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Astron. Astrophys. 321, 323-329 (1997)

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3. Angular rotation rate

3.1. Global rotation

The differential rotation rate is calculated for each [FORMULA] latitude bin and is plotted in Fig. 1, along with the 1- [FORMULA] deviation from the mean. These deviations are of the order of [FORMULA] /day. The rotation appears rigid over a broad range of latitudes. This rigidity may be due to a deep anchoring of young sunspots, if we may indeed assume such is the case, because our tracking procedure preferentially selects these young sunspots. To compare with other data, we began by performing a fourth-degree Legendre polynomial fit, using a least-squares minimization of the observed data, averaged over [FORMULA] latitude bins. Since we observed no strong north-south asymmetry in rotation rates, we kept only the even Legendre coefficients, which are then converted to A, B, and C, such that the rotation rate is modeled by


where [FORMULA] is the latitude. Fig. 2 compares the polynomial fit performed on cycle 19 and 21 sunspots. The rotation during cycle 21 seems to be a little more rigid than that of cycle 19, and exhibits a slightly smaller equatorial rotation rate, though the differences are not significant. This rigidity has also been observed by Antonucci et al. (1990) on large-scale photospheric features.

[FIGURE] Fig. 1. Sidereal rotation rate of Meudon sunspot data, in [FORMULA] latitude bins, for the period 1957 to 1962. Errors are at the 1- [FORMULA] confidence level
[FIGURE] Fig. 2. Comparison of cycle 19 (solid line) and 21 (dashed line) sunspot rotation rates using a fourth-degree polynomial fit on the averaged data

3.2. Cycle variability

We now investigate the variability of the rotation rate throughout the cycle. Table 2 shows the expansion (1) coefficients during cycle 19. The period ranging from 1957 to 1959 is one of intense activity, while the minimum occurred in 1963. Errors at the 1- [FORMULA] confidence level are computed using a Monte Carlo simulation.


Table 2. Parameters of expansion [FORMULA], where [FORMULA] is the latitude, for cycle 19 sunspots (in [FORMULA] rad/s) and cycle 21 (Paper I). Errors are at the 1- [FORMULA] confidence level

The errors in the Legendre polynomial coefficients are usually determined by computing the covariance matrix (see Press et al. 1986, p521ff). However, this method assumes that when the fit parameters are displaced from their optimum values by a deviation


then the resulting deviation in [FORMULA] is quadratic:


When higher order terms are significant, the covariance matrix becomes unreliable, and we turn to Monte Carlo simulations to estimate realistic errors (Press et al. 1986, p529ff). We found that errors computed with the two methods are different, so we adopted the Monte Carlo method to determine them.

Based on these errors, we conclude that the variations of the coefficients A, B, and C over the five year duration of the data set were not significant.

3.3. North-south asymmetry

To investigate the north-south asymmetry of sunspot rotation rates, we now consider the odd coefficients of the polynomial expansion


where [FORMULA], [FORMULA], and [FORMULA] are the coefficients A, B, and C of Eq. 1. [FORMULA] and [FORMULA] represent the north-south asymmetry of the rotation rate at low and high latitudes, respectively. [FORMULA] and [FORMULA] are shown in Table 3 for cycles 19 and 21. Let us note that [FORMULA] is not reliable because there are too few sunspots at high latitudes, but a second-degree polynomial fit would provide a poor fit of the data. Both cycles exhibit no significant north-south asymmetry.


Table 3. North-south asymmetry coefficients for cycle 19 and 21, with their 1- [FORMULA] errors

3.4. Age and magnetic polarity dependence

We then examined the variability of the rotation rates with sunspot age. For this purpose, two classes of sunspots were considered: those four days old or less (759 sunspots) and those more than eight days old (194 sunspots). We also considered "complex sunspot" rotation rates. Complex sunspots belong to groups in which it is not possible to determine their age. Fig. 3.2 shows the rotation rates for the three classes and compares their respective polynomial fits. Young sunspots show a slightly more rigid rotation than old ones. The uncertainty on the coefficients A, B and C are large, due to the small sampling. However, the same trend was observed in cycle 21 sunspots, which gives some confidence in the result. It should be noted that other sunspot data sets (Mount Wilson) and sunspot-group data sets (Greenwich) also show that sunspots rotation is age-dependent (e.g. Ward, 1965; Gilman & Howard, 1983). However, these authors could not conclude that young sunspots were deeply anchored: the rigidity of young sunspot rotation suggests a deep anchorage, at a place where the convective layers show little differential rotation.

[FIGURE] Fig. 3. Sidereal rotation rate of cycle 19 Meudon sunspots, according to age. Top left: sunspots younger than 4 days. Top right: sunspots older than 8 days. Bottom left: complex sunspots. Bottom right: polynomial fits for young (solid line), old (dashed line), and complex (dotted line) sunspots. Error bars are at the 1- [FORMULA] confidence level
[FIGURE] Fig. 4. Sidereal rotation rate of cycle 19 Meudon leading and following sunspots. Top left: leaders. Top right: followers. Bottom: polynomial fit for leaders (solid line) and followers (dashed line). Error bars are at the 1- [FORMULA] confidence level

A bipolar group consists of a pair of sunspots of opposite polarity, with one leading and one following. The rotation rates of leaders (432 points) and followers (175 points) is studied separately (Fig. 3.4). However, the error bars are too large to detect any significant differences between these two sunspot classes, as the one observed by Gilman & Howard (1985) and in Paper II (faster rotation of leaders).

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998