Cycle Dependence of
Sunspot Properties
Diplomarbeit
Christoph Kiess
June 2, 2013
0.1. ZUSAMMENFASSUNG
0.1
i
Zusammenfassung
Die Aktivität der Sonne variiert mit einer Periodendauer von etwa elf Jahren. Historisch wurde dieser Zyklus in der Änderung der Anzahl sichtbarer Sonnenflecken entdeckt. Sein Ursprung liegt im solaren magnetischen
Feld und dem Dynamoprozess der es erzeugt. Das Studium von Sonnenflecken ermöglicht es uns nicht nur hydrodynamische Vorgänge in der Photosphäre und der Atmosphäre der Sonne zu verstehen, es ist insbesondere auch
von Bedeutung, da sich Änderungen im solaren Dynamo in Eigenschaften
von Sonnenflecken widerspiegeln könnten. In jüngster Vergangenheit wurden von verschiedenen Autoren Unregelmäßigkeiten im Verhalten der Sonne
registriert. Die daraus resultierende Debatte, ob die Sonne auf ein säkulares Intensitätsminimum hinsteuert gibt weiteren Ansporn Sonnenflecken
eingehend zu studieren.
Wir untersuchen Korrelationen zwischen der Größe von Umbra und Penumbra, Umbraintensität, magnetischer Feldstärke, sowie die Abhängigkeit
dieser Parameter vom magnetischen Fluss. Insbesondere konzentrieren wir
uns auch auf die Größenverteilung von Umbrae.
Daten beziehen wir vom Helioseismic and Magnetic Imager (HMI), einem
Instrumentes des Solar Dynamics Observatory (SDO) SDO ist ein Satellit
der NASA der sich seit Frühjahr 2010 in einem geosynchronen Orbit um die
Erde befindet. Wir analysieren in den 2,5 Jahren zwischen 1. Mai 2010 und
31. Oktober 2012 einen Datensatz pro Tag. In 4229 Sonnenflecken verwenden
wir teils manuelle und teils automatisierte Methoden um die Umbrae zu
studieren. HMI Daten haben die Vorteile, dass alle sichtbaren Sonnenflecken
enthalten sind und dass es nahezu keine Lücken in der Zeitreihe gibt. HMI
stellt dabei ausreichend spektropolarimetrische Informationen bereit, um
eine Rekonstruktion des Magnetfeldes in Sonnenflecken zu ermöglichen.
Wir finden, dass eine Lognormalverteilung geeignet ist das Spektrum
von Umbragrößen zu beschreiben. Insbesondere im Bereich von großen Sonnenflecken besteht jedoch eine Abweichung zwischen angepasster Kurve und
Histogramm. Des weiteren hat sich herausgestellt, dass die Methode, also
die verwendete Definition der Umbra, einen Einfluss auf die Breite der Verteilung hat.
Die Zusammenhänge zwischen Größe, Intensität, magnetischer Feldstärke
und magnetischem Fluss sind wie erwartet. Die Größe ist streng korreliert
mit dem gesamten magnetischen Fluss durch einen Sonnenfleck. Je größer
ein Fleck ist, desto stärker ist sein magnetisches Feld und desto dunkler
die Umbra. Diese Zusammenhänge sind statistischer Natur, es gibt eine signifikante Streuung. Zudem sind sie meist nicht linear, wir finden Hinweise
für eine maximale magnetische Feldstärke und eine minimale Intensität der
Umbra.
Wir finden einen linearen Zusammenhang zwischen den Größen von Umbra und Sonnenfleck. Zudem steigt das Verhältniss von magnetischem Fluss
ii
in Umbra zu Penumbra mit zunehmendem totalem Fluss. Insbesondere sind
alle beobachteten Zusammenhänge untereinander konsistent. Eine signifikante zeitliche Änderung einer der gemessenen Größen konnte in den 2.5
Jahren die wir untersuchen nicht festgestellt werden. Es finden sich jedoch
Hinweise, dass es bei geringer Sonnenaktivität der Anteil großer Sonnenflecken bezogen auf die Gesamtzahl der Flecken kleiner ist.
Unsere Ergebnisse sind mit den Resultaten anderer Autoren mit denen
wir vergleichen im Wesentlichen konsistent. Ein genauer Vergleich von Fit
Parametern ist nicht möglich, wenn Daten an verschiedenen Wellenlängen
oder mit verschiedenem Streulichtanteil erhoben wurden. Letztlich finden
wir kein ungewöhnliches Verhalten der Parameter die wir untersuchen. Daraus bedeutet nicht, dass sich die Sonne derzeit nicht aussergewöhnlich verhält.
0.2
Abstract
Solar activity varies with a period of roughly eleven years. This cycle has
been discovered in the variation of the number of visible sunspots. The origin of those spots lays in the magnetic field and the dynamo process that
is creating it. The study of sunspots not only tells us about the hydrodynamical processes in the photosphere and solar atmosphere, it is also of
interest since changes in the solar dynamo could be reflected in properties
of sunspots. Recently various authors recognized irregularities in the longterm behavior of the sun. The resulting debate whether there might be a
grand minimum in solar activity in near future further motivates the close
study of sunspots.
We study correlations between umbral and penumbral size, umbral intensity, magnetic field strength and size as well as the correlation of those
parameters with total magnetic flux. We are also interested in the temporal
variation of those parameters. We focus in particular on the size distribution
of sunspot umbrae.
We use data recorded by the Helioseismic and Magnetic Imager (HMI)
onboard the Solar Dynamics Observatory (SDO), a satellite that was launched
by NASA in early 2010. During the 2.5 years between May 1st , 2010 and October 31st , 2012 we analyze one dataset per day. For 4229 selected sunspots
we use manual and automatic methods to study the umbrae. The advantages of HMI data are that the full disk images contain all visible sunspots
and that there are almost no interruptions in the time series. Further HMI
provides enough spectropolarimetric information to be able to reconstruct
magnetic fields in sunspots.
A lognormal function fits the umbral size distribution quite well. Especially for large sunspots there is a deviation between fit and histogram. We
noticed that the selection method, i.e. the definition of an umbra, can have
a significant influence on the width of the size distribution.
0.2. ABSTRACT
iii
The correlations between radius, intensity, magnetic field strength of
umbrae and total flux are as expected. Large umbrae tend to be darker and
show stronger magnetic field. Umbral size and total flux of sunspots are
strictly correlated. The larger a spot is, the stronger is its magnetic field
and the darker is its umbra. Those relations are of statistical nature, there
is a significant scattering. Most of the correlations are not linear, we find
evidence for a maximum magnetic field strength and a minimum intensity
in umbrae.
We find a linear trend between umbra and sunspot size. We further
find that the ratio of umbral to penumbral flux increases with increasing
total flux. All those correlations are consistent with each other. Within
the 2.5 years we analyze it is not possible to conclude a temporal trend of
any parameter we observe. We find evidence for proportionally less large
sunspots during periods of low solar activity.
Our results are consistent with the findings of other authors. It is hardly
possible to compare fit parameters if the data was taken at a different wavelength or with different stray light contamination. We can not conclude an
unusual behavior in any of the umbral properties we observe. This does not
mean that the sun does not behave strange recently.
iv
0.3
Acknowledgements
The data used in this thesis are courtesy of NASA/SDO and the HMI science team.
I want to thank Dr. Reza Rezaei and Prof. Dr. Wolfgang Schmidt for the
supervision and help. I also want to thank the scientific stuff and the other
students at the KIS for helpful suggestions and a great time at the institute.
Christoph Kiess
Contents
0.1
0.2
0.3
Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . .
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . .
i
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1 Introduction
1.1 Recent Scientific Context . . . . . . . . . . . . . . . . . . . .
1.2 Influnce on Climate . . . . . . . . . . . . . . . . . . . . . . . .
1
1
3
2 Theory
2.1 The Sun . . . . . . . . . . . . . . . .
2.1.1 Inner Solar Structure . . . . .
2.1.2 Solar Rotation . . . . . . . .
2.1.3 The Photosphere . . . . . . .
2.1.4 Atmosphere and Solar Wind
2.1.5 Solar Activity . . . . . . . . .
2.1.6 Sunspots . . . . . . . . . . .
2.1.7 The Solar Cycle . . . . . . .
2.1.8 Long Term Solar Variability .
2.1.9 Solar Magnetism . . . . . . .
2.2 Physical Background . . . . . . . . .
2.2.1 Classical Optics . . . . . . . .
2.2.2 Radiative Transfer . . . . . .
2.2.3 Spectral Lines . . . . . . . . .
2.3 Lognormal Distribution . . . . . . .
2.3.1 Definition . . . . . . . . . . .
2.3.2 Motivation . . . . . . . . . .
2.3.3 Limitations . . . . . . . . . .
2.3.4 Chi Square . . . . . . . . . .
2.4 The Instrument . . . . . . . . . . . .
2.4.1 The Spacecraft . . . . . . . .
2.4.2 HMI . . . . . . . . . . . . . .
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vi
CONTENTS
3 Data Analysis
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Algorithms and Methods . . . . . . . . . . . . . . . . .
3.2.1 Correction for Limb Darkening . . . . . . . . .
3.2.2 Center of Gravity . . . . . . . . . . . . . . . . .
3.2.3 Separation of Distinct Umbrae . . . . . . . . .
3.2.4 Umbral Size Distribution and Lognormal Fit .
3.3 Manual Sunspot Selection . . . . . . . . . . . . . . . .
3.3.1 Disadvantages om Manually Selected Data . .
3.4 Automatic Threshold Method . . . . . . . . . . . . . .
3.4.1 Main Curve . . . . . . . . . . . . . . . . . . . .
3.4.2 Time and Hemisphere Separation . . . . . . . .
3.4.3 Mean Umbral Size . . . . . . . . . . . . . . . .
3.5 Reduced Dataset . . . . . . . . . . . . . . . . . . . . .
3.5.1 Sunspot Tracking Algorithm . . . . . . . . . .
3.5.2 Large Subset . . . . . . . . . . . . . . . . . . .
3.5.3 Comparison Between Umbral Size Distributions
3.5.4 Small Subset . . . . . . . . . . . . . . . . . . .
3.5.5 Inversion of the Magnetic Field . . . . . . . . .
3.5.6 Calculation of Sunspot Properties . . . . . . .
3.5.7 Temporal and Hemisphere Variation . . . . . .
3.5.8 Size-Intensity-Magnetic-Field Relationship . . .
3.5.9 Magnetic Flux . . . . . . . . . . . . . . . . . .
3.5.10 Relation between Umbral and Sunspot Size . .
3.5.11 Comparison of the Data . . . . . . . . . . . . .
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4 Discussion
4.1 Umbral Size Distribution . . . . . . . . . . . . . . . . .
4.1.1 Manual Dataset . . . . . . . . . . . . . . . . . . .
4.1.2 Threshold Method . . . . . . . . . . . . . . . . .
4.1.3 Reduced Dataset . . . . . . . . . . . . . . . . . .
4.1.4 Comparison Between Different Size Distributions
4.1.5 Comparison to other Results . . . . . . . . . . .
4.2 Heliocentric Angle Dependence . . . . . . . . . . . . . .
4.2.1 Umbral Size . . . . . . . . . . . . . . . . . . . . .
4.2.2 Umbral Intensity . . . . . . . . . . . . . . . . . .
4.3 Temporal Variation . . . . . . . . . . . . . . . . . . . . .
4.3.1 Full Dataset . . . . . . . . . . . . . . . . . . . . .
4.3.2 Reduced Dataset . . . . . . . . . . . . . . . . . .
4.4 Correlations Between Parameters . . . . . . . . . . . . .
4.4.1 Intensity, Radius and Magnetic Field . . . . . . .
4.4.2 Umbral Size and Sunspot Size . . . . . . . . . . .
4.4.3 Magnetic Flux . . . . . . . . . . . . . . . . . . .
4.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . .
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vii
CONTENTS
4.6
4.5.1 Umbral Threshold Intensity . . .
4.5.2 Automatizing Sunspot Selection
4.5.3 Distribution of Magnetic Flux . .
Conclusions . . . . . . . . . . . . . . . .
5 Appendix
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i
Chapter 1
Introduction
This introduction is meant to motivate this thesis and answer the question why a statistical analysis of sunspots is important. Therefore a brief
overview about recent results in this topic is given. We do not claim this to
be complete.
Chapter 2 contains an overview about the sun, sunspots and the solar
cycle. Further some mathematics and the instruments that were used for
observation are described. In Chapter 3 we describe our data analysis in detail. The selection of sunspots and calculations are described in this chapter.
It also contains all important figures and results. Those results are discussed
in Chapter 4, it also contains an outlook for possible future work in this field
of research and the conclusions of this thesis.
1.1
Recent Scientific Context
Since the last (23rd) solar cycle an unusual behavior of the
sun compared to previous cycles
has been recognized by various
authors.
Figure 1.1 shows the
monthly and the monthly smoothed
SIDC sunspot number since the
last solar minimum (SIDC-team
1750 - 2013). Smoothing has been
done by averaging over thirteen
months with a different weight. The
maximum of this current solar cycle
Figure 1.1: SIDC monthly sunspot
is expected to be significantly lower
number. Smoothed values are shown
than in previous cycles (Kane 2013).
in gray.
1
2
CHAPTER 1. INTRODUCTION
Livingston (2002) find sunspots to be brighter and have a weaker magnetic field in year 2000-2001 compared to data from 1990-1991. The 10.7 cm
radio flux is highly correlated to the sunspot number Usoskin (2008) and can
be used as a proxy for solar activity. There is a growing deviation between
the index depending on the sunspot number and the index depending on
radio flux (Tapping & Valdés 2011; Clette & Lefèvre 2012). The latter authors also claim that the ratio of small to big sunspots decreased from solar
cycle 22 to 23 (Lefèvre & Clette 2011). There is a debate whether this could
indicate a grand minimum in near future. Another indication that the sun
behaves currently unusual is the significant phase difference in the sunspot
number between northern and southern hemisphere (McIntosh et al. 2013).
The last minimum of the solar cycle (cycle 23 to 24) lasted unusually
long. This can also qualitatively be seen in the sunspot number (Fig. 2.1) and
the butterfly diagram (Fig.2.2) of the last decades. Further Tripathy et al.
(2010) find in helioseismic data a different behavior in the last minimum
(cycle 23 to 24) compared to the minimum before.
Because of the unusual solar behavior it is of interest whether it is still
possible to confer earlier findings regarding sunspot parameters with recent data. Bogdan et al. (1988) and Schad & Penn (2010) report about the
distribution of umbral sizes between 1917 and 1982 and during solar cycle
no. 23, respectively. Both find this distribution to be lognormal. If there is
a change in the ratio of small to big umbrae as mentioned above, this might
be reflected in the shape or width of the size distribution.
Many publications discuss the correlations of sunspot properties like umbral intensity, size or magnetic field strength. Because of a high intrinsic
scattering in those parameters it is necessary to analyze a sufficiently large
set of sunspots. Kopp & Rabin (1992) analyze only six spots and find the
magnetic field strength in to be stronger in bigger umbrae. Livingston (2002)
observe a trend between magnetic field strength and intensity in umbrae.
Darker umbrae have stronger fields. Those trends are consistent with the
findings of Mathew et al. (2007) who analyze more than 160 sunspots and
describe the correlation between umbral radius and intensity. They approximate the dependence with a power law function. Those relations between
maximum magnetic field strength, minimum intensity and size of umbrae
are also confirmed by Schad & Penn (2010) and by Rezaei et al. (2012).
To contribute to the understanding of the solar cycle it is important to
know which parameters are periodic and which are constant.
Albregtsen & Maltby (1978) report a temporal trend in the umbral intensity. This could not be confirmed with more recent measurements by
Mathew et al. (2007), Wesolowski et al. (2008) and Rezaei et al. (2012), who
found the intensities to scatter around a constant value. The latter find a
temporal trend in intensity and magnetic field strength of umbrae.
3
1.2. INFLUNCE ON CLIMATE
1.2
Influnce on Climate
In a private context, as a student of physics, I have been occasionally asked
about the concrete benefit of the scientific research I am doing.
Already in 1801 Wiliam Herschel, one of the pioneers in modern astronomy, had the idea that the the activity of the sun might influence the
climate of earth on a short time scale. In a time when the Schwabe cycle
(Schwabe 1844) was not jet known, Herschel distinguished between intervals with higher and lower sunspot number and tried to find a correlation
between between solar activity and climate. As a climate proxy he decided
to use the price of wheat, since it is correlated to the quality of the harvest.
His conclusion is as follows:
“The result of this review of the foregoing five periods is, that,
from the price of wheat, it seems probable that some temporary
scarcity or defect of vegetation has generally taken place, when
the sun has been without those appearances which we surmise
to be symptoms of a copious emission of light and heat.”
–Herschel (1801, page 316)
From a present point of view, almost twenty solar cycles later, the interaction between solar activity and earths climate is still not sufficiently understood. The total solar radiation varies with the solar cycle. There is also
indication for a long term dependence on solar activity (eg. Shapiro et al.
2011). Although the immediate temperature variation on earth caused due
to variations in the solar constant is minor, solar activity in general might
have a significant influence on climate. There are several possible ways how
the sun might influence the temperature on earth (e.g Haigh 2007): The
thickness of the ozone layer depends on the solar (ultraviolet) radiation.
Further, solar activity modulates the cosmic radiation, and with less radiation there is less ionization in the upper atmosphere. This might influence
cloud formation processes, since the ions can serve as condensation nuclei.
Clouds have an influence on temperature in various ways. Earths albedo,
the greenhouse effect, or the energy transport in the atmosphere are effected by them. Whether such processes are significant or negligible is not
understood.
Further, solar activity has also a more direct influence on earth and
civilization on a short time scale. The intense X-ray radiation of flares
can change the total electron content of the ionosphere and disturb modern electronic communication (e.g. GPS, Afraimovich 2000). Changes in
the solar wind pressure due to coronal mass ejections can weaken earths
magnetosphere and create geomagnetic storms strong enough to induce significant currents in large conductors like railways, pipelines or power lines.
This had even led to blackouts in the electronic power supply in the past
(Boteler et al. 1998).
4
CHAPTER 1. INTRODUCTION
Chapter 2
Theory
This chapter starts with an overview about the sun with focus on sunspots
and the solar cycle. We also give a brief overview about optics, needed to
understand the instrumentation used for observation. This instrumentation
is described in the end of this chapter.
2.1
The Sun
In stellar astronomy stars are categorized using their electromagnetic spectrum. Our sun is a typical main branch star, spectral type G2V. The approximation of the solar electromagnetic spectrum with a black body spectrum
leads to an effective temperature of about 5770 Kelvin. The diameter of
the sun is almost 1.4 million kilometers, the distance from sun to earth is
about 150 million kilometers. This overview about the internal solar structure and the layers of its atmosphere mainly borrows from Stix (2004) and
Weigert et al. (2009).
The internal regions of the sun are not transparent for electromagnetic
waves, with telescopes it is only possible to observe the solar surface and
atmosphere. Oscillations on the solar surface are the result of acoustic and
magnetoacoustic waves traveling through the sun. This opens the field of
helioseismology and gives the possibility to examine for example the solar
density in different depths via measuring the sound speed.
The only particles able to reach observers on the earth directly from
solar interior are neutrinos. They originate in nuclear fusion processes like
the proton-proton chain or in the so-called CNO cycle. Since the neutrinos
emitted by different reactions show different energy spectra, and since the
probabilities for those reactions have different temperature dependencies,
the spectrometric study of neutrinos helps to constrain the solar temperature
very precisely.
5
6
2.1.1
CHAPTER 2. THEORY
Inner Solar Structure
Inside the sun there is an equilibrium between gas pressure and gravitative
pressure. Both temperature and pressure decrease with radius. We can
distinguish between three different regions inside the sun.
Core The temperature in the solar center is of the order 107 K, the density
is around 105 kg/m3 . This environment provides a sufficient probability for
proton-proton collisions. Nuclear fusion processes are the primary energy
source of the sun. Despite the production of helium and neutrinos the
processes in the inner core create photons in the MeV regime. This heats
up the solar core. The solar core is limited to roughly r/RSUN ≈ 0.25.
Radiative Zone Beyond the solar core the energy thermalizes and photons start a diffusive motion through the sun. They get absorbed and reemitted. This radiative zone is in local thermodynamic equilibrium (LTE)
and the gradient in the temperature leads to an outward directed effective
energy transport. Because the density gradient is high enough there is no
large scaled plasma motion in the radiative zone. This changes at the so
called tachocline at roughly r/RSUN ≈ 0.7
Convective Zone The tachocline is about 200 000 km below the solar
surface. Above this layer radiative energy transport is not sufficient to bring
the heat to the outer layers of the sun: the equilibrium between absorbed
and emitted photons becomes unstable. Thus parcels of the gas get buoyant
and the motion of the plasma becomes overturning. The convective nature
of the sub photosphere layers can be seen directly in intensity and in Doppler
images of the solar surface. The resulting structures in the photosphere are
called solar granulation. The convective zone rotates differentially, both in
radius and in latitude. This rotation has a major role in the solar dynamo
process that leads to the solar cycle and to sunspots.
2.1.2
Solar Rotation
The sun rotates with respect to the fix stars in our galaxy. The solar interior
below the tacholine rotates approximately like a solid body, the angular
velocity of the convection zone is both a function of latitude and of radius,
which means the sun performs a rather complicated differential rotation.
On the surface and at the equator the sidereal period (with respect to the
stars) is 25.34 days, the sun rotates slower at the poles. Active regions rotate
slightly slower than the quiet solar plasma.
Richard Carrington defined the synodal period to be exactly 27.2753 days
and started counting solar rotations on November 9th, 1853. The Carringon
7
2.1. THE SUN
rotation number is a commonly used coordinate to describe features visible
on the sun.
In this thesis we used the empirical approximation (Howard et al. 1984)
Ωsideral [deg / day] = 14.522 − 2.84 sin2 (θ)
(2.1)
to correct for differential rotation of sunspots depending on the latitude θ.
2.1.3
The Photosphere
The region where the solar opacity drops from nontransparent to transparent
is called photosphere. It is therefore the deepest directly observable layer,
most emitted sunlight originates in the solar photosphere. The photosphere
is only a little more than one hundred kilometers thick and is often referred
to as the solar surface. Its distance from the center is about 6.96 × 105 km,
the solar radius. The temperature of the photosphere decreases in radial direction from around 6500 K, where the sun begins to be transparent, down to
a minimum of roughly 4200 K. This negative gradient is the reason why photospheric spectral lines can be seen in absorption. The photosphere shows,
in addition to granulation, so called active regions which are related to magnetic fields. Those magnetic structures will be discussed in section 2.1.5 in
detail.
Limb Darkening
Although the solar surface is thin, it has a finite thickness and the emitted
light of the photosphere is a line of sight integral. This integral reaches from
the outer photosphere down to a certain geometrical depth which depends on
the viewing angle. If the sun is observed at disk center, i.e. perpendicular to
the solar surface, photons from relative deep layers can reach the observer.
An observation close to solar limb, in contrast, results in photons with a
higher origin. Since the photosphere is hotter in deeper layers the solar
image is brightest at disk center and becomes darker closer to the limb. This
effect is called limb darkening. Since the opacity is a function of wavelength
it depends on the observed spectral regime.
2.1.4
Atmosphere and Solar Wind
The optically thin layers above the photosphere build the outer solar atmosphere. It can be divided in the chromosphere, the transition region and
the solar corona. The solar atmosphere merges into the solar wind, a particle steam which reaches out up to edge of our solar system. Above the
photosphere the temperature increases again.
8
CHAPTER 2. THEORY
Chromosphere Between the temperature minimum of around 4200 K and
the layers with roughly 25 000 K the temperature increases relative slowly
and steadily. Because of the colors that can be seen in this region during
a total eclipse it is called called solar chromosphere. The density is much
lower in the chromosphere compared the photosphere below.
Transition Region Above the chromosphere there is a drastic temperature increase and density decrease. The part of the solar atmosphere where
this change takes place is called transition region.
Corona The hot region beyond the transition region is named solar corona.
It has temperatures of several millions Kelvin and therefore contains highly
ionized elements. The possible atomic transitions in those ions lie at relative
high energies, the corona radiates in the ultra violet regime. With increasing
distance from the solar center the density of the plasma decreases while the
temperature stays rather constant. The solar corona merges into a radially
outward directed particle stream, the solar wind. It reaches out beyond the
planets, up to the heliosphere at a distance of roughly 100 AU. The corona
and the solar wind are not constant over short time scales, their intensities
are correlated with solar activity. The processes heating the plasma are not
sufficiently understood, but are surely related to the solar magnetism.
2.1.5
Solar Activity
The solar surface shows dark structures with diameters of sometimes more
than an arcminute. Those so called active regions can under ideal conditions
be seen with the naked eye and are known for a long time. With the invention
of the telescope in the early 17th century their scientific study began.
Active regions consist of one or more sunspots or pores. Often multiple
active regions can be visible on the solar surface. Sunspots have a dark inner
part, the umbra, that is surrounded by a brighter envelope, the penumbra.
Those penumbrae are still slightly darker than the quiet solar surface. Pores
are in general smaller than sunspots, have similar intensities than small
umbrae and do not show a penumbra.
In a small sample of sunspots their lateral position, total number or size
seems to be rather random within certain boundaries. Sunspots do not occur
in latitudes greater or roughly 45◦ and do rarely have radii larger than an
arcminute. A sufficient amount of data shows that the mean number and
lateral position of sunspots vary with a period of roughly eleven years, the
solar cycle(Sec. 2.1.7)
Another milestone of solar physics was the discovery of a Zeeman splitting in an umbral spectral line by George Ellery Hale. It showed that magnetic fields are the origin of solar activity. Since those early findings sunspots
2.1. THE SUN
9
have been a subject of a more detailed research. In the following description
of sunspots we follow mainly Solanki (2003).
2.1.6
Sunspots
Sunspots can have various shapes. In general they consist of an umbra that
is surrounded by a penumbra. The largest and most circular configurations
appear to have the longest life times with up to several weeks. Small spots,
in contrast, can vanish already a few hours after they occur.
Bipolarity Since magnetic field lines are always closed, the magnetic flux
piercing the solar surface and creating a sunspot has to return back into the
sun again. This leads to spots with different magnetic polarity in the same
active region. In general those bipolar structures contain one precursive spot
of one polarity followed by spots of the other polarity and the leading spot
is more stable. The polarity is the same for almost all active regions on one
hemisphere and opposite on the other. The polarity configuration switches
after each solar cycle.
Umbrae
The umbra is the inner dark part of a sunspot. In this thesis we find that
umbrae have sizes up to roughly 200 microhemispheres (MHS). One MHS is
10−6 of the visible solar hemisphere and corresponds to roughly 3.05 Mm2
or 5.79 arcsec2 . Since it is a quotient of two sizes it has no dimension. The
sizes of sunspot umbrae start at 0.5 to a few MHS. The reason why sunspot
umbrae are dark is that they are relatively cool, they have an effective
temperature of roughly 4000 K. This is because the almost vertical magnetic
field of up to several thousand Gauss (G, 1 G ≡ 10−4 Tesla) suppresses the
convection and there is less heat transported in umbrae compared to the
quiet sun.
This energy transport in umbrae is still a field of current research: Simulations show that umbrae should be cooler as they actually are. The answer
to this question might lay within the umbral substructure. Under a higher
resolution umbrae are not homogeneously dark, they show small bright features called umbral dots.
Pores The sizes of umbrae and pores overlap. The common definition is
that sunspots show penumbrae and pores do not. Pores can evolve into
spots and spots can loose their penumbra and decay into pores. This makes
it hard to differentiate between them in some cases. Especially young and
evolving spots might have only partial penumbrae. The pores bigger than
the smallest spots are not stable, they start to form a penumbra.
10
CHAPTER 2. THEORY
Light Bridges A single sunspot can have multiple distinct umbrae which
can divide or merge together as it evolves. Rather faint bright structures
between umbrae are called light bridges. It somehow depends on the definition of an umbra if one speaks of two distinct umbrae or of one umbra
with a lightbridge in it. This transition might also be fluent. Some sunspots
have umbae with partial light bridges, that do not split the whole umbra,
for example.
Penumbrae
The penumbra surrounding an umbra is slightly darker than the quiet sun.
It has a magnetic origin as well, the mean penumbral field is radially outward
directed from the sunspot and gives the penumbra a filamental structure.
Penumbral light is highly polarized.
Wilson Depression
Another experimental finding is that spots close to the solar limb often
show a wider penumbra at the side of the umbra directing towards solar
limb compared to the disk center side. This was first noticed by Wilson in
1769 and therefore is called Wilson effect. It is a projection effect and is due
to a geometrical depression (the Wilson depression) in the layers of constant
optical depth in sunspots. In the umbra we can see between several hundred
to a few thousand kilometers deeper compared to the quiet sun.
Hydrodynamic Flows in Sunspots
Sunspots are no static objects, the plasma within them is in a dynamical
equilibrium. There is a radial outward directed lateral plasma motion in
the order of a few kilometers per second in the penumbra, the Evershed
flow. Since the lateral motion at large heliocentric angles results in a net
motion of the plasma relative to the observer, it can be observed via analyzing the Doppler shift of penumbral spectral lines of sunspots close to
solar limb. The speed of this motion depends on the geometrical height. In
the chromosphere above the penumbra the direction of the motion of the
plasma is even reversed. Further upflows and downflows within the penumbra and umbra can be observed on different scales. Hence with modern
telescopes and advancing image reconstruction techniques the quality of the
three dimensional understanding of the motions within sunspots constantly
improves.
Subsurface Structure of Sunspots
Since direct observations are not possible and since the analysis of helioseismic data does not provide us with sufficient spatial resolution we have
2.1. THE SUN
11
relatively little knowledge about the structure of sunspots underneath the
surface. Basically the only possibility to understand sunspots below the
photosphere is to simulate data and compare those simulation to observations. Especially for penumbral filaments and umbral bright dots it is not
known for sure how far down they reach. Light bridges as well might be just
photospheric phenomena or they might reach down and split flux tubes in
deeper layers.
Sunspot Formation and Evolution
Active regions and sunspot are the result of magnetic flux tubes permitting
through the solar photosphere (Sec. 2.1.9). It is believed that flux tubes
have their origin at the bottom of the convection zone. They get buoyant
and rise to the solar surface. The formation of a sunspot can take between
some hours and several days until the umbra reaches its maximum size. In
general first a pore appears and them the penumbra starts to form around
it. Shortly after a new sunspot arises it can start to decay. This process is
still not fully understood, it can be very slow so spots can live for several
weeks. There have to be processes that somehow transport magnetic flux
away from the sunspot. Occasionally sunspots coalesce and become one
larger spot or split up into multiple fragments.
2.1.7
The Solar Cycle
The sun is not a static object. Several of its parameters change on various
time scales reaching from a temperature increase over billions of years down
to oscillations with periods of several minutes. Since Schwabe (1844) it is
known that the mean number of visible sunspots changes with a period of
roughly eleven years. This period is called or Schwabe cycle or just solar
cycle. Rudolph Wolf introduced the relative sunspot number (Wolf 1850),
also called Zürich number, as ten times the number of visible active regions
plus the number of visible sunspots. Although this value depends on the
observation (with a better telescope one might resolve more sunspots) it is
still of great value because it has been cataloged for multiple centuries. Figure 2.1 shows the variation of the SIDC monthly smoothed sunspot number
(SIDC-team 1750 - 2013) since 1750. See also figure 1.1 in the introduction
for the recent variation of the sunspot number.
There are more parameters known to follow this eleven year cycle than
just the raw number of spots. The polarity of the leading sunspots in bipolar
groups changes after each solar cycle. Hale et al. (1919) therefore concluded,
that the full period of the magnetic cycle is 22 years. Further the mean position of sunspots varies with time. In the beginning of a cycle magnetic
features occur on relative high latitudes. Later in the cycle the position
moves towards the equator, an effect that leads to so called butterfly di-
12
CHAPTER 2. THEORY
Figure 2.1:
Monthly smoothed SIDC sunspot number.
Source:
http://sidc.oma.be/ See also figure 1.1 for the latest development.
agrams (Fig. 2.2). The are of flares (Flare index) and the 10.7 cm radio
flux (F 10 index) changes with the period of the Schwabe cycle (Covington
1969). The solar intensity also varies by roughly 0.1% (Willson & Hudson
1988), the sun being brighter in phases of higher solar activity.
2.1.8
Long Term Solar Variability
From the variation of the sunspot number we know that the length and
amplitude of the solar cycle vary over several cycles (Fig. 2.1). The period
of the cycles varies by around ±1 year and the Wolf number in the order
of a factor 2. It is not understood to what amount such variations are of a
stochastic nature or how deterministic this variation is. The mean intensity
variation over multiple cycles suggests another intensity cycle in a time scale
of of 80 to 100 years (Gleissberg 1967). This Gleissberg cycle seems not to
be strictly periodic. There is a phase shift between the solar cycle on the
northern and southern solar hemisphere. Zolotova et al. (2010) for example
find that the hemisphere with the leading phase changes after several cycles.
Currently (cycle no. 24) the solar cycle on the northern hemisphere is ahead
of the southern solar cycle by several months (eg. McIntosh et al. 2013).
Between roughly 1675 and 1715 almost no sunspots were visible on the sun.
This period is called the Maunder minimum and is the only directly observed
grand minimum.
Despite the direct measurement (with sunspot number or F 10 index for
example) there is the possibility for an indirect reconstruction of historical
solar activity. The production rate of radionuclide in the earth atmosphere
depends on and solar activity (eg. Stuiver 1961). From the analysis of Be10
2.1. THE SUN
13
Figure 2.2: Butterfly Diagram from USAF/NOAA sunspot data. The color
index shows the relative area covered with sunspots in MHS, binned in 3◦
latitude. Source: http://solarscience.msfc.nasa.gov/
atoms in polar ice shields and C14 atoms in tree rings it is possible to reconstruct the production rate of those nuclei. Although such data is very
noisy and the eleven year cycle can not be resolved, there are clear hints
for long term activity variations and historical Maunder-like minima in radionuclide data (Usoskin 2008). Hence it is likely that some time in future
another solar grand minima will occur again. Since we neither understand
the processes leading to such variations, nor know if they are of stochastic
or (quasi-)periodic nature it is very hard to predict future solar activity.
2.1.9
Solar Magnetism
Hale (1908) found for the first time a Zeeman splitting of absorption lines
in sunspot umbrae. He concluded that within sunspots a strong magnetic
field is present. Today we know that the magnetic field strength in umbrae
reach from two to three thousand Gauss, and that not only sunspots are
magnetic features, but that the whole sun has a magnetic field as well.
Following Ampères law an electric current induces a magnetic field. Since
the sun consists to large parts of ionized gas, the rotating solar plasma is
in principle capable of creating strong magnetic fields. In order to create theoretical models of the sun including magnetism one has to calculate
electrodynamics in a fluid conductor. This opens the field of magnetohydrodynamics (MHD, Spruit 2013). From the electrodynamic point of view,
Maxwells equations have to be satisfied, from the hydrodynamic side the
Navier Stokes equations constrain the motion of the plasma. Magnetic fields
are important in various fields in astrophysics, MHD waves for example can
14
CHAPTER 2. THEORY
carry energy over long distances. It is believed that magnetic mechanisms
are partly responsible for coronal heating. In order to understand sunspots
it is crucial to discuss a central result of solar MHD in some detail: the
concept of magnetic flux tubes.
Field Lines and Flux Tubes
The Lorentz force is perpendicular to the direction of magnetic field and perpendicular to the direction of motion of a charged particle. This constrains
moving charges in magnetic fields on helical trajectories along the magnetic
field lines. For high magnetic field strengths and low plasma densities the
plasma is bound to the magnetic field. This can be quite impressive in the
solar atmosphere where plasma follows loops of magnetic field and those can
be seen as spicules or prominences in emission.
If, on the other hand, the plasma density is high and the energy of the
plasma is high compared to the energy in the magnetic field as well, the
field lines have to follow the motion of the medium. This is the case in the
interior of the sun. In this case it is justified to speak of ”frozen” magnetic
field, since the plasma motion dictates the motion of magnetic field lines.
Bundles of parallel field lines are called magnetic flux tubes. The magnetic flux Φ is defined as the integral over the magnetic field B through a
surface S.
I
~ dS
~
(2.2)
Φ= B
S
Alfvéns theorem says that magnetic flux is conserved (dΦ/dt = 0) and
founds on the assumption of infinite conductivity which is sufficiently satisfied in the solar interior. Because the magnetic pressure pm = B 2 /(8π) contributes to the total pressure, magnetic flux tubes have a lover gas pressure
and therefore a lower density than the surrounding unmagnetized plasma.
This leads to a buoyant rise of flux tubes.
Since magnetic field lines are always closed, the ones permitting the solar
surface have to re-enter the sun somewhere again. This leads to the bipolar
structure of active regions as discussed above.
The Solar Dynamo
One of the most important open questions in modern solar physics is how the
solar dynamo works in detail. If magnetic field lines are frozen, the moving
plasma can deform the field and the differential rotation (Ω) of the solar
plasma is capable of winding up magnetic field lines in the solar interior.
This mechanism can create torodial magnetic flux tubes out of a polodial
(dipol-like) magnetic field. Those flux tubes are believed to be close to the
tachocline. If they get buoyant and rise through the surface, they can form
15
2.2. PHYSICAL BACKGROUND
active regions. This part of the solar dynamo is called the Ω-effect and can
be reproduced with simulations.
We observe a periodic variation of the number of sunspots, however.
Ergo there has to be an inverse effect, that forms a polodial configuration
out of torodial initial conditions. Several of such so called α-effects or αmechanisms have been proposed. There is so far no dynamo model that
describes the eleven year solar cycle while being consistent with all observations. Also see Spruit (2012) for a critical view on this topic.
2.2
2.2.1
Physical Background
Classical Optics
This paragraph gives an elementary overview about the propagation of light
and therefore provides the basic physical principles to understand the optical
instruments explained later. For the description of the instrumentation used
in this thesis it is sufficient to describe light as classical electromagnetic wave.
The formation of spectral lines and radiative transfer can be understood
sufficiently if we describe light as photons. This introduction makes use of
both the particle and the wave picture of light and does not go into the
details of the correct quantum mechanical description.
Electromagnetic Waves
The electrical field component of an electromagnetic wave propagating in ~ez
direction is
~ t) = Ex (z, t)~ex + Ey (z, t)~ey .
E(z,
(2.3)
λ
Ei (t) = Ei cos(2π t + ϕi )
c
i = x, y
(2.4)
λ is the wavelength and c the speed of light. The phases ϕx and ϕy determine
the polarization state of the wave.
In reality, the light that arrives at the instrument is not fully coherent,
but is rather described as a superposition of many such electric waves with
different lengths λj and phases ϕj .
The intensity of the light, the physical quantity which we are going to
measure in our detector, is proportional to the square of the electric field
strength. This is the reason why it is possible to build filters as described
in sections 2.4.2 and 2.4.2.
Propagation of Light in Media
The speed of light in vacuum is c = 299 792 458 m/s. If an electromagnetic
wave enters a medium, its speed cmed and its wavelength λ = νc will be
16
CHAPTER 2. THEORY
reduced. The factor c/cmed = nmed is called refraction index, which in
general is a function of temperature and wavelength.
Snell’s law
If a beam of light propagates from one medium with refraction index n1 into
another with n2 it will be refracted, i.e. the ray changes its direction. This
can be expressed via Snell’s law.
n1 sin(θ1 ) = n2 sin(θ2 )
(2.5)
where Θ is the angle between beam and surface normal vector.
Birefringence
There are media, typically crystals and liquid crystals, where different directions in the material have different refraction indexes. Those materials
are called birefringent. Various optical instruments (e.g. waveplates) use
this effect to manipulate the polarization state of the incoming light. If, for
example a linear polarized wave of the form of equation 2.3 enters a birefringent material with refraction indexes nx and ny in x and y direction,
respectively, and this material has a thickness d, it will induce a phase shift
δ between the two polarization states.
δ=
d
d
d
−
= (nx − ny )
λx λy
λ
(2.6)
Further the two polarization states will be refracted in different angles
following equation 2.5.
Stokes formalism
To describe the polarization state of light we define a vector with four compo~ This vector contains the complete polarimetric
nents called Stokes vector S.
information of the light.
 
I
Q
~= 
(2.7)
S
U 
V
I = hEx2 + Ey2 i
Q = hEx2 − Ey2 i
U = h2Ex Ey cos(δ)i
V = h2Ex Ey sin(δ)i
(2.8)
2.2. PHYSICAL BACKGROUND
17
The Stokes parameter I is the intensity of the electromagnetic wave.
Q and U refer to the linear polarization of the light while V carries the
information about circular polarization. The brackets hi denote an averaging
over time. This can not be omitted since a CCD sensor detects intensity.
The degree of polarization of an electromagnetic wave is
r
Π=
Q2 + U 2 + V 2
I2
(2.9)
Π = 0 stands for unpolarized light. For Π = 1 the electromagnetic wave is
fully polarized.
2.2.2
Radiative Transfer
In vacuum light can in principle travel an arbitrary distance without extinction. In a medium with density ρ we have to introduce the spectral
absorption coefficient κν which leads to an intensity (Iν ) decrease with distance r.
In addition to this absorption a medium with temperature T > 0 emits
light. If we describe this spectral emission with the source function Sν the
change in intensity can be written as
dIν
= −κν ρ(Iν − Sν ).
dr
(2.10)
In local thermodynamical equilibrium (LTE), which is approximately the
case in the solar interior, Sν can be approximated with the Planck function.
This means we assume the the spectrum of the object to be that of a black
body radiator of the temperature T .
B(λ, T ) =
1
2hc2
5
λ exp(hc/λkB T ) − 1
(2.11)
Photons in the solar interior have a relative short free mean path. Radiative energy transport through the sun therefore takes much time, photons
get absorbed, re-emitted and scattered. They perform a diffusive motion
through the plasma.
We define the spectral optical depth τν as
dτν = −κν ρdr.
(2.12)
The signals we measure are always the result the integrated emissions and
absorptions along the line of sight (cf. limb darkening, Sec. 3.2.1).
18
2.2.3
CHAPTER 2. THEORY
Spectral Lines
Atoms, ions and molecules have discrete energy levels. If the change in
the energy state goes along with the absorption or emission of a photon,
its wavelength is characteristic for a given element. Astrophysical objects
have therefore not the perfect Planck spectra of black body radiators. The
solar spectrum shows many absorption and emission lines. It is possible do
extract a lot of information out of the intensities, shapes and positions of
those spectral lines.
The ionization states of a hot gas, like in the solar atmosphere, follow
the Saha equation, the excitation states of the atoms can be described via
the Boltzmann distribution. The hotter the gas, the higher it is ionized or
excited. Therefore specific excitation (and ionization) states only exist in a
relatively narrow temperature regime. This makes it possible to investigate
the solar atmosphere in a specific height by observing the spectral lines of
an ion (or atom or molecule) that forms in that height.
Line Sifts and Line Broadening
There are different processes that influence the shape of a spectral line. Its
exact position and shape can be used to study various characteristics of
astrophysical objects.
First, there is a natural line broadening because of the principle uncertainty of the energy levels participating in the transition (quantum mechanical energy-time uncertainty).
There is a broadening of spectral lines due to collision of the atoms. It
depends on density and temperature of the radiating gas. Further there
is the a line shift and a line broadening because of the optical Doppler
effect. On a microscopic scale there is a thermal Doppler broadening because
the particles follow a velocity distribution depending on their temperature.
There is also a line broadening because of spatially unresolved turbulent
motions of the plasma. The line shift due to a relative motion between
emitter end observer can be used to detect oscillations, large scaled plasma
motions, or the rotation speed on the solar surface.
If the observer can control and understand the line broadening effects
mentioned above, it is possible to draw conclusions about the electromagnetic field in the observed regions. This is because many atomic energy
levels can be influenced by such a field. A electric field can cause spectral
lines to split up due to the linear and quadratic Stark effect. Electric fields
in the sun are small, more important is the Zeeman effect which describes
the splitting of energy levels in a magnetic field.
2.2. PHYSICAL BACKGROUND
19
Zeeman Effect
The quantum mechanical Zeeman effect is the result the influence of magnetic fields on energy levels in an atom. A magnetic field removes the degeneracy of the energy states with different magnetic quantum number in
the atom. In the ideal case, if the line is magnetically sensitive enough, this
leads to a split of the spectral line and multiple peaks can be observed. In
solar physics the magnetic field strengths are often not strong enough to
observe the individual Zeeman components. The result is that the Zeeman
effect contributes to the broadening of the absorption line.
In the simplest case, the splitting of a line into a Zeeman triplet, there are
two shifted σ components and one π component. Those different Zeeman
components show different polarization states. This polarization depends
on the viewing angle with respect to the direction of the magnetic field.
If the observer looks perpendicular to the magnetic field direction, the π
component would be linearly polarized perpendicular to the field and the σ
components would be polarized linearly parallel to the field.
Magnetic Field Inversion
The Zeeman effect in sunspots in the absorption line used in this thesis
(Fe I 617.33 nm) is not strong enough to go the straightforward way and
measure the magnetic field directly from the line multiplet. The way to go
is to simulate what we are expecting to measure, i.e. solving the radiative
transfer equation for given parameters (temperature, magnetic field, etc.)
and compare the resulting Stokes Vector with the measurement. Those
parameters then are iteratively adjusted in order to obtain a best fit between
simulation and data. If the observation includes all four Stokes parameters
it is possible to invert not only the magnitude, but the direction of the
magnetic field vector as well. We then assume resulting magnetic field vector
to represent the real magnetic field in the sun.
In this thesis we make use of the VFISV (very fast inversion of the
stokes vector, Borrero et al. 2007, 2011) code to invert the magnetic field.
The computation the magnetic field needs recourses sine for each pixel many
synthetic Stokes profiles have to be created in order to find the best fit. This
inversion code is optimized for HMI data, it makes use of a fast polynomial
approximation for the Stokes profiles which is slightly inaccurate. Further
the line shift of the Zeeman components of the HMI Fe I line is directly
implemented, the code avoids calculations involving the quantum numbers
of the energy levels participating in the line transition. The VFISV therefore
can not be used on all magnetic sensitive spectral lines, but it has the big
advantage that it is able to invert data fast enough to keep up with the high
data rate of HMI.
20
2.3
2.3.1
CHAPTER 2. THEORY
Lognormal Distribution
Definition
We define the lognormal distribution ρ (e.g. Sachs 1999) for positive x values
(ρ = 0 for x ≦ 0).
(ln x − µ)2
1
(2.13)
= exp −
ρ(x) = √
2σ 2
2πxσ
Later we are going to fit this distribution to the histogram of umbral sizes.
The two free parameters, σ and µ have no direct physical meaning. The
2
median of the distribution is eµ , the mean is e(µ+σ /2) and the variance is
2
2
e(2µ+σ /2) (eσ − 1). With this definition the function is normalized to its
area.
Bogdan et al. (1988) and Schad & Penn (2010) study the distribution
umbral sizes from 1917 to 1982 and during solar cycle no. 23 respectively.
Both groups use a different definition for the lognormal function.
dN
dN
(ln A − lnhAi)2
.
(2.14)
=−
ln
+ ln
dA
2 ln σA
dA max
Here (dN/dA) is the density function, hAi the mean and σA the width of
the function. Both definitions are mathematically equivalent. Equation 2.14
can be normalized by choosing
dN
= (2πhAi2 σA ln σA )−1/2 .
(2.15)
dA max
The relations
σ=
p
ln σA
and
µ = lnhAi + ln σA
(2.16)
allow to convert equation 2.14 into equation 2.13.
2.3.2
Motivation
The lognormal distribution is related to the normal distribution. A random
variable is distributed lognormally when its logarithm is distributed normally. This motivates to define the lognormal distribution as exponential
of equation 2.14. From this definition it is also evident that the graph of
the lognormal function has the shape of a parabola if it is plotted with two
logarithmic axes (cf. e.g. Fig. 3.8).
We are going to use the first definition of the lognormal distribution
(Eq. 2.13) because the normalization is mathematically simpler and since it
is commonly used in the math literature.
The motivation of the usage of a lognormal function to describe the size
spectrum of sunspot umbrae lies within its connection to independent fragmentation processes (Bogdan et al. 1988, and references therein). Imagine
21
2.3. LOGNORMAL DISTRIBUTION
an object of size A0 which fragments into two fractions of the sizes A0 x1
and A0 (1 − x1 ), where x1 ∈ [0, 1] is a random. After n such fragmentations
the size of one of one fragment is
An = A0
n
Y
xi .
(2.17)
i=1
Therefore the logarithm of the relative size An /A0 is
ln
An
A0
=
n
X
ln (xi ) .
(2.18)
i=1
Following the central limit theorem, for large n, the sum over random variable is normal distributed. Therefore the sum of the logarithm of those
variables has to follow a lognormal distribution. If the sizes of the flux tube
bundles forming sunspots are the result of such fragmentation processes,
then they might be lognormally distributed in size.
2.3.3
Limitations
The idea of the fit of the lognormal distribution to a histogram (Sec. 3.2.4)
needs some discussion. Let x be the umbral area in MHS and xi be discrete
values for the bins of the histogram h(xi ). ρ(x) is the assumed size distribution (Eq. 2.13). We make the constraint that we normalize the histogram
to its area and fit a normalized distribution.
Z ∞
X
ρ(x) dx = 1
(2.19)
h(xi ) =
i
0
This induces a small systematical error because the width of the histogram
is finite while the lognormal distribution is defined for all x > 0. Thus the
histograms has systematically larger values compared to the fit. The correct
way would be to normalize the distribution not to its integral from zero to
infinity, but to the size spectrum the histogram actually covers. This means
we would renormalize ρ such that
Z xmax
X
h(xi ) =
ρ(x) dx = 1.
(2.20)
i
xmin
This makes things way more complicated since thisR problem can only
R xmaxbe
∞
solved numerically. In practice the difference between 0 ρ(x) dx and xmin ρ(x) dx
is only a few percent. We therefore stick to the analytical normalization.
It is not verified whether a lognormal function is really the size distribution which underlies the size distribution of rising flux tubes or flux tube
bundles (i.e. umbrae). Another point of view would be to interpret the
22
CHAPTER 2. THEORY
lognormal function just as a analytical function with two parameters which
fits the data quiet well. Consider the graph of the histogram of umbral sizes
(e.g. Fig. 3.5): a simple power law function is obviously not sufficient to fit
the data since it can not reproduce the curvature of the graph as log log
plot. One can therefore interpret the lognormal function as a second order
approximation. From this point of view the value of the fit is that it is a
simple approximation of the distribution. The result are two fit parameters
σ and µ that can be compared to other findings (Table 3.1).
We can tell for sure that the assumption of lognormal distributed sunspot
umbrae has to fail somehow for large umbrae, since there are no arbitrary
large sunspots possible (the solar surface is finite). A lognormal distribution
predicts a finite possibility for umbrae to occur in all size ranges greater zero.
2.3.4
Chi Square
If there are n measurements yi with uncertainties yi,err at the positions xi ,
then the reduced chi square χ2 as of a function f (x) with j free parameters
and those measurement is defined as
1 X yi − f (xi ) 2
2
.
(2.21)
χ =
n−j n
yi,err
The χ2 is commonly used in physics do decide how good an analytical function fits data. In curve fitting routines the free parameters of f are then
adjusted to find the best fitting function which is the one with the least
possible χ2 . If the data has been randomly drawn from the function and
the errors were chosen correct the χ2 should have a value close to one.
We mention this here explicitly because we are going to use the χ2 for
another purpose as well. It will be used to compare the curve fits belonging
to different datasets. The χ2 is then a proxy for the deviation between two
datasets resulting of two different measurements.
2.4. THE INSTRUMENT
2.4
23
The Instrument
Astronomical observatories in space have several advantages compared to
ground based observatories. First, there is not necessarily a restriction of
observing time due to day and night, depending on the orbit of the spacecraft. Second, there is no atmosphere between the instrument and the target
one is observing. Therefore it is possible to observe in wavelengths where
the earths atmosphere is optically thick. In the optical regime clouds do
not disturb the observation. Because of absence of atmospheric seeing (limitation in the resolution due to refraction in the air) there is not only no
need for a complicated adaptive optics system in telescopes on spacecrafts,
the quality of the data is also more constant over time. Telluric absorption
lines are absent in satellite data. Although his can be an advantage if one
is interested in molecular lines, the downside is that it is not possible to use
them as a reference for calibration. Further disadvantages of space astronomy are the high costs, the long lead time and the limited data rate. It is
also almost impossible to repair or modify the instruments.
2.4.1
The Spacecraft
The Solar Dynamics Observatory
(SDO: Pesnell et al. 2012) is part
of NASAs ”living with a star” program. It carries three instruments,
AIA, HMI and EVE, with different scientific purposes. SDO was
launched by NASA in February
2010 and cycles the earth in a
geosynchronous orbit. Such an orbit has the advantage that there is a
better data download possible compared to more distant satellites, like
the Solar and Heliospheric Observatory (SOHO) in the Lagrange point
L1.
Figure 2.3: SDO
AIA The Atmospheric Imaging Source: http://sdo.gsfc.nasa.gov/
Assembly takes full disk images of
the sun at nine different wavelength
from optical to far ultraviolet. It
is mainly designed to study the
corona.
24
CHAPTER 2. THEORY
EVE The Extreme Ultraviolet Variability Experiment is a spectrometer
with high temporal and spectral resolution. It observes wavelengths between
0.1 and 105 nm.
HMI This thesis uses data taken by the Helioseismic and Magnetic Imager, an instrument designed to do polarimetric and Doppler observation in
the solar photosphere. HMI will be discussed in more detail in the following
section.
2.4.2
HMI
The Helioseismic and Magnetic Imager1 (Schou et al. 2012; Couvidat et al.
2012) observes the neutral iron line at 617.33 nm which forms in the low
photosphere (Norton et al. 2006; Fleck et al. 2011). All four Stokes parameters are recorded at six wavelength points covering the spectral line. With
those measurements the full polarimetric information of the light is given
and using all 24 images of data one can reconstruct the solar magnetic field
via inversion techniques (Sec. 2.2.3, 3.5.5). This set of full stokes images
is provided every 12 minutes, others are more frequent. The full disk images taken by HMI have 4096 × 4096 pixels. The spatial resolution of the
telescope is roughly one arcsecond, the size of each pixel is half an arcsecond.
HMI uses a refractory with Galilei design that has an effective focal
length of 2468 mm. The effective focal length of the whole system is 4953 mm.
To avoid unwanted light in the instrument the front window of HMI is a
50 Å bandpass filter.
This section also describes the central instruments that are used in the
optical system of the HMI. Their description follows Schou et al. (2012),
Couvidat et al. (2012) and Stix (2004). The calculations in this section do
not go into detail. The equations refer to a ray entering the optical system
parallel its optical axis. This approximation is sufficient to understand the
physical principles the instrument is working with. For an exact calculation
one would have to take the angle between ray and optical axis into account.
This leads to trigonometric correction in most giving phase differences δ.
Polarization Selectors
Waveplates are thin films of a birefringent material that can modify the
polarization state of light that passes them. HMI uses λ/4 and λ/2 wave
plates that induce phase shifts of π/2 and π. A λ/2 wave plate can be used
to rotate the direction of linear polarized light, a λ/4 waveplate converts
linear polarized light into elliptical polarized light and vice versa.
1
HMI data is public available at http://jsoc.stanford.edu/
2.4. THE INSTRUMENT
25
After the telescope a λ/2, a λ/4 and a λ/2 waveplate are mounted to
select the polarization state of interest and convert it into linear polarized
light.
Fabry-Pérot-Interferometer
A Fabry-Pérot-Interferometer (FPI) is an interference filter made of two parallel, partially transmitting mirrors. The light that transmits the instrument
can interfere with the light reflected twice and then escapes the instrument.
If the distance between the mirrors is d, then all rays transmitted by the
FPI have a phase shift of δ = d/λ. Hence, constructive interference only
occurs for
δ = kλ, k ∈ N.
(2.22)
ISS
In the optical path, before the spectroscopic instruments, an Image Stabilization System (ISS) to finetune the position of the solar image and to
remove jitter (motions of the spacecraft on a short time scale) is included.
Lyot Filter
A Lyot filter makes use of a birefringent material to create interference.
First a polarization filter makes sure, that linear polarized light enters the
instrument. Then a birefringent material induces a phase shift between
ordinary and extraordinary beam (Eq. 2.6). Since this phase depends on
the wavelength, for certain wavelengths the direction of linear polarization
is rotated while for other wavelengths linear polarized light is converted to
elliptical polarized light.
It is now possible to use a polarization filter in the optical path after the
birefringent material that fully transmits a certain wavelength λ0 (because it
is linear polarized) and absorbs parts of elliptical polarized waves. The result
is a wavelength filter for λ0 called Lyot Element. A Lyot Filter typically
consists of multiple such Lyot elements of different lengths.
In HMI the telecentric lense is followed by an interferometric blocking
filter to further reduce the width of spectrum of the incoming light. It is
acting like a FPI and has a with full width half maximum (FWHM) of 8 Å.
The blocking filter is followed by tunable devices used to select a narrow
wavelength and the polarization states of the light. First the wavelength is
selected with a Lyot interferometer. It has a 2:4:8:16:1 design where only
the narrowest filter element is made tunable. The FWHM of the Lyot elements reach from 5.52 Å for the widest element to 344 mÅ for the narrowest
element.
After the Lyot elements again waveplates and a polarizer are mounted.
26
CHAPTER 2. THEORY
Michelson Interferometers (MI)
A MI is an optical instrument that retards parts of the transmitted light
to create interference. It consists of a polarizing beam splitter S0 and two
mirrors S1 and S2. The beam splitter is put with an angle of 45◦ in the
optical path and reflect half of the incoming light. Both, transmitted and
reflected light are then redirected back by mirrors S1 and S2 so both beams
meet again at the beam splitting mirror. S0 reflects incoming light from
S2 (which was previously transmitted) and transmits light from S1 (which
was previously reflected). These two beams now interfere. The distances
from S0 to S1 (d1 ), and from S0 to S2 (d2 ) are not the same in a Michelson
Interferometer, the phase difference between the two polarization states is
now δ = (d1 − d2 )/λ. Only for light that fulfills the criterion δ = kλ, ∈ N
there is no destructive interference.
The HMI optical system makes use of two tunable Michelson Interferometers. The wideband MI has a FWHM of 172 mÅ and the FWHM of the
narrowband MI is 86 mÅ. The resulting FWHM of the HMI filter system is
612 mÅ for the untuned part and 76 mÅ for the tuned part.
CCD
A beamspliter spreads the light to two CCD detectors. One of them measures the line of sight magnetograms and velocity, it is comparable to the
MDI instrument on SOHO (Scherrer et al. 1995). The other CCD returns
data for the vector magnetic field.
This brief description of HMI gives an idea of how the instrument works
in principle. It is not meant to be exact or complete. For example the design
of the Lyot filter is in reality much more complicated as described here. See
also http://hmi.stanford.edu/ for more information about the instrument.
Chapter 3
Data Analysis
3.1
Introduction
The goal of this thesis is a statistical analysis of sunspot properties recorded
with the HMI instrument onbord the SDO spacecraft. We use 2.5 years of
data from May 1st , 2010 to October 31st , 2012. This is basically the rising
phase of solar cycle no. 24. The cadence used for the analyses is roughly 24
hours.
The selection of sunspots can be divided into three parts: First each
sunspot and umbra were marked manually. This had the disadvantage that
there was no unified criterion for umbrae. We therefore applied an automatic
threshold technique to create a second dataset of sunspot umbrae. Finally
this data was reduced manually to avoid duplication in the data.
The result of the data selection were four sets of binary masks for
sunspots and umbrae:
• The first one, created with manual selection of sunspots and manual
selection of umbrae.
• A second one by applying a threshold method on each sunspot in the
first set to select the umbra.
• Two small sets with unique spots.
Section 3.2 introduces methods that were used in the analysis in detail.
Sections 3.3,3.4 and 3.5 describe the creation of the four data sets mentioned
above and the analysis carried out. This chapter further contains all plots
and tables with fit parameters and other results. The scientific meaning of
the results will then be discussed in Chapter 4.
Calculations in this thesis were carried out with interactive data language
(IDL). While there are other computing languages which are more efficient,
IDL can be very useful since reading data, simple calculations and the export
of figures is very straightforward. IDL is commonly used in astrophysics.
Hence, many routines exist in libraries and do not have to be implemented.
27
28
3.2
3.2.1
CHAPTER 3. DATA ANALYSIS
Algorithms and Methods
Correction for Limb Darkening
If we want to analyze intensities at different heliocentric angles we have to
correct the data for limb darkening 2.1.3. This can be done by normalizing
the measurements to the local mean quiet sun intensity. A more sophisticated way is to correct each pixel for the limb darkening at its heliocentric
angle. With this method we remove of the intensity gradient towards the
limb. This is especially important for a sunspot selection with universal
threshold.
In order to correct the data for limb darkening, an empirical relation
for mean intensity as function of heliocentric angle has been derived. We
chose twelve full disk images with no visible active regions and calculated
the cosine of the heliocentric angle, µ for each pixel in each of those images.
d
.
(3.1)
µ = cos Θ = cos arcsin
r
The coordinates of the solar center and the solar radius r were taken from
the header of the data file. d is the distance between the pixel of interest
and the disk center. We normalized each intensity map of those twelve days
to the mean intensity of a square of 200 pixels width at disk center. We then
sorted the intensity measurements with respect to their heliocentric angle
and binned this data into 500 bins. The black curve in figure 3.2 shows this
empirical limb darkening function. The gray dots are a sample of intensity
measurements, each referring to one pixel. We overplotted a limb darkening
curve following Neckel (1996) for λ = 617.33 nm for comparison in red.
Figure 3.2 shows two full disk images at November 11th, 2011. The left
panel is not corrected for limb darkening. In the right one each pixel in each
image was corrected by dividing the measured intensity by the intensity
of the empirical limb darkening function at the corresponding heliocentric
angle. Pixels with µ < 0.17 were set to the value 2 which is higher than
the brightest regions visible on the solar disk and therefore shown in white.
This simplifies further analysis since all dark pixels in the data then belong
to magnetic features.
3.2. ALGORITHMS AND METHODS
29
Figure 3.1: The limb darkening function. The black curve shows our empirical function. The gray dots are a sample of single intensity measurement.
Each dot shows the intensity of a single pixel. The scattering is due to
intensity differences between granules and intergranular plasma. The red
curve shows a polynomial approximation for λ = 617.33 nm following Neckel
(1996).
Figure 3.2: Original and limb darkening corrected full disk image. Both
images were normalized to their mean at disk center. In the corrected image
values with µ < 0.17 were set to white to simplify further analysis.
30
3.2.2
CHAPTER 3. DATA ANALYSIS
Center of Gravity
We need to calculate the position of a sunspot in order to be able to deproject
its area. This position is later used to track sunspots (Sec. 3.5). We decided
to identify the position of a sunspot with the center of gravity (COG) of its
polarization map (cf. Eq. 2.9), and the position of an umbra with its COG.
Since umbrae are darker than the surrounding sun we apply the COG
method to one minus the normalized umbral intensity map. Penumbra and
quiet sun have been excluded from the calculation by setting their intensities
to zero. Figure 3.3 shows an example of an inverted umbral intensity map
(AR11131). It also shows the intensity profile for a cut thought the umbra.
Figure 3.7 shows in the right panel the intensity map of this spot with umbral
contour.
Figure 3.3: The left panel shows an inverted intensity map of the umbra of
AR11131 (see the right panel of Fig. 3.7 for the original map). The threshold
value chosen to separate the umbra from the penumbra is 0.6 relative to the
quiet sun (i.e. 0.4 in the inverted image). The red line shows a cut through
the image. The intensities along this cut are plotted in black in the right
panel. The gray curve also shows penumbral and quiet sun intensity. A
center of gravity method was applied to the inverted intensity map to find
the center of the umbra (red cross). The reason that inverted quiet sun
intensity does not scatter around zero is that the map in this example has
not been corrected for limb darkening.
3.2.3
Separation of Distinct Umbrae
The separation of sunspots was done manually with the mouse (Sec. 3.3).
The separation of multiple distinct umbrae was done with the automatic
method described below. The first step is to create a mask for all umbrae
within one sunspot by finding all pixels darker than a particular threshold.
3.2. ALGORITHMS AND METHODS
31
We start with this binary map which may contain multiple distinct umbrae.
The result will be multiple binary maps, each containing one umbra.
The straightforward way to go was to use the standard IDL programs
CONTOUR.PRO and POLYFILLV.PRO. CONTOUR.PRO returns the coordinates of closed paths around regions below a given threshold value.
Those contour paths were then filled by POLYFILLV.PRO, it returns the
pixels inside each contour. Unfortunately the masks returned by POLYFILLV.PRO were not exactly identical with the input. The reason lies probably in the limitation due to numerical representation of floating numbers.
We programmed an algorithm capable of separating regions of given
intensity without using CONTOUR.PRO. This algorithm operates on the
discrete array (the the pixels of the image), in contrast to CONTOUR.PRO,
which uses normalized coordinates (relative positions on the map as a floating value).
Algorithm
The program which separates the umbrae starts with a binary mask of multiple distinct umbrae, the master. It assigns the first umbral pixel in the
master the value u1 , this is the index of the umbra it belongs to.
The algorithm checks all neighbor pixels of the master if they are umbral
pixels. If this is the case they are also marked with the value u1 since they
belong to the same umbra. This is recursively repeated on those pixels, i.e.
the neighbors of the neighbors are checked as well. If no other umbral pixel
of the master touches a pixel with value u1 , the first umbra is complete and
the loop terminates.
The program continues the same algorithm with the next umbral pixel
(that does not belong to u1 ) of the master. Since this pixel has to belong
to another umbra, is labeled u2 . The algorithm described in the previous
paragraph is applied on u2 , the second umbra is marked this way. The
program continues with all remaining pixels in the master until all umbrae
are separated.
3.2.4
Umbral Size Distribution and Lognormal Fit
We analyzes the distribution of umbral sizes with multiple datasets. We
used the method described in this section for all size distributions in this
thesis.
Histogram
Since there are many more small umbrae than large ones, the bin width was
chosen to increase exponentially. We show all graphs of size distributions
as log-log-plots (e.g. Fig. 3.8). As position of the bin we chose its middle
value in logarithmic space, which is the middle of each bin in the plots. This
32
CHAPTER 3. DATA ANALYSIS
is lower than the arithmetic mean of lower and upper boundary of the bin.
The largest umbrae have a size around 200 MHS. The upper boundary of
the histogram was chosen to be equal the largest umbra in the data. It is
somehow arbitrary to chose the lower boundary of the histogram. We start
at 0.5 MHS as explained below.
We assume an uncertainty of one pixel for the umbral radius and identify
the umbral radius with the radius of a circle with the same area as the umbra
(umbral equivalent radius). The smallest umbrae therefore have the largest
relative uncertainties in their area. The width of the first (and smallest) bin
of the histogram depends on the overall number of bins and on the lower
boundary on the histogram. Those two values were chosen in a way that
this minimal bin width is about the same size as the uncertainty of the size
of the umbrae it contains. For a histogram with 50 bins this minimum lower
size is about 0.5 MHS. For a histogram with more bins this value would have
been higher.
The histogram was then normalized to its area by normalizing each bin.
For all bins the number of counts in the bin was divided by the total number of counts and multiplied with the relative width of the bin. The relative
width is the width of the bin with respect to the whole histogram. This
procedure normalized the histogram to its integral and kept the shape invariant.
Lognormal Fit
A normalized histogram has the advantage that it is possible to fit the
normalized lognormal distribution (Eq.2.13), an analytic function with only
two free parameters.
To weight the bins the errors were taken proportional to the inverse of the
square root of the number of counts in each bin. We did not consider errors in
the size of sunspot umbrae (i.e. the position of the bin). Although the errors
in the lower and upper part of the size distribution are relative large (cf. e.g.
Fig. 3.5), the rather constant error for the medium sized spots justifies the
exponential increasing binwidth. We fit the lognormal distribution with a
Levenberg-Marquardt least square method to the histogram.
Characteristic Quantities
Mean and Standard Deviation We calculate the mean value and the
standard deviation for each dataset where we fitted the lognormal curve.
The values calculated from the data depend on the lower boundary (0.5 MHS)
of the histogram. We also calculate the mean and the square root of the
variance of the lognormal curve from the fit parameters. Those parameters are not exactly comparable since the analytic curve is defined for all
x < 0. The fitted parameters σ and µ have no physical dimension, Mean
3.3. MANUAL SUNSPOT SELECTION
33
and square root of the variance can be interpreted as position and width of
the distribution in MHS.
Large Umbrae The relative number of umbrae decreases rapidly with
size. Because there are little large umbrae in our data, the error bars of the
the upper part of the histogram are large and the least square algorithm
is not very sensitive to them. It might be that an absence of large umbrae
is not reflected in the fitted curve. We therefore investigate large umbrae
separately. We define a ”large umbra” to be three times larger than the
mean umbral size. If the distribution of umbral sizes in invariant, then the
fraction of large umbrae should not change.
3.3
Manual Sunspot Selection
The first step was to mark all sunspots manually for each continuum intensity map. The progress was as follows: The full disk image of the sun
(4k×4k pixels) was loaded, rebinned for visualization and displayed. We
then selected an active region manually. This returns a sub image with an
edge length of about 200 to 500 pixels on each side. We displayed the sub
image in full size.
In order to correct for limb darkening we selected the quiet sun close
to the sunspot of interest and normalized each sub image to the mean of
this local continuum. Next, we defined an intensity threshold to select the
sunspot penumbra. This threshold was adjusted manually for each sunspot.
We excluded other surrounding sunspots and pores of the same active region
with the mouse.
In the cases where it was not possible to select the sunspot penumbra via
threshold we used the mouse to select it manually. Often the selection of the
penumbra was a combination of manual and threshold method. The result
is a binary mask for the sunspot, the gray mask in figure 3.4 for example.
This sunspot mask was then used to select the umbra with a threshold.
By this definition each pixel within the mask which is darker than this
threshold belongs to the umbra. The threshold was chosen individually
for each spot. We applied the center of gravity method as described in
section 3.2.2 to calculate the position of umbra and sunspot. The manually
selected sunspot masks are the foundation for this thesis. We used them in
most further data analysis to create different datasets of umbrae.
34
CHAPTER 3. DATA ANALYSIS
Figure 3.4: Binary masks for sunspot and umbra (combined) and normalized
intensity map of the sunspotspot (AR11131). Figure 3.7 shows the same
sunspot with intensity contour levels.
Figure 3.5: Umbral size distribution with lognormal fit. The histogram includes a total of 4229 sunspots. The umbrae have been selected with an
individual threshold for each sunspot. Distinct umbrae within one sunspot
have not been counted separately. The fit parameters can be found in table,3.1.
3.3. MANUAL SUNSPOT SELECTION
3.3.1
35
Disadvantages om Manually Selected Data
The dataset with manually marked umbrae bears some problems. First the
umbral sizes are somehow subjective and not reproducible. Further, we
normalized the sunspot intensity to the local continuum close to the spot.
We did not correct each pixel individually for limb darkening. Figure. 3.6
shows this local mean intensity plotted against the cosine of the heliocentric
angle of the spot it belongs to. The scatter in this plot is because the local
mean had not been chosen at the exactly same heliocentric angle as the
umbral position. Since the intensities of sunspot umbrae are normalized
to those intensities shown in the plot, the scattering of the plot propagates
into the umbral intensity statistics and induces errors. 1 We can avoid those
errors with the proper limb darkening correction as described in section 3.2.1.
Figure 3.6: Quiet sun limb darkening. Each black dot shows the normalized
continuum intensity of a small region close to a sunspot. Those intensities
are plotted against the µ-value of the umbral centers of the sunspots. The
scattering of these values encouraged us to carry out a more careful limb
darkening correction (described in section 3.2.1), leading to the empirical red
limb darkening curve (see also Fig. 3.1). Most dots are situated above that
curve, this indicates that the local continuum had been chosen systematically
closer to disk center compared to the sunspot position.
1
The manually selected umbrae were not used for an intensity analysis.
36
3.4
CHAPTER 3. DATA ANALYSIS
Automatic Threshold Method
We corrected all intensity maps with the algorithm described in section 3.2.1
and continued the analysis with those images. We applied a threshold
method which selects umbrae automatically. Multiple thresholds have been
tested, (Fig 3.7). See also section (Sec. 3.4.1) for a more detailed discussion. For the analysis of the size distribution we sicked to the rather high
threshold of 0.6.
The automatic threshold method makes use of the previously saved
sunspot masks. We open those masks and the corresponding limb darkening corrected images. The regions darker than the threshold are selected
as umbra. We further apply the algorithm described in section 3.2.3 to distinguish between distinct umbrae in the same sunspot. This was not the
case with the first (manual) dataset.
Figure 3.7: The left image shows a complex sunspot configuration
(AR11476), the spot on the right is a simple circular spot (AR11131). Contours of two different threshold values (0.6 and 0.52) with respect to mean
quiet sun intensity ave been overplotted in red and blue respectively. The
left image shows pores aswell.
3.4. AUTOMATIC THRESHOLD METHOD
3.4.1
37
Main Curve
We have manually marked a total of 4229 sunspots as described in section 3.3. We applied the threshold algorithm as described in section 3.4 on
the limb darkening corrected maps using the mask of those sunspots. We
found 6892 distinct umbrae among 4194 sunspots. The difference in the
number of sunspot is because the umbrae in 35 spots were brighter than the
threshold and therefore those spots were ruled out. Each of the 6892 umbrae
has an area larger than 0.5 MHS. We created two histogram of umbral sizes
(Fig. 3.8), one with all umbrae counted separately and one with the total
umbral area of each sunspot. We fitted lognormal distributions as described
in section 3.2.4. Fit parameters can be found in table 3.1.
Figure 3.8: Umbral size distribution of 4194 sunspots containing 6892 umbrae. In the left panel distinct umbrae within one sunspot were counted
separately, the right panel shows a histogram of the total umbral size per
sunspot. The data has been binned in 50 bins with exponentially increasing
width. Errors in x direction mark the width of the bins, the y errors are
taken proportional to the number of counts in each bin. The black line is
a Levenberg-Marquardt least square fit. The dashed curves give the one
sigma variation of the fit. Parameters for the lognormal function can be
found in table 3.1. The distribution and fit are compared to other findings
in section 3.5.3 and in figure 3.16.
We use the dataset containing separated umbrae for a more detailed
analysis on the umbral size distribution. Figure 3.8 can therefore be interpreted a main result, from now on we are analyzing variations or subsets of
this data. The two histograms of figure 3.8 are shown again in figure 3.16 in
the end of this section where we compare different results.
38
CHAPTER 3. DATA ANALYSIS
Influence of Threshold
The easiest way to select a sunspot umbra is by applying an intensity threshold. Choosing a particular intensity threshold is often somehow arbitrary.
Figure 3.3 shows a sunspot intensity profile. In this example possible values
for thresholds range roughly from 0.3 to 0.6 of mean quiet sun intensity.
This profile is just an example and there are much smaller spots than this
one. Smaller spots tend to have brighter umbrae than large ones (cf. e.g.
Fig 3.19, although we discuss minimum intensity in that case) and therefore
might require a higher threshold to be properly selected. Since we use an
universal threshold for all sunspots, we decided to stick to the rather hight
value of of 0.6. With values higher than this we would have partly selected
penumbra as well.
Figure 3.7 shows a rather complex and a simple sunspot with two different threshold contours. With relative low thresholds one might not detect
small and bright umbrae. Higher thresholds on the other hand are not capable of separating umbrae divided by umbral light bridges (Muller 1979).
Using a high threshold should also result in larger umbra sizes.
The left panel of figure 3.9 shows the umbral size distribution for two
different intensity thresholds. Fit parameters are listed in table 3.1.
Figure 3.9: The left plot shows the umbral size distribution of umbrae selected with different thresholds. The black and red curve and histogram
correspond to a threshold of 0.6 and 0.5 relative to quiet sun intensity, respectively. Error bars have been omitted for clarity. Figure 3.8 shows the
black curve with errors. The right panel shows the umbral size distribution
for smoothed data. Fit parameters can be found in table 3.1.
Influence of Resolution
The spatial resolution of the telescope might have a similar influence on
the measured size distribution of sunspot umbrae as the threshold. With a
blurred image it might not be possible to resolve light bridges. We carry
3.4. AUTOMATIC THRESHOLD METHOD
39
out the same analysis as described in section 3.4.1, once with smoothed data
and once with a different threshold. Smoothing has been done by folding
the intensity profile with a box car function of three times three pixels.
We show the size distribution for the smoothed data in figure 3.9 and
the fit parameters in table 3.1.
3.4.2
Time and Hemisphere Separation
In order to understand the physics of the solar cycle it is of great interest to
know which parameters underlay periodic changes and which stay constant.
Although our data covers only a time span of 2.5 years it is still interesting
whether the size distribution is invariant during this time. There might be a
phase difference in the solar cycle between northern and southern hemisphere
(Temmer et al. 2002).
The analysis of the size distribution as described in section 3.2.4 has been
carried out for different subsets: We separated the data in time into two
parts (divided at August 1st , 2011) and in hemisphere. Figure 3.10 shows
the histograms for time separation and for hemisphere separation. Further
we analyzed four divided subsets where we separated for both hemisphere
and time. The results of all fits can be found in table 3.1, the fit is in the
Appendix (Fig 5.1) .
Figure 3.10: The right figure shows the umbral size distribution for the first
and second half of the data, separated at August 1st , 2011. The graph contains a plot of the data separated by hemispheres. Lognormal functions have
been fitted to the histograms as described in section 3.2.4. Fit parameters
can be found in table 3.1. Error bars have been omitted for clarity.
40
3.4.3
CHAPTER 3. DATA ANALYSIS
Mean Umbral Size
We analyzed the temporal variation and the variation with heliocentric angle
of the mean unbral size and intensity during the rising part of solar cycle
no. 24. The mean intensity and the mean size of the 6892 umbrae were
binned in 5 bins each covering six months. Figure 3.11 shows this data and
a linear least square fit.
Figure 3.12 shows the mean umbral size and averaged mean umbral intensity plotted against the heliocentric angle. The data has been binned in
eight bins each covering 10◦ .
Figure 3.11: Mean umbral intensity and umbral radius of 6892 versus time.
The data was binned in five bins each covering six months. The error bars in
radius and intensity represent the one sigma error of the mean values. The
error bars in time mark the width of the bins and were not included in the
linear fit (solid line). Dashed lines give the one sigma uncertainties of the
fit. The black curve in the right panel shows the umbral minimum intensity
which was not corrected for limb darkening, corrected minimum intensity is
shown in red and corrected mean intensity in blue. Fit parameters can be
found in table 3.3.
3.5. REDUCED DATASET
41
Figure 3.12: Heliocentric angle dependence of umbral radius and umbral
intensity. Data of 6892 umbrae was binned in eight bins of 10◦ width each.
The error bars in x direction mark the width of the bins, the error bars in y
direction represent the one sigma error of the mean values. Colors are chosen
as in figure 3.11. Since there is probably no real variation with heliocentric
angle, the figure primarily tells us about systematic errors due to projection
effects and about limb darkening.
3.5
Reduced Dataset
In the previous sections we analyzed one full disk image per day. Hence, it
is possible for a long living sunspot to be recorded up to 14 times. To avoid
such duplication Bogdan et al. (1988) limited their data to ±7.5◦ around
the central median (cf. Fig. 3.13). While this is an easy way to omit most
of the duplication one does also miss many short lived sunspots. Further
without taking differential rotation properly into account there might still
be some duplicates in the data.
We applied a manual reduction of the data to avoid counting sunspots
multiple times (Figs. 3.13, 3.14). This has the large advantage to not only
avoid duplication while still counting all sunspots. Further since we choose
manually which spot we want to use we can sick to young sunspots while
not counting decaying active regions. This is of interest because the decay
and evolution of spots might influence the umbral size distribution. If we
stick only to young spots we can omit such effects at least partly.
3.5.1
Sunspot Tracking Algorithm
The pixel coordinates of the 4194 sunspots in the data have been transformed into a heliographic coordinate system (longitude Φ with respect to
the central meridian and latitude Θ with respect to the solar equator). For
this coordinate transformation two angles are needed: the inclination of the
solar rotation axis with respect to the ecliptic and the orientation of the
instruments CCD chip with respect to the central meridian of the sun. The
42
CHAPTER 3. DATA ANALYSIS
Figure 3.13: Heliographic coordiante system, the dotted grid for longitude
and lattitude has a width of 15◦ , solar radius is chosen as 960 arcsec. In the
left panel the heliocentric angles µ = 0.5 (60◦ ) and µ = 0.17 (≈ 80◦ ) are
highligted in blue and orange respectively. The solid black lines are at ±7.5◦
around the central meridian, the sun rotates roughly 15◦ per day. The red
line shows an approximation for differential rotation. It corresponds to the
left black line after 24 h following equation 2.1. The right coordinate system
visualizes is tilt of the solar coordinates with respect to the observer. The
inclinations are overstated. We correct for this tilt and for the differential
rotation when we track sunspots.
right panel in figure 3.13 shows the tilted coordiante system. Both angles
and the current solar radius in arcseconds were taken from the header of the
data file.
Starting with the first sunspot in the data its theoretical positions on
the 13 succeeding days have been calculated. We used the empirical formula
given in the Chapter 2 to correct for differential rotation (Eq. 2.1).
We then visualized sub images of the sun (centered on the calculated theoretical sunspot position) for those 14 days. All spots close to this position,
”close” was chosen to be less than 15◦ distance in latitude and longitude,
were highlighted. Figure 3.14 shows such a time series, exemplary for four
successive days, not for all 14. With this algorithm it was possible to track
all spots in an active region for as long as they were visible on the sun. The
spots of interest then have been marked manually and the rest were deselected. This algorithm was then used successively on the remaining spots in
the data.
The algorithm just described was applied twice with different selection
criteria. This resulted in large subset containing 488 sunspots (Sec. 3.5.2)
and a small a small subset with 205 sunspots (Sec. 3.5.4).
3.5. REDUCED DATASET
43
Figure 3.14: Visual tracking of the active region AR11260. The figure gives
an impression of how the algorithm that was used to track sunspots works.
The sub images show the same region on the sun on four successive days.
The algorithm locked on the left sunspot in the first date, the theoretical
position of this spot on successive days was calculated and nearby spots
were highlighted. The sunspots marked with green rectangles were selected
manually, red ones were discarded. This figure is just for explanation, the
analysis was carried out for all appearances of an active region (maximum
fourteen) and not just four.
44
3.5.2
CHAPTER 3. DATA ANALYSIS
Large Subset
For the first subset of data we selected each sunspot once. The main criteria
for selection were a full evolved umbra and a relative young age of the spot.
Spots in decaying phase were omitted. None of the selected spots has a
heliocentric angle larger than 60◦ (see the blue line in figure 3.13). This
subset was used for statistical analysis of umbral sizes. We used a threshold
of 0.6 to select the umbrae.
The dataset of 488 spots (910 distinct umbrae) was used to create an
umbral size distribution as described in section 3.2.4. See figure 3.15 for the
plot and table 3.1 for the fit parameters.
Figure 3.15: Umbral size distribution with log normal curve fit. The histogram consists of 910 umbrae out of 488 sunspots. The selection of sunspots
as described in section 3.3. In this distribution duplication has been omitted. We therefore expect less large umbrae compared with the histograms
in figure 3.8, since large sunspots trend to live longer than small ones. Fit
parameters can be found in table 3.1.
Fig.
3.5
3.8
3.8
3.10
3.10
3.10
3.10
5.1
5.1
5.1
5.1
3.9
3.9
3.5
Data
Manual
Combined
Separated
N
S
First
Second
N, First
N, Second
S, First
S, Second
Smoothed
Th = 0.5
Unique
N
4229
4194
6892
2677
4215
1999
4893
1341
2874
658
2019
5972
5148
910
Nlarge
228
274
567
203
364
137
430
106
258
31
172
474
432
65
Nlarge /N
0.054
0.065
0.082
0.076
0.086
0.068
0.088
0.079
0.090
0.047
0.085
0.079
0.084
0.071
X̄
18
21
13
13
13
12
13
13
13
10
13
15
12
13
s
18
25
20
19
20
16
21
18
20
13
21
21
17
20
σ
0.969
1.143
1.54
1.51
1.54
1.50
1.55
1.49
1.55
1.43
1.51
1.47
1.56
1.54
σerr
0.015
0.017
0.04
0.05
0.04
0.06
0.04
0.06
0.05
0.11
0.05
0.03
0.05
0.09
µ
2.47
2.59
1.56
1.61
1.53
1.56
1.57
1.66
1.48
1.39
1.68
2.00
1.61
1.64
µerr
0.02
0.02
0.05
0.07
0.06
0.08
0.06
0.08
0.07
0.15
0.07
0.04
0.06
0.12
χ2
4.13
3.01
10.5
3.00
4.47
2.28
6.30
1.01
3.00
0.87
2.00
7.01
8.52
1.10
Mean(ρ)
18.9
25.6
15.5
15.6
15.1
14.7
16.0
16.0
14.6
11.2
16.8
21.8
16.9
16.8
sqrt(Var(ρ))
18.7
30.3
26.8
26.2
26.0
24.3
27.8
26.2
25.4
17.4
28.1
35.1
29.6
29.1
3.5. REDUCED DATASET
Table 3.1: Parameters and errors for the fitted lognormal distribution (Eq. 2.13). The second row indicates which data was
used. ”First” means the histogram and fit contain only data from May 5th ,2010 until July 31st , 2011. ”Second” refers to
the interval between August 1st , 2011 and October 31st , 2012. ”N” and ”S” refer to northern and southern hemisphere
respectively, the row indicated with ”Smoothed” was derived from a smoothed data. ”Th = 0.5” gives the value of the
threshold used to select sunspot umbrae. If not mentioned otherwise this threshold is chosen as 0.6. X̄ is the mean umbral
size, ”large” umbrae are larger than three times this value. s is the standard deviation of the data. The last to columns give
mean and square root of the variance of the fit.
45
46
3.5.3
CHAPTER 3. DATA ANALYSIS
Comparison Between Umbral Size Distributions
We compare the different umbral size distributions described in the previous
sections with each other. Further we compare them to results published by
Bogdan et al. (1988) and Schad & Penn (2010).
Bogdan et al. (1988) limited their data to ±7.5◦ around the central
meridian and therefore omit the influence of limb variation, projection effects and Wilson depression on the umbral sizes (Sec. 4.2). They fit umbral
sizes with a rectangle.
Schad & Penn (2010) used an automatic threshold method on one image
per day as we do. There is a difference between their and our data, they
used a different instrument observing another wavelength (they use an Fe I
line at 868.8 nm) and chose a slightly higher threshold. They also do not
manually select the sunspots and started their fit at 1.5 MHS (we started at
0.5 MHS). This means that they include pores in their data.
The upper panel of figure 3.16 shows our histogram of 6892 umbrae (same
as Fig. 3.8, left) plus the the size distribution of our reduced dataset and
findings for rising phase, cycle maximum and total data of solar cycle no. 23
by Schad & Penn (2010). The lower panel shows our histogram and fit
for combined umbrae (same as Fig. 3.8, left). It also contains the distribution combined umbrae, of manually selected umbrae and the findings from
Bogdan et al. (1988).
3.5. REDUCED DATASET
47
Figure 3.16: Histograms of the sizes of separated umbrae (upper panel) and
combined umbrae (lower panel), same as in figure 3.8. We adapted fits form
Bogdan et al. (1988) and Schad & Penn (2010) and normalized them with
equation 2.15. Further the fit to manually selected umbrae and the fit the
reduced dataset with unique umbrae (Figs. 3.5, 3.15 are shown in blue. Each
panel contains the fit to histogram in the other one (red curve).
48
CHAPTER 3. DATA ANALYSIS
Chi Square Method
In order to be able to compare different size distributions more qualitatively
than just to plott the the curves in the same graph and check visually, we
decided to calculate a proxy that describes the deviance of a curve to the
data. The quality of a fit can be described using the χ2 (Eq. 2.21). We
already made use of this when we calculated the best possible fit to different
datasets. We now calculate the χ2 between our data and the curves described
in the previous section. This χ2 is not the lest possible χ2 , but the smaller
it is, the better does a particular curve fit our data. This method makes it
possible to decide which curve published by the other authors fits our data
best.
Table 3.2: Deviation between our data and different lognormal distributions. The columns χ2S and χ2C refer to the histograms with separated and
combined umbrae respectively. The values are computed with equation 2.21.
Results adopted from Schad & Penn (2010) and Bogdan et al. (1988) were
normalized with equation 2.15. The values in this table can be used as a
proxy how good other curves fit our data. The Schad & Penn (2010) results
are listed chronologically starting with the declining phase of solar cycle
no. 22. The least possible χ2 is the value belonging to the curve that was
actually fitted to the data (bold). Figure 3.16 shows the two histograms and
most of the fits.
Fig. Fit
χ2S
χ2C
3.8
Separated Umbrae
10.5 76.1
3.8
Combined Umbrae
190 3.01
3.5
Manually Selected Umbrae
256 13.1
3.15 Reduced Dataset
11.0 68.6
Schad & Penn Falling
18.5 51.7
Schad & Penn Minimum
40.7 78.7
Schad & Penn Rising
26.3 44.1
Schad & Penn Maximum
37.5 59.2
Schad & Penn Falling
38.2 63.0
Schad & Penn All
30.2 52.4
Bogdan et al. Lower
119.8
215
Bogdan et al. Upper
155.9
249
3.5. REDUCED DATASET
3.5.4
49
Small Subset
For the second subset we focused only on fully evolved sunspots. In order
to reduce scatter no complicated spots were selected. This led to a small
subset of 205 sunspots which was used to analyze the correlation between
umbral properties (Sec. 3.5.8).
The penumbrae of the sunspots were carefully selected by applying an
individual threshold on the intensity images. Bright penumbral parts which
were completely surrounded by penumbra darker than the threshold were
still counted as penumbra. In some complicated cases the penumbral mask
had manually been corrected via mouse fore example when different sunspots
touched.
In this subset we used a threshold of 0.52 (cf. Fig. 3.7) to select the
umbra since we did not include very small spots and since we want to make
sure to omit penubral light in the umbra when we analyze mean intensity
and field strength.
The dataset with 205 sunspots was used to analyze correlations between
umbral and sunspot size, magnetic flux, and umbral magnetic field strength.
3.5.5
Inversion of the Magnetic Field
We used the VFISV code (Sec. 2.2.3) to invert the magnetic field of all
sunspots in the small subset. This calculation was performed by one of
the authors supervisors, Dr. Reza Rezaei, who applied the code using the
sunspot maps defined by the author. Figure 3.17 shows contains a sample
of an inversion result and the continuum intensity for AR11166.
3.5.6
Calculation of Sunspot Properties
The binary masks for umbrae and sunspot were used on the continuum
intensity and the inverted maps to calculate mean and minimum intensity,
mean and maximum magnetic field and magnetic flux. Further equivalent
radii for spot and umbra were calculated from a circle of the same size as
the area of the mask.
Uncertainties
HMI measures the total intensity and the intensities of different polarization
states of the light. Uncertainties in the Stokes parameters propagate in all
umbral parameters. We assume the errors of the intensity measurements to
be small.
The measurement of the umbral size further depends on the threshold we
choose. The judgment of the observer is in principle a source for systematical
errors. The uncertainty is at least in the order of the resolution of the
telescope. The uncertainty of the magnetic field depends, beside of the
50
CHAPTER 3. DATA ANALYSIS
Figure 3.17: Example of an inversion result (AR11166). The four panels
show the continuum intensity and the magnitude, azimuth and inclination
of the magnetic field.
3.5. REDUCED DATASET
51
accuracy of the Stokes vector, on the reliability of the inversion code. All
those uncertainties finally propagate into the uncertainty of the magnetic
flux. Unfortunately this makes the physically most fundamental parameter
in our discussion also the one with the largest error.
We used a measurement uncertainty of 0.001 for the relative intensity,
150 G for the magnetic field strength and 0.5 Mm for the equivalent radius.
The error for the flux is propagated form the error of the size plus an error
of 5% for the magnetic field for each umbra.
3.5.7
Temporal and Hemisphere Variation
We analyzed the temporal variation of the umbral radius and the minimum
intensity with the dataset containing 205 sunspots. We separated the data
by hemispheres and averaged the measurements so each bin but the last
one includes ten sunspots. The separation in hemispheres was done because
the a phase difference in the solar cycle could cover a temporal variation.
Figure 3.18 shows a scatter plot with linear least square fits for this data. We
discuss our findings in section 4.3.1. Compare also with section 3.4.3, where
we carried out a similar analysis with the dataset containing all spots.
Figure 3.18: Temporal variation of umbral radius and minimum intensity.
Each data point is the average of minimum ten sunspots (the last ones
contain more). The data has been separated in hemispheres since there is
a phase shift in sunspot number between them. The solid lines show linear
fits, the dashed lines are the one sigma errors of the fit. Fit parameters can
be found in table 3.3.
52
CHAPTER 3. DATA ANALYSIS
Table 3.3: Fit parameters and one sigma errors for temporal variation of
umbral intensity and size. We fitted the function y = ax + b to the data. x
is the time in days starting with (May 1st , 2010) = 0. Umbral radius and
intensity are labeled with R and I, northern and southern hemisphere with
N and S, respectively. The index N.C. indicates that the measurements were
for not corrected for limb darkening.
Fig.
3.11
3.11
3.11
3.11
3.18
3.18
3.18
3.18
3.5.8
y
R
IMIN, N.C.
IMIN
IMEAN
RN
RS
IMIN, N
IMIN, S
a
2.95
0.288
0.344
0.470
5.33
4.22
0.172
0.198
aerr
0.16
0.010
0.008
0.006
0.46
0.48
0.016
0.019
−0.3
1.6
0.7
0.7
−0.8
12
2.3
−1.2
b
10−4
10−5
10−5
10−5
10−4
10−4
10−5
10−5
3.1
2.0
1.6
1.2
8.4
7.4
3.0
2.9
berr
10−4
10−5
10−5
10−5
10−4
10−4
10−5
10−5
Size-Intensity-Magnetic-Field Relationship
We expect the umbral intensity, magnetic field strength and size to be correlated. Figures 3.20,3.19 and 3.21 show scatter plots of those parameters for
both mean values and the maximum field and minumum intensity. We fitted
linear and power law functions to each dataset. The linear approach is the
most simple fit. We provide linear curves especially to be able to compare
our data to the findings of other authors. See section 3.5.11 for a detailed
discussion about the comparison of different intensity measurements.
Both intensity and magnetic field level off with increasing umbral size.
This motivates us to fit a power law, especially since it has only two free
parameters in contrast to second order polynomials which would have been
another possibility. Nonlinear dependencies in the radius-field and in the
radius-intensity correlations imply that the relation between intensity and
magnetic field is probably not linear as well. We therefore chose to include
a power law fit in figure 3.21.
3.5. REDUCED DATASET
53
Figure 3.19: Umbral intensity as function of umbral equivalent radius.
Error bars are shown in gray. Power law and linear least square fit are
overplotted in blue and red respectively, the fit parameters and correlation
coefficients are given in table 3.4. The dashed lines show the one sigma
errors of the fit. The selection criteria for the 205 sunspots are described in
section 3.5.4.
Figure 3.20: Umbral magnetic field strength as function of umbral equivalent radius. See also description in figure 3.19.
Figure 3.21: Relation between umbral intensity and magnetic field strength.
See also description in figure 3.19.
54
3.5.9
CHAPTER 3. DATA ANALYSIS
Magnetic Flux
Magnetic flux through the solar surface is the reason for sunspots. It is
therefore interesting to see how the parameters of the spot, namely the
umbral size, intensity and magnetic field strength depend on the total flux
through the spot. Figures 3.22, and 3.23 show scatter plots of those values.
For the intensity and maximum magnetic field strength we fitted power law
functions. In the flux-size relation (Fig. 3.23) we used a linear fit as first order
approximation. From the physical point of view this is not really justified.
Since flux is basically the product of the size and field strength in the spot
(see Eq. 2.2) it should scale linear with area, not with radius. Plotting the
area of the umbra would not be consistent with the other discussion in this
section. The left panel of figure 3.23 shows the ratio of umbral to penumbral
flux versus the total flux thought the sunspot. We fitted a linear function
to the relation. All fit parameters, errors and correlation coefficients can be
found in table 3.4.
Figure 3.22: Maximum umbral magnetic field strength and minimum umbral intensity with respect to the total magnetic flux through sunspots. In
both cases power law curves were fit with a least square procedure. Fit
parameters and correlation coefficients are given in table 3.4.
3.5. REDUCED DATASET
55
Figure 3.23: The left panel shows the variation of the ratio of flux through
the umbra and penumbra depending on total flux. The single spot that
shows a exceptional hight value turned out to have a weak penumbra. The
right panel shows dependence of the size of umbrae on the total magnetic
flux through the sunspot. In both cases a linear function was fitted with a
least square procedure. Fit parameters are given in table 3.4.
3.5.10
Relation between Umbral and Sunspot Size
We studied the relation between total umbral size and sunspot size for the
small dataset. Figure 3.24 shows the scatter plot of those quantities and a
linear least square fit.
Figure 3.24: Sunspot radius versus umbral radius. The solid line shows a
linear least square fit, one sigma errors are plotted dashed. Fit parameters
are given in table 3.4.
CHAPTER 3. DATA ANALYSIS
56
Table 3.4: Fit parameters for figures 3.19-3.24. R stands for radius in Mm, I for relative intensity, B for magnetic field in G
and Φ for magnetic flux in 1021 Mx. The linear equation has the form y = a + bx and the power law function has the form
y = cxd . χ2 values are given for the fits. CP and CS are Pearson (linear) and Spearman (rank) correlation coefficient of the
data (e.g. Sachs 1999). Parameters labeled with U refer to the umbra, P stands for penumbra, and parameters with S refer
to the whole sunspot. Not explicitly labeled parameters belong to the umbra.
Fig.
3.19
3.19
3.20
3.20
3.21
3.21
3.22
3.22
3.23
3.23
3.24
x
RU
RU
RU
RU
IMIN
IMEAN
ΦS
ΦS
ΦS
ΦS
RU
y
IMIN
IMEAN
BMAX
BMEAN
BMAX
BMEAN
BMAX
IMIN
ΦU /ΦP
RU
RS
a
0.418
0.539
1920
1633
3248
3208
−
−
0.327
2.36
1.44
aerr
0.006
0.006
34
33
34
77
−
−
0.007
0.07
0.27
b
0.0449
0.0377
116
68
3913
3556
−
−
0.075
0.763
2.25
berr
0.0011
0.0010
6
6
171
221
−
−
0.002
0.019
0.05
χ2lin
5.77
6.82
1.27
0.80
0.46
0.16
−
−
10.7
1.04
0.65
c
0.83
0.580
1739
1489
1609
1052
2182
0.369
−
−
−
cerr
0.05
0.013
48
45
33
41
29
0.012
−
−
−
d
−0.93
−0.33
0.23
0.18
−0.26
−0.59
0.124
−0.586
−
−
−
derr
0.03
0.01
0.02
0.02
0.01
0.04
0.009
0.017
−
−
−
χ2pl
1.78
3.13
0.87
0.68
0.44
0.14
1.13
4.42
−
−
−
CP
−0.76
−0.69
0.73
0.65
−0.92
−0.94
0.65
−0.68
0.57
0.95
0.97
CS
−0.78
−0.69
0.73
0.62
−0.93
−0.94
0.67
−0.72
0.55
0.96
0.97
57
3.5. REDUCED DATASET
3.5.11
Comparison of the Data
Stray Light
The contamination of stray light is a problem that needs some discussion.
First of all, diffraction limits the resolution of an optical instrument. The
resolution is defined as the distance from the central maximum to the first
minimum of the point spread function of the telescope. For HMI this is
about 1”, the critical sampling is about 0.5”. Further the point spread
function is wider than its first minimum and has side lobes. Point sources
therefore always illuminate multiple pixels on the detector.
Light can be scattered or reflected while passing the optical path of the
instrument. Inhomogeneities on some mirrors or lenses for example can
cause stray light. In theory it is possible to correct for stray light with
a deconvolution with an appropriate function. Such a function could be
derived from planetary transits or eclipses (eg. Mattig 1971).
Such a stray light correction has not been carried out in this thesis. If
one compares intensity measurements of different instruments the data is
affected by different amounts of stray light. This is especially important if
one compares with ground based observations which are additionally effected
by atmospheric seeing. In particular the correlation between umbral size and
intensity (cf. Fig. 3.19) is probably also because of stray light.
Data Measured at Different Wavelengths
It is far from trivial to compare the intensity measurement of in different
spectral regimes. The opacity of the photosphere is a function of wavelength.
In the infrared we are able to see into deeper layers than in shorter wavelengths. The intensity contrast of the umbra to the bright sun is decreasing
with wavelength (e.g. Maltby 1970).
If we assume local thermodynamic equilibrium (LTE), i.e. we expect
the umbra and the quiet sun to be black body radiators each, the Planck
function (B(λ, T ), eq. 2.11) describes the spectral intensity. We define the
brightness temperature of the quiet sun Tqs = 6000 K.
If we want to compare intensities at different wavelengths, one way to go
would be to normalize the measured continuum intensity at λ2 to the quiet
sun intensity at λ1 . This had been done by Rezaei et al. (2012) for example.
Another possibility is, if LTE holds, to calculate the theoretical intensity
ratio Iλ2 as follows. We first calculate the umbral brightness temperature
Tu , for a given relative umbral intensity Iλ1 for the wavelength λ1 .
Iλ1 =
hc
Tu (λ1 , Tqs ) =
λ1 k B
ln 1 +
B(λ1 , Tu )
B(λ1 , Tqs )
1
Iλ1 (exp(hc/λ1 kB Tqs ) − 1)
(3.2)
−1
(3.3)
58
CHAPTER 3. DATA ANALYSIS
With this temperature we can now calculate the relative umbral intensity
at a different wavelength λ2 .
Iλ2 =
B(λ2 , Tu )
B(λ2 , Tqs )
(3.4)
This method gives us the chance to compare measurements in different spectral regimes, it was also used by e.g. Schad & Penn (2010).
Comparison to other Work
We compared findings published by Kopp & Rabin (1992), Mathew et al.
(2007), Schad & Penn (2010) and Rezaei et al. (2012) to our data. We used
equation 3.4 to shift their fits into the wavelength regime of HMI. The figures
are in the Appendix (Figs 5.2, 5.3 and 5.3). Despite the different umbral
contrast at different wavelengths, the individual spectral lines have different
formation heights. The magnetic field in those different heights can be
expected to be different therefore as well. We do not apply a correction for
the magnetic field.
Chapter 4
Discussion
4.1
Umbral Size Distribution
The fit parameters to umbral size distributions are given in table 3.1.
4.1.1
Manual Dataset
Figure 3.5 shows the umbral size distribution of 4229 umbrae and a lognormal curve fit. With a χ2 of 4.13 the lognormal function fits the data
relatively well. We see a clear curvature in the histogram, a simpler approach, a power law function for example, would have been insufficient to
fit the data.
This is as expected, Bogdan et al. (1988) and Schad & Penn (2010) find
this trend as well. Although we confirm the lognormal trend, it was not
possible to reproduce the their fit parameters with the dataset of manually
selected umbrae. This does not mean that the size of sunspots changed
drastically over the last decades, the reason for the deviation lies within the
different methods of umbra selection.
In the manual method we did not count distinct umbrae within one
sunspot separately. This had been the reason to carry out an automated
threshold method to investigate whether we can reproduce the curves other
authors found.
Maximum of the Size Distribution
The size distribution has a clear maximum. Umbrae smaller than this maximum occur less frequently. This is a new finding, the other authors mentioned above did not report a maximum.
Since we do not count pores in our statistics it is not possible to identify
the lower part of the umbral size distribution with the distribution of all
magnetic flux elements in the solar photosphere. Therefore the decrease of
the histogram for small features does not tell us much about the behavior
59
60
CHAPTER 4. DISCUSSION
of small flux tubes. We can learn from the maximum, that there is a typical
size needed to form a penumbra and that we find little sunspots smaller
than this size.
A maximum in the size distribution of umbrae could a be a constraint
for theories that aim to describe sunspot formation processes.
4.1.2
Threshold Method
With the threshold method we find a total of 6892 umbrae within 4194
sunspots. Figure 3.8 shows two size distributions of those umbrae, one with
total umbral size per sunspot and one where we count disjunct umbrae
within one sunspot individually. Again, it is justified to fit the histograms
with a lognormal curve.
There is quite a difference between both size distributions. We conclude
that the definition of the umbra, i.e. if we count total umbra per spot or if
we treat each umbral feature individually is very important.
If we analyze magnetic features on the solar surface there is some kind of
a hierarchy: On a large scale there are whole active regions. Then there are
individual sunspots and on a smaller scale there are individual umbrae within
those sunspots. On all three levels size distributions of the magnetic features
might tell us something about the processes that lead to the diversity in
sizes. We did not study the distribution of entire active regions in this
thesis thought.
The distribution of the total umbral size per sunspot relates to the second
level. We see a maximum in the distribution of umbral sizes. This is analog
to the finding of a maximum with the manual selection discussed above.
The size distribution of separated umbrae does not show a maximum
which means that we do not find a minimum umbral size. Umbrae within
sunspots might divide in arbitrary small segments above our lower boundary
of 0.5 MHS.
In both histograms the lognormal function does not perfectly fit the distribution. There is a deviation between lognormal fit and histogram for
umbrae larger than roughly 100 MHS. There are few really large sunspots
and the error bars are large in this regime. But since the fit is above the histogram for all those bins, the assumption of lognormally distributed umbral
sizes probably fails in this part of the size spectrum.
Influence of Threshold and Resolution
Figure3.9 shows the umbral size distribution for umbrae selected with different thresholds and a size distribution for umbrae from smoothed data.
The lognormal size distribution of sunspot umbrae does not change much
for different umbral thresholds. Also the smoothing does not affect the fit
significantly. We can expect that the instrument has only a minor influence
4.1. UMBRAL SIZE DISTRIBUTION
61
on the shape of the size distribution. This justifies the comparison of the
result to other work, provided the methods to select umbrae are the same.
Time and Hemisphere Separation
In addition to discussing the physical meaning of the umbral size spectrum,
we want to study whether it is invariant over time or changes with the solar
activity cycle. We further distinguish between northern and southern solar
hemisphere because of a possible phase difference between them. Figure 3.10
shows histograms for time and hemisphere separation of the data with lognormal fits, in table 3.1 the fit parameters are listed. We do not find any
significant variation neither with time, nor with hemisphere.
When we separate the data in both, time and in hemisphere we see less
large sunspot umbrae in the first part in the southern hemisphere (cf. Table 3.1 and Appendix Fig. 5.1). This is also reflected in the relatively small
standard deviation of the data and variance of the fit for the first southern distribution. We find relatively few large umbrae in this first southern
dataset. Since the cycle in the northern hemisphere is currently ahead of
the southern cycle this is evidence that less large umbrae exist during cycle
minima.
4.1.3
Reduced Dataset
The umbral size distribution of the reduced dataset containing 910 umbrae
out of 488 sunspots is shown in figure 3.15. Since we use less sunspots than in
other datasets the errors of the bins, especially for narrow bins and for large
umbral sizes, are relative large. As in the size distribution for 6892 umbrae,
the lognormal curve fits the data. We see a similar deviation between large
umbrae and the fit as in the full dataset.
4.1.4
Comparison Between Different Size Distributions
Figure 3.16 shows the two size distributions of sunspot umbrae obtained with
the threshold method. Several fits of other size distributions are overplotted.
Table 3.2 contains the χ2 proxies that measure the deviation between those
fits and the histogram.
Manual and Automatic Selection
The manual selected umbral size distribution is a distribution of of total
umbral sizes per sunspot. We therefore compare it to the threshold-selected
distribution of total umbral sizes (Fig. 3.16, lower panel). We find the threshold distribution to be wider with respect to the manual distribution. This
means that we count more small and more large umbrae. We conclude
that the manually chosen threshold is higher for small umbrae and lower for
62
CHAPTER 4. DISCUSSION
large umbrae with respect to the universal threshold of 0.6 in this dataset.
35 sunspots in the manual selected set have a threshold even higher than
0.6 and were ruled out in the analysis with threshold.
This finding suggests that an automatic threshold method is not necessarily ideal to select sunspot umbrae. On the other hand the threshold
allows to reproduce the data selection. We found that umbral mean and
minimum intensity are correlated with umbral size, this is discussed in section 4.4.1. No matter whether this correlation is because of stray light or
real, it might exist for the umbral threshold intensity as well. A variable
umbral to penumbral threshold intensity should have an influence on the
umbral to sunspot size ratio discussed later.
Reduced Dataset and Full Dataset
In the reduced dataset of 488 sunspots multiple umbrae within one sunspot
have been counted individually. We therefore compare the result with the
histogram of separated umbrae (Fig. 3.16). There is only a very small difference between both distributions. The fit parameters are the same within the
errors, the χ2 between the fit to the reduced dataset and the histogram of all
data is very close to the least possible χ2 . We conclude that the reduction
of the data did not change the size distribution and it sufficient to study the
distribution of umbral sizes with the full dataset.
The similarity of both distributions is unexpected because of several
reasons. We expected to find less large umbrae in the reduced set, since
large sunspots live longer and therefore should have been counted more
often in the full set. We also expected to find a trend towards small umbrae
in the reduced set because we limited the data to µ = 0.5 (60◦ ) in the small
set while the large set goes to µ = 0.17 (80◦ ) form disk center (Fig. 3.12).
Further there is the possibility for a deviation between the size distributions
because of sunspot evolution. In the reduced set we limited our data to
young spots, the full set contains spots of every observable stage of sunspot
evolution.
Quality of the Fits
Imagine we draw a random sample from a given distribution and then fit
the histogram of this sample with an analytic function. If we use the correct
function then the χ2 of the fit should be close to one, given the errors we
assumed were correct.
In the various fits to the umbral size distribution of umbrae selected
with threshold method we find a χ2 between 0.87 and 10.5. This tells us
that our assumptions were roughly valid. We calculate the smallest χ2 for
the smaller subsets (the manually selected umbrae and the umbrae on the
southern hemisphere for example), the larger our dataset, the worse gets
4.1. UMBRAL SIZE DISTRIBUTION
63
the χ2 . This is a hint that there is something wrong: either our errors
decrease to fast with increasing amount of data (we chose them inverse to
the square root of number of counts in each bin), or our assumption of a
lognormal distribution is not entirely correct. See also the discussion about
the motivation of the lognormal distribution in the introduction, section 2.3.
We take the increasing χ2 as a hint, that the lognormal distribution is not
the perfect density function do describe umbral sizes. This is in agreement
with the finding, that there is a deviation of large umbrae and our fit.
4.1.5
Comparison to other Results
Bogdan et al. (1988) and Schad & Penn (2010) also analyzed the size distribution of sunspot umbrae (Sec. 3.5.3). The first group used photographic
plates of Mount Wilson white light data covering a time span from 1917 to
1982, the latter analyzed data from the Kitt Peak Vacuum Telescope during solar cycle no. 23. Both groups use different methods to select umbrae
compared to us.
Bogdan et al. The size distribution evaluated by Tom Bogdan and coworkers deviates a lot from our result (Fig. 3.16, Table 3.2). Although the
authors find a lognormal shape of the size distribution, we think it is not
possible to compare the results because of differences in the methods.
Schad and Penn The threshold method used by Schad & Penn (2010) is
similar to ours (Sec. 3.5.3). We show some of their renormalized lognormal
fits in figure 3.16. If there is a real physical variation of the distribution of
umbral sizes with the solar cycle, and if this variation is described correctly
by Schad & Penn (2010), then we would expect that their fit to the rising
phase to cycle no. 23 would fit our data best (compared to other curves
published those authors). Table 3.2 shows the χ2 between our data and
those different fits. The fit to the rising phase of the last cycle fits our data
only slightly better than most other fits.
The generally good alignment between our data and the distributions
published by Schad & Penn (2010) is remarkable. This suggests that the
distribution of umbral sizes did not change between the previous and the
current solar cycle (23 and 24).
Clette and Lefèvre We do not find change in the ratio of small to large
umbrae as observed by Clette & Lefèvre (2012) from cycle 22 to 23. The
umbral size distributions of solar cycle no. 23 and 24 are similar.
64
4.2
CHAPTER 4. DISCUSSION
Heliocentric Angle Dependence
We used the full dataset of 6892 umbrae to analyse the dependence of umbral
equivalent radius and intensity on their distance from disk center.
4.2.1
Umbral Size
We find a trend in the variation of the mean umbral size with heliocentric
angle. Figure 3.12 shows in the left panel the mean umbral radius, binned
in eight bins with a width of 10◦ each. The mean umbral radius increases
from 2.8 Mm close do disk center up to 3.5 Mm close to limb.
A real variation of any parameter with longitude can be excluded because
of the rotational symmetry of the sun. There might be a size dependence
on latitude, however. Further it is possible that the Wilson effect or the
umbral limb darkening affects the measured size. The main reason for this
behavior is probably the decreasing spatial resolution towards to the solar
limb. Close to the limb small umbrae might not be resolved and therefore
are not counted in this statistics. We conclude that it is important for the
comparison of umbral size distributions to which heliocentric angles umbrae
were included in the analysis.
4.2.2
Umbral Intensity
The right panel of figure 3.12 shows the averaged umbral intensity versus heliocentric angle, normalized to the mean intensity at disk center. The black
symbols show the umbral minimum intensity (the darkest pixel), not corrected for limb darkening. We see an umbral limb darkening, as expected.
In blue and red the corrected mean and minimum intensities are shown,
respectively. A variation in this data would suggest a different limb darkening law in the umbra compared to the quiet sun. This is not observed
in the averaged mean umbral intensities. The minimum intensity shows a
slight increase towards solar limb. This increase is small compared to the
uncertainty of the mean value. Further the umbral core intensity is affected
by projection effects (a pixel close to limb covers a larger area on the solar
surface than a pixel close to disk center). We conclude that we do not find
the limb darkening law of the umbra to be different compared to the relation
of in the quiet sun for the HMI wavelength.
4.3
Temporal Variation
The fit parameters to relations discussed in this section can be found in
table 3.3.
4.4. CORRELATIONS BETWEEN PARAMETERS
4.3.1
65
Full Dataset
Figure 3.11 shows the temporal variation of the mean umbral radius in the
left and intensity in the right panel. In all cases the measurements of 6892
umbrae were binned in five bins each covering six months and a linear curve
was fitted. We see neither in size or in the average mean intensity a temporal
trend.
4.3.2
Reduced Dataset
We address the same topic with the 205 well developed sunspots, but study
the minimum intensity. We find different temporal trends in the hemispheres
for both radius and intensity (Fig. 3.18). Those relations are different than
the findings of Mathew et al. (2007), who carry out a similar analysis with
160 spots from the rising phase of the previous cycle (no. 23). Regarding
the large scattering in our data the trends are hardly significant and more
data is needed.
The increasing umbral size in the southern hemisphere (red curve in
Fig. 3.18, left) is consistent with the finding that the mean value of umbral
size is relative small in the first half of our data in the southern hemisphere
(Table 3.1). This is a slight evidence that less large umbrae exist during
solar minima.
4.4
Correlations Between Parameters
Fit parameters and correlation coefficients for the relations discussed in this
section can be found in table 3.4.
4.4.1
Intensity, Radius and Magnetic Field
Figures 3.20,3.19 and 3.21 show scatter plots of umbral equivalent radius,
intensity, and magnetic field strength. We see a significant correlation between all properties. Large umbrae have high magnetic field strengths and
are dark. Dark umbrae have high field strengths.
The correlation between intensity and field is higher compared to the correlation of those parameters with radius. This might be because we discuss
an equivalent radius and did not limit the data to circular sunspots. The
minimum intensity and maximum field strength show higher correlations
with size than the averaged umbral values.
The power law fits show a significantly lower χ2 compared to the linear
fits (Table 3.4). This confirms that intensity and field strength level off with
increasing sunspot size. This is not the case for the correlation between
intensity and field strength, it is rather linear.
66
4.4.2
CHAPTER 4. DISCUSSION
Umbral Size and Sunspot Size
We observe a linear trend between the radius of the total umbral size and
the total sunspot size in the small dataset covering 205 sunspots (Fig. 3.24).
Although the strong correlation between those values is no surprise, it is
remarkable how good the linear approach fits the data. Neither umbral nor
penumbral size levels off with respect to the other. This means that both
are, within the the sunspot sizes we observe, not limited.
4.4.3
Magnetic Flux
The total magnetic flux through a sunspot depends on the size of the spot.
The strong correlation we see between those values (Fig. 3.23) is therefore
no surprise. The same is true for intensity and magnetic field, they show
the same correlation with flux as they do with size (Fig. 3.22.
If we plot against flux the nonlinearity is even more evident compared
to the plots versus radius (Figs. 3.19, 3.20).
We find the ratio of umbral to penumbral flux to increase with increasing
total flux (Fig. 3.23). Although scattering in this data is high we fit a linear
curve for this relation to highlight the correlation.
Since we found a constant size ratio between umbra and penumbra, the
variation in the ratio of umbral to penumbral flux is remarkable. It is consistent with the finding that the umbral magnetic field increases with umbral
size. With increasing total flux, the magnetic properties change (the field
becomes stronger) but the spatial relations we study are invariant.
Comparison to Other Authors
The figures 5.2, 5.3 and 5.4 in the Appendix show that we find the same
trends in magnetic field, intensity and size relations of umbrae as other
authors. When we convert the umbral intensity contrast of their fits to
the HMI wavelength the deviation between their and our findings generally
decreases. We do not correct for differences in the magnetic field due to
different formation heights of spectral lines thought. This partly explains
the offset between our findings and adapted curves, another possible reason
is stray light. We therefore can not compare the exact fit parameters, the
important result is that the overall relations between umbral field strength,
intensity and size are the same for the Fe I 617.33 nm, the Ni I 676.8 nm, the
Fe I 868.8 nm, and the Fe I 1565 nm lines.
4.5. OUTLOOK
4.5
67
Outlook
We have some ideas about what could be done in future regarding the statistical analysis of sunspots.
4.5.1
Umbral Threshold Intensity
It would be interesting to study umbrae selected with another criterion for
the umbral penumbral boundary other than a universal threshold . This
would give the possibility to study the variation of the umbral penumbral
threshold intensity with umbral size. One possibility might be to identify
the region where the sunspot intensity gradient is largest with the umbral to
penumbral boundary and then somehow fit the individual umbral threshold
intensity.
4.5.2
Automatizing Sunspot Selection
Space missions like SDO or SOHO provide a large amount of data. Although
we limited our data to one image per day in this thesis, the manual selection
of over 4000 sunspots took much time. Further the selection is not exactly
reproducible. Both problems could be solved with an implementation of a
robust algorithm which is capable of selecting sunspots automatically.
Recent publications often focus on either a sample of multiple sunspots
(like this thesis) or on the temporal evolution of single sunspots. With
automatized selection methods it would be possible to bring those topics
together and make full use of the high cadence of HMI data.
4.5.3
Distribution of Magnetic Flux
When we study sunspots, we study the observable result of a magnetic flux
tube piercing the solar surface. The theoretical considerations about sunspot
sizes, as for example in section 2.3, focus on the magnetic flux itself. Our
analysis targets on the umbra.
With modern computers it would be possible in a reasonable time to invert enough sunspots to investigate the distribution in magnetic flux, rather
than sunspot size.
68
CHAPTER 4. DISCUSSION
4.6
Conclusions
We can draw following conclusions from the analysis of the sunspots in 2.5
years of HMI data.
• The lognormal function is a valid approximation for the distribution
of umbral sizes. For large umbrae we find a deviance between the fit
and the data.
• The analysis of automatically selected umbrae led to a wider size distribution compared to manual selection. This is evidence that the
umbral threshold intensity depends on the umbral size.
• The average umbral size increases with heliocentric angle. This is
because small umbrae can not be resolved close to the solar limb.
• The limb darkening corrected umbral size stays constant with heliocentric angle. This means we find the same limb darkening law in the
umbra as in the quiet sun.
• We do not find a temproal variation of umbral intensity or size in 2.5
years. We find evidence for less large umbrae during solar activity
minima.
• The umbral magnetic field strength and umbral intensity increases
(decreases) with umbral size. Both properties level off with size.
• The umbral magnetic field strength and umbral intensity are highly
correlated and can be approximated with a linear fit.
• The umbral size and field strengh increase with total magnetic flux,
the intensity decreases. This is consistent with the relations of size,
intensity and field strength.
• The ratio of umbral to penumbal flux increases with total flux.
• The umbral size increases linearly with the sunspot size.
Chapter 5
Appendix
This chapter contains some of the figures of Chapter 3 with additional fits
for comparison.
i
ii
CHAPTER 5. APPENDIX
Figure 5.1: Fits to umbral size distributions where the data was separated
in time and hemisphere. Only the first half of the southern hemisphere
(the part with the least solar activity) deviates from our other findings. See
table3.1 for the fit parameters.
iii
Figure 5.2: Minimum umbral intensity versus umbral radius (same as in figure 3.19). We overplotted fits by Schad & Penn (2010) (cyan, Fe I 868.8 nm)
and Mathew et al. (2007) (green, Ni I 676.8 nm). The solid lines are their
findings, the dashes lines are shifted to the HMI wavelength (617.33 nm)
with equation 3.4.
iv
CHAPTER 5. APPENDIX
Figure 5.3: Umbral minimum intensity versus maximum field strength (same
as in figure 3.21). We overplotted fits by Schad & Penn (2010) (cyan, Fe I
868.8 nm) and Rezaei et al. (2012) (orange, Fe I 1565 nm). The solid lines
are their findings, the dashes lines are shifted to the HMI wavelength
(617.33 nm) with equation 3.4.
v
Figure 5.4: Umbral radius versus maximum field strength (same as in figure 3.20). We overplotted fits by Schad & Penn (2010) (cyan, Fe I 868.8 nm)
and Kopp & Rabin (1992) (yellow, Fe I 1565 nm).
vi
CHAPTER 5. APPENDIX
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Ich erkläre hiermit dass alle ich diese Arbeit selbstständig verfasst und
nur mit den angegebenen Hilfsmitteln angefertigt habe. Alle Stellen, die dem
Wortlaut oder dem Sinne nach anderen Werken entnommen sind wurden
durch die Angabe der Quellen als Entlehnungen kenntlich gemacht.
Christoph Kiess