Long-Term Modulation of Cosmic-Ray Intensity and Correlation Analysis Using Solar and Heliospheric Parameters

Based on the monthly sunspot numbers (SSNs), the solar-flare index (SFI), grouped solar flares (GSFs), the tilt angle of heliospheric current sheet (HCS), and cosmic-ray intensity (CRI) for Solar Cycles 21?–?24, a detailed correlation study has been performed using the cycle-wise average correlation (with and without time lag) method as well as by the “running cross-correlation” method. It is found that the slope of regression lines between SSN and SFI, as well as between SSN and GSF, is continuously decreasing from Solar Cycle 21 to 24. The length of regression lines has significantly decreased during Cycles 23 and 24 in comparison to Cycles 21 and 22. The cross-correlation coefficient (without time lag) between SSN–CRI, SFI–CRI, and GSF–CRI has been found to be almost the same during Cycles 21 and 22, while during Cycles 23 and 24 it is significantly higher between SSN–CRI and HCS–CRI than for SFI–CRI and GSF–CRI. Considering time lags of 1 to 20 months, the maximum correlation coefficient (negative) amongst all of the sets of solar parameters is observed with almost the same time lags during Cycles 21?–?23, whereas exceptional behaviour of the time lag has been observed during Cycle 24, as the correlation coefficient attains its maximum value with two time lags (four and ten months) in the case of the SSN–CRI relationship. A remarkably large time lag (22 months) between HCS and CRI has been observed during the odd-numbered Cycle 21, whereas during another odd cycle, Cycle 23, the lag is small (nine months) in comparison to that for other solar/flare parameters (13?–?15 months). On the other hand, the time lag between SSN–CRI and HCS–CRI has been found to be almost the same during even-numbered Solar Cycles 22 and 24. A similar analysis has been performed between SFI and CRI, and it is found that the correlation coefficient is maximum at zero time lag during the present solar cycle. The GSFs have shown better maximum correlation with CRI as compared to SFI during Cycles 21 to 23, indicating that GSF could also be used as a significant solar parameter to study the cosmic-ray modulation. Furthermore, the running cross-correlation coefficient between SSN–CRI and HCS–CRI, as well as between solar-flare activity parameters (SFI and GSF) and CRI is observed to be strong during the ascending and descending phases of solar cycles. The level of cosmic-ray modulation during the period of investigation shows the appropriateness of different parameters in different cycles, and even during the different phases of a particular solar cycle. We have also studied the galactic cosmic-ray modulation in relation to combined solar and heliospheric parameters using the empirical model suggested by Paouris et al. (Solar Phys.280, 255, 2012). The proposed model for the calculation of the modulated cosmic-ray intensity obtained from the combination of solar and heliospheric parameter gives a very satisfactory value of standard deviation as well as \(R^{2}\) (the coefficient of determination) for Solar Cycles 21?–?24.     

Study of Multiple Coronal Mass Ejections at Solar Minimum Conditions

Solar activity alternates between active and quiet phases with an average period of 11?years, and this is known as the Schwabe cycle. Additionally, solar activity occasionally falls into a prolonged quiet phase (grand solar minimum), as represented by the Maunder Minimum in the 17th century, when sunspots were almost absent for 70?years and the length of the Schwabe cycle increased to 14?years. To examine the consistency of the cycle length characteristics during the grand solar minima, the carbon-14 contents in single-year tree rings were measured using an accelerator mass spectrometer as an index of the solar variability during the grand solar minimum of the 4th century BC. The signal of the Schwabe cycle was detected with a statistical confidence level of higher than 95?% by wavelet analysis. This is the oldest evidence for the Schwabe cycle at the present time, and the cycle length is considered to have increased to approximately 16?years during the grand solar minimum of the 4th century BC. This result confirms the association between the increase of the Schwabe cycle length and the weakening of solar activity, and indicates the possible prolonged absence of sunspots in the 4th century BC as during the Maunder Minimum. Theoretical implications from solar dynamo theory are discussed in order to identify the trigger of prolonged sunspot absence. A possible association between the long-term solar variation around the 4th century BC and terrestrial cooling in this period is also discussed.     

A Reconstruction of Ultraviolet Spectral Irradiance During the Maunder Minimum

We present a reconstruction of the solar spectrum in the near and mid-ultraviolet spectral range during the Maunder Minimum, a period of strongly suppressed magnetic activity spanning the second half of the 17th century. This spectral reconstruction is based on an extension of the Monte Carlo Solar Spectral Irradiance Model (MOCASSIM). The new version of the model, documented in this paper, extends its spectral range down to 150 nm, its temporal range back to 1610, includes a secular modulation of the quiet-Sun emissivity based on a total solar irradiance reconstruction, and uses the Atmospheric Laboratory for Applications and Science-3 (ATLAS-3) spectrum as a reconstruction baseline. The model is validated against the ATLAS-1 spectrum for 29 March 1992, showing a general agreement varying from ～?1 % in the 300?–?400 nm range, up to 3?–?5 % below 200 nm, the largest discrepancies occurring in emission lines formed in the chromosphere and transition region. We also reconstruct ultraviolet spectra for May 2008 and March 2009, spanning the extended phase of low activity separating Cycles 23 and 24. Our results suggest that despite the unusually long temporal extent of this activity minimum, the ultraviolet emission still remained slightly higher than during the Maunder Minimum, due to the lingering presence of decay products from active regions having emerged in the late descending phase of Cycle 23.     

Identification of Gleissberg Cycles and a Rising Trend in a 315-Year-Long Series of Sunspot Numbers

We show in this short note that the method of singular spectrum analysis (SSA) is able to clearly extract a strong, clean, and clear component from the longest available sunspot (International Sunspot Number, ISN) time series (1700?–?2015) that cannot be an artifact of the method and that can be safely identified as the Gleissberg cycle. This is not a small component, as it accounts for 13% of the total variance of the total original signal. Almost three and a half clear Gleissberg cycles are identified in the sunspot number series. Four extended solar minima (XSM) are determined by SSA, the latest around 2000 (Cycle 23/24 minimum). Several authors have argued in favor of a double-peaked structure for the Gleissberg cycle, with one peak between 55 and 59 years and another between 88 and 97 years. We find no evidence of the former: solar activity contains an important component that has undergone clear oscillations of \(\approx90\) years over the past three centuries, with some small but systematic longer-term evolution of “instantaneous” period and amplitude. Half of the variance of solar activity on these time scales can be satisfactorily reproduced as the sum of a monotonous multi-secular increase, a \(\approx90\)-year Gleissberg cycle, and a double-peaked (\(\approx10.0\) and 11.0 years) Schwabe cycle (the sum amounts to 46% of the total variance of the signal). The Gleissberg-cycle component definitely needs to be addressed when attempting to build dynamo models of solar activity. The first SSA component offers evidence of an increasing long-term trend in sunspot numbers, which is compatible with the existence of the modern grand maximum.     

The Interaction Between Coronal Mass Ejections and Streamers: A Statistical View over 15 Years (1996 – 2010)

We report on the statistical analysis of the interaction between coronal mass ejections (CMEs) and streamers based on 15 years (from 1996 to 2010 inclusive) of observation of the solar corona with the LASCO-C2 coronagraph. We used synoptic maps and improved the method of analysis of past investigations by implementing an automatic detection of both CMEs and streamers. We identified five categories of interaction based on photometric and geometric variations between the pre- and post-CME streamers: “brightening”, “dimming”, “emergence”, “disappearance”, and “deviation”. A sixth category, “no change”, included all cases where none of the above variations is observed. A “global set” of 21?242 CMEs was considered as well as a subset of the 10 % brightest CMEs (denoted “top-ten”) and three typical periods of solar activity: minimum, intermediate, and maximum. We found that about half of the global population of CMEs are not associated with streamers, whereas 93 % of the 10 % brightest CMEs are associated. When there is a CME-streamer association, approximately 95 % of the streamers experience a change, either geometric or radiometric. The “no change” category therefore amounts to approximately 5 %, but this percentage varies from 1?–?2 % during minimum to 7?–?8 % during intermediate periods of activity; values of 3?–?5 % are recorded during maximum. Emergences and disappearances of streamers are not dominant processes; they constitute 16?–?17 % of the global set and 23 % (emergence) and 28 % (disappearance) of the “top-ten” set. Streamer deviations are observed for 57 % and 70 % of, respectively, the global set and “top-ten” CMEs. The cases of dimming and brightening are roughly equally present and each case constitutes approximately 30?–?35 % of either set, global or “top-ten”.     

Magnetic Flux Transport Simulations of Solar Surface Magnetic Distributions During a Grand Minimum

It is well known that magnetic activity on the Sun modulates from one cycle to the next. The most striking occurrence of this is called a grand minimum where magnetic activity all but disappears. The latest grand minimum occurred between the years 1645 and 1715 and is called the Maunder minimum. In this paper magnetic flux transport simulations are used to consider what type of surface magnetic field configurations may be produced both during and after a grand minimum depending on how the grand minimum occurs. It is shown that the surface configurations during and after a grand minimum strongly depend on the phase of the cycle in which the grand minimum starts and whether it lasts for an odd or even number of cycles. If the grand minimum starts around cycle minimum then a significant amount of large-scale magnetic flux may persist on the Sun at high latitudes during the grand minimum. In contrast, if it starts at cycle maximum during the grand minimum it is possible for there to be essentially zero large-scale magnetic flux over the entire surface of the Sun. It is shown that for a single grand minimum event the reversal of the polar fields at the presently observed time in the solar cycle is only reproduced if the event starts at cycle minimum and extends over an even number of cycles. In contrast, if the grand minimum runs for an odd number of cycles it is possible for there to be no reversal of the polar fields or for the reversals to occur at times inconsistent with our present understanding of the solar cycle. Consequences of the assumptions made in the modelling are discussed and the significance of the simulations for direct modelling of events such as the Maunder minimum are considered.     

Study of Geomagnetic Storms and Solar and Interplanetary Parameters for Solar Cycles 22 and 24

The aim of this paper is to investigate the association of geomagnetic storms with the component of the interplanetary magnetic field (IMF) perpendicular to the ecliptic (\(Bz\)), the solar wind speed (\(V\)), the product of solar wind speed and \(Bz\) (VBz), the Kp index, and the sunspot number (SSN) for two consecutive even solar cycles, Solar Cycles 22 (1986?–?1995) and 24 (2009?–?2017). A comparative study has been done using the superposed epoch method (Chree analysis). The results of the present analysis show that \(Bz\) is a geoeffective parameter. The correlation coefficient between Dst and \(Bz\) is found to be 0.8 for both Solar Cycles 22 and 24, which indicates that these two parameters are highly correlated. Statistical relationships between Dst and Kp are established and it is shown that for the two consecutive even solar cycles, Solar Cycles 22 and 24, the patterns are strikingly similar. The correlation coefficient between Dst and Kp is found to be the same for the two solar cycles (?0.8), which clearly indicates that these parameters are well anti-correlated. For the same studied period we found that the SSN does not show any relationship with Dst and Kp, while there exists an inverse relation between Dst and the solar wind speed, with some time lag. We have also found that VBz is a more relevant parameter for the production of geomagnetic storms, as compared to \(V\) and \(Bz\) separately. In addition, we have found that in Solar Cycles 22 and 24 this combined parameter is more relevant during the descending phase as compared to the ascending phase.     

Reconstruction of Subdecadal Changes in Sunspot Numbers Based on the NGRIP 10Be Record

Sunspot observations since 1610 A.D. show that the solar magnetic activity displays long-term changes, from Maunder Minimum-like low-activity states to Modern Maximum-like high-activity episodes, as well as short-term variations, such as the pronounced 11-year periodicity. Information on changes in solar activity levels before 1610 relies on proxy records of solar activity stored in natural archives, such as ¹⁰Be in ice cores and ¹⁴C in tree rings. These cosmogenic radionuclides are produced by the interaction between Galactic cosmic rays (GCRs) and atoms in the Earth’s atmosphere; their production rates are anti-correlated with the solar magnetic activity. The GCR intensity displays a distinct 11-year periodicity due to solar modulation of the GCRs in the heliosphere, which is inversely proportional to, but out of phase with, the 11-year solar cycle. This implies a time lag between the actual solar cycles and the GCR intensity, which is known as the hysteresis effect. In this study, we use the North Greenland Ice Core Project (NGRIP) records of the ¹⁰Be flux to reconstruct the solar modulation strength (Φ), which describes the modulation of GCRs throughout the heliosphere, to reconstruct both long-term and subdecadal changes in sunspot numbers (SSNs). We compare three different approaches for reconstructing subdecadal-scale changes in SSNs, including a linear approach and two approaches based on the hysteresis effect, i.e. models with ellipse–linear and ellipse relationships between Φ and SSNs. We find that the ellipse approach provides an amplitude-sensitive reconstruction and the highest cross-correlation coefficients in comparison with the ellipse–linear and linear approaches. The long-term trend in the reconstructed SSNs is computed using a physics-based model and agrees well with the other group SSN reconstructions. The new empirical approach, combining a physics-based model with ellipse-modeling of the 11-year cycle, therefore provides a method for reconstructing SSNs during individual solar cycles based on ¹⁰Be in ice cores. This, in turn, represents a new window for studying short-term changes in solar activity on unprecedented timescales, which may help improve our understanding of the solar dynamo.     

Prediction of Solar Cycle 24 Using Sunspot Number near the Cycle Minimum

Correlations between monthly smoothed sunspot numbers at the solar-cycle maximum [R _max] and duration of the ascending phase of the cycle [T _rise], on the one hand, and sunspot-number parameters (values, differences and sums) near the cycle minimum, on the other hand, are studied. It is found that sunspot numbers two?–?three years around minimum correlate with R _max or T _rise better than those exactly at the minimum. The strongest correlation (Pearson’s r=0.93 with P<0.001 and Spearman’s rank correlation coefficient r _S=0.95 with P=9×10^?12) proved to be between R _max and the sum of the increase of activity over 30 months after the cycle minimum and the drop of activity over 30 or 36 months before the minimum. Several predictions of maximal amplitude and duration of the ascending phase for Solar Cycle 24 are given using sunspot-number parameters as precursors. All of the predictions indicate that Solar Cycle 24 is expected to reach a maximal smoothed monthly sunspot number (SSN) of 70?–?100. The prediction based on the best correlation yields the maximal amplitude of 90±12. The maximum of Solar Cycle 24 is expected to be in December 2013?–?January 2014. The rising and declining phases of Solar Cycle 24 are estimated to be about 5.0 and 6.3 years, respectively. The minimum epoch between Solar Cycles 24 and 25 is predicted to be at 2020.3 with minimal SSN of 5.1?–?5.4. We predict also that Solar Cycle 25 will be slightly stronger than Solar Cycle 24; its maximal SSN will be of 105?–?110.     

An Early Prediction of the Amplitude of Solar Cycle 25

A “Solar Dynamo” (SODA) Index prediction of the amplitude of Solar Cycle 25 is described. The SODA Index combines values of the solar polar magnetic field and the solar spectral irradiance at 10.7 cm to create a precursor of future solar activity. The result is an envelope of solar activity that minimizes the 11-year period of the sunspot cycle. We show that the variation in time of the SODA Index is similar to several wavelet transforms of the solar spectral irradiance at 10.7 cm. Polar field predictions for Solar Cycles 21?–?24 are used to show the success of the polar field precursor in previous sunspot cycles. Using the present value of the SODA index, we estimate that the next cycle’s smoothed peak activity will be about \(140 \pm30\) solar flux units for the 10.7 cm radio flux and a Version 2 sunspot number of \(135 \pm25\). This suggests that Solar Cycle 25 will be comparable to Solar Cycle 24. The estimated peak is expected to occur near \(2025.2 \pm1.5\) year. Because the current approach uses data prior to solar minimum, these estimates may improve as the upcoming solar minimum draws closer.     

Clustering of Emerging Magnetic Flux

We report observations of the large-scale spatial dependence of the Sun's luminosity variations over the period 1993–1995. The measurements were made using a new scanning disk solar photometer at Big Bear Solar Observatory, specially designed to measure large-scale brightness variations at the 10^–4 level. Since the level of solar activity was very low for the entire observation period, the data show little solar cycle variation. However, the residual brightness signal I/I (after subtracting the mean, first, and second harmonics) does show a strong dependence on heliocentric angle, peaking near the limb. This is as one would expect if the residual brightness signal (including the excess brightness coming from the active latitudes) were primarily facular in origin. Additional data over the next few years, covering the period from solar minimum to maximum, should unambiguously reveal the large-scale spatial structure of the solar cycle luminosity variations.     

Redefining the limit dates for the Maunder Minimum

The Maunder Minimum corresponds to a prolonged minimum of solar activity a phenomenon that is of particular interest to many branches of natural and social sciences commonly considered to extend from 1645 until 1715. However, our knowledge of past solar activity has improved significantly in recent years and, thus, more precise dates for the onset and termination of this particularly episode of our Sun can be established. Based on the simultaneous analysis of distinct proxies we propose a redefinition of the Maunder Minimum period with the core “Deep Maunder Minimum” spanning from 1645 to 1700 (that corresponds to the Grand Minimum state) and a wider “Extended Maunder Minimum” for the longer period 1618–1723 that includes the transition periods.     

Study of cosmic ray intensity in relation to the interplanetary magnetic field and geomagnetic storms for solar cycle 24

In the present study, we investigate the association of cosmic ray intensity (CRI) with various solar wind parameters (i.e. solar wind speed V, plasma proton temperature, plasma proton density), interplanetary magnetic field (IMF B), geomagnetic storms (GSs), averaged planetary A-index (Ap index) and sun spot number (SSN) for the period 2009–2016 (solar cycle 24) by using their daily mean average. To find the association of CRI with various solar wind parameters, GSs, IMF B, Ap index and SSN, we incorporate the analysis technique by superposed-epoch method. We have observed that CRI decreases with the increase in IMF B. Moreover the time-lag analysis has been performed by the method of correlation coefficient and observed a time lag of 0 to 2 day between the decrease in CRI and increase in IMF B. In addition, we show that the CRI is found to decrease in a similar pattern to disturbance storm time (Dst index) for most of the period of solar cycle 24. The high and positive correlation is found between CRI and Dst index. The CRI and Ap index are better anti-correlated to each other than CRI and IMF. CRI and SSN are positively correlated with each other. Solar wind parameters such as solar wind speed V is a CR-effective parameter while plasma proton temperature and plasma proton density are not CR-effective parameters. The indicated parameters such as Dst index, Ap index, IMF B and solar wind parameters such as solar wind speed V, plasma proton temperature, plasma proton density shows a kind of irregular variations for solar cycle 23 and 24 while CRI and SSN shows distinct behaviour for the two cycle.     

Short-Term Variations in the Equatorial Rotation Rate of Sunspot Groups

We have detected several periodicities in the solar equatorial rotation rate of sunspot groups in the catalog Greenwich Photoheliographic Results (GPR) during the period 1931?–?1976, the Solar Optical Observing Network (SOON) during the period 1977?–?2014, and the Debrecen Photoheliographic Data (DPD) during the period 1974?–?2014. We have compared the results from the fast Fourier transform (FFT), the maximum entropy method (MEM), and the Morlet wavelet power-spectra of the equatorial rotation rates determined from SOON and DPD sunspot-group data during the period 1986?–?2007 with those of the Mount Wilson Doppler-velocity data during the same period determined by Javaraiah et al. (Solar Phys. 257, 61, 2009). We have also compared the power-spectra computed from the DPD and the combined GPR and SOON sunspot-group data during the period 1974?–?2014 to those from the GPR sunspot-group data during the period 1931?–?1973. Our results suggest a ～?250-day period in the equatorial rotation rate determined from both the Mt. Wilson Doppler-velocity data and the sunspot-group data during 1986?–?2007. However, a wavelet analysis reveals that this periodicity appears mostly around 1991 in the velocity data, while it is present in most of the solar cycles covered by the sunspot-group data, mainly near the minimum epochs of the solar cycles. We also found the signature of a period of ～?1.4 years in the velocity data during 1990?–?1995, and in the equatorial rotation rate of sunspot groups mostly around the year 1956. The equatorial rotation rate of sunspot groups reveals a strong ～?1.6-year periodicity around 1933 and 1955, a weaker one around 1976, and a strong ～?1.8-year periodicity around 1943. Our analysis also suggests periodicities of ～?5 years, ～?7 years, and ～?17 years, as well as some other short-term periodicities. However, short-term periodicities are mostly present at the time of solar minima. Hence, short-term periodicities cannot be confirmed because of the larger uncertainty in the data.     

Heliospheric Modulation of Galactic Cosmic Rays During Solar Cycle 23

Galactic cosmic rays (GCRs) encounter an outward-moving solar wind with cyclic magnetic-field fluctuation and turbulence. This causes convection and diffusion in the heliosphere. The GCR counts from the ground-based neutron monitor stations show intensity changes that are anti-correlated with the sunspot numbers with a lag of a few months. GCRs experience various types of modulation from different solar activity features and influence space weather and the terrestrial climate. In this work, we investigate certain aspects of the GCR modulation at low cut-off rigidity (R _c≈1 GV) in relation to some solar and geomagnetic indices for the entire solar cycle 23 (1996?–?2008). We separately study the GCR modulation during the ascending phase of cycle 23 including its maximum (1996?–?2002) and the descending phase including its minimum (2003?–?2008). We find that during the descending phase, the GCR recoveries are much faster than those of the solar parameters with negative time-lag. The results are discussed in light of modulation models, including drift effects and previous results.     

Long-Term Solar Cycle Evolution: Review of Recent Developments

The sunspot number series forms the longest directly observed index of solar activity and allows one to trace its variations on the time scale of about 400 years since 1610. This time interval covers a wide range from seemingly vanishing sunspots during the Maunder minimum in 1645–1700 to the very high activity during the last 50 years. Although the sunspot number series has been studied for more than a century, new interesting features have been found even recently. This paper gives a review of the recent achievements and findings in long-term evolution of solar activity cycles such as determinism and chaos in sunspot cyclicity, cycles during the Maunder minimum, a general behaviour of sunspot activity during a great minimum, the phase catastrophe and the lost cycle in the beginning of the Dalton minimum in 1790s and persistent 22-year cyclicity in sunspot activity. These findings shed new light on the underlying physical processes responsible for sunspot activity and allow a better understanding of such empirical rules as the Gnevyshev–Ohl rule and the Waldmeier relations.     

Cosmic Ray Intensity Variations in Relation with Solar Flare Index and Sunspot Numbers

From the monthly data of cosmic ray intensity (CRI), sunspot numbers (SSN) and solar flare index (SFI), an attempt has been made to study the relationship between CRI and solar activity (SA) parameters SSN and SFI. The correlation between SA parameters and CRI for different neutron monitoring stations having low, middle and high cut-off rigidity has been investigated. The anti-correlation between SA and CRI is found to exist with some time lag. Based on the method of minimizing correlation coefficient and time-delayed component method, the observed time-lag between SA parameters (SSN and SFI) and CRI has been found to be large for odd solar cycles in comparison to even solar cycles. The results of time-lag analysis between CRI and SSN and between CRI-SFI have also been compared. The findings of correlative study between CRI and SSN are in agreement with earlier results, while the CRI-SFI relationship provides new insights to understand the solar modulation of cosmic rays.     

Evidence for the effect of sunspot activity on the El Niño/Southern Oscillation

The El Niño No. 3 area index (5°S∼ 5°N, 150°W∼ 90°W) and yearly sunspot number (SSN) from 1408 to 1978 are used to investigate the influence of solar activity on the El Niño/Southern Oscillation (ENSO), through periodicity analysis, cross wavelet transform (XWT), cross correlation and ensemble empirical mode decomposition (EEMD) analyses. The solar activity period, the Hale period, and the Gleissberg period are determined in the El Niño index time series, but of weak statistical significance. Cross correlation analysis of the index with SSN, and that of its low-frequency components decomposed by EEMD clearly indicate that solar activity may take effect on the ENSO, and such an impact should undergo an accumulation procedure (phase delay). XWT also indicates the existence of the impact. It is found that the index is negatively correlated with SSN when SSN is large during a certain long-term interval, and positively when SSN is small. Strong El Niño is inferred to be taken place in decade(s) to come.     

Tilt of Sunspot Bipoles in Solar Cycles 15 to 24

We use recently digitized sunspot drawings from Mount Wilson Observatory to investigate the latitudinal dependence of tilt angles of active regions and its change with solar cycle. The drawings cover the period from 1917 to present and contain information as regards polarity and strength of magnetic field in sunspots. We identified clusters of sunspots of same polarity, and used these clusters to form “bipole pairs”. The orientation of these bipole pairs was used to measure their tilts. We find that the latitudinal profile of tilts does not monotonically increase with latitude as most previous studies assumed, but instead, it shows a clear maximum at about 25?–?30 degree latitudes. Functional dependence of tilt (\(\gamma\)) on latitude (\(\varphi\)) was found to be \(\gamma= (0.20\pm0.08) \sin(2.80 \varphi) + (-0.00\pm0.06)\). We also find that latitudinal dependence of tilts varies from one solar cycle to another, but larger tilts do not seem to result in stronger solar cycles. Finally, we find the presence of a systematic offset in tilt of active regions (non-zero tilts at the equator), with odd cycles exhibiting negative offset and even cycles showing the positive offset.