Dances with Wolfs: A Short History of Sunspot Indices

Contributed by Carl E. Feehrer (FEEC)

Revised August 2000


Efforts to monitor the amount and variation of the sun's activity by counting spots on its disk have a long and rich history. Relatively complete visual estimates of daily activity date back to 1818, monthly averages can be extrapolated back to 1749, and estimates of annual values can be similarly determined back to 1700. Finally, although much less precisely and primarily for determination of maxima and mimima over long periods of time, sunspot counts can even be established for observations dating from the time of Galileo (1610).

As equipment and methods available for sunspot monitoring have evolved and the numbers and geographic distributions of observers have grown, different accounting procedures have been formulated. Each has aimed at achieving the best and most consistent record of activity while at the same time maintaining continuity with older records. The most prominent of these have aimed at utilizing a "relative sunspot number", that is, a weighted index whose component values scale approximately with the apparent areas of the sun's disk actually covered by spots at any given time.

As the sections below suggest, the major indices spring from a common ancestor, and each has at its heart the simple formula developed by the originator of the "relative sunspot number" concept. The key differences among them lie in the ways in which they and the organizations that use them control for variations in observing skills, location/seeing conditions, and equipment associated with the sometimes diverse bodies of observers who contribute data. As might be expected, the various indices sometimes produce different results, at least in near-term estimates of sunspot activity.

It is important for readers of the following sections to recognize that, in collapsing approximately 150 years of history, many interesting events have necessarily been omitted and some important details have been "leveled" in the interest of concise presentation. Those interested in gaining a more complete perspective are referred to books, journal articles, and Internet sources cited at the end of this introduction and elsewhere in the manual.


The modern era of sunspot counting began in the mid-1800s with the research of Bern Observatory director Rudolf Wolf, who introduced what he called the "Universal Sunspot Number" as an estimate of the sun's activity. This investigator, motivated by H. H. Schwabe's discovery of an apparent 10-year periodicity--"sunspot cycle"--in the frequency with which spots seemed to appear over the years, sought to develop an index by which long-term trends could be monitored and periodicity verified and studied [1].

It is said that Wolf would have preferred to measure the areas covered by the sunspots rather than their number (Waldmeier, 1961), but the methods and equipment of the day were not adequate for this task [2]. As an alternative, he developed an index based on the number of spots and spot groups (clusters of related spots). Working with a polarizer-equipped 8cm, f/14 refractor at 64x and recognizing that groups of related spots were more closely tied to his desired area measure than individual spots, he chose to weight the groups more heavily than the spots in his index. He also chose to exclude in his daily observations those small spots and pores that were visible only under excellent seeing conditions.

The final form of Wolf's index, R, produced in 1848, assigns a weight of 10 to each cluster of related spots (g) and unit value to each individual spot (f):

R=(10g + f)

Thus we have, for a day in which 8 groups can be identified, along with 47 individual spots, the Wolf number,


The reader should note that the Wolf number scale contains a gap between zero and 11. The minimum value is, of course, zero if no spots are found that satisfy the counting criterion; there are, therefore, no groups. But given one spot, the next possible value is 11--the result of assigning 10 to the group and one to the spot. We shall mention this characteristic again in the next section.


After leaving Bern, Wolf became director of the Swiss Federal Observatory in Zurich, where his work was eventually taken over by a series of successors, Wolfer, Bruner, and, most recently, Waldmeier. Over the course of time, more observatories were enlisted in the task in order to combat the variable seeing conditions at particular locations, thereby preserving continuity in the daily counts, and the seeds of an international effort were sown. Also during this time, an important modification was made to what has come to be called the "relative sunspot number".

It will be recalled from our earlier discussion that Wolf did not include in his daily counts small spots and pore groups that were visible only under excellent seeing conditions. With a larger and better-dispersed group of observers, average seeing conditions improved, and Wolf's counting convention was changed. The new convention required observers to count all of the groups and all of the spots visible on a given day through Wolf's now-standard telescope setup.

In addition to this modification of the Wolf number, the Observatory developed a weighting scheme in which values ranging from one to five were assigned to spots on the basis of size and structure. Although important at the time for introducing a measure of "prominence" into the index, this addition typically produces only a small difference in the resulting number and is not employed in most of the indices used today.

Given these modifications in the original formulation, how could one then expect to create and maintain a record that was continuous with Wolf's observations? A solution to this problem was found by comparing the results obtained by Wolfer during 16 years of observations carried on in parallel with Wolf and introducing a coefficient, sometimes called a k-factor, into the Wolf index,

k(10g + f)

It was determined that, assuming a value of 1.0 for Wolf's observations, a coefficient of 0.6 for Wolfer's observations would place the indices on approximately the same scale, thus maintaining the desired continuity. This value of k is now referred to as the "Zurich Reduction Coefficient", and the resulting index is identified as Rz [3].

For the example in Section 2 above and ignoring a "size" weighting, we have as a result (with rounding),

Rz =0.6((10x8)+47)= 76

In our discussion of the Wolf index above, we noted the lack of relative sunspot number values between zero and eleven. What Rz value would be assigned when only one spot was observed? Remaining consistent with the convention, Zurich would assign the (rounded) value,

Rz =0.6((10x1)+1))»7

This outcome reflects a basic incompatibility between the two scales and makes direct conversions of relative sunspot numbers from one index to the other extremely difficult, particularly when the spot count is very low. Despite this, Wolf and Zurich numbers provide the bases for essentially all long-term estimates of solar activity through the years that utilize sunspot counts (Taylor, 1991) .


With the retirement in 1980 of Waldmeier as director of the Zurich Observatory, formal responsibility for assessment of sunspot activity was assumed by the Royal Observatory in Brussels, Belgium. The Sunspot Index Data Center (SIDC) located at that site now produces a broad range of daily, monthly, and annual reports, as well as predictions of future near- and intermediate-term sunspot activity. The number of observers contributing to SIDC's operations has grown substantially since the transition from Zurich and now includes a mix of amateur and professional observers distributed over 40 individual stations.

Because of the size and diversity of its observer base, SIDC necessarily conducts extensive statistical treatments on incoming data that are aimed at reducing unwanted variation and maintaining quality control. Although the details of these treatments are beyond the scope of this introduction, it is of interest to note that they involve a series of iterations in which station reports and a reference standard produced in Locarno, Switzerland are compared with respect to increasingly stringent criteria. The iterations enable progressive elimination of observer reports that significantly exceed the mean estimate ("outliers") and result in an estimate with satisfactory accuracy and reliability.

Initially, Belgium's estimates were made with a k-factor of 1.0, presenting a departure from the Zurich value of 0.6 [4]. Since that time, it has varied from that value and is currently approximately 0.9. It is designated by the International Astronomical Union (IAU) as Ri and is considered to be one of the "official" relative sunspot numbers computed today.


A second "official" estimate is provided by the Boulder index, which is computed by the Space Environment Center of the National Oceanic and Atmospheric Administration (NOAA). Like the other indices, this number employs the Wolf formulation, includes all visible spots, and utilizes a k-factor to assure conformity. Data gathered from observatories different from those reporting to SIDC provide the basis for estimates, and it has been reported that the Boulder number is typically 25% higher than the SIDC number on a given occasion.


The formal involvement of American observers in international sunspot estimation activities began during World War II with an effort to circumvent long delays associated with the receipt of critically-needed data from Zurich. Beginning with two observers who provided data that could be used to establish monthly trends in activity, responsibility for the effort was transferred to the Solar Division of AAVSO in December 1944. This resulted in an increase in the observer base of between twenty and forty.

Working under the auspices of the Department of Terrestrial Magnetism (DTM) of the Carnegie Institution of Washington, the Division proceeded with the announced goal "... [not of supplanting] the Zurich statistics, but to give a good, up-to-date estimate of current solar activity measured by sunspot-numbers for application to long-range ionospheric predictions" (Shapley, 1946). Although it has now operated for many years under a continuing grant from the National Oceanic and Atmpospheric Administration (NOAA), pursuit of the original goal continues to motivate the operations of the AAVSO's Solar Division.

An immediate challenge faced by the Division was to devise adequate means for maintaining day-to-day continuity with the Zurich index in the absence of parallel data from Switzerland. To appreciate the problem, the reader has only to consider how, using the Zurich convention but without the ability to perform timely calibrations, estimates of sunspot activity might differ substantially with variation in the value assigned to the k factor; further, to consider how very reasonable it would be to expect such variation given the diversity of skills, equipment, observing frequencies, and seeing conditions presented by the corps of AAVSO volunteers.

The problem was tackled in mid-1945 by Alan Shapley, who developed statistical procedures for rationalizing records of the observations of two experienced "standard observers", Neal Heines, then chairperson of the Solar Division, and the Mount Holyoke (MA) observatory with Zurich data from corresponding periods--a strategy not unlike that pursued by Wolfer and colleagues to "close the loop" with the original Wolf series. The procedures were conducted until 1949 when it became apparent that the American Relative Sunspot Numbers, now identified as Ra, had drifted from by-then-available Zurich numbers. Efforts to restore conformity with Rz, including the formulation and application of revised statistical treatments aimed at maintaining long-term quality control were undertaken at that time (Shapley, 1949).

For many years, the Solar Division applied Shapley's revised statistical treatments without modification to incoming data. At intervals, the historical performance of each qualified observer was captured by two statistics, one a k-factor and a second, called a "w-factor". Together , it was felt, these adequately described the typical extent of an observer's variation from an AAVSO-maintained reference standard, as well as the consistency of that variation. At the conclusion of a reporting month, observers' k- and w-factors were applied to the estimates they made each day, reducing the variance of individual reports from the mean estimate for that day. The revised estimates were then averaged to produce a final daily number.

In recent times, Shapley's approach has been modified. Following a suggestion by Taylor that drifts in Ra were more likely to be minimized by more frequent re-evaluation of k-factors than by placing confidence in (infrequently computed) reliability weightings (w), the new procedure requires updating of k-factors at least annually. At this writing, the coefficients are updated every six months and each is based on a full year of a given observer's observations, provided that these total at least 85. Only two values of w are used, one of which is used with experienced observers and a second of which is used with new observers who have not yet completed a sufficient number of observations. It is believed that this procedure, coupled with the large number of observers in the AAVSO group leads to highly reliable and stable estimates of Ra.

When deviations in the Ra index are discovered (see, for example, Schaefer, 1997a, b) steps are taken to restore the required integrity (Foster, 1997; Foster, 1999). Very recently and in a renewed effort to control against excessive variation in estimating the numbers of sunspot groups, the chairperson of the Division has provided explicit guidelines to its contributors (Lawrence, 1999). The guidelines stress the importance of multiple scans at low, medium,and high magnifications and inclusion of all visible spots in the count, as well as the importance of frequent observation and familiarity with sunspot classification systems as aids to accurate grouping. Finally, an extensive study by Schaefer (1993, 1997) indicates that telescope apertures should be reduced to approximately 5 cm in order to insure that the diffraction disc, rather than seeing conditions, places the limit on what can be resolved by the eye.

After all statistical procedures have been completed--about the middle of the month following the month of observation--the adjusted daily and monthly mean numbers are distributed electronically and by regular mail to the Association's monthly Solar Bulletin subscribers and to the National Geophysical Data Center (NGDC) at NOAA. Along with the Ri relative sunspot numbers, they are maintained by this organization for research purposes and for distribution to other interested parties. Smoothed means of Ra numbers are computed at intervals with the aid of an algorithm developed by Waldmeier and are published in the Bulletin and/or in the Journal of the American Association of Variable Star Observers (JAAVSO).

It would not be appropriate to end this section without mentioning an additional Solar Division initiative to which some sunspot observers also contribute. This initiative, undertaken in 1958, is concerned with the monitoring of Sudden Enhancements of Signal(s) (SES). Using special receivers tuned to Very Low Frequency (VLF) stations located around the world whose transmissions bounce off the ionosphere, these observers monitor changes in signal strength that occur as a result of solar flare activity, which tends to peak around the time of sunspot maximum [5]. As with the sunspot data, these data are checked for accuracy, reported along with the daily Ra values in the Solar Bulletin, and sent to NGDC.


The similarities and differences obtaining among indices discussed above are summarized in Table 1 [NOTE: Because of its similarity to the International Number, the Boulder Number is not included in the comparison]. Viewed in this format, the reader may be tempted to ask whether or not the relative sunspot numbers determined with the various conventions are coincident, that is, whether they provide substantially the same information. The answer is both "yes" and "no". Although each may provide a unique value on any given occasion, all support the finding of long-term periodicity in the sun's behavior and all provide information concerning the relative magnitudes of maxima and minima during its various cycles.

Table 1. Summary Comparison of Sunspot Indices

American (Ra)  
Spot Count Discipline exclude small pores count all count all count all
Group Multiplier 10 10 10 10
Standard Magnification 64 64 64 scan 1:40x-50x
scan 2:50x-70x
scan 3:80x-100x
"Reference Observer" n/a Wolf Zurich
Scale factor to match Wolf (R) n/a 0.6 0.6 n/a
Average k-coefficient 1.0
0.6 .9

Despite general agreement on these issues, however, different indices can and occasionally do lead to quite different conclusions regarding the specific lengths and actual minima and maxima of particular historical cycles. They may also produce different judgments regarding the extent of the sun's activity now as opposed to earlier times, predictions as to whether such activity will be greater or less in the future, etc. Given the different networks of observers, equipment, counting conventions, and statistical treatments, one could hardly expect otherwise.

If the indices produce different results, can it be said that one is better than the rest? Some equivocation is probably also in order here, for as Coffey et al. (1999) suggest, "Each piece of data has its own story." However, it is clear that the Wolf (R) index and any other index that does not include all of the spots and pores that are visible at magnifications of 64x and above underestimates the amount sunspot activity. Further and by the same reasoning, one might expect that the Zurich index and any other index that followed those conventions would provide a better estimate than R ,if for no other reason than that all spots that can be detected are included. Finally, as we pointed out in Section 3.0, direct conversions between Wolf numbers and the Zurich numbers are problematical. Because they also incorporate all the the spots in their indices, Ri and Ra numbers also cannot be converted directly to the Wolf index.

Regarding Ri and Ra, Coffey et al. report a high correlation (0.983) between the two over a period of 53.5 years, with average annual relative sunspot numbers of 76.08 (±57.02) and 75.26 (±58.15), respectively. They also report a strong correlation (0.980) between the Ri numbers and solar radio flux, an objective measure of the sun's activity. Although an equivalent analysis of Ra and solar flux is not reported, one might expect a correspondingly high correlation between these two indices over the same period of time.


Despite the fact that they aid in ensuring the continuity of long-term historical records of sunspot activity, it is frequently noted that estimates based on (10g + f) exhibit high degrees of variability. Those observations argue for the formulation of indices that are less sensitive to human judgment and seeing conditions.

At least two different, though complementary, approaches to the problem can be identified. One takes human judgment out of the loop entirely, substituting objective, physical (e.g., solar flux) measurements of activity for the more subjective techniques associated with visual observation [6]. The second seeks ways to put the classic visual methods on a more stable footing. An example of the latter approach is presented below.

It is almost always true that groups of spots are more easily detected than the individual spots which comprise them, and it is usually the tallying of the individual spots that provides the more difficult reporting task. Given those facts, would it make sense to base an index on the group component and not bother counting individual spots?

An extensive study of historical records recently conducted by Hoyt and Schatten (1998).suggests that an index based entirely on a count of groups might present a desirable alternative to the (10g + f) formulation. Using only the numbers of sunspot groups reported by observers, they were able to derive much-improved estimates of daily, monthly, and yearly sunspot activity in the period from 1610 to 1995. This "Group Sunspot Number" provided a greater degree of internal self-consistency than the classic alternative and enabled several important conclusions to be drawn:

  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 in 1805, so there is no long anomalous cycle of 17 years as reported in the Wolf Sunspot Numbers. The longest cycle [in the record] 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."


In addition to modifying the formula for computing relative sunspot numbers, the Zurich organization developed a taxonomy for classifying sunspot groups on the basis of evolutionary changes in their extent and visual appearance. The scheme associates a letter from A to J (excluding I) with each characteristically different developmental phase. For example, a group classified as "D" in this system signifies a group that exhibits penumbra on both leading and following spots and that has an (heliographic) extent of less than 10 degrees. This scheme, which covers the growth pattern of a group from a single spot to a fully developed, large, bipolar collection of spots ranks as one of the two chief systems for characterizing the appearance of sunspots.

Two other prominent taxonomic schemes for sunspot classification exist. One of these, developed by P.S. McIntosh in 1971, is actually an extension of the original Zurich system. With it, the observer classifies a sunspot group on each of three dimensions, each one of which is assigned a range of letters. The first of these, with a few differences, is similar to the Zurich convention. The second dimension is concerned with the extent and type of penumbra (if any) surrounding the largest spot contained in a group,and the third is concerned with the distribution of spots within a group. An example of a typical classification in the McIntosh system might be "Dsi", meaning that both leading and following spots have penumbra, that the penumbra of the largest spot is circular in appearance and has a heliographic extent equal to or less than 2.5 degrees, and that a few individual spots lie between the leader and follower.

The second major system is that developed by George Ellery Hale at Mt. Wilson in 1919. Unlike the Zurich and McIntosh systems and not an alternative to either, this scheme is aimed at characterizing the magnetic properties of sunspots rather than their visual appearance. Thus a group in the Mt. Wilson system might be classified as a "beta" group, signifying one in which at least two primary spots or magnetic concentrations of opposite but equal polarity are related.




I wish to express my deep appreciation to Casper Hossfield, a former chairman of AAVSO's Solar Division, for his willingness to review and comment on drafts of this section and for his assistance in helping me identify pertinent literature, particularly that relating to the early history of AAVSO's Solar Division. The exposition of important historical threads is very much better than it might have been as a result of his energetic participation.

Carl E. Feehrer


[1] The length of the sunspot cycle, is now regarded to be on the order of 11 years, although much variation can be found. From an historical perspective, much of this variation is due to differences in observers, equipment, counting methods, and frequency of observations. Undoubtedly some is also due to real variation in cycle time. The interested reader is referred to a paper related to this variation by Hoyt and Schatten (1992), who recently compared estimates of cycle length made by Wolf and others with observations made by the eminent astronomer Sir William Herschel at a time when the sun was behaving in an anomalous fashion.

[2] Interestingly, although means for measuring areas covered by sunspots now exist, it is possible to generate simple, first-order approximations to the areas of the disk that are covered by using counts alone (Bray and Loughhead, 1981) and to estimate the lifetime of an individual spot (Waldmeier, 1955, cited in Taylor, 1991):

A = 16.7Ri

where A is expressed in millionths of the visible hemisphere, corrected for foreshortening (see Section 4.0 for definition of Ri)

    t= 0.1A(max)

    where t is expressed as time in days, and A(max) refers to the maximum area in millionths of the visible hemisphere attained by a spot during its growth cycle

      [3]Although the two are usually collapsed into a single entity called a k-factor, there is a subtle difference between a coefficient that is intended to reduce observations made on one scale to their equivalents on another scale, and a coefficient(s) that is intended to achieve the best possible agreement among observers using the same scale (i.e., an "observatory constant"). The Zurich value of 0.6 represents an attempt to accomplish the first of these; all other references to k-factors represent attempts to accomplish the second.

      [4] Hossfield (1996) suggests that the adoption by Belgium of a k-factor value of 1.0 while Zurich's stood at 0.6 is suspect and has had considerable impact on the resulting numbers. He points out that, in order for both values to be "correct", 40% of the spots Waldmeier and the Zurich organization had typically counted would have had to have been left out in the Belgian count on any given occasion. This could have resulted from Belgium's inadvertent restoration of the original Wolf Number Index through use of magnifications that were too low to reveal small groups and/or failure to recognize that Zurich's number was, in fact, a scale factor and not an observatory constant (see Footnote [3] above).

      [5] A solar flare is a burst of light that occurs in the chromosphere in close vertical alignment with a highly active sunspot group. The "life cycle" of a flare begins with a very rapid rise to peak intensity which is maintained for a brief time and is followed by a relatively slow decay. Though variable in duration, a flare typically lasts about 30 minutes. Flare events are usually detectable visually only with the aid of certain filters (e.g., H-alpha) although occasionally one can be seen in white light.

      A statistical relationship exists between the number of flares per solar rotation and the mean relative sunspot number associated with that rotation. The relationship is given as Nf = a(Ravg - 10), where Nf is the number of flares, Ravg is the mean sunspot number, and a is a constant (Chernan, 1978).

      [6] It is possible to work "backwards" from data relating to magnetic activity on the sun to an approximate sunspot number. The best estimate of this activity, detected at a frequency of 2.8 GHz, is called the 10.7 cm flux. Adjusted for the distance between the earth and the sun, it can provide an estimate of the sunspot number via the relation (Tapping, 1999):

      Sunspot number = 1.14 (10.7cm flux) - 73.21



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