Figure 4(c) is obtained to minimize error in the flux and the result in Figure 4(d) is obtained to minimize error in the separation. Magnetic flux and polarity separation are all parameters of region importance. However, what is the real relationship between them?
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Figure 3. The relationship between the tilt angle (ψ) and the magnetic flux (φ). The points denote values of 517 active regions, and the diamonds denote the mean value in 0.1 bin of the flux as unit of 1022 Mx in (a) and (b). The solid lines denote least-squares polynomial fits to the data. φ20 denotes the magnetic flux in unit of 1020 Mx. In (c) and (d), the stars denote the logarithms of the mean values in 0.1 width bins and solid lines denote the least-squares polynomial fits to the data. The error bars correspond to 1σ .
3.4. T HE TILT- ANGLE DEPENDENCE ON THE MAGNETIC POLARITY SEPARATION
In this subsection, we study how the tilts depend on the magnetic polarity separation ‘d’. In Figures 5(a) and 5(b), the ‘points’ denote values of 517 active regions; the ‘diamonds’ denote mean values in the separation of 5.0 Mm. The lines are leastsquares polynomial fits to 517 ARs. Figures 5(c) and 5(d) show the logarithms of the mean values and the least-squares polynomial fit to them. The error bars are 1σ . From the results shown in Figures 5(d), we find a different tilt dependence on the separation when the log(d) is smaller and larger than 1.9, which corresponds to the separation of 80 Mm. Therefore, we obtain that the tilt angles are increased
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Figure 4. The statistical relationship between the flux (φ) and the polarity separation (d). The solid lines denote the least-squares linear fits to the data.
(decreased) with increasing separation when the separations are smaller (larger) than 8 × 109 cm. They are d < 80 Mm (4) sin ψ ∼ d ±K sin θ d > 80 Mm, where the values of ‘K’ are approximately the same on either side of 80 Mm, as shown in Figure 5(d). However, more data are needed in which the separation of two polarities is larger than 140 Mm. 3.5. F LUCTUATION OF THE TILT ANGLE OF BIPOLAR ACTIVE REGIONS From Figures 1, 2, 3, and 5, it is noted that there is a huge scatter of individual active region tilt to the average behavior of least-squares fit lines. Fisher, Fan, and Howard (1995) found the same phenomenon and that the degree of tilt scatter was independent of latitude, but depends on the separation, roughly as d −3/4 . After
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Figure 5. The relationship between the tilt angles (ψ) and the polarity separation. The points denote values of 517 active regions, the diamonds denote the mean value in the separation of 5.0 Mm in (a) and (b). The solid lines denote least-squares polynomial fits to the data. In (c) and (d), the stars denote the logarithms of mean values and solid lines denote the least-squares polynomial fits to the data. The error bars correspond to 1σ . ’Co’ denotes correlation coefficient; ‘K’ denotes slopes of linear fit.
investigating the measurement error in the tilt angle, they found that measurement errors were significantly smaller than the levels of scatter shown by the spot group data set, and concluded that the tilt scatter has a physical origin. Longcope and Fisher (1996) performed a detailed analysis and found that the ‘d’ dependence of the tilt fluctuations from the theoretical models could be closely matched to the observed tilt scatter found by Fisher, Fan, and Howard (1995). Here, we show a similar behavior of the tilt angle in Figure 6. The tilt fluctuation is calculated as the absolute value of the difference of an individual active region tilt from the leastsquares fit. We found that the tilt scatter was indeed independent of the latitude, but a bit dependent on the separation, as shown in Figures 6(b) and 6(d). However, it seems not to depend on flux, as shown in Figure 6(c).
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Figure 6. The relationship of the tilt fluctuations, |ψ| = |ψi − ψfit |, dependence on latitude, flux and separation. Points denote 517 active regions, lines are the least-squares fit.
3.6. I MBALANCE OF THE MEASURED MAGNETIC FLUX OF THESE ACTIVE REGIONS
The imbalance of measured magnetic flux in the photosphere would be an extremely valuable piece of information about the difference between the leading and following polarities of active regions, and might reveal information about the nature of the morphological differences between leading and following polarities, or possibly indicate calibration problems with the magnetic measurements. It has been observed in both large and small active regions. Livi, Wang, and Martin (1985) found that the flux imbalance ranges up to 30% for ephemeral active regions in areas of the quiet Sun. In early observations of sunspot groups, comparably large imbalances were seen by Sheeley (1966) and Stenflo (1968). The flux imbalance in
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Figure 7. The histogram of the distribution function of the ratio of net magnetic flux to total magnetic flux. Magnetic flux imbalance is shown.
56 δ sunspots seems much more severe, up to 70% (Shi and Wang, 1994). Choudhary, Venkatakrishnan, and Gosain (2002) studied the magnetic flux imbalance of 137 active regions, including both those of high- and low-flare production as well as young and small dipoles. They found that the maximum and median values of the flux imbalance are about 62% and 9.5%. More recently, Tian et al. (2002) found that the imbalance of magnetic flux were even much more severe, up to more than 1021 Mx when they statistically investigated the 25 most severe super active regions with major flares and/or major solar storms in the 22nd and 23rd cycles. Here are 517 samples with simpler magnetic configuration. We calculated the preceding and following flux when those active regions were located near central meridian and where Bz > 20 gauss in each magnetogram. The imbalance of magnetic flux is studied by showing the distribution function of the ratio of net magnetic flux to total magnetic flux, i.e., (|φp | − |φf |)/(|φp | + |φf |) in Figure 6. From this histogram, we find that this distribution function is centered almost around zero net flux. About 47% of active regions are in 10% flux imbalance and 31% of active regions are in flux imbalance between 10% and 20%. Only 22% of active regions have a flux imbalance larger than 20%. From this study, we infer that active regions with simpler magnetic configuration tend to smaller imbalance of magnetic flux of the proceeding and following polarity. Otherwise, the more complex the magnetic configuration is, the larger the imbalance of the magnetic flux (Tian, Lin, and Wang, 2002; Choudhary, Venkatakrishnan, and Gosain, 2002).
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4. Conclusions and Discussion From this statistical study, we have obtained the following results: (1) The tilt angle (ψ) is a function of the latitude (θ). Our observed result, sin ψ = 0.5 sin θ, is in good agreement with that obtained by Wang and Sheeley (1991). (2) The tilt angle is a function of the magnetic flux. The tilt angle is increased (decreased) with flux increasing when the flux is smaller (larger) than 5 × 1021 Mx. (3) The tilt angle is a function of the magnetic polarity separation. The tilt angle is increased (decreased) with separation increasing when the separation is smaller (larger) than 8 × 109 cm. (4) The magnetic flux (φ in ‘1020 Mx’) is correlated to the magnetic polarity separation (d in ‘Mm’), following φ20 ∼ d 1.15 . The result is close to the observed result of Wang and Sheeley (1989). (5) The tilt fluctuations are independent of the latitude, but depend on the separation, which is similar to the result obtained by Fisher, Fan, and Howard (1995). (6) The distribution function of (|φp | − |φf |)/(|φp | + |φf |) is centered almost around zero net flux. The imbalance of magnetic flux is lower than 10% for 47% of our 517 samples with simpler magnetic configuration; 31% of active regions are in imbalance of magnetic flux between 10% and 20%; only 22% of active regions have an imbalance of magnetic flux larger than 20%. We know that in sunspot penumbras the field is tilted away from the vertical. Since the observations by HSOS include sunspot fields, there will be an artifact in flux calculations. On the other hand, the threshold effect is also a source of artifacts in flux determination. The projection effect (solar rotation) may produce some bias, but not significantly, because we only chose active regions located in the central meridian. Because of these additional uncertainties of flux calculation, it seems difficult to get fine results of the magnetic flux.
Acknowledgements The authors thank the anonymous referee very much for his/her valuable and helpful suggestions and comments to improve this work. This research is supported by NSFC Grant 19973009 and NKBRSF G20000784 in China and Y. Liu was supported by NASA NAG5-3077. They are grateful to the HSOS team for good observations of vector magnetic fields.
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