# Revealing Coupled Periodicities in Sunspot Time Series Using Bispectrum--An Inverse Problem
## 1 Introduction
The results indicate that the century-type cycle of Gleissberg exhibits a double structure with periodicities of 50-80 and 90-140 years, while the Suess cycle has a less complex structure with a variation period of 170-260 years. [PERSON] et al. [8] reported correlations between direct solar activity indices and planetary configurations. While no successful physical mechanism was proposed to explain these correlations, the potential link between planetary motion and solar activity has been largely overlooked. Despite energy considerations indicating that planets cannot directly cause solar activity, the question of whether planets can perturb the solar dynamo operation remains open. The authors used a 9400-year solar activity reconstruction from cosmogenic radionuclides to explore this hypothesis. They found a remarkable agreement between long-term cycles in solar activity proxies and periodicities in planetary torque. Some periodicities remained phase-locked over the 9400-year period. This led to the hypothesis that long-term solar magnetic activity is modulated by planetary effects, with potential implications for solar physics and the solar-terrestrial connection. Galactic cosmic rays produce cosmogenic radionuclides, such as 10 Be and 14C, when entering Earth's atmosphere. The solar magnetic field in the heliosphere modulates cosmic rays, influencing the production rate of these radionuclides. Various periodicities, ranging up to 2200 years, have been reported based on radionuclides [8]. The idea that planetary motions can influence solar activity has a long history, dating back to the late 1850s with [PERSON]. While energy considerations rule out planets as the direct cause of solar activity, the question of whether planets can affect the solar dynamo operation remains an open area of investigation [8]. However, in [9], [PERSON] et al. found no support for the hypothesis of a planetary influence on solar activity and raised the question of whether the centennial periodicities of solar activity observed during the Holocene were representative of solar activity variability in general. Their results do suggest, however, that it is important to test other proxy records of solar variability for periods other than the Holocene. If the observed periodicities in variability are not constant and stationary over time, this rules out any regular forcing factor, such as planetary positions, having a significant influence on solar activity. In [10], the authors used a more conservative non-parametric random-phase method and found that the long-period coherence between planetary torque and heliospheric modulation potential became insignificant. Thus, they concluded that the considered hypothesis of planetary tidal influence on solar activity was not based on solid ground. [PERSON] [11] suggested that there could be an influence of some specific configurations of the giant planets in the origin of the 12-year and 50-year periodicities of the north-south asymmetry on solar activity. [PERSON] and [PERSON] [12] showed that the statistical test presented in [8] to demonstrate a causal link between the planetary orbits and the level of solar activity was conceptionally flawed and biased. Furthermore, their execution of the test contained severe technical errors. A corrected test revealed that the period coincidences reported by [8] were statistically insignificant. [PERSON], in [13], reviewing the 11-year cycle of solar activity, commented that it was characterized by the rise and fall in the numbers and surface area of sunspots. Furthermore, a number of other solar activity indicators also vary in association with the sunspots, including the 10.7 cm radio flux, the total solar irradiance, the magnetic field, flares and coronal mass ejections, geomagnetic activity, galactic cosmic ray fluxes, and radioisotopes in tree rings and ice cores. The study by [PERSON] [14] used the evolution of an index that characterized the relative position of the planets Venus, Earth, and Jupiter. Their linear configurations had an 11-year cycle.