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.     

Prediction of the Maximum Amplitude and Timing of Sunspot Cycle 24

Precursor techniques, in particular those using geomagnetic indices, often are used in the prediction of the maximum amplitude for a sunspot cycle. Here, the year 2008 is taken as being the sunspot minimum year for cycle 24. Based on the average aa index value for the year of the sunspot minimum and the preceding four years, we estimate the expected annual maximum amplitude for cycle 24 to be about 92.8±19.6 (1-sigma accuracy), indicating a somewhat weaker cycle 24 as compared to cycles 21 – 23. Presuming a smoothed monthly mean sunspot number minimum in August 2008, a smoothed monthly mean sunspot number maximum is expected about October 2012±4 months (1-sigma accuracy).     

太阳黑子月平均面积数活动周及其特征

我们对第12周至第22周的太阳黑子月平均面积数进行统计分析,并与相应的太阳黑子月平均数相比较,结果表明太阳黑子月平均面积数活动周与太阳黑子月平均数活动周有一定的关系。在多数情况下,太阳黑子出现最大值的时间与太阳黑子面积数出现最大值的时间上不一致;太阳黑子平滑月平均数活动周上升期与太阳黑子平滑月平均面积数上升期在大多数情况下不相同;太阳黑子平滑月平均数活动周平均效果的瓦德迈尔效应(Waldmeiereffect)一般要比太阳黑子平滑平均面积数的活动周明显;文中还对太阳黑子平滑月平均面积数活动周的特征进行了分析。     

A Kalman Filter Technique for Improving Medium-Term Predictions of the Sunspot Number

In this work we describe a technique developed to improve medium-term prediction methods of monthly smoothed sunspot numbers. Each month, the predictions are updated using the last available observations (see the monthly output in real time at ). The improvement of the predictions is provided by applying an adaptive Kalman filter to the medium-term predictions obtained by any other method, using the six-monthly mean values of sunspot numbers covering the six months between the last available value of the 13-month running mean (the starting point for the predictions) and the “current time” (i.e. now). Our technique provides an effective estimate of the sunspot index at the current time. This estimate becomes the new starting point for the updated prediction that is shifted six months ahead in comparison with the last available 13-month running mean, and it provides an increase of prediction accuracy. Our technique has been tested on three medium-term prediction methods that are currently in real-time operation: The McNish–Lincoln method (NGDC), the standard method (SIDC), and the combined method (SIDC). With our technique, the prediction accuracy for the McNish–Lincoln method is increased by 17 – 30%, for the standard method by 5 – 21% and for the combined method by 6 – 57%.     

The Shape of Solar Cycle Described by a Modified Gaussian Function

The shape of each sunspot cycle is found to be well described by a modified Gaussian function with four parameters: peak size A, peak timing t _m, width B, and asymmetry α. The four-parameter function can be further reduced to a two-parameter function by assuming that B and α are quadratic functions of t _m, computed from the starting time (T ₀). It is found that the shape can be better fitted by the four-parameter function, while the remaining behavior of the cycle can be better predicted by the two-parameter function when using the data from a few (about two) months after the starting time defined by the smoothed monthly mean sunspot numbers. As a new solar cycle is ongoing, its remaining behavior can be constructed by the above four- or two-parameter function. A running test shows that the maximum amplitude of the cycle can be predicted to within 15% at about 25 months into the cycle based on the two-parameter function. A preliminary modeling to the first 24 months of data available for the current cycle indicates that the peak of cycle 24 may probably occur around June 2013±7 months with a size of 72±11. The above results are compared to those by quasi-Planck functions.     

An Estimate for the Size of Sunspot Cycle 24

For the sunspot cycles in the modern era (cycle?10 to the present), the ratio of R _Z(max)/R _Z(36th month) equals 1.26±0.22, where R _Z(max) is the maximum amplitude of the sunspot cycle?using smoothed monthly mean sunspot number and R _Z(36th month) is the smoothed monthly mean sunspot number 36 months after cycle?minimum. For the current sunspot cycle?24, the 36th month following the cycle?minimum occurred in November 2011, measuring?61.1. Hence, cycle?24 likely will have a maximum amplitude of about 77.0±13.4 (the one-sigma prediction interval), a value well below the average R _Z(max) for the modern era sunspot cycles (about 119.7±39.5).     

What the Sunspot Record Tells Us About Space Climate

The records concerning the number, sizes, and positions of sunspots provide a direct means of characterizing solar activity over nearly 400 years. Sunspot numbers are strongly correlated with modern measures of solar activity including: 10.7-cm radio flux, total irradiance, X-ray flares, sunspot area, the baseline level of geomagnetic activity, and the flux of galactic cosmic rays. The Group Sunspot Number provides information on 27 sunspot cycles, far more than any of the modern measures of solar activity, and enough to provide important details about long-term variations in solar activity or “Space Climate.” The sunspot record shows: 1) sunspot cycles have periods of 131± 14 months with a normal distribution; 2) sunspot cycles are asymmetric with a fast rise and slow decline; 3) the rise time from minimum to maximum decreases with cycle amplitude; 4) large amplitude cycles are preceded by short period cycles; 5) large amplitude cycles are preceded by high minima; 6) although the two hemispheres remain linked in phase, there are significant asymmetries in the activity in each hemisphere; 7) the rate at which the active latitudes drift toward the equator is anti-correlated with the cycle period; 8) the rate at which the active latitudes drift toward the equator is positively correlated with the amplitude of the cycle after the next; 9) there has been a significant secular increase in the amplitudes of the sunspot cycles since the end of the Maunder Minimum (1715); and 10) there is weak evidence for a quasi-periodic variation in the sunspot cycle amplitudes with a period of about 90 years. These characteristics indicate that the next solar cycle should have a maximum smoothed sunspot number of about 145 ± 30 in 2010 while the following cycle should have a maximum of about 70 ± 30 in 2023.     

预报第23周平滑月平均太阳黑子数的一种方法

利用已知的22个完整太阳活动周平滑月平均黑子数的记录，对正在进行的太阳周发展趋势给出了预测方法，并应用于第23周，同时与其他预报方法的结果进行了比较。     

Sunspot Group Decay

We examine daily records of sunspot group areas (measured in millionths of a solar hemisphere or μHem) for the last 130 years to determine the rate of decay of sunspot group areas. We exclude observations of groups when they are more than 60° in longitude from the central meridian and only include data when at least three days of observations are available following the date of maximum area for a group’s disk passage. This leaves data for over 18 000 measurements of sunspot group decay. We find that the decay rate increases linearly from 28 μHem day⁻¹ to about 140 μHem day⁻¹ for groups with areas increasing from 35 μHem to 1000 μHem. The decay rate tends to level off for groups with areas larger than 1000 μHem. This behavior is very similar to the increase in the number of sunspots per group as the area of the group increases. Calculating the decay rate per individual sunspot gives a decay rate of about 3.65 μHem day⁻¹ with little dependence upon the area of the group. This suggests that sunspots decay by a Fickian diffusion process with a diffusion coefficient of about 10 km² s⁻¹. Although the 18 000 decay rate measurements are lognormally distributed, this can be attributed to the lognormal distribution of sunspot group areas and the linear relationship between area and decay rate for the vast majority of groups. We find weak evidence for variations in decay rates from one solar cycle to another and for different phases of each sunspot cycle. However, the strongest evidence for variations is with latitude and the variations with cycle and phase of each cycle can be attributed to this variation. High latitude spots tend to decay faster than low latitude spots.     

Dependence of SSNM on SSNm – a Reconsideration for Predicting the Amplitude of a Sunspot Cycle

An improved correlation between maximum sunspot number (SSN_M) and the preceding minimum (SSN_m) is reported when the monthly mean sunspot numbers are smoothed with a 13-month running window. This relation allows prediction of the amplitude of a sunspot cycle by making use of the sunspot data alone. The estimated smoothed maximum sunspot number (126±26) and time of maximum epoch (second half of 2000) of cycle 23 are in good agreement with the predictions made by some of the precursor methods.     

Group Sunspot Numbers: Sunspot Cycle Characteristics

We examine the `Group' sunspot numbers constructed by Hoyt and Schatten to determine their utility in characterizing the solar activity cycle. We compare smoothed monthly Group sunspot numbers to Zürich (International) sunspot numbers, 10.7-cm radio flux, and total sunspot area. We find that the Zürich numbers follow the 10.7-cm radio flux and total sunspot area measurements only slightly better than the Group numbers. We examine several significant characteristics of the sunspot cycle using both Group numbers and Zürich numbers. We find that the `Waldmeier Effect' – the anti-correlation between cycle amplitude and the elapsed time between minimum and maximum of a cycle – is much more apparent in the Zürich numbers. The `Amplitude–Period Effect' – the anti-correlation between cycle amplitude and the length of the previous cycle from minimum to minimum – is also much more apparent in the Zürich numbers. The `Amplitude–Minimum Effect' – the correlation between cycle amplitude and the activity level at the previous (onset) minimum is equally apparent in both the Zürich numbers and the Group numbers. The `Even–Odd Effect' – in which odd-numbered cycles are larger than their even-numbered precursors – is somewhat stronger in the Group numbers but with a tighter relationship in the Zürich numbers. The `Secular Trend' – the increase in cycle amplitudes since the Maunder Minimum – is much stronger in Group numbers. After removing this trend we find little evidence for multi-cycle periodicities like the 80-year Gleissberg cycle or the two- and three-cycle periodicities. We also find little evidence for a correlation between the amplitude of a cycle and its period or for a bimodal distribution of cycle periods. We conclude that the Group numbers are most useful for extending the sunspot cycle data further back in time and thereby adding more cycles and improving the statistics. However, the Zürich numbers are slightly more useful for characterizing the on-going levels of solar activity.     

Analysis of Sunspot Activity Cycles

A nonlinear analysis of the daily sunspot number for each of cycles 10 to 23 is used to indicate whether the convective turbulence is stochastic or chaotic. There is a short review of recent papers considering sunspot statistics and solar activity cycles. The differences in the three possible regimes – deterministic laminar flow, chaotic flow, and stochastic flow – are discussed. The length of data sets necessary to analyze the regimes is investigated. Chaos is described and a chronology of recent results that utilize chaos and fractals to analyze sunspot numbers follows. The parameters necessary to describe chaos – time lag, phase space, embedding dimension, local dimension, correlation dimension, and the Lyapunov exponents – are determined for the attractor for each cycle. Assuming the laminar regime is unlikely if chaos is not indicated in a cycle by the calculations, the regime must be stochastic. The sunspot numbers in each of cycles 10 to 19 indicate stochastic behavior. There is a transition from stochastic to chaotic behavior of the sunspot numbers in cycles 20, 21, 22, and 23. These changes in cycles 20 – 23 may indicate a change in the scale of turbulence in the convection zone that could result in a change in the convective heat transfer and a change in the size of the convection region for these four cycles.     

The Behavior of Sunspot Contrast during Cycle 23

Results are presented from a study of various sunspot contrast parameters in broadband red (672.3 nm) Cartesian full-disk digital images taken at the San Fernando Observatory (SFO) over eight years, 1997 – 2004, of the twenty-third sunspot cycle. A subset of over 2700 red sunspots was analyzed and values of average and maximum sunspot contrast as well as maximum umbral contrast were compared to various sunspot parameters. Average and maximum sunspot contrasts were found to be significantly correlated with sunspot area (r _s=− 0.623 and r _s=− 0.714, respectively). Maximum umbral contrast was found to be significantly correlated with umbral area (r _s=− 0.535). These results are in agreement with the works of numerous other authors. No significant dependence was detected between average contrast, maximum contrast, or maximum umbral contrast during the rising phase of the solar cycle (r _s=0.024, r _s=0.033, and r _s=0.064, respectively). During the decay phase, no significant correlation was found between average contrast or maximum contrast and time (r _s=− 0.057 and r _s=0.009, respectively), with a weak dependence seen between maximum umbral contrast and cycle (r _s=0.102).     

Correlation Function Analysis between Sunspot Cycle Amplitudes and Rise Times

The running cross-correlation coefficient between solar-cycle amplitudes and rise times at a certain cycle lag is found to vary in time, when using the smoothed monthly-mean sunspot group numbers available for 1610 – 1995. It may be negative or positive for different periods of time. The Waldmeier effect (in which the rise times decrease with amplitude) is also found to be very weak for some cycles. This result represents an observational constraint on solar-dynamo models and can help us better understand the long-term evolution of solar activity.     

Prediction of Solar Cycle 24 Using Sunspot Number near the Cycle Minimum

Correlations between monthly smoothed sunspot numbers at the solar-cycle maximum [R _max] and duration of the ascending phase of the cycle [T _rise], on the one hand, and sunspot-number parameters (values, differences and sums) near the cycle minimum, on the other hand, are studied. It is found that sunspot numbers two?–?three years around minimum correlate with R _max or T _rise better than those exactly at the minimum. The strongest correlation (Pearson’s r=0.93 with P<0.001 and Spearman’s rank correlation coefficient r _S=0.95 with P=9×10^?12) proved to be between R _max and the sum of the increase of activity over 30 months after the cycle minimum and the drop of activity over 30 or 36 months before the minimum. Several predictions of maximal amplitude and duration of the ascending phase for Solar Cycle 24 are given using sunspot-number parameters as precursors. All of the predictions indicate that Solar Cycle 24 is expected to reach a maximal smoothed monthly sunspot number (SSN) of 70?–?100. The prediction based on the best correlation yields the maximal amplitude of 90±12. The maximum of Solar Cycle 24 is expected to be in December 2013?–?January 2014. The rising and declining phases of Solar Cycle 24 are estimated to be about 5.0 and 6.3 years, respectively. The minimum epoch between Solar Cycles 24 and 25 is predicted to be at 2020.3 with minimal SSN of 5.1?–?5.4. We predict also that Solar Cycle 25 will be slightly stronger than Solar Cycle 24; its maximal SSN will be of 105?–?110.     

The shape of the sunspot cycle

The temporal behavior of a sunspot cycle, as described by the International sunspot numbers, can be represented by a simple function with four parameters: starting time, amplitude, rise time, and asymmetry. Of these, the parameter that governs the asymmetry between the rise to maximum and the fall to minimum is found to vary little from cycle to cycle and can be fixed at a single value for all cycles. A close relationship is found between rise time and amplitude which allows for a representation of each cycle by a function containing only two parameters: the starting time and the amplitude. These parameters are determined for the previous 22 sunspot cycles and examined for any predictable behavior. A weak correlation is found between the amplitude of a cycle and the length of the previous cycle. This allows for an estimate of the amplitude accurate to within about 30% right at the start of the cycle. As the cycle progresses, the amplitude can be better determined to within 20% at 30 months and to within 10% at 42 months into the cycle, thereby providing a good prediction both for the timing and size of sunspot maximum and for the behavior of the remaining 7–12 years of the cycle. The U.S. Government right to retain a non-exclusive, royalty free licence in and to any copyright is acknowledged.     

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.     

Fluctuations of Solar Activity during the Declining Phase of the 11-Year Sunspot Cycle

The number of coronal mass ejections (CMEs) erupting from the Sun follows a trend similar to that of sunspot numbers during the rising and maximum phase of the solar cycle. In the declining phase, the CME number has large fluctuations, dissimilar to those of sunspot numbers. In several studies of solar – interplanetary and solar – terrestrial relationships, the sunspot numbers and the 2800-MHz flux (F10) are used as representative of solar activity. In the rising phase, this may be adequate, but in the declining phase, solar parameters such as CMEs may have a different behaviour. Cosmic-ray Forbush decreases may occur even when sunspot activity is low. Therefore, when studying the solar influence on the Earth, one has to consider that although geomagnetic conditions at solar maximum will be disturbed, conditions at solar minimum may not be necessarily quiet.     

A Comparison of Wolf's Reconstructed Record of Annual Sunspot Number with Schwabe's Observed Record of ‘Clusters of Spots’ for the Interval of 1826–1868

Samuel Heinrich Schwabe, the discoverer of the sunspot cycle, observed the Sun routinely from Dessau, Germany during the interval of 1826–1868, averaging about 290 observing days per year. His yearly counts of ‘clusters of spots’ (or, more correctly, the yearly number of newly appearing sunspot groups) provided a simple means for describing the overt features of the sunspot cycle (i.e., the timing and relative strengths of cycle minimum and maximum). In 1848, Rudolf Wolf, a Swiss astronomer, having become aware of Schwabe's discovery, introduced his now familiar ‘relative sunspot number’ and established an international cadre of observers for monitoring the future behavior of the sunspot cycle and for reconstructing its past behavior (backwards in time to 1818, based on daily sunspot number estimates). While Wolf's reconstruction is complete (without gaps) only from 1849 (hence, the beginning of the modern era), the immediately preceding interval of 1818–1848 is incomplete, being based on an average of 260 observing days per year. In this investigation, Wolf's reconstructed record of annual sunspot number is compared against Schwabe's actual observing record of yearly counts of clusters of spots. The comparison suggests that Wolf may have misplaced (by about 1–2 yr) and underestimated (by about 16 units of sunspot number) the maximum amplitude for cycle 7. If true, then, cycle 7's ascent and descent durations should measure about 5 years each instead of 7 and 3 years, respectively, the extremes of the distributions, and its maximum amplitude should measure about 86 instead of 70. This study also indicates that cycle 9's maximum amplitude is more reliably determined than cycle 8's and that both appear to be of comparable size (about 130 units of sunspot number) rather than being significantly different. Therefore, caution is urged against the indiscriminate use of the pre-modern era sunspot numbers in long-term studies of the sunspot cycle, since such use may lead to specious results.     

The Relative Phase Asynchronization between Sunspot Numbers and Polar Faculae

The monthly sunspot numbers compiled by Temmer et al. and the monthly polar faculae from observations of the National Astronomical Observatory of Japan, for the interval of March 1954 to March 1996, are used to investigate the phase relationship between polar faculae and sunspot activity for total solar disk and for both hemispheres in solar cycles 19, 20, 21 and 22. We found that (1) the polar faculae begin earlier than sunspot activity, and the phase difference exhibits a consistent behaviour for different hemispheres in each of the solar cycles, implying that this phenomenon should not be regarded as a stochastic fluctuation; (2) the inverse correlation between polar faculae and sunspot numbers is not only a long-term behaviour, but also exists in short time range; (3) the polar faculae show leads of about 50–71 months relative to sunspot numbers, and the phase difference between them varies with solar cycle; (4) the phase difference value in the northern hemisphere differs from that in the southern hemisphere in a solar cycle, which means that phase difference also existed between the two hemispheres. Moreover, the phase difference between the two hemispheres exhibits a periodical behaviour. Our results seem to support the finding of Hiremath (2010).