The sunspots-number-for-25th-cycle

University of Karachi
Department of Physics
The Sunspots number for 25th
cycle
Prepared by:
Sehrosh Ahmed,
Enrollment# SCI/PHY/KU-36700/2012
Waqas Qamar,
And Waseem Shah
Supervisor:
Dr. Zaheer Uddin
Assistant Professor

Certificate
This is to certify that the work presented in this dissertation entitled “The
Sunspots number for 25th cycle” has been carried out by Sehrosh Ahmed, Waqas
Qamar and Waseem Shah (enrollment# SCI/PHY/KU-36700/2012, SCI/PHY/KU-
36721/2012 and SCI/PHY/KU-36720/2012 respectively), under our supervision and is
accepted in its present form by the department of physics, University of Karachi as
satisfying the dissertation requirement for the degree of Master in Physics.
________________
Dr. Zaheer Uddin
Assistant Professor
Department of Physics
University of Karachi
__________________________
External

Acknowledgment
I am deeply grateful to almighty Allah for His help and blessing that always
involved in the completion of this study, because without His blessing and help this
research was impossible for me. I would also like to thank all of them who gave support
at various stages of this study.
I am greatly thankful to my research supervisor Dr. Zaheer Uddin who granted
his precious time to me; provided his professional guidance and continually motivated
me by his verbal utterances and supervision in this research work.
Finally I will appreciate my beloved parents, brothers, sister, and every one whom
prayers became a source of inspiration for me.

CHAPTER # 01: INTRODUCTION TO SUN:---------------1
1.1. INTRODUCTION:---------------------------------------------------1
1.2. BASIC PARAMETERS OF SUN:------------------------------------1
1.2.1. MASS OF SUN:--------------------------------------------------------------1
1.2.2. DISTANCE:------------------------------------------------------------------2
1.2.3. RADIUS:---------------------------------------------------------------------2
1.2.4. LUMINOSITY:---------------------------------------------------------------3
1.3. LAYERS OF SUN:---------------------------------------------------4
1.3.1. THE SUN’S CORE:----------------------------------------------------------4
1.3.2. THE RADIATIVE ZONE:----------------------------------------------------5
1.3.3. THE CONVECTION ZONE:-------------------------------------------------6
1.3.4. THE PHOTOSPHERE:--------------------------------------------------------6
1.3.5. THE CHROMOSPHERE:-----------------------------------------------------7
1.3.6. THE CORONA:--------------------------------------------------------------7
CHAPTER # 02: SOLAR ACTIVITIES:----------------------8
2.1. SOLAR FLARES:----------------------------------------------------8
2.2. SOLAR WIND:------------------------------------------------------8
2.3. SOLAR PROMINENCE:---------------------------------------------9
2.4. CORONAL MASS EJECTION:--------------------------------------9
CHAPTER #03: THE SUNSPOTS:---------------------------11
3.1. THE SUNSPOTS:--------------------------------------------------11
3.2. MAXIMUM AND MINIMUM:-------------------------------------13
CHAPTER #04: SOLAR CYCLE CURVE FITTED USING
GAUSSIAN DISTRIBUTION:-----------------------------16
4.1. GAUSSIAN DISTRIBUTION:--------------------------------------16
4.2. GAUSSIAN PARAMETERS:---------------------------------------20
CHAPTER #05: SOLAR CYCLE CURVE FITTED USING
WEIBULL DISTRIBUTION:------------------------------23
5.1. WEIBULL DISTRIBUTION:--------------------------------------23

5.2. PROBABILITY DENSITY FUNCTION (PDF):------------------23
5.3. CUMULATIVE DENSITY FUNCTION (CDF):------------------23
5.4. PARAMETERS:----------------------------------------------------23
CHAPTER # 06: PREDICTION TECHNIQUES----------29
6.1. PREDICTION OF NEXT SUNSPOTS CYCLE:-------------------29
6.2. POLYNOMIAL REGRESSION METHOD:-----------------------29
6.3. NEXT SOLAR CYCLE BY GAUSSIAN DISTRIBUTION:--------30
6.4. NEXT SOLAR CYCLE BY WEIBULL DISTRIBUTION:---------34
6.5. AVERAGES:-------------------------------------------------------38
CONCLUSION:--------------------------------------------------42
REFERENCES:--------------------------------------------------43

Abstract:
The Sunspots are the darkest places on the sun’s photosphere. They
are produced due to the strong magnetic field. Solar wind and solar flares
are produced due to Sunspots which have great effect on atmosphere of
space and our earth’s environment. The Sunspots reach at its peak in
every 11 years and then decrease gradually which makes a solar cycle.
Sunspots data are being recorded since 1749. So, yet 24 solar cycles have
been completed. Considering this data, Sunspots for the 25th
cycle is
predicted. In this work we used three methods for the predictions. Each
cycle was simulated using Gaussian distribution, Weibull distribution.
The simulated values are used to predict Sunspots number for the next
cycle. The average of both distribution is also used for the predictions.
All these methods give the shape of the distribution. In order to find the
maximum of the 25th
cycle we used polynomial fitting.

1
Chapter # 01: Introduction to Sun
1.1. Introduction:
The Sun is the only star in our solar system. It is the brightest and biggest
star in our solar system. It has very importance because the weather of space
and solar wind depend on its activity and these make effect on our Earth’s
environment and human life. It is also a valuable source of heat energy and light
for life on the Earth.
The Sun is made up of Hydrogen (70%), Helium (28%), and Carbon,
Nitrogen, Oxygen (1.5%) that’s why it is also called hot gas ball. It has 99% of
the solar system mass with radius 700,000 km (more than a hundred times larger
than the Earth’s). It is 93 million miles away from the Earth. Light need 8
minutes to reach the Earth surface from the Sun surface. Basic parameter will
be discussed in next heading.
1.2. Basic Parameters of Sun:
Some basics parameters of the Sun are discussed which are shown in
table 1.1.
1.2.1. Mass of Sun:
Mass of Sun is measured by Kepler’s law which gives Gmʘ. Mass of Sun
within an error of about 0.15 percent is:
𝑚 𝑜 = (1.989 ± 0.003) × 1030
𝑘𝑔
Due to radiation and solar wind Sun losses 4x109
kg per second and 109
kg per second respectively which are negligible. The approximate value of mass
of the Sun is:
𝑚 𝑜 = 2 × 1030
𝑘𝑔
Which is 333,000 times the mass of the Earth.

2
1.2.2. Distance:
The distance between the Sun and the Earth is not constant because the
orbit of the Earth around the Sun is an ellipse. The distance between the Sun
and the Earth is at minimum, about 1.471x1011
m, around 3 January and is at
maximum, about 1.521x1011
m, around 3 July. The average distance between
Sun and Earth is:
𝐴 𝑜 = 1.5 × 1011
𝑚
The average time “τo” for Sunlight to reach the Earth surface is:
𝜏 𝑜 = 500 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
1.2.3. Radius:
The angular diameter of solar distance is not constant it ranges from 31.6ʹ
to 32.7ʹ. The average value is 32ʹ which is 0.533o
. The radius of Sun is then:
𝑟𝑜 = 696000𝑘𝑚
The average volume of Sun can be calculated using radius:
𝑣 𝑜 =
4
3
𝜋𝑟𝑜
3
𝑣 𝑜 =
4
3
𝜋(696 × 106
)3
𝑣 𝑜 = 1.412 × 1027
𝑚3
The average density of the Sun:
𝜌 𝑜 =
𝑚 𝑜
𝑣 𝑜
𝜌 𝑜 =
2 × 1030
1.412 × 1027
𝜌 𝑜 = 1360.544 𝑘𝑔/𝑚3
𝑜𝑟 1.408 𝑔/𝑐𝑚3

3
1.2.4. Luminosity:
The luminosity is defined as “the average radiation power of the
Sunlight outside the atmosphere of Earth”. The total radiation power “Lʘ” of
the Sun can be determined by using two parameters, solar constant “S” and
average distance between the Sun and the Earth “Aʘ”. The luminosity is given
by:
𝐿ʘ = 4𝜋𝐴ʘ
2
𝑆
S represents the total irradiance at the mean distance of the Earth and is
called solar constant but solar constant is not constant, the value of solar
constant is:
𝑆 = 1366 ± 3 𝑤/𝑚2
The luminosity of the Sun is:
𝐿ʘ = 4𝜋(1.5 × 1011)2
(1366)
𝐿ʘ = 3.86 × 1026
𝑤
Table 1.1: Basic Parameters of Sun
Parameters Approximate value
 Mass of Sun 2×1030
kg
 Distance between Sun and Earth 1.5×1011
m
 Radius of Sun 696×106
m
 Volume of Sun 1.412×1027
m3
 Density of Sun 1.408 g/m3
 Luminosity of Sun 3.86×1026
w
 The Sun age 4.5×109
years

4
1.3. Layers of Sun:
We observed that the Sun is made up of gases but it has different layers
which is shown in fig 1.1. The layers have different densities and different
temperature as we moved from core to outermost surface of the Sun. These
layers are made of hot gases and they are not solid like Earth’s layers. Usually
whole Sun divide into six layers with respect to the different temperature,
densities and energy transformation phenomenon.
1. The Core.
2. The Radiative Zone.
3. The Convection Zone.
4. The Photosphere.
5. The Chromosphere.
6. The Corona.
Fig 1.1: Layers of Sun.
http://ivansc663universe.weebly.co
m/uploads/1/3/3/0/13305579/10832
03.jpg?855
1.3.1. The Sun’s Core:
It is the inner most layer of the Sun in which thermonuclear reactions
take place which creating very high temperature, all energy of the Sun produced
in its core. It has very high density (which is 150 times dense as compared to
water). Energy moves from the core to outward surface of the Sun and
surrounding atmosphere. The temperature of this region is about 27 million
degrees Fahrenheit.

5
The core is a combination of two properties which is good for nuclear
reactions. In core the energy of Hydrogen atom converts in to Helium and it is
possible due to high temperature and pressure. Core is the only part of the Sun
in which energy is produced through fusion 99% of the Sun’s energy is
produced in the core and other parts of the Sun are heated by energy transferred
by layers.
In core intense heated story the internal structure of the atom and all
atoms are broken in to their constituent parts. An atom is made of electron,
proton and neutron. Neutron have no charge therefore neutron do not interacts
with surrounding. The proton and electron remain in the core and derive the
reactions.
The high temperature provides a large amount of energy to the electrons
and protons. And in a result electrons and protons move very quickly. Due to
this motion the electrons and protons are combined with the high density of
plasma. And the particles continuously slam in to one another and create nuclear
reaction. It is fusion combination of particles provides the energy source to the
Sun.
1.3.2. The Radiative Zone:
The outer layer of the core is known as Radiative Zone. This layer is less
dense as compared to core. Here the temperature drops below 3.5 million
degrees Fahrenheit and it is cooler than the core of the Sun.
When the energy is produced in the core of the Sun it needs a way to
travel from the core to the outer region. Therefore, heat energy is transform in
form of radiation. In radiative zone of the Sun temperature is cooler as
compared to the core that’s why some atoms are able to remain intact. These
atoms are able to absorbed energy for a while and after some time emit that
energy. In this way the energy transformation takes place in the form of

6
interaction of surrounding atoms, in the result energy moves slowly outwards.
It takes more than 170,000 years to get out from the radiation zone.
1.3.3. The Convection Zone:
The layer after Radiative zone is known as Convection zone. It is the final
30% of the Sun’s radius. The energy continue to moves outward but the
temperature of this layer is 2 million degrees Fahrenheit and energy comes from
the layers whose temperature is 4.5 million degrees Fahrenheit due to this
abrupt change in temperature atoms absorb energy but the layer is dense that’s
why the atoms do not release it readily and the significant of transfer of energy
by radiation slows down and the energy transform in the form of convection
current of heated and cooled gas toward the surface.
In this zone the bottom of the convection zone which is near to radiative
zone is hotter and the top of the convection zone is cooler. The hotter material
of radiative zone moves upward from bottom of the convection zone to the top
of the convection zone. And at the top of the convection zone temperature is
low that’s why the hotter material is cool down and come back to the bottom
and heat up again. In this manner a rolling motion is produced. In this layer the
energy is transferred much faster as compared to radiative zone.
1.3.4. The Photosphere:
The next layer of the Sun is known as photosphere. In this layer the gases
are thin enough and less dense that’s why we can see it and it is a visible layer
of Sun. The energy comes on the Earth’s surface is release by this layer. The
transformation of energy is so fast and it takes 8 minutes to come on the Earth’s
surface. The Sunspots also appears on this layer which is the cooler regions than
the whole. The temperature of the layer is approx. 10,000 degrees Fahrenheit
and the Sunspots are cooler about 7,800 degrees Fahrenheit. Energy is
transported through the photosphere once again by radiation.

7
1.3.5. The Chromosphere:
The chromosphere is above layer of the photosphere. It is relatively thin
layer of the Sun which is about 2000 km thick. The temperature of this layer is
approximately 50000 degrees Fahrenheit. It is sculpted by magnetic field lines
that restrain the electrically charged solar plasma. In the other parts of the Sun
as we move upward from the core temperature decrease but in chromosphere as
we move upward the temperature increase which is unique and things are heat
up again here in the result huge solar flares and loops of hot gases are produce
which goes up to thousands of miles.
In this layer the transformation of energy is continue in upward direction
in the form of radiation. Hydrogen atoms absorb energy from photosphere and
most of the energy is then emitted as red light. From this region sometimes
material ejection away from the Sun.
1.3.6. The Corona:
After Chromosphere the layer is named as Corona, it is the outermost
layer of the Sun. It is very thin layer therefore it is difficult to observe from the
Earth. Normally we can see it during a total solar eclipse. The ionized elements
within the corona glow in the X-rays and extreme ultraviolet wavelengths by
using this we can also see the images of this layer and make it visible for us. In
during a total solar eclipse its looks like a crown that’s why it is known as
corona. It is very hot layer and the temperature of it is about 4 million degrees
Fahrenheit. The corona stretches far out into space, and in fact sometimes
particles from it reach the Earth’s orbit. It is very thin layer of the Sun and it
start around 10,000 km above the solar photosphere.
The Corona varies in shape and the outward flowing plasma of the corona
is shaped by magnetic field lines called coronal streamers, they extend millions
of miles into space.

8
Chapter # 02: Solar Activities
2.1. Solar Flares:
Solar Flares are rapid localized brightening in the lower atmosphere.
They are storms that appear as explosive bright spots on the surface of the Sun
and come from the release of the magnetic energy associated with Sunspots.
Most of the flares occur in active regions where magnetic fields concentrate and
are complex. They are located at where the polarity of magnetic fields reverses.
Flares occur when considerable amounts of magnetic field energy are suddenly
converted to heat and light energy. Solar flare is also a kind of blast in which
electrically charged particles (electrons, protons and heavier particles) are
accelerated outward in the solar wind. They can last from minutes to hours. The
radiation from one of these blasts can have 1027
- 1032
ergs energy in tens of
minutes which is equivalent of ten million volcanic eruptions.
A flares produces enhanced emission in all wavelengths across the EM
spectrum, including radio, optical, ultraviolet, soft X-rays, hard X-rays and ɣ-
rays.
Solar flares are the largest explosion in the Sun and are so powerful that
it can disturb our electronic devices and signals on Earth.
X-rays from flares can interfere with radio communications and UV
radiation can make on satellites in orbit. In 1989 a solar storm effected the
telecommunication on the Earth.
2.2. Solar Wind:
The Sun gradually losses of its mass in the form of high speed protons
and electrons from the outer layers. This flux of particles is called solar wind.
This flux varies constantly in speed, temperature and density. Solar winds are

9
caused by variance in the magnetic field of the Sun. The solar wind is visible
when the light of the Sun is blocked such as during eclipse.
The outer layers of the Sun reach a temperature up to 2 million degree
Fahrenheit (1.1 million Celsius). At this temperature the Sun’s gravity cannot
hold the fast moving particles (electrons and protons) and these particles escape
from Sun.
Affecting on Earth:
The solar wind is emitted in all direction. Some of the solar wind reach
our planet. The Earth’s magnetic field behaves like a shell and redirect the
particles of solar wind.
Increase in solar flares and in solar wind increase the radiation levels and
the satellites radiations. Even some radiations can also interfere the electrons
and mobile phone communications on Earth.
2.3. Solar Prominence:
A Solar prominence (also known as a filament) is a large, bright feature
extending outward from the Sun’s surface. Solar prominences are the arc of gas
that erupts from the surface of the Sun. When they are observed outside of the
solar limb, they appear as bright features because they reflect Sunlight towards
us. Prominences are anchored to the Sun’s surface in the photosphere, and
extend outwards into the Sun’s corona. While the corona consists of extremely
hot ionized gases, known as plasma, that do not emit much visible light,
prominences contain much cooler plasma. They have relatively cool gas
(60,000 – 80,000 K) compared with the low density gas of the corona.
2.4. Coronal mass ejection:
The magnetic field lines which form solar flares sometimes become so
wrapped like rubber bands under tension. They snap and break and then

10
reconnect at the other points. Due to which gaps form which cannot hold the
Sun’s plasma on its surface. The plasma explodes from the surface which is
known as coronal mass ejection.
It takes several hours to come out from the Sun but after escaping it
moves at speed up to 1000km (more than 7 million per hour). One of the fastest
CMEs is recorded in 2012 which travelled at a speed of 6.48 million to 7.92
million mph.
Sun ejects CME in all directions. But when these ejections are directly
towards Earth, it looks like a roughly circular halo surrounding the Sun. This
type of CME is called Halo CME. Halo CME are shoot out at right angles to
Earth and have more chances to impact the Earth.

11
Chapter #03: The Sunspots
3.1. The Sunspots:
The Sun may look unchanged, but it exhibited different phenomena such
as the prominence, flares, coronal mass ejection, solar wind, and the Sunspots.
The Sunspots are one of the important phenomenon in all of these, because all
phenomena depend on it.
The Sunspots are the dark place on the Sun’s photosphere. They are the
regions where most intense magnetic fields are present on the Sun’s surface.
The Sunspots are cooler than the whole region of the photosphere, they are
about 1500 k cooler so appear to be darker than photosphere. They are the only
magnetic structures on the Sun’s surface which can visible to naked eye.
The magnetic field lines in the Sun emerge at the surface of the Sun as
small bundles called pores. The two or more than two pores interact with each
other and squeeze the plasma between them and form a long structure called
light bridge. These bridges merge to form a single long bridge which is called
Sunspot.
Sunspots are temporary phenomenon on the photosphere of the Sun. A
given Sunspots can have a lifetime ranging from a few hours to a few months.
They consist of a dark umbra at a temperature T~4500 K which is less than in
the photosphere (T~6000K).
The Sunspots look dark because the strong magnetic fields block the
upward flow of energy to the surface and prevents hotter gas from entering these
region, and they are radiating less energy into space. They are usually appeared
in pairs. The two Sunspots of a pair have different polarities, one would be a
magnetic north and the other is a magnetic south, and can be joined by magnetic
lines. The Sunspots are the magnetic regions of the Sun which are the thousands
of time stronger than the Earth’s magnetic field. The Sunspot can be very large,

12
up to 50,000 kilometers in diameter. The field is strongest in the darkest part of
the Sunspots which is called Umbra. And in the lighter part field is weak which
is called Penumbra. Over all Sunspots have a magnetic field of 1000 times
stronger compared to surrounding.
The rotation of the Sun was first detected the motion of these Sunspots
was observed. More than 150 years ago it was found that their number varies
with a period of about eleven years. This Sunspots cycle is nowadays called
solar activity cycle because all energetic phenomena on the Sun vary with this
cycle (at least their occurrence).
The Sunspots are one of the strongest pieces of evidence for the solar
cycle which describes a variation in solar activity over an 11 years’ period.
The graph 3.1a and 3.1b show the 24 solar cycles of the Sunspots. We
have used unsmoothed and smoothed data for these graphs respectively.
We have calculated smoothed data from the unsmoothed data by using
formula as given in Hathaway Wilson Reichmann 1999 (2),
𝑅 𝑛 =
1
24
∑ 𝑅 𝑛+𝑖 +
1
24
∑ 𝑅 𝑛+𝑖
6
𝑖=−5
5
𝑖=−6
Graph 3.1a: Unsmoothed graph.

13
Graph 3.1b: Smoothed graph.
3.2. Maximum and Minimum:
With the help of data, we have found maximum and minimum Sunspots
number which are represented by Emax and Emin and given in table 3.1.
Table 3.1a: Maximum Sunspots number.
Cycle Year Month Emax
1 1761 5 178.7
2 1769 10 263.7
3 1778 5 398.2
4 1787 12 290
5 1804 10 103.8
6 1817 3 160.3
7 1830 4 177.3
8 1836 12 343.8
9 1847 10 300.6
10 1860 7 221.9
11 1870 5 293.6
12 1882 4 159.6

14
13 1893 8 215.4
14 1907 2 180.3
15 1917 8 257.7
16 1929 12 179.9
17 1938 7 275.6
18 1947 5 285
19 1957 10 359.4
20 1969 3 192.3
21 1979 9 266.9
22 1989 6 284.5
23 2000 7 244.3
24 2014 2 146.1
Table 3.1b: Minimum Sunspots number.
Cycle Year Month Emin
1 1754 1 0
2 1766 6 5
3 1775 2 0
4 1784 7 10
5 1798 7 0
6 1813 1 0
7 1824 11 0
8 1833 6 1.7
9 1843 2 5.9
10 1855 9 0
11 1867 1 0
12 1879 3 0
13 1889 11 0.3

15
14 1902 4 0
15 1913 6 0
16 1924 1 0.8
17 1933 12 0.4
18 1944 4 0.4
19 1954 6 0.4
20 1964 7 4.8
21 1976 7 2.9
22 1986 6 0.6
23 1996 10 0.7
24 2009 8 0
25 2013 9 54.5

16
Chapter #04: Solar Cycle Curve Fitted Using Gaussian
Distribution
4.1. Gaussian Distribution:
Gaussian Distribution is very commonly used in statistics to represent
real valued random variables and is normally called “Normal Distribution”
and also called “Bell Curve” because the probability density function has bell
shape Fig 4.1.
Fig 4.1: Normal distribution has bell shape curve.
A random variable X is normally distributed with mean µ and variance
σ2
if its probability distribution function is;
𝑓(𝑥) =
1
√2𝜋𝜎2
exp [
−(𝑥 − 𝜇)2
2𝜎2
]
Where:
 µ is the mean of x and is given by:

17
𝜇 =
∑ 𝑓𝑥
∑ 𝑓
𝜇 = ∫ 𝑥𝑓(𝑥)𝑑𝑥
∞
−∞
 σ2
is the variance of x and is given by:
𝜎2
=
∑ 𝑓(𝑥 − 𝜇)2
∑ 𝑓
𝜎2
= ∫ (𝑥 − 𝜇)2
𝑓(𝑥)𝑑𝑥
∞
−∞
Here we have tried to fit Gaussian distribution in solar cycle curve. We
have fitted Gaussian distribution in all 23 periods of solar cycle. With the help
of Gaussian distribution we will try to predict the next cycle maxima. In Fig 4.2
the graphs of solar cycle and Gaussian distribution are given in these graphs.
There are two series red and blue. Blue series indicates solar cycle, where red
series indicates Gaussian distribution:
Fig 4.2: Graphs between solar cycle and probability density function.

20
4.2. Gaussian Parameters:
The mean “µ” and the variance “σ2
” are the parameters of the Gaussian
distribution function. The Gaussian parameters will help us to predict the 24th
solar cycle maxima. We have done some work to find the relation between solar
cycle and Gaussian parameters. The Gaussian parameters for all 23 solar cycle
are given in Table 4.1. In Fig 4.3 the graphs between Gaussian parameters.
Table 4.1: Gaussian parameters
Periods Year
Durations
(Years) Mean (µ)
Standard
Deviation
(σ)
Variance
(σ2
)
1 1755 11 75.12 28 784
2 1766 9 56.45 22.8 519.84
3 1775 10 43 11 121
4 1785 13 39 31 961
5 1798 12 71.63 31 961
6 1810 13 80 29.9 894.01
7 1823 11 77.24 27.29 744.95
8 1834 9 40 16 256
9 1843 13 68 26 676
10 1856 11 51 21 441
11 1867 12 49 21 441
12 1879 11 53.5 25 625
13 1890 11 45 22 484

21
14 1901 12 70 26.82 719.3
15 1913 10 59 21 441
16 1923 10 62.12 25.07 628.75
17 1933 11 60 20 400
18 1944 10 52 22 484
19 1954 10 50 21 441
20 1964 12 62 28 784
21 1976 10 56 21 441
22 1986 10 55 22 484
23 1996 12 63 27 729
Sum 253 1338.07 545.89 13460.84
Averages 11 58.18 23.73 585.25
Fig 4.3: Graphs between Gaussian parameters.
Fig 4.3a: Graph between Periods and Mean “µ”.
Fig 4.3b: Graph between Periods and σ.

22
Fig 4.3c: Graph between Periods and Variance “σ2
”.
Fig 4.3d: Graph between Periods, µ and σ.
Fig 4.3e: Graph between Periods, µ and σ2
.

23
Chapter #05: Solar Cycle Curve Fitted Using Weibull
Distribution
5.1. Weibull Distribution:
Waloddi Weibull, Swedish physicist introduced the distribution function
in 1939 known as Weibull Distribution Function which is the most popular
model in statistics and is the most widely used distribution.
5.2. Probability Density Function (PDF):
The random variable “X” with parameters k and λ is said to be Weibull
distributed if its probability density function is:
𝑓(𝑥; 𝜆, 𝑘) = {
𝑘
𝜆
(
𝑥
𝜆
)
𝑘−1
𝑒
−(
𝑥
𝜆
)
𝑘
, 𝑥 ≥ 0
0, 𝑥 < 0
5.3. Cumulative Density Function (CDF):
The random variable “X” with parameters k and λ is said to be Weibull
distributed if its cumulative distribution function is:
𝑓(𝑥; 𝜆, 𝑘) = {1 − 𝑒
−(
𝑥
𝜆
)
𝑘
, 𝑥 ≥ 0
0, 𝑥 < 0
5.4. Parameters:
The distribution function has two parameters “k” and “λ”. They are shape
factor and scale factor respectively. Fig 5.1 shows the effects of the parameter.
Shape parameter has the greatest effect on the distribution. Scale parameter
effects the height and width of the distribution.

24
Fig 5.1a: Effects of shape factor on
distribution.
Fig 5.1b: Effect of scale factor on
distribution.
Note: If we increase the scale factor “λ” the height will decrease and width will increase and
if we decrease the scale factor “λ” the height and width will increase and decrease respectively.
Now, we have fitted Weibull distribution in all 23 periods of solar cycle.
Fig 5.2 shows the fitting of Weibull distribution and solar cycle.
Fig 5.2: Graphs of Weibull distribution fitted in solar cycle.

27
In Table 5.1, the Weibull distribution parameters from the solar cycle are
given.
Table 5.1: Parameters of Weibull distribution.
Periods Year
Durations
(Years)
Scale
Factor (ʎ)
Shape
Factor (k)
1 1755-1766 11 85 3
2 1766-1776 10 61 3
3 1776-1785 9 37 3
4 1785-1797 12 42 3
5 1797-1811 14 92 3
6 1811-1823 12 79 3
7 1823-1835 12 88 3
8 1835-1844 9 30 3
9 1844-1856 11 69 3
10 1856-1868 12 62 3
11 1868-1879 11 42 3
12 1879-1890 11 62 3
13 1890-1902 12 50 3
14 1902-1914 12 69 3
15 1914-1923 9 53 3
16 1923-1935 12 70 3
17 1934-1946 12 65 3
18 1944-1956 12 62 3
19 1954-1966 12 62 3
20 1965-1977 12 62 3
21 1975-1987 12 78 3
22 1987-1999 13 50 3

28
23 1997-2011 14 61.8 3
Sum 266 1431.8 69
Averages 11.56 62.25 3
Note: The shape factor remain same through all cycles.
Fig 5.3: Graph between solar periods and scale factors.

29
Chapter # 06: Prediction Techniques
6.1. Prediction of Next Sunspots Cycle:
There are various methods to predict the number of Sunspots but we have
used polynomial regression method to predict maximum Sunspots number and
then with the help of Gaussian and Weibull distribution function we have tried
to predict the Sunspots cycle.
6.2. Polynomial Regression Method:
By using 4th
order polynomial, we have work on all 24 maximum
Sunspots number, by using all we have tried to predict 25th
maxima and we
found Emax=100.16 which will prominent in next cycle, according to our
prediction. Which shown in table 6.1.
Table 6.1: Predicted Sunspots number
Cycle Year Month Emax Predicted
1 1761 5 178.7 226.87
2 1769 10 263.7 249.71
3 1778 5 398.2 260.82
4 1787 12 290 263.2
5 1804 10 103.8 259.49
6 1817 3 160.3 252.04
7 1830 4 177.3 242.86
8 1836 12 343.8 233.63
9 1847 10 300.6 225.68
10 1860 7 221.9 220.04
11 1870 5 293.6 217.40
12 1882 4 159.6 218.13
13 1893 8 215.4 222.24

30
14 1907 2 180.3 229.46
15 1917 8 257.7 239.16
16 1929 12 179.9 250.38
17 1938 7 275.6 261.85
18 1947 5 285 271.95
19 1957 10 359.4 278.74
20 1969 3 192.3 279.97
21 1979 9 266.9 273.02
22 1989 6 284.5 254.99
23 2000 7 244.3 222.61
24 2014 2 146.1 172.3
25 2024 11 100.16
6.3. Next Solar Cycle by Gaussian distribution:
In chap 4 by using Gaussian distribution on solar cycle we found that the
average mean and average variance are 58.177 and 585.25 respectively, by
using these two parameters we have predicted Sunspots number for next solar
cycle which is shown in graph 6.1 and the predicted Sunspots number for next
cycle are given in table 6.2.
Graph 6.1: Graph of predicted solar cycle.

31
Table 6.2: Predicted Sunspots number for next cycle.
Year Month
Sunspots
Number
2016 12 5.501777
2017 1 6.084133
2017 2 6.716198
2017 3 7.400776
2017 4 8.140669
2017 5 8.93865
2017 6 9.797445
2017 7 10.7197
2017 8 11.70797
2017 9 12.76468
2017 10 13.89207
2017 11 15.09221
2017 12 16.36696
2018 1 17.7179
2018 2 19.14633
2018 3 20.65322
2018 4 22.23919
2018 5 23.90448
2018 6 25.64889
2018 7 27.4718
2018 8 29.37207
2018 9 31.34808
2018 10 33.3977
2018 11 35.51821
2018 12 37.70636
Year Month
Sunspots
Number
2019 1 39.95833
2019 2 42.26968
2019 3 44.63543
2019 4 47.04998
2019 5 49.50719
2019 6 52.00033
2019 7 54.52215
2019 8 57.06487
2019 9 59.62025
2019 10 62.17958
2019 11 64.73376
2019 12 67.27332
2020 1 69.78852
2020 2 72.26935
2020 3 74.70562
2020 4 77.08707
2020 5 79.40334
2020 6 81.64415
2020 7 83.7993
2020 8 85.85879
2020 9 87.81288
2020 10 89.65214
2020 11 91.36758
2020 12 92.9507
2021 1 94.39353

32
Year Month
Sunspots
Number
2021 2 95.68874
2021 3 96.82968
2021 4 97.81043
2021 5 98.62588
2021 6 99.27175
2021 7 99.74462
2021 8 100.042
2021 9 100.1623
2021 10 100.1048
2021 11 99.87
2021 12 99.45899
2022 1 98.874
2022 2 98.11811
2022 3 97.19531
2022 4 96.11041
2022 5 94.86907
2022 6 93.47767
2022 7 91.94331
2022 8 90.27374
2022 9 88.47728
2022 10 86.56277
2022 11 84.53948
2022 12 82.41705
2023 1 80.20539
2023 2 77.91464
2023 3 75.55508
Year Month
Sunspots
Number
2023 4 73.13702
2023 5 70.67079
2023 6 68.1666
2023 7 65.63453
2023 8 63.08442
2023 9 60.52586
2023 10 57.96807
2023 11 55.41989
2023 12 52.88976
2024 1 50.38562
2024 2 47.9149
2024 3 45.48452
2024 4 43.10084
2024 5 40.76964
2024 6 38.49612
2024 7 36.28492
2024 8 34.14007
2024 9 32.06504
2024 10 30.0627
2024 11 28.13542
2024 12 26.28499
2025 1 24.5127
2025 2 22.81937
2025 3 21.20534
2025 4 19.67052
2025 5 18.21442

33
Year Month
Sunspots
Number
2025 6 16.8362
2025 7 15.53466
2025 8 14.30831
2025 9 13.1554
2025 10 12.07394
2025 11 11.06172
2025 12 10.11639
2026 1 9.235437
2026 2 8.416245
2026 3 7.656113
2026 4 6.952282
2026 5 6.301957
2026 6 5.702332
2026 7 5.15061
2026 8 4.644017
2026 9 4.179824
2026 10 3.755357
2026 11 3.368011
2026 12 3.015261
2027 1 2.694668
2027 2 2.40389
Year Month
Sunspots
Number
2027 3 2.140687
2027 4 1.90292
2027 5 1.688562
2027 6 1.495694
2027 7 1.322505
2027 8 1.167296
2027 9 1.028475
2027 10 0.904556
2027 11 0.794156
2027 12 0.695995
2028 1 0.608884
2028 2 0.531732
2028 3 0.463532
2028 4 0.403362
2028 5 0.350381
2028 6 0.303819
2028 7 0.262977
2028 8 0.227222
2028 9 0.19598
2028 10 0.168733
2028 11 0.145017

34
The all 24 cycles with predicted cycle are shown in graph 6.2.
Graph 6.2a: Unsmoothed graph of
all 24 cycles with predicted cycle.
Graph 6.2b: Smoothed graph of all
24 cycles with predicted cycle.
6.4. Next Solar Cycle by Weibull distribution:
In chap 5 by using Weibull distribution on solar cycle we found that the
average scale and average shape factor are 62.25217 and 3 respectively, by
using these two parameters we predicted Sunspots number for next solar cycle
which is shown in graph 6.3.
The predicted Sunspots number for next cycle by using Weibull
distribution are given in table 6.3.

35
Table 6.3: Predicted Sunspots number for next cycle.
Year Month
Sunspots
Number
2016 12 0.065975
2017 1 0.263894
2017 2 0.593714
2017 3 1.05533
2017 4 1.648536
2017 5 2.372996
2017 6 3.228211
2017 7 4.213487
2017 8 5.3279
2017 9 6.570269
2017 10 7.939125
2017 11 9.43268
2017 12 11.0488
2018 1 12.78497
2018 2 14.63829
2018 3 16.60541
2018 4 18.68258
2018 5 20.86554
2018 6 23.14957
2018 7 25.52947
2018 8 27.9995
2018 9 30.55345
2018 10 33.18455
2018 11 35.88557
2018 12 38.64873
Year Month
Sunspots
Number
2019 1 41.46577
2019 2 44.32794
2019 3 47.22603
2019 4 50.15039
2019 5 53.09095
2019 6 56.03727
2019 7 58.97859
2019 8 61.90382
2019 9 64.80166
2019 10 67.6606
2019 11 70.46902
2019 12 73.2152
2020 1 75.88745
2020 2 78.47413
2020 3 80.96374
2020 4 83.34499
2020 5 85.60689
2020 6 87.73882
2020 7 89.73058
2020 8 91.57252
2020 9 93.25557
2020 10 94.77132
2020 11 96.11211
2020 12 97.27109
2021 1 98.24227

36
Year Month
Sunspots
Number
2021 2 99.02056
2021 3 99.60188
2021 4 99.98313
2021 5 100.1623
2021 6 100.1384
2021 7 99.9115
2021 8 99.48295
2021 9 98.85502
2021 10 98.03115
2021 11 97.01584
2021 12 95.81463
2022 1 94.43406
2022 2 92.88164
2022 3 91.16575
2022 4 89.29562
2022 5 87.28125
2022 6 85.13329
2022 7 82.86301
2022 8 80.48217
2022 9 78.00292
2022 10 75.43774
2022 11 72.79932
2022 12 70.10043
2023 1 67.35387
2023 2 64.57234
2023 3 61.76835
Year Month
Sunspots
Number
2023 4 58.95412
2023 5 56.14151
2023 6 53.34193
2023 7 50.56627
2023 8 47.82482
2023 9 45.1272
2023 10 42.48236
2023 11 39.89847
2023 12 37.38294
2024 1 34.94236
2024 2 32.58253
2024 3 30.30839
2024 4 28.12407
2024 5 26.0329
2024 6 24.03741
2024 7 22.13935
2024 8 20.33976
2024 9 18.63896
2024 10 17.03665
2024 11 15.53188
2024 12 14.12319
2025 1 12.8086
2025 2 11.58568
2025 3 10.45162
2025 4 9.403257
2025 5 8.437176

37
Year Month
Sunspots
Number
2025 6 7.549718
2025 7 6.737052
2025 8 5.995225
2025 9 5.320202
2025 10 4.707915
2025 11 4.154296
2025 12 3.655318
2026 1 3.207027
2026 2 2.805563
2026 3 2.447192
2026 4 2.128321
2026 5 1.845515
2026 6 1.595512
2026 7 1.375227
2026 8 1.181766
2026 9 1.012421
2026 10 0.864677
2026 11 0.736208
2026 12 0.624871
2027 1 0.528705
2027 2 0.445924
Year Month
Sunspots
Number
2027 3 0.374906
2027 4 0.314187
2027 5 0.262451
2027 6 0.21852
2027 7 0.181347
2027 8 0.150001
2027 9 0.12366
2027 10 0.101604
2027 11 0.083201
2027 12 0.0679
2028 1 0.055224
2028 2 0.044759
2028 3 0.036152
2028 4 0.029099
2028 5 0.023339
2028 6 0.018654
2028 7 0.014856
2028 8 0.011789
2028 9 0.009321
2028 10 0.007343
2028 11 0.005764

38
The all 24 cycles with predicted cycle are shown in graph 6.4.
Graph 6.4a: Unsmoothed graph of
all 24 cycles with predicted cycle.
Graph 6.4b: Smoothed graph of all
24 cycles with predicted cycle.
6.5. Averages:
We have taken the averages of Gaussian and Weibull for the next cycle,
the average values are show in table 6.4.
Table 6.4: Averages of Gaussian and Weibull.
Year Month
Sunspots
Number
2016 12 2.783876
2017 1 3.174014
2017 2 3.654956
2017 3 4.228053
2017 4 4.894602
2017 5 5.655823
2017 6 6.512828
2017 7 7.466595
2017 8 8.517937
2017 9 9.667472
2017 10 10.9156
2017 11 12.26245
2017 12 13.70788
Year Month
Sunspots
Number
2018 1 15.25143
2018 2 16.89231
2018 3 18.62931
2018 4 20.46088
2018 5 22.38501
2018 6 24.39923
2018 7 26.50063
2018 8 28.68578
2018 9 30.95076
2018 10 33.29113
2018 11 35.70189
2018 12 38.17755
2019 1 40.71205

39
Year Month
Sunspots
Number
2019 2 43.29881
2019 3 45.93073
2019 4 48.60018
2019 5 51.29907
2019 6 54.0188
2019 7 56.75037
2019 8 59.48435
2019 9 62.21096
2019 10 64.92009
2019 11 67.60139
2019 12 70.24426
2020 1 72.83799
2020 2 75.37174
2020 3 77.83468
2020 4 80.21603
2020 5 82.50512
2020 6 84.69148
2020 7 86.76494
2020 8 88.71566
2020 9 90.53422
2020 10 92.21173
2020 11 93.73985
2020 12 95.1109
2021 1 96.3179
2021 2 97.35465
2021 3 98.21578
Year Month
Sunspots
Number
2021 4 98.89678
2021 5 99.39408
2021 6 99.70505
2021 7 99.82806
2021 8 99.76247
2021 9 99.50865
2021 10 99.068
2021 11 98.44292
2021 12 97.63681
2022 1 96.65403
2022 2 95.49987
2022 3 94.18053
2022 4 92.70302
2022 5 91.07516
2022 6 89.30548
2022 7 87.40316
2022 8 85.37795
2022 9 83.2401
2022 10 81.00026
2022 11 78.6694
2022 12 76.25874
2023 1 73.77963
2023 2 71.24349
2023 3 68.66171
2023 4 66.04557
2023 5 63.40615

40
Year Month
Sunspots
Number
2023 6 60.75427
2023 7 58.1004
2023 8 55.45462
2023 9 52.82653
2023 10 50.22521
2023 11 47.65918
2023 12 45.13635
2024 1 42.66399
2024 2 40.24872
2024 3 37.89646
2024 4 35.61246
2024 5 33.40127
2024 6 31.26677
2024 7 29.21214
2024 8 27.23992
2024 9 25.352
2024 10 23.54968
2024 11 21.83365
2024 12 20.20409
2025 1 18.66065
2025 2 17.20253
2025 3 15.82848
2025 4 14.53689
2025 5 13.3258
2025 6 12.19296
2025 7 11.13585
Year Month
Sunspots
Number
2025 8 10.15177
2025 9 9.237801
2025 10 8.390925
2025 11 7.608008
2025 12 6.885854
2026 1 6.221232
2026 2 5.610904
2026 3 5.051653
2026 4 4.540301
2026 5 4.073736
2026 6 3.648922
2026 7 3.262918
2026 8 2.912892
2026 9 2.596123
2026 10 2.310017
2026 11 2.05211
2026 12 1.820066
2027 1 1.611687
2027 2 1.424907
2027 3 1.257796
2027 4 1.108553
2027 5 0.975507
2027 6 0.857107
2027 7 0.751926
2027 8 0.658648
2027 9 0.576068

41
Year Month
Sunspots
Number
2027 10 0.50308
2027 11 0.438679
2027 12 0.381947
2028 1 0.332054
2028 2 0.288246
2028 3 0.249842
2028 4 0.216231
Year Month
Sunspots
Number
2028 5 0.18686
2028 6 0.161236
2028 7 0.138916
2028 8 0.119505
2028 9 0.10265
2028 10 0.088038
2028 11 0.075391
We have shown all 24 cycles with average predicted cycles in graph 6.5.
Graph 6.5a: Unsmoothed graph. Graph 6.5b: Smoothed graph.

42
Conclusion:
The Sunspots is the most important phenomena which have great effects
on our Earth. There is no formula which can give accurate values of Sunspots
because it is a random process depends on various conditions on surface of Sun.
However, various forecasting methods have been used to predict Sunspots
number. In this work we used simulation technique to simulate each cycle using
two different distributions. By using Weibull distribution, we conclude that one
factor which is shape factor is found to be constant, and its constant value is 3;
only scale parameter changes depending upon the maximum Sunspots value in
that particular cycle. Once the maximum Sunspots number for next cycle is
predicted the shape can be determined by simulated Weibull parameter. In this
way we found the Sunspots number for cycle 25.
We also used Gaussian distribution to simulate the Sunspots cycle,
contrary to Weibull distribution both the parameter of Gaussian distribution
vary in each cycle. We also used polynomial fitting to predict the Sunspots
number, we tried 2nd to 8th degree polynomial but none of them gave better
result. In last we used 𝑥𝑒
(−
𝑥
3000
)
2
that gave us better result. Using the same
function we predicted Sunspots number for 25th cycle using Weibull, Gaussian
and average of both the distribution which are shown in Tables 6.2, 6.3 and 6.4.
We suggest further work to be done to find more accurate function for
the predictions.

43
References:
1. David H, Hathaway, Robert M. Wilson, and Edwin J, Reichmann, A
Synthesis Of Solar Cycle Prediction Techniques, Journal Of
Geophysical, Research, Vol, 104. NO. A 10, October 1, 1999.
2. Yingna Su, Magnetic Shear In Two-Ribbon Solar Flares, M.S.,
Purple Mountain Observatory (2004).
3. Eduard Kontar, Solar Flares and Particle Acceleration, STFC
Summer School, Armagh, 2012.
4. Mahmoud E. Yousif, the Sunspots Mechanism, PACS No: 96.60.-J,
96.60. Qd, 96.60.Hv.
5. Peter V.Foukal, the Variable Sun, Scientific American February
1990, P26 To 27.
6. Arnold Hanslmeier, the Sun and Space Weather, Univ.-Platz 5, A-
8010 Graz, P6 To 7.
7. Mursula K, Manoharan P, Nandy D, Tanskanen E & Verronen P:
Long-Term Solar Activity And Its Implications To The Heliosphere,
Geomagnetic Activity, And The Earth’s Climate. J. Space Weather
Space Clim., 2013, 3, A21, Published By EDP Sciences 2013.
8. Aswathy Mary Prince *, Dr. Sanish Thomas **, Er. Ravi Jon **, Dr.
D.P Jayapandian, A Study On Midrange Periodicity Of Sunspots
Number During Solar Cycles 21, 22, 23 & 24, International Journal
Of Scientific And Research Publications, Volume 3, Issue 2,
February 2013 ISSN 2250-3153.
9. Nestor Lasso, What Is The Sun Made Of, AST1022L – Spring 2012
University Of Florida.
10. Demystifying Scientific Data: RET 2006, Rev 3, Solar Activity And
Space Weather: Introduction To The Sun, Solar Activity, And The
SRT.

44
11. NASA’S HELIOPHYSICS DIVISION OF THE SCIENCE MISSION
DIRECTORATE, Mysteries Of The Sun, National Aeronautics And
Space Administration.
12. Www.Eyeonthesky.Org/Activities_Pdf/XxSun_LayersoftheSunws.Pdf
13. C. Julian Chen ,John Wiley & Sons, Inc., Hoboken, New Jersey,
"Physics of Solar Energy", 2011, [Chap#3, Page#67-69].
14. Hannu Koskinen and Rami Vainio, Spring 2011, "Lectures on Solar
Physics: From the core to the heliopause" [Chap#1, Page#7-11]
15. Maria Isabel Ribeiro, February 2004, "Gaussian Probability
Density Functions".
16. The Fidelis Group, LLC, March 6, 2014, "The Weibull Distribution'.
17. http://solar.physics.montana.edu
18. http://image.gsfc.nasa.gov/poetry/ask/a11534.html
19. http://Earthobservatory.nasa.gov/Features/SolarMax/solarmax_2.ph
p
20. http://curious.astro.cornell.edu/about-us/50-our-solar-system/the-
Sun/solar-cycle-and-Sun
21. http://www.popsci.com/what-is-Sunspots
22. http://www.thetimenow.com/astronomy/solar-wind.php

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