Solar Physics

, Volume 278, Issue 2, pp 457–469 | Cite as

Sunspot Time Series – Relations Inferred from the Location of the Longest Spotless Segments

Open Access
Article

Abstract

Spotless days (i.e., days when no sunspots are observed on the Sun) occur during the interval between the declining phase of the old sunspot cycle and the rising phase of the new sunspot cycle, being greatest in number and of longest continuous length near a new cycle minimum. In this paper, we introduce the concept of the longest spotless segment (LSS) and examine its statistical relation to selected characteristic points in the sunspot time series (STS), such as the occurrences of first spotless day and sunspot maximum. The analysis has revealed statistically significant relations that appear to be of predictive value. For example, for Cycle 24 the last spotless day during its rising phase should be about August 2012 (± 9.1 months), the daily maximum sunspot number should be about 227 (± 50; occurring about January 2014±9.5 months), and the maximum Gaussian smoothed sunspot number should be about 87 (± 25; occurring about July 2014). Using the Gaussian-filtered values, slightly earlier dates of August 2011 and March 2013 are indicated for the last spotless day and sunspot maximum for Cycle 24, respectively.

Keywords

Spotless days Longest spotless segment Prediction Cycle 24 

1 Introduction

The activity of the Sun is expressed through various solar indices, but the daily International Sunspot Number (ISN) is the key indicator used due to the length of the available record (Hathaway 2010). Until 1980 the ISN was compiled by the Swiss Federal Observatory, the ISN being better known as the Wolf or Zürich number. Since 1981 the Royal Observatory of Belgium (Solar Influences Data Center – SIDC) has computed ISN (sidc.omea.be).

In 1844, Schwabe, after 18 years of observations of the number of sunspot groups and spotless days, found that his data indicated the presence of sunspot periodicity, measuring about 11 years in length (Schwabe 1844). The process of determining the dates and values describing solar cycles depends on the methods and input data used to find them (Hathaway 2010). The idea of using spotless days to find the minimum of activity appeared in the papers of Waldmeier (1961) and McKinnon (1987). In 1995, Wilson (1995) proposed to use the first spotless day as a predictor for the sunspot minimum. The possible connections of spotless days with the timing and size of the solar cycle were more accurately examined by Wilson and Hathaway (2005, 2006a, 2007). In 2006, Hamid and Galal (2006) proposed to use the number of spotless events prevailing in the minimum preceding the new cycle as a prediction precursor of new cycle characteristics. Recently, Nielsen and Kjeldsen (2011) discussed the ongoing accumulation of spotless days in different solar cycles.

In this work we use the daily ISN series covering the period between January 1818 and May 2011 provided by the SIDC as the basis for characterizing intervals with low solar activity. This period included the decline phase of Cycle 6, Cycles 7 – 23 and the initial rise of Cycle 24. The intrinsic nature and accuracy for four main eras of sunspot number observations are different in the ISN time series (Clette et al.2007). For example, Cycles 7 – 9 include the era of Schwabe’s records with a large number of days without observations (Wilson 1998). Cycles 10, 11 and the rise of Cycle 12 belong to Wolf’s era (years 1848 – 1882), while those of Cycles 13 – 21 belong to the Zürich era. Since 1981, when the IAU World Data Center for sunspot numbers was transferred from the Zürich Observatory to Brussels, a new approach for calculation of the sunspot number has been established (Clette et al.2007). However, in the papers of Hathaway, Wilson, and Reichmann (2002), Wilson and Hathaway (2006b, 2008), Li and Liang (2010), these authors determined that the ISN data are reliable from Cycle 12 to the present. Comparison of the ISN time series with sunspot group numbers, devised by Hoyt and Schatten (1998), indicates only ∼ 1% discrepancy between them for the period 1874 – 1995 (Hathaway and Wilson 2004; Hathaway 2010; Usoskin 2008).

The main purpose of this work is to study relations determined from the position of the longest spotless segment (LSS, the longest sequence of consecutive days when no spots were observed) with respect to locations of some characteristic points in the ISN series. We analyze these relations for three different sets of data. The first set includes Cycles from 8 to 23 (all cycles present in the SIDC daily ISN series; Cycle 7 is excluded because there are too many days without data in its minimum), the second spanning Cycles 10 to 23, and the last covering Cycles 13 to 23. As both the position and the length of the LSS are observables, which can be determined without doing any averaging of the data, we use these parameters to determine various predictive characteristics of the sunspot cycles. Using these preferential relations, we also make some predictions regarding Cycle 24.

2 Definitions of Some Characteristic Intervals

In our previous paper (Zięba et al.2006) we introduced the concept of the passive interval, which we defined as the time distance (denoted d00) from the first spotless day after an old cycle maximum to the last spotless day before the next new cycle maximum. All spotless days occur within the passive intervals. For each passive interval we have a minimum of activity (cycle minimum) and occurrence of the LSS. The position of the LSS can then be determined relative to the positions of different distinctive points within the ISN time series.

We measure distances to the middle of each LSS from the beginning of its passive interval and relative to two differently defined maxima (daily peak maximum and the daily peak Gaussian maximum) located before the first spotless day. The peak maximum is identified as the day having the maximal sunspot number, while the Gaussian maximum is identified as the day having the maximal sunspot number after smoothing the ISN time series using an 810-day Gaussian filter.

The Gaussian filter, having a full width at half maximum (FWHM) equal to 810 days, effectively removes short-term variations of solar activity on time scales of about two years that can produce double peaked maxima. The size of this filter is similar to 24-month Gaussian filter used when monthly averaged sunspot numbers are considered (Hathaway, Wilson, and Reichmann 1999; Hathaway 2010).

Definitions of all the time intervals (distances) used in our paper are given in Table 1. They are calculated as differences between the aforementioned characteristic points of solar cycles. Figure 1 shows these points for the passive interval 16, while the computed values of the relevant distances are given in the last column of Table 1. Knowing these distances, the following six ratios can be calculated for passive interval 16: r0s=d0s/d00=0.605, rps=dps/dpx=0.681, rGs=dGs/dGx=0.612, r0m=d0m/d00=0.497, rpm=dpm/dpx=0.609, and rGm=dGm/dGx=0.544.
Figure 1

The daily values of ISN drawn for the time interval between Solar Cycles 15 and 16. The black curve represents the data smoothed with the Gaussian filter of full width at half maximum (FWHM) of 810 days. Positions of the characteristic extreme points near sunspot minimum and maxima are denoted as the numbers enumerated from 1 January 1918 (for dates see Table 2).

Table 1

Definitions of the time intervals used. The last column presents values of the relevant distances calculated for the passive interval 16.

Interval (distance)

Definition

Interval 16 [days]

d00

Passive interval – the time distance from the first spotless day after a cycle maximum to the last spotless day before the next cycle maximum.

2291

d0s

The time distance from the first spotless day after a cycle maximum to the middle day of LSS.

1387

d0m

The time distance from the first spotless day after a cycle maximum to the day of minimum given by the 810 Gaussian filter (Gaussian minimum).

1138

dpx

The time distance between the two successive peak maxima that includes the related passive interval.

3469

dps

The time distance from the peak maximum to the middle day of LSS.

2363

dpm

The time distance from the peak maximum to the Gaussian minimum.

2114

dGx

The time distance between the two successive Gaussian maxima that includes the related passive interval.

3652

dGs

The time distance from the Gaussian maximum to the middle day of LSS.

2234

dGm

The time distance from the Gaussian maximum to the Gaussian minimum.

1985

The analyzed ISN time series allows us to determine the positions of the characteristic points within each solar cycle, thereby allowing for a determination of the distances and ratios for the 16 known passive intervals (Cycles 8 – 23). These values are summarized in Table 2. Figure 2 displays the values of the calculated ratios (r0s, r0m, rps, rpm, rGs, and rGm). Because the ratios display rather small fluctuations over time (i.e., the passive intervals), this suggests that possible linear relations exist between the various related distances that could prove to be of predictive value (e.g., determination of the expected times for the last spotless day and the maximum for an ongoing sunspot cycle).
Figure 2

Variation of six ratios (a) r0s=d0s/d00, r0m=d0m/d00, (b) rps=dps/dpx, rpm=dpm/dpx, (c) rGs=dGs/dGx, rGm=dGm/dGx describing the position of LSS (d0s, dps, dGs) or the Gaussian minimum (d0m, dpm, dGm) with respect to relevant distances (d00, dpx, dGx) for the passive intervals from 8 to 23. The mean value (m) and standard deviation (sd) are also presented.

Table 2

Dates defining passive intervals, positions of LSS, cycles maxima and minima together with computed characteristic distances and ratios r0s=d0s/d00, r0m=d0m/d00, rps=dps/dpx, rpm=dpm/dpx, rGs=dGs/dGx, rGm=dGm/dGx. Days are calculated from 1 January 1818.

N

Passive interval

First day date

Last day date

Dist. d00

Day

Date

Dist.

d0s

d0m

Rat.

r0s

r0m

Day

Date

Value

Dist.

dpx

dGx

Dist.

dps

dpm

Dist.

dGs

dGm

Rat.

rps

rpm

Rat.

rGs

rGm

LSS

Gaussian min.

Peak maximum

Gaussian maximum

7

50

19.02.1818

3696

1828

02.01.1823

1778

0.48

4198

29.06.1829

231

3746

03.04.1828

?1900

15.03.1823

1850

0.50

4299

08.10.1829

66.4

8

4407

24.01.1830

1972

5958

24.04.1834

1551

0.79

6992

21.02.1837

261

2794

1760

1659

0.63

0.60

6379

19.06.1835

5742

20.09.1833

1335

0.68

7054

24.04.1837

127.2

2755

1544

1443

0.55

0.52

9

8141

15.04.1840

2636

9661

13.06.1844

1520

0.58

10820

16.08.1847

254

3828

2669

2607

0.70

0.64

10777

04.07.1847

9357

14.08.1843

1216

0.46

11136

27.06.1848

113.2

4082

2365

2303

0.62

0.56

10

11456

13.05.1849

3249

13764

07.09.1855

2308

0.71

15520

28.06.1860

220

4700

2944

2628

0.63

0.61

14705

05.04.1858

13899

20.01.1856

2443

0.75

15434

03.04.1860

90.7

4298

3079

2763

0.66

0.64

11

15987

08.10.1861

2836

17914

17.01.1867

1927

0.68

19231

26.08.1870

317

3711

2394

2480

0.65

0.63

18823

14.07.1869

17913

16.01.1867

1926

0.68

19348

21.12.1870

119.2

3914

2393

2479

0.64

0.63

12

20228

19.05.1873

3781

22354

15.03.1879

2126

0.56

24116

10.01.1884

166

4885

3123

3006

0.64

0.64

24009

25.09.1883

22125

29.07.1878

1897

0.50

24054

09.11.1883

63.9

4706

2894

2777

0.59

0.59

13

24483

11.01.1885

2531

26266

29.11.1889

1783

0.70

27616

10.08.1893

237

3500

2150

2212

0.61

0.61

27014

17.12.1891

26064

11.05.1889

1581

0.62

27674

07.10.1893

80.5

3620

1948

2010

0.56

0.56

14

28438

10.11.1895

3547

30420

15.04.1901

1982

0.56

32091

11.11.1905

182

4475

2804

2746

0.63

0.60

31985

28.07.1905

30542

15.08.1901

2104

0.59

32269

08.05.1906

59.2

4595

2926

2868

0.65

0.62

15

32431

17.10.1906

3638

34842

24.05.1913

2411

0.66

36378

07.08.1917

268

4287

2751

2573

0.64

0.61

36069

02.10.1916

34665

28.11.1912

2234

0.61

36507

14.12.1917

86.8

4238

2574

2396

0.60

0.57

16

37354

09.04.1920

2291

38741

26.01.1924

1387

0.61

39847

05.02.1927

173

3469

2363

2234

0.68

0.61

39645

18.07.1926

38492

22.05.1923

1138

0.50

40159

14.12.1927

71.1

3652

2114

1985

0.61

0.54

17

41168

18.09.1930

1775

42330

23.11.1933

1162

0.65

43494

30.01.1937

233

3647

2483

2171

0.68

0.59

42943

29.07.1935

42214

30.07.1933

1046

0.59

43828

30.12.1937

107.1

3669

2367

2055

0.65

0.56

18

45236

07.11.1941

1408

46147

06.05.1944

911

0.65

47261

25.05.1947

323

3767

2653

2319

0.70

0.62

46644

15.09.1945

46023

03.01.1944

787

0.56

47570

29.03.1948

140.5

3742

2529

2195

0.67

0.59

19

48566

20.12.1950

1763

49842

18.06.1954

1276

0.72

51127

24.12.1957

355

3866

2581

2272

0.67

0.63

50329

18.10.1955

49704

31.01.1954

1138

0.65

51180

15.02.1958

185.8

3610

2443

2134

0.63

0.59

20

52535

01.11.1961

1743

53591

22.09.1964

1056

0.61

55207

24.02.1969

215

4080

2464

2411

0.60

0.60

54278

10.08.1966

53602

03.10.1964

1067

0.61

55227

16.03.1969

106.0

4047

2475

2422

0.61

0.60

21

56817

23.07.1973

1456

57910

20.07.1976

1093

0.75

59118

10.11.1979

302

3911

2703

2683

0.69

0.66

58273

18.07.1977

57766

27.02.1976

949

0.65

59321

31.05.1980

150.4

4094

2559

2539

0.65

0.62

22

60591

22.11.1983

1330

61364

03.01.1986

773

0.58

63420

21.08.1991

300

4302

2246

2043

0.52

0.57

61921

14.07.1987

61424

04.03.1986

833

0.63

62918

06.04.1990

148.3

3597

2306

2103

0.54

0.58

23

64375

02.04.1994

1378

65291

04.10.1996

916

0.66

66675

19.07.2000

246

3255

1871

2373

0.57

0.60

65753

09.01.1998

65129

25.04.1996

754

0.55

66853

13.01.2001

112.4

3935

1709

2211

0.53

0.56

24

67962

27.01.2004

?

69991

17.08.2009

2029

?

?

?

?

?

3316

3138

?

?

?70718

14.08.2011

69656

16.09.2008

1694

?

?

?

?

?

2981

2803

?

?

Figures 3, 4 and 5, respectively, show the scatter plots of d00 v. d0s, dpx v. dps and dGx v. dGs. Also shown are the inferred linear regressions for the three different cycle groupings (All, > 9 and > 12) and the position along the x-axis for Cycle 24.
Figure 3

Scatter plot of distances between the first and the last spotless days (d00) versus the elapsed time from the first spotless day to LSS (d0s) for the indicated intervals. The determined equation of the best linear fit, the correlation coefficient (r) and the standard error of estimation (SEE) for each of the discussed data sets are also given.

Figure 4

Scatter plot of distances between the two successive peak maxima (dpx) versus the elapsed time from the first maximum to LSS (dps) for the indicated intervals. The determined equation of the best linear fit, the correlation coefficient (r) and the standard error of estimation (SEE) for each of the discussed data set are also given.

Figure 5

Scatter plot of distances between the two successive Gaussian maxima (dGx) versus the elapsed time from the first maximum to LSS (dGs) for the indicated intervals. The determined equation of the best linear fit, the correlation coefficient (r) and the standard error of estimation (SEE) for each of the discussed data set are also given.

The relations d00 v. d0s (Figure 3) and dGx v. dGs (Figure 5) do not indicate any significant statistical differences among the best linear fits to the three considered data sets. However, in the case of the relation dpx v. dps (Figure 4) the best linear fit to the data set “> 12” (Cycles 13 – 23) deviates clearly from those obtained for the other two. The deviation is caused mainly by the data points for Cycle 22. When we ignore Cycle 22, which has three practically equal peak maxima 300, 295 and 296 occurring almost at yearly intervals, the correlation coefficient between dpx and dps increases from 0.62 to 0.84.

Figure 6 displays similar scatter plots, but using the position of the Gaussian minimum (d0m, dpm and dGm). All three relations d00 v. d0m, dpx v. dpm, and dGx v. dGm are statistically significant. The best agreement among the fits for the three discussed sets of data is seen in the case of the relation dGx v. dGm (Figure 6(c)). Again, the position along the x-axis for Cycle 24 is shown. Previously, Wilson and Hathaway (2005) have analyzed a relation similar to d00 v. d0m using the smoothed monthly mean sunspot number and found these variables strongly correlated (correlation coefficient r=0.95 with the standard error of estimate (SEE) equal to 9.9 months for the data set covering Cycles 10 – 23). Our data give for these cycles the correlation coefficient between d00 and d0m equal to 0.94 and SEE=321 days. Its value grows to 0.98 with SEE=159 days for Cycles 13 – 23.
Figure 6

The plots in panels (a), (b) and (c) are similar to those in Figures 3, 4 and 5, respectively, but instead of considering positions of LSS inside of sunspot series positions of Gaussian minima are used.

3 Results and Discussion

Using the linear relations depicted in Figures 3, 4 and 5, we find that the occurrences of the last spotless day and sunspot maximum for an ongoing sunspot cycle can be predicted given determination of the LSS for the ongoing cycle. The same is possible using the relations given in Figure 6 based on the 810-day Gaussian minimum. However, as the position of the LSS is known earlier than the Gaussian minimum and its location does not result from any smoothing procedure (thus, reflecting a real physical process), we will concentrate on using those relations based on LSS rather than on using those relations based on the Gaussian minimum.

As we do not find any significant statistical differences among relations calculated for the three discussed data sets, in further work we concentrate on relations obtained from the longest data set (Cycles 8 – 23). In Table 3 we present the parameters of these simple linear regressions and the successive steps leading to determination of the occurrence time of the last spotless day and the maximum of the ongoing Cycle 24. Predictions coming from positions of LSS and Gaussian minima are given.
Table 3

Parameters of the simple linear regression (SEE – standard error of estimation) and predictions for the occurrence time of some extreme points of the ongoing Cycle 24.

Variables

correlation

Intercept

Slope

Variables

+ reference point (see Table 2)

Predictions for Cycle 24

month

90% confidence interval

x

y

d0s, d00

r=0.953

−29±217

1.56±0.14

SEE=277

d0s

2029

d00

3136

+67962=71098

The last spotless day

August 2012

May 2011 – December 2013

dps, dpx

r=0.859

728±512

1.27±0.20

SEE=287

dps

3316

dpx

4939

+66675=71614

The peak maximum

January 2014

September 2012 – May 2015

dGs, dGx

r=0.965

538±246

1.40±0.10

SEE=125

dGs

3138

dGx

4931

+66853=71784

The Gaussian maximum

July 2014

December 2013 – February 2015

d0m, d00

r=0.896

308±241

1.44±0.16

SEE=343

d0m

1694

d00

2747

+67962=70709

The last spotless day

August 2011

January 2010 – March 2013

dpm, dpx

r=0.914

1050±344

1.20±0.14

SEE=228

dpm

2981

dpx

4627

+66675=71302

The peak maximum

March 2013

February 2012 – April 2014

dGm, dGx

r=0.956

1094±234

1.23±0.10

SEE=140

dGm

2803

dGx

4542

+66853=71395

The Gaussian maximum

June 2013

October 2012 – February 2014

According to Table 3 the last spotless day for Cycle 24 should be about August 2012, derived from the d00 v. d0s relation. However, using the relation d00 v. d0m, the last spotless day is expected to occur about August 2011. The overlap between the 90% confidence intervals of these two estimates gives for the last spotless day occurrence the time interval between May 2011 and March 2013.

Applying Wilson and Hathaway’s (2005) relation between the time from the first to the last spotless day and the time elapsed from the first spotless day to sunspot minimum occurrence (for Cycle 24 this is 60 months), the last spotless day is expected to occur about July 2011. This is in agreement with our prediction (August 2011) based on d00 v. d0m relation, which is similar to those found by Wilson and Hathaway. Also, according to Nielsen and Kjeldsen (2011), who discuss the ongoing accumulation of spotless days in different solar cycles, the last spotless day before the maximum of Cycle 24 will be about December 2012. This prediction too is close to our result calculated from the d00 v. d0s relation. Up to now the last spotless day for Cycle 24 occurred on 14 August 2011.

The occurrence of the maximum daily sunspot number for Cycle 24 is predicted to be about January 2014 and the occurrence of the peak Gaussian maximum is predicted to be about July 2014, these dates having an uncertainty of about 287 and 125 days, respectively. Using the relations shown in Figure 7 between maximal sunspot values of an ongoing cycle and the relevant passive interval d00 it is possible to estimate how large these maxima might be. A similar relation (having r=−0.67), but one based on the smoothed monthly mean sunspot number, was given by Wilson and Hathaway (2005) for Cycles 10 – 23.
Figure 7

Scatter plot of the maximal sunspot numbers (peak – square points and Gaussian – diamonds points) versus the time distance from the first to the last spotless day (d00) for the indicated intervals. The regression equations, the correlation coefficients (r) and standard errors of estimation (SEE) are also given.

The d00 value of the passive interval 24, according to Table 3, equals 3136 days, so the sunspot number peak maximum for Cycle 24 is predicted to be about 227±50, and about 87±25 for Cycle 24’s Gaussian maximum. The inferred correlations between maximal sunspot numbers and the d00 time from the first to the last spotless day (see Figure 7) are not particularly strong, with regressions explaining only a part of the observed variance (∼ 25% for the peak maximum and ∼ 50% for the Gaussian one). Consequently, the occurrence of the maximal daily value can vary over a considerable range. Assuming that 14 August 2011 is the last spotless day for Cycle 24, d00 cannot be shorter than 2756 days, leading us to infer that there is only a 5% chance that its peak maximum and the Gaussian maximum will exceed 321 and 139, respectively. Because large values of d00 are associated with weaker cycles, the predicted maxima for Cycle 24 seems likely to be located near the maximal values of Cycles 10, 14 and 15. Usoskin (2008) has suggested that Cycles 14 and 15 are associated with the time of the modern minimum in solar activity. Since Cycle 24 appears likely to be of a similar nature as Cycles 14 and 15, could this be a strong indication that Cycle 24 heralds the start of another minimum in solar activity in the sunspot record?

Figure 8 (left) shows the temporal plots of mGx and mpx, both displaying similar behavior. While true, even- and odd-cycle differences are more clearly discerned in the peak maxima than in the Gaussian maxima. All values, except for even-odd-cycle pairs 8 – 9 and 22 – 23, are found to follow the Gnevyshev–Ohl rule, which states that the odd-following cycle tends to be the stronger cycle (Gnevyshev and Ohl 1948; Kopecky 1950). Presuming the Gnevyshev–Ohl rule still applies, we infer that Cycle 25 should be somewhat stronger than Cycle 24. The right portion of Figure 8 presents ratios between Gaussian and peak maximal sunspot values for Cycles 8 – 23 and the best-fitted sinusoid for these data (coefficient of determination, R-squared=0.901). The value of this ratio for Cycle 24 calculated from the best-fitted sinusoid equals 0.414, while that computed from the predicted peak and Gaussian maximal sunspot numbers for Cycle 24 (Figure 8 left) is 0.383±0.19. As the two different and independent ways used to calculate the ratio between Gaussian and peak maximal sunspot numbers for Cycle 24 give almost the same value, we suppose that our predictions for maximal sunspot numbers are highly probable.
Figure 8

(Left) Scatter plot of maximal sunspot numbers for peak and Gaussian cycle maxima, (right) the plot of ratios between Gaussian to peak maximal sunspot numbers and the best-fitted sinusoid for data of Cycles 8 – 23. For Cycle 24 the predicted values are drawn.

The best-fitted sinusoid indicates a period of ∼ 150 years, which can be associated with the upper limit of the Gleissberg-cycle period (Gleissberg 1939; Ogurtsov et al.2002; Duhau 2003; Duhau and de Jager 2008; De Jager, Duhau, and van Geel 2010). The predicted value of Cycle 24 suggests that the Sun may be at the start of a new Gleissberg minimum.

The agreement of conclusions coming from plots presented in Figure 8 with some characteristic features of solar activity known from various review papers (Usoskin 2008; Hathaway 2010; De Jager and Duhau 2011) indicates that the parameters we used might be useful for studying physical processes responsible for solar variability. Future studies of passive interval properties continue.

4 Conclusions

The results obtained in this work can be summarized as follows:
  1. i)

    Recognizing the longest spotless segment (LSS) localized somewhere in an epoch of solar minimum we have attained three new linear relations, shown in Figures 3, 4 and 5. The strong correlation (r=0.96, Figure 5) between the time elapsed from the previous maximum to LSS and the time distance between successive maxima including the LSS suggests that the LSS can provide insight toward understanding variations in solar activity.

     
  2. ii)

    All inferred relations are statistically significant and allow the time of the last spotless day and the maximum of the ongoing cycle to be predicted on the basis of identifying the LSS.

     
  3. iii)

    The inferred relations, calculated independently for three different groupings of solar cycles (Cycles 8 – 23, 10 – 23 and 13 – 23), do not differ significantly, thereby indicating that the daily ISN time series gives consistent results from Cycle 8 to the present.

     
  4. iv)

    For Cycle 24, we predict the last spotless day to occur in August 2012 and the epoch of sunspot maximum to occur during the first half of 2014 based on the non-Gaussian-filtered data.

     
  5. v)

    The maximal daily sunspot number during the epoch of sunspot maximum should be about 227±50, and the smoothed maximal value using the 810-day Gaussian filter should be about 87±25.

     
  6. vi)

    Using the Gaussian-filtered values, we predict August 2011 and March 2013 for the occurrences of the last spotless day and sunspot maximum for Cycle 24, respectively.

     
  7. vii)

    Our predictions for Cycle 24 are consistent with those recently published by Wilson (2011) and allow us to conclude that Cycle 24 will be a low solar activity cycle (Pesnell 2008).

     
  8. viii)

    Cycle 24 likely represents the start of another minimum in solar activity, like Cycles 14 and 15, which occurred early in the twentieth century.

     

Notes

Acknowledgements

We thank the anonymous referee for constructive comments and suggestions which much improved the original version of the manuscript. We also deeply appreciate the reviewer’s help in English-language correction.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Astronomical ObservatoryJagiellonian UniversityKrakówPoland
  2. 2.Institute of PhysicsJagiellonian UniversityKrakówPoland

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