The Mathematical Characteristics of Sunspot Variations.
Abstract
This paper mathematically defines the course of the Wolf sunspot numbers during the sixteen completed cycles since 1755. The usual periodic or "superposition" point of view is rejected in favor of the "outburst" hypothesis of Haim and Waldmeier, which is here independently confirmed in detail. Each successive cycle or "outburst" is re garded as a fresh phenomenon, following, however, a standard pattern. As a new addition to the "Wolf" sunspot data published by W. Brunner in the Zurich statistics, the zero, first, and second moments for each cycle are calculated about the times of minimum, and all the observed data are employed to show that the em pirical form R = FOaebO represents, to a good first approximation, the course of spot numbers during a single outburst or cycle (of, on the average, eleven years). Here R is the Wolf number, 0 is the interval in years after the time of beginning, s, of the out burst, and F, a, and b are parameters which differ for different cycles but remain con stant for a single cycle. Values of these four parameters, s, a, b, and F, are tabulated for each cycle, and the goodness of fit is discussed. There are small random discrepancies, year by year, averaging no more than halfa dozen Wolf numbers, between observed and computed spot numbers; and an important systematic discrepancy which is particularly evident when the interval between the time of maximum and of centroid is examined. According to the form suggested, the centroid should always come after the maximum, but actually this is the case oniy for cycles of higher maximum. For the group of cycles of low maximum the observed cen troid precedes the maximum; the interval between the two shows a strong correlation with the height of maximum. This means that in a second approximation the suggested formula for R will require modification. The computed parameters are examined for possible mutual dependence, and the indication is that F and b can be taken as rough functions of a. The evennumbered cycles have smaller values of a than any oddnumbered cycle. In view of the strong 1 family resemblance shown by this study among apparently rather dissimilar cycles, the conventional periodogram analysis of spot numbers definitely must be abandoned, because such resemblance would be too improbable as the result of the superposition of independent periodic factors. Longrange prediction of spots will remain impractij~ cable, but shortrange prediction will be facilitated by the methods outlined in this paper. * The extensive contributions to this study by Mr. Panofsky represent his Senior thesis as a student of astronomy, toward his A.B., Princeton, 1938. 38
 Publication:

The Astrophysical Journal
 Pub Date:
 November 1938
 DOI:
 10.1086/143994
 Bibcode:
 1938ApJ....88..385S