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Sunspot area in relation to the newly revised sunspot number.

ABSTRACT

On the basis of yearly means of the total corrected sunspot area (SSA) and the newly revised sunspot number (SSN) for the interval 1875-2015, it is shown that the total SSA is simply related to SSN as SSA = (9.1 [+ or -] 2.1) SSN, this being the [+ or -] 1 standard deviation (sd) prediction interval about the mean. Instead, based on the least-squares linear fit of SSA to SSN, one determines the inferred regression equation to be y = -101.8 + 11.3x, where y is SSA and x is SSN, the inferred regression having a coefficient of correlation r = 0.986 and standard error of estimate se = 126 millionths of a solar hemisphere. For all sunspot cycles (SCs) SC12-SC24, the minimum yearly ratio (= SSA/SSN) has always occurred within [+ or -]1 year of SSN minimum, while the maximum yearly ratio often has occurred post-SSN maximum during the declining portion of the SC. SC24, the current SC, had minimum values in SSN (= 4.2), SSA (= 22.8), and ratio (= 5.4) in 2008 and maximum values (= 113.3, 1,252.2, and 11.1, respectively) in 2014, presuming that a later, larger ratio maximum does not occur.

INTRODUCTION

Two important measures of the long-term behavior of solar activity are (1) Wolf s relative SSN (also called Zurich SSN but now called the International SSN) and (2) the total corrected area of sunspots (i.e., the measured area of sunspots corrected for foreshortening due to the position of the spots on the solar disk). Regarding the first measure of solar activity, Johann Rudolf Wolf--a professional Swiss astronomer from the Bern Observatory and later at Zurich--introduced his familiar relative SSN R in 1849, formulating it as R = k (10g + f), where k is a correction factor used to compensate for observing conditions, the size of the telescope, the method of observation, and the individual observer; g is the number of sunspot groups; and/is the number of individual sunspots discernible on the solar disk. In 1882, Wolfs method for computing SSN was changed to compensate for the inclusion of smaller sunspots, which Wolf originally had neglected, changing the k factor from 1 to 0.6. The Swiss Federal Observatory at Zurich continued to provide SSN until 1981, when the task of determining SSN became the responsibility of the Royal Observatory of Belgium's Solar Influences Data Analysis Center (SIDC). Effective July 1, 2015, the International SSN was again revised, primarily to reincorporate a k factor equal to 1, because a number of inhomogeneities were found to exist in the SSN series. This caused the amplitudes of the individual SCs to increase (Clette et al. 2015, Hathaway 2015, Wilson 2015).

According to Waldmeier (1961), Wolf originally wanted to use SSA as his preferred measure for solar activity. However, because the technique of photography was still in its early infancy (Hirsch 2000), and limited instrumental equipment did not allow him to draw sunspots to accurate scale, Wolf posited his rather arbitrary relative SSN formula. Later, when reliable SSA determination became possible, a comparison of the yearly means of the relative SSN to SSA revealed that the two measures of solar activity generally behaved in a proportional way.

Regarding the second measure of solar activity, Richard Christopher Carrington introduced the technique of photography for measuring SSA in 1874 at the Royal Greenwich Observatory (RGO) in England. Furthermore, he established a systematic method for determining SSA (Kiepenheuer 1953). In particular, the position and area of sunspots were measured directly from photographic plates by means of reticules divided into small squares, all measured relative to Sun center. The photographic plates were obtained using telescopes primarily at Greenwich, England; Cape Town, South Africa; and Kodaikanal, India. The RGO measurements of the position and area of sunspots continued for more than 100 years, from May 1874 to December 1976. In order to extend the record beyond 1976, the Solar Observing Optical Network (SOON) of the United States Air Force and the National Oceanic and Atmospheric Administration has been used to provide the necessary data. The current method of a real determination, however, is one based on visual determination from drawings rather than directly from photographic plates (Baranyi et al. 2001, Wilson and Hathaway 2005, 2006).

A comparison of same-day visually determined SSAs from SOON against photographically determined SSAs from the Rome Observatory (Wilson and Hathaway 2005) reveals that the visually determined SSAs appear to be underestimated. Likewise, based on comparisons against Mt. Wilson photographic plate data (Howard et al. 1984), the visually determined SSAs are also found to be underestimated. The amount of underestimate appears to be about 40% (see description given in http://solarscience.msfc.nasa.gov/greenwch.shtml). Therefore, the record of SSAs maintained at the National Aeronautics and Space Administration's (NASA's) Marshall Space Flight Center (MSFC) include a correction factor of 1.4 after 1976.

The proportional nature between SSA and SSN often has been described by the relationship A = 16.7R, where A represents SSA and R the relative SSN, when using yearly means (e.g., Kiepenheuer 1953, de Jager 1959, Bray and Loughhead 1964, Tandberg-Hanssen 1967, Zirin 1988). Unfortunately, this simple formula belies the true variation between the two parameters, especially over the solar cycle, from SC minimum to the next SC minimum. Likewise, because SSN recently has been revised, the old formulation clearly is in error. In this paper, examination of the relationship between SSA and SSN is revisited based on the use of yearly means.

METHODS AND MATERIALS

Annual values of SSA and SSN were taken from two sources: (1) SSA from MSFC (http://solarscience.msfc.nasa.gov/greenwch.shtml) and (2) SSN from the Royal Observatory of Belgium's SIDC (http://sidc.oma.be/silso/DATA/SN_y_tot_V2.0.txt). The yearly and 10-year moving average (10-yma) variations of the two parameters and their ratio (= SSA/SSN) are displayed for the interval 1875-2015. Also, the yearly values of the ratio relative to the epochs of SC minimum (Emin) and maximum (Emax) are examined. The 10-yma values are given simply to indicate the inferred trend in the parametric yearly values. Yearly values of SSA are determined from monthly values of the total corrected SSA in millionths of a solar hemisphere.

RESULTS AND DISCUSSION

Figure 1 displays the variation of (a) SSN, (b) SSA, and (c) ratio for the interval 1875-2015. The thin, jagged lines represent the yearly means, and the thick, smoothed lines represent the 10-yma values. The largest yearly SSN (= 269.3) and SSA (= 3,048.5) occur in 1957, associated with SCI9. The largest yearly ratio (= 14.1) occurs in 1982, associated with the maximum SSA (= 2,283.3) in SC21, while the smallest yearly ratio (= 3.1) occurs in 1913, associated with the minimum SSN and SSA in SCI5.

For the interval 1875-1976, the mean and sd of the ratio measures 8.85 and 2.05, respectively, while for the interval 1977-2015, they measure 9.66 and 1.91, respectively. The t statistic for independent samples (Lapin, 1978) is computed to be 2.12, which by hypothesis testing suggests that the difference in sample means is not statistically important at the 2% level of significance (i.e., the 98% confidence level). Hence, one accepts that the two samples can be combined into a single sample, with the mean ratio being 9.1 (sd = 2.1). Therefore, based on the yearly means, SSA, indeed, is found to be crudely proportional to SSN, given by the formula SSA = (9.1 [+ or -] 2.1) SSN, this being the [+ or -] 1 sd prediction interval about the mean.

The 10-yma values of SSN and SSA are shown to increase over time, from about SCI4 to a peak in SCI9, then to drop substantially in SC20 before rebounding somewhat in SC21-SC23 and then decreasing again in SC24, the current SC. The 10-yma value of ratio is found to be below the long-term mean of 9.1 during SC12-SC15, above the mean in SC16-SC19, below the mean in SC20, and above the mean in SC21-SC23. The 10-yma of ratio appears likely to fall below the mean again in SC24.

Figure 2 depicts the scatterplot of the yearly means of SSA versus SSN. Based on linear regression analysis, the two parameters are found to be strongly linearly related, having a coefficient of correlation r = 0.986, a coefficient of determination [r.sup.2] = 0.97 (meaning that the inferred regression can explain about 97% of the variance in SSA, and vice versa), and an se = 126.0. The inferred regression is given as y = -101.8 + 11.3x, where y represents SSA and x represents SSN.

Four points are denoted in Fig. 2: namely, the points marked 1917, 1918, 1959, and 1982. These particular points (years) are found to have the greatest deviation from their inferred values as given by the regression equation, with 1917 and 1918 being lower by 2.6 and 2.4 se, respectively, and 1959 and 1982 being higher by 3.4 and 4.4 se, respectively. This suggests, perhaps, that the SSA during 1917 and 1918 (associated with SCI5) possibly is due to a smaller percentage of large area sunspots (individual sunspots having SSA > 1,000), while SSA during 1959 (SC19) and 1982 (SC21) possibly is due to a larger percentage of large area sunspots. Indeed, examination of the SSA record reveals that the percentage of large area spots relative to the total number of daily spots observed was only 1.5% for 1917 (54/3,510) and 1% for 1918 (29/2,905), while it was 5.2% for 1959 (234/4,514) and 2.3% for 1982 (86/3,752).

Table 1 summarizes the SSN, SSA, and ratio characteristics for SC12-SC24. Noticeable is that, for all cycles, the minimum values of SSN and SSA occurred in the same year, except for SC21, when the minimum SSA preceded the minimum of SSN by 1 year. Also, for all cycles, the maximum values of SSN and SSA occurred in the same year, except for SC20, SC21, and SC23 when the maximum SSA followed the maximum in SSN by 2-3 years. For the ratio, the minimum always has occurred within [+ or -] 1 year of the minimum in SSN, with the minimum in the ratio preceding the minimum in SSN in SC14, SC21, and SC23 and the minimum in the ratio following the minimum in SSN in SCI2, SCI7, and SCI8. Also, for the ratio, the maximum preceded the maximum in SSN in SCI6 (by 2 years) and SC20 (by 1 year) and followed the maximum in SSN in SCI3 (by 4 years), SCI5 (by 5 years), SCI7 (by 1 year), SCI9 (by 2 years), SC21 (by 3 years), and SC23 (by 5 years). Only during SCI2, SCI4, SCI8, SC22, and possibly SC24 has the maximum in the ratio occurred in the same year as the maximum in SSN. (Since SSN max for SC24 occurred in 2014 and six of the SCs had a delayed maximum in the ratio relative to Emax based on SSN, it remains to be seen whether or not a later occurring ratio maximum might be seen for SC24, indicating the occurrence of a larger percentage of large area spots.)

Figure 3 shows the results of an epoch analysis of the yearly variation in the ratio relative to (a) Emin and (b) Emax, where these epochs are determined using SSN. Plotted are the means (the thick, dotted lines), the [+ or -]1 sd error bars about the means, and the largest observed individual cycle ratio values for t in years from the epoch of choice. Regarding the variation of ratio relative to Emin, clearly the smallest ratio occurs in close association to Emin; i.e., the average area per SSN is smallest near SSN minimum, indicating that spots occurring near SC minimum generally are smaller in a real size. Following Emin, the ratio quickly rises, attaining a maximum mean value at t = 6 years, or some 2 years following the SSN maximum mean occurrence (see Table 1); i.e., the average SSA per SSN often is largest between SSN maximum to a few years following SSN maximum during the declining portion of the SC. This suggests that the largest spots of a SC often occur between SSN maximum to a few years following SSN maximum. Regarding the variation of ratio relative to Emax, clearly the largest ratio occurs near SSN maximum to a few years following SSN maximum during the declining portion of the solar cycle. The largest observed value (= 14.1) is found to have occurred at t = 3 years relative to Emax.

Tables 2 and 3 provide the ratio values relative to Emin and Emax, respectively. SC24, the current ongoing SC, has known ratio values only through t = 7 years (relative to Emin) and t = 1 year (relative to Emax), having a minimum ratio (= 5.4) in 2008 and maximum ratio (=11.1), thus far, in 2014.

In conclusion, SSA, indeed, is found to be proportional to SSN, given as SSA = (9.1 [+ or -] 2.1) SSN when using yearly means. However, because SSA and SSN vary over the solar cycle, a better description of the relationship between SSA and SSN is the inferred regression y = -101.8 + 11.3x, where y is SSA and x is SSN. The inferred linear regression equation has r = 0.986 and se = 126 millionths of a solar hemisphere. While there is a strong linear correlation between the two parameters, statistically important deviations from the inferred values have occurred, in particular in 1917 and 1918 when the observed SSA is considerably lower than the inferred SSA, given the SSN, and in 1959 and 1982 when the observed SSA is considerably higher than the inferred SSA, given the SSN.

LITERATURE CITED

Baranyi, T., L. Gyori, A. Ludmany, and H. E. Coffey 2001. Comparison of Sunspot Area Data Bases, Monthly Notices of the Royal Astronomical Society 323, pp. 223-230.

Bray, R. J. and R. E. Loughhead 1964. Sunspots, John Wiley & Sons, New York, p. 237. Clette, F., L. Svalgaard, J. M. Vaquero, and E. W. Cliver 2015. Revisiting the Sunspot Number: A 400-Year Perspective on the Solar Cycle, Space Sciences Series of ISSI 53, The Solar Activity Cycle: Physical Causes and Consequences, A. Balogh, H. Hudson, K. Petrovay, and R. Von Steiger (eds.), Springer-Verlag, New York, pp. 35-103.

de Jager, C. 1959. Structure and Dynamics of the Solar Atmosphere, in Encyclopedia of Physics, S. Fltigge (ed.), Vol. LII, Astrophysics III, The Solar System, Springer-Verlag, Berlin, p. 322.

Hathaway, D. H. 2015. The Solar Cycle, Living Rev. Solar Phys. 12, DOI 10.1007/Irsp-2015-4, 87 pp.

Hirsch, R. 2000. Seizing the Light: A History of Photography, McGraw-Hill, 530 pp.

Howard, R., P. A. Gilman, and P. I. Gilman 1984. Rotation of the Sun Measured from Mount Wilson White-Light Images, The Astrophysical Journal 283, pp. 373-384.

Kiepenheuer, K. 0.1953. The Sun, Chapter 6. Solar Activity, G. P. Kuiper (ed.), The Solar System, Vol. I, The University of Chicago Press, Chicago, pp. 322-465.

Lapin, L. L. 1978. Statistics for Modern Business Decisions, 2nd edition, Harcourt Brace Jovanovich, Inc., New York, p. 486.

Tandberg-Hanssen, E. 1967. Solar Activity, Blaisdell Publishing Company, Walthem, MA, p. 181. Waldmeier, M. 1961. The Sunspot-Activity in the Years 1610-1960, Schulthess and Company, Zurich, 171 pp.

Wilson, R. M. 2015. Sunspot Cycle Characteristics based on the Newly Revised Sunspot Number, Journal of the Alabama Academy of Science 86, Numbers 3-4, July/October, pp. 203-221.

Wilson, R. M. and D. H. Hathaway 2005. A Comparison of Rome Observatory Sunspot Area and Sunspot Number Determinations with International Measures, 1958-1998, NASA/TP-2005-214191, NASA Marshall Space Flight Center, Huntsville, AL, 20 pp. <http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20060022159.pdf>.

Wilson, R. M. and D. H. Hathaway 2006. On the Relation between Sunspot Area and Sunspot Number, NASA/TP--2006-214324, NASA Marshall Space Flight Center, Huntsville, AL, 20 pp. http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20060020186.pdf.

Zirin, H. 1988. Astrophysics of the Sun, Cambridge University Press, Cambridge, p. 303.

Robert M. Wilson

NASA Marshall Space Flight Center, NSSTC, Huntsville, Alabama

Corresponding: Robert M. Wilson (robert.m.wilson@nasa.gov)

Table 1. Summary of sunspot number (SSN) and sunspot
area (SSA) characteristics for sunspot cycles (SC) 12-24

                               SSN

SC      min      max    Emin   Emax   ASC    DES     PER

12      5.7    106.1    1878   1883     5      6      11
13     10.4    142.0    1889   1893     4      8      12
14      4.6    105.5    1901   1905     4      8      12
15      2.4    173.6    1913   1917     4      6      10
16      9.7    129.7    1923   1928     5      5      10
17      9.2    190.6    1933   1937     4      7      11
18     16.1    214.7    1944   1947     3      7      10
19      6.6    269.3    1954   1957     3      7      10
20     15.0    150.0    1964   1968     4      8      12
21     18.4    220.1    1976   1979     3      7      10
22     14.8    211.1    1986   1989     3      7      10
23     11.6    173.9    1996   2000     4      8      12
24      4.2    113.3    2008   2014     6     --      --
mean    9.9    169.2                  4.0    7.0    10.8
sd      5.1     50.6                  0.9    1.0     0.9

                                 SSA

SC       min       max    Emin   Emax   ASC    DES     PER

12      22.2    1148.9    1878   1883     5      6      11
13      76.7    1460.6    1889   1893     4      8      12
14      27.9    1195.9    1901   1905     4      8      12
15       7.5    1533.9    1913   1917     4      6      10
16      54.7    1388.9    1923   1928     5      5      10
17      91.3    2072.8    1933   1937     4      7      11
18     124.7    2634.1    1944   1947     3      7      10
19      34.6    3048.5    1954   1957     3      7      10
20      53.9    1601.3    1964   1970     6      5      11
21     166.4    2283.3    1975   1982     7      4      11
22     124.7    2579.2    1986   1989     3      7      10
23       81.9   1828.7    1996   2002     6      6      12
24      22.8    1252.2    2008   2014     6     --      --
mean    68.4    1848.3                  4.6    6.3    10.8
sd      48.2     620.7                  1.3    1.2     0.8

       Ratio          E (Ratio)

SC     min     max    min    max

12     3.6    10.8    1879   1883
13     7.4    11.7    1889   1897
14     4.7    11.3    1900   1905
15     3.1    10.6    1913   1922
16     5.6    11.9    1923   1926
17     8.1    11.0    1934   1938
18     7.7    12.3    1945   1947
19     5.2    12.8    1954   1959
20     3.6    11.4    1964   1967
21     7.4    14.1    1975   1982
22     8.4    12.2    1986   1989
23     6.4    11.8    1995   2005
24     5.4     111    2008   2014
mean   5.9    11.8
sd     1.8     0.9

ASC = ascent duration, the elapsed time in years from Emin to
Emax.

DES = descent duration, the elapsed time in years form Emax
(cycle n) to Emin (cycle n + 1).

PER = period, the elapsed time in years from Emin (cycle n) to
Emin (cycle n + 1).

Table 2. Ratio values relative to the epoch of sunspot number
minimum (Emin) for elapsed time t in years from Emin

SC       -5      -4      -3      -2      -1       0       1       2

12       --      --     7.5     5.8     4.5     3.9     3.6     8.3
13      9.8     9.4     8.9     8.1     7.8     7.4     8.4     9.5
14      7.4    11.7     8.4     5.4     4.7     6.1     7.0     8.3
15      8.6     9.4     8.6     6.8     6.2     3.1     9.5     8.8
16      8.3    10.0     9.8     9.6    10.6     5.6    10.0    11.2
17     10.7    11.5     8.7     8.0     8.8     9.9     8.1    10.3
18     10.7     9.2     8.3     8.4    11.0     7.7     7.7    11.8
19     11.2    10.3    11.5     9.0     7.2     5.2    10.2    11.9
20     12.8    10.3     8.0     8.7     7.2     3.6     5.2     8.9
21      9.3     9.4     8.5     8.1     7.4      9.2    8.8    10.4
22     11.4    14.1    10.4    13.4     8.7     8.4     8.8    10.9
23     12.2    10.1     9.1     7.6     6.4     7.1     7.3     8.6
24     11.1    10.5    11.8     9.9    10.6     5.4     5.5     8.6
mean   10.3    10.5     9.2     8.4     7.8     6.4     7.7     9.8
sd      1.6     1.4     1.3     2.0     2.1     2.1     2.0     1.3

SC        3       4       5       6       7       8       9

12      7.5     9.8    10.8     9.8     9.4     8.9     8.1
13     10.0    10.3     9.9     9.1     7.9    11.7     8.4
14      7.0    11.3     8.6    10.6     8.6     9.4     8.6
15      7.6     8.8     8.3    10.0     9.8     9.6    10.6
16     11.9     9.2    10.7    11.5     8.7     8.0     8.8
17      8.6    10.9    11.0    10.7     9.2     8.3     8.4
18     12.3    10.2    11.2    10.3    11.5     9.0     7.2
19     11.3    11.5    12.8    10.3     8.0     8.7     7.2
20     11.4    10.5     9.7    10.8     9.3     9.4     8.5
21     10.0     9.9    11.4    14.1    10.4    13.4     8.7
22     12.2    10.7    12.2    10.1     9.1     7.6     6.4
23      8.5     9.3    10.0    11.2    11.1    10.5    11.8
24      9.3     9.4     9.2    11.1     8.9      --      --
mean    9.8    10.1    10.4    10.7     9.4     9.5     8.6
sd      1.9     0.8     1.2     1.2     1.1     1.6     1.5

t is the elapsed time in years, where (= 0 is the reference epoch year.

Table 3. Ratio values relative to the epoch of sunspot number maximum
(Emax) for elapsed time t in years from Emax

SC     -4      -3      -2      -1      0       1

12      3.6     8.3     7.5     9.8    10.8     9.8
13      7.4     8.4     9.5    10.0    10.3     9.9
14      6.1     7.0     8.3     7.0    11.3     8.6
15      3.1     9.5     8.8     7.6     8.8     8.3
16     10.0    11.2    11.9     9.2    10.7    11.5
17      9.9     8.1    10.3     8.6    10.9    11.0
18     11.0     7.7     7.7    11.8    12.3    10.2
19      7.2     5.2    10.2    11.9    11.3    11.5
20      3.6     5.2     8.9    11.4    10.5     9.7
21      7.4     9.2     8.8    10.4    10.0     9.9
22      8.7     8.4     8.8    10.9    12.2    10.7
23      7.1     7.3     8.6     8.5     9.3    10.0
24      8.6     9.3     9.4     9.2    11.1     8.9
mean    7.2     8.1     9.1     9.7    10.7    10.0
sd      2.5     1.7     1.2     1.6     1.0     1.0

SC     2       3       4       5       6       7

12      9.4     8.9     8.1     7.8     7.4     8.4
13      9.1     7.9    11.7     8.4     5.4     4.7
14     10.6     8.6     9.4     8.6     6.8     6.2
15     10.0     9.8     9.6    10.6     5.6    10.0
16      8.7     8.0     8.8     9.9     8.1    10.3
17     10.7     9.2     8.3     8.4    11.0     7.7
18     11.2    10.3    11.5     9.0     7.2     5.2
19     12.8    10.3     8.0     8.7     7.2     3.6
20     10.8     9.3     9.4     8.5     8.1     7.4
21     11.4    14.1    10.4    13.4     8.7     8.4
22     12.2    10.1     9.1     7.6     6.4     7.1
23     11.2     11.1   10.5    11.8     9.9    10.6
24       --      --      --      --      --      --
mean   10.7     9.8     9.6     9.4     7.7     7.5
sd      1.2     1.7     1.2     1.7     1.6     2.2

t is the elapsed time in years, where
t = 0 is the reference epoch year.


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