Sun's retrograde motion and violation of even-odd cycle rule in sunspot activity

Abstract

The sum of sunspot numbers over an odd-numbered 11-yr sunspot cycle exceeds that of its preceding even-numbered cycle, and it is well known as Gnevyshev and Ohl rule (or G–O rule) after the names of the authors who discovered it in 1948. The G–O rule can be used to predict the sum of sunspot numbers of a forthcoming odd cycle from that of its preceding even cycle. However, this is not always possible because occasionally the G–O rule is violated. So far, no plausible reason is known either for the G–O rule or for the violation of this rule. Here, we show the epochs of the violation of the G–O rule are close to the epochs of the Sun's retrograde orbital motion about the centre of mass of the Solar system (i.e. the epochs at which the orbital angular momentum of the Sun is weakly negative). Using this result, it is easy to predict the epochs of violation of the G–O rule well in advance. We also show that the solar equatorial rotation rate determined from sunspot group data during the period 1879–2004 is correlated/anticorrelated to the Sun's orbital torque before/after 1945. We have found the existence of a statistically significant ∼17-yr periodicity in the solar equatorial rotation rate. The implications of these findings for understanding the mechanism behind the solar cycle and the solar-terrestrial relationship are discussed.

Sun: magnetic fields, Sun: rotation, sunspots, Solar system: general

1 Introduction

Solar activity varies on many time-scales. It can impact climate and the near-earth space environment (e.g. Hoyt & Schatten 1997; Hathaway, Wilson & Reichmann 1999; Rozelot 2001; Hiremath & Mandi 2004; Georgieva et al. 2005). Therefore, prediction of the amplitudes of the variations in solar activity will greatly help the society. Sunspots are the earliest observed phenomenon of solar activity. The sunspot cycles are numbered from the cycle that began in the year 1755 (cycle 1). The current sunspot cycle, which began in the year 1996, is an odd-numbered cycle (cycle 23). The well-known Gnevyshev-Ohl rule or G–O rule (Gnevyshev & Ohl 1948) states that the sum of sunspot numbers (R_sum) over an odd-numbered sunspot cycle exceeds that of its preceding even-numbered sunspot cycle. By using the G–O rule, it is possible to predict the R_sum of an odd-numbered cycle from that of its preceding even-numbered cycle with a reasonable accuracy (Wilson 1988). However, some pairs of the even-odd cycles violated the G–O rule, i.e. in such pairs the R_sum of the odd-numbered cycle is less than that of its preceding even-numbered cycle (e.g. cycles' pairs 4, 5 and 22, 23). So far, no plausible reason is known either for the G–O rule or for its violation. In order to predict the amplitude of an odd cycle by using the G–O rule, it is necessary and essential to know in advance whether the even-odd numbered cycles' pair will satisfy the G–O rule or not. However, there is no method available for predicting the violation of the G–O rule. Predictions on the basis of precursor technique, the G–O rule and the statistical analysis of preceding cycles indicated a high R_sum for the current cycle 23, similar to or exceeding that in cycle 22 (Joselyn et al. 1997). The prediction of the violation of the G–O rule by the cycles' pair 22, 23 based on the long-term trends in sunspot activity (Schove 1955; Komitov & Bonev 2001; Javaraiah 2003b) seems to be right. However, the epoch of the next violation of the G–O rule is not yet predicted, and the available sunspot data may be inadequate to use this method.

There are two main approaches for explaining the mechanism of the solar cycle: one is based on a turbulent dynamo operating in or immediately below the solar convection envelope, and the other is a large-scale oscillation, superposed on a fossil magnetic field in the radiative core. According to the turbulent dynamo theory, the solar differential rotation produces a toroidal field (east-west component) by continuously winding up a poloidal field (north-south component), induction effect of cyclonic turbulence regenerates the poloidal field, and the excess poloidal and toroidal fields are removed by the enhancement of diffusion by convective turbulence. A sufficiently detailed and realistic model of the dynamo process to account for all the different aspects of the solar magnetism is not yet available. The available turbulent dynamo models have several difficulties. For example, in these models the role of the differential rotation in the cyclic variation of the solar activity is not clear; the reason for the cycle-to-cycle modulations of solar activity is not yet found, and have no predictive power. The basic idea of the magnetic oscillator models is to consider the observed oscillating large-scale solar magnetic field as an effect of periodic amplification of the primordial fields due to oscillations in the differential rotation rate of the solar interior. The main difficulty in the oscillator models is regarding energetics. No oscillator model offers the means of maintaining the oscillations against dissipation of velocity and magnetic fields (see reviews by Rosner & Weiss 1992; Ossenderijver 2003). In this regard, it may be worthwhile to investigate whether the Solar system dynamics could influence the internal dynamics of the Sun (Gokhale & Javaraiah 1995).

The idea that the gravity of the planets might be the cause of the solar cycle dates back at least to Carrington (Brown 1900). Subsequently, many scientists suggested a possibility of the tidal forces due to planets or the rate of change of the Sun's orbital angular momentum about the centre of mass of the Solar system (barycentre) having a role in the mechanism of solar activity. Such an idea of a role of the Solar system dynamics has been doubted because (see Ferris 1969): (i) the energy of the tidal force due to the planets is small compared to the Sun's surface gravity; and (ii) the Sun's centre of mass is in free fall in the sum-total gravitational field of all the planets. Nevertheless, the hypothesis of a relationship between the Sun's motion about the barycentre and the solar activity is supported by a growing number of studies indicating that something must be true in the ‘planetary hypothesis’ (Jose 1965; Wood & Wood 1965; Blizard 1983, 1989; Fairbridge & Shirley 1987; Sperber & Fairbridge 1990; Gokhale 1996; Zaqarashvili 1997; Landscheidt 1999; Charvátová 2000; Juckett 2000, 2003).

The Sun wobbles about the Solar system barycentre with the distance varying up to two times its radius. The Sun's spin momentum contributes 1–2 per cent to the total angular momentum of the Solar system. Jose (1965) showed the existence of a relationship between a Hale cycle and the changes in the angular momentum of the Sun's motion about the barycentre. Recently, Zaqarashvili (1997) and Juckett (2000) found that the Sun's motion about the barycentre is having a role even in the cause of the solar differential rotation. The configurations and the directions of alignments of the major planets are considerably different during the even- and the odd-numbered cycles (Mörth & Schlamminger 1979). The differential rotation analysis of Javaraiah & Gokhale (1995) and Javaraiah (1996, 2003a) revealed the frequencies that are compatible with the frequencies of the specific alignments of two or more planets. The existence of a relationship, which is similar to the G–O rule in sunspot activity, is also found between the differences in the differential rotation during the odd- and the even-numbered cycles (Javaraiah, Bertello & Ulrich 2005a). Therefore, one would reasonably expect that the violation of the G–O rule and the variations in the differential rotation are probably having a relationship with the Sun's motion about the barycentre. We have investigated this in the present paper.

In Section 2 we describe the data and the analysis. In Section 3, we show the existence of a relationship between the violation of the G–O rule in sunspot activity and the Sun's retrograde motion about the centre of mass of the Solar system. In Section 4, we show the existence of coupling in the Sun's spin and orbital motion. In Section 5, we discuss about the implication of these results for understanding the long-term variations in the solar activity (including the Maunder minimum) and the solar-terrestrial relationship.

2 Data and Analysis

Dr Ferenc Varadi kindly provided us the values of the distance (R) of the Sun's centre from the Solar system barycentre, the orbital velocity of the Sun (V), the orbital angular momentum of the Sun (L) and the rate of change of the orbital angular momentum (orbital torque dL/dt), for each interval of length 10 d during the period 1600–2099. He determined these values using the recent Jet Propulsion Laboratory (JPL) DE405 ephemeris (Seidelmann 1992; Standish 1998) for the period 1600–2100.

The solar differential rotation can be determined from the full disc velocity data using the standard polynomial expansion (Howard & Harvey 1970),

(1)

while for sunspot data it is sufficient to use only the first two terms of the expansion (Newton & Nunn 1951), i.e.

(2)

where ω(φ) is the solar sidereal angular velocity at latitude φ, the coefficients A represents the equatorial rotation rate and B and C measure the latitudinal gradient in the rotation rate, with B representing mainly low latitudes and C largely higher latitudes (C is too small to be determined from sunspot data). (Note that the above equations have no theoretical foundation, but fit very well to the corresponding data, said above.)

In this analysis, the sunspot data and reduction are same as in Javaraiah & Gokhale (1995) and Javaraiah (2003a, b). We have used the Greenwich data on sunspot groups during the period 1879–1976 and the spot group data from the Solar Optical Observing Network (SOON) during the period 1977–2004 (available at ftp://ftp.science.mfsc.nasa.gov/ssl/pad/solar/greenwich.htm). The data consist of the observation time, heliographic latitude and longitude, central meridian distance (CMD), etc., for each spot group on each day of its observation. The sidereal rotation velocities (ω) have been computed for each pair of consecutive days in the life of each spot group using its longitudinal and temporal differences between these days. We have not used the data corresponding to the |CMD| > 75° on any day of the spot group lifespan and the displacements exceeding 3° in the longitude or 2° in the latitude per day. We determined the annual variations in the coefficients A and B by fitting each year's spot group data to equation (2).

3 Violation of the G–O Rule

Fig. 1 shows the variations in the R, V, L and dL/dt, determined from the planetary data available for every 10 d, during the period 1600–2099. In this figure, we also show the variations in the yearly mean values of the sunspot numbers during the period 1600–2004 (bottom panel). The epochs 1632, 1811 and 1990, when the Sun's motion about the barycentre was retrograde (i.e. when L was changed from positive to a weakly negative; Jose 1965), are indicated by the dotted vertical lines. The other two epochs of the big drops in L were at 1672 and 1851, and the expected next epoch of such a big drop in L will be at 2030. All these are indicated by the dashed vertical lines. For obvious reasons, near each of the big drops in L there is a big drop of dL/dt. We have given in Table 1 the values of R, L, V and dL/dt, and the values of the ecliptic longitudinal positions of the giant planets at these six epochs. The phase of dL/dt is leading the phase of L by about 4 yr near the dotted lines and about 5 yr near the dashed lines. This is expected because the former is the force and the latter is the motion and both have a main period of 19.86 yr, the conjunction period of Jupiter and Saturn. At the epochs where the steep decreases in L are indicated by the dotted vertical lines, the decrease in R is more steeper than the decrease in V. It is opposite in case of the epochs that are marked by the dashed vertical lines. The gap between the consecutive dotted lines and also between the consecutive dashed lines is about 179 yr, i.e. the period of the well-known 179-yr cycle in the Sun's motion related to the Solar system barycentre. It is also the period of an interval between alignments of all the outer planets in a same configuration and in a same direction in space. It is approximately equal to the nine conjunction periods of Jupiter and Saturn (Jose 1965). The gap between a dotted line and its neighbour dashed line is about 43 yr, the conjunction period of Saturn and Uranus.

Figure 1.

Values of R, V, L and dL/dt in each 10-d interval during the period 1600–2099 and the annual mean of the number of sunspots during the period 1600–2004. The units of R, V, L and dL/dt are au, au d⁻¹, M_⊙× (au)² d⁻¹ and M_⊙× (au)² d⁻², respectively, where M_⊙ is the mass of the Sun and ‘au’ is the astronomical unit. Near the peak of each sunspot cycle the corresponding Waldmeir sunspot cycle number is marked. The epochs, 1632, 1811 and 1990, at which the Sun's orbital motion was retrograde are indicated by the dotted vertical lines. The other three epochs, 1672, 1851 and 2030, where L is steeply decreased are marked by the dashed vertical lines. The horizontal lines represent the mean values.

Table 1.

The values of dL/dt, L, R, V and the ecliptic positions (in degree) of the giant planets — Jupiter (J), Saturn (S), Uranus (U) and Neptune (N) — at the epochs for which the values of L are given and marked by the dotted and dashed vertical lines in Fig. 1. The units of R, V, L and dL/dt are au, au d ⁻¹, M_⊙× (au)² d⁻¹ and M_⊙× (au)² d⁻², respectively, where M_⊙ is mass of the Sun and ‘au’ is the astronomical unit.

The G–O rule was violated by the sunspot cycles' pair 4, 5 at the beginning of the Dalton minimum, and it is most likely to be violated by the cycles' pair 22, 23 (e.g. Javaraiah 2003b). It seems that near the end of the cycle, which was just one cycle before the cycle at the beginning of the Maunder minimum, the G–O rule was violated (by the cycles' pair −12, −11, say). (Beer et al. 1990 have shown the existence of cyclic behaviour during the Maunder minimum.) Interestingly, each of these cycles' pairs is close to an epoch at which the Sun's orbital motion is retrograde, which is indicated (Fig. 1) by the dotted vertical lines. The peak value of sunspot cycle 8 is higher than that of cycle 9. By virtue of this difference, the cycle pair 8, 9 violated the G–O rule. The temporal behaviours of L and dL/dt suggest that such a situation might have occurred in the year 1672, and the next such situation may occur near the year 2030 (however, at this epoch the drop in L will be relatively small; see Fig. 1 and Table 1). These findings indicate the existence of a relationship between the violation of the G–O rule and the Sun's retrograde motion about the centre of mass of the Solar system. In Table 1, it can be seen that the epochs at which L was steeply decreased, Saturn was aligned approximately in opposition to Jupiter, and Uranus and Neptune were nearer to Saturn (i.e. Jupiter leads by about 180° with respect to the other three giant planets). Obviously, such configurations of the major planets are responsible for the Sun's retrograde motion about the barycentre, which in turn seems to be responsible for the violation of the G–O rule. Since the planetary configurations and the Sun's retrograde motion can be computed well in advance, it is possible to know the epochs of violations of the G–O rule well in advance. Therefore, the G–O rule is expected to be violated by the Hale cycle, which will include (or end at) the year 2169, i.e. only after a gap of about eight Hale cycles after the current Hale cycle 11 (comprises cycles' pair 22, 23). However, the violation by virtue of the difference in the heights of the peaks of the cycles — like the cycles' pair 8, 9 near the year 1851 — is expected to be happening near the year 2030, i.e. by the cycles' pair 26, 27.

4 Sun's Spin-Orbit Coupling

Fig. 2 shows variations in the annual average values of R, V, L, dL/dt and the values of the differential rotation parameters A and B determined from the yearly sunspot group data during the period 1879–2004. The error bars are ±1σ (standard deviation) values. Due to the reduced number of sunspot groups the values have large errors at the cycles' minima. We corrected the time series of A and B by replacing the values having error larger than three times the median error by the values simulated from the linear fits [a similar correction was applied in an earlier paper by Javaraiah & Komm (1999)]. In Fig. 2, the solid curves in the lower two panels connected the points of the corrected data and the dotted curves connected the uncorrected data. In this figure, the variations in the solar equatorial rotation rate, A, look to be largely similar to the variations of dL/dt. After 1945, the variations of both A and dL/dt have somewhat large amplitudes. During this time, the mean level of activity is also relatively large (see Fig. 1). The epoch, 1990–1991, at which the Sun's orbital motion is retrograde, the value of A is low and dL/dt is almost zero. The correlation between A and dL/dt is positive before around 1945 and negative after that time (correlation coefficient r= 40 and −50 in intervals of about 50 yr before and after 1945, respectively). These results indicate the existence of a relationship between A and dL/dt. The orbital angular momentum might have been transferred to the spin momentum for about 50 yr before 1945, and the reverse might have happened in the latter 50 yr (Juckett 2000).

Figure 2.

Variations in the yearly mean values of R, V, L, dL/dt, A and B. The units of R, V, L and dL/dt are the same as in Fig. 1 and Table 1. In case of A and B, the solid and dashed curves represent the corrected and the uncorrected data, respectively, and the error bars are 1σ values. The epoch 1990, at which the orbital motion of the Sun was retrograde, is indicated by the dotted vertical line. The horizontal lines represent the mean values.

The correlation between the latitudinal gradient of rotation (B) and dL/dt is weak. The signs of correlations (r≈−20 to +25) between (B) and dL/dt are found to be opposite during the aforesaid epochs of the positive and the negative correlations between A and dL/dt (it should be noted here that there exists a considerable phase difference between A and B variations, e.g. Javaraiah et al. 2005a).

Javaraiah & Gokhale (1995) and Javaraiah & Komm (1999) found the ∼18.3-yr, ∼8-yr, and a few other short periodicities in B. Fig. 3 shows the fast Fourier transform (FFT) spectra of the annual variations of dL/dt, A and B. From this figure it can be seen that both spectra of A and B have the dominant peaks at frequency 1/18 yr⁻¹, which are significant on 3.6σ and 5.5σ levels, respectively. The corresponding periodicities in A and B match approximately with that of the main periodicity in dL/dt (the peak at 1/21 yr⁻¹, 6.6σ). We repeated the FFT analysis by extending the time series from the original 126 data points to 1024 data points by padding the time series with zeros. The values of the aforesaid main periodicities in A, B and dL/dt are found to be 17.1, 18.29 and 19.69 yr, respectively.

Figure 3.

FFT power spectra of dL/dt (dashed curve), A (dotted curve) and B (solid curve). The power values are normalized to unity. Near the tops of the dominant peaks, which are significant on > 3σ (particularly in A and B), the values of the corresponding periods are shown.

Note that there is about 1-yr difference between the aforesaid main periodicities of A and B. This may be explained as follows. It is believed that the magnetic structures of the active regions originate near the base of the convection zone (about 200 000 km below the surface) and the magnetic buoyancy causes them to rise through the convection zone and emerge on the surface. The rotation rates of spot groups depend on their lifespans and age. This can be interpreted as the rotation rates of the magnetic structures of spot groups vary as their anchoring depths vary during their lifespans (Javaraiah & Gokhale 1997; Gokhale & Javaraiah 2002; Hiremath 2002; Sivaraman et al. 2003). Fig. 4 shows the FFT spectra of A and B determined from the first 2-d data (young groups) of the spot groups of lifespan 7–12 d. To have adequate data we have used the moving time intervals of sizes 5 yr (same as in Javaraiah & Gokhale 1995; Javaraiah 1998). In Fig. 4, we have also showed the FFT spectrum determined from the 5-yr smoothed time series of dL/dt. In this figure it can be seen that the dominant peaks in the spectra of A and B are well coincided. We also repeated the FFT analysis for these smoothed time series by extending them as described above. The dominant peaks in the spectra of the extended time series of both A and B are found to be at 1/17.4 yr⁻¹. This indicates that the 18.3-yr periodicity — found in B derived from the combined data (dominated by small and short-lived groups) of the spot groups of different lifespans and age — may correspond to slightly shallower layers. The 17.1-yr periodicity may correspond to slightly deeper layers (also see Javaraiah 1998).

Figure 4.

Same as Fig. 3, but A and B are determined from only the first 2-d (young groups) data of the spot groups of lifespan 7–12 d. To have adequate spot group data, 5-yr moving time intervals are used. In case of dL/dt, 5-yr smoothed time series is used.

The 17.1-yr periodicities of both A and B closely match with the 17.5-yr period found in the mixing of the low-frequency components of L and the instantaneous spin projection vector (Juckett 2000). In addition, it seems that there exists a good agreement between the amplitudes of the variations in the Sun's spin and the orbital angular momenta, particularly at the common epochs of the steep decreases in both L and A. At these epochs, L decreases by an amount approximately equal to the mean value of it. For example, at the epoch 1990.97 the amount of the steep decrease in L is about −2.1 × 10⁴⁷ g cm² s⁻¹. At this epoch, the amount of the drop in A is about 1 per cent and the corresponding spin momentum is found to be approximately −1.1 × 10⁴⁷ g cm² s⁻¹. (The mean yearly value of A is 14.505 ± 0.008° d⁻¹. The uncertainty, 1σ value, in this mean value suggests that the mean amplitude of the yearly variation in the solar equatorial rotation rate during the period 1879–2004 is about 0.056 per cent only. Overall, about 0.1 per cent difference is found between the mean equatorial rotation rates during the even and the odd cycles; Javaraiah 2003a.)

The above results provide a direct observational support to the models of the spin-orbit coupling of an oblate Sun (e.g. Juckett 2000). The results also indicate that the perturbations required for maintaining the oscillations in the solar differential rotation and the solar magnetic field as the participants in the mechanism of solar cycle are coming from the Solar system dynamics.

5 Discussion

Usoskin, Mursula & Kovaltsov (2001) argued that between sunspot cycles 4 and 5 the data are sparse and unreliable, and interpreted the very long cycle 4 as consisting of two short cycles. If this interpretation is correct, then it seems that the G–O rule was not violated by Hale cycle 2. By comparing the sunspot observations of the aforesaid period with those at other times, and also by analysing other proxies of solar activity, Krivova, Solanki & Beer (2002) showed that no cycle was missed at the end of the 18th century and the official sunspot cycle numbering and parameters are correct. Usoskin, Mursula & Kovaltsov (2003) argued that the statistical analysis performed in the paper by Krivova et al. (2002) was not validated by quantitative tests, and even contains several errors. Hence, whether the G–O rule was violated during the Hale cycle 2 or there was an additional weak cycle in 1790s, is yet to be confirmed. The result found in Section 3 essentially strongly suggests that the G–O rule was indeed violated by the Hale cycle 2.

Blizard (1989) found a depressed level of activity in a few cycles that follow the epochs of the retrograde orbital motion of the Sun. This can be seen in Fig. 1, i.e. the level of activity is relatively low during at least a few cycles that follow the dotted vertical lines. (Around the years 1730 and 1900 the level of activity was considerably low, whereas the sizes of the drops in L and R were not large, but there were considerably large drops in V. In 1900 there was opposition alignment, about 25°, of all the other major planets with Neptune.) As already mentioned in Section 3, the expected violation of the G–O rule close to the dotted vertical line at 1632 is followed by the Maunder minimum, and the G–O rule violation close to the dotted vertical line at 1811 is followed by the Dalton minimum. Therefore, violation of the G–O rule close to the dotted vertical line at 1990 is also expected to be followed by a Maunder/Dalton-like minimum in activity. That is, the present trend of the relatively low level of sunspot activity in the current cycle 23, which follows the dotted vertical line at 1990, may continue for a few more sunspot cycles. Such an indication is also found in the recent studies of the long-term variations in sunspot activity (e.g. Bonev, Penev & Sello 2004; Hathaway & Wilson 2005) and solar equatorial rotation rate (Javaraiah 2003b; Javaraiah, Bertello & Ulrich 2005b). In addition, a number of authors predicted a weak activity during the next cycle 24, using a large number of techniques (e.g. Kane 1999; Echer et al. 2004; Svalgaard, Cliver & Kamide 2005). Therefore, violation of the G–O rule may be an indication of onset of a Maunder/Dalton-like minimum in activity. All these results seem to be consistent with the result, the existence of a 179-yr cycle coherence relationship between the solar magnetic activity and the Solar system dynamics, by Jose (1965). If we consider that the latest three consecutive 179-yr cycles in sunspot activity began around the years 1632, 1811 and 1990, the first half (may represent a Gleissberg cycle) of each of the first two cycles is weaker than the corresponding second half, suggesting that the current half cycle (includes about 80 per cent of current century) would be weaker than the later half. Within this half 179-yr cycle the average activity during the first half (the double Hale cycle that comprises the cycles 22–25) may be weaker than that during the second half (Javaraiah 2003b). [Note that two major drops in L occur within the first quarter of a 179-yr cycle, in a time gap of about 43 yr. In fact, the current epoch of sunspot activity seems to be at the declining phase of the Gleissberg cycle whose minimum is expected to occur near the end of cycle 25 (Javaraiah et al. 2005b).]

The Maunder minimum (1645–1715) and other such low-activity epochs were also explained on the basis of variations in the Sun's motion about the centre of mass of the Solar system (Fairbridge & Shirley 1987; Charvátová 2000; Juckett 2000). One of the reasons often quoted for rejecting a role of the Solar system dynamics in the mechanism of the solar activity is that sunspot activity was absent during the Maunder minimum, but the planetary configurations were present (e.g. Smythe & Eddy 1977). This argument does not seem to be valid because the interval between the alignments of all the outer planets in the same configuration and in the same direction in space is about 179 yr, and no alignment of the major planets repeats exactly (Jose 1965, and also see Table 1). Considerable information on the solar rotation rate during the Maunder minimum is available. Eddy, Gilman & Trotter (1976) analysed the sunspot drawings made by J. Hevelius during the period 1662–1664, i.e. just before the begin of the Maunder minimum, and found that the equatorial rotation rate was about 4 per cent higher than the value during the modern time. Abarbanell & Wöhl (1981) analysed the same data and found that during the period 1662–1664, the equatorial rotation rate was same as that during the modern time. Ribes & Nesme-Ribes (1993) analysed a unique collection of sunspot observations recorded at the Observatoire de Paris from 1660 to 1719, and found that the equatorial rotation was about 2 per cent lower than that during the modern time. Recently, Vaquero, Sánchez-Bajo & Gallego (2002) analysed the observations of a sunspot carried out by Flamsteed (1684) from 1684 April 25 to 1684 May and found that during the deep Maunder minimum (1666–1700) the rotation rate near the equator was about 5 per cent lower than that of the modern time. Overall, these results suggest existence of a large drop in the equatorial rotation rate during the deep Maunder minimum. The large drop in the equatorial rotation rate during the deep Maunder minimum might be related to the steep decreases in L at 1632 and 1671 (see Fig. 1), and obviously to the configurations of the major planets at these epochs (see Table 1). The effect of the large drop in A near these epochs might have been persisted throughout the Maunder minimum and caused the nearly complete absence of activity (Javaraiah 2003b). It is interesting to note that the beginning of the well-known Spörer minimum (1450–1550) was approximately about 179 yr ago from the year 1632. Therefore, the cause of the Spörer minimum may be also same as that of the Maunder minimum, as suggested above.

We have used here the data on the giant planets only. However, the inner planets may also be important because of their proximity to the Sun. Their tidal forces on the Sun are larger than those of the outer planets (except Jupiter). Therefore, when they are closely aligned with the Jupiter, the combined effect may cause ‘jerks’ (rate of change of acceleration) in the orbital motion of the Sun (Wood & Wood 1965). There are some spikes in the variations in A and B. Particularly around some minimum years, the values of B are almost zero or even positive (see Fig. 2). Many of these spikes may be resulted because the sizes of the spot group data are small during these years. The spikes in the variation of B at the years 1887, 1962 and 1996 have considerable influence on the significance levels of the derived periodicities in B (Javaraiah & Komm 1999). However, during these years the statistics are sufficiently good and the values have errors less than three times the median error. Moreover, we found the similar abnormal behaviours in the values of B during 1962 from the Mt Wilson Observatory and the Kodaikanal Observatory sunspot data (available at ). We confirmed the abnormal behaviour of B during the year 1996 using the data of sunspot drawings of Mt Wilson Observatory (available at ). Kambry & Nishikawa (1990) also derived a similar value of B during the year 1962 from the spot group data measured in the National Astronomical Observatory of Japan. Hence, the aforesaid abnormal behaviour of B seems to be a real property of the rotations of the sunspots during the aforesaid years. Incidentally, on 1962 February 5 the five naked-eye planets plus the Sun and Moon were aligned within 15.8°, and there was a solar eclipse at the same time (Mosely 1996).

In view of the existence of a statistically significant ∼18.3 periodicity in B, it is interesting to note that the major droughts in the world (Hoyt & Schatten 1997) and even the major earthquakes in California () seem to have occurred in gaps of about 18 yr. A similar periodicity may exist in the Earth's rotation (Kirov, Georgieva & Javaraiah 2002). The precession period of the Moon is also 18.6 yr. In addition, there were severe droughts in 1886/1887, 1962/1963 and 1995/1996 (Foweler & Kilsby 2000), when the values of the coefficient B are abnormal. Therefore, in view of the results in Sections 3 and 4 it may be worthwhile to investigate whether the variations in the internal dynamics of the Sun and the Earth, and the terrestrial phenomena are all governed by the Solar system dynamics. However, it should be noted here that so far no convincing evidence is found for the influence of the planetary dynamics on terrestrial phenomena, climate, dynamics of the Earth and/or earthquakes. Gribbin & Plagemann (1974) described the 1982 alignment of all the nine planets as a superconjunction, with all of them on the same side of the Sun. They had predicted that this alignment would cause a massive earthquake in 1982 and a major disaster in Los Angeles. Fortunately, that prediction failed. In the 1982 alignment, the planets spread out over 98° (DeYoung 1979).

It should also be noted here that some of the relatively short-term predictions of the solar activity that were made based on the hypothesis of a role of Solar system dynamics in the mechanism of solar activity have failed (Meeus 1991; Li, Yun & Gu 2001). A reason for this may be that the underlying physics is not clear. On the other hand, the inclinations of the orbital planes of the planets and the Sun's equator to the ecliptic (or to the invariable plane) seem to be important (Blizard 1983; Javaraiah 1996, 2003a; Juckett 2000), but they were not taken into account in most of the earlier investigations.

6 Summary and Conclusion

We have shown that the epochs of the violations of the well-known G–O rule in pairing of sunspot cycles are close to the epochs of the Sun's retrograde orbital motion about the centre of mass of the Solar system. From this result it is easy to know well in advance the epochs of violations of the G–O rule. The G–O rule is expected to be violated by the Hale cycle, which will include (or end at) the year 2169, i.e. only after a gap of about eight Hale cycles after the current Hale cycle 11. However, the violation of the G–O rule by virtue of the difference in the values of the peaks of the cycles' pair — like the cycles' pair 8, 9 near the year 1851 — is expected to be happening near the year 2030, i.e. by the cycles' pair 26, 27. We also showed that the solar equatorial rotation rate determined from the sunspot group data during the period 1879–2004 correlates to the Sun's orbital torque, positively before 1945 and negatively after that time. The equatorial rotation has a dominant periodicity at ∼17 yr. These results are well consistent with the results in the model of the spin-orbit coupling of an oblate Sun by Juckett (2000), and may provide a direct observational support to the hypothesis of a role of solar dynamics in the internal dynamics of the Sun and in the variations of solar activity.

Acknowledgments

The author is thankful to Dr Ferenc Varadi for providing the entire planetary data used here and for a fruitful discussion on the results. The author also thanks Prof Roger K. Ulrich for comments and an anonymous referee for useful suggestions. The author is presently working for the Mt Wilson Archive Digitization project at UCLA, funded by NSF grant ATM-0236682.

Author notes

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Permanent address: Indian Institute of Astrophysics, Bangalore-560 034, India (E-mail: )