Counting Sunspots and Sunspot Inflation

Guest post by Dr. Leif Svalgaard

The official sunspot number is issued by SIDC in Brussels http://sidc.be/sunspot-data/ . The [relative] sunspot number was introduced by Rudolf Wolf http://en.wikipedia.org/wiki/Rudolf_Wolf  in the middle of the 19^th century. He called it the ‘relative’ number because it is rather like an index instead of the actual number of spots on the Sun. Spots occur in groups [which we today call ‘active regions’] and Wolf realized that the birth of a new group was a much more significant event than the emergence of just a single new spot within a group, so he designed his index, R, [for any given day] to be a weighted sum of the number of spots, S, and the number of groups, G, giving the groups a weight of 10: R = S + 10*G. The number of 10 was chosen because on average a group contains about 10 spots, and also because it is a convenient number to multiply by.

Later, Wolf introduced the so-called ‘k-factor’ to compensate for differences in the size of telescope, precise counting method, observer acuity, etc, in order to bring the relative sunspot number determined by another observer on to the same scale as Wolf’s: R = k (10*G + S), where k is 1 for Wolf himself using his ‘standard telescope’ [Figure 1a,b] and his rules [not counting the smallest spots] for counting spots. From the 1860s Wolf had to travel extensively and he used exclusively [for the rest of his life] a much smaller telescope [Figure 1c]. With a smaller telescope Wolf, obviously’ saw fewer spots [and groups!], so he used k = 1.5 to convert his counts to the scale of the standard telescope.

Wolf’s successor, Alfred Wolfer, thought [rightfully] that the rule of ‘not counting the smallest spots’ was too vague and advocated to count all spots and groups that could be seen. This, of course, made his count larger than Wolf’s, so based on overlapping counts during 1876-1893, determined that to place his [Wolfer’s] relative number on to the Wolf scale he should multiply by 0.6 [one could say that his k-factor was 0.6]. This conversion factor of 0.6 has been adopted by all [Zurich] observers ever since. Adopted, not measured, as Wolf is not around any more. SIDC adopts that same factor, thus striving to stay on the Zurich scale.

So far, so good. But at some point in the 1940s, the Zurich observers began to ‘weight’ sunspots according to size and complexity, such that large spots would not be counted just once [as Wolf and Wolfer did], but up to five times, i.e. given a weight of five. There is nothing wrong with that, if one then also adjusts the k-factor to reflect this new way of counting. The director of the Zurich observatory from 1945-1979, Max Waldmeier, may have thought [?] that the weighting was introduced a long time ago [he mentions ‘about 1882’] so that no change of k-factor would be needed. Waldmeier set up a station in Locarno in southern Switzerland [as the weather on the other side of the Alps is often complimentary to that in Zurich] to provide observations when it was cloudy in Zurich. The observers in Locarno [Sergio Cortesi began in 1957 and is still at it] were instructed to use the same weighting scale as Waldmeier in Zurich. Because SIDC to this day normalize all observations they collect from a network of 60-70 observers to the count from Locarno, the weighting scheme carries over unchanged to the modern sunspot number.

We know that Wolfer did not weight the spots [contrary to Waldmeier’s assertion], because Wolfer himself explicitly [in 1907] stated that each ‘spot is counted only once, regardless of size’, and also because Wolfer’s counts as late as in 1924 when compared to other observers’ simply show that single spots are counted only once no matter how large.

To get a feeling for how the weighting works, try to count the spots on the Locarno drawing for today http://www.specola.ch/drawings/2013/loc-d20130104.JPG and compare your counts with the values given for each numbered group in the little table at the upper right.

(Note: I did this exercise, and found that my layman’s count was much lower than the “official” count, lending credence to Leif’s premise. Try it! – Anthony)

Marco Cagnotti’s [from Locarno] count is 11 groups and 53 ‘weighted’ spots. My count of the actual number of spots is 23. Try it for yourself. Your count may differ by about one from mine, but that does not change the fact that the weighted relative number 10*11+53=163 is about 23% larger than the ‘raw’, simple count of 10*11+23=133 that Wolfer and Wolf would have reported. For the whole of 2012 the ‘over count’ was 18%. So, it seems that the relative sunspot number suffered a 20% inflation because Waldmeier did not change his k-factor to compensate for the weighting.

Can we verify any of this? Well, one verification you can do yourself: just count the spots. But a better test is to ask the Locarno observers to report two numbers: the weighted count as usual and the unweighted count, where each spot is counted just once. Such a test has been [as is being] done. Figure 2 shows the effect of the weighting. Blue symbols show the official weighted count, and red symbols show Marco and my raw counts. The conclusion should be obvious.

Can we check when the inflation actually began? At the Royal Greenwich Observatory photographs of the Sun have been taken ‘every’ day since 1874 until they stopped in 1975. From these photographs the RGO determined the area of all sunspots for every day. It turns out that there is a very good relationship [as you might expect – more spots, more area covered with spots] between the sunspot area SA and the sunspot number R. For the interval before Waldmeier that relationship is closely described by R = 0.3244 * SA^0.732 [for monthly values].

The top panel of Figure 3 shows how well the sunspot number calculated from this formula matches that reporter by the Zurich observers.

Applying the same formula to data after 1945 gives us the lower panel. Under the assumption that the Sun did not know about Waldmeier we would expect the same relationship to hold, but in fact there is an abrupt change of the observed vs. the expected sunspot numbers between 1946 and 1947 of [you guessed it] 20%. Several other solar indicators give the same result. So there are several smoking guns.

What to do about this? One obvious thing would be to simply to remove the inflation [dividing the modern sunspot number by 1.20] and to stop weighting the spots. This turns out to be a bad idea, at least users of the sunspot numbers complain that they do not want to change the modern numbers as they are used in operational programs. The next-best thing is to adjust the old numbers before 1947 by multiplying them by 1.20. This is what we have decided to do [at least for now]. Who are ‘we’? You can see that here http://ssnworkshop.wikia.com/wiki/Home

There is a precedent for this [with the same ‘solution’]. In 1861 Wolf had published his first list of relative sunspot numbers, which he then updated every year after that. But about 1875 he realized that he had underestimated Schwabe’s counts [which formed the backbone of the list before Wolf’s own observations began in 1849]. Consequently, Wolf increased wholesale all the published sunspot numbers before 1849 by 25%. So we are in good company.

A somewhat disturbing [to many people] consequence of the correction of the official sunspot number is that there is now no evidence for a Modern Grand Maximum [‘the largest in 8000 years’ or some such].

NOTE: Figure 4 added 1/5/13 at Leif’s request