Kelvin H. Moresi^{1}, Sung-Hong Park^{2}, Aimee A. Norton^{2}

^{1} Palo Alto High School, 50 Embarcadero Rd, Palo Alto, CA 94301, USA

^{2} W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305-4085, USA

Solar active region (ARs) contain a group of sunspots distributed on the photospheric surface at different degrees of spacing. Compactness is one geometric property of a sunspot group which is independent of scale and orientation. Among the three components of the McIntosh sunspot classification, there is one called ‘c’ that categorizes sunspots into four subgroups based on the distribution of spots in the interior of the group. Meanwhile, in the Mount Wilson (or Hale) classification, ARs are assigned to β if they have two sunspots of opposite polarity with a simple and distinct division, while the δ class exhibits a sunspot group that consists of umbrae of opposite magnetic polarities within a common penumbra^{[1]}. δ-spots are known to produce most of large flares^{[2]} and expected to be more compact than β-spots. Here we develop a simple but practical way of measuring the compactness of ARs and apply the developed method to a set of β- and δ-spots to examine their difference in compactness and to find any relations of compactness with AR magnetic properties.

For a two-dimensional shape under consideration, its compactness can be quantified as the ratio of the area of the shape compared to the area of the smallest bounding circle in which all points of the shape lie (refer to Figure 1). This ratio is in the range of 0 (lowest compactness) to 1 (highest compactness). Our measure uses a bounding circle, but alternatively one can use any shape of their choosing, such as a bounding box or convex hull. The Spaceweather HMI Active Region Patch (SHARP) continuum intensity images mapped into cylindrical equal area projection (hmi.sharp_cea_720s) were used for the calculation of compactness. One can extend this compactness analysis with historical sunspot data obtained by intensity images and even hand-drawn sketches. We also examine the compactness in relation to two SHARP magnetic parameters: (1) total unsigned magnetic flux of a given AR and (2) R-value which is the sum of the weighted, unsigned flux along the magnetic polarity inversion lines^{[3]}. Both magnetic parameters are strong indicators of flaring, making their correlation with the compactness relevant.

Figure 1. Left: a grid representation of a sunspot group consisting of umbral (black) and penumbral (gray) pixels with all distances between pairs of the minimum and maximum (x,y) coordinates of the sunspot pixels (red lines). The smallest bounding circle (green) is defined as having a diameter of the largest distance (d, blue line) between the sunspot pixels. Compactness is determined as the ratio of the total area of the sunspot pixels divided by the area of the bounding circle. As this was a trial analysis to explore compactness, there may be errors associated with the area determination of a circle in the cylindrical equal area projection. This calculation needs to be more carefully implemented in future work. Right: HMI continuum intensity images of (top) a δ-spot (NOAA 11263, SHARP 754) observed at 20:36 TAI on 2011-08-01 with compactness of 0.3, and (bottom) a β-spot (NOAA 11450, SHARP 1528) at 10:36 TAI on 2012-03-31 with compactness of 0.1.

We first compared the compactness with the total unsigned flux and R-value of the AR NOAA 12916 over its passage within -55 and 55 degrees in longitude at 2-hour cadence (see Figure 2). The compactness varies in a similar manner to the flux and R-value. We also analyzed 17 β- and 15 δ-spots at the time of their maximum umbral flux. Figure 3 shows scatter plots of the compactness versus the total unsigned flux (left) and R-value (right). We find that the δ-spots are more compact than the β-spots. The average compactness of the δ-spots is 0.17±0.08, while it is 0.08±0.04 for the β-spots. The compactness is found to be positively correlated with the total unsigned flux and R-value, although there is significant scatter. We speculate that the different formation mechanisms and categories of δ-spots^{[4]} may cause this scatter. We therefore classified δ-spots as quadrupoles, bipoles or collisions. A collision of flux systems means the δ was formed by bipoles emerging at distinctly different times. NOAA 12192, the largest flux region of Cycle 24, was not observed during its emergence, so we cannot know if it is a colliding system, but it is more bipolar than quadrupolar. There is a trend that δ-spots with quadrupoles are less compact than bipolar or collisional δ-spots. A comprehensive study of compactness with a large set of ARs may help enlighten us to any characteristic evolution patterns and formation processes of sunspot groups in different Hale classes.

Figure 2. Top left: continuum intensity images of NOAA 12916 (SHARP 7890) with a bounding circle overlaid. Clockwise from top left: time series of the compactness, the total unsigned flux and the R-value with the vertical line corresponding to the time of the intensity image.

Figure 3. Compactness of 32 ARs are plotted as a function of the total unsigned flux (left) and the R-value (right). The symbols represent β-spots (red), quadrupolar δ-spots (blue), and δ-spots that are bipolar or formed by collisions (orange).

**References**

[1] Kunzel, H. 1965, *AN*, **288**, 177

[2] Guo, J., Lin, J., & Deng, Y. 2014, *MNRAS*, **441**, 2208, 497

[3] Schrijver, K., 2007, *ApJ*, **L117**

[4] Toriumi, S., Schrijver, C. J., Harra, L. K., Hudson, H., & Nagashima, K. 2017, *ApJ*, ** 834**, 56