Fine Structure in the Sunspot Record

Abstract

A new method for the analysis of time-series, using the ideas of Maximum Entropy, has been applied to the yearly mean records of sunspot numbers. These numbers are regarded as reflecting an underlying magnetic cycle in the sun. The actual time-series considered consists of the square roots of the Wolf numbers, alternated every half cycle, so as to give a roughly sinusoidal shape with zero mean and a clear period of about 22 years. This device yields a much simpler spectrum than does an application of the method to the raw data.

The most probable spectrum is very clean, with the following features. Firstly, we see a group of four strong spectral lines, equally spaced, and with the dominant peak at 0.045 cycles per year (c/y). This corresponds to a period. of 22.1 years, the well-known magnetic cycle. The next marked feature is a cluster of lines at approximately three tunes the frequency of the first group and therefore a prime candidate for a third harmonic. The last significant components of the spectrum are two incompletely resolved sets of lines on either side on the main group.

All this spectral information, which includes phases as well as amplitudes, will provide clues to the mechanism of the solar cycle. Here we consider chiefly the third harmonic. By taking cuts in the spectrum we can reconstruct time-series implied by selected frequency ranges and compare them. Doing this for the main group around 22 years, and for the possible third harmonic, we find that there is a component in the series having an envelope proportional to the cube of the size of the dominant signal, a local frequency three times that of the large component and a constant time delay of one and a half years. We have found that the spectral details of this third harmonic can be accounted for using a cubic function of the main signal and its time derivative.

The sunspot record shows an anomaly at the end of the 18th century. When this is removed, the spectrum becomes even cleaner. The confused group between 0.06 and 0.09 c/y now resolves into a set of sharp, regularly spaced peaks whose frequencies have a surprising ratio to those of the main group. This talk develops ideas first reported at MaxEnt 91.