Critical Analysis of a Hypothesis of the Planetary Tidal Influence on Solar Activity

The present work is a critical revision of the hypothesis of the planetary tidal influence on solar activity published by Abreu et al. (Astron. Astrophys. 548, A88, 2012; called A12 here). A12 describes the hypothesis that planets can have an impact on the solar tachocline and therefore on solar activity. We checked the procedure and results of A12, namely the algorithm of planetary tidal torque calculation and the wavelet coherence between torque and heliospheric modulation potential. We found that the claimed peaks in long-period range of the torque spectrum are artefacts caused by the calculation algorithm (viz. aliasing effect). Also the statistical significance of the results of the wavelet coherence is found to be overestimated by an incorrect choice of the background assumption of red noise. Using a more conservative non-parametric random-phase method, we found that the long-period coherence between planetary torque and heliospheric modulation potential becomes insignificant. Thus we conclude that the considered hypothesis of planetary tidal influence on solar activity is not based on a solid ground.

The authors acknowledge Jose Abreu and Jürg Beer for providing the original data, details of the algorithm of planetary torque computation and for the stimulating discussion.

Authors and Affiliations

1. University of Oulu, Oulu, Finland

   S. Poluianov & I. Usoskin

Corresponding author

Correspondence to S. Poluianov.

Appendix: On the Significance of Coherence Between Narrow- and Wide-Band Signals

It is important to note that one must be careful when computing the coherence between narrow- and wide-band signals. An incorrectly assessed significance of the coherence can produce false physical conclusions.

We illustrate it by a simple and clear numerical example. Let us generate two independent and non-coherent narrow- and wide-band signals and call them x(t) and y(t), respectively. The first one is a purely harmonic signal x(t)=cos(2πf ₀ t), where f ₀ is the frequency and t is the time. The second one y(t) is white noise with normal distribution, zero mean and unity standard deviation. The signals and their Fourier spectra are shown in Figure 7.

An illustration of computation of coherence between narrow- and wide-band signals. The considered signals x(t) and y(t) are given in the top panel and their Fourier spectra S _x (t) and S _y (t) are in the bottom panel.

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Now let us calculate the formal coherence of two signals by the following formula:

where S _xy is the cross-spectrum of x(t) and y(t), and S _x and S _y are Fourier spectra of x(t) and y(t), respectively. A cross-spectrum is defined as

where the asterisk means a complex conjugate. The product given by Equation (9) extracts a narrow frequency range from the wide-band signal y(t) by the narrow-band signal x(t). The coherence defined by the cross-spectrum [Equation (8)] has non-zero values only near the frequency of the narrow-band signal x(t). Thus, a formal non-zero coherence exists between the two unrelated signals.

The described feature exists not only for Fourier analysis but for other kinds of spectra including wavelet analysis as well.

The result of calculation of the wavelet coherence for the two synthetic signals is presented in Figure 8. The contours that indicate statistical significance areas are based on the autoregressive model AR(1) (red noise) in the top panel and on the non-parametric random-phase method in the bottom panel. There is some coherence between the signals x(t) and y(t). Since two signals are non-coherent by definition, the computed coherence should not be statistically significant. However, one can see that the AR(1)-method estimates the coherence as significant. The non-parametric random-phase method estimates coherence between x(t) and y(t) as insignificant for the same conditions. It corresponds to the initial properties of the signals.

An illustration of computation of coherence between narrow- and wide-band signals. The wavelet coherence between signals x(t) and y(t) with statistical significance is calculated by two different methods: autoregressive model AR(1) (red noise, in the top panel) and non-parametric random-phase method (in the bottom panel). Significance is shown as black contours.

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The present illustration is close to the case of computation of coherence between heliospheric modulation potential and planetary torque. The former one has wide-band spectrum while the spectrum of the latter consists of a few narrow peaks. It leads to the described effect and explains the derived coherence spots and their statistical insignificance in Figure 4.

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