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The solar cycle: a modified Gaussian function for fitting the shape of the solar cycle and predicting cycle 25

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The smoothed monthly sunspot numbers of the new version in each solar cycle can be well fitted by a four-parameter modified Gaussian function, with a correlation coefficient around 0.986. A longer tail tends to be related to a larger amplitude, a narrower width, and a shorter rise time of the shape. This function can be simplified to a two-parameter function using the Waldmeier effect and the anticorrelation between the asymmetry and width of the shape, and can also be reduced into a one-parameter function with the peak value as the only free parameter. The two- and one-parameter functions are much superior to the four-parameter function in predicting the maximum amplitude (\(S_{\mathrm{m}}\)) and rise time (\(T_{\mathrm{a}}\)) of the cycle, although the former two are slightly inferior to the latter in fitting the shape of the cycle (\(r\approx 0.958\) and 0.949). The one-parameter function is similar to the two-parameter function in predicting \(S_{\mathrm{m}}\) and \(T_{\mathrm{a}}\). For the two-parameter function, the mean relative prediction errors of \(S_{\mathrm{m}}\) over the last ten cycles are 15.2%, 11.9%, 11.6%, 10.7%, 7.9%, and 4.8% at \(m=18\), 24, 30, 36, 42, and 48 months into the cycle, respectively, and the MSE skill score is larger than 0.7 if \(m\geqslant 22\). The mean absolute (relative) prediction error of \(T_{\mathrm{a}}\) is around 5.5 months (11%) since \(m=24\) and the MSE skill score is around 0.58. Using the data available at the current time, cycle 25 is predicted to peak around October (\(\pm 7\) months) 2024 with a maximum amplitude of \(124\pm 30\).

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The data used in this study are downloaded from the Sunspot Index and Long-term Solar Observations website (




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We are grateful to the anonymous reviewer for valuable suggestions that have improved this manuscript.


This work was supported by National Key R&D Program of China under grant 2021YFA1600504 and the National Science Foundation of China (NSFC) under grants 11873060, 11790305 and U1831121.

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Data analysis and manuscripts were completed by DZL.

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Correspondence to Z. L. Du.

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Du, Z.L. The solar cycle: a modified Gaussian function for fitting the shape of the solar cycle and predicting cycle 25. Astrophys Space Sci 367, 20 (2022).

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