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ISSN 0016 7932, Geomagnetism and Aeronomy, 2014, Vol. 54, No. 8, pp. 1000­1005. © Pleiades Publishing, Ltd., 2014.

Interrelation between the Amplitude and Length of the 11 Year Sunspot Cycle
E. V. Miletsky and V. G. Ivanov
Main Astronomical (Pulkovo) Observatory, Russian Academy of Sciences, Pulkovskoe sh. 65, St. Petersburg, 196140 Russia e mail: eugenm@gao.spb.ru
Received February 8, 2014

Abstract--Alternative procedures for determining the length (duration) of the 11 year sunspot cycle are con sidered. For cycles 12­23 it is shown that the cycle and the cycle growth branch latitudinal lengths are much more closely related to the cycle amplitude than similar traditional lengths, if the time when the exponent, describing the average sunspot latitude drift, reaches the "reference" latitude value (the latitude phase refer ence time) is selected as a cycle starting time. Two relationships are obtained. The first relationship makes it possible to rather accurately determine the cycle amplitude based on information about two cycle time intervals: the shift in the latitude phase reference point and the cycle latitudinal length. The second relationship relates the value of the interval between the above exponents of adjacent cycles to the cycle amplitudes. The found relationships between the important amplitude, latitude, and time parameters of the 11 year solar magnetic cycle should be taken into account in the construction of adequate physical models. DOI: 10.1134/S0016793214080118

1. INTRODUCTION The length of the 11 year solar (sunspot) activity cycle is the main cycle characteristic (together with amplitude) and is among the key parameters when dynamo models of the solar magnetic cycle are con structed (Karak and Choudhuri, 2011; Nandy et al., 2011). Cycle length is determined as a time interval between the initial and final reference points of the cycle phase. For obvious reasons, times of successive cyclic minimums are usually selected as phase refer ence points for the 11 year sunspot cycle (Waldmeier, 1961; Wilson, 1987; Solanki et al., 2002; Du et al., 2006; Richards et al., 2009). The selection of such minimums evidently depends on the activity index used and the initial data processing algorithm. Since the cycles evolve in time (the Schwabe­Wolf law) and space (the SpÆrer law), the sunspots of a new cycle usually appear at middle heliolatitudes, whereas the sunspot activity of an old cycle still continues at low heliolatitudes. Thus, adjacent cycles overlap in time, and such an overlapping can reach two years. Therefore, a minimum characterizes only the time of minimal total activity of such cycles. However, it is also difficult to distinguish the time of minimum (e.g., during multimonth periods without sunspots) in the absence of such overlapping. To overcome these difficulties, some researchers (Hathaway et al., 1994; Roshchina and Sarychev, 2011) selected the so called "starting" time, which is

obtained from the description of the curve of this cycle using the parametric functional dependence where this starting time is among the dependence parame ters, as a reference point for the 11 year cycle phase. We (Ivanov and Miletsky, 2014) proposed another approach, according to which the cycle latitudinal phase reference point (LPRP) is selected as a cycle phase reference point, for which the exponent describing the sunspot average latitudes drift reaches a certain "reference" latitude value (26.6° in magni tude) in a cycle. Note that the exponent on average reaches this value of latitude at times of minimums (for cycles 12­23). However, in each specific cycle, this time is shifted relative to a minimum toward advance or delay. At such an approach, the LPRP is determined by the time of the exponent approximat ing the average latitudes cyclic drift. 2. DATA AND THEIR PROCESSING We used the data on sunspots presented in the Greenwich catalog and its continuation NOAA/ USAF (http://solarscience.msfc.nasa.gov/greenwch. shtml) for 1874­2013. We formed a series of rotation averages of sunspot latitudes and the sunspot group number index (G). The times and values of cyclic min imums (TGmin, Gmin) and maximums (TGmax, Gmax) are calculated based on the average values for a rotation, smoothed by a 13 point window with sinuso idal weights for the entire Sun and independently for its hemispheres.

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Table 1. Traditional (TTmin) and latitudinal (TTlat) lengths of the 11 year sunspot activity cycles for the entire solar disk and separately for the hemispheres (cycles 12­23) Northern Hemisphere Cycle TTmin 12 13 14 15 16 17 18 19 20 21 22 23 10.37 12.4 10.61 11.35 10.00 10.54 9.78 10.38 11.65 9.78 10.83 11.05 TTlat 11.58 12.92 12.04 11.26 10.60 10.29 10.00 9.52 11.55 9.92 9.75 11.04 TTmin 11.50 11.21 12.17 9.86 9.93 10.83 10.15 10.60 10.39 11.20 9.93 12.62 TTlat 11.47 11.47 12.18 10.07 10.23 10.03 10.59 10.23 10.15 9.50 9.52 11.89 TTmin 11.12 11.28 12.10 9.86 10.38 10.45 10.08 10.31 11.57 10.16 10.15 12.47 TTlat 11.85 12.02 12.68 10.65 10.74 10.15 10.31 9.69 11.41 9.90 9.36 12.03 Southern Hemisphere Entire disk

Let us assume that TTmin = TGmin(n + 1) ­ TGmin(n) is a traditional cycle length in the nth cycle. We proposed the TTlat(n) = Tmin(n + 1) ­ Tlst(n) value as the cycle latitudinal length. Here Tlst(n) is LPRP. The TTmst(n) = Tlst(n) value indicates the
TTmst(19N) = 0.8 year 35 30 25 20 15 10 5 0 12 10 8 G 6 4 2 0 1956 1954 Tmin(19N) = 1954.2 1958 1954

LPRP value and shift sign relative to a minimum. The TTmin and TTlat values for cycles 12­23 are pre sented in Table 1. For example, for the Northern (N) Hemisphere of cycle 19 presented in Fig. 1, we have: TTmin(19N) =
TTlat(19N) = 9.6 year

Latitude, deg

TTlst(19N) = 1955.0 19N 1956 1958 1960 1962 1964 1966

1960 Years

1962

1964 1966 Tmin(20N) = 1964.6

Fig. 1. Time variations (average for rotation and smoothed values) in average latitudes (top panel) and sunspot group number index G (bottom panel) for the Northern (N) Hemisphere in cycle 19. Time intervals are for the Northern (N) Hemisphere in cycle 19. TTmin(19N) and TTmin(20N) are times of minimums. Traditional cycle length is TTmin = TTmin(20N) ­ TTmin(19N). Latitudinal phase reference point is Tlst(19N). Cycle latitudinal length is TTlat(19N) = Tmin(20N) ­ Tlst(19N). GEOMAGNETISM AND AERONOMY Vol. 54 No. 8 2014


1002 R = ­0.71
13N 14S 14S 14N 12N 12S 16N 16S 20S 17S 15S 18S 19S 17N 18N 21N 21S

MILETSKY, IVANOV N & S (K = 24) TTlat, years TTlatN 13 12 11 TTlatS 10 9 2 3 4 5 6 7 Gmax 8 9 10 12 14 16 18 Cycles 20 22 24

13 TTlat, years 12 11 10 9 0

23S 13S 20N 15N 23N

23S 24N

19N

Fig. 2. Left panel: the dependence of the cycle latitudinal length (TTlat) on the cycle amplitude (Gmax) with regard to hemi spheres N and S (cycle numbers are shown by numerals with letters). Right panel: the dependence of the cycle latitudinal lengths TTlatN (solid line and circles) and TTlatS (dotted line and asterisks) for hemispheres N and S, respectively, on the cycle number.

Tmin(20N) ­ Tmin(19N) = 1964.6 ­ 1954.2 = 10.4 yr, TTmst(19N) = Tlst(19N) ­ Tmin(19N) = 1955.0 ­ 1954.2 = 0.8 yr, TTlat(19N) = Tmin(20N) = Tlst(19N) = 1964.6 ­ 1955.0 = 9.6 yr. 3. RESULTS As is known, the amplitude and length of the 11 year cycle vary from cycle to cycle. The interrela tion between these parameters is a problem of prime importance. Studies (Dicke, 1978; Hoyng, 1993) indi cate that the correlation between the cycle amplitude and traditional length (TTmin) is negative and insig nificant (from ­0.27 to ­0.35). Thus, a weak tendency toward cycle shortening with increasing cycle ampli tude is detected. We now determine the degree of interrelation between the amplitude (Gmax) and the traditional (TTmin) and latitudinal (TTlat) lengths for cycles 12­ 23, which are presented in Table 1. The resultant correlation coefficients (R) between the cycle amplitude (Gmax) and the traditional (Gmax, TTmin) and latitudinal (Gmax, TTlat) lengths are R(Gmax, TTmin) = ­0.55 (the confidence level is CL = 93.5%) and R(Gmax, TTlat) = ­0.83 (CL = 99.9%), respectively. When the hemispheres are considered separately (k = 24), the correlation coeffi cients are R(Gmax, TTmin) = ­0.32 (CL = 85.1%) and R(Gmax, TTlat) = ­0.71 (CL = 99.98%) (Fig. 2, left hand panel). The corresponding regression equa tions have the following form: for the entire disk, TTlat = A0 ­ A1*Gmax, where A0 = (13.9 ± 0.7), A1= (0.29 ± 0.06), (k = 12, R(Gmax, TTlat) = ­0.83). When the hemispheres are considered separately, TTlat = A0 ­ A1*Gmax, where A0 = (13.0 ± 0.5), A1 = (0.40 ± 0.09), (N & S, k = 24, R(Gmax, TTlat) = ­0.71. In all cases the cycle length decreases with increas ing cycle amplitude. The value of the interaction

(expressed by negative correlation) is pronouncedly larger for the latitudinal length (TTlat), when LPRP is taken as a cycle phase reference point. Thus, the most significant "inverse" interrelation was revealed between the cycle amplitude and latitudi nal length. In addition, the activity secular trend is clearly defined in (very similar) dependences of cycle lengths TTlatN and TTlatS (solar hemispheres N and S) on the cycle number (Fig. 2, right hand panel). We now elucidate the character and value of the interrelation between the cycle amplitude (Gmax) and other time intervals of the 11 year cycle. Thus, the inverse correlation between the cycle growth phase length and amplitude at a maximum (the Waldmeier rule) is known (Waldmeier, 1935). The Wolf number is used to calculate the corresponding values. In this case the correlation is ­0.65 for cycles 12­23 correspond ing to the Greenwich catalog epoch. Let us assume that TTmimx(n) = TGmax(n) ­ TGmin(n) is the tra ditional length of the cycle growth branch in the nth cycle, and TTltmx(n) = TGmax(n) ­ Tlst(n) is the lat itudinal length of the cycle growth branch (by analogy with the cycle latitudinal length introduced above). The values of these parameters for cycles 12­23 are presented in Table 2. For cycles 12­23 for the entire disk (k = 12), the correlations between the cycle amplitude (Gmax) and the traditional (Gmax, TTmimx) and latitudinal (Gmax, TTltmax) lengths of the cycle growth branch are R(Gmax, TTmimx) = ­0.63 (CL = 97.1%) and R(Gmax, TTltmax) = ­0.86 (CL = 99.96%), respec tively. When the hemispheres are taken into account (k = 24), the corresponding correlation coefficients are R(Gmax, TTmimx) = ­0.25 (CL = 75.5%) and R(Gmax, TTlat) = ­0.56 (CL = 99.6%). In all cases the cycle growth branch length (as well as the total length considered above) is inversely pro portional to the amplitude of this cycle. In this case the interrelation is pronouncedly stronger when LPRP,
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Table 2. Traditional (TTmimx) and latitudinal (TTltmx) growth branches of the 11 year sunspot activity cycles for the entire solar disk and separately for the hemispheres (cycles 12­23) Northern Hemisphere Cycle TTmimx 12 13 14 15 16 17 18 19 20 21 22 23 2.68 4.93 4.34 5.08 5.97 3.51 5.45 5.15 2.69 3.28 3.66 4.85 TTltmx 3.89 5.45 5.77 4.99 6.57 3.26 5.67 4.29 2.59 3.42 2.58 4.84 TTmimx 5.46 3.44 7.02 4.18 3.96 5.08 3.36 3.51 5.31 3.88 4.93 5.97 TTltmx 5.43 3.70 7.03 4.39 4.26 4.28 3.80 3.14 5.07 2.18 4.52 5.24 TTmimx 5.08 3.51 4.86 4.18 5.07 3.65 3.44 3.74 5.75 3.36 3.43 5.60 TTltmx 5.81 4.25 5.44 4.97 5.43 3.35 3.67 3.12 5.59 3.10 2.64 5.16 Southern Hemisphere Entire disk

rather than the cycle amplitude minimum, is taken as a cycle phase reference point, which pronouncedly increases the significance of the latitudinal version of the Waldmeier rule. Since we established that the 11 year cycle ampli tude (Gmax) depends on TTlat (see above) and TTmst (see (Ivanov and Miletsky, 2014)), we could obtain the linear regression relationship simultaneously relating amplitude Gmax to TTmst and TTlat (Gmax = f(TTmst, TTlat)) based on the data for cycles 12­23 for the entire solar disk (K = 12). For the entire solar disk (K = 12), we have: Gmax = A0 + A1* TTmst ­ A2 *TTlat, where A0 = (29.2 ± 6.1), A1 = (2.0 ± 1.1), A2 = (1.7 ± 0.6), (R = 0.88, k = 12, SD = 1.6). With regard to the solar hemispheres N & S (K = 24), we have: Gmax = A0 + A1 *TTmst ­ A2* TTlat, where A0 = (15.2 ± 3.2), A1 = (0.8 ± 0.4), A2 = (0.9 ± 0.3), (R = 0.76, k = 24, rms is SD = 1.2). The obtained relationship makes it possible to determine the 11 year cycle amplitude by dividing the time interval between two cyclic minimums (the clas sical cycle length) into two intervals, i.e., the LPRP shift relative to a minimum (TTmst) and the cycle lat itudinal length (TTlat), using the curve of average lat itudes. We should note that the Gmax representation accuracy will remain unchanged if we select, instead of TTmst, a shift of the exponent approximating the curve of average latitudes relative to the nearest (previ ous) minimum that is proportional to TTmst as the first interval at any other fixed latitude and the corre sponding interval of this exponent up to the next min imum as the second interval (at the same latitude). These time intervals factually characterize the time position of the mean latitude curve relative to adjacent minimums. Thus, the considered cycle length division variant, when the first interval (TTmst) is specified
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equal to the LPRP shift relative to the nearest mini mum and the second interval is specified equal to the cycle latitudinal length (TTlat), is an important but particular case, when a reference latitude value equal to 26.6° in magnitude is selected as a fixed latitude (see Fig. 1). The problem of the time interval (we denote it TTcst (n, n + 1)) between the curves of average lati tudes (more exactly, the exponents approximating these curves) in adjacent cycles numbered n and n + 1 (see Fig. 4) is also very interesting. This time interval can be represented as a sum of two intervals: the cycle n
7 R = ­0.86 N + S = Total (K = 12) 6 TTtmx, year 5 4 3 2 4 6 8 10 12 GmaxTot 14 16

Fig. 3. Dependence of the latitudinal length of the cycle growth branch (TTltmx) on the cycle amplitude (GmaxTot) for the entire disk. 2014


1004

MILETSKY, IVANOV Gmax(n) 35 30 25 Latitude, deg 20 TTcst 15 10 5 0 1953 min R = ­0.83 TTlat(n) Gmax(n + 1) R = +0.69 TTmst(n + 1)

1956

1959

1962

1965 1968 Year

1971

1974

1977

Fig. 4. A sample of determination of the latitudinal distance (TTcst) between adjacent cycles n and n + 1 and the latitudinal time intervals of the cycle: the latitudinal length for cycle n (TTlat(n)) and the LPRP shift relative to a minimum for cycle n + 1 (TTmst(n + 1)). TTcst(n, n + 1) = TTlat(n) + TTmst(n + 1).

latitudinal length TTlat(n) and the interval of the LPRP shift relative to a cycle n + 1 minimum TTmst(n + 1). We previously established that each interval sum mand is related to the amplitude of the corresponding cycle. On this basis, we obtained the regression equation relating the TTcst(n, n +1) value (based on the data for the entire solar disk) to the amplitudes of adjacent cycles Gmax(n) and Gmax(n + 1): TTcst(n, n + 1) = A0­ A1*Gmax(n) + A2*Gmax(n + 1), where A0 = (11.9 ± 0.7), A1 = (0.19 ± 0.06), A2 = (0.08 ± 0.06), (R = 0.77, k = 11, SD = 0.52). 4. CONCLUSIONS For cycles 12­23, we studied the interrelation between the cycle amplitude (Gmax) and the tradi tional and latitudinal lengths, as well as the traditional and latitudinal lengths of the growth branches. We proposed selecting the latitudinal phase reference point (LPRP) as a cycle phase reference point. In all cases the cycle and cycle growth branch lengths decrease with increasing cycle amplitude. The interrelation value (expressed by negative cor relation) is pronouncedly larger for the cycle latitudi nal length and the growth branch latitudinal length than for similar traditional lengths. Thus, these inter relations become stronger and the significance of the latitudinal version of the Waldmeier rule increases if LPRP is selected as a cycle phase reference point.

We obtained an equation that makes it possible to determine the 11 year cycle amplitude based on infor mation about two cycle time intervals: the LPRP shift (TTmst) and the cycle latitudinal length (TTlat), which correlate with two successive cyclic minimums. We established that the length of the time interval, which separates the exponents approximating curves of average latitudes of adjacent cycles, can be repre sented as an expression relating the interval to these cycle amplitudes. The discovered interrelations between the 11 year cycle amplitude and the time intervals characterizing the curves of average latitudes time position relative to adjacent minimums show that important amplitude, latitude, and time parameters of the 11 year cycle cor relate with one another and should be taken into account when physical models of the solar magnetic cycle are constructed. ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research (project no. 13 02 00277), NSh 1625.2012.2 grant, and the Presidium of the Russian Academy of Sciences (programs 21 and 22). REFERENCES
Du, Z.L., Wang, H.N., and He, X.T., A new method to determine epochs of solar cycle extrema, Chin. J. Astron. Astrophys., 2006, vol. 6, pp. 338­344.
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INTERRELATION BETWEEN THE AMPLITUDE AND LENGTH Hathaway, D.H., Wilson, R.M., and Reichmann, R.J., The shape of the sunspot cycle, Sol. Phys., 1994, vol. 151, pp. 177­190. Hoyng, P., Helicity fluctuations in mean field theory: An explanation for the variability of the solar cycle?, Astron. Astrophys., 1993, vol. 272, pp. 321­329. Ivanov, V.G. and Miletsky, E.V., The SpÆrer law and link between latitude and amplitude characteristics of solar activity, Geomagn. Aeron., 2014, vol. 54, no. 7, pp. 907­ 915. Karak, B.B. and Choudhuri, A.R., The Waldmeier effect and the flux transport solar dynamo, Mon. Not. R. Astron. Soc., 2011, vol. 410, pp. 1503­1512. Nandy, D., Munoz Jaramillo, A., and Martens, P.C.H., The unusual minimum of solar cycle 23 caused by changes in the Sun's meridional plasma flows, Nature, 2011, vol. 471, pp. 80­82.

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Richards, M.T., Rogers, M.L., and Richards, D.S., Long term variability in the length of the solar cycle, Publ. Astron. Soc. Pacific, 2009, vol. 121, pp. 797­809. Roshchina, E.M. and Sarychev, A.P., Approximation of 11 year solar cycles, Sol. Syst. Res., 2011, vol. 45, no. 6, pp. 539­545. Solanki, S.K., Krivova, N.A., Schussler, M., and Fligge, M., Search for a relationship between solar cycle amplitude and length, Astron. Astrophys., 2002, vol. 396, pp. 1029­1035. Waldmeier, M., Neue Eigenschaften der Sonnenflecken kurve, Astron. Mitt., 1935, vol. 14, pp. 105­130. Waldmeier, M., The sunspot activity in the years 1610­1960, Zurich: Zurich Schult. Co., 1961. Wilson. R.M. On the distribution of sunspot cycle periods, J. Geophys. Res., 1987, vol. 92, pp. 10 101­10 104.

Translated by Yu. Safronov

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