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 Magnetic Activity and Total Irradiance Since the Maunder Minimum

We develop a model for estimating solar total irradiance since 1600 AD using the sunspot number record as input, since this is the only intrinsic record of solar activity extending back far enough in time. Sunspot number is strongly correlated, albeit nonlinearly with the 10.7-cm radio flux (F _10.7), which forms a continuous record back to 1947. This enables the nonlinear relationship to be estimated with usable accuracy and shows that relationship to be consistent over multiple solar activity cycles. From the sunspot number record we estimate F _10.7 values back to 1600 AD. F _10.7 is linearly correlated with the total amount of magnetic flux in active regions, and we use it as input to a simple cascade model for the other magnetic flux components. The irradiance record is estimated by using these magnetic flux components plus a very rudimentary model for the modulation of energy flow to the photosphere by the subphotospheric magnetic flux reservoir feeding the photospheric magnetic structures. Including a Monte Carlo analysis of the consequences of measurement and fitting errors, the model indicates the mean irradiance during the Maunder Minimum was about 1 ± 0.4 W m⁻² lower than the mean irradiance over the last solar activity cycle.     

The Maunder Minimum and the Sun as the Possible Source of Particles Creating Increased Abundance of the <Superscript>14</Superscript>C Carbon Isotope

The Maunder Minimum was the time during the second part of the 17th century, nominally from 1645 to 1717 AD, when unusually low numbers of sunspots were observed. On the basis of numerous recorded observations of auroras in the early 18th century, the end of the Minimum could be regarded as around 1700, but details of sunspot observations by Jan Heweliusz (Heweliusz, Machina Coelestis, 1679), John Flamsteed and Philippe de La Hire in 1684 allow us to interpret the Maunder Minimum as the period without a significant cessation of activity. This Minimum was also recognized in ¹⁴C data from trees which grew during the second part of 17th century. The variation in the production rate of radioactive carbon isotope ¹⁴C is due to modulation of the cosmic ray flux producing it by the changing level of solar activity and solar magnetic flux. Stronger magnetic fields in the solar wind make it more difficult for cosmic rays to reach the Earth, causing a drop in the production rate of ¹⁴C. However, more detailed analyses of ¹⁴C data indicate that the highest isotope abundances do not occur at the time of sunspot minima, as would be expected on the basis of modulation of the cosmic ray flux by the solar magnetic field, but two years after the sunspot number maximum. This time difference (or phase delay) can be accounted for if in fact there are both solar and non-solar cosmic ray contributions. Solar flares could also contribute high-energy particles and produce ¹⁴C and are generally not most frequent at the time of the highest sunspot numbers in the cycle.     

A Note on Solar Cycle Length Estimates

Recently, new estimates of the solar cycle length (SCL) have been calculated using the Zurich Sunspot Number (R_Z) and the Regression-Fourier-Calculus (RFC)-method, a mathematically rigorous method involving multiple regression, Fourier approximation, and analytical expressions for the first derivative. In this short contribution, we show estimates of the solar cycle length using the RFC-method and the Group Sunspot Number (R_G) instead the R_Z. Several authors have showed the advantages of R_G for the analysis of sunspot activity before 1850. The use of R_G solves some doubtful solar cycle length estimates obtained around 1800 using R_Z.     

Could a Hexagonal Sunspot Have Been Observed During the Maunder Minimum?

The Maunder Minimum is the period between 1645 and 1715. Its main characteristic is abnormally low and prolonged solar activity. However, some authors have doubted the low level of solar activity during that period by questioning the accuracy and objectivity of the observers. This work presents a particular case of a sunspot observed during the Maunder Minimum with an unusual shape of its umbra and penumbra: a hexagon. This sunspot was observed by Cassini in November 1676, just at the core of the Maunder Minimum. This historical observation is compared with a twin case that occurred recently in May 2016. The conclusion reached is that Cassini’s record is another example of the good quality of the observations that were made during the Maunder Minimum, showing the meticulousness of the astronomers of that epoch. This sunspot observation made by Cassini does not support the conclusions of Zolotova and Ponyavin (Astrophys. J. 800, 42, 2015) that professional astronomers in the seventeenth century only registered round sunspots. Finally, a discussion is given of the importance of this kind of unusual sunspot record for a better assessment of the true level of solar activity in the Maunder Minimum.     

Observations of sunspots by Flamsteed during the Maunder Minimum

In the bookHistoria Coelestis Brittannica, John Flamsteed (1725) lists his daily solar observations from 1676 onwards. Coupled with his comments in thePhilosophical Transactions of the Royal Society and his letters to William Derham in the Cambridge University Library, it is possible to reconstruct a daily chronology of his solar and sunspot observations from 1676 to 1700. These observations are important because, coupled with daily logs of observations by Picard, La Hire, Eimmart, and others, a detailed record of the observations during a portion of the Maunder Minimum can be constructed. For example, for 1691, a typical year, the longest gap between observations is only four days. Flamsteed's observations are also important because they add to the data gathered by Wolf, Spoerer, Maunder, Eddy, and others in their study of solar activity in the seventeenth century. Flamsteed's observations are summarized here and a sample of his observations is presented.     

Can the Solar Cycle Amplitude Be Predicted Using the Preceding Solar Cycle Length?

In this work, the evolution of the relationship between Solar Cycle Length of solar cycle n (SCL _n ) and Solar Cycle Amplitude of the solar cycle n+1 (SCA _n+1) is studied by using the R _Z and R _G sunspot numbers. We conclude that this relationship is only strongly significant in a statistical sense during the first half of the historical record of R _Z sunspot number whereas it is considerably less significant for the R _G sunspot number. In this sense we assert that these simple lagged relationships should be avoided as a valid method to predict the following solar activity amplitude.     

Redefining the limit dates for the Maunder Minimum

The Maunder Minimum corresponds to a prolonged minimum of solar activity a phenomenon that is of particular interest to many branches of natural and social sciences commonly considered to extend from 1645 until 1715. However, our knowledge of past solar activity has improved significantly in recent years and, thus, more precise dates for the onset and termination of this particularly episode of our Sun can be established. Based on the simultaneous analysis of distinct proxies we propose a redefinition of the Maunder Minimum period with the core “Deep Maunder Minimum” spanning from 1645 to 1700 (that corresponds to the Grand Minimum state) and a wider “Extended Maunder Minimum” for the longer period 1618–1723 that includes the transition periods.     

Overlooked sunspot observations by Hevelius in the early Maunder Minimum, 1653–1684

In the bookMachina Coelestis (1679), Johannes Hevelius lists his daily solar observations from 1653 to 1679. He mentions 19 sunspot groups during this interval, of which 14 are unique to Hevelius and five are confirmed by other observers. There are an additional 9 sunspot groups during this interval that were not observed by Hevelius. In five cases he was not observing, but in the other four cases he did observe but failed to comment upon sunspots. The spots he missed or failed to observe tend to occur near the end of his career. This suggests Hevelius occasionally missed sunspots but usually was a reliable observer. These observations are important because they provide us the only known daily listing of solar observations during the early years of the Maunder Minimum. They are also important because they were overlooked by Wolf, Spoerer, Maunder, Eddy, and others in their study of solar activity in the seventeenth century. They provide us the best record of the sunspot maximum of 1660 when one sunspot lasted at least 86 days as it traversed the solar disk four times. The same region was active for seven solar rotations.     

Cosmogenic Isotope Variability During the Maunder Minimum: Normal 11-year Cycles Are Expected

The amplitude of the 11-year cycle measured in the cosmogenic isotope ¹⁰Be during the Maunder Minimum is comparable to that during the recent epoch of high solar activity. Because of the virtual absence of the cyclic variability of sunspot activity during the Maunder Minimum this seemingly contradicts an intuitive expectation that lower activity would result in smaller solar-cycle variations in cosmogenic radio-isotope data, or in none, leading to confusing and misleading conclusions. It is shown here that large 11-year solar cycles in cosmogenic data observed during periods of suppressed sunspot activity do not necessarily imply strong heliospheric fields. Normal-amplitude cycles in the cosmogenic radio-isotopes observed during the Maunder Minimum are consistent with theoretical expectations because of the nonlinear relation between solar activity and isotope production. Thus, cosmogenic-isotope data provide a good tool to study solar-cycle variability even during grand minima of solar activity.     

Modulation of Atmospheric 14C Concentration by the Solar Wind and Irradiance Components of the Hale and SCHWABE Solar Cycles

¹⁴C abundance on the Earth can be modulated by both the solar wind and irradiance components of the solar cycle. The magnetic field component of the solar wind modulates ¹⁴C production whereas the irradiance component can result in a change in the exchange rate between the various reservoirs of the carbon biogeochemical cycle. The effects would be nearly synchronous and difficult to separate. The 0.1% amplitude of irradiance variation during the two most recent solar cycles is well known. A 22-yr cycle exists also in the measured global temperature record.We have divided the University of Washington high-precision data on¹⁴C in tree rings into three 91-yr intervals: AD 1540–1630, 1630–1720 and 1715–1805, before, during and after the Maunder Minimum. Unfortunately the AD 1540–1630 interval includes part of the Spörer Minimum as well as the intermediate interval of high solar activity. These data were analyzed by the DFT, MEM and MTM methods of spectral time series analysis. The ca. 22-yr cycle is prominent during the Maunder Minimum, whereas the 11-yr cycle is most prominent after the Maunder Minimum but totally suppressed during the Maunder Minimum. The lesser amplitude of the 11-yr cycle before the Maunder Minimum is most probably due to overlap with the Spörer Minimum.Vasiliev and Kocharov VK83 have previously suggested that the 22-yr cycle persists through the Maunder Minimum whereas the 11-yr cycle is suppressed. Our calculations show that irradiance forcing of the carbon cycle during the 11-yr cycle is negligible, so the observed 11-yr cycle in¹⁴ C must be the result of production rate changes. The presence of the 22-yr cycle and suppression of the 11-yr cycle during the Maunder Minimum is in accord with a model by Jokipii Jok91.     

A Simple Method to Check the Reliability of Annual Sunspot Number in the Historical Period 1610 – 1847

A simple method to detect inconsistencies in low annual sunspot numbers based on the relationship between these values and the annual number of active days is described. The analysis allowed for the detection of problems in the annual sunspot number series clustered in a few explicit periods, namely: i) before Maunder minimum, ii) the year 1652 during the Maunder minimum, iii) the year 1741 in Solar Cycle −1, and iv) the so-called “lost” solar cycle in the 1790s and the subsequent onset of the Dalton Minimum.     

Solar Rotation in the 17th century

Sunspot drawings made by Galileo Galilei in 1612 are used to derive the law of differential rotation at that time. The main interest of the work is during the time of observations, just at the beginning of telescopic observations and some decades before the Maunder Minimum (1645 – 1715), a period where the sunspots almost disappeared from the solar surface. For this purpose we have carried out careful corrections of the different sources of errors derived from the observing technique. By comparing with other results of the same century, a significant difference is only detected by comparing with data corresponding to the deep Maunder Minimum (Paris Observatory drawings). The characteristics of the solar differential rotation, and extrapolating the behavior of solar activity, did not differ before or after the Maunder Minimum. We also include an analysis of hitherto ignored sunspot drawings by N. Bion made in October and November 1672.     

A Preliminary Estimate of the Size of the Coming Solar Cycle 24, based on Ohl’s Precursor Method

For many purposes (e.g., satellite drag, operation of power grids on Earth, and satellite communication systems), predictions of the strength of a solar cycle are needed. Predictions are made by using different methods, depending upon the characteristics of sunspot cycles. However, the method most successful seems to be the precursor method by Ohl and his group, in which the geomagnetic activity in the declining phase of a sunspot cycle is found to be well correlated with the sunspot maximum of the next cycle. In the present communication, the method is illustrated by plotting the 12-month running means aa(min ) of the geomagnetic disturbance index aa near sunspot minimum versus the 12-month running means of the sunspot number Rz near sunspot maximum [aa(min ) versus Rz(max )], using data for sunspot cycles 9 – 18 to predict the Rz(max ) of cycle 19, using data for cycles 9 – 19 to predict Rz(max ) of cycle 20, and so on, and finally using data for cycles 9 – 23 to predict Rz(max ) of cycle 24, which is expected to occur in 2011 – 2012. The correlations were good (∼+0.90) and our preliminary predicted Rz(max ) for cycle 24 is 142±24, though this can be regarded as an upper limit, since there are indications that solar minimum may occur as late as March 2008. (Some workers have reported that the aa values before 1957 would have an error of 3 nT; if true, the revised estimate would be 124±26.) This result of the precursor method is compared with several other predictions of cycle 24, which are in a very wide range (50 – 200), so that whatever may be the final observed value, some method or other will be discredited, as happened in the case of cycle 23.     

Recurrent magnetic activity,sunspot number and its rate of decline

Power spectral densities computed from low-latitude horizontal intensity of the Earth's magnetic field over two-year periods of declining phases of solar cycles 16 to 19 show a close relationship with the maximum relative sunspot number of the following solar cycles. The maximum sunspot number shows an exponential rise with the power density near 1/27 cd^?1; maximum R _z,however, increases linearly with power density near 1/14 cd^?1. It is also shown that the rate of decline of sunspot number in a solar cycle is almost exactly related, linearly, to power spectral density for the preceding solar cycle. Power densities near 1/27 and 1/14 cd^?1 in declining phase of solar cycle appear to be satisfactory indices for the maximum relative sunspot number of the following cycle and its rate of decline thereafter.     

The Solar Spectral Irradiance as a Function of the Mg <Emphasis Type="SmallCaps">ii</Emphasis> Index for Atmosphere and Climate Modelling

We present a new method to reconstruct the solar spectrum irradiance in the Ly α – 400 nm region, and its variability, based on the Mg ii index and neutron-monitor measurements. Measurements of the solar spectral irradiance available in the literature have been made with different instruments at different times and different spectral ranges. However, climate studies require harmonised data sets. This new approach has the advantage of being independent of the absolute calibration and aging of the instruments. First, the Mg ii index is derived using solar spectra from Ly α (121 nm) to 410 nm measured from 1978 to 2010 by several space missions. The variability of the spectra with respect to a chosen reference spectrum as a function of time and wavelength is scaled to the derived Mg ii index. The set of coefficients expressing the spectral variability can be applied to the chosen reference spectrum to reconstruct the solar spectra within a given time frame or Mg ii index values. The accuracy of this method is estimated using two approaches: direct comparison with particular cases where solar spectra are available from independent measurements, and calculating the standard deviation between the measured spectra and their reconstruction. From direct comparisons with measurements we obtain an accuracy of about 1 to 2%, which degrades towards Ly α. In a further step, we extend our solar spectral-irradiance reconstruction back to the Maunder Minimum introducing the relationship between the Mg ii index and the neutron-monitor data. Consistent measurements of the Mg ii index are not available prior to 1978. However, we remark that over the last three solar cycles, the Mg ii index shows strong correlation with the modulation potential determined from the neutron-monitor data. Assuming that this correlation can be applied to the past, we reconstruct the Mg ii index from the modulation potential back to the Maunder Minimum, and obtain the corresponding solar spectral-irradiance reconstruction back to that period. As there is no direct measurement of the spectral irradiance for this period we discuss this methodology in light of the other proposed approaches available in the literature. The use of the cosmogenic-isotope data provides a major advantage: it provides information about solar activity over several thousands years. Using technology of today, we can calibrate the solar irradiance against activity and thus reconstruct it for the times when cosmogenic-isotope data are available. This calibration can be re-assessed at any time, if necessary.     

Variation of solar irradiance and mode frequencies during Maunder minimum

Using the sunspot numbers reported during the Maunder minimum and the empirical relations between the mode frequencies and solar activity indices, the variations in the total solar irradiance and 10.7 cm radio flux for the period 1645 to 1715 is estimated. We find that the total solar irradiance and radio flux during the Maunder minimum decreased by 0.19% and 52% respectively, as compared to the values for solar cycle 22. This revised version was published online in July 2006 with corrections to the Cover Date.     

Solar cycles during the Maunder minimum

We have obtained new consistent versions of the 400-yr time series of the Wolf sunspot number W, the sunspot group number G, and the total sunspot area S (or the total sunspot magnetic flux Φ). We show that the 11-yr cycle did not cease during the Maunder minimum of solar activity. The characteristics of the extrema of individual 11-yr cycles in 1600–2005 have been determined in terms of the total sunspot area index. We provide arguments for using alternating (“magnetic”) time series of indices in investigating the solar cyclicity.     

A comparison of the spectral characteristics of the Wolf Sunspot Number (RZ) and Group Sunspot Number (RG)

The objective of this paper is to compare the spectral features of the recently derived Group Sunspot Numbers (R _G) and the traditional Wolf Sunspot Numbers (R _Z) for the 1700–1995 period. In order to study the spectral features of both time series, two methods were used, including: (a) the multitaper analysis and (b) the wavelet analysis. Well-known features of the solar variability, such as the 98.6-yr (Gleissberg cycle), 10–11-yr (Schwabe cycle) and 5-yr (second solar harmonic) periodicities were identified with high confidence using the multitaper analysis. Also observed was a larger amount of power spread in high frequencies for R _Z than for R _G spectra. Furthermore, a multitaper analysis of two subsets, A (1700–1850) and B (1851–1995), has indicated that the main differences occurred in the first subset and seem to be due to uncertainties in the early observations. The wavelet transform, which allows observing the spectra evolution of both series, showed a strong and persistent 10–11-yr signal that remained during the whole period. The Meyer Wavelet Transform was applied to both R _Z and R _G. This study indicates that the main spectral characteristics of both series are similar and that their long-term variability has the same behavior.