Long-Term Variations in the Solar Differential Rotation

Using Greenwich data (1879–1976) and SOON/NOAA data (1977–2002) on sunspot groups we found the following results: (i) The Sun's mean (over all the concerned cycles during 1879–1975) equatorial rotation rate (A) is significantly larger (≈0.1%) in the odd-numbered sunspot cycles (ONSCs) than in the even-numbered sunspot cycles (ENSCs). The mean rotation is significantly (≈10%) more differential in the ONSCs than in the ENSCs. North–south difference in the mean equatorial rotation rate is larger in the ONSCs than in the ENSCs. North–south difference in the mean latitude gradient of the rotation is significant in the ENSCs and insignificant in the ONSCs. (ii) The known very large decrease in A from cycle 13 to cycle 14 is confirmed. The amount of this decrease in the mean A was about 0.017 μrad s⁻¹. Also, we find that A decreased from cycle 17 to cycle 18 by about 0.008 μrad s⁻¹ and from cycle 21 to cycle 22 by about 0.016 μrad s⁻¹. From cycle 13 to cycle 14 the decrease in A was more in the northern hemisphere than in the southern hemisphere, it is opposite in the later two epochs. The time gap between the consecutive drops in A is about 44 years, suggesting the existence of a `44-yr' cycle or `double Hale cycle' in A. The time gap between the two large drops, viz., from cycle 13 to cycle 14 and from cycle 21 to cycle 22, is about 90 years (Gleissberg cycle). We predict that the next drop (moderate) in A will be occurring from cycle 25 to cycle 26 and will be followed by a relatively large-amplitude `double Hale cycle' of sunspot activity. (iii) Existence of a 90-yr cycle is seen in the cycle-to-cycle variation of the latitude gradient (B). A weak 22-yr modulation in B seems to be superposed on the relatively strong 90-yr modulation. (iv) The coefficient A varies significantly only during ONSCs and the variation has maximum amplitude in the order of 0.01 μrad s<sup>−1</sup> around activity minima. (v) There exists a good anticorrelation between the mean variation of B during the ONSCs and that during the ENSCs, suggesting the existence of a `22-yr' periodicity in B. The maximum amplitude of the variation of B is of the order of 0.05 μrad s⁻¹ around the activity minima. (vi) It seems that the well-known Gnevyshev and Ohl rule of solar activity is applicable also to the cycle-to-cycle amplitude modulation of B from cycle 13 to cycle 20, but the cycles 12 (in the northern hemisphere, Greenwich data) and 21 (in both hemispheres, SOON/NOAA data) seem to violate this rule in B. And (vii) All the aforesaid statistically significant variations in A and B seem to be related to the approximate 179-yr cycle, 1811–1989, of variation in the Sun's motion about the center of mass of the solar system.     

Long-Term Variations in Solar Differential Rotation and Sunspot Activity

The solar equatorial rotation rate, determined from sunspot group data during the period 1879–2004, decreased over the last century, whereas the level of activity has increased considerably. The latitude gradient term of the solar rotation shows a significant modulation of about 79 year, which is consistent with what is expected for the existence of the Gleissberg cycle. Our analysis indicates that the level of activity will remain almost the same as the present cycle during the next few solar cycles (i.e., during the current double Hale cycle), while the length of the next double Hale cycle in sunspot activity is predicted to be longer than the current one. We find evidence for the existence of a weak linear relationship between the equatorial rotation rate and the length of sunspot cycle. Finally, we find that the length of the current cycle will be as short as that of cycle 22, indicating that the present Hale cycle may be a combination of two shorter cycles. Presently working for the Mt. Wilson Solar Archive Digitization Project at UCLA.     

Long-Term Variations of Solar Differential Rotation and Sunspot Activity: Revisited

Long-term variations of solar differential rotation and sunspot activity are investigated through re-analyzing the data on parameters of the differential-rotation law obtained by Makarov, Tlatov, and Callebaut (Solar Phys. 170, 373, 1997), Javaraiah, Bertello, and Ulrich (Astrophys. J. 626, 579, 2005a; Solar Phys. 232, 25, 2005b), and Javaraiah et al. (Solar Phys. 257, 61, 2009). Our results indicate that the solar-surface-rotation rate at the Equator (indicated by the A-parameter of the standard solar-rotation law) shows a secular decrease since Cycle 12 onwards, given by about 1?–?1.5×10^?3 (deg?day^?1?year^?1). The B-parameter of the standard differential-rotation law seems to also show a secular decrease since Cycle 12 onwards, but of weak statistical significance. The rotation rate averaged over latitudes 0^°?–?40^° does not show a secular trend of statistical significance. Moreover, the average sunspot area shows a secular increase of statistical significance since Cycle 12 onwards, while a negative correlation is found between the level of sunspot activity (indicated by the average sunspot area) and the solar equatorial rotation on long-term scales.     

Solar-Cycle-Related Variations in the Solar Differential Rotation and Meridional Flow: A Comparison

We have analysed a large set of sunspot group data (1874 – 2004) and find that the meridional flow strongly varies with the phase of the solar cycle, and the variation is quite different in the northern and the southern hemispheres. We also find the existence of considerable cycle-to-cycle variation in the meridional velocity, and about a 11-year difference between the phases of the corresponding variations in the northern and the southern hemispheres. In addition, our analysis also indicates the following: (i) the existence of a considerable difference (about 180^°) between the phases of the solar-cycle variations in the latitude-gradient terms of the northern and the southern hemispheres’ rotations; (ii) the existence of correlation (good in the northern hemisphere and weak in the southern hemisphere) between the mean solar-cycle variations of meridional flow and the latitude-gradient term of solar rotation; (iii) in the northern hemisphere, the cycle-to-cycle variation of the mean meridional velocity leads that of the equatorial rotation rate by about 11 years, and the corresponding variations have approximately the same phase in the southern hemisphere; and (iv) the directions of the mean meridional velocity is largely toward the pole in the longer sunspot cycles and largely toward the equator in the shorter cycles.     

Cosmic-Ray Decreases and Interplanetary Disturbances During the Solar Magnetic Polarity Reversal in Solar Cycle 20

Using the cosmic-ray intensity data recorded with ground-based monitors at Mt. Washington and Deep River, and with cosmic-ray telescopes on Pioneer 8 and 9 spacecraft as well as the 2-hour averages of the IMF (magnitude and direction) and the solar wind bulk speed and density at 1 AU, the cosmic-ray decreases and interplanetary disturbances, that occurred during the period of solar magnetic polarity reversal in solar cycle 20, were investigated.We observed a two-step Forbush decrease on 22–23 November 1969, and a Forbush decrease on 26 November 1969, which are respectively consistent with the model of Barnden (1973), and of Parker (1963) and Barnden (1973). Only one Forbush decrease event was observed in December 1969, a period during which there was a solar magnetic polarity reversal; the Forbush decrease was attributed to a long-lived corotating high-speed solar wind stream. This is indicative that at heliolongitudes from 43° E to 70° W of S–E radial, covered by the observations, the solar magnetic polarity reversal in solar cycle 20 was not carried by, nor related to, individual transient structures, and that the reversal most probably evolved gradually.     

On the Long-Term Modulation of Solar Differential Rotation

Long-term modulation of solar differential rotation was studied with data from Mt. Wilson and our original observations during Solar Cycles 16 through 23. The results are that i) the global B-value (i.e. latitudinal gradient of differential rotation) is modulated with a period of about six or seven solar cycles, ii) the B-values of the northern and southern hemispheres are also modulated with a period similar to the global one, but iii) they show quasi-oscillatory behavior with a phase shift between them. We examined the yearly fluctuations of the B-values in every solar cycle with reference to the phase of the sunspot cycle and found that the B-values in the sunspot-minimum years show large and erratic variations, while those in the sunspot-maximum years show small fluctuations. Positive correlation between the former B-values and the latter was found. We discuss the independent long-term behavior of solar differential rotation between the northern and southern solar hemispheres and the implication for the solar dynamo.     

Temporal Variations of the Solar Rotation Determined by Sunspot Groups

The extended Greenwich data set consisting of positions of sunspot groups is used for the investigation of cycle-related variations of the solar rotation in the years 1874–1981. Applying the residual method, which yields a single number for each year describing the average deviation from the mean value of the solar rotation, the dependence of the rotation velocity residual on the phase of the solar cycle is investigated. A secular deceleration of the solar rotation was found: the slope being statistically significant at the 3σ level. Periods of 33, 22, 11, 5.2, and 3.5 years can be identified in the power spectra. The rotation velocity residuals were averaged for all years with the same solar cycle phase relative to the nearest preceding sunspot minimum. The variation pattern reveals a higher than average rotation velocity in the minimum of activity and, to a lesser extent, also around the maximum of activity. The analysis was repeated with several changes in the reduction method, such as elimination of the secular trend, application of statistical weights, different cutoffs of the central meridian distance, division of the latitude into subregions and treating data from the years of activity minima separately. The results obtained are compared with those from the literature, and an interpretation of the observed phenomena is proposed.     

A Check on the Variations of Earth‘s Rotation with an Ancient Solar Eclipse

We address the relation between an ancient total eclipse, which occurred on A.D.1542 August 11 and the variation of Earth‘‘s rotation. The total eclipse was recorded in some ancient Chinese books, especially in local chronicles. Some of the documents include useful information for determining the location of the totality zone. The parameters of the eclipse are calculated by using the DE406 Ephemeris.A high-precision value of AT which expresses the variation of the Earth‘‘s rotation,of about 300 ～ 380 s, is obtained.     

Iodine-Cell Spectroscopy Applied to Investigating the Solar Differential Rotation

In an attempt to examine whether the spectroscopic Doppler method with an iodine cell (which is known to be successful for precise radial-velocity determinations in stellar astronomy) could be effective for investigating the solar differential rotation, we carried out intensive observations to collect spectra at a large number of points on the solar disk by using the Domeless Solar Telescope along with the horizontal spectrograph of the Hida Observatory. Having converted the resulting line-of-sight velocity component into the angular rotational rate (ω), we derived a differential rotation law, w_sidereal  (deg day^-1) = 14.03 (±0.06)-1.84 (±0.57) sin²y-1.92 (±0.85) sin⁴y\omega_{\mathrm{sidereal}}\; (\mathrm{deg}\,\mathrm{day}^{-1}) =14.03 (\pm0.06)-1.84 (\pm0.57) \sin^{2}\psi-1.92 (\pm0.85) \sin^{4}\psi (ψ: heliographic latitude), which is reasonably consistent with other spectroscopic determinations published so far. Our analysis also revealed several practical points to note for successful application (e.g., exclusion of those data that are not well distant from the meridian; mutual data subtraction/averaging for symmetric counterparts at the eastern and western hemisphere). Considering its easiness and cheapness, this iodine-cell-featured spectroscopic method may be regarded as an effective and practical tool for studying the differential rotation of the Sun.     

Polarity Reversal Time of the Magnetic Dipole Component of the Sun in Solar Cycle 24

The Sun’s general magnetic field has shown polarity reversal three times during the last three solar cycles. We attempt to estimate the upcoming polarity reversal time of the solar magnetic dipole by using the coronal field model and synoptic data of the photospheric magnetic field. The scalar magnetic potential of the coronal magnetic field is expanded into a spherical harmonic series. The long-term variations of the dipole component ( $g^{0}_{1}$ ) calculated from the data of National Solar Observatory/Kitt Peak and Wilcox Solar Observatory are compared with each other. It is found that the two $g^{0}_{1}$ values show a similar tendency and an approximately linear increase between the Carrington rotation periods CR 2070 and CR 2118. The next polarity reversal is estimated by linear extrapolation to be between CR 2132.2 (December 2012) and CR2134.8 (March 2013).     

Differential Rotation of the Solar Corona from Magnetic Field Data

A method for investigating the differential rotation of the solar corona using the coronal magnetic field as a tracer is proposed. The magnetic field is calculated in the potential approximation from observational data at the photospheric level. The time interval from June 24, 1976, to December 31, 2004, is considered. The magnetic field has been calculated for all latitudes from the equator to ±75? with a 5? step at distances from the base of the corona 1.0 R_⊙ to 2.45 R_⊙ near the source surface. The coronal rotation periods at 14 distances from the solar center have been determined by the method of periodogram analysis. The coronal rotation is shown to become progressively less differential with increasing heliocentric distance; it does not become rigid even near the source surface. The change in the coronal rotation periods with time is considered. At the cycleminimumthe rotation has been found to bemost differential, especially at small distances from the solar center. The change in coronal rotation with time is consistent with the tilt of the solar magnetic equator. The results from the magnetic field are compared with those obtained from the brightness of the green coronal Fe XIV 530.3 nm line. The consistency between these results confirms the reliability of the proposed method for studying the coronal rotation. Studying the rotation of the coronal magnetic field gives hope for the possibility of using this method to diagnose the differential rotation in subphotospheric layers.     

Magnetic Field as a Tracer for Studying the Differential Rotation of the Solar Corona

The characteristics of differential rotation of the solar corona for the period 1976?–?2004 were studied as a function of the distance from the center of the Sun. For this study, we developed a method using the coronal magnetic field as a tracer. The field in a spherical layer from the base of the corona up to the source surface was determined from photospheric measurements. Calculations were performed for 14 heliocentric distances from the base of the corona up to 2.45 \(R_{\odot }\) solar radii (the vicinity of the source surface) and from the equator to \(\pm 75^{\circ }\) of latitude at \(5^{\circ }\) steps. For each day, we calculated three spherical components, which were then used to obtain the field strength. The coronal rotation periods were determined by the periodogram method. The rotation periods were calculated for all distances and latitudes under consideration. The results of these calculations make it possible to study the distribution of the rotation periods in the corona depending on distance, time, and phase of the cycle. The variations in the coronal differential rotation during the time interval 1976?–?2004 were as follows: the gradient of differential rotation decreased with the increase of heliocentric distance; the rotation remaining differential even in the vicinity of the source surface. The highest rotation rates (shortest rotation periods) were recorded at the cycle minimum at small heliospheric distances, i.e. small heights in the corona. The lowest rotation rate was observed at the middle of the ascending branch at large distances. At the minimum of the cycle, the differential rotation is most clearly pronounced, especially at small heliocentric distances. As the distance increases, the differential rotation gradient decreases in all phases. The results based on magnetic data and on the brightness of the coronal green line 530.3 nm Fe xiv used earlier show a satisfactory agreement. Since the rotation of the magnetic field at the corresponding heights in the corona is probably determined by the conditions in the field generation region, an opportunity arises to use this method for diagnostics of differential rotation in the subphotospheric layers.     

Rotation of the Solar Equator

Regular measurements of the general magnetic field of the Sun, performed over about half a century at the Crimean Astrophysical Observatory, the J. Wilcox Solar Observatory, and five other observatories, are considered in detail for the time 1968?–?2016. They include more than twenty-six thousand daily values of the mean line-of-sight field strength of the visible solar hemisphere. On the basis of these values, the equatorial rotation period of the Sun is found to be 26.926(9) d (synodic). It is shown that its half-value coincides within error limits with both the main period of the magnetic four-sector structure, 13.4577(25) d, and the best-commensurate period of the slow motions of the major solar system bodies, 13.479(22) d (sidereal). The probability that the two periods coincide by chance is estimated to be about \(10^{-7}\). The true origin of this odd resonance is unknown.     

The Energy Lost by Differential Rotation in the Generation of the Solar Toroidal Magnetic Field

The integrals, I_i(t) = _GL u_i j × B _i dv over the volume GL are calculated in a dynamo model of the Babcock–Leighton type studied earlier. Here, GL is the generating layer for the solar toroidal magnetic field, located at the base of the solar convection zone (SCZ); i=r, , , stands for the radial, latitudinal, and azimuthal coordinates respectively; j = (4)^-1 × B, where B is the magnetic field; u_r,u are the components of the meridional motion, and u is the differential rotation. During a ten-year cycle the energy _cycle I(t)dt needs to be supplied to the azimuthal flow in the GL to compensate for the energy losses due to the Lorentz force. The calculations proceed as follows: for every time step, the maximum value of |B| in the GL is computed. If this value exceeds B_cr (a prescribed field) then there is eruption of a flux tube that rises radially, and reaches the surface at a latitude corresponding to the maximum of |B| (the time of rise is neglected). This flux tube generates a bipolar magnetic region, which is replaced by its equivalent axisymmetric configuration, a magnetic ring doublet. The erupted flux can be multiplied by a factor F_t, i.e., by the number of eruptions per time step. The model is marginally stable and the ensemble of eruptions acts as the source for the poloidal field. The arbitrary parameters B_cr and F_t are determined by matching the flux of a typical solar active region, and of the total erupted flux in a cycle, respectively. If E(B) is the energy, in the GL, of the toroidal magnetic field B = B sin cos , B (constant), then the numerical calculations show that the energy that needs to be supplied to the differential rotation during a ten-year cycle is of the order of E(B_cr), which is considerably smaller than the kinetic energy of differential rotation in the GL. Assuming that these results can be extrapolated to larger values of B_cr, magnetic fields 10⁴ G, could be generated in the upper section of the tachocline that lies below the SCZ (designated by UT). The energy required to generate these 10⁴ G fields during a cycle is of the order of the kinetic energy in the UT.     

On Rotation of the Solar Corona

We have studied the rotation of the solar corona using the images taken at a 9.4?nm wavelength by the AIA 094 instrument on board the Solar Dynamics Observatory (SDO) satellite. Our analysis implies that the solar corona rotates differentially. It appears that ??, the angular rotation velocity of the solar corona, does not only depend on heliographic latitude but is also a function of time, while the nature of the latter dependence remains unclear. Besides measurement errors, deviations ???? from the mean rotational speed are also caused by proper motion of the observed point source (the tracer) with respect to its surroundings. The spread in ?? values at a particular heliographic latitude is a real effect, not caused by measurement errors. Most of the observations carry relative error less than 1?% in???.     

Study of Cosmic-Ray Intensity Variations Associated with Anomalous,Long-Duration High-Speed Solar Wind Streams in 2003

High-speed solar wind streams (HSWS) were identified for solar cycles 22 and 23 (up to 2004). Preliminarily, HSWS were classified in three groups according to their continuous period of occurrence. In the declining phase of solar cycle 23, 2003 is found to be anomalous, showing a very large number of HSWS events of long duration (> ten days). We have studied the effect of HSWS on the cosmic-ray intensity as well as their relationship with geomagnetic disturbance index Ap on yearly, daily, and hourly bases. The yearly average of solar-wind speed was also found to be maximum in 2003. Being within the declining phase of solar activity, the occurrence of solar flares in 2003 is quite low. In particular during HSWS, no solar flares have been observed. Associations with cosmic-ray changes do not support the notion that the HSWS are usually effective in producing significant cosmic-ray decreases. Out of 12 HSWS events observed during the period 2002 (December) to 2003, four events of significant cosmic-ray decreases at all the stations have been selected for further analysis. The cosmic-ray intensity has been found to decrease during the first phase of the event (first five days of HSWS) at all three neutron-monitor stations situated at different latitudes with different cutoff rigidities. The rigidity spectra of observed decreases in cosmic-ray intensity for these four cases have been found to be significantly different than that of Fds (Forbush decrease). In two cases the spectra are softer, whereas in the other two they are harder than that of Fds. However, if the average of all four events is considered together then the spectra of the decrease in cosmic rays during HSWS exactly match that of Fds. Such a result implies that initially individual events should be considered, instead of combining them together, as was done earlier. The Ap index is also found to generally increase in the first phase of the event. However, the four events selected on the basis of cosmic-ray decrease are not always associated with enhanced values of the Ap index. As such, the significance of our study is that further detailed investigations for much longer periods and on an event-by-event basis is required to understand the effect of coronal-hole-associated HSWS.     

Variations of the Chromospheric Network with the Solar Cycle

We have digitized a small sample of Caii K spectroheliograms from the photographic archive of Arcetri Observatory for the period 1950–1970. For each digitized plate, active regions have been automatically identified and masked by applying an algorithm based on the geometrical connection between active pixels. Contrast histograms have been derived for the full disk and for quiet regions. Both of them show a neat Gaussian profile on the dark side and a significant tail on the bright side, due to the brightness enhancements associated with magnetic fields. A solar cycle dependence of the histogram asymmetry is clearly evident in quiet region data, thus confirming that the network component might provide a significant contribution to long-term variations of the total irradiance.     

Reversal of Heliospheric Magnetic Field Polarity: Theoretical Model

A simple analytical model of the reversal of the heliospheric magnetic field is suggested. The shape of the heliospheric current sheet is found for each instant of time using a kinematic approximation. Calculation results are illustratively presented in graphic and animated forms, showing a 3-D dynamic picture of the reversal of the heliospheric magnetic field throughout a 22-year solar cycle.     

On the Differential Rotation of Stars

We discussmethods for analyzing the observational manifestations of differential rotation (DR). Based on the literary sources and our estimates (for 4 cool dwarfs), a list containing 75 stars was compiled. Using an example of analysis of the ‘ΔΩ–T_eff’, ‘ΔΩ–rotation period’ and ‘ΔΩ–Rossby number’ diagrams we compared the determined parameters of DR stars with the results of theoretical studies. Particular attention is paid to the problems of measurements of DR parameters in low-mass dwarf stars, including the completely convective ones.We analyzed the measured parameters of solar-type DR stars. The objects having anti-solar DR (α < 0), and possiblemethods allowing to determine the sign of α are considered.We distinguish the areas of research that in the future may expand our understanding of DR manifestations: a study of DR in the inner regions of stars and an analysis of manifestations of the rotational brightness modulation caused by the spottedness of components in eclipsing-variable systems (subsynchronized systems).