A study of solar cycle
Author:
Sourabh Singh Chauhan
School of Physcial Sciences
NISER,Bhubneswar
Supervisor:
Dr. Dipankar Banarjee
Associate Professor
Indian Institute of
Astrophysics, Bangalore
A study of solar cycle
Sourabh Singh Chauhan
Project guide:- Dr. Dipankar Banarjee
Abstract
Solar actvity is one of the widely observed phenomena since the beginning
of astronomy. Theoretical understanding of the solar activity comes for the dynamo models. In the initial part of we disucss basics of Magnetohydrodynamics
and kinetic dynamo model. Observations have shown that sunspot area is a better
proxy for understanding the solar activity than sunspot number. From the recently
digitized photographic images of Kodaikanal data (arXiv:1212.4776) we have analyzed the sunspot area data obtained from six solar cycles (i.e. from year 1923 to
1986). We have obtained high correlation of daily averaged sunspot area, monthly
averaged sunspot area (0.93) and yearly sunspot area(0.96) with Greenwich observatory data. Along with these regularities many irregularities exist in solar cycle.
Irregularities in solar cycle arise due to fluctuations in meridional circulation and
fluctuations in Babcock Leighton mechanism. In the present paper we will discuss
one irregularity named as Waldmeier effects (WE1 and WE2). Observations hint
on validation of high diffusivity model (arXiv:1008.0824). According to high diffusivity model longer cycle allows diffusivity to act for a longer time which results
in low amplitude and weaker cycle.
1
Introduction
Astronomical observations of the sky had led to many new discoveries in Science. For
several years astronomers have been pointing their telescopes to the full of several objects
like galaxies, globular clusters, nebulae, Magellanic cloud, pulsars etc. Careful observation of data obtained and appropriate interpretation of data has given us opportunity
to answer fundamental questions like how universe was created and how it evolves over
the passage of time. In order to understand how galaxies evolve, we need to understand
how stars within the galaxy evolve. By observing several stars standard models of stellar
evolution have been made which makes predictions quite accurately. In order to understand stars in more details we should first study our closest star -the sun and generalize
results for other stars. Sun is hot plasma confined within a volume. Scientists are trying
to confine plasma to use it as a energy source. The sun is an observatory for them to
study confine plasma in such a large volume. A complete study of the sun requires combination of several areas of physics. It requires the understanding of nuclear processes to
understand nuclear synthesis happening in the core of sun, understanding of seismology
to determine the structure of solar interior, knowledge of spectroscopy to understand
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the behavior sun’s plasma at different levels, understanding of magnetohydrodynamics
to understand plasma dynamics and many other fields.
Here we present some theoretical understanding of the sun and solar cycle accompanied
with results of observations.
2
Theoretical backgrounds
The sun is a star and consists of several charged, neutral particles moving in specific way
to give an organized structure and complex dynamics to sun. Motion of these particles
leads to several unique features on the sun. Although most of the observed features of sun
can be (except plasma oscillations and other small length scale and time scale properties)
described by macroscopic theories, in order to understand the dynamics completely one
should have an understanding of microscopic theory as well. In this section we will give
a brief introduction to macroscopic theory of charged fluids which will be useful later
when we try to explain features of sunspots.
Macroscopic description of magneto-fluid is called magnetohydrodynamics. If we observe a system in length scales larger than Debye length and time scales much larger
than the inverse of plasma frequency, we can consider an assembly of electrons, ions
and neutrals as continuous entity known as magneto fluid. MHD description seems to
require a lot of extra conditions but we will later see that most of the astronomical
observations can be described by that. In order to describe the the state of the a fluid in
MHD in addition to density ρ(x), T (x) we need magnetic field (B(x)).(As E(x)) is not
an independent variable. Therefore in Navier stokes equation we need to add magnetic
body force which leads to
1
B2
(B.∇)B
∂v
+ v.(∇v) = F − ∇(p +
)+
+ ν∇2 v
∂t
ρ
8π
4πρ
In energy equation an ohmic term is added. However this equation is not important
if we have a system where compressibility does not play much role.
Maxwell’s equation leads to induction equation:∂B
= ∇ × (v × B) + λ∇2 B
∂t
where
c2
4πσ
is magnetic diffusivity. For astrophysical length and time scales diffusivity is negligible
(ideal MHD limit). Leading to the equation
λ=
∂B
= ∇ × (v × B)
∂t
2
Z
d
B.dS = 0
dt S
This is known as Alfen’s flux freezing theorem. According to this theorem if two fluid
elements are connected through a field line, they always remain connected by the field
line for ideal MHD fluid. A region of concentrated magnetic field with little magnetic
field outside is called flux tube. Flux tubes follow Alfen’s theorem. Flux tubes are used
extensively in solar physics to explain many phenomena.
Sunspots:- In the photosphere dark spots are observed in white light images. These
dark spots are known as sunspots. They occur in pair. They have a large magnetic field
(3000G) and size can be even up to 10000km diameter. Their life time on sun’s surface
varies from 1 day to 28 days.
Figure 1: Sunspots and magnetic buoyancy
Regions on the sun where two large sunspots are observed are often called bipolar
magnetic region. The formation of these spots can be described by concept of magnetic
buoyancy. Let us say pressure inside the tube is pi and pe outside. For balancing pressure
across boundary
B2
pe = pi +
8π
By ideal gas equation with temperature T we get
ρe − ρi
B2
=
ρe
8πpe
Therefore internal part of flux tube is lighter and must be buoyant. This is known
as magnetic buoyancy. If middle part of flux tube is buoyant, we get sunspots on the
surface (as the tube has risen above the surface).
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2.1
Parker’s dynamo theory
In order to describe the mechanism of solar cycle,one of the first attempt was made
by Parker. If we describe magnetic field of sun as poloidal (azimuthal component) and
toroidal field component. Generation of toroidal field from poloidal field was well to come
from differential rotation of the sun. While he was the first one to give a mechanism
to generate poloidal field from toroidal field. He gave an idea that the stretching field
lines due to differential rotation can be twisted due to helical turbulence. In a rotating
frame spreading gives rise to vorticity in the plasma blob due to convection. Several
such loops combine to produce poloidal field in meridional plane. This idea can be put
mathematically by using a theory of mean field MHD.
Figure 2: Different stages of dynamo theory
Mean field MHD
Consider that turbulence is produced as fluctuation to ensemble average of field.
Then for any fluctuating quantity:′
a = ā + a
Where ā is the ensemble average of property of a. where a1 is ensemble average.
On putting fluctuating field to induction equation and then taking difference with the
average of this induction we get:∂ B̄
= ∇ × (v̄ × B̄) + ∇ × E + λ∇2 B̄
∂t
′
′
where E = v × B is mean emf. By smoothing approximation and assuming turbulence
to be isotropic we get for mean emf:E = αB̄ − β∇ × B̄
4
1
α = − (v ′ .(∇ × v ′ ))τ
3
1
β = (v ′ .v ′ )τ
3
Here τ is correlation time of fluctuations. Finally we ger
∂ B̄
= ∇ × (v̄ × B̄) + ∇ × (αB̄) + (λ + β)∇2 B̄
∂t
Here α is the measure of average helical motion in fluid. This coefficient describes
the production of poloidal component from toroidal component in helical turbulence. β
is for turbulent diffusion which is quite larger in comparison to molecular diffusion λ.
Therefore in this kinetic dynamo if we know v̄ and statistical properties of fluid we can
determine evolution of magnetic field.
The kinetic dynamo theory was accepted for a long time until it was realized that
the magnetic field at tachocline is of the order of 105 gauss which can’t be twisted to
produce poloidal field. Now accepted mechanism is Babcock-Leighton mechanism which
produces flux transport dynamo.
Figure 3: Turbulent dynamo
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3
Observational features of the sun
The sun is a star consisting of magnetize plasma. As it is not a solid body, rotation of
sun is quite strange. The sun rotates faster near equator than the poles. Helioseismology
has revealed that it has a differential rotation in terms of height also. Rotation speed
varies as we go deeper up to tachocline. Below Tachocline it has more or less same
rotation speed with depth.
3.1
Structure of the sun
Figure 4: Structure of the sun
3.1.1
Core
The core of sun is extremely dense (150gm/cm3 ) plasma which goes extends from centre
to 0.2-0.25 solar radius. Its temperature is nearly 15.7 million kelvin1 . 99 % of heat of
the sun is generated in the core of sun. Energy is produced by nuclear fusion through
p-p chain reaction which converts Hydrogen to Helium. Energy produced in the core is
transfered to different layers of sun through processes of convection. Gamma radiation
produced here is absorbed and re emitted so it takes a lot of time nearly 10000 yrs for
photons to reach the surface of sun. In the energy transport process of sun matter has
to remain thermal equilibrium with photons. This leads to long time scale of energy
transport in sun.
1
http://en.wikipedia.org/wiki/Sun
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3.1.2
Radiative zone
This zone extends from up to 0.7 solar radii and temperature drops from 7 million K to
2 million K. As the temperature gradient is not large enough to start convection, energy
is transferred through radiation process. Density drops to 0.2 gm/cm3 near the end
of radiative zone. On the top of radiation zone a separating layer called Tachocline is
observed. It is after this layer differential rotation is observed at different heights from
sun.
3.1.3
Convective zone
It is the region above radiation zone. In this region atoms are not fully ionized to
give radiation and density is appropriate for convection processes. So hot material
from radiative zone is taken up to the surface through Rayleigh-Benard convection.
Due to density variations several thermal cells are formed which takes the material to
photosphere. These thermal cells are observed as granules on the surface of sun. In this
zone meridional circulation takes place which takes hot plasma from equator to poles in
a periodic manner.
3.1.4
Photosphere
This is the layer which is visible to us. Below this region visible photons are absorbed by
H − ions because they have a large density below this surface. The spectrum of sunlight
shows a black-body curve with temperature about 5777K. The sunspots are observed
on this layer as a dark spots. The upper part of the sun is less brighter than the central
part leading to limb darkening.
3.1.5
Chromosphere
Chromosphere is lower part of sun’s atmosphere. It is dominated by emission lines
mainly Hα line of wavelength 656.3nm. It is about 2000 km deep. Temperature rises
from 4400K on top of photosphere to 25000K at the top of chromosphere. Chromosphere
heating is not yet fully understood. One of the explanation is magnetic reconnection
happening on top of the photosphere takes hot plasma to the top of chromosphere leading
to higher temperature. Density of 104 order less than the density at photosphere.
3.1.6
Corona
It is outer atmosphere of sun extending up to millions of kilometers. Its density is very
less (10−12 kg/m3 ) and high temperature (5 × 106 K). The question of how coronal is
heated from chromosphere is still a debatable question. Its outer edges transport charge
particles in the form of solar wind. In corona open magnetic loop structures like coronal
holes (giving rise to high speed solar wind) and closed flux tubes like coronal loops
(giving rise to CME when breaks).
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3.2
Various phenomena on sun
Due to change in magnetic activity several features 2 are observed on the surface and in
the atmosphere of the sun. Most of these phenomena can be explained using the concept
of magnetic flux tubes of various size.
3.2.1
Filaments and prominences
Filaments are the big loops of magnetic fields suspended above the surface of sun. They
extend from thousands of kilometers to millions of kilometers. Their stability varies from
several hours to days. Filaments appear dark in Hα light. Filaments when observed
near the limb are called prominences. Prominences appear as bright objects because of
background absorption.
3.2.2
Solar flares
Flares are phenomena of huge outbursts of solar material. It ejects clouds of electrons,
ions and atoms through the corona into space. They are formed when two flux tubes
come together and they reconnect to form new magnetic structure. Reconnection leads
to unconnected magnetic field which accelerates particles leading to flares. Most of the
flares are followed by coronal mass ejection.
3.2.3
Solar wind
Solar winds consists of charged particles having energy range of 1. to 10 KeV. It is
observed in outer atmosphere of the sun. Fast components of solar wind follow magnetic field lines of coronal hole. Faster solar wind is produced from polar region than
equator.When solar wind reaches magnetosphere of any planet, particles get deflected
due to Lorenz force. Some particles penetrate the magnetosphere at higher latitudes
leading to aurora.
3.2.4
CME
As the name suggests coronal mass ejection is the phenomena in which a large plasma
material is thrown in the space through corona. Most of the CMEs are triggered by
flares. One of the mechanism is that a flare near the bottom of large flux tube can
produce waves on the bottom surface leading to instability and the kink oscillations in
flux tube. This eventually leads to breaking of the flux tube and ejection of large mass
to corona. One more mechanism is magnetic reconnection at the bottom of flux tube.
The cause of CME is also magnetic reconnection.
3.2.5
Plage
Plage is bright region in chromosphere of the sun. They are found near sunspots. They
map closely to faculae in photosphere. Magnetic field diffuses away from plage like
2
http : //burro.astr.cwru.edu/stu/advanced/sunp henomena.html
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structure following a network known as active networks.
3.3
Sunspots and Solar cycle
Sunspots have been observed ever since the time of Galileo. In 1844 it was Schwabe who
first reported a period of about 10 years in sunspot number. Later wolf pointed out that
we should have different weight-age for number in case of sunspot groups leading to a
sunspot number called wolf number3 defined as
R = k(10g + n)
Where k is correction factor for observer, g is the no of sunspot groups and n is the no.
of individual sunspot number. Data for daily sunspot numbers have been available since
1874. In order to have consistency a standard smoothed sunspot number was defined
which is a 13 month running average of total daily sunspot number. It shows a period
of about 11 years.
However later it was found out that the study of sunspot area will be a better observation than sunspot number. Data for sunspot areas is available for 100 yrs. Several
features of solar cycle and sunspots have been well studied.
• Sunspots tend form to only in the lower latitudes ranging up to 30-40 degrees.
• Sunspot appear in pairs and pair has a tilt towards the equator (Joy’s law).
• It is also seen that sunspots appear at higher latitude in the beginning of cycle (at
around 25-30) and move towards the equator as cycle progresses (Sporer’s law
of zones).
• Solar cycle also shows a strong correlation with the 10.7 cm radio flux coming from
the sun.
• Variation in no. of CME and total solar irradiance is observed depending upon
minimum and maximum of solar cycle.
• The study of polarity of magnetic field of sunspots shows that binary group of
sunspot has opposite polarity. Sunspots in northern hemisphere have opposite polarity to corresponding sunspots in Southern hemisphere. The polarity of sunspots
is reversed after every solar cycle (Hale’s polarity law).
• Rise time of solar cycles has anti correlation with the strength of cycle (Amplitude
of solar cycle). (Waldmeier effect)
• Polar fields reverse polarity during each cycle at about the time of cycle minumum.
3
Living reviews in Solar physics,7,(2010),1) by David H. Hathaway
9
4
Observations and results
For 100 years Kodaikanal observatory has been photographic image of sun in different
wavelengths (WL, Hα , Ca − IIK). In order to understand solar activity this data can
be used. Here we are using WL images to study the solar cycle and features of sunspots.
4.1
Digitization and data calibration
First data is digitized using 4K × 4K format CCD digitizer unit. Blackening of the
plate depends on the exposure time. Black area on plate corresponds to optical density
of layer. Plate density is related to log of plate exposure through product of incident
intensity and time. So we can convert plate density to solar intensity values. However
plate intensity curves were not available for years before 1969. Therefore relative plate
density was used. Using haugh transform the center of solar disk was found out and
it was made to coincide with the center of plate and aligned in such a way north is up
and south is down in image. Images were saved in FITS format with proper header
information.
4.2
Identification of sunspots
Sunspots in the image are found out based on the variation of pixel values. The area
where a large pixel variation is observed a contour is made to cover that area (corrected
to 3D projection effect). Then centroid of contour and hence position on disk are found
out. Area of contours is in millionth of hemisphere area. All this is done IDL scripts.
4.3
Sunspot area variations
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Figure 5: Daily average of total sunspot area
Daily averaged sunspot area
Daily avg area(in millionth of solar hemishphere area)
12000
Kodaikanal data
Greenwich data
10000
8000
6000
4000
2000
0
-2000
1920
1930
1940
1950
1960
TIme in yrs
11
1970
1980
1990
These images have been removed (privacy issues)
• Monthly average of total sunspot area along with a comparison with Greenwich
data correlation =0.93
• Correlation between monthly average sunspot area of Kodaikanal data and Greenwich data
• Yearly average data for sunspot area with correlation 0.96.
• Butterfly diagram
• Monthly average of total sunspot area in northern and southern hemisphere showing asymmetry.
4.4
Waldmeier effect
There are two kinds of Waldmeier effect.
• WE1–There is an anti correlation between the rise time and amplitude of solar
cycle
• WE2–There is correlation between the rise rate and amplitude of solar cycle
In order to observe Waldmeier effect proper definition of rise time and rise rate is required. Here we have taken it as–
• Rise time- Time taken between the area to reach from 0.2P to 0.8P. (P=amplitude
of cycle)
• Rise rate- Slope between two points after the cycle crosses 0.2P.
For analysis we have taken yearly average of monthly sunspot area for analysis and
smoothening is done by using a Gaussian filer of FWHM of 1 yr.
Here we have verified both the effects (fig). A positive correlation for WE2 (R =
−0.33)and negative correlation for WE1 (R = 0.77) verifies the result.
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Figure 6: WE2 showing correlation between rise rate and amplitude of cycle
WE2 in for sunspot area
Amplitude of cycle(in millionth of solar hemisphere area)
3000
correlation=0.77
f(x)=1.99*x+422.74
2800
2600
2400
2200
2000
1800
1600
1400
1200
1000
800
200
300
400
500
600
Rise rate (in area/yr)
700
800
900
Waldmeier effect is an irregularity in solar cycle. Irregularities in solar cycle arises due
to fluctuations in meridional circulation and fluctuations in Babcock Leighton mechanism. WE1 is not observed in low diffusivity dynamo model. However WE2 is observed
in both high and low diffusivity models. According to high diffusivity model longer cycle
allows diffusivity to act for a longer time which results in low amplitude and weaker cycle. That is why we observe WE1 effect in high diffusivity model. On the other hand for
a longer rise time means a longer time for toroidal field to get developed (in expense of
differential rotation) and attain a higher value. This contradicts the Waldmeier effect1.
Therefore high diffusivity model is more accurate here in these observations.
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Figure 7: All 6 cycles smoothed yearly average data
All cycles togther
3000
’cycle1.txt’
’cycle2.txt’
’cycle3.txt’
’cycle4.txt’
’cycle5.txt’
’cycle6.txt’
sunspot area (yrly avg)
2500
2000
1500
1000
500
0
0
5
2
4
6
yrs
8
10
12
Further works and conclusion
• Study of Waldmeier effect can be improved by checking smoothing with different
filters (different FWHM).
• By following the motion of individual sunspot one can determine the rotation speed
of the sun as a function of lattitude.
• A comparison with dynamo model prediction will give better understanding of the
phenomena.
• Wavelet transform of the time series gives various periodicity. A rigorous study of
these periods is required.
• Yearly sunspot area shows double peaked maximum. One can look theoretical
explanation of this phenomena based on dynamo models.
• Finding out the correlation dimension will help us to quantify the nonlinearity in
the time series.
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Figure 8: WE1 showing anti correlation between rise time and amplitude of cycle
WE1 in for sunspot area
Amplitude of cycle(in millionth of solar hemisphere area)
3000
correlation=-0.33
f(x)=-375.733*x+2293.35
2800
2600
2400
2200
2000
1800
1600
1400
1200
1000
800
1
6
1.2
1.4
1.6
1.8
2
Rise time (in yrs)
2.2
2.4
2.6
2.8
Acknowledgement
I did this summer project under the guidance of Dr. Dipankar Banarjee as a part of 3rd
year summer project at Indian Institute of Astrophysics, Bangalore during May 2013 to
July 2013. I am highly thankful for his proper guidance. I would also like to say thanks
to his Phd. students Vaibhav, Sudeep; his summer students Pavithraa, Prabhakaran
and scietific officers Nazia, Muthupriya for having healthy discussions through out the
project. At last I would like to say thanks to IIA for providing me this opportunity to
do the summer project.
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References
• Physics of fluids and plasmas:
Rai Choudhuri
An introduction for astrophysics by Arnab
• ‘‘The Waldmeier effect and the flux transport solar dynamo’’ Bidya Binay
Karak, Arnab Rai Choudhuri
• ‘‘Digitized archive of the Kodaikanal images: Representative results
of solar cycle variation from sunspot area determination’’ by B. Ravindra,
15
T. G. Priya, K. Amareswari, M. Priyal, A. A. Nazia, D. Banerjee
• Wikipedia
• Living reviews in Solar physics,7,(2010),1) by David H. Hathaway
• http://www.physics.iisc.ernet.in/~arnab/lectures.pdf
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