The Schwabe and Gleissberg Periods in the Wolf Sunspot Numbers and the Group Sunspot Numbers

Three wavelet functions: the Morlet wavelet, the Paul wavelet, and the DOG wavelet have been respectively performed on both the monthly Wolf sunspot numbers (R_z) from January 1749 to May 2004 and the monthly group sunspot numbers (R_g) from June 1795 to December 1995 to study the evolution of the Gleissberg and Schwabe periods of solar activity. The main results obtained are (1) the two most obvious periods in both the R_z and R_g are the Schwabe and Gleissberg periods. The Schwabe period oscillated during the second half of the eighteenth century and was steady from the 1850s onward. No obvious drifting trend of the Schwabe period exists. (2) The Gleissberg period obviously drifts to longer periods the whole consideration time, and the drifting speed of the Gleissberg period is larger for R_z than for R_g. (3) Although the Schwabe-period values for R_z and R_g are about 10.7 years, the value for R_z seems slightly larger than that for R_g. The Schwabe period of R_z is highly significant after the 1820s, and the Schwabe period of R_g is highly significant over almost the whole consideration time except for about 20 years around the 1800s. The evolution of the Schwabe period for both R_z and R_g in time is similar to each other. (4) The Gleissberg period in R_z and R_g is highly significant during the whole consideration time, but this result is unreliable at the two ends of each of the time series of the data. The evolution of the Gleissberg period in R_z is similar to that in R_g.     

太阳黑子的世纪周期及对24、25活动周的预报

太阳活动除了具有公认的11 a周期以外,还存在着一个80～120 a变化的世纪周期,也称为Gleissberg周期.使用傅里叶变换和小波分析的方法,分析了1700～2008年的年均黑子数世纪周期的变化规律.得到结果:在太阳活动世纪周期的低谷期,所对应11 a太阳周的极大年和极小年的黑子数目都比其他太阳周的低.在这300多年里,世纪周期的周期长度也有变化.由世纪周期的变化趋势,预测第24、25太阳活动周将处于世纪周期的低谷期.通过对以前3个世纪周期的谷期黑子数求平均的方法,得到第24,25太阳周极大年年均黑子数为63.6±21.1,极小年的为2.2±2.1.这些结果有助于理解当前太阳活动反常宁静这一现象.     

Synchronization of Sunspot Numbers and Sunspot Areas

Sunspot activity is usually described by either sunspot numbers or sunspot areas. The smoothed monthly mean sunspot numbers (SNs) and the smoothed monthly mean areas (SAs) in the time interval from November 1874 to September 2007 are used to analyze their phase synchronization. Both the linear method (fast Fourier transform) and some nonlinear approaches (continuous wavelet transform, cross-wavelet transform, wavelet coherence, cross-recurrence plot, and line of synchronization) are utilized to show the phase relation between the two series. There is a high level of phase synchronization between SNs and SAs, but the phase synchronization is detected only in their low-frequency components, corresponding to time scales of about 7 to 12 years. Their high-frequency components show a noisy behavior with strong phase mixing. Coherent phase variables should exist only for a frequency band with periodicities around the dominating 11-year cycle for SNs and SAs. There are some small phase differences between them. SNs lag SAs during most of the considered time interval, and they are in general more asynchronous around the minimum and maximum times of a cycle than at the ascending and descending phases.     

Time Series Modeling of Sunspot Numbers Using Long-Range Cyclical Dependence

This paper deals with the analysis of the sunspot number time series using a new technique based on cyclical long-range dependence. The results show that sunspot numbers have a periodicity of 130 months but, more importantly, that the series is highly persistent, with an order of cyclical fractional integration slightly above 0.30. That means that the series displays long memory, with a large degree of dependence between the observations that tends to disappear very slowly in time.     

Tilt of Sunspot Bipoles in Solar Cycles 15 to 24

We use recently digitized sunspot drawings from Mount Wilson Observatory to investigate the latitudinal dependence of tilt angles of active regions and its change with solar cycle. The drawings cover the period from 1917 to present and contain information as regards polarity and strength of magnetic field in sunspots. We identified clusters of sunspots of same polarity, and used these clusters to form “bipole pairs”. The orientation of these bipole pairs was used to measure their tilts. We find that the latitudinal profile of tilts does not monotonically increase with latitude as most previous studies assumed, but instead, it shows a clear maximum at about 25?–?30 degree latitudes. Functional dependence of tilt (\(\gamma\)) on latitude (\(\varphi\)) was found to be \(\gamma= (0.20\pm0.08) \sin(2.80 \varphi) + (-0.00\pm0.06)\). We also find that latitudinal dependence of tilts varies from one solar cycle to another, but larger tilts do not seem to result in stronger solar cycles. Finally, we find the presence of a systematic offset in tilt of active regions (non-zero tilts at the equator), with odd cycles exhibiting negative offset and even cycles showing the positive offset.     

Reconstruction of a Monthly Homogeneous Sunspot Area Series Since 1832

In this work, a procedure to elaborate a homogeneous sunspot area series using the Royal Greenwich Observatory/USAF/NOAA data (from 1874 to the present) and the De la Rue and co-workers data (from 1832 to 1868) is presented. These two data series correspond to time intervals that do not overlap and a direct comparison between them could not be carried out. We used the International Sunspot Number (R_i) and the Group Sunspot Number (R_G) as a link between the two original series. Thus, two homogeneous sunspot area series have been built using a simple mathematic procedure based on linear relations.     

Duration of Polar Activity Cycles and Their Relation to Sunspot Activity

We have defined the duration of polar magnetic activity as the time interval between two successive polar reversals. The epochs of the polarity reversals of the magnetic field at the poles of the Sun have been determined (1) by the time of the final disappearance of the polar crown filaments and (2) by the time between the two neighbouring reversals of the magnetic dipole configuration (l=1) from the H synoptic charts covering the period 1870–2001. It is shown that the reversals for the magnetic dipole configuration (l=1) occur on an average 3.3±0.5 years after the sunspot minimum according to the H synoptic charts (Table I) and the Stanford magnetograms (Table III). If we set the time of the final disappearance of the polar crown filaments (determined from the latitude migration of filaments) as the criterion for deciding the epoch of the polarity reversal of the polar fields, then the reversal occurs on an average 5.8±0.6 years from sunspot minimum (last column of Table I). We consider this as the most reliable diagnostic for fixing the epoch of reversals, as the final disappearance of the polar crown filaments can be observed without ambiguity. We show that shorter the duration of the polar activity cycle (i.e., the shorter the duration between two neighbouring reversals), the more intense is the next sunspot cycle. We also notice that the duration of polar activity is always more in even solar cycles than in odd cycles whereas the maximum Wolf numbers W _\max is always higher for odd solar cycles than for even cycles. Furthermore, we assume there is a secular change in the duration of the polar cycle. It has decreased by 1.2 times during the last 120 years.     

Modeling of Sunspot Numbers by a Modified Binary Mixture of Laplace Distribution Functions

This paper presents a new approach for describing the shape of 11-year sunspot cycles by considering the monthly averaged values. This paper also brings out a prediction model based on the analysis of 22 sunspot cycles from the year 1749 onward. It is found that the shape of the sunspot cycles with monthly averaged values can be described by a functional form of modified binary mixture of Laplace density functions, modified suitably by introducing two additional parameters in the standard functional form. The six parameters, namely two locations, two scales, and two area parameters, characterize this model. The nature of the estimated parameters for the sunspot cycles from 1749 onward has been analyzed and finally we arrived at a sufficient set of the parameters for the proposed model. It is seen that this model picks up the sunspot peaks more closely than any other model without losing the match at other places at the same time. The goodness of fit for the proposed model is also computed with the Hathaway – Wilson – Reichmann measure, which shows, on average, that the fitted model passes within 0.47 standard deviations of the actual averaged monthly sunspot numbers.     

Relative Sunspot Numbers in the First Half of the Seventeenth Century

We derived daily relative sunspot numbers and their monthly and annual means in the first half of the seventeenth century. The series of observations collected by Wolf were recorded in the years 1611–1613 and 1642–1644. We used a nonlinear two-step interpolation method derived earlier (Letfus, 1996, 1999) to enlarge the number of daily data. Before interpolation the relative monthly frequency of observations in 24 months of the first time interval 1611–1613 was 49.4% and in 22 months of the second interval 1642–1644 was 49.9%. After interpolation the relative frequency increased in the first time interval to 91.3%, in the second time interval to 82.6%. Most data series in the years 1611–1613 overlap one another and also overlap with a series, for which Wolf estimated a scaling factor converting relative sunspot numbers on the Zürich scale. We derived the scaling factors of all individual series of observations also from the ratios of observed numbers of sunspots to the numbers of sunspot groups (Letfus, 2000). The differences between almost all scaling factors derived in one and the other way are not substantial. All data series were homogenized by application of scaling factors and parallel data in the overlapping parts of data series were averaged. Resulting daily relative sunspot numbers and their monthly and annual means in the years l61l–1613 are given in Table I and those in the years 1642–1644 in Table II. The annual means of these data are compared with analogous data obtained otherwise.     

A Comparison of the 10.7-cm Radio Flux Values and the International Sunspot Numbers for Solar Activity Cycles 19, 20, and 21

A nonlinear analysis of the daily 10.7-cm radio flux values for each of Solar Cycles 19, 20, and 21 is used to determine if the results match those of the International Sunspot Numbers for each of these cycles. Fractals and chaos are described and a brief review of utilizing fractals and chaos is given. The origin of the 10.7-cm radio flux is discussed and a short review of recent work discussing its measurement and its relation to the international sunspot number and other proxies for solar activity cycles given. The parameters used to describe chaos for the 10.7-cm radio flux are discussed. The length of the data sets for either statistical analysis or nonlinear analysis of the 10.7-cm radio flux values is considered. These results indicate that the 10.7-cm radio flux values appear to be stochastic for Cycle 19 and chaotic for Cycles 20 and 21. The International Sunspot Numbers show similar behavior for these three cycles. A day-by-day comparison of the dimensionless 10.7-cm radio flux values and the dimensionless International Sunspot Numbers differences shows a linear trend. The results remain consistent in that the 10.7-cm radio flux values indicate, as did the International Sunspot Numbers, that there is a transition from stochastic behavior for Cycle 19 to chaotic behavior in Cycles 20 and 21. The day-by-day comparison of the 10.7-cm radio flux values and the International Sunspot Numbers emphasizes that the 10.7?cm radio flux values are responding to the magnetic field associated with the sunspots.     

Changing Relationships Between Sunspot Number,Total Sunspot Area and $F_{10.7}$ in Cycles 23 and 24

This article is an update of a study (Tapping and Valdès in Solar Phys. 272, 337, 2011) made in the early part of Cycle 24 using an intercomparison of various solar activity indices (including sunspot number and the 10.7 cm solar radio flux), in which it was concluded that a change in the relationship between photospheric and chromospheric/coronal activity took place just after the maximum of Cycle 23 and continued into Cycle 24. Precursors (short-term variations) were detected in Cycles 21 and 22. Since then the sunspot number index data have been substantially revised. This study is intended to be an update of the earlier study and to assess the impact of the revision of the sunspot number data upon those conclusions. This study compares original and revised sunspot number, total sunspot area, and 10.7 cm solar radio flux. The conclusion is that the transient changes in Cycles 21 and 22, and the more substantial change in Cycle 23, remain evident. Cycle 24 shows indications that the deviation was probably another short-term one.     

Cosmic Ray Intensity Variations in Relation with Solar Flare Index and Sunspot Numbers

From the monthly data of cosmic ray intensity (CRI), sunspot numbers (SSN) and solar flare index (SFI), an attempt has been made to study the relationship between CRI and solar activity (SA) parameters SSN and SFI. The correlation between SA parameters and CRI for different neutron monitoring stations having low, middle and high cut-off rigidity has been investigated. The anti-correlation between SA and CRI is found to exist with some time lag. Based on the method of minimizing correlation coefficient and time-delayed component method, the observed time-lag between SA parameters (SSN and SFI) and CRI has been found to be large for odd solar cycles in comparison to even solar cycles. The results of time-lag analysis between CRI and SSN and between CRI-SFI have also been compared. The findings of correlative study between CRI and SSN are in agreement with earlier results, while the CRI-SFI relationship provides new insights to understand the solar modulation of cosmic rays.     

Wavelet Analysis of Several Important Periodic Properties in the Relative Sunspot Numbers

We investigate the wavelet transform of yearly mean relative sunspot number series from 1700 to 2002. The curve of the global wavelet power spectrum peaks at 11-yr, 53-yr and 101-yr periods. The evolution of the amplitudes of the three periods is studied. The results show that around 1750 and 1800, the amplitude of the 53-yr period was much higher than that of the the 11-yr period, that the ca. 53-yr period was apparent only for the interval from 1725 to 1850, and was very low after 1850, that around 1750, 1800 and 1900, the amplitude of the 101-yr period was higher than that of the 11-yr period and that, from 1940 to 2000, the 11-yr period greatly dominates over the other two periods.     

Reconstruction of Monthly and Yearly Group Sunspot Numbers From Sparse Daily Observations

Some periods before 1820 are poorly covered by sunspot observations. In addition to apparent, long observational gaps, there are also periods when there are only few sparse daily sunspot observations during a long time. It is important to estimate the reliability of the monthly and yearly mean sunspot values obtained from such sparse daily data. Here we suggest a new method to estimate the reliability of individual monthly means. The method is based on comparing the actual sparse data (sample population) to the well-measured sunspot data in 1850–1996 (reference population), and assumes that the statistical properties of sunspot activity remain similar throughout the entire period. For each sample population we first found those months in the reference population that contain the same data set, and constructed the statistical distribution of the corresponding monthly means. The mean and standard error of this distribution represent the mean and uncertainty of a monthly mean sunspot number reconstructed from sparse daily observations. The simple arithmetic mean of daily values can be adequately applied for months which contain more than 4–5 evenly distributed daily observations. However, the reliability of monthly means for less covered months has to be estimated more carefully. Using the estimated, new monthly values, we have also calculated the weighted annual sunspot numbers.     

The 27-Day Signal in Sunspot Number Series and the Solar Dynamo

We analyze the Wolf number daily series WN (1849 to present) as well as two other related series characterizing solar activity. Our analysis consists in computing the amplitude of a given Fourier component in a sliding time window and examining its long-term evolution. We start with the well-known 27.03- and 27.6-day periods and observe strong decadal variations of this amplitude as well as a sharp increase of the average value starting around 1905. We then consider a packet of 31 lines with periods from 25.743 to 28.453 days, which is shown to be a better representation of the synodic solar rotation. We first examine the temporal evolution of individual lines, then the energy of the packet. The energy of the packet increases sharply at the beginning of the 20th century, leading by more than two decades the well-known increase of the Wolf number. The nonaxisymmetry of sunspots increases before the total increase of activity and may be considered as a precursor. We discuss briefly and tentatively this observation in terms of solar dynamo theory.     

Identification of AlF Molecular Lines in Sunspot Umbral Spectra

High resolution FTS sunspot umbral spectra from NSO/Kitt Peak were used to detect rotational lines due to 11 electronic transitions of the molecule AlF, in the wavelength region 4400 – 9000 Å. The presence of lines due to bands C – A (0,0), (0,1), (0,2) and (1,2), D – A (0,0) and (0,1), F – A (0,0), G – A (0,0), and F – B (0,0) is confirmed. Further, the presence of lines due to C – A (1,0) and (1,1) transitions is treated as doubtful because of heavy blending in the region, by rotational lines of TiO. Among the identified lines, those which are well resolved were selected for measurement of equivalent widths. The measured values for the bands D – A (0,0) and F – A (0,0) were used to estimate the respective effective rotational temperatures to be 1240 ± 120 K and 2390 ± 400 K.     

Reconstruction of Subdecadal Changes in Sunspot Numbers Based on the NGRIP 10Be Record

Sunspot observations since 1610 A.D. show that the solar magnetic activity displays long-term changes, from Maunder Minimum-like low-activity states to Modern Maximum-like high-activity episodes, as well as short-term variations, such as the pronounced 11-year periodicity. Information on changes in solar activity levels before 1610 relies on proxy records of solar activity stored in natural archives, such as ¹⁰Be in ice cores and ¹⁴C in tree rings. These cosmogenic radionuclides are produced by the interaction between Galactic cosmic rays (GCRs) and atoms in the Earth’s atmosphere; their production rates are anti-correlated with the solar magnetic activity. The GCR intensity displays a distinct 11-year periodicity due to solar modulation of the GCRs in the heliosphere, which is inversely proportional to, but out of phase with, the 11-year solar cycle. This implies a time lag between the actual solar cycles and the GCR intensity, which is known as the hysteresis effect. In this study, we use the North Greenland Ice Core Project (NGRIP) records of the ¹⁰Be flux to reconstruct the solar modulation strength (Φ), which describes the modulation of GCRs throughout the heliosphere, to reconstruct both long-term and subdecadal changes in sunspot numbers (SSNs). We compare three different approaches for reconstructing subdecadal-scale changes in SSNs, including a linear approach and two approaches based on the hysteresis effect, i.e. models with ellipse–linear and ellipse relationships between Φ and SSNs. We find that the ellipse approach provides an amplitude-sensitive reconstruction and the highest cross-correlation coefficients in comparison with the ellipse–linear and linear approaches. The long-term trend in the reconstructed SSNs is computed using a physics-based model and agrees well with the other group SSN reconstructions. The new empirical approach, combining a physics-based model with ellipse-modeling of the 11-year cycle, therefore provides a method for reconstructing SSNs during individual solar cycles based on ¹⁰Be in ice cores. This, in turn, represents a new window for studying short-term changes in solar activity on unprecedented timescales, which may help improve our understanding of the solar dynamo.     

Evolution of the Level of Sunspot Activity in Solar Cycles I. Evolution in the Descending Phase

Taking the 13-point smoothed monthly sunspot number, Ri, and the deviation of the 13 associated monthly sunspot numbers from the smoothed one, D_i, as a number-pair describing the global level of sunspot activity, the evolution of the level is statistically studied for the period from the month which is just 48 months before the minimum to the minimum in the descending phase, using the observed data of Solar Cycles 10 to 22. Our results show (1) for 46 months (94%) of the studied 49 months it is found that for a given month, the distribution of the 13 pairs which come from the 13 solar cycles on a log Ri-D_i plane may be fitted by a straight line with a correlation coefficient larger than the critical one at confidence level α= 5%, and for 36 months (73%) the fitting is even better, for α= 1%;(2) time variations of these two parameters and their correlations in the studied period can be described respectively by functions of time, whose main trends may be expressed by a linear or simple curvilinear function; (3) the evolutionary path of the level of sunspot activity may be represented by a logarithmic function as log R_i=0.704 In D_i-0.291.