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Astron. Astrophys. 345, 1006-1010 (1999)

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2. Basis function and cycle shape parameters

For our analysis, we use quarterly averages of the sunspot areas. They extend from the year 1875 to 1996 and consist of two parts having different reliability. The first part comes from the Greenwich Photoheliographic Results, extending from 1875 to 1971, but the graph given by White & Trotter (1976) was used as we have no primary digital data. Five points divided one year of the abscissa into four equidistance parts in the enlarged graph, then we regarded the average of coordinate values of adjacent two points as quarterly average of the corresponding time. The rest comes from sunspot observations of Yunnan Observatory (Hong & Wang, 1988). The Greenwich Photoheliographic Results are available until the year 1976, a comparison of the first part with the second was done for the overlapping period (1971 to 1976). Deviation between the two parts is less than [FORMULA], and for most values of sunspot areas it is less than [FORMULA]. These quarterly averages are plotted in Fig. 1 for the entire interval (1875-1996).

[FIGURE] Fig. 1. Quarterly averages of the sunspot areas for 1875 to 1996. This illustrates the cyclic behavior of the sunspot areas and shows the variations in size and shape of each sunspot cycle. A comparison of the sunspot data and the two-parameter functional fit for the cycles 12-22 is given too (details are given late in the text). The thin line represents the data, the thick dotted line represents the two-parameter fit for each cycle.

Inspection of Fig. 1 reveals that individual cycles show a wide range of temporal behavior. For example, some cycles (like 12-16, and 20) are small in amplitude (about half the size of cycle 19), while others are considerably larger. Most cycles show substantial asymmetry with the rise to maximum being faster than the fall to minimum. Although the average cycle length (minimum to minimum) is about 11 years, individual cycles vary in length from about 9 to 14 years. All these properties are similar to the characteristics of the solar cycles described by the international sunspot relative numbers. Initiated by the work of Hathaway, et al. (1994), a function of the form


is proposed to describe the shape of the sunspot area cycles, where parameter a represents the amplitude, b is related to the time from minimum to maximum, c gives the asymmetry of the cycle, [FORMULA] denotes the starting time. Here we use a linear term of [FORMULA] in the exponential function of the equation instead of the quadratic term used by Hathaway, et al. (1994). This function reproduces both the rise and decay portions of the sunspot cycle. We determined the best-fit parameters for each cycle using a nonlinear least-squares fitting algorithm in which all four parameters can vary. In attempting to fit each cycle with the four-parameter basic function we found that the process initially gave parameters with large uncertainties and the parameter b could be fixed at a constant value for the 11 complete cycles. Although slight variations of the values of the parameter b occur from cycle to cycle, they appear to be unrelated to the other parameters and the results are consistent with taking a simple value of [FORMULA] for all cycles. This effectively reduces the number of parameters by one and also stabilizes the fitting procedure by removing one degree of freedom. Eq. (1) becomes a function of three parameters. Then we fix the parameter [FORMULA] to reproduce the shape of the sunspot area cycle again. Table 1 gives summary of best-fit parametric values for the three-parameter function. Examination of the best-fit parametric values for each cycle shows that parameters a and c are related. Fig. 2 shows parameter c as a function of parameter a for all 11 cycles. A strong relationship between a and c is clearly seen, and the best fit to this relationship is given by a two-order polynomial expression of the form


[FIGURE] Fig. 2. Parameter c plotted as a function of parameter a (amplitude) for all the cycles 12-22. The functional fit given by Eq. (2) is represented by the thick line through the data points marked by symbol [FORMULA].


Table 1. Summary of the cycle shape parametric values

The correlation coefficient [FORMULA]. This effectively reduces again the number of parameters by one, leaving only two. Then the two-parameter fit is attempted to be done for all 11 cycles. Table 1 also lists the best-fit parametric values for the two-parameter function. Fig. 1 shows a comparison of the sunspot area data and the two-parameter functional fit for the years 1875 to 1996 (cycles 12-22). A measure of the goodness-of-fit is given in the fifth and eighth columns respectively for the three-parameter functional fit and the two-parameter functional fit, which is defined by [FORMULA], where [FORMULA] is the quarterly-averaged value of the sunspot area, [FORMULA] the functional fit value, and N the number of quarters in the cycle. The three-parameter function usually gives a slightly better fit than the two-parameter function, and the best fit of the above three kinds is given by the four-parameter function. These differences are not, however, considered significant. The more important result is that sunspot area data can be adequately fit with a relatively simple function of only two parameters.

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999