Vol.0 (20xx) No.0, 000–000

11institutetext: Department of Physics, Shivaji University, Kolhapur, India
22institutetext: Physical Research Laboratory, Navrangpura, 380009 Ahmedabad;
<EMAIL_ADDRESS>
33institutetext: CAS Key Laboratory of Geospace Environment, Department of
Geophysics and Planetary Sciences, University of Science and Technology of
China, Hefei 230026, China
44institutetext: CAS Key Laboratory of Solar Activity, National Astronomical
Observatories, Beijing 100101, China
55institutetext: National Solar Observatory, 3665 Discovery Drive, Boulder, CO
80303, United States
66institutetext: Goddard Space Flight Centre, NASA, Washington, USA
77institutetext: Department of Physics, Govt. P. G. College, Satna, M. P.,
India,
88institutetext: Department of Physics, Govt. PG Autonomous College, Datiya,
M. P, India
99institutetext: Sh. M. M. Patel Institute of Sciences and Research, Kadi
Sarva Vishwavidyalaya, Gandhinagar, India
Received 20xx month day; accepted 20xx month day

# Spotless days and geomagnetic index as the predictors of solar cycle 25

Dipali S. Burud 1199 Rajmal Jain 22 Arun K. Awasthi 3344 Sneha Chaudhari 99
Sushanta C. Tripathy 55 N. Gopalswamy 66 Pramod Chamadia 77 Subhash C. Kaushik
88 Rajiv Vhatkar 11

###### Abstract

We study the sunspot activity in relation to spotless days (SLDs) during the
descending phase of solar cycle $11$–$24$ to predict the amplitude of sunspot
cycle $25$. For this purpose, in addition to SLD, we also use the geomagnetic
activity (aa index) during the descending phase of a given cycle. A very
strong correlation of the SLD (R=$0.68$) and aa index (R=$0.86$) during the
descending phase of a given cycle with the maximum amplitude of next solar
cycle has been estimated.The empirical relationship led us to deduce the
amplitude of cycle $25$ to be 99.13$\pm$ 14.97 and 104.23$\pm$ 17.35 using SLD
and aa index, respectively as predictors.Both the predictors provide
comparable amplitude for solar cycle $25$ and reveal that the solar cycle $25$
will be weaker than cycle $24$. Further we derive that the maximum of cycle
$25$ is likely to occur between February and March 2024. While the aa index
has been used extensively in the past, this work establishes SLDs as another
potential candidate for predicting the characteristics of the next cycle.

###### keywords:

Sun: activity — Sun: sunspots — Sun: rotation — methods: statistical

##  1 Introduction

The magnetic field is generated in the solar interior through the dynamo
mechanism via the motion of plasma and seen as a dark spot with strong
magnetic field. These observed sunspots are varying in number between $0$
(lowest) and $250$ (highest) in approximately $11$ years period well known as
Schwabe cycle. The solar cycles are characterized by their length, amplitude,
rise time and fall time etc. The solar activities such as the sunspot number,
sunspot areas, total solar irradiance,Coronal Mass Ejection (CME), solar
flares, facular area and $10.7$ cm solar radio flux ( $F10.7$) etc, show
cyclic behavior as observed by the various researcher (Jain 1986, 1997;
Krivova & Solanki 2002; Braun et al. 2005; Ataç et al. 2006; Kilcik et al.
2010, 2012; Hathaway et al. 2002; Javaraiah 2019; El-Borie et al. 2020). The
period of Schwabe cycle varies between $7.4$ and $14.8$; with the ascending
and descending phase of $4$-$5$ and $5$-$6$ years, respectively (Usoskin &
Mursula 2003).These types of periodicity are very useful in understanding the
solar phenomena which has the long-term change in climate and short-term
changes in space weather (Nandy & Martens 2007).

In a given solar cycle, there are many days without sunspots, which are known
as spotless days (SLDs). The number of SLDs in a year or in a given cycle can
be obtained from the web site (www.sidc.be/silso/datafiles) which covers the
period $01$ January $1818$ to December $2020$. The SLDs are generally noted
during the descending phase and reach to maximum in the sunspot minimum year.
SLDs are also found to be present during the maximum year. The highest number
of SLDs ($311$) occurred in the year $1913$ of cycle $14$.The maximum number
of SLDs ($973$) in a given cycle was found in cycle $12$. The total numbers of
SLD in cycle $23$ (August $1996$ to $31$ December $2019$) are found to be
$573$, similar to cycle $15$ ($556$) and cycle $16$ ($541$). However the cycle
$24$, showed a total $913$ SLDs untilDecember $2019$ (Burud et al. $2021$, in
preparation).

The highest value of SLD in a given cycle has been used to define the minimum
phase of the corresponding cycle by (Waldmeier 1961) and (McKinnon & Waldmeier
1987). The authors suggested that the date of sunspot minimum can be decided
by the number of SLD. (Wilson 1995) used the timing of occurrence of the first
observed SLD in a given cycle with various other timings such as time period
between the solar minimum to the first SLD observation after the solar
maximum, time period between the solar maximum to the first SLD observation
after the solar maximum etc. to predict the minimum period of solar cycle
$22$. Similar result has been observed by (Hamid & Galal 2006) and (Carrasco
et al. 2016). (Harvey & White 1999) showed that the SLD is not the only
parameter to decide the minimum of a solar cycle but other parameters such as
the monthly averaged sunspot number, the number of regions (total, new- and
old-cycle) etc. need to be considered. (Helal & Galal 2013a) indicates that
the relation between the SLDs along two years intervals around the preceding
minimum can be used to predict the amplitude of the next solar cycle and
obtained a value of $118.2$ for cycle $25$.

Further, the geomagnetic activity (aa index) during the descending phase of a
given sunspot cycle provides an asymptomatic prediction of the magnitude of
annual mean sunspot number of the next sunspot cycle (Jain 1997). Employing aa
index as predictor, (Jain 1997) estimated amplitude of the solar cycle $23$ to
be $166.2$, which is found in agreement with the observed value of $173.4$.
However, it may be noted that (Jain 1997) extended the period of descending
phase to five years (minimum plus previous four years) to predict the
amplitude and timing of the next solar cycle as predictor. Further, it would
be interesting to explore the relationship between the two predictors: SLD and
aa index.

In this paper, we carry out a statistical analysis of SLD and aa index during
descending phase of the solar cycle $11$ to $24$, to investigate the
characteristics of the solar cycle $25$. This paper is organized as follows:
in section $2$ we give sources and description of the data, analysis and
results are presented in section $3$ and discussions and brief conclusions are
presented in section $4$ and $5$ respectively.

##  2 Sources of Data

We consider the sunspot number (R) that has recently been published and obtain
from SILSO web site to study the physical properties of SLD from solar cycle
$06$ to $24$ (Wilson et al. 1996; Li et al. 2005). The SLD data are taken for
the period January, $1868$ to December $2019$ covering cycle $11$ to $24$
since the aa index is available for the period. In order to minimize the
inconsistency in calendar years related to sunspots numbers (see Petrovay
2020) a $13$-month boxcar averages of monthly mean sunspot numbers is
calculated as follows:

$R\textsubscript{m}=\frac{1}{24}(R\textsubscript{i,-6}+2\sum_{j=-5}^{j=5}R\textsubscript{i,j}+R\textsubscript{i,6})$

where, $R\textsubscript{m}$ is the $13$-month smoothed maximum annual mean
sunspot number which is calculated by using the mean of monthly sunspot
numbers (Rj) over $13$ months centered on the corresponding month (Ri).

The aa index has been taken from British geological survey where the data is
available from $1868$ at
(www.geomag.bgs.ac.uk/data_service/data/magnetic_indices/aaindex). The aa
index provides a daily average level of the geomagnetic activity measured in
nT. Eight datasets of $3$-hourly values of aa index are averaged to obtain the
daily aa index for a given day. As mentioned earlier, the aa index is taken
for the period January, $1868$ to December $2019$, representing cycle $11$ to
$24$.

##  3 Analysis and results

###  3.1 Prediction of the 13-month smoothed maximum annual mean sunspot
number

The SLD data is taken from website maintained by Solar Influences Data Center
( SIDC), Royal observatory of Belgium. It is also observed that the SLD can
occur between two days when the sunspots have been located on the solar disk.
Generally, during the descending phase of the solar cycle, we have observed
that there are many continuous days without sunspots. We define such
continuous period of days observed as consecutive spotless days. Our data of
SLD show that the consecutive days between $1$ to $92$ days, as presented in
Table $1$. This table allows us to study occurrence frequency of SLD
distribution with respect to number of consecutive days. As shown in figure
$1$, we find the distribution follows a negative exponential function.

Table 1: Occurrence frequency of SLD for descending phase of solar cycle $11$-$24$ Consecutive Days |  Occurrence frequency |  Consecutive Days |  Occurrence frequency |  Consecutive Days |  Occurrence frequency
---|---|---|---|---|---
1 |  264 |  16 |  9 |  31 |  4
2 |  140 |  17 |  7 |  36 |  2
3 |  113 |  18 |  7 |  37 |  2
4 |  86 |  19 |  9 |  39 |  1
5 |  60 |  20 |  7 |  40 |  2
6 |  50 |  21 |  6 |  42 |  1
7 |  40 |  22 |  4 |  43 |  1
8 |  38 |  23 |  2 |  44 |  1
9 |  31 |  24 |  5 |  45 |  1
10 |  25 |  25 |  3 |  47 |  1
11 |  17 |  26 |  6 |  49 |  1
12 |  17 |  27 |  6 |  69 |  1
13 |  22 |  28 |  1 |  92 |  1
14 |  15 |  29 |  1 |  |
15 |  20 |  30 |  3 |  |
Figure 1: Occurrence frequency of SLD as a function of number of consecutive
SLD during the descending phase of solar cycle 11 to 24.

The table $2$ represents the yearwise comparison of parameters such as, 13
month smoothed annual mean sunspot number ( Rm), number of SLDs (SLD) and sum
of aa index $(\sum{aa})$ over the desending phases of cycle $21$, $22$, $23$
and $24$.

The Table $2$ reveals that the total numbers of SLD ($\sum{SLD}$) of the
descending phase of cycle $22$ is $1.06$ times higher relative to cycle $21$,
and, similarly cycle $23$ is 2.07 times larger relative to cycle $22$. This
trend of incrmeant in the value of SLD has also been observed in the current
cycle $24$, in which the SLD of its descending phase is 1.18 times higher
relative to cycle 23. This indicates the upward trend in the number of SLD
while,13-month smoothed monthly sunspot number (Rm) and sum of aa index
$(\sum{aa})$ shows downward trend for the long term cycle has begun from cycle
21.This suggests that Sun is heading towards modern minimum of Gleissberg
cycle (Feynman & Gabriel 1990).

Table 2: Yearwise comparison of Rm, no. of SLD and sum of aa index $(\sum{aa})$ over descending phase of solar cycle $21$,$22$, $23$ and $24$ cycle $21$ |  cycle $22$ |  cycle $23$ |  cycle $24$
---|---|---|---
Year |  Rm |  SLD |  $\sum{aa}$ |  Year |  Rm |  SLD |  $\sum{aa}$ |  Year |  Rm |  SLD |  $\sum{aa}$ |  Year |  Rm |  SLD |  $\sum{aa}$
1982 |  114.28 |  0 |  98782 |  1992 |  93.85 |  0 |  79691 |  2004 |  41.90 |  3 |  67475 |  2015 |  72.63 |  0 |  64945
1983 |  74.68 |  4 |  86134 |  1993 |  55.56 |  0 |  74251 |  2005 |  28.93 |  13 |  67653 |  2016 |  41.51 |  27 |  58315
1984 |  42.20 |  13 |  84635 |  1994 |  30.23 |  19 |  85515 |  2006 |  16.06 |  65 |  47211 |  2017 |  21.33 |  96 |  56635
1985 |  17.88 |  83 |  65833 |  1995 |  17.27 |  61 |  63906 |  2007 |  7.98 |  163 |  43628 |  2018 |  8.68 |  208 |  40438
1986 |  13.77 |  129 |  61308 |  1996 |  9.10 |  165 |  54244 |  2008 |  2.86 |  265 |  41277 |  2019 |  3.52 |  273 |  36404

Further in this study, following the technique proposed in (Jain 1997), (Wang
& Sheeley 2009), and (Bhatt et al. 2009), we define the descending phase of a
given nth cycle to be spanning over a period of five years, which comprises of
the minimum year and preceding four years.Subsequently, representative aa
index and SLD values of the nth cycle have been estimated by integrating over
the entire descending phase duration of five years due to the following
reason. Based on the examination of the polarity of the active regions and
identification of the location of the ephemeral regions, (Harvey 1992)
proposed that the overlap between the two cycle is five years. Further, based
on a statistical investigation of the time-series observations of $128$ years,
(Jain 1997) found that the aa index averaged over the period of five years is
the effective precursor for estimating the characteristics of the next solar
cycle. A total number of aa index $(\sum aa\textsubscript{dsc})$ and SLDs
$(\sum SLD\textsubscript{dsc}$ ), integrated over the descending phase of the
nth cycle for the cycle $11$ to $24$ is given in table $3$.

Table 3: Parameters of solar cycle $11$-$24$

Cycle no. |  Descending years |  $(\sum{SLD}$)dsc |  $\sum{SLD}$ |  $(\sum{aa}$)dsc | Rmax
---|---|---|---|---|---
|  |  |  |  |  Observed |  Calculated by SLD (error in %) |  Calculated by aa (error in%)
11 |  1874-1878 |  753 |  806 |  151672 |  140 |  - |  -
12 |  1886-1890 |  699 |  973 |  221763 |  72.5 |  85.42(17.8) |  67.8(6)
13 |  1898-1902 |  845 |  909 |  141825 |  87.9 |  90.39(2.8) |  91.7(4)
14 |  1909-1913 |  846 |  903 |  199465 |  64.2 |  76.96(20) |  64.5(0.5)
15 |  1919-1923 |  387 |  556 |  250574 |  105.4 |  76.86(27) |  84.1(20)
16 |  1929-1933 |  394 |  541 |  293386 |  78.1 |  119.09(52) |  101.5(30)
17 |  1940-1944 |  253 |  427 |  333868 |  119.2 |  118.45(0.6) |  116.06(2.6)
18 |  1950-1954 |  398 |  414 |  352692 |  151.7 |  131.42(13.4) |  129.8(14)
19 |  1960-1964 |  149 |  197 |  337074 |  201.3 |  118.08(41) |  136.22(32)
20 |  1972-1976 |  247 |  325 |  362000 |  110.6 |  140.9(27) |  130.9(18)
21 |  1982-1986 |  229 |  254 |  396692 |  164.5 |  131.97(19.7) |  139.3(15)
22 |  1992-1996 |  245 |  289 |  357607 |  158.5 |  133.63(15.6) |  151.2(4.6)
23 |  2004-2008 |  509 |  573 |  267244 |  120.8 |  132.16(9.4) |  137.8(14)
24 |  2015-2019 |  604 |  913 |  256737 |  110.44 |  107.87(2.3) |  107.17(2.9)
25a |  |  |  |  |  |  99.13 ± 14.97 |  104.23± 17.35

Note: a \- Represent the value calculated by both precursors: SLD and aa index

Shown in Figure $2$ is the 13 month smoothed maximum annual mean sunspots
number ($R_{max}$) of (n+1)th cycle plotted as a function of observed number
of total SLD during the descending phase $(\sum SLD\textsubscript{dsc}$ ) of
the n th cycle. The best $\chi^{2}$ linear fit to the data shows correlation
coefficient of 0.68, which led us to derive an empirical relation between
these two indices for cycles $11$ \- $24$ as follows.

${(R_{max})\textsubscript{n+1}=154.70-0.092*(\sum
SLD\textsubscript{n})\textsubscript{dsc}\pm 14.97}$ (1)

where, $(R\textsubscript{max})\textsubscript{n+1}$ is the predicted 13 month
smoothed maximum annual mean sunspot number of the (n+1)th cycle.

From figure 2, it is evident that point #18 (corresponding to Rmax of cycle 19
and SLDs in the descending phase of cycle 18) is an obvious outlier and thus
been excluded in obtaining the empirical relation (1). Similar behavior of
point #18 is also noted in the figure 3, in which the correlation of Rmax of
preceding cycle is obtained against integrated aa index. This may be
attributed to either an underestimation of SLD and aa index, or an unusually
high magnetic field activity in the cycle 19 ( Rmax=201) compared to all of
the investigated cycles where the maximum value of Rmax has reached only to a
level of 164.5. The reason for this anomalously activity exhibited by
predictors of cycle 18 has been investigated in detail by (Wang & Sheeley
2009). They attributed an underestimation of the aa index (and radial IMF
strength which is derived from the aa index) to cause this unusual behavior.
Based on higher polar field strengths as estimated by Sheeley Jr (2008), and a
low-level of sunspot activity in 1954 (end of cycle 18), they have further
argued that radial IMF strength and thus, aa index has been underestimated.
This led to the exclusion of point #18 from the correlation trend. In another
investigation by Podladchikova et al. (2017), who established a correlation
between the maximum amplitude of the solar cycle and the temporal behaviors of
the sunspot number during the declining phase of the preceding cycle in the
form of “integral activity” which is the area under the sunspot cycle curve.
After studying solar cycle 10-23, they found that a slow declining trend of
the sunspot number, which exhibited a sudden increase during the year
1950-1954 (i.e. descending phase of cycle 18). Since the rate of declination
of sunspots has decreased in this duration, they attributed this to cause
higher value Rmax for cycle 19.This unusual behavior of cycle 19 led to its
exclusion from further investigations by other investigators Petrovay (2010)
and Clilverd et al. (2006).

The total number 0f observed SLD in the descending phase of cycle 24 are
$(\sum SLD\textsubscript{24})\textsubscript{dsc}$ = 604, which yields
$(R\textsubscript{max})\textsubscript{25}$ =99.13 $\pm$ 14.97 using equation
1. This indicates that the amplitude of cycle $25$ will be lower than the
cycle $24$ ($R\textsubscript{max}$)24 =$110.44$.

Figure 2: The 13 month smoothed maximum annual mean sunspot number (Rmax) of
(n+1)th cycle plotted as a function of ($\sum$SLD)dsc of the nth cycle. The
data point corresponding to sum of the SLDs in the desending phase of cycle 18
$(\sum SLD)\textsubscript{18}$ and maximum amplitude of sunspot number (Rmax)
of cycle 19, marked with point #18 has been excluded from fitting equation.

Further, aa index during the descending phase of a given cycle has been found
to be a good predictor of the amplitude of the following cycle (Ohl 1966; Jain
1997; Lantos & Richard 1998; Hathaway et al. 1999; Hathaway 2009; Wang &
Sheeley 2009; Bhatt et al. 2009). In order to apply the prediction scheme
similar to that employed in the previous section on SLDs, we calculate the
total value of aa index $(\sum aa)\textsubscript{dsc}$ over the descending
phase of the given nth cycle. We provide the result in Table $3$ for the solar
cycles $11$ to $24$.

Figure 3: The 13 month smoothed maximum annual mean sunspot numbers
$(R_{max})$ of (n+1)th cycle plotted as a function of ($\sum$aa)dsc of the
$n^{th}$ cycle.The correlation between them is (0.86). The data point
corresponding to sum of the aa index in the desending phase of cycle 18 $(\sum
aa)\textsubscript{18}$ and maximum amplitude of sunspot number (Rmax) of cycle
19, marked with point #18 has been excluded from fitting equation.

We show in Figure 3 the relation between $13$ month smoothed maximum annual
mean sunspot numbers $(R\textsubscript{max})$ of (n+1)th cycle and $(\sum
aa)\textsubscript{dsc}$ of the nth cycle. The linear fit with a correlation
coefficient ($0.86$) is found with following empirical relation.

$(R_{max})^{n+1}=3.4*10^{-4}*(\sum
aa\textsubscript{n})\textsubscript{dsc}+16.96)\pm 17.35$ (2)

where, $(R\textsubscript{max})\textsubscript{n+1}$ is the predicted $13$ month
smoothed maximum annual mean sunspot numbers for the (n+1)th cycle.

Considering the cycle $24$ to have ended in December $2019$, reference:
(www.sidc.be/silso/datafiles/node/166) we obtain $(\sum
aa\textsubscript{dsc})$= $256737$. Using this value in the relation ($2$) we
obtain $(R\textsubscript{max})\textsubscript{25}$ = 104.23 $\pm$ 17.35. This
value of Rmax, derived from the aa index as the predictor agrees well with
that derived using SLD as a predictor (99.13 $\pm$ 14.97). Therefore, this
work identifies the potential of SLD as the precursor of next solar cycle.
Further, we have been able to confirm that the cycle $25$ will be weaker than
the cycle $24$.

The table $3$ gives a comparison between the observed and calculated values of
the $13$ month smoothed maximum annual mean sunspot numbers for both the
precursor methods. The comparative analysis indicates that in some cases, the
calculated values from relation 1 agree well (relative to that calculated from
relation 2) with the observed values of Rmax (for cycle $23$ and $24$) whereas
vice-versa for others. Therefore, we find that, in general,the calculated
values of Rmax from both the relations agree well with the observed values.

###  3.2 Prediction of the Maximum Period of Solar Cycle 25

Following Waldmeier effect (Waldmeier 1935) and our findings that the solar
cycle $25$ will be weaker it is clear that the ascending phase of cycle $25$
is expected to be longer in duration. Thus it is important to predict the
ascending period of cycle $25$. For this purpose we explore the relationship
between the ascending phase period in years,
$(P\textsubscript{asc})\textsubscript{n}$ and
$(R\textsubscript{max})\textsubscript{n}$ . Shown in Figure 4 is the plot
between $(P_{asc})_{n}$ and $(R_{max})_{n}$ for the cycles $11$-$24$. The best
fit line shows a correlation value of $-0.78$. The trend reveals that
$(P\textsubscript{asc})\textsubscript{n}$ of a solar cycle decreases with the
increase in $(R\textsubscript{max})\textsubscript{n}$, representing
manifestation of the “Waldmeier effect”. The linear fit is expressed in the
form of the following empirical relation:

$(P\\_{asc})_{n}=5.7-0.015\texttimes(R\textsubscript{max})\textsubscript{n}\pm
0.43$ (3)

On substituting the value of $(R\textsubscript{max})\textsubscript{25}$ =
99.13 $\pm$ 14.97 , derived from the SLD technique, we obtain
$(P\textsubscript{asc})\textsubscript{25}$ = 4.21 $\pm$ 0.43 years. On the
other hand, by substituting $(R\textsubscript{max})\textsubscript{25}$ =
104.23$\pm$ 17.35, obtained from the aa method, we estimate
$(P\textsubscript{asc})\textsubscript{n}$ = 4.14 $\pm$ 0.43 years. Results
from both techniques reveal that ascending period for cycle $25$ is likely to
be around 4.17 $\pm$ 0.43 years, ( average of both techniques), which suggests
the peak amplitude of the cycle $25$ is likely to occur between February and
March $2024$.

Figure 4: The plot shows the ascending phase of the nth cycle as a function of
13 month smoothed maximum annual mean sunspot numbers (Rmax) of the same
cycle. The data point representing the cycle 19 (marked by #19) has been
excluded from fitting equation.

##  4 Discussion

In the current investigation we use the spotless days (SLDs) during the
descending phase of a given cycle to predict the sunspot amplitude of the next
cycle. Based on this technique we obtained maximum amplitude
$(R\textsubscript{max})$ for cycle $25$ to be 99.13 $\pm$ 14.97. In addition
to SLD as a predictor of the sunspot cycle characteristics, we also used sum
of aa index as another solar activity proxy and found amplitude to be 104.23
$\pm$ 17.35. This is in good agreement with the amplitude estimated using SLD.
Our predicted amplitude of sunspot number 99.13 ± 14.97 for cycle 25 is in
agreement with the recent predictions made by few other investigators (within
the range of 82-140); (Upton & Hathaway 2018; Jiang et al. 2018; Bhowmik &
Nandy 2018; Petrovay et al. 2018; Bisoi et al. 2020; Sello 2019; Dani &
Sulistiani 2019; Covas et al. 2019; Labonville et al. 2019; Miao et al. 2020).
On the other hand, some studies (Sarp et al. 2018; Li et al. 2018) contrarily
predict the cycle 25 to have a higher value (150-170). This difference may be
attributed to different predictors as well as numerical scheme used (such as
non-linear prediction algorithm (154) and bimodal distribution (168)). Thus,
we find the amplitude of solar cycle $25$ to be weaker than the current cycle
$24$. Further, the period of the peak amplitude of cycle $25$ is predicted to
be occurring between February and March $2024$.

Table 4: Prediction for cycle $25$ (Author year) |  Method |  Maximum amplitude
---|---|---
(Javaraiah 2017) |  Cosine function+ linear fit |  30 - 40
(Javaraiah 2015) |  Maximum entropy method (MEM), and Morlet wavelet analysis. |  50±10
(Covas et al. 2019) |  Neural networks Spatiotemporal |  57 ±17
(Kakad et al. 2017) |  Shannon entropy model |  63±11.3
(Labonville et al. 2019) |  Solar dynamo models |  89(+/-29/14)
(Attia et al. 2013) |  Neural networks Neuro-fuzzy |  90.7 ± 8
(Gopalswamy et al. 2018) |  Microwave imaging observations |  89 and 59
(Singh & Bhargawa 2019) |  Extrapolation method |  89 ± 9 and 78 ± 7
(Kitiashvili 2016) |  Data assimilation method |  90 ± 15
(Jiang & Cao 2018) |  SFT model |  93 to 125
(Chowdhury et al. 2021) |  Precursor method Ap index |  100.21 ± 15.06
(Singh & Bhargawa 2017) |  Hurst exponent |  103 ± 25
(Hiremath 2006) |  Forced and damped harmonic oscillator model |  110 (± 11)
(Pishkalo 2019) |  Polar field as precursor |  116±12
(Hawkes & Berger 2018) |  Magnetic helicity |  117
(Helal & Galal 2013b) |  Precursor technique of spotless |  118.2
(Pesnell & Schatten 2018) |  Employing solar dynamo index |  135 ±25
(Du 2020) |  Logarithmic relationship between Rm and aamin |  151.1±16.9
(Sarp et al. 2018) |  Non-linear prediction algorithm |  154± 12

We have also shown in Table $2$ that numbers of SLD are continuously
increasing while sum of aa index and sunspot numbers are decreasing from cycle
21 . A total 509 SLD were observed during the descending phase of solar cycle
$23$, significantly higher than the value of $229$ and $245$ for solar cycles
$21$ and $22$ respectively. On the other hand, in the solar cycle $24$, an
extremely low sunspot activity during ascending ($2009$-$2012$) as well as
descending phase ($2015-2019$) have been reported, and 604 SLDs have been
observed. The continuous increases of numbers of SLD and sum of the aa index
from cycle 21 and inferences from the current investigation reveal that the
cycle $24$ and $25$ are likely to be the part of the series of $2$-$4$
successive weak and long cycles. This may suggest that the sun has been
progressing towards or in the Gleissberg minimum or grand solar minimum
(Petrovay et al. 2018; Yousef 2003; Feynman & Gabriel 1990). Considering low
amplitude of cycle $25$ and decreasing trend of solar surface magnetic field,
it is likely that cycle $26$ may also be a weak cycle and if this trend
continues then it appears that Sun may be heading towards next Grand Minimum.
However, while our sunspot number predictions are in agreement with that
obtained by several investigators in the past (see Table 4), they do not
necessarily concur on our inference on Sun to be heading towards grand solar
minimum. For example,(Bhowmik & Nandy 2018) predicted the maximum amplitude of
the 25th cycle to be 118 with a peak in 2024,which is in agreement with our
prediction. However, their ensemble forecast model indicates the reversal of
weakening trend of solar activity.

It may be noted that the last Maunder Minimum was during $1645$-$1715$ AD, and
the end of upcoming cycle $26$ would be nearly in $2043$ AD, almost $400$
years after the Maunder Minimum. Our above conclusions is further strengthened
from the investigation carried out by (Cameron et al. 2016) who showed that
the measurements of the dipole moment of about 2G at the end of cycle $23$, is
close to simulated value of about 2.5 $\pm$ 1.1 G for cycle $25$ in $2020$.
Based on the best correlation between the dipole moment during the solar
minimum and the strength of the next solar cycle, (Cameron et al. 2016)
further suggested that cycle $25$ will be of moderate amplitude, not much
higher than that of the current cycle. It is widely known from several
evidences that the strength of the sun’s polar fields near the time of a
sunspot cycle minimum determines the strength of the following solar activity
cycle. (Hathaway & Upton 2016) applied their Advective Flux Transport code,
with flows well constrained by observations, to simulate the evolution of the
sun’s polar magnetic fields from early $2016$ to the end of $2020$, near the
expected time of cycle $24/25$ minimum. Considering various limitations and
uncertainties in simulations they find that the average strength of the polar
fields near the end of cycle $24$ will be similar to that measured near the
end of cycle $23$, indicating that cycle $25$ will be similar in strength to
the current cycle $24$. Further, (Macario-Rojas et al. 2018) also showed that
the cycle $25$ will be a weak solar cycle with slow rise time (-14.4 per cent)
and maximum activity ($\pm$19.5 per cent) with respect to the current cycle
$24$. Table $4$ summarizes the various predictions of several investigators
for the amplitude of cycle $25$. The table contains the author as well as year
of publication, prediction methods and predicted values in column $1$, $2$,
and $3$ respectively. From the table, it is clear that our results concur with
the most of previous investigators in the sense that the solar cycle $25$ has
amplitude lower than the amplitude of solar cycle $24$. On the other hand,
there are few authors who claim that the cycle $25$ may be of similar or
higher amplitude than the cycle $24$. Moreover, our results from both
techniques (SLD and aa) fall within the uncertainty of the NOAA/NASA consensus
predictions on December $9$, $2019$ that strength of upcoming solar cycle $25$
would be comparable to cycle $24$ (https:// www. swpc. noaa. Gov/ news /solar-
cycle-25-forecast-update).

##  5 Conclusions

We have applied the available data for the SLD from cycle $11$ to $24$
($1874$-$2019$). The occurrence frequency of SLD shows negative exponential
distribution with respect to the consecutive days. A new prediction technique
is proposed based on the number of SLD observed in the declined phase of solar
cycle which shows that the maximum of the $13$ month smoothed maximum annual
mean sunspot number for cycle $25$ will be 99.13 $\pm$ 14.97. The SLD
occurrence density per cycle is found to be increase from cycle $21$ while sum
of aa index and 13-month smoothed monthly sunspot number are decreasing during
the descending phase of a cycle in particular (cf. table $2$). This trend is
indicative of weakening in solar surface magnetic field of the sun. In view of
of our analysis which resulted in the fact that solar cycle $25$ is likely to
be weaker in strength relative to the current cycle $24$, we conclude that the
sun may be proceeding towards next grand minimum.

###### Acknowledgements.

Authors received the data for Sunspot Index and Long-term Solar Observations
(SILSO) site of SIDC, WDC, Royal Observatory, Belgium. Cooperation received
from President, KSV is sincerely acknowledged. RJ was invited by GSFC/ NASA to
work on this investigation. AKA acknowledges the funding support from
NSFC-11950410498 and KLSA-202010 grants. We earnestly thanks the referee for
valuable remarks which has improved the paper.

## References

  * Ataç et al. (2006) Ataç, T., Özgüç, A., & Rybak, J. 2006, Sol. Phys., 237, 433, doi: 10.1007/s11207-006-0017-5
  * Attia et al. (2013) Attia, A.-F., Ismail, H. A., & Basurah, H. M. 2013, Ap&SS, 344, 5, doi: 10.1007/s10509-012-1300-6
  * Bhatt et al. (2009) Bhatt, N. J., Jain, R., & Aggarwal, M. 2009, Sol. Phys., 260, 225, doi: 10.1007/s11207-009-9439-1
  * Bhowmik & Nandy (2018) Bhowmik, P., & Nandy, D. 2018, Nature Communications, 9, 5209, doi: 10.1038/s41467-018-07690-0
  * Bisoi et al. (2020) Bisoi, S. K., Janardhan, P., & Ananthakrishnan, S. 2020, Journal of Geophysical Research (Space Physics), 125, e27508, doi: 10.1029/2019JA027508
  * Braun et al. (2005) Braun, H., Christl, M., Rahmstorf, S., et al. 2005, Nature, 438, 208, doi: 10.1038/nature04121
  * Cameron et al. (2016) Cameron, R. H., Jiang, J., & Schüssler, M. 2016, ApJ, 823, L22, doi: 10.3847/2041-8205/823/2/L22
  * Carrasco et al. (2016) Carrasco, V. M. S., Aparicio, A. J. P., Vaquero, J. M., & Gallego, M. C. 2016, Sol. Phys., 291, 3045, doi: 10.1007/s11207-016-0998-7
  * Chowdhury et al. (2021) Chowdhury, P., Jain, R., Ray, P., Burud, D., & Chakrabarti, A. 2021, Solar Physics, 296, 1
  * Clilverd et al. (2006) Clilverd, M. A., Clarke, E., Ulich, T., Rishbeth, H., & Jarvis, M. J. 2006, Space Weather, 4, S09005, doi: 10.1029/2005SW000207
  * Covas et al. (2019) Covas, E., Peixinho, N., & Fernandes, J. 2019, Sol. Phys., 294, 24, doi: 10.1007/s11207-019-1412-z
  * Dani & Sulistiani (2019) Dani, T., & Sulistiani, S. 2019, in Journal of Physics Conference Series, Vol. 1231, Journal of Physics Conference Series, 012022, doi: 10.1088/1742-6596/1231/1/012022
  * Du (2020) Du, Z. L. 2020, Ap&SS, 365, 104, doi: 10.1007/s10509-020-03818-1
  * El-Borie et al. (2020) El-Borie, M. A., El-Taher, A. M., Thabet, A. A., et al. 2020, ApJ, 898, 73, doi: 10.3847/1538-4357/ab9d21
  * Feynman & Gabriel (1990) Feynman, J., & Gabriel, S. B. 1990, Solar Physics, 127, 393
  * Gopalswamy et al. (2018) Gopalswamy, N., Mäkelä, P., Yashiro, S., & Akiyama, S. 2018, Journal of Atmospheric and Solar-Terrestrial Physics, 176, 26, doi: 10.1016/j.jastp.2018.04.005
  * Hamid & Galal (2006) Hamid, R. H., & Galal, A. A. 2006, in Solar Activity and its Magnetic Origin, ed. V. Bothmer & A. A. Hady, Vol. 233, 413–416, doi: 10.1017/S1743921306002390
  * Harvey (1992) Harvey, K. L. 1992, in Astronomical Society of the Pacific Conference Series, Vol. 27, The Solar Cycle, ed. K. L. Harvey, 335
  * Harvey & White (1999) Harvey, K. L., & White, O. R. 1999, J. Geophys. Res., 104, 19759, doi: 10.1029/1999JA900211
  * Hathaway (2009) Hathaway, D. H. 2009, Solar Cycle Forecasting, Vol. 32, 401, doi: 10.1007/978-1-4419-0239-9_20
  * Hathaway & Upton (2016) Hathaway, D. H., & Upton, L. A. 2016, Journal of Geophysical Research (Space Physics), 121, 10,744, doi: 10.1002/2016JA023190
  * Hathaway et al. (1999) Hathaway, D. H., Wilson, R. M., & Reichmann, E. J. 1999, J. Geophys. Res., 104, 22375, doi: 10.1029/1999JA900313
  * Hathaway et al. (2002) —. 2002, Sol. Phys., 211, 357, doi: 10.1023/A:1022425402664
  * Hawkes & Berger (2018) Hawkes, G., & Berger, M. A. 2018, Sol. Phys., 293, 109, doi: 10.1007/s11207-018-1332-3
  * Helal & Galal (2013a) Helal, H. R., & Galal, A. A. 2013a, Journal of Advanced Research, 4, 275, doi: 10.1016/j.jare.2012.10.002
  * Helal & Galal (2013b) —. 2013b, Journal of Advanced Research, 4, 275, doi: 10.1016/j.jare.2012.10.002
  * Hiremath (2006) Hiremath, K. M. 2006, A&A, 452, 591, doi: 10.1051/0004-6361:20042619
  * Jain (1986) Jain, R. 1986, MNRAS, 223, 877, doi: 10.1093/mnras/223.4.877
  * Jain (1997) —. 1997, Sol. Phys., 176, 431, doi: 10.1023/A:1004973827442
  * Javaraiah (2015) Javaraiah, J. 2015, New A, 34, 54, doi: 10.1016/j.newast.2014.04.001
  * Javaraiah (2017) Javaraiah, J. 2017, Solar Physics, 292, 1
  * Javaraiah (2019) Javaraiah, J. 2019, Sol. Phys., 294, 64, doi: 10.1007/s11207-019-1442-6
  * Jiang & Cao (2018) Jiang, J., & Cao, J. 2018, Journal of Atmospheric and Solar-Terrestrial Physics, 176, 34, doi: 10.1016/j.jastp.2017.06.019
  * Jiang et al. (2018) Jiang, J., Wang, J.-X., Jiao, Q.-R., & Cao, J.-B. 2018, ApJ, 863, 159, doi: 10.3847/1538-4357/aad197
  * Kakad et al. (2017) Kakad, B., Kakad, A., & Ramesh, D. S. 2017, Sol. Phys., 292, 181, doi: 10.1007/s11207-017-1207-z
  * Kilcik et al. (2010) Kilcik, A., Özgüç, A., Rozelot, J. P., & Ataç, T. 2010, Sol. Phys., 264, 255, doi: 10.1007/s11207-010-9567-7
  * Kilcik et al. (2012) Kilcik, A., Yurchyshyn, V. B., Rempel, M., et al. 2012, ApJ, 745, 163, doi: 10.1088/0004-637X/745/2/163
  * Kitiashvili (2016) Kitiashvili, I. N. 2016, The Astrophysical Journal, 831, 15
  * Krivova & Solanki (2002) Krivova, N. A., & Solanki, S. K. 2002, A&A, 394, 701, doi: 10.1051/0004-6361:20021063
  * Labonville et al. (2019) Labonville, F., Charbonneau, P., & Lemerle, A. 2019, Sol. Phys., 294, 82, doi: 10.1007/s11207-019-1480-0
  * Lantos & Richard (1998) Lantos, P., & Richard, O. 1998, Sol. Phys., 182, 231, doi: 10.1023/A:1005087612053
  * Li et al. (2018) Li, F. Y., Kong, D. F., Xie, J. L., Xiang, N. B., & Xu, J. C. 2018, Journal of Atmospheric and Solar-Terrestrial Physics, 181, 110, doi: 10.1016/j.jastp.2018.10.014
  * Li et al. (2005) Li, K.-J., Gao, P.-X., & Su, T.-W. 2005, Chinese J. Astron. Astrophys., 5, 539, doi: 10.1088/1009-9271/5/5/011
  * Macario-Rojas et al. (2018) Macario-Rojas, A., Smith, K. L., & Roberts, P. C. E. 2018, MNRAS, 479, 3791, doi: 10.1093/mnras/sty1625
  * McKinnon & Waldmeier (1987) McKinnon, J. A., & Waldmeier, M. 1987, Sunspot numbers, 1610-1985 : based on “The sunspot activity in the years 1610-1960”
  * Miao et al. (2020) Miao, J., Wang, X., Ren, T.-L., & Li, Z.-T. 2020, Research in Astronomy and Astrophysics, 20, 004, doi: 10.1088/1674-4527/20/1/4
  * Nandy & Martens (2007) Nandy, D., & Martens, P. C. H. 2007, Advances in Space Research, 40, 891, doi: 10.1016/j.asr.2007.01.079
  * Ohl (1966) Ohl, A. 1966, Soln. Dannye, 12, 84
  * Pesnell & Schatten (2018) Pesnell, W. D., & Schatten, K. H. 2018, Sol. Phys., 293, 112, doi: 10.1007/s11207-018-1330-5
  * Petrovay (2010) Petrovay, K. 2010, Living Reviews in Solar Physics, 7, 6, doi: 10.12942/lrsp-2010-6
  * Petrovay (2020) —. 2020, Living Reviews in Solar Physics, 17, 2, doi: 10.1007/s41116-020-0022-z
  * Petrovay et al. (2018) Petrovay, K., Nagy, M., Gerják, T., & Juhász, L. 2018, Journal of Atmospheric and Solar-Terrestrial Physics, 176, 15, doi: 10.1016/j.jastp.2017.12.011
  * Pishkalo (2019) Pishkalo, M. I. 2019, Sol. Phys., 294, 137, doi: 10.1007/s11207-019-1520-9
  * Podladchikova et al. (2017) Podladchikova, T., Van der Linden, R., & Veronig, A. M. 2017, The Astrophysical Journal, 850, 81
  * Sarp et al. (2018) Sarp, V., Kilcik, A., Yurchyshyn, V., Rozelot, J. P., & Ozguc, A. 2018, MNRAS, 481, 2981, doi: 10.1093/mnras/sty2470
  * Sello (2019) Sello, S. 2019, arXiv e-prints, arXiv:1902.05294.  https://arxiv.org/abs/1902.05294
  * Sheeley Jr (2008) Sheeley Jr, N. 2008, The Astrophysical Journal, 680, 1553
  * Singh & Bhargawa (2017) Singh, A. K., & Bhargawa, A. 2017, Ap&SS, 362, 199, doi: 10.1007/s10509-017-3180-2
  * Singh & Bhargawa (2019) —. 2019, Ap&SS, 364, 12, doi: 10.1007/s10509-019-3500-9
  * Upton & Hathaway (2018) Upton, L. A., & Hathaway, D. H. 2018, Geophys. Res. Lett., 45, 8091, doi: 10.1029/2018GL078387
  * Usoskin & Mursula (2003) Usoskin, I. G., & Mursula, K. 2003, Sol. Phys., 218, 319, doi: 10.1023/B:SOLA.0000013049.27106.07
  * Waldmeier (1935) Waldmeier, M. 1935, Astronomische Mitteilungen der Eidgenössischen Sternwarte Zurich, 14, 105
  * Waldmeier (1961) —. 1961, The sunspot-activity in the years 1610-1960
  * Wang & Sheeley (2009) Wang, Y. M., & Sheeley, N. R. 2009, ApJ, 694, L11, doi: 10.1088/0004-637X/694/1/L11
  * Wilson (1995) Wilson, R. M. 1995, Sol. Phys., 158, 197, doi: 10.1007/BF00680842
  * Wilson et al. (1996) Wilson, R. M., Hathaway, D. H., & Reichmann, E. J. 1996, J. Geophys. Res., 101, 19967, doi: 10.1029/96JA01820
  * Yousef (2003) Yousef, S. 2003, in ESA Special Publication, Vol. 535, Solar Variability as an Input to the Earth’s Environment, ed. A. Wilson, 177–180