SOLAR ACTIVE LONGITUDES FROM KODAIKANAL WHITE-LIGHT DIGITIZED DATA

Carrington maps are the Mercator-projected synoptic charts of the spherical Sun in the Carrington reference frame (Harvey & Worden 1998). We have used the daily detected sunspot images (as shown in panel (b) of Figure 1) to construct the Carrington maps. A longitude band of 60° (−30° to +30° in heliographic coordinates) is selected for each image to construct these maps (following Sheeley et al. 2011). This involves stretching, ${B}_{\circ }$ angle correction (the B₀ angle defines the tilt of the solar north rotational axis toward the observer, and it can also be interpreted as the heliographic latitude of the observer or the center point of the solar disk), a shift in the Carrington grid, and additions. One Carrington map has been constructed considering a full 360° rotation of the Sun in 27.2753 days. In order to correct for the overlaps, we have used the "streak map" (Sheeley et al. 2011) for every individual Carrington map and divided the original maps with them. Data gaps occur as black longitude bands in these maps. The whole procedure is shown in the panels of Figure 2. Here we must emphasize the fact that in our Kodaikanal data we have some missing days (the complete list of missing days has been published with Paper I). In order to increase our confidence in the obtained results, we have not considered any Carrington map in our analysis that has one or more missing days.

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First we follow the "rectangular grid" method (following Berdyugina & Usoskin 2003), where a full Carrington map has been divided into 18 rectangular strips, each of 20° longitudinal width. We then compute a quantity "weight" (W) defined as

$$\begin{eqnarray}&&{W}_{i}=\displaystyle \frac{{S}_{i}}{{\displaystyle \sum }_{j=1}^{18}{S}_{j}}\end{eqnarray} \tag{ 1 }$$

where ${S}_{i}$ is the total sunspot area in the ith bin. We note the longitudes of the highest and the second highest active bins and calculate the separation between them (afterward referred to as "longitude separation"). We impose a minimum of a 20% peak ratio between the second highest and highest peak in order to avoid any sporadic detection. Two such representative Carrington maps from the Kodaikanal white-light data archive are shown in panels (a) and (b) of Figure 3 and their corresponding bar plots in panels (c) and (d). For the two representative cases shown in the figure, we notice that for CR number 1799 the longitude separation is 180°, whereas for CR 1980 the difference is 20°. We compute such longitude separations for each and every Carrington map for the whole hemisphere (referred to as "full disk" henceforth) and for individual northern and southern hemispheres. Histograms constructed using these separation values for each of the three mentioned cases are shown in different panels of Figure 4. In all three cases (panels (a)–(c) in Figure 4) we see that the maximum occurrence is for the 20° separation. Apart from that, we also notice peaks at ∼120° and at ∼200° for the full-disk case, whereas these peaks shift a little bit for the northern and southern hemisphere cases. Apart from these mentioned peaks, for the northern and southern hemispheres, we also see weak bumps in the histograms at ∼160° and ∼270°. In an earlier work using Greenwich data, Berdyugina & Usoskin (2003) had reported a phase difference of 0.5 (180° in terms of longitude) between the two most active longitude bins.

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The reason for this high number of occurrences of the longitude separations at ∼20° is probably related to the longitudinal extent of the sunspots compared to the chosen bin width of 20°. To be specific, there are often cases when the largest sunspot or sunspot groups are shared by two consecutive longitude bins. Now these occurrences are on a statistical basis, and thus increasing the bin size only shifts the highest peak to the chosen bin value (e.g., for a 40° bin size we find the maximum at 40°). Since this effect is related to the sunspot sizes, we use area-thresholding on the sunspots and note the longitude separations as described in the next section.

3.1.1. Area-thresholding and Active Longitude

We now use the area-thresholding method on the sunspots found in every Carrington map. One such illustrative example is shown in Figure 5. We again use CR 1980 for demonstration, as it has sunspots of various sizes. Different panels in Figure 5 show the Carrington map before and after doing different area-thresholdings.

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After performing the area-thresholding, we follow the same procedure as described in the earlier subsection to find the longitude separation between the two most active bins.

Following the extracted sunspot area distribution from the Kodaikanal white-light data (as shown in Figure 7 in Mandal et al. 2016), we chose two kinds of area-thresholding values. The first set of values correspond to sunspots with smaller sizes. These values range from 10 to 100 μHem as shown in panels (a)–(d) in Figure 6. From this figure, we immediately notice that the height of the histogram peak at ∼20° decreases progressively with the decrease in sunspot sizes. This is explained by the fact that, as sunspot sizes go down, the probability of a sunspot being shared by two longitude bins is also reduced. This results in a lower peak at ∼20°. Also, in every case, we notice prominent peaks in the histograms at ∼90° and ∼180° separations.

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We next investigate the longitude separation for the sunspots with larger sizes. In this case, the thresholding values range from 100 to 650 μHem. In Figures 6(e)–(h) we show the histograms of the longitude difference for these different thresholds. We see two noticeable differences in this case compared to the former one of small sunspot sizes. First, the peak height of the histograms ∼20° increases (relative to the other peak heights) as we move toward the larger sunspots, which is expected for the reason we discussed earlier. Along with that we notice that there is no peak near 90° as found earlier, but a peak at ∼180° is still present along with other new peaks. Here we should highlight the fact that with higher sunspot area-thresholding, the statistics become poor and the peaks become less significant statistically.

From our previous analysis we saw that the discreteness introduced by the longitude bins has a definitive effect on the calculated longitude separation. Therefore, we explore the other method, the "bolometric curve" method (Berdyugina & Usoskin 2003), which produces a smooth curve as described below.

In order to generate the smooth bolometric profile, we first invert the intensities of the Carrington maps to make the white background black with the sunspot as a bright feature. Next, the map is stretched to convert sine latitude into latitude (Figure 7(a)). We then generate a limb-darkening profile with the expression profile = 0.3 + 0.7μ in latitude and longitude, where μ is the cosine of the heliocentric angle (Figure 7(b)). We then create an intermediate map by shifting the limb-darkening profile, multiplying, and adding with the intensity-inverted Carrington map along every longitude (Figure 7(c)).

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In the end, this intermediate map is added along latitude for each longitude to generate a factor, called f. The f curve is converted to a bolometric magnitude curve (m) (Figure 7(d)) using the expression

$$\begin{eqnarray}&&m=-2.5\mathrm{log}\left[\displaystyle \frac{(1-f){{T}_{\mathrm{ph}}}^{4}+f\times {T}_{\mathrm{sp}}^{4}}{{{T}_{\mathrm{ph}}}^{4}}\right]\end{eqnarray} \tag{ 2 }$$

where the bright surface temperature T_ph = 5750 K and the sunspot temperature T_sp = 4000 K.

Figure 8 shows the two representative plots of this bolometric method. We chose the same two Carrington maps as shown in Figure 3 for easy comparison between the two methods. Now we see that for CR 1799 the bolometric curve basically traces the active bars (as shown in panel (c) of Figure 3); that is, the minimums of the bolometric curve represent the locations of maximum spot concentrations. We notice that the separation in this case (179°) equals the separation obtained previously (180°), but for CR 1980, the difference in this case is ∼150°, whereas the previously obtained value was 20°. This is because the bolometric curve takes into account close spot concentrations contradictory to the fixed longitudinal bins as defined in the rectangular grid method. In principle if we smooth out the peaks shown in panel (d) of Figure 3, we should then arrive at a curve similar to the bolometric one, but the amount of smoothing is subjective and may be different for different Carrington maps. However, in the case of the bolometric method, we must emphasize that the bolometric curve has been generated using a fixed prescription (Equation (2)) and thus is free from any subjectivity issues. Similar to the earlier method, in this case we have also calculated the longitude separation between the two most active spot concentrations for every Carrington map and plotted the histograms as shown in Figure 9. We can clearly see that for every case (full disk and northern and southern hemispheres) the histograms peak at ∼180°. In each case, the histogram distribution looks similar to a bell-shaped curve. We thus fit every distribution with a Gaussian function, as shown by the solid black lines in Figure 9. The centers of the fitted Gaussians for the three cases are at 178°, 180°, and 176°. This agrees well with the results found by Berdyugina & Usoskin (2003) and Usoskin et al. (2005). Apart from the well-structured peak at ∼180°, we also highlight the two other peaks (though considerably weaker) at ∼90° and ∼270° by two arrows in Figure 9. We remind the reader here that these peaks were also found with the rectangular grid method (Figure 4). These two peaks at ∼90° and ∼270° probably arise due to the dynamic nature of the active longitude locations. Apart from that, there could also have been some contributions from the different sunspot sizes on the active longitude separations (Ivanov 2007).

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Next we investigate the occurrences of the peaks found in Figure 9 for every individual solar cycle (cycles 16–23). The different panels of Figure 10 show the longitude separation histograms for the full period as well as for the individual cycles. We notice that the separation peaks at 180° for every cycle, and the height of this peak follows the cycle strengths; the strongest cycle, cycle 19 in this case, has the maximum number of occurrences at 180° and so on. Also we notice that the two other peaks (at ∼90° and ∼270°) are also present in most of the cycles, though with lesser strengths. Thus we confirm that these active longitudes persist for the whole 90 years of data analyzed in this paper.

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Previous studies have shown that the migration of active longitudes is governed by the solar differential rotation. According to Berdyugina & Usoskin (2003) and Usoskin et al. (2005), the migration pattern can be easily explained if one uses the differential rotation profile suitably. Thus, we move over to a dynamic reference frame defined by solar differential rotation as described below.

The rotation rate of the longitude of activity for the ith Carrington rotation can be expressed as

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{i}={{\rm{\Omega }}}_{0}-B{\sin }^{2}\langle \theta {\rangle }_{i},\end{eqnarray} \tag{ 3 }$$

where $\langle \theta {\rangle }_{i}$ denotes the sunspot area weighted latitude with ${{\rm{\Omega }}}_{0}$ and B being 1433 day⁻¹ and 3.40, respectively (following Usoskin et al. 2005). Using this rotation rate, the longitudinal position of active longitude in the Carrington frame for the ith rotation (${{\rm{\Lambda }}}_{i}$) can be calculated from the same at the ${N}_{0}^{\mathrm{th}}$ rotation (${{\rm{\Lambda }}}_{0}$) through the relation

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{i}={{\rm{\Lambda }}}_{0}+{T}_{C}\displaystyle \sum _{j={N}_{0}+1}^{i}({{\rm{\Omega }}}_{C}-{{\rm{\Omega }}}_{j})\end{eqnarray} \tag{ 4 }$$

with ${{\rm{\Omega }}}_{C}=\tfrac{360\,^\circ }{{T}_{C}}$ and T_C = 25.38 days. From the longitudes, the phases are calculated as $\phi =\tfrac{{\rm{\Lambda }}}{360\,^\circ }$. These phases are made continuous (Figure 13) by minimizing $| {\phi }_{i+1}+N-{\phi }_{i}|$ with N spanning positive and negative integers. Here ${\phi }_{i+1}$ is replaced by ${\phi }_{i+1}+N$ for the N that gives the minimum absolute difference mentioned. We calculate the missing phases to fill the gaps occurring due to missing Carrington maps by interpolating over $\langle \theta {\rangle }_{i}$. From Figure 13, we see quite a few distinct features. Immediately we recover the 11 year period of the solar cycle. Also, we see that for an individual cycle the curve first steepens and then dips toward the later half of the cycle. This is explained by considering the fact that, in the beginning of a solar cycle, sunspots appear at higher latitudes where the rotation rate is quite different from the Carrington rotation rate (which is basically the rotation rate at ≈15° latitude). As the cycle progress, sunspots move down toward the equator, and the curves then tend to flatten.

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Though we call it a "theoretical curve," we want to remind the reader that the area weighted latitude information is extracted for the generated Carrington maps, and thus it will be appropriate to call it a "data-driven theoretical curve."

We use this "theoretical curve" to demonstrate the association of the migration of the active longitudes with the solar differential rotation. In the top panel of Figure 14 we plot the full-disk bolometric profiles (as obtained previously) and stack them over the Carrington rotation for the period 1954–1965 (which corresponds to the 19th cycle). The two dark curves are the manifestation of the two dips corresponding to two active longitudes. To highlight this trend more, we use sigma-thresholding (i.e., mean+σ) on the original image and plotted it in the bottom panel of Figure 14. We then generate the theoretical curves, corresponding to two active longitudes as obtained from each bolometric profile and overplotted them. A good match between the theoretical curve and the obtained active longitude positions confirms the fact that the migration of these active longitudes is indeed dictated by the solar differential rotation. Here we again highlight the fact that the missing phases have been filled using the interpolation method. Since the current phase has contributions from the previous phases (see Equation (4)), we could not match every detail of the observed pattern for all of the cycles. The small discrepancy between the theoretical curve and the data could also be due to the fixed values of the differential rotation parameters used in this study, which may not be suitable for all cycles.

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