Relative sunspot numbers in the first half of eighteenth century

We revised relative sunspot numbers in the time interval 1700–1748 for which Wolf derived their annual means. The frequency of daily observations, counting simultaneously the number of sunspots and the number of sunspot groups necessary for determinating Wolf's relative sunspot numbers, is in this time interval very low and covers, on average, 4.8% of the number of all days only. There also exist incomplete observations not convenient to determine relative sunspot numbers. To enlarge the number of daily relative sunspot numbers we used the nonlinear, two-step interpolation method derived earlier by Letfus (1996, 1999). After interpolation, the mean value increased to 13.8%. Waldmeier (1968) found that the scaling factor k can be derived directly from the observed number of spots f and from the number of sunspot groups g. From the observations made at Zürich (Wolf and his assistants, Wolfer), at Peckeloh, and at Moncalieri during the years 1861–1928, we derived a new, more correct empirical relation. The resulting annual relative sunspot numbers are given in Table II. However, only for 26 years (53.0%) from the total number of 49 years was it possible to derive annual relative sunspot numbers. The observations were missing for the other years. This corresponds with results of Wolf, which gives the annual relative sunspot numbers for all 49 years. For the years when the data were missing, he marked these values as interpolated or very uncertain ones. Most of the observations originate from two data series (Kirch, Plantade), for which Wolf derived a higher scaling factor (k=2.0) than followed from the newly derived relation (k=1.40). The investigated time interval covers four solar cycles. After our results, the height of the first cycle (No. –4), given by Wolf, should be lowered by about two-thirds, the following two cycles (Nos. –3 and –2) lowered by one-third, as given by Wolf, and only the height of the fourth one (No. –1) should be unchanged. The activity levels of the cycles, as represented by group sunspot numbers, are lower by about one-fourth and, in the case of the first one (No. –4) even by two-thirds of the levels derived by us. The group sunspot numbers, derived from a much greater number of observations, have also greater credibility than other estimates. The shapes of the cycles, as given by Wolf, can be considered only as their more or less idealized form.     

Daily relative sunspot numbers 1749–1848: reconstruction of missing observations

A great part of missing daily relative sunspot numbers in the time interval 1749–1848 was reconstructed by nonlinear two-step method of interpolation. In the first step gaps of missing observations not longer than five days were directly interpolated. In the second step data were sorted to so-called Bartels scheme, i.e., to rows of the length of 27 days subsequently ranged in a matrix. In this step the missing value at any position was interpolated from the data at the same position of preceding and following rows. The interpolation was limited to sequences of no more than four missing data. The procedure enables to interpolate long gaps and simultaneously to respect the 27-day variation of solar activity. Monthly and annual means of relative sunspot numbers are presented. The differences between monthly and annual means of the primary observations and of the data completed by interpolation fluctuate around zero. The amplitude of fluctuations depends inversely on the frequency of observations. Most conspicuous are the deviations in the time interval 1776–1795 where the frequency of observations is very low or almost zero. The average dispersion of monthly differences is ±11.5 R and that of annual differences is ±7.8 R. The two-step method of interpolation was tested on the series of daily data in the time interval 1918–1948. The sequence of missing daily data in the years 1818–1848 represents a masking function. This function was applied to the continuous data series in the time interval 1918–1948 and then the modified series was reconstructed. The differences between the monthly and annual means of primary and reconstructed data are small with fluctuations around zero and with dispersion for monthly differences ±2.7 R a for annual differences ±0.6 R. Corresponding dispersions of the data differences for monthly means in the time interval 1818–1848 are ±4.3 R and ±1.1 R for annual means. The small dispersion values and small differences among them give evidence about the applicability and the effectiveness of the nonlinear two-step method of interpolation and also about high credibility of relative sunspot numbers after reconstruction.     

New Evidence for Long-Term Persistence in the Sun's Activity

Possible persistence of sunspot activity was studied using rescaled range and detrended fluctuation analyses. In addition to actual Wolf numbers (1700–2000 A.D.), two solar proxies were used in this research, viz., an annual sunspot proxy obtained for 1090–1700 A.D. and sunspot numbers reconstructed from the decadal radiocarbon series (8005 B.C. – 1895 A.D). The reconstruction was made using a five-box carbon exchange model. Analyses showed that in all cases the scaling exponent is significantly higher than 0.5 in the range of scales from 25 yr up to 3000 yr. This indicates the existence of a long-term memory in solar activity, in agreement with results obtained for other solar indices.     

Mathematical modelling of the sunspot cycle

The sunspot record for the time interval 1749–1977 can be represented conveniently by an harmonic model comprising a relatively large number of lines. Solar activity can otherwise be considered as a sequence of partly overlapping events, triggered periodically at intervals of the order of 11 years. Each individual cycle is approximated by a function of the Maxwell distribution type; the resulting impulse model consists of the superposition of the independent pulses. Application of these two models for the prediction of annual values of the Wolf sunspot numbers leads to controversial results. Mathematical modelling of the sunspot time series does not give an unambiguous result.     

A new look at wolf sunspot numbers in the late 1700's

Long-term homogeneous observations of solar activity or many solar cycles are essential for investigating many problems in solar physics and climatology. The one key parameter used in most long-term studies is the Wolf sunspot number, which is susceptible to observer bias, particularly because it is highly sensitive to the observer's ability to see the smallest sunspots. In this paper we show how the Wolf sunspot number can be derived from the number of sunspot groups alone. We utilize this approach to obtain a Group Wolf number. This technique has advantages over the classical method of determining the Wolf number because corrections for observer differences are reduced and long-term self-consistent time series can be developed. The level of activity can be calculated to an accuracy of ± 5% using this method. Applying the technique to Christian Horrebow's observations of solar cycles 1, 2, and 3 (1761–1777), we find that the standard Wolf numbers are nearly homogeneous with sunspot numbers measured from 1875 to 1976 except the peak of solar cycle 2 is too low by 30%. This result suggests that further analyses of early sunspot observations could lead to significant improvements in the uniformity of the measurements of solar activity. Such improvements could have important impacts upon our understanding of long-term variations in solar activity, such as the Gleissberg cycle, or secular variations in the Earth's climate.     

Prediction of the maximum annual mean sunspot number in the coming solar maximum epoch

Using an earlier correlation analysis between the annual sunspot numbers at sunspot maximum epochs and the minimum annual aa index in the immediately preceding years, the minimum annual aa index (21.6) during 1985–86 implies a maximum annual sunspot number of about 190±40 in the coming solar maximum epoch, in about 1988–89.     

Reconstruction of Wolf Sunspot Numbers on the Basis of Spectral Characteristics and Estimates of Associated Radio Flux and Solar Wind Parameters for the Last Millennium

A reconstruction of sunspot numbers for the last 1000 years was obtained using a sum of sine waves derived from spectral analysis of the time series of sunspot number R _z for the period 1700–1999. The time series was decomposed in frequency levels using the wavelet transform, and an iterative regression model (ARIST) was used to identify the amplitude and phase of the main periodicities. The 1000-year reconstructed sunspot number reproduces well the great maximums and minimums in solar activity, identified in cosmonuclides variation records, and, specifically, the epochs of the Oort, Wolf, Spörer, Maunder, and Dalton Minimums as well the Medieval and Modern Maximums. The average sunspot number activity in each anomalous period was used in linear equations to obtain estimates of the solar radio flux F _10.7, solar wind velocity, and the southward component of the interplanetary magnetic field.     

Solar rotation during the Maunder Minimum

We have measured solar surface rotation from sunspot drawings made in a.d. 1642–1644 and find probable differences from present-day rates. The 17th century sunspots rotated faster near the equator by 3 or 4%, and the differential rotation between 0 and ±20° latitude was enhanced by about a factor 3. These differences are consistent features in both spots and groups of spots and in both northern and southern hemispheres. We presume that this apparent change in surface rotation was related to the ensuing dearth of solar activity (the Maunder Minimum) which persisted until about 1715.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Solar activity in the sixteenth and seventeenth centuries (a revision)

Maximum relative sunspot numbers for the 16th and 17th century were computed by means of the dependence of the maximum relative sunspot numbers on the solar cycle rise time and on the cycle asymmetry. In these dependencies four separate modes of relations, two for odd and two for even cycles, were identified. These modes are coupled two and two in even-odd cycle pairs. The rise times and the asymmetries of solar cycles in the 16th and 17th centuries were taken from cycle extreme estimates by Schove (1979), from auroral and telescopic sunspot observations during this period, but with some necessary corrections. Annual relative sunspot numbers and decade averages were estimated from the cycle maxima and the epochs of extremes. In addition, the efficiency of auroral records in latitudes lower than 55 deg was computed for the time interval 1500–1868. For this purpose the dependence of occurrence numbers of aurorae on the cycle and decade means of the relative sunspot numbers was derived.     

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.     

To the description of long-term variations in the solar magnetic flux: The sunspot area index

We show that the Wolf sunspot numbers W and the group sunspot numbers GSN are physically different indices of solar activity and that it is improper to compare them. Based on the approach of the so-called “primary” indices from the observational series of W(t) and GSN(t), we suggest series of yearly mean sunspot areas beginning in 1610 and monthly mean sunspot areas beginning in 1749.     

The Intermediate-Term Quasi-Cycles of Wolf Number and Group Sunspot Number Fluctuations

I apply spectral and auto-correlation analyses to the monthly Wolf number fluctuations for 22 solar cycles and to the group sunspot number fluctuations for 18 solar cycles and find the existence of an 11-month quasi-periodicity in these data. Its strength correlates very well (ρ ⑈ 0.8) with the variance of fluctuations. Moreover, for both Wolf and group sunspot indexes I divide a stationary version of fluctuation time series into two parts: those from periods of low and high solar activity. I find statistically significant quasi-periodicity (9 months) in both high- and low-activity data sets. I also find the quasi-period of about 15 months in the time series of high-activity periods.     

The Transitional Time Scale from Stochastic to Chaotic Behavior for Solar Activity

Following the progression of nonlinear dynamical system theory, many authors have used varied methods to calculate the fractal dimension and the largest Lyapunov exponent ₁ of the sunspot numbers and to evaluate the character of the chaotic attractor governing solar activity. These include the Grassberger–Procaccia algorithm, the technique provided by Wolf et al., and the nonlinear forecasting approach based on the method of distinguishing between chaos and measurement errors in time series described by Sugihara and May. In this paper, we use the Grassberger–Procaccia algorithm to estimate the other character of the chaotic attractor. This character is time scale of a transition from high-dimensional or stochastic at shorter times to a low-dimensional chaotic behavior at longer times. We find that the transitional time scale in the monthly mean sunspot numbers is about 8 yr; the low-dimensional chaotic behavior operates at time scales longer than about 8 yr and a high-dimensional or stochastic process operates at time scales shorter than about 8 yr.     

Comments on the so-called Maunder minimum

Arguments are presented in favour of the operation of the 11-yr cycle during the Maunder minimum and before it. The laws of differential rotation (1642–1644 and 1899–1901) before the low cycle are shown to differ insignificantly. It is suggested that the Maunder minimum was a result of the 600-yr cycle effect (during its epoch of minimum) on the 80–90 yr cycle.     

Coronal Activity During the 22-Year Solar Magnetic Cycle

The existence of a 22-year heliomagnetic cycle was inferred long ago not only from direct measurements of the solar magnetic field but also from a cyclic variability of a number of the solar activity phenomena. In particular, it was stated (a rule derived after Gnevyshev and Ohl (1948) findings and referenced as the G–O rule in the following) that if sunspot number Rz cycles are organized in pairs of even–odd numbered cycles, then the height of the peak in the curve of the yearly-averaged sunspot numbers Rz-y is always lower for a given even cycle in comparison with the corresponding height of the following odd cycle. Exceptions to this rule are only cycles 4 and 8 which, at the same time, are the nearest even cycles to the limits of the so-called Dalton minimum of solar activity (i.e., the 1795–1823 time interval). In the present paper, we are looking for traces of the mentioned G–O rule in green corona brightness (measured in terms of the Fexiv 530.3 nm emission line intensity), using data covering almost five solar cycles (1943–1994). It was found that the G–O rule seems to work within the green-line corona brightness, namely, when coronal intensity measured in an extended solar middle-latitude zone is considered separately from the rest of the solar surface. On the other hand, the same G–O rule is valid at the photospheric level, as the heliographic latitudinal dependence of sunspot numbers (1947–1984) shows.     

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.     

A revised listing of the number of sunspot groups made by Pastorff, 1819 to 1833

J. W. Pastorff of Drossen, Germany, made about 1477 observations of sunspots between 1819 and 1833. These observations were erroneously interpreted by A. C. Ranyard in 1874 and then used by Rudolf Wolf in his calculations of the Wolf Sunspot Numbers. The result is a noisier daily time series and overestimation of the monthly and yearly means for these years. Pastorff was actually a very good observer. In this paper, Pastorff's original observations are reexamined and more nearly correct values for the number of sunspot groups are tabulated. We show some examples of the problems created by Ranyard's interpretation and the consequences for the history of solar activity that a correct interpretation of Pastorff's observations will have. Pastorff's observations provide valuable information on the first strong cycle after the Dalton Minimum (1795–1823).     

Group Sunspot Numbers: A New Solar Activity Reconstruction

In this paper, we construct a time series known as the Group Sunspot Number. The Group Sunspot Number is designed to be more internally self-consistent (i.e., less dependent upon seeing the tiniest spots) and less noisy than the Wolf Sunspot Number. It uses the number of sunspot groups observed, rather than groups and individual sunspots. Daily, monthly, and yearly means are derived from 1610 to the present. The Group Sunspot Numbers use 65941 observations from 117 observers active before 1874 that were not used by Wolf in constructing his time series. Hence, we have calculated daily values of solar activity on 111358 days for 1610–1995, compared to 66168 days for the Wolf Sunspot Numbers. The Group Sunspot Numbers also have estimates of their random and systematic errors tabulated. The generation and preliminary analysis of the Group Sunspot Numbers allow us to make several conclusions: (1) Solar activity before 1882 is lower than generally assumed and consequently solar activity in the last few decades is higher than it has been for several centuries. (2) There was a solar activity peak in 1801 and not 1805 so there is no long anomalous cycle of 17 years as reported in the Wolf Sunspot Numbers. The longest cycle now lasts no more than 15 years. (3) The Wolf Sunspot Numbers have many inhomogeneities in them arising from observer noise and this noise affects the daily, monthly, and yearly means. The Group Sunspot Numbers also have observer noise, but it is considerably less than the noise in the Wolf Sunspot Numbers. The Group Sunspot Number is designed to be similar to the Wolf Sunspot Number, but, even if both indices had perfect inputs, some differences are expected, primarily in the daily values.     

Group Sunspot Numbers: A New Solar Activity Reconstruction

In this paper, we construct a time series known as the Group Sunspot Number. The Group Sunspot Number is designed to be more internally self-consistent (i.e., less dependent upon seeing the tiniest spots) and less noisy than the Wolf Sunspot Number. It uses the number of sunspot groups observed, rather than groups and individual sunspots. Daily, monthly, and yearly means are derived from 1610 to the present. The Group Sunspot Numbers use 65941 observations from 117 observers active before 1874 that were not used by Wolf in constructing his time series. Hence, we have calculated daily values of solar activity on 111358 days for 1610–1995, compared to 66168 days for the Wolf Sunspot Numbers. The Group Sunspot Numbers also have estimates of their random and systematic errors tabulated. The generation and preliminary analysis of the Group Sunspot Numbers allow us to make several conclusions: (1) Solar activity before 1882 is lower than generally assumed and consequently solar activity in the last few decades is higher than it has been for several centuries. (2) There was a solar activity peak in 1801 and not 1805 so there is no long anomalous cycle of 17 years as reported in the Wolf Sunspot Numbers. The longest cycle now lasts no more than 15 years. (3) The Wolf Sunspot Numbers have many inhomogeneities in them arising from observer noise and this noise affects the daily, monthly, and yearly means. The Group Sunspot Numbers also have observer noise, but it is considerably less than the noise in the Wolf Sunspot Numbers. The Group Sunspot Number is designed to be similar to the Wolf Sunspot Number, but, even if both indices had perfect inputs, some differences are expected, primarily in the daily values.     

A Fractal Structure of the Time Series of Global Indices of Solar Activity

The structure of time series of daily global indices of solar activity is investigated: the sunspot numbers for the time interval between the years 1854 and 1996, the Greenwich total sunspot area for 1874–1983, the radio-flux at 10.7 cm (F10.7) for 1964–1996, and the Stanford mean solar magnetic field for 1975–1996. The fractal dimensions are determined by two fractal and spectral methods. The identified three time-scale ranges, 2 days–2 months, 2 months–2 years, 2 years–8 and more years, with the fractal dimensions 1.4–1.6, 2, 1.2–1.6, respectively, show perhaps some fractal structure of time series of global indices. The first time-scale range may correspond to ordinary brownian noise and the second to flicker noise. The solar rotation influence of the value of the fractal dimensions at the time range close to the rotational period is studied.