4.5 Cycle shape

Sunspot cycles are asymmetric with respect to their maxima (Waldmeier, 1935Jump To The Next Citation Point). The elapsed time from minimum up to maximum is almost always shorter than the elapsed time from maximum down to minimum. An average cycle can be constructed by stretching and contracting each cycle to the average length, normalizing each to the average amplitude, and then taking the average at each month. This is shown in Figure 24View Image for cycles 1 to 22. The average cycle takes about 48 months to rise from minimum up to maximum and about 84 months to fall back to minimum again.
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Figure 24: The average of cycles 1 to 22 (thick line) normalized to the average amplitude and period. The average cycle is asymmetric in time with a rise to maximum over 4 years and a fall back to minimum over 7 years. The 22 individual, normalized cycles are shown with the thin lines.

Various functions have been used to fit the shape of the cycle and/or its various phases. Stewart and Panofsky (1938Jump To The Next Citation Point) proposed a single function for the full cycle that was the product of a power law for the initial rise and an exponential for the decline. They found the four parameters (starting time, amplitude, exponent for the rise, and time constant for the decline) that give the best fit for each cycle. Nordemann (1992) fit both the rise and the decay with exponentials that each required three parameters – an amplitude, a time constant, and a starting time. Elling and Schwentek (1992Jump To The Next Citation Point) also fit the full cycle but with a modified F-distribution density function which requires five parameters. Hathaway et al. (1994Jump To The Next Citation Point) suggested yet another function – similar to that of Stewart and Panofsky (1938Jump To The Next Citation Point) but with a fixed (cubic) power law and a Gaussian for the decline. This function of time

( )3 [ ( )2 ] −1 F(t) = A t −-t0- exp t −-t0 − c (6 ) b b
has four parameters: an amplitude A, a starting time t0, a rise time b, and an asymmetry parameter c. The average cycle is well fit with A = 193, b = 54, c = 0.8, and t0 = 4 months prior to minimum. This fit to the average cycle is shown in Figure 25View Image. Hathaway et al. (1994Jump To The Next Citation Point) found that good fits to most cycles could be obtained with a fixed value for the parameter c and a parameter b that is allowed to vary with the amplitude – leaving a function of just two parameters – amplitude and starting time.
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Figure 25: The average cycle (solid line) and the Hathaway et al. (1994Jump To The Next Citation Point) functional fit to it (dotted line) from Equation (6View Equation). This fit has the average cycle starting 4 months prior to minimum, rising to maximum over the next 54 months, and continuing about 18 months into the next cycle.

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