1.1 The sunspot number

Despite its somewhat arbitrary construction, the series of relative sunspot numbers constitutes the longest homogeneous global indicator of solar activity determined by direct solar observations and carefully controlled methods. For this reason, their use is still predominant in studies of solar activity variation. As defined originally by Wolf (1859), the relative sunspot number is
RW = k(10 g + f), (1 )
where g is the number of sunspot groups (including solitary spots), f is the total number of all spots visible on the solar disc, while k is a correction factor depending on a variety of circumstances, such as instrument parameters, observatory location, and details of the counting method. Wolf, who decided to count each spot only once and not to count the smallest spots, the visibility of which depended on seeing, used k = 1. The counting system employed was changed by Wolf’s successors to count even the smallest spots, attributing a higher weight (i.e., f > 1) to spots with a penumbra, depending on their size and umbral structure. As the new counting naturally resulted in higher values, the correction factor was set to k = 0.6 for subsequent determinations of RW to ensure continuity with Wolf’s work, even though there was no change in either the instrument or the observing site. This was followed by several further changes in the details of the counting method (Waldmeier, 1961; see Kopecký et al., 1980, Hoyt and Schatten, 1998Jump To The Next Citation Point, and Hathaway, 2010bJump To The Next Citation Point for further discussions on the determination of RW).

In addition to introducing the relative sunspot number, Wolf (1861Jump To The Next Citation Point) also used earlier observational records available to him to reconstruct its monthly mean values since 1749. In this way, he reconstructed 11-year sunspot cycles back to that date, introducing their still universally used numbering. (In a later work he also determined annual mean values for each calendar year going back to 1700.)

In 1981, the observatory responsible for the official determination of the sunspot number changed from Zürich to the Royal Observatory of Belgium in Brussels. The website of the SIDC (originally Sunspot Index Data Center, recently renamed Solar Influences Data Analysis Center), External Linkhttp://sidc.oma.be, is now the most authoritative source of archive sunspot number data. But it has to be kept in mind that the sunspot number is also regularly determined by other institutions: these variants are informally known as the American sunspot number (collected by AAVSO and available from the National Geophysical Data Center, External Linkhttp://www.ngdc.noaa.gov/ngdc.html) and the Kislovodsk Sunspot Number (available from the web page of the Pulkovo Observatory, External Linkhttp://www.gao.spb.ru). Cycle amplitudes determined by these other centers may differ by up to 6 – 7% from the SIDC values, NOAA numbers being consistently lower, while Kislovodsk numbers show no such systematic trend.

These significant disagreements between determinations of RW by various observatories and observers are even more pronounced in the case of historical data, especially prior to the mid-19th century. In particular, the controversial suggestion that a whole solar cycle may have been missed in the official sunspot number series at the end of the 18th century is taken by some as glaring evidence for the unreliability of early observations. Note, however, that independently of whether the claim for a missing cycle is well founded or not, there is clear evidence that this controversy is mostly due to the very atypical behaviour of the Sun itself in the given period of time, rather than to the low quality and coverage of contemporary observations. These issues will be discussed further in Section 3.2.2.

Given that RW is subject to large fluctuations on a time scale of days to months, it has become customary to use annual mean values for the study of longer term activity changes. To get rid of the arbitrariness of calendar years, the standard practice is to use 13-month boxcar averages of the monthly averaged sunspot numbers, wherein the first and last months are given half the weight of other months:

( i=∑5 ) R = 1-- Rm, −6 + 2 Rm,i + Rm,6 , (2 ) 24 i=−5
UpdateJump To The Next Update Information Rm,i being the mean monthly value of RW for ith calendar month counted from the present month. It is this running mean R that we will simply call “the sunspot number” throughout this review and what forms the basis of most discussions of solar cycle variations and their predictions.

In what follows, (n) R max and (n) R min will refer to the maximum and minimum value of R in cycle n (the minimum being the one that starts the cycle). Similarly, t(nm)ax and t(mni)n will denote the epochs when R takes these extrema.

1.1.1 Alternating series and nonlinear transforms

Instead of the “raw” sunspot number series R (t) many researchers prefer to base their studies on some transformed index R ′. The motivation behind this is twofold.

(a) The strongly peaked and asymmetrical sunspot cycle profiles strongly deviate from a sinusoidal profile; also the statistical distribution of sunspot numbers is strongly at odds with a Gaussian distribution. This can constitute a problem as many common methods of data analysis rely on the assumption of an approximately normal distribution of errors or nearly sinusoidal profiles of spectral components. So transformations of R (and, optionally, t) that reduce these deviations can obviously be helpful during the analysis. In this vein, e.g., Max Waldmeier often based his studies of the solar cycle on the use of logarithmic sunspot numbers R ′ = log R; many other researchers use R ′ = R α with 0.5 ≤ α < 1, the most common value being α = 0.5.

(b) As the sunspot number is a rather arbitrary construct, there may be an underlying more physical parameter related to it in some nonlinear fashion, such as the toroidal magnetic field strength B, or the magnetic energy, proportional to B2. It should be emphasized that, contrary to some claims, our current understanding of the solar dynamo does not make it possible to guess what the underlying parameter is, with any reasonable degree of certainty. In particular, the often used assumption that it is the magnetic energy, lacks any sound foundation. If anything, on the basis of our current best understanding of flux emergence we might expect that the amount of toroidal flux emerging from the tachocline should be ∫ |B − B0|dA where B0 is some minimal threshold field strength for Parker instability and the surface integral goes across a latitudinal cross section of the tachocline (cf. Ruzmaikin, 1997). As, however, the lifetime of any given sunspot group is finite and proportional to its size (Petrovay and van Driel-Gesztelyi, 1997Henwood et al., 2010), instantaneous values of R or the total sunspot area should also depend on details of the probability distribution function of B in the tachocline. This just serves to illustrate the difficulty of identifying a single physical governing parameter behind R.

One transformation that may still be well motivated from the physical point of view is to attribute an alternating sign to even and odd Schwabe cycles: this results in the the alternating sunspot number series R±. The idea is based on Hale’s well known polarity rules, implying that the period of the solar cycle is actually 22 years rather than 11 years, the polarity of magnetic fields changing sign from one 11-year Schwabe cycle to the next. In this representation, first suggested by Bracewell (1953), usually odd cycles are attributed a negative sign. This leads to slight jumps at the minima of the Schwabe cycle, as a consequence of the fact that for a 1 – 2 year period around the minimum, spots belonging to both cycles are present, so the value of R never reaches zero; in certain applications, further twists are introduced into the transformation to avoid this phenomenon.

After first introducing the alternating series, in a later work Bracewell (1988) demonstrated that introducing an underlying “physical” variable RB such that

R± = 100(RB ∕83 )3∕2 (3 )
(i.e., α = 2∕3 in the power law mentioned in item (a) above) significantly simplifies the cycle profile. Indeed, upon introducing a “rectified” phase variable1 ϕ in each cycle to compensate for the asymmetry of the cycle profile, RB is a nearly sinusoidal function of ϕ. The empirically found 3/2 law is interpreted as the relation between the time-integrated area of a typical sunspot group vs. its peak area (or peak RW value), i.e., the steeper than linear growth of R with the underlying physical parameter RB would be due to the larger sunspot groups being observed longer, and therefore giving a disproportionately larger contribution to the annual mean sunspot numbers. If this interpretation is correct, as suggested by Bracewell’s analysis, then RB should be considered proportional to the total toroidal magnetic flux emerging into the photosphere in a given interval. (But the possibility must be kept in mind that the same toroidal flux bundle may emerge repeatedly or at different heliographic longitudes, giving rise to several active regions.)
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