Comparison of the shape and temporal evolution of even and odd solar cycles

Abstract

Aims. We study the difference in the shape of solar cycles for even and odd cycles using the Wolf sunspot numbers and group sunspot numbers of solar cycles 1−23. We furthermore analyse the data of sunspot area sizes for even and odd cycles SC12−SC23 and sunspot group data for even and odd cycles SC8−SC23 to compare the temporal evolution of even and odd cycles. Methods. We applied the principal component analysis (PCA) to sunspot cycle data and studied the first two components, which describe the average cycle shape and cycle asymmetry. We used a distribution analysis to analyse the temporal evolution of the even and odd cycles and determined the skewness and kurtosis for even and odd cycles of sunspot group data. Results. The PCA confirms the existence of the Gnevyshev gap (GG) for solar cycles at about 40% from the start of the cycle. The temporal evolution of sunspot area data for even cycles shows that the GG exists at least at the 95% confidence level for all sizes of sunspots. On the other hand, the GG is shorter and statistically insignificant for the odd cycles of aerial sunspot data. Furthermore, the analysis of sunspot area sizes for even and odd cycles of SC12−SC23 shows that the greatest difference is at 4.2−4.6 years, where even cycles have a far smaller total area than odd cycles. The average area of the individual sunspots of even cycles is also smaller in this interval. The statistical analysis of the temporal evolution shows that northern sunspot groups maximise earlier than southern groups for even cycles, but are concurrent for odd cycles. Furthermore, the temporal distributions of odd cycles are slightly more leptokurtic than distributions of even cycles. The skewnesses are 0.37 and 0.49 and the kurtoses 2.79 and 2.94 for even and odd cycles, respectively. The correlation coefficient between skewness and kurtosis for even cycles is 0.69, and for odd cycles, it is 0.90. Conclusions. The separate PCAs for even and odd sunspot cycles show that odd cycles are more inhomogeneous than even cycles, especially in GSN data. Even cycles, however, have two anomalous cycles: SC4 and SC6. The variation in the shape of the early sunspot cycles suggests that there are too few and/or inaccurate measurements before SC8. According to the analysis of the sunspot area size data, the GG is more distinct in even than odd cycles. This may be partly due to sunspot groups maximizing earlier in the northern than in the southern hemisphere for even cycles. We also present another Waldmeier-type rule, that is, we find a correlation between skewness and kurtosis of the sunspot group cycles.