# Long–Term Variations in Solar Differential Rotation and Sunspot Activity,
II: Differential Rotation Around the Maxima and Minima of Solar Cycles 12 – 24

J. Javaraiah∗ Bikasipura, Bengaluru-560 111, India.
∗(Formerly working at Indian Institute of Astrophysics, Bengaluru-560 034,
India.)
email: jajj55<EMAIL_ADDRESS>

###### Abstract

Studies of variations in the solar differential rotation are important for
understanding the underlying mechanism of solar cycle and other variations of
solar activity. We analyzed the sunspot-group daily data that were reported by
Greenwich Photoheliogrphic Results (GPR) during the period 1874 – 1976 and
Debrecen Photoheliographic Data (DPD) during the period 1977 – 2017. We
determined the equatorial rotation rate [$A$] and the latitude gradient [$B$]
components of the solar differential rotation by fitting the data in each of
the 3-year moving time intervals (3-year MTIs) successively shifted by one
year during the period 1874 – 2017 to the standard law of differential
rotation. The values of $A$ and $B$ around the years of maxima and minima of
Solar Cycles 12 – 24 are obtained from the 3-year MTIs series of $A$ and $B$
and studied the long–term cycle-to-cycle modulations in these coefficients.
Here we have used the epochs of the maxima and minima of the Solar Cycles 12 –
24 that were recently determined from the revised Version-2 international
sunspot-number series. We find that there exits a considerably significant
secular decreasing-trend in $A$ around the maxima of solar cycles. There exist
no secular trends in both $A$ and $B$ around the minima of solar cycles. The
secular trend in $B$ around the maxima of solar cycles is also found to be
statistically insignificant. We fitted a cosine function to the values of $A$,
and also to those of $B$, after removing the corresponding linear trends. The
cosine-fits suggest that there exist $\approx$54-year ($\approx$94-year) and
$\approx$82-year ($\approx$79-year) periodicities in $A$ ($B$) around the
maxima and minima of solar cycles, respectively. The amplitude of the cosine-
profile of $A$ ($B$) around the minima of solar cycles is about 41 % (65 %)
larger than that of $A$ ($B$) of around the maxima. In addition, the cosine
profiles of $A$ and $B$ suggest a large (up to $180^{\circ}$) phase difference
between the long-term variations of $A$, and also between those of $B$, of
around maxima and minima of solar cycles. Implications of all these results
are discussed briefly.

###### keywords:

Sun: Dynamo – Sun: surface magnetism – Sun: activity – Sun: sunspots

##  1 Introduction

Solar dynamo models explain several properties of solar cycle (bab61; dg06;
camer17). The solar differential rotation is an important ingredient in these
models. Solar activity vary in many time scales. Therefore, studies of
variations in solar differential rotation are important for better
understanding the physical mechanism of solar cycle and other variations of
solar activity. Variations in the differential rotation of the Sun’s outer
layers have been known for several decades. Many authors studied the solar
cycle variation and other short- and long–term variations in the differential
rotation by using different data and techniques (hr76; gh84; sh85; jsb85;
balth86; kn90; snod92; khh93; yk93; jg95; mak97; jj03, 2005, 2011, 2013;
gupta99; jk99; kgk02; braj06; ju06; svand08; jub09; cvi10; li14, jb16; os16;
bo17; ruz17; roudi18). However, so far only the 11-year period torsional
oscillation of photospheric layers was discovered by hl80 and lh82 is
established. Helioseismic techniques have revealed that the torsional
oscillation pattern extends inward up to about one-third thickness of the
Sun’s convection zone (howe00). Other results to some extent established are:
both equatorial rate and latitude gradient of rotation are large at solar
minimum than at maximum, and we have a secular decrease of the solar
equatorial rotation with secular increase of activity (see ruz17 and the
references therein).

jbu05a (hereafter Article-I) analyzed the Greenwich and Solar Optical
Observing Network (SOON) sunspot-group data during the period 1874 – 2007 and
determined the average values of the coefficients of differential rotation
during each of the Solar Cycles 12 – 22 to study the cycle-to-cycle modulation
in these coefficients and found the secular decrease (about $-0.01^{\circ}$
day-1 cycle-1) of the cycle-to-cycle modulation in the equatorial rotation
rate. balth86, braj06, and li14 have also studied in detail the cycle-to-cycle
variations in the solar differential rotation. In Article-I cosine functions
to the values of the coefficients of differential rotation were fitted and we
found the existence of a 79-year cycle (Gleissberg cycle) in the latitude
gradient of rotation. suz14 found that a period of about 6–7 solar cycles
exits in long-term modulation of the latitude gradient of differential
rotation. It should be noted here a number of authors have found changes in
the period (60–130-year) of the Gleissberg cycle (garc98; hath99; roze94;
ogurt02; hath15; jj17). Recently, jj19 determined cycle-to-cycle variations in
the sunspot activity at the minima and the maxima of Solar Cycles 12 – 24 and
noticed that there exists a difference in the long-term periodicities of the
activity at solar minimum and maximum. The rotation rates of magnetic active
regions (sunspot groups, etc.) vary with life times and sizes (areas) of the
active regions. Large/long-lived sunspot groups rotate more slowely than
small/short-lived sunspot groups (war65, 1966; hgg84; balth86). This could
have to do with the magnetic structures of large and small sunspot groups
anchoring at deep and shallow layers of the solar convection zone,
respectively (jg97; hi02; siva03). The average size of sunspot groups in the
minimum of a solar cycle is smaller than that of the sunspot groups in the
maximum. Hence, the study of solar differential rotation of the sunspot groups
at solar minimum and maximum may be useful for better understanding the
physical mechanism of the solar long-term variability. In the present analysis
we determined the values of the coefficients of differential rotation in the
3-year MTIs during the period 1874 – 2017. From the obtained time series we
determined the values of the coefficients of differential rotation of the
sunspot groups around the minima and maxima of Solar Cycles 12 – 24 and
separately study the long-term cycle-to-cycle modulations in the coefficients
of the differential rotation around the maxima and minima of solar cycles.

In the next section we describe the data and method of analysis. In Section 2
we present results and in Section 3 we present conclusions and discuss upon
them briefly.

##  2 Data and Method of Analysis

Here we have used the daily sunspot-group data reported in GPR during the
period April 1874 – December 1976 and DPD during the period January 1977 –
June 2017. These data are downloaded from the website
fenyi.solarobs.unideb.hu/pub/DPD/. The details about these data can be found
in gyr10, bara16, and gyr17. These data contain the date and time of
observation (we converted these into date with fractional day, [$t$]),
heliographic latitude [$\lambda$] and longitude [$L$], corrected whole-spot
area, and central meridian distance [CMD], etc. of a sunspot group for each of
the days it was observed. Here each disk passage of a recurrent sunspot group
is treated as an independent sunspot group and in order to reduce the
foreshortening effect (if any), we have not used the data in any day of the
sunspot group life-time/disk passage ([$\tau$], its values are 2, 3,$\dots$,12
days). We have determined the equatorial rotation rate [$A$] and the
latitudinal gradient [$B$] of the solar differential rotation by fitting the
data corresponding to a specified time-interval to the standard law of
differential rotation:

$\omega(\theta)=A+B\sin^{2}(\theta),$ $None$

where $\omega(\theta)=\frac{L_{i}-L_{i-1}}{t_{i}-t_{i-1}}+14^{\circ}.18$ is
the solar sidereal rotation rate at the latitude $\theta=\lambda_{i-1}$, $i=$
2, 3,$\dots$,$\tau$, and $14^{\circ}.18$ is the Carrington rigid body rotation
rate of the Sun. In all our earlier analyzes we have assigned the velocity
value to the mean of $\lambda_{i-1}$ and $\lambda_{i}$. Following the
suggestion by ok05 here we assigned the velocity value to the $\lambda_{i-1}$
(also see sudar14). The data correspond to $\omega>3^{\circ}$ day-1 and the
latitudinal drift $\frac{\lambda_{i}-\lambda_{i-1}}{t_{i}-t_{i-1}}>2^{\circ}$
day-1 are excluded. This considerably reduced the uncertainties (standard
deviations) in the derived values of $A$ and $B$ (war65, 1966; jg95). The data
corresponding $t_{i}-t_{i-1}>2$ days (non-consecutive days) are also excluded.
This further reduced the uncertainties in $A$ and $B$. In order to have a
better statistics (particularly during solar minimum) first we have fitted
Equation 1 to the data in the 3-year MTIs 1874 – 1876, 1875 –
1877,$\dots$,2015 – 2017 during the period 1874 – 2017. Northern- and
southern-hemispheres’ data are combined. The values of $A$ and $B$ around the
years of maxima and minima ($i.e.$ in the 3-year intervals in which the years
of maxima and minima are middle years) of the Solar Cycles 12 – 24 are
obtained from the 3-year MTIs series of $A$ and $B$. We have used the epochs
of the maxima and the minima, cycle-lengths, $etc.$ of the Solar Cycles 12 –
24 that were recently determined by pesnell18 using the revised Version-2
international sunspot-number (SN) series.

##  3 Results

In Table 1, we have given the values of mean sidereal angular velocity
($\omega$, in degree day-1) and corresponding uncertainty (standard error,
$\delta\omega$) in different $2^{\circ}$ latitude ($\theta$) intervals,
determined from the sunspot-group data around the minima ($i.e.$ from the
combined data in the intervals 1877–1879, 1889–1891, 1901–1903, 1912–1914,
1922–1924, 1932–1934, 1943–1945, 1953–1955, 1975–1977, 1985–1987, 1995–1997,
and 2007–2009) and around the maxima (i.e. from the combined data in the
intervals 1882–1884, 1893–1895, 1905–1907, 1916–1918, 1927–1929, 1936–1938,
1946–1948, 1957–1959, 1967–1969, 1978–1980, 1988–1990, 2000–2002, and
2013–2015) of Solar Cycles 12–24. In this table the corresponding number ($N$)
of the velocity values in each latitude interval is also given. Northern- and
southern-hemispheres’ data are folded. In many latitude intervals $N$ is
considerably large. Hence, the corresponding uncertainties in the values of
the mean angular velocities are reasonably small.

Table 1.: Values of mean ($\omega$, in degree day-1) and corresponding standard error ($\delta\omega$) of sidereal angular velocity in different $2^{\circ}$ latitude ($\theta$) intervals determined from the sunspot-group data around the minima ($i.e.$ from the combined data in the intervals: 1877–1879, 1889–1891, 1901–1903, 1912–1914, 1922–1924, 1932–1934, 1943–1945, 1953–1955, 1975–1977, 1985–1987, 1995–1997, and 2007–2009), and around the maxima (i.e. from the combined data in the intervals: 1882–1884, 1893–1895, 1905–1907, 1916–1918, 1927–1929, 1936–1938, 1946–1948, 1957–1959, 1967–1969, 1978–1980, 1988–1990, 2000–2002, and 2013–2015) of Solar Cycles 12–24. The corresponding number ($N$) of the velocity values in each latitude interval is also given. Northern- and southern-hemispheres’ data are folded. Latitude | Around minimum |  | Around maximum
---|---|---|---
interval | $\omega$ | $\delta\omega$ | $N$ |  | $\omega$ | $\delta\omega$ | $N$
$0^{\circ}$–$2^{\circ}$ | 14.57 | 0.05 | 318 |  | 14.56 | 0.04 | 550
$2^{\circ}$–$4^{\circ}$ | 14.50 | 0.04 | 473 |  | 14.56 | 0.02 | 1452
$4^{\circ}$–$6^{\circ}$ | 14.56 | 0.03 | 830 |  | 14.52 | 0.02 | 3393
$6^{\circ}$–$8^{\circ}$ | 14.46 | 0.03 | 875 |  | 14.46 | 0.01 | 5412
$8^{\circ}$–$10^{\circ}$ | 14.47 | 0.03 | 1094 |  | 14.45 | 0.01 | 7627
$10^{\circ}$–$12^{\circ}$ | 14.39 | 0.03 | 805 |  | 14.40 | 0.01 | 9350
$12^{\circ}$–$14^{\circ}$ | 14.44 | 0.04 | 542 |  | 14.38 | 0.01 | 10290
$14^{\circ}$–$16^{\circ}$ | 14.38 | 0.04 | 642 |  | 14.33 | 0.01 | 10136
$16^{\circ}$–$18^{\circ}$ | 14.30 | 0.04 | 608 |  | 14.28 | 0.01 | 9575
$18^{\circ}$–$20^{\circ}$ | 14.28 | 0.03 | 866 |  | 14.25 | 0.01 | 8459
$20^{\circ}$–$22^{\circ}$ | 14.21 | 0.03 | 1147 |  | 14.18 | 0.01 | 6528
$22^{\circ}$–$24^{\circ}$ | 14.22 | 0.03 | 1081 |  | 14.11 | 0.01 | 5086
$24^{\circ}$–$26^{\circ}$ | 14.09 | 0.04 | 796 |  | 14.05 | 0.02 | 3914
$26^{\circ}$–$28^{\circ}$ | 14.02 | 0.04 | 700 |  | 13.96 | 0.02 | 2681
$28^{\circ}$–$30^{\circ}$ | 14.06 | 0.05 | 520 |  | 13.90 | 0.02 | 1792
$30^{\circ}$–$32^{\circ}$ | 13.75 | 0.05 | 357 |  | 13.86 | 0.03 | 1224
$32^{\circ}$–$34^{\circ}$ | 13.67 | 0.07 | 212 |  | 13.74 | 0.04 | 811
$34^{\circ}$–$36^{\circ}$ | 13.88 | 0.12 | 104 |  | 13.69 | 0.05 | 492
$36^{\circ}$–$38^{\circ}$ | 13.68 | 0.14 | 59 |  | 13.71 | 0.06 | 321
$38^{\circ}$–$40^{\circ}$ | 13.42 | 0.17 | 37 |  | 13.39 | 0.09 | 154

Figure 1 shows the mean latitude dependence of the solar sidereal rotation
rate [$\omega(\theta)$] around the minima and around the maxima of Solar
Cycles 12 – 24 determined from the sunspot-group data of these cycles. In this
figure we have shown the profiles of the differential rotation deduced from
the values of the coefficients $A$ and $B$ of Equation 1, and also the values
of $\omega(\theta)$ determined by averaging the data into $2^{\circ}$ latitude
intervals, which are also given in Table 1. Northern- and southern-
hemispheres’ data are folded. As can be seen in this figure, as already known
($e.g.$ balth86; khh93; gupta99; jj03; braj06; ruz17), the mean angular
velocity of the sunspot groups around the minima of the solar cycles is
slightly larger than that of the sunspot groups around the maxima in all
latitudes (see the dashed and the continuous curves). Around the minima of the
solar cycles the mean equatorial velocity $A$ is about 0.16 % larger than that
around the maxima (this difference is statistically significant on at least
1.9$\sigma$ level, where $\sigma$ is standard deviation of $A$ around minima).
The mean latitude gradient $B$ around the maxima is about 4.4 % larger than
that around the minima (it is consistent with the fact that dynamo action is
strong during maximum and weak during minimum), but this difference is
statistically highly insignificant, i.e. only 1.2$\sigma$ level (around the
minima the number of data points is about 86 % lower than that of the around
the maxima). On the other hand, there exist considerable cycle-to-cycle
variations in both $A$ and $B$ around the maxima as well as those around the
minima of Solar Cycles 12 – 24 (see below).

Figure 1.: The mean latitude dependence of the solar sidereal rotation rate
[$\omega(\theta)$] around the minima and around the maxima ($i.e.$ in the
3-year intervals in which the epochs of minima and maxima are middle years) of
Solar Cycles 12 – 24. The continuous curve (black) and dashed curve (red) show
the corresponding profiles of the differential rotation deduced from the
values of the coefficients $A$ and $B$ in Equation 1. These values of the
coefficients and the number of velocity values ($N$) gone in the calculations
are also shown. Northern- and southern-hemispheres’ data are combined. The
filled circles (black) and open circles (red) represent the values of
$\omega(\theta)$ that are given in Table 1 and determined by averaging the
data into $2^{\circ}$ latitude intervals.

Figure 2.: The values (open circles conected by dotted curve) of $A$ and $B$ determined from the sunspot group data in the 3-year MTIs 1874 – 1876, 1875 – 1877,$\dots$,2015 – 2017 during the period 1874 – 2017 versus the middle years 1975, 1876,$\dots$,2016 of the intervals. Northern- and southern-hemispheres’ data are combined. The values at the epochs (middle years of the corresponding 3-year intervals) of minimum and maximum are indicated with the filled-square and filled-circle, respectively. The Waldmeier numbers of the solar cycles and the corresponding epochs of the minimum and maximum are also shown by the symbols m and M, respectively. The horizontal continuous-line represents the mean value over the whole period 1874 – 2017 (mean $A=14^{\circ}.51$ day-1 and mean $B=-2^{\circ}.42$ day-1). In b the continuous curve represents the variation in yearly mean total sunspot number during 1875–2016. Table 2.: Values of $A$ and $B$ (in degree day-1) in the 3-year intervals at the epochs $T_{\rm m}$ and $T_{\rm M}$ (middle years of the corresponding 3-year intervals) of the minima and maxima (indicated by the suffixes m and M, respectively) of Solar Cycles 12 – 24. The corresponding number ($N$) of velocity values is given. Northern- and southern-hemispheres’ data are combined. The minimum ($R_{\rm m}$) and maximum ($R_{\rm M}$) Version-2 sunspot numbers are also given. a implies tentative (see text). Cycle | $R_{\rm m}$ | Interval | $T_{\rm m}$ | $N$ | $A_{\rm m}$ | $B_{\rm m}$
---|---|---|---|---|---|---
12 | 3.7 | 1877–1879 | 1878 | 314 | $14.470\pm 0.054$ | $-1.838\pm 0.588$
13 | 8.3 | 1889–1891 | 1890 | 1188 | $14.651\pm 0.060$ | $-2.834\pm 0.454$
14 | 4.5 | 1901–1903 | 1902 | 654 | $14.604\pm 0.091$ | $-3.058\pm 0.754$
15 | 2.5 | 1912–1914 | 1913 | 429 | $14.515\pm 0.086$ | $-2.736\pm 0.577$
16 | 9.4 | 1922–1924 | 1923 | 928 | $14.447\pm 0.040$ | $-2.105\pm 0.305$
17 | 5.8 | 1932–1934 | 1933 | 680 | $14.558\pm 0.041$ | $-2.267\pm 0.394$
18 | 12.9 | 1943–1945 | 1944 | 1346 | $14.424\pm 0.045$ | $-1.915\pm 0.311$
19 | 5.1 | 1953–1955 | 1954 | 1221 | $14.479\pm 0.051$ | $-2.369\pm 0.292$
20 | 14.3 | 1963–1965 | 1964 | 1135 | $14.630\pm 0.040$ | $-2.211\pm 0.334$
21 | 17.8 | 1975–1977 | 1976 | 1288 | $14.514\pm 0.039$ | $-2.431\pm 0.272$
22 | 13.5 | 1985–1987 | 1986 | 1215 | $14.526\pm 0.040$ | $-3.277\pm 0.259$
23 | 11.2 | 1995–1997 | 1996 | 1204 | $14.571\pm 0.041$ | $-2.102\pm 0.330$
24 | 2.2 | 2007–2009 | 2008 | 517 | $14.384\pm 0.056$ | $-0.580\pm 0.555$
|  | 2008–2010 |  2009a | 784 | $14.524\pm 0.071$ | $-2.321\pm 0.4734$
| $R_{\rm M}$ |  | $T_{\rm M}$ |  | $A_{\rm M}$ | $B_{\rm M}$
12 | 124.4 | 1882–1884 | 1883 | 4043 | $14.598\pm 0.021$ | $-2.976\pm 0.281$
13 | 146.5 | 1893–1895 | 1894 | 5694 | $14.548\pm 0.019$ | $-2.175\pm 0.226$
14 | 107.1 | 1905–1907 | 1906 | 4037 | $14.490\pm 0.024$ | $-2.122\pm 0.321$
15 | 175.7 | 1916–1918 | 1917 | 5998 | $14.504\pm 0.020$ | $-2.475\pm 0.221$
16 | 130.2 | 1927–1929 | 1928 | 5171 | $14.495\pm 0.019$ | $-2.368\pm 0.245$
17 | 198.6 | 1936–1938 | 1937 | 6870 | $14.544\pm 0.018$ | $-3.042\pm 0.154$
18 | 218.7 | 1946–1948 | 1947 | 8006 | $14.487\pm 0.017$ | $-2.620\pm 0.144$
19 | 285.0 | 1957–1959 | 1958 | 10379 | $14.467\pm 0.014$ | $-2.767\pm 0.104$
20 | 156.6 | 1967–1969 | 1968 | 6367 | $14.503\pm 0.020$ | $-2.379\pm 0.177$
21 | 232.9 | 1978–1980 | 1979 | 9186 | $14.516\pm 0.017$ | $-2.373\pm 0.127$
22 | 212.5 | 1988–1990 | 1989 | 7853 | $14.530\pm 0.019$ | $-2.572\pm 0.129$
23 | 180.3 | 2000–2002 | 2001 | 8849 | $14.492\pm 0.016$ | $-2.245\pm 0.151$
24 | 116.4 | 2013–2015 | 2014 | 7003 | $14.439\pm 0.021$ | $-2.322\pm 0.250$

Figure 3.: Latitude dependence of angular velocity ($\omega$) determined from
the daily data of sunspot groups around the minima (left panels) and around
the maxima (right panels) of Solar Cycles 23 (top panels) and 24 (middle
panels). The bottom panel represents the latitude dependence of $\omega$ in
interval 2008–2010. Northern- and southern-hemispheres’ data are combined. The
continuous curve (red) represents the profile of the differential rotation
deduced from the corresponding values of the coefficients $A$ and $B$ given in
Table 2.

.

Figure 5.: The filled circle-dashed curves represents the cycle-to-cycle
variations in the equatorial rotation rates [$A_{\rm M}$] and [$A_{\rm m}$],
and the latitude gradients [$B_{\rm M}$] and [$B_{\rm m}$], of solar rotation
determined from the sunspot-group data around the maxima and minima (indicated
by the suffixes M and m, respectively) of Solar Cycles 12 – 24. The values of
$A_{\rm m}$ and $B_{\rm m}$ in interval 2008–2010 are also shown (red-filled-
square). Northern- and southern-hemispheres’ data are combined. The horizontal
dotted-line represents the mean. The continuous line represents the linear
best-fit (red-long-dashed line represent the best-fit when 2009 is considered
for minimum epoch of Cycle 24). The values of the slope and the correlation
coefficient are also shown.

Figure 2 shows variations in $A$ and $B$ determined from the sunspot group
data (northern- and southern-hemispheres’ data are combined) in the 3-year
MTIs 1875 – 1877, 1876 – 1878, $\dots$, 2015 – 2017 during the period 1874 –
2017. In Table 2 we have given the values of $A$ and $B$ around the maxima and
minima of Solar Cycles 12 – 24 obtained from the aforementioned 3-year MTIs.
These values are indicated in Figure 2 by the filled-circles and the filled-
squares, respectively. We have also shown the values of $A$ and $B$ obtained
from the data in the interval 2008–2010. The reason for this is given below.
The description of the yearly variations of $A$ and $B$ can be found in our
earlier papers, jg95 and jk99, and in the paper by ruz17. Here we concentrated
on cycle-to-cycle modulations in $A$ and $B$. ruz17 have noted that when a
significant peak of equatorial rotation velocity is observed during minimum of
activity, the strength of the next maximum is smaller than the previous one.
This is not found from Figure 2.

Equation 1 is found to be not fitted well to the data in around minima of
cycles. This is, obviously, because of insufficient data caused by large
scarcity of sunspots at high latitudes of sunspot belt. As can be seen in
Figure 2 around the minimum of Solar Cycle 24 the values of $A$ and $B$ are
abnormally low. In Figure 3 we show the latitude dependence of angular
velocity during the 3-year intervals that comprise the epochs of minima and
the maxima of Solar Cycles 23 and 24. Northern- and southern-hemispheres’ data
are combined. In this figure the continuous curves represent the corresponding
profiles of the differential rotation deduced from the values of the
coefficients $A$ and $B$ in Equation 1, which are also given in Table 2. As
can be seen in this figure, the coefficients $A$ and $B$ reasonably well
determined, $i.e.$ Equation 1 is reasonably well fitted to the data around the
maxima of Solar Cycles 23 and 24 and even around the minimum of Solar Cycle
23. The corresponding fit is not good around the minimum of Solar Cycle 24,
$i.e.$ in the interval 2007–2009, because there is a large scarcity of data in
higher latitudes of the sunspot latitude-belt (less overlap of Solar Cycles 23
and 24). Therefore, we have also checked the results by considering
tentatively 2009 for minimum epoch of Solar Cycle 24 and in Figure 3 we have
also shown the latitude dependence of the velocity in the interval 2008–2010
(middle year is 2009). The fit of Equation 1 to the data of this interval is
reasonably good (Note: in the 13-month smoothed monthly mean of the revised
SN, 2008.958, $i.e.$ around 15 November 2008, is the minimum epoch of Solar
Cycle 24.)

Figure 4 shows the cycle-to-cycle variations in the values of $A_{\rm M}$,
$A_{\rm m}$, $B_{\rm M}$, and $B_{\rm m}$, $i.e$ the values of $A$ and $B$
around the maxima and around the minima of Solar Cycles 12–24 given in Table 2
(the suffixes M and m indicate epochs of sunspot maximum and minimum, $i.e.$
the middle years of the 3-year intervals that comprise these years,
respectively). For the reason given above, besides the values of $A$ and $B$
in the interval 2007–2009 that comprises 2008 in its middle, we have also
shown the corresponding values in the interval 2008–2010 that comprises 2009
in its middle. The linear least-square fit correspond to each of these
parameters is done by taking uncertainty in the parameter, $i.e.$ ${\rm
weight}=\frac{1}{\sigma^{2}}$ is used. In Article-I the existence of a secular
decreasing trend was found in the cycle-to-cycle modulation of $A$ determined
from the sunspot group data (over the whole cycle) of Solar Cycles 12 – 22,
but it was not found in the corresponding modulation in $B$. As can be seen in
Figure 4 there exist no notable secular trends in both $A_{\rm m}$ and $B_{\rm
m}$, whereas there is a considerably significant secular decreasing-trend in
$A_{\rm M}$ (the slope is about five times larger than its standard
deviation). The decreasing trend of $A_{\rm M}$ is slightly less steeper
(slope $-0.006\pm 0.001$) than that of the average $A$ over the whole cycle
(slope $-0.01\pm 0.001$) found in Article-I. There is a secular decreasing
trend in $B_{\rm M}$ ($i.e.$ the secular decreasing trend in the latitudinal
gradient around maxima of solar cycles), but it is not well defined (the slope
is only about 1.5 times of its standard deviation). We find the following
correlations ($r$ represents the correlation coefficient and the values given
within parentheses are found when 2009 is considered for the minimum epoch of
Solar Cycle 24):
$r=0.35$ ($0.11$) between $A_{\rm m}$ and $A_{\rm M}$,
$r=0.33$ ($0.48$) between $A_{\rm m}$ and $B_{\rm M}$,
$r=-0.65$ ($-0.46$) between $A_{\rm m}$ and $B_{\rm m}$,
$r=-0.45$ between $A_{\rm M}$ and $B_{\rm M}$,
$r=-0.36$ ($0.02$) between $A_{\rm M}$ and $B_{\rm m}$, and
$r=-0.12$ ($-0.39$) between $B_{\rm m}$ and $B_{\rm M}$.
The anticorrelation between $A$ and $B$ implies that an increase in the
equatorial rotation rate with an increase in the latitude gradient of
rotation, and vise-versa (it should be noted that $B$ is negative).

We find no significant correlation between sunspot number (either the value of
sunspot minimum [$R_{\rm m}$] or of maximum [$R_{\rm M}$]) and $A$ or $B$ (the
highest correlation, $r=-0.39$, is found between $B_{\rm M}$ and $R_{\rm M}$).
However, the existence of secular decrease of $A$ with secular increase of
solar activity seems to be some extent established (see li14; os16; ruz17, and
the references therein). The existence of a positive correlation between $A$
determined from the whole-cycle data and length of cycle is known (mend99). In
Article-I a relatively weak correlation ($r=0.57$) was found between $A$ and
cycle-length from the data of Solar Cycles 12 – 22 and based on that a short
length was predicted for Cycle 23. That prediction is incorrect (Cycle 23 is a
long cycle). Here the correlation between $A_{\rm M}$ and cycle-length is
found highly insignificant. The following reasonably large correlations
($r>0.6$) are found (the values given within parentheses are found when 2009
is considered for the minimum epoch of Solar Cycle 24):
$r=0.67$ ($0.67$) between $A_{\rm m}$ and cycle length,
$r=0.77$ ($0.77$) between $A_{\rm m}$ and cycle declining time, and
$r=0.64$ ($0.44$) between $B_{\rm m}$ and cycle rising time.

Using the value of $A_{\rm m}$ of Solar Cycle 24 in the linear relationship
correspond to the aforementioned highest correlation between $A_{\rm m}$ and
cycle-declining time (determined from the data of Cycles 12 – 23), we obtained
$5.69\pm 0.42$ years for the declining time of Cycle 24, which suggests that
Solar Cycle 24 will end at $2019.98\pm 0.42$, $i.e.$ within the interval July
2019 – May 2020. This may be close to reality. However, the aforementioned
correlations may not strong enough for prediction purpose.

Due to the large equatorial rotation rate there could be a strong Coriolis
force at the minimum of a solar cycle, which greatly affects (decreases) the
emergence of toroidal magnetic flux at low latitudes (chg87). Depending on the
strength of $A_{\rm m}$ this effect might persist throughout the corresponding
solar cycle. In fact, a reasonable correlation is also exist between $A_{\rm
m}$ and cycle-length. Coriolis forces act more effectively on small magnetic
features, that may reduce the rate of emergence of toroidal flux at low
latitudes (the effect of magnetic buoyancy force is large on large magnetic
structures and plasma drag force is large on small magnetic structures).
Hence, there is a reasonably good correlation between $A_{\rm m}$ and the
declining time of the corresponding cycle, suggesting that a large $A_{\rm m}$
might cause a long declining time. It should be noted that a large absolute
value of $B$ means a large latitude gradient, $i.e.$ more differential
rotation (strong dynamo). The existence of a reasonably good correlation
between $B_{\rm m}$ and rising time of a cycle may imply that more
differential rotation during the minimum of a cycle might cause the cycle to
rise fast. This in turn related to the amount of toroidal magnetic flux
production depending on the strength of the differential rotation during the
minimum of the cycle. However, the physical relationships of all the
aforementioned correlations need investigations.

Figure 6.: The dashed curve represents the best-fit cosine function to the
values (filled-circles) of (a) $A_{\rm M}$, (b) $B_{\rm M}$, (c) $A_{\rm m}$,
and (d) $B_{\rm m}$, after subtracting the corresponding linear-fits. The
corresponding values of root-mean-square-deviation (rmsd) are 0.018, 0.22,
0.06, and 0.44, respectively, and the the corresponding values of $\chi^{2}$
are 11.53, 16.35, 23.22, and 16.95, respectively (21.026 is the 5 % level
significant value of $\chi^{2}$). The obtained periods of $A_{\rm M}$, $B_{\rm
M}$, $A_{\rm m}$, and $B_{\rm m}$ are 4.9, 8.5, 7.4 and 7.2 (in number of
solar cycles), $i.e.$ $54.05\pm 5.58$-year, $93.75\pm 10.03$-year, $81.62\pm
8.73$-year, and $79.42\pm 8.5$-year, respectively (Note: the average period of
solar cycles is $11.03\pm 1.18$-year; see Pesnell, 2018.) The dotted curve
(red) represents the one-rmsd level.

Figure 7.: The dashed curve represents the best-fit cosine function to the
values (filled-circles) of (a) $A_{\rm m}$ and (b) $B_{\rm m}$, by replacing
the values of these parameters in interval 2007–2009 with that in interval
2008–2010. The corresponding values of root-mean-square-deviation (rmsd) are
0.06 and 0.29, respectively, and the corresponding values of $\chi^{2}$ are
19.9 and 10.14 respectively. The obtained periods of $A_{\rm m}$ and $B_{\rm
m}$ are 7.9 and 7.6 (in number of solar cycles), $i.e.$ $87.14\pm 9.32$-year
and $83.83\pm 9.0$-year, respectively. The dotted curve (red) represents the
one-rmsd level.

Figure 5 shows the best-fits of cosine functions to the values of $A_{\rm M}$,
$B_{\rm M}$, $A_{\rm m}$, and $B_{\rm m}$ that are given in Table 2, after
subtracting the corresponding linear trends (in the cosine-fits ${\rm
weight}=\frac{1}{\sigma^{2}}$ is used). The cosine-fit to the data of most of
the above parameters is reasonably good. Except in the case of the cosine-fit
of $A_{\rm m}$, the $\chi^{2}$ of the cosine-fit to the data of each of the
remaining parameters is reasonably small. In each of these cases, most of the
data points are within the one-rmsd level. The obtained periods of $A_{\rm
M}$, $B_{\rm M}$, $A_{\rm m}$, and $B_{\rm m}$ are 4.9, 8.5, 7.4 and 7.2 (in
number of solar cycles), $i.e.$ $54.05\pm 5.58$-year, $93.75\pm 10.03$-year,
$81.62\pm 8.73$-year, and $79.42\pm 8.5$-year, respectively (Note: the average
period of a solar cycle is $11.03\pm 1.18$-year; see pesnell18.) Considerable
differences exist in the periodicities of $A$, and also in those of $B$, that
correspond to the epochs of maxima and minima of solar cycles. The amplitude
($0.058^{\circ}$ day-1) of the cosine profile of $A_{\rm m}$ is about 41 %
larger than that ($0.034^{\circ}$ day-1) of $A_{\rm M}$, whereas the amplitude
($0.6^{\circ}$ day-1) of the cosine profile of $B_{\rm m}$ is about 65 %
larger than that ($0.21^{\circ}$ day-1) of $B_{\rm M}$. The cosine profiles of
$A_{\rm M}$ and $A_{\rm m}$ suggest the existence of a large (up to 180∘)
phase (initial) difference (approximate anticorrelation) between $A_{\rm M}$
and $A_{\rm m}$. The cosine profiles of $B_{\rm M}$ and $B_{\rm m}$ also
suggest the existence of a large (up to 180∘) phase (initial) difference
(approximate anticorrelation) between $B_{\rm M}$ and $B_{\rm m}$. The cosine
profiles of $A_{\rm m}$ and $B_{\rm M}$ seem to be approximately in phase (a
positive correlation). The cosine profile of $A_{\rm M}$ indicates that in the
case around maxima of Solar Cycles 12–24, the equatorial rotation rate is
maximum at Cycles 12, 17, and 22, and minimum at Solar Cycles 14, 19, and 24.
The cosine profile of $A_{\rm m}$ indicates that in the case of around minima
of Solar Cycles 12–24 the equatorial rotation rate is maximum at Solar Cycles
13/14 and 21, and minimum at Solar Cycles 17/18 and 24. The cosine profile of
$B_{\rm M}$ indicates that in the case around maxima of Solar Cycles 12–24,
the latitude gradient of rotation is maximum at Solar Cycle 18 and minimum at
Solar Cycles 14 and 22. The cosine profile of $B_{\rm m}$ indicates that in
the case of around minima of Solar Cycles 12–24 the latitude gradient of
rotation is maximum at Solar Cycles 14 and 21/22, and minimum at Solar Cycle
18 (it should be noted that $B_{\rm M}$ and $B_{\rm m}$ are having negative
sign, a large negative value implies more differential rotation).

For the reason given above, we have repeated the calculations of the cosine
fits for $A_{\rm m}$ and $B_{\rm m}$ by replacing the values of these
parameters in interval 2007–2009 with that in interval 2008–2010, $i.e.$ by
considering the 2009, instead of 2008, for the minimum epoch of Solar Cycle
24. Figure 6a and 6b show the corresponding best-fits of cosine functions,
which are more accurate than the corresponding ones shown in Figures 5c and
5b. That is, in the case of the cosine fits of both $A_{\rm m}$ and $B_{\rm
m}$ that are shown in Figures 6a and 6b the values of $\chi^{2}$ are smaller,
and in the case of $B_{\rm m}$ the value of rmsd is also smaller than the
corresponding fits shown in Figures 5c and 5d. The cosine fits shown in Figure
6 indicate the periodicities $87.14\pm 9.32$-year and $83.83\pm 9.0$-year for
$A_{\rm m}$ and $B_{\rm m}$, respectively, which are slightly longer than
those indicated by the corresponding fits shown in Figure 5c and 5d. Except
these all the suggestions made above remain the same.

Cent percent anticorrelation between $A$ and $B$ implies variations in
equatorial rotation rate and magnitude of latitude gradient are perfectly in
phase. Because of a large difference between the periods of $A_{\rm M}$ and
$B_{\rm M}$, there could be a large collapse (catastrophic) in the
corresponding phases during Solar Cycles 12 – 24. The relatively less
difference between the periods of $A_{\rm m}$ and $B_{\rm m}$ and the
existence of an anticorrelation between the profiles of these parameters may
suggest that around the minimum of a solar cycle the angular momentum
transport from higher latitudes to the equator may be more efficient. The
cosine profiles of $A_{\rm m}$ and $A_{\rm M}$ look to be in phase and in
anti-phase during the alternate intervals of about 40 – 50-years (there exist
a very weak correlation between $A_{\rm m}$ and $A_{\rm M}$; see above),
causing reduction and improvement of flux emergence in the respective
alternate intervals. Since as mentioned above the effect of $A_{\rm m}$
persists throughout the cycle, there exist an approximate coherence in the
cosine profiles of $A_{\rm m}$ and $B_{\rm M}$, which my have a role in the
toroidal flux production rate causing variations in the amplitudes of
Gleissberg cycles.

##  4 Conclusions and Discussion

Studies of variations in the solar differential rotation are important for
understanding the underlying mechanism of solar cycle and also other
variations of solar activity. We analyzed GPR and DPD daily sunspot group data
during the period 1874 – 1917 and determined the equatorial rotation rate
[$A$] and the latitude gradient [$B$] of the solar differential rotation by
fitting the data in each 3-year MTI during the period 1874 – 1917 to the
standard law of differential rotation. The values of $A$ and $B$ around the
years of maxima and minima of Solar Cycles 12 – 24 are obtained from the
3-year MTIs series of $A$ and $B$ and studied the long-term cycle-to-cycle
modulations in these coefficients. Here we have used the epochs of the maxima
and minima of the Solar Cycles 12 – 24 that were recently determined from the
revised Version-2 international sunspot-number (SN) series. We find that there
exits a considerably significant secular decreasing-trend in $A$ around the
maxima of solar cycles. There exist no secular trends in both $A$ and $B$
around the minima of solar cycles. The secular trend in $B$ around the maxima
of solar cycles is also found to be statistically insignificant. We fitted a
cosine function to the values of $A$, and also to those of $B$, after removing
the corresponding linear trends. The cosine-fits suggest that there exist
$\approx$54-year ($\approx$94-year) and $\approx$82-year ($\approx$79-year)
periodicities in $A$ ($B$) around the maxima and minima of solar cycles,
respectively. The amplitude of the cosine-profile of $A$ ($B$) around the
minima of solar cycles is about 41 % (65 %) larger than that of $A$ ($B$) of
the maxima. In addition, the cosine profiles of $A$ and $B$ suggest a large
(up to $180^{\circ}$) phase difference between the long-term variations of
$A$, and also between those of $B$, around the maxima and minima of solar
cycles.

In Article-I the combined Greenwich and SOON sunspot-group data during Solar
Cycles 12–23 were analyzed and shown the existence of a $7.4\pm 0.5$-cycle in
$B$ by fitting a cosine-fit to the cycle-to-cycle modulation of $B$. The
similar periodic modulations were also seen in the values of $B$ determined
from the sunspot-group data of both the northern and the southern hemispheres
(jbu05b). In jbu05b the cycle-to-cycle modulations in whole cycles $A$ and $B$
determined in Article-I were compared with the modulations found in $A$ and
$B$ determined using the Kodaikanal Observatory (Solar Cycles: 15–21), Mt.
Wilson Observatory (Solar Cycles: 16–21), Solar-Observatory Kanzelhöhe (Solar
Cycles: 18–21), and National Observatory of Japan (Solar Cycles: 19-21)
sunspot data bases and pointed out the differences/similarities in the results
and some plausible causes for the differences. suz14 analyzed the sunspot-
group data measured in Mt. Wilson Observatory and their own measurements
during Solar Cycles 16–23 and found the existence of a 6–7-cycle in the cycle-
to-cycle modulation of $B$ (absolute values) determined from the whole-sphere
data and also in the modulations of $B$ determined from the northern- and
southern-hemispheres’ data. The pattern of the modulation of $B$ during Solar
Cycles 16–23 that is found by suz14 closely similar to the corresponding
portion (from Cycle 16 to Cycle 23) of the modulation (of $|B|$) during Solar
Cycles 12–23 found in Article-I, as well as to that of $B$ around maxima of
the cycles in the current analysis. Also, it seems to closely fit to the
corresponding portion of the cosine-curve of $B$ of whole cycles in Article-I
and also to that of $B$ around the maxima of the cycles in the present
analysis. Overall, all these results seem to strongly confirm the existence of
about a 7-cycle modulation in $B$ determined from sunspot data. In suz14 it is
found that in Solar Cycle 16 the northern hemisphere value of $B$ is
significantly smaller than that of the southern-hemisphere and almost
monotonic decrease in the magnitude of $B$ in both the hemispheres from Solar
Cycles-17 to 21. The overall pattern seems to be suggesting the existence of a
phase-shift (occurred in Cycle 16) between the northern and southern
hemispheres. In jbu05b it is found that except in Cycle 12 the magnitude of
the northern-hemisphere $B$ is highly significantly larger than that of the
southern hemisphere, in each of the remaining cycles the difference between
the northern- and southern-hemispheres’ values of $B$ is statistically
insignificant. That is, no phase difference is found between the cycle-to-
cycle modulations in $B$ of northern and southern hemispheres. In the present
analysis we have not studied the north–south asymmetry of the modulations in
$A$ and $B$ around the maxima and the minima of solar cycles. In suz14 error
bars of $B$ values are relatively small. However, surprisingly in the largest
Cycle 19 the error bar of $B$ seems to be largest. Particularly, the error bar
of the northern-hemisphere’s value of $B$ of this cycle is much larger than
that of the southern hemisphere and in fact, larger than those of both the
northern- and southern-hemisphere’s values of the remaining all cycles. In
Cycle 19 activity in northern hemisphere was larger that of southern
hemisphere ($e.g.$ tem06; jj19). The gaps in Greenwich data due to missing
days of observations are filled with the data from other observatories. Such
an update of Mt. Wilson sunspot-group data may be necessary. suz14 found that
the modulation in $B$ determined from the data around the middle years
(unequal intervals) of the solar cycles is almost the same as that of $B$
determined from the data of whole cycles. The arguments above on the
modulation in $B$ of whole cycles also apply to the modulation in $B$
determined from the data around the middle years of the cycles. The
modulations in $A$ and $B$ around the maxima of cycles found here are also
closely similar to those of $A$ and $B$ determined from the data of the whole
cycles shown in Article-I. This could be due to a large portion of the data of
a cycle are coming from around the maximum of the cycle, $i.e.$ the ratio of
the size of a whole-cycle’s data to that of the data around the maximum is
considerably small. In many studies of solar cycle-variations of $A$ and $B$
it has been found that the magnitudes of $A$ and $B$ are much larger, and
larger fluctuations in the yearly variations of $A$ and $B$, during the minima
of many solar cycles than during the maxima ($e.g$. balth86; jg95, jj03).
suz14 found the same in the yearly variations of $B$ and also found that if
the value of $B$ at minimum/maximum of a cycle is large, then the cycle-
average value of $B$ is large. In spite of we have used here the 3-year
intervals, there exist considerably large fluctuations in the values of $A$
and $B$ around the minima of some cycles (see Figure 2). We find the existence
of an anticorrelation between the cycle-to-cycle modulations of $B$ around the
maxima and the minima of cycles and for the reason mentioned above the
modulation in $B$ around the maxima is almost the same as that of $B$ of the
whole cycle. The average value of $B$ over all cycles maxima is found slightly
larger that of over all cycles minima. However, the amplitude of the cosine-
fit of $B$ around the minima of cycles is much larger than that of $B$ around
the maxima.

The $\approx$54-year periodicity in $A$ around the maxima of solar cycles
found here might confirm the $\approx$55-year grand cycle of $A$ was found by
yk93 using National Astronomical Observatory of Japan (NAOJ) sunspot-group
data during the period 1948 – 1987. The known $\approx$55-year periodicity of
sunspot activity ($e.g.$ tan11; gao16; komit16) may be related to the
$\approx$54-year period of $A$. The well-known Gleissberg cycle of solar
activity may be related to the $\approx$94-year period in $B$ around maxima of
solar cycles and to both the 82-year and the 79-year periods in $A$ and $B$,
respectively, around minima of solar cycles. That is, causes of the
periodicities of solar activity may be have influences of the periodicities in
$A$ and $B$ (also see Article-I; jb16). However, what causes the variations in
$A$ and $B$ is not known yet, except that there are speculations for the
existence of the Sun’s spin–orbit coupling (wood65; juc03; jj05).

The maximum and minimum epochs of a solar cycle comprise relatively large and
small numbers of large sunspot groups, respectively. The magnetic structures
of large and small sunspot groups may anchor at relatively deep and shallow
layers of the solar convection zone, respectively. Hence, the periodicities in
$A$ and $B$ around the maxima and minima of solar cycles are different and
might originate at relatively deep and shallow layers of the solar convection
zone, respectively. However, the observed rotation rate of a sunspot group (in
general a magnetic tracer) does not need to be the rotation rate of plasma at
the anchoring depth of the sunspot group. The difference between the plasma
and sunspot-group rotation rates depends upon the effects of the dynamical
forces like magnetic buoyancy, Coriolis force, and drag force on the magnetic
structure of the sunspot group ($e.g.$ dh94). Therefore, the cause of the
differences in the periodicities of $A$ and $B$ around the maxima and minima
of solar cycles still needs to be established.

##  5 Acknowledgments

The author thanks the anonymous referee for the critical review and for useful
comments and suggestions. The author acknowledges the work of all the people
contribute and maintain the GPR and DPD sunspot databases. The sunspot-number
data are provided by WDC-SILSO, Royal Observatory of Belgium, Brussels.

##  6 Conflict of interest

The author declares to have no conflicts of interest.

## References

  * Babcock (1961) Babcock, H.W.: 1961, Astrophys. J. 133, 572. DOI: 10.1086/14706
  * Badalyan and Obridko (2017) Badalyan, O.G., Obridko, V.N.: 2017, Mon. Not. Roy. Astron. Soc. 466, 4535. DOI: 10.1093/mnras/stx134
  * Balthasar, Vázquez, and Wöhl (1986) Balthasar, H., Vázquez, M., Wöhl, H.: 1986, Astron. Astrophys. 155, 87. ADS: http://adsabs.harvard.edu/abs/1986A%26A…155…87B
  * Baranyi, Győri, Ludmány (2016) Baranyi, T., Győri, L., Ludmány, A.: 2016, Solar Phys. 291, 3081. DOI: 10.1007/s11207-016-0930-1
  * Brajša, Ruždjak, and Wöhl (2006) Brajša, R., Ruždjak, D., Wöhl, H.: 2006, Solar Phys. 237, 365. DOI: 10.1007/s11207-006-0076-7
  * Cameron, Dikpati, and Brandenburg (2017) Cameron, R.H., Dikpati, M., Brandenburg, A.: 2017 Space Sci. Rev. 210, 367. DOI: 10.1007/s11214-015-0230-3
  * Chandra, Vats, and Iyer (2010) Chandra, S., Vats, H.O., Iyer, K.N.: 2010, Mon. Not. Roy. Astron. Soc. 407, 1108. DOI: 10.1111/j.1365-2966.2010.16947.x
  * Choudhuri and Gilman (1987) Choudhuri, A.R., Gilman, P.A.: 1987, Astrophys. J., 316, 788. DOI: 10.1086/165243
  * Dikpati and Gilman (2006) Dikpati, M., Gilman, P.A.: 2006, Astrophys. J. 649, 498. DOI: 10.1086/506314
  * D’Sliva and Howard (1994) D’Silva, S., Howard, R.F.: 1994, Solar Phys. 151, 213. DOI: 10.1007/BF00679072
  * Gao (2016) Gao, P.X.: 2016, Astrophys. J., 830, 140. DOI: 10.3847/0004-637X/830/2/140
  * Garcia and Mouradin (1998) Garcia, A., Mouradian, Z.: 1998, Solar Phys. 180, 495. DOI: 10.1023/A:1005018712900
  * Gilman and Howard (1984) Gilman, P.A., Howard, R.: 1984, Astrophys. J. 283, 385. DOI: 10.1086/162316
  * Gupta, Sivaraman, and Howard (1999) Gupta, S.S., Sivaraman, K.R., Howard, R.: 1999, Solar Phys. 188, 225\. DOI: 10.1023/A:1005229124554
  * Győri, Baranyi, and Ludmány (2010) Győri, L., Baranyi, T., Ludmány, A.: 2010, in Proc. Intern. Astron. Union 6, Sympo. S273, 2011, 403. DOI: 10.1017/s174392131101564X
  * Győri, Ludmány, and Baranyi (2017) Győri, L., Ludmány, A., Baranyi, T.: 2017, Mon. Not. Roy. Astron. Soc. 465, 1259. DOI: 10.1093/mnras/stw2667
  * Hathaway (2015) Hathaway, D.H.: 2015, Living Rev. Solar Phys. 12, N0.4. DOI:10.1007/lrsp-2015-4
  * Hathaway, Wilson, and Reichmann (1999) Hathaway, D.H., Wilson, R.M., Reichmann, E.J.: 1999, J. Geophys. Res. 104, 22375. DOI: 10.1029/1999JA900313
  * Hiremath (2002) Hiremath, K.M.: 2002, Astron. Astrophys. 386, 674. DOI: 10.1051/0004-6361:20020276
  * Howard (1976) Howard, R.: 1976, Astrophys. J. Lett. 93, L159. DOI: 10.1086/182328
  * Howard, Gilman, and Gilman (1984) Howard, R., Gilman, P.I., Gilman, P.A.: 1984, Astrophys. J. 283, 373\. DOI: 10.1086/162315
  * Howard and LaBonte (1980) Howard, R., LaBonte, B. J.: 1980, Astrophys. J., 239, L33. DOI: 10.1086/183286
  * Howe et al. (2000) Howe, R., Christensen-Dalsgaard, J., Hill, F., Komm, R.W., Larsen, R.M., Schou, J., Thompson, M.J., Toomre, J.: 2000, Astrophys. J. 533, L163. DOI: 10.1086/312623
  * Javaraiah (2003) Javaraiah, J.: 2003, Solar Phys. 212, 23. DOI: 10.1023/A:1022912430585
  * Javaraiah (2005) Javaraiah, J.: 2005, Mon. Not. Roy. Astron. Soc. 362, 1311. DOI: 10.1111/j.1365-2966.2005.09403.x
  * Javaraiah (2011) Javaraiah, J.: 2011, Adv. Space Res. 48, 1032. DOI: 10.1016/j.asr.2011.05.004
  * Javaraiah (2013) Javaraiah, J.: 2013, Solar Phys., 287, 197. DOI: 10.1007/s11207-013-0345-1
  * Javaraiah (2017) Javaraiah, J.: 2017, Solar Phys. 292, 172. DOI: 10.1007/s11207-017-1197-x
  * Javaraiah (2019) Javaraiah, J.: 2019, Solar Phys. 294, 64. DOI: 10.1007/s11207-019-442-6
  * Javaraiah and Bertello (2016) Javaraiah, J., Bertello, L.: 2016, Solar Phys. 291, 3485. DOI: 10.1007/s11207-016-1003-1
  * Javaraiah, Bertello, and Ulrich (2005a) Javaraiah, J., Bertello, L., Ulrich, R.K.: 2005a (Article-I), Solar Phys. 232, 25. DOI: 10.1007/s11207-005-8776-y
  * Javaraiah, Bertello, and Ulrich (2005b) Javaraiah, J., Bertello, L., Ulrich, R.K: 2005b, Astrophys. J. 626, 579. DOI: 10.1086/429898.
  * Javaraiah and Gokhale (1995) Javaraiah, J., Gokhale, M.H.: 1995, Solar Phys. 158, 173. DOI: 10.10.1007/BF00680841
  * Javaraiah and Gokhale (1997b) Javaraiah, J., Gokhale, M.H.: 1997, Astron. Astrophys. 327, 795.
  * Javaraiah and Komm (1999) Javaraiah, J., Komm, R.W.: 1999, Solar Phys. 184, 41. DOI: 10.1007/s11207-006-0130-5
  * Javaraiah and Ulrich (2006) Javaraiah, J., Ulrich, R.K.: 2006, Solar Phys. 237, 245. DOI: 10.1007/s11207-006-0130-5
  * Javaraiah et al. (2009) Javaraiah, J., Ulrich, R.K., Bertello, L., Boyden, J.E.: 2009, Solar Phys. 257, 61. DOI: 10.1007/s11207-009-9342-9
  * Juckett (2003) Juckett, D.A.: 2003, Astron. Astrophys. 399, 731. DOI: 10.1051/0004-6361:20021923
  * Kambry and Nishikawa (1990) Kambry, M.A., Nishikawa, J.: 1990, Solar Phys. 126, 89. DOI: 10.1007/BF00158300
  * Khutsishvili, Gigolashvili, and Kvernadze (2002) Khutsishvili, E.V., Gigolashvili, M.SH., Kvernadze, T.M.: 2002, Solar Phys. 206, 219. DOI: 10.1023/A:1015068629350
  * Komitov et al. (2016) Komitov, B., Sello, S., Duchlev, P., Dechev, M., Penev, K., Koleva, K.: 2016, Bulg. Astronomic. J. 25, 78.
  * Komm, Howard, and Harvey (1993) Komm, R.W, Howard, R.F., Harvey, J.W.: 1993, Solar Phys., 143, 19. DOI: 10.1007/BF00619094
  * LaBonte and Howard (1982) LaBonte, B.J., Howard, R.: 1982, Solar Phys. 75, 161. DOI: 10.1007/BF00153469
  * Li et al. (2014) Li, K.J., Feng, W., Shi, X.J., Xie, J.L., Gao, P.X., Liang, H.F: 2014, Solar Phys. 289, 759. DOI: 10.1007/s11207-013-0369-6
  * Makarov, Tlatov, and Callebaut (1997) Makarov, V.I., Tlatov, A.G., Callebaut, D.K.: 1997, Solar Phys. 170, 373\. DOI: 10.1023/A.1004995826593
  * Mendoza (1999) Mendoza, B.: 1999, Solar Phys. 188, 237. DOI: 10.1023/A:1005256728266
  * Obridko and Shelting (2016) Obridko, V.N., Shelting, B.D.: 2016, Astron. Lett. 42, 631. DOI: 10.1134/S1063773716080041
  * Ogurtsov et al. (2002) Ogurtsov, M.G., Nagovitsyn, Y.A., Kocharov, G.E., Jungner, H.: 2002, Solar Phys. 211, 371. DOI: 10.1023/A:1022411209257
  * Olemskoy and Kitchatinov (2005) Olemskoy, S.V., Kitchatinov, L.L.: 2005, Astron. Lett. 31, 706. DOI: 10.1134/1.2075313
  * Pesnell (2018) Pesnell, W.D.: 2018, Space Weather 16, 1997. DOI: 10.1029/2018SW002080
  * Roudier et al. (2018) Roudier, Th., Švanda, M., Ballot, J., Malherbe, J.M., Rieutord, M.: 2018, Astron. Astrophys. 611, A92. DOI: 10.1051/0004-6361/201732014
  * Rozelot (1994) Rozelot, J.P.: 1994, Solar Phys. 149, 149. DOI: 10.1007/BF00645186
  * Ruždjak et al. (2017) Ruždjak, D., Brajša, R., Sudar, D., Skokić, I., Poljančić-Beljan, I.: 2017, Solar Phys. 292, 179. DOI 10.1007/s11207-017-1199-8
  * Singh and Prabhu (1985) Singh, J., Prabhu, T.P.: 1985, Solar Phys. 97, 203. DOI: 10.1007/BF00152989
  * Sivaraman et al. (2003) Sivaraman, K.R., Sivaraman, H., Gupta, S.S., Howard, R.: 2003, Solar Phys. 214, 65. DOI: 10.1023/A:1024075100667
  * Snodgrass (1992) Snodgrass, H.B.: 1992, In: Harvey, K.L.(ed.) The Solar Cycle, CS-27, Astron. Soc. Pac., San Francisco, 205.
  * Snodgrass and Howard (1985) Snodgrass, H.B., Howard, R.: 1985, Solar Phys. 95, 221. DOI: 10.1007/BF00152399
  * Sudar et al. (2014) Sudar, D., Skokć, Ruždjak, D., Brajsa, R., Wöhl, H.: 2014, Mon. Not. Roy. Astron. Soc. 439, 1471. DOI: 10.1093/mnras/stu099
  * Suzuki (2014) Suzuki, M.: 2014, Solar Phys. 289, 4021. DOI: 10.1007/s11207-014-0576-9
  * Švanda et al. (2008) Švanda, M., Klvǎna, M., Sobotka, M., Bumba, V.: 2008, Astron. Astrophys. 477, 285. DOI: 10.1051/0004-6361:20077718
  * Tan (2011) Tan, B.: 2011, Astrophys. Space Sci., 332, 65. DOI: 10.1007/s10509-010-0496-6
  * Temmer et al. (2006) Temmer, M., Rybb́ak, J., Bendík, Veronig, A., Vogler, F., Otruba, W., Pötzi, W., Hanslmeier, A.: 2006, Astron. Astrophys. 447, 735. DOI: 10.1051/0004-6361:20054060
  * Ward (1965) Ward, F.: 1965, Astrophys. J. 141, 534. DOI: 10.1086/148143
  * Ward (1966) Ward, F.: 1966, it Astrophys. J. 145, 416. DOI: 10.1086/148783
  * Wood and Wood (1965) Wood, R.M., Wood, K.D.: 1965, Nature 208, 129. DOI: 10.1038/208129a0
  * Yoshimura and Kambry (1993) Yoshimura, H., Kambry, M.A.: 1993, Solar Phys., 143, 205. DOI: 10.1007/BF00646482