3 Results

In the following analysis we use the combined data of 1874-1976, for the whole region of heliographic latitudes of $0^{\rm o}$ to $40^{\rm o}$in both the hemispheres. This will ensure better statistics i.e., more data and hence less uncertainty in the results of rotation rates and rate of change of rotation rates. Another aspect of taking combined data in this region ($0^{\rm o}$ to $40^{\rm o}$) is that we want to compare the variation of rotation rates of the sunspot groups with the radial variation of rotation inferred by helioseismology. Though solar rotation varies from the equator to the latitude zone of $40^{\circ}$ ($\sim$1 degree/day corresponding to $\sim$30 nHz; see Balthasar et al. 1986; Zappala & Zuccarello 1991), the radial variation of the gradient of rotation (inferred by helioseismology) in the belt of sunspot latitude zone is similar. Hence we combined the data set of rotation rates of the sunspot groups for the latitude zone of $0^{\rm o}$ to $40^{\rm o}$. It is to be noted that, in addition to Greenwich data available from 1874 to 1976, data are also available up to the present date on the web site. However, for the sake of easy computer programming and in order to keep the continuity of group numbers, we use the data from 1874-1976 only. For the sake of comparison with the solar internal rotation, as inferred by helioseismology, rotation rates of sunspot groups are computed in units of nHz. Errors quoted in the following rotation rates are computed as: $\sigma \over N^{1/2}$, where $\sigma$ is the standard deviation and N is the number of data points.

3.1 The results of $\omega$ and AD independent of life span of the sunspot groups

Firstly we classify the data into two data sets (<100 mh and >100 mh) and using Eqs. (1) and (2), we compute $\omega$and AD. In Fig. 1a, we present the number of spot groups with respect to their age. The dotted line represents the area <100 mh and dashed line represents area >100 mh. In Fig. 1b, we present average rotation rates $\omega$ with the age of sunspot groups that are independent of their life spans. By dividing the convective envelope from the base ( $0.695\,R_{\odot}$) to the surface ( $1\,R_{\odot}$) into equally 11 parts and with an equal interval of $0.03\,R_{\odot}$, in the same Fig. 1b, we plot the internal rotation for the latitude of $15^{\rm o}$ as inferred by helioseismology. The reason that we have to select a rotation profile at the latitude of $15^{\rm o}$ only is as follows. The radial variation of the gradient of rotation profile inferred by helioseismology is similar for the latitude zones $0^{\rm o}$ to $40^{\rm o}$ and the maximum number of sunspot groups during the solar cycle emerge near this belt. We present the rate of change of rotation rates ADin Fig. 2a. The results presented in Fig. 1b are in agreement with the results of previous studies (Balthasar et al. 1986; Touminen & Virtanen 1987) that the old spot groups rotate slower than the young spot groups resulting in a net deceleration (see Fig. 2a) of rotation rates during their lifetimes. It is crucial to note that in Fig. 1b both the curves of rotation rates of spot groups, except rotation rates of spot groups which have an age $\le$4 days, are similar to the curve of radial variation of the internal rotation as inferred by helioseismology. The disagreement of rotation rate of spot groups below $\le$4 days with the helioseismologically inferred rotation is due to mixture of life spans of all the spot groups. The large contribution of spot groups which have very small ($\le$3 days) life spans and which may be rotating faster than the very low contribution slow rotating spot groups of long lifetime ($\ge$4 days) may be the reason for the aforementioned discrepancy. This hypothesis is justified by the following analysis wherein we compute initial rotation rates with respect to their different life spans. The aforementioned discrepancy will disappear in the following analysis.

3.2 Determination of ${\omega _{1}}$ and AD₁ with respect to sunspot groups life span

After classifying the data into two groups of areas (<100 mh and >100 mh) and using Eqs. (1) and (2), we compute initial rotation rates ${\omega _{1}}$ and rate of change of initial rotation rates AD₁ of sunspot groups with respect to their life spans. By computing these parameters we may get an idea at what depths sunspots may be originating and in turn we may get information on radial variation of rotation of the solar plasma beneath the surface. The number of spot groups with different life spans is presented in Fig. 2b. In Fig. 3a, we present ${\omega _{1}}$ with respect to life span of sunspot groups for the two sets of areas. On the same curves, we plot radial variation of rotation inferred by helioseismology. Except in the case of large spot groups which are rotating slightly faster than the small spot groups, we get the same results as those obtained by JG97. Although large spot groups rotate faster than the solar plasma, the nature of curves of rotation rates of sunspot groups and rotation of the sun's internal plasma as inferred from helioseismology are similar. Now we justify our hypothesis as proposed in the previous section. Irrespective of their sizes, from Figs. 2b and 3a, it is clear that the spot groups having small ($\le$4 days) life spans (in large numbers) rotate faster than the spot groups with long life spans. Thus this important fact may be the reason for the results presented in the previous section that the profile of rotation rates (with respect to their age) of spot groups whose lifetime is $\le$4 days is different than the rotation inferred from helioseismology. The large error bars, for the spot groups having small ($\le$4 days) life spans and areas <100 mh, may be due to greater dispersion in the rotation rates.

Figure 3: a) Initial rotation rate ${\omega _{1}}$ of the sunspot groups, with respect to their life span. Continuous curve is helioseismologically inferred radial variation of rotation. Dotted curve represents the rotation rates of spot groups having areas <100 mh and dashed curve represents the rotation rates of spot groups having areas >100 mh. In the same plot, radius values are plotted along the top x axis. b) Rotation inferred by helioseismology versus initial rotation rates of the sunspot groups. Signs of square values represent the areas having >100 mh and triangles represent the areas having <100 mh.

Figure 4: a) Rate of change of initial rotation rates AD1 of the sunspot groups for different life spans. The dotted line shows data on sunspots having areas <100 mh and the dashed line represents having areas >100 mh. b) Rate of change of initial rotation rates AD1 of the sunspot groups for different life spans. The dotted line shows data on sunspots having areas <100 mh. Continuous line is a straight line fit to the AD1 values.

As in the analysis of JG97, we find a strong correlation between the two curves of rotation rates of sunspot groups and the solar internal rotation. The correlation coefficient for the area <100 mh is 0.8668 and that for the area >100 mh is found to be 0.7999. The high correlation between the curves of initial rotation rates of different sunspot groups and the curve of solar internal rotation inferred by helioseismology appears to indicate that rotation rates determined by sunspot groups really represent the solar internal plasma rotation. On the other hand, a very high correlation may sometimes be misleading in interpreting the results and initial rotation rates of sunspot groups need not represent actual plasma rotation. Thus after finding a correlation, one has to apply statistical as well as physical tests (see the following subsections).

For further statistical tests, in Fig. 3b, we plot the rotation inferred by helioseismology versus the initial rotation rates obtained from the analysis of sunspot groups. These results satisfy almost all the statistical criteria (Elmore & Woehlke 1996) including the criterion of homogeneity of the data which is clear from Fig. 3b. Moreover, the probability of correlation coefficients are found to be significant below less than 1$\%$ level (Fisher 1930). Thus, determined high correlation coefficients may not be due to a casual effect, it must be due to some causal effect. Applying a physical test in the following section, we delineate the causal effect.

Figure 5: a) Rate of change of initial rotation rates AD1 of the sunspot groups for different life spans. The dashed line shows data on sunspots having areas >100 mh. Continuous line is a straight line fit to the AD1 values. b) Rate of change of solar internal plasma rotation which is depicted as a continuous line. Where as AD1 values presented in Fig. 4a are superposed on this figure.

3.3 Clues to the sites of origin of different sunspot groups

For the physical test, let us assume that sunspots of different sizes and of different life spans may originate near the base of the convection zone and rise towards the surface. We also assume that while the rise of the sunspots is balanced by the Coriolis force, buoyancy force, drag force, etc., the sunspots are mainly influenced by the ambient plasma rotation. Hence a naive conclusion is that the rising path of the flux tubes and thus rotation rates of sunspot groups represent the radial variation of solar internal plasma rotation. That means if we compute the rate of change of initial rotation rates ${\omega _{1}}$, the spot groups that may originate in the belt of sunspot latitude zones and near the region of the base of convection zone up to $0.935\,R_{\odot}$(see Figs. 1b and 3a, where we present radial variation of rotation curve inferred by helioseismology), we should get a positive value of AD₁ and for the spot groups that originate in the region between $0.935\,R_{\odot}$ to $1.0\,R_{\odot}$ we should get a negative value of AD₁. This is due to the fact that, in the radial variation of rotation profile inferred by helioseismology, we get a positive gradient of rotation from the base of the convection zone up to $0.935\,R_{\odot}$ and a negative gradient of rotation from $0.935\,R_{\odot}$ towards the surface. Thus the initial rotation rates of sunspot groups for different life spans may indeed represent the radial variation of the rotation of the solar plasma. In Fig. 4a, we present the results of rate of change of initial rotation rates AD₁ of sunspot groups which are similar to our expectation.

Once we establish that variation of the initial rotation rates of sunspot groups with respect to their life spans indeed represent the radial variation of solar internal plasma rotation, the question that arises then is do all the spot groups of different sizes originate near the base of the convection zone or in different places of the convective envelope or do different sizes of sunspot groups originate at different regions of the convective envelope. We therefore fit a straight line of the form $AD_1 = A + B\tau$, (where A and B are coefficients to be determined and $\tau$ is the life span of the spot group) to the results presented in Fig. 4a. By fitting a straight line to these AD₁ values, for two different sets of areas, we present the results in Figs. 4b and 5a. Note that in Fig. 5a, we have not included the point for the life span equal to 3 days; this is due to the fact that we get large uncertainties in the coefficients (A and B) of the least square fit compared to the coefficients of the least square fit obtained by the data with life spans $\ge$4 days. From the straight line fit and, using determined coefficients A (= -7.2293 $\pm$ 0.0150 nHz/day for the small spot groups and -8.8676 $\pm$ 0.9454 nHz/day for the large spot groups) and B (= 1.1997 $\pm$ 0.0430 nHz/day for the small spot groups and 1.0252 $\pm$ 0.0433 nHz/day for the large spot groups), we can know the life span of spot groups of different sizes whose AD₁ values remain zero. For example, by taking a hint from the helioseismic rotation curve, one would expect that near $0.935\,R_{\odot}$ where the rotation curve is almost constant along the radius and the spot groups neither accelerate nor decelerate, the value of AD₁ must be zero. A least square fit to the data yields that for the small spot groups of areas <100 mh and for the life span of $6.03\pm0.23$ day, we get that AD₁ is zero. On the other hand the spot groups whose area is >100 mh have a zero value of AD₁for the life span of $8.65\pm1.3$ day. This indicates that spot groups of area <100 mh whose life span is $\le$6 days may originate in the region of the convective envelope from $0.935\,R_{\odot}$to $1.0\,R_{\odot}$ and life spans greater than 6 days may originate below the region of $0.935\,R_{\odot}$. As for the areas >100 mh, spot groups which have a life span $\le$9 days may originate in the region of $0.935\,R_{\odot}$ to $1.0\,R_{\odot}$ and life span greater than 9 days may originate below the region of $0.935\,R_{\odot}$.