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This is to certify that the

dissertation entitled

"Precision Infrared and Visible Solar Photometry

A Photometric Study of the Solar Irradianoe Variation"

presented by

Haosheng Lin

has been accepted towards fulfillment
of the requirements for

 

Ph . D . degree in Physics

 

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IUL M

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MSU is An Affirmative Action/Equal Opportunity Institmion
chS-o.‘

 

 

 

 

PRECISION INFRARED AND VISIBLE SOLAR PHOTOMETRY

A PHOTOMETRIC STUDY
OF
THE SOLAR IRRADIANCE VARIATION

By

Haosheng Lin

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1992

ABSTRACT

PRECISION INFRARED AND VISIBLE SOLAR PHOTOMETRY
A PHOTOMETRIC STUDY
OF
THE SOLAR IRRADIANCE VARIATION

By

Haosheng Lin

The solar irradiance was observed to be varying with time scale of days to the
11 years solar cycle time scale. The daily variations were well correlated with the
passage of the dark sunspots and bright facular area on the photosphere. However,
the cause of the small irradiance variation with amplitude of 10‘“3 of the total solar
irradiance correlated with the solar cycle is not well understood. Several scenarios
with observable surface effects were suggested by various authors to explain this
phenomenon, but no observational confirmation is available up to date. Thus, a
precision Solar Photometric Telescope (SPT) was constructed to study the large
scale brightness structure of the solar photosphere. This instrument does full disk,
broad band (10 nm FWHM), two colors (500 nm and 650 nm) imaging of the solar
photosphere. Photometry with a newly constructed Infrared Array Camera (IAC)
extends these measurements to 1.2, 1.6 and 2.2 micron. This dissertation reports the
observations done with these instruments, which revealed low contrast brightness
structures with scale ranging from 104 km to some fraction of the solar disk on the
solar photosphere. The contribution of these structures to the total solar irradiance

were discussed.

For Peiling

iii

ACKNOWLEDGEMENTS

Living in a foreign land is no easy task for anybody. Studying in a foreign
university is even harder. I would like to thank the many people who helped to

make it easier, and turn it into a wonderful adventure.

First I would like to thank my wife, Peiling, who shared all the frustrations and
joys with me, and provided a home full of love. She also shielded me from all the

distractions in life so I could finish this dissertation.

Dr. Jeffrey R. Kuhn, with whom I learned all the fun of the astrophysics of
the sun and astronomy in general, also gave the living example of a great scientist
at work. He was also the first person to greet us with warm welcome. All the
astronomy folks in the north wing of the third floor, especially Debbie Benedict,
provided a homelike atmosphere in which I can do my work and sometimes just

hang around.

I am grateful to the observing staffs of the National Solar Observatory at Sacra-
mento Peak, New Mexico, who helped with the infrared observations that comprise
half of the data presented in this dissertation. The hospitality of the whole Sac.

Peak community also made our three months’ stay a memorable experience.

Finally, I would like to thank my parents and my brothers and sisters for their

love and support, without which I won’t have been able to finish my study.

iv

TABLE OF CONTENTS

INTRODUCTION

1.1 Motivation

1.2 The Physics
1.2.1 Magnetohydrodynamics
1.2.2 The Surface Magnetic Field
1.2.3 The Subsurface Magnetic Field

1.3 This Thesis

1.4 Related Subjects

INSTRUMENTATION
2.1 General Considerations
2.2 The Solar Photometric Telescope
2.2.1 Optical System
2.2.2 The Shutter Assembly
2.2.3 The Computer Control and Data Acquisition System
2.3 The Infrared Array Camera

OBSERVATIONS AND DATA REDUCTION
3.1 Observing Procedure
3.2 The Visible Wavelength Data
3.3 The Infrared Data
3.4 Data Reduction

11
15
17
20

24
24
26
3O

34
36

39
39
40
42
42

3.4.1 Dark Subtraction and F latfielding
3.4.2 Calibration of Solar Disk Center
3.4.3 Image Shift and Rotation

3.4.4 Image Distortion Correction

3.4.5 Image Averaging

3.4.6 Fractional Residual Image Construction

PROPERTIES OF PHOTOSPHERIC
NETWORK AND FACULAE
4.1 Introduction
4.2 Noise Estimate
4.3 Properties of The Photospheric Network
4.3.1 Cross-Correlation Between Different Timesteps and Colors
4.3.2 Power Spectrum Analysis
4.3.3 Cross-Correlation With the Chromospheric Network
4.3.4 Cross-Correlation With the Supergranulation Velocity Field
4.3.5 Line Contribution to the Network Contrast
4.3.6 Photospheric Network Contrast Near Disk Center
4.3.7 Center-to—Limb Variation of Network Contrast
4.3.8 Temperature Structure of Supergranulation
4.4 Properties of Photospheric F aculae
4.4.1 Spatial Distribution of Faculae
4.4.2 Faculae Contrast Near Disk Center and its

Center-To—Limb Variation

THE SOLAR IRRADIAN CE VARIATIONS
AND CONCLUSIONS

vi

44
56
57
58
61
61

63
63
67
72
72
73
81
82
85
88
89
96
96
96

97

105

5.1 Introduction

5.2 Irradiance Contribution of F aculae and Sunspots
5.3 Energy Balance of Active Regions

5.4 The Average Photosphere

5.5 Interpretations

5.6 Conclusions

APPENDICES

Appendix A On The Gauge Freedom of the

Gain Iterating Algorithm

Appendix B Least Square Algorithm to Find the
Disk Center Coordinates of Full Disk Solar Images

Bibliography

vii

105
107
109
113
117
122

123

127

131

LIST OF TABLES

Table 3.1 Summary of Visible Wavelength Observations 40
Table 3.2 Summary of Infrared Wavelength Observations 42

Table 5.1 The Sunspots and Faculae Irradiance Contributions

of 1991 Data 108

viii

LIST OF FIGURES

Figure 1.1 Solar Irradiance Data from ACRIM and ERB Experiments
Figure 2.1 System Diagram of the Solar Photometric Telescope
Figure 2.2 The Heliostat of the SPT
Figure 2.3 The Optical System of the SPT
Figure 2.4 Ray-Tracing Diagram of the Optics
of the SPT in Low Resolution Mode.
Figure 2.5 Ray-Tracing Diagram of the Optics
of the SPT in High Resolution Mode.
Figure 2.6 The Shutter Assembly and the Filter Wheel
Figure 2.7 Experimental Setup of the Infrared Array Camera
Figure 3.1 Raw Solar Image in the Green Bandpass
Figure 3.2 Raw Solar Image in the Infrared H Band
Figure 3.3 Green Bandpass Flatfield
Figure 3.4 Infrared H Band Flatfield
Figure 3.5 Green Time-Averaged Solar Image
Figure 3.6 H Band Time-Averaged Solar Image
Figure 3.7 Time-Averaged Residual Image in the Green Bandpass
Before Distortion Correction
Figure 3.8 Time-Averaged Residual Image in the Green Bandpass
Figure 3.9 Time-Averaged Residual Image in the Red Bandpass

Figure 3.10 Time-Averaged Residual Image in the Infrared J band

ix

27
28
29

31

33
35
38
41
43
45
46
47
48

49
50
51
52

Figure 3.11 Time-Averaged Residual Image in the Infrared H band

Figure 3.12 Residual Image in the Infrared K band

Figure 3.13 A2(r) Optical Distortion Amplitude

Figure 4.1 Histograms of Contrast from Quiet and Active Regions

Figure 4.2 Difference Between Active and Quiet Region Contrast histograms

Figure 4.3 Red Contrast Image with Large Scattered Light Contamination

53
54
60
65
66
71

Figure 4.4 Filtered Green Band Contrast Image Showing Photospheric Network 74

Figure 4.5 Filtered Red Band Contrast Image Showing Photospheric Network 75

Figure 4.6 Cross-Correlation of Visible Wavelengths Images
Figure 4.7 Cross-Correlation of Infrared wavelengths Images at Disk Center
Figure 4.8 Comparison of Faculae in J and H Band Near Disk Center

Figure 4.9 Average Power Spectrum of Photospheric Network

76
77
78
79

Figure 4.10 Cross-Correlation Between Photospheric Network and Ca K Network 83

Figure 4.11 Cross—Correlation Between Supergranulation

and Photospheric Network
Figure 4.12 Transmission Function of the Red Filter
Figure 4.13 Transmission Function of the Green Filter
Figure 4.14 Photospheric Network Contrast Near Disk Center
Figure 4.15 Network Contrast Center-to-Limb Variation in Green Band
Figure 4.16 Network Contrast Center-to-Limb Variation in Red Band
Figure 4.17 Network Contrast Center-to-Limb Variation in Infrared H Band
Figure 4.18 Network Contrast Center-to~Limb Variation in Infrared K Band
Figure 4.19 Center-to—Limb Variation of Faculae Contrast
Figure 5.1 Sunspots and F aculae Irradiance Contribution
Figure 5.2 The Average Photosphere in Green Bandpass

Figure 5.3 The Average Photosphere in Red Bandpass

84
87
90
86
91
92
93
94
103
110
115
116

Figure 5.4 Limb Brightness Profile of the Average Photosphere 118

Figure 5.5 Brightness Temperature Profile of The Average Photosphere 119

xi

CHAPTER 1

INTRODUCTION

1.1 Motivations

The Active Cavity Radiometer Irradiance Monitor (ACRIM) experiment on
board the Solar Maximum Mission (SMM) satellite (Willson and Hudson 1988),
and the Earth Radiation Budget (ERB) experiment on board the Nimbus-7 satellite
revealed that the solar irradiance is variable (Figure 1.1). The time scale of this
variation ranges from days to years. On the time scale of days, the irradiance is
observed to vary with amplitude of a few tenths of a percent. On the long term, the
irradiance increases and decrease with sunspot number, being higher by 1 x 10"3
during the solar maximum. The integrated-light observations of ACRIM and ERB
do not provide us information on how the sun modulates its radiative output, nor
does it tell us if it is the total luminosity that is changing. However, correlations

suggest that the mechanism is related to the magnetic activity on the sun.

Solar magnetic activity manifests itself as magnetic flux tubes with various sizes
in the solar photosphere. Brightness changes are always associated with these mag-
netic flux tubes. The sunspots are the best known. They appear as dark spots
on the Solar photosphere, and reduce the total Solar irradiance. Faculae are also
associated with the surface magnetic flux tubes. They are the most visible bright
feature in a whitelight Solar image. However, their brightness is not exceedingly

high when observed with moderate spatial resolution. Their brightness is the great-

Solar Irradiance Time Seires

 

    

 

 

 

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Figure 1.1 Solar Irradiance Data From ACRIM and ERB Experiments. The offset
of about 3 W / m*m is due to a real difference of the absolute calibration between the
two instruments. The large fluctuations of the irradiance with short timescales, as
well as the slow decrease toward 1986 and the subsequent increase of the irradiance
are truly of solar origin. This is evident from the good agreement between the two

data sets.

est near the Solar limb, and falls off rapidly as they are rotated into disk center.
Since sunspots and faculae are the most visible brightness components on the Solar
surface, it is only natural to try to model the Solar irradiance variations with only
the contributions from these two components. The brightness of a large sunspot
can be as low as about 20 % of the quiet photosphere. Indeed, the passage of large
sunspot groups across the observable solar disk was found to be well correlated with

large dips in the ACRIM and ERB data.

An early difficulty was that there were no ground based observations with pho-
tometric accuracy comparable to the satellite experiments to directly measure the
irradiance contributions of each magnetic element. Thus, sunspot and facular con-
tributions were modeled in terms of the Photometric Sunspot Index (PSI) and
Photometric Faculae Index (PFI). That is, given the measured contrast of sunspot
and faculae as a function of p, the cosine of the angle between our line-of-sight
to the disk center and the radius vector of the sunspot /faculae, one can infer the
irradiance contribution of the sunspot / faculae from its area. The daily sunspot and
facular area can be measured from the synoptic measurements being carried out at
the major observatories. Calculations of this type were able to model the short-term
irradiance variations reasonably well (Willson, Gulkis, J anssen, Hudson, and Chap-
man 1981). However, people also realized that the 11-year irradiance variations
cannot be accounted for with just the sunspot and facular contributions (see the
review articles of,e.g. Chapman 1987, Hudson 1988, and Fr6hlich, Foukal, Hickey,
Hudson, and Willson 1991).

In an attempt to identify the brightness structure that is modulating the so-
lar irradiance, Foukal and Lean (1988) compared the Data from ACRIM and the
ERB experiment with the HeI A10830 A index, and CaII K plage data. Both the

HeI z\10830 A and CaII K are resonance absorption lines in the solar Fraunhofer
spectrum. The electron configuration of the CaII ion is 1322.322p63323p6431. The
transitions between the 2s and 2p states form a doublet at A = 3934 A and A = 3968
A, known as the CaII K and H lines, respectively. The CaII K line has a total an-
gular momentum of J = 3/ 2, and is the stronger of the doublet. It is formed in the
chromosphere, and its brightness was found to be well correlated with the magnetic
field strength (F azier, 1970). The full-disk integrated CaII K flux thus measures the
magnetic activity of the sun, and spatially resolved CaII K spectroheliograms are
used to measure the area of faculae, or the CaII K plage index. The HeI A10830 A
line is the result of the transition between the 23 and 2p states of neutral helium.
Its strength is increased in magnetic elements like plages, and prominences, and
its equivalent width was found to be well correlated with global magnetic activity
(Harvey 1984, Donnelly, Harvey, Heath, and Repoff 1985), and Foukal et al. used
it as a measurement of the combined area of the faculae and network. They found
that while the short term irradiance variation is well modeled by the sunspot index
and the Call K plage index, the long-term irradiance variations must be modeled
with the HeI A10830 A index. This they argued is the proof of the need of a third
component, the ”active network” contribution, to account for the observed total

solar irradiance variation.

The limb brightness measurement (Kuhn, Libbrecht, and Dicke 1988) provided
more evidence of a third contribution to the solar irradiance. In this experiment,
Kuhn et al. showed evidence of a variable limb brightness temperature excess at
active latitudes with an amplitude (peak-to~peak) of 7 K correlated with the SMM
measurements. Assuming that this limb brightness excess is a thermal temperature

excess, integration over the Solar disk reproduced the solar cycle irradiance variation

observed by SMM. Another important result of this experiment is that the polar
flux did not change as the active latitudes flux did. This implies that the Solar

luminosity is indeed varying with the Solar cycle.

The ”active network model” of F oukal interprets the solar irradiance variation
as the result of the solar cycle modulation of the brightness of the surface magnetic
elements, and it favors the surface magnetic field to modulate the solar irradiance
output. The limb brightness observations of Kuhn et a1. may not be incompatible
with the ”active network” model at first glance. Indeed, the higher photospheric
brightness along the active latitudes may be due to a higher active network con-
trast, which is unresolved in the limb brightness observations and is included in the
quiet photosphere components. However, Kuhn et al. argued that it is due to a real
increase in the photospheric temperature along the solar active latitudes, and that
a deep-seated horizontal magnetic field under the active latitudes is the modulat-
ing mechanism of the observed solar irradiance variations. Thus, there are subtle
differences between the two models. Which is right? Unfortunately, there are not
enough observations to prove or disprove either model. The satellite measurements
of the irradiance are spatially unresolved, and they don’t provide a clear picture of
what is causing the total irradiance variation. Modeling with proxy measurements
relies on our knowledge of the sunspot and facular contrast center-to-limb varia-
tions which may be enough to model variations in the total solar irradiance, but
they don’t tell us much about the sun. The limb brightness observations did provide
some spatial resolution, but it still is not enough for a full understanding for the
irradiance modulating mechanism at work. Thus, the next logical step would be a
precise, spatially resolved full-disk photometric measurement to try to resolve the

contributions of the various brightness components on the Solar photosphere, and

this is the motivation for this research.

Before large format CCD detectors were made generally available to the astron-
omy community, photoelectric imaging was done by scanning with single element
detectors, or with linear array detectors. The Extreme Limb Photometer (ELP) of
Chapman and Meyer (1986) and the observations of Hirayama, Hamana, and Mizu-
gaki (1985) are two examples. This type of observation requires a long scanning
process to obtain large area coverage, and the fluctuating atmosphere of the earth
limits the large-scale photometric sensitivity. The ELP and the photometer of Hi-
rayama et al. produced some useful results on the facular contrast measurements,
however, no measurements of the total solar irradiance variations were achieved

with this type of instruments.

The first attempt to do precise full disk photometric observation of the Sun with
CCD array detectors was reported by Nishikawa ( 1990a, 1990b) and N ishikawa and
Hirayama (1990). These experiments employed two telescopes equipped with video
CCD cameras to obtain simultaneous 2-color measurements of the Sun. Nishikawa
was able to reproduce the short-term solar irradiance variation measured by ACRIM,
taking into account only the contributions from faculae and sunspots. However, the
author concluded that the large scale noise in their data was too great and it was
useless for the study of large scale brightness structure contributions to the total

irradiance.
1.2 The Physics

We have learned a lot from observations about the Solar irradiance variations.
However, as discussed above, these observations not only did not provide clear-cut
answers to the questions we had, but raised more new questions. On the theoretical

side, the situation isn’t any better. We know that the Solar magnetic field is at

work, but exactly how is not clear. In the outer 25 ‘70 or so of the solar radius,
convection is the dominant mechanism of energy transport. The full hydrodynamic
fluid equations of convection are complicated and analytical solution is impossible.
Mixing—Length theory is the best model we have to treat this problem. The Sun is
composed mainly of hydrogen and helium. Hydrogen accounts for about 91 ‘70 of the
solar materials, and helium accounts for 8 %. The remaining 1 ‘70 is a mixture of the
heavier elements (Zirin 1988). The ionization energy of hydrogen atom is 13.6 ev,
and the hydrogen atoms can easily be ionized at 15000 K. Thus, the majority of the
hydrogen atoms are ionized except for the outermost layer where the temperature of
the solar plasma drops below the hydrogen ionization temperature. The boundary
of the hydrogen ionization layer is about 2000 km below the photosphere. Since the
solar plasma is highly ionized, the convective efficiency may be strongly influenced

by the presence of magnetic fields.

The solar magnetic field on the surface manifests itself in the form of sunspots,
faculae and network. Deeper into the interior, we believe there exists a global-scale
magnetic field. The effect of the surface magnetic field is readily observable, and
there are models explaining qualitatively the brightness of faculae and darkness of
sunspots (Spruit, 1977, Deinzer, Hensler, Schiissler, and Weisshaar, 1984, Schatten,
Mayr, Omidvar, and Maier 1986, Knolker, Schiissler, and Weisshaar, 1988). The
effects of the subsurface magnetic field is less certain, because the opacity of the
solar plasma limits our ability to probe deep into the solar interior. Despite these
limitations, we will review theories related to this subject and will try to patch

together a picture of how the magnetic fields modulate the solar radiative output.

1.2.1 The Magnetohydrodynamics

Although the MHD problem of the solar plasma is complicated and there are few

analytical solutions, we can still gain some insight by examining some special cases
for which analytical solutions are possible. An important concept, the ”frozen—in”
field, is the basis of almost all the modeling concerning solar magnetic phenomenon.
It is a special case when the magnetic Reynolds number R M of the solar plasma is
large and the magnetic field is carried with the fluid flow, or ”frozen” into the fluid

flow. Its physics is described below.

The behavior of plasma in a magnetic field is governed by the Maxwell equations
and the hydrodynamic equations. Since the solar plasma is highly ionized, it can
be considered as a good conductor and the Maxwell’s displacement current can be
ignored. Thus, we have the two curl equations for the magnetic and electric field in

cgs unit as

4 1013
v E=———— 1—1
X cat ( )
-O 4-0
anzlj (1—2)
C

In the presence of magnetic field, the generalized Ohm’s law includes the Lorentz

force,

i=a(f3+ xB) (1—3)

Old;

where a is the conductivity of the plasma. When combined, these three equations
give the induction equation of B that describes the time variation of the magnetic

field:

=Vx(6xB)+nV2B (1—4)

where 17 = c2/47r0 is the magnetic diffusivity. In the special case with 17 = 0,

equation (14) becomes a diffusion equation.

8; =17V2B (1—5)

Equation (1-5) indicates that the initial configuration of magnetic field will decay

in a diffusion time scale of

47raL2
TD:

 

2 (1 — 6)
c

where L is the scale length of the spatial variation of B. The typical diffusion time
scale in the solar environment for magnetic structure like sunspot is of order of 102

years.

A different limiting case is that the plasma is in motion and has negligible

electrical resistance. Then equation (1-4) becomes

%tl_3.=Vx(27xB) (1—7)
This equation says that the magnetic field flux through any loop moving with the

local fluid velocity is constant in time, or, the magnetic field line is frozen into the

fluid and are carried along with it.

10

In general, both the diffusion term and the transport term in the right-hand-side
of equation (1-4) are not negligible, and the time—variation of B is the combination
of the two effects. Simple dimensional considerations show that when the product
of the characteristic size and velocity of the system is large compared with the
diffusion constant 17, then the transport term dominates the diffusion term. Hence,

we can define a magnetic Reynolds number R M given by

RM=— (1‘8)

where L and V are the scale length, and characteristic velocity of the fluid, respec—
tively. If R M is large compared with unity, then the transport term dominates the
diffusion term. If RM is small, then the diffusion term dominates. In the solar
environment, the magnetic Reynolds number is usually large, due to the large size
and velocity of the system involved. Thus, the frozen-in field approximation is a

good working assumption.

The predominant magnetic field configuration in the sun is the magnetic flux
tube, at least near the surface. In the solar convection zone, the convective flow
tends to carry the magnetic flux tube with it, due to the large magnetic Reynolds
number RM. However, the magnetic flux tubes of sunspot and faculae do not
just float around, but are anchored at the base of the convection zone. Thus they
will only be bent, instead of been carried away. The magnetic energy 32/87r, and
magnetic tension 82/471r along the direction of the field lines associated with any
magnetic field, may affect fluid motion. The mechanical effect of the magnetic en-
ergy is the magnetic pressure, and can be neglected when considering the interaction

between the magnetic field and the fluid flow. Its effect is to reduce the thermal

11

pressure inside the flux tube, since the pressure balance between the ambient non-
magnetic plasma and the magnetic flux tube must be maintained. The magnetic
tension tends to resist the bending, or in other words, obstruct the fluid motion

perpendicular to the direction of the field.

1.2.2 The Surface Magnetic Fields

The surface magnetic field influences Solar luminosity by changing the effective
surface emissivity of the magnetic elements. Sunspots appear dark, hence emit less
energy, while faculae and network are bright compared with the surrounding pho—
tosphere, and emit more energy than the ”normal” photosphere with equal surface
area. Sunspots and faculae are different manifestation of slender magnetic flux tubes
perpendicular to the solar surface localized near the photosphere (Observations have
shown that faculae and network are composed of small magnetic flux tubes, and
the term ”faculae” refers to both faculae and network flux tubes in the rest of this
thesis, unless otherwise specified). The reason for their radically different behavior
is apparently the different size of the flux tubes. The size of the sunspots range
from a few thousand km for small spots to fractions of the solar radius. The size
of the faculae flux tube, however, is small. It is equal to or beyond the best spatial
resolution achievable from ground based observations. The generally accepted value
is about 200 km in diameter. In both cases, the strong magnetic field of the flux
tubes (1 to 3 KG for sunspots, and 1.5 KG for faculae. See, e.g., McPherson, Lin,
and Kuhn 1992, Solanky, Ruedi, and Livingston 1992, and Rabin 1992) prevent
convective energy transport across the flux tube, but the small size of the facular
flux tube makes it possible for the radiative energy transport to achieve thermal
balance between the solar plasma in and outside of the flux tube. Since the lifetime

of the faculae flux tube is 5 minutes and is much longer then the dynamical time

12

scale for the solar plasma to adjust to any pressure perturbation, magnetohydro-
static equilibrium is also achieved between the facular flux tube and the surrounding
photosphere. The presence of magnetic pressure inside the facular flux tube reduces
the thermal pressure. This requires a decrease of flux tube plasma density since the
temperature is balanced in and outside of the tube, and results in a lower opacity
inside. Thus, optical density unity is reached in a deeper layer of atmosphere inside
the flux tube where the plasma temperature is higher. Thus it appears brighter
than the surrounding nonmagnetic photosphere even near disk center. Near the
limb, the brightness of the faculae is largely due to the ”hot wall” effect. Because of
the depression of the geometrical height of the layer with unit optical density inside
the flux tube, we see the wall of the flux tube nearly face on when observing near
the limb. Since the limb-darkening of the photosphere is a strong effect, the faculae
appear with high contrast near the limb. The faculae thus act as a ”leak” where
the energy of the deeper atmosphere can escape more easily. However, Kuhn (1992)
argued that, in a model where energy is transported by diffusion and the photon
mean-free-path is decreasing exponentially, the horizontal energy transport is much
more effective than the vertical energy transport. Eventually the excess flux of the
facular flux tube comes from the surrounding layers. Thus this ”window” effect

does not lead to overall higher luminosity.

Sunspots, on the other hand, block the convective energy transport and reduce
the emitted flux by as much as 75 ‘70 for a large spot. This is due to the large diam-
eter of the flux tube which prevents radiative energy transport to from penetrating
deep into the flux tube. However, magnetohydrostatic equilibrium must hold, and
the presence of the strong magnetic field requires a lower thermal pressure of the

plasmas. The plasma temperature inside the sunspots is thus lower. The cooler

l3

sunspot plasma thus results in a reduced energy flux in the direction vertical to
the solar surface. Where does the blocked energy go? This is still a controversial
subject. Some believe that the blocked energy is re—radiated away by the accompa-
nying faculae with a timescale comparable to the lifetime of the active regions, while
others believe that the solar convection is very effective in mixing any local excess
energy throughout the entire convection zone within a short timescale. Kuhn (1992)
argued that photospheric magnetic fields produce anisotropic radiation fields, as is
evident in the case of the faculae. Thus, the sunspot blocked energy is redirected
to the direction parallel to the solar surface. He also showed some evidence of this
anisotropic sunspot radiation field in the autocorrelation of the ACRIM data. In

this case the sunspots do not decrease the solar luminosity on a long timescale.

The ”hot wall” model of faculae described above is plausible; however, alterna-
tive explanations exist. In the ”hillock” model, the excess facular brightness can be
due to non-thermal heating by, e.g., acoustic wave, hydromagnetic waves, electri-
cal current dissipation, or by systematic upflows of hot plasma from deeper layers
(Vainstein, Kuklin, and Maksimov 1977 and references therein). Indeed, the ”hot
wall” effect alone cannot produce the observed facular contrast. For example, the
calculation of Deinzer, Hensler, Schiissler, and Weisshaar (1984) cannot account for
the observed high contrast value at the extreme limb, and they concluded that some
heating mechanism in addition to the ”hot wall” effect is needed to explain observa-
tions. Schatten, Mayr, Omidvar, and Maier (1986) described a dynamical model in
which faculae are the result of systematic upflows along the facular magnetic field
line. In this model, the facular flux tubes carry ionized Hydrogen from the ionization
layer 2000 km below the surface upward. Thus, the material inside the facular flux

tube contains more energy (the Hydrogen recombination energy) compared with the

14

surrounding nonmagnetic photosphere composed of neutral hydrogen. The facular
flux tube thus acts as a channel within which the hotter material can be transported
to the surface more easily. The hotter material results in a uplifted solar surface in
the flux tubes. The heating and the geometry of the ”hillock” as seen from different

angle may produce the observed facular center—to-limb contrast variation.

Whatever the model of the faculae and sunspots is, they operate in the outer
layer of the Sun. The limb-brightness observations of Kuhn et al showed that the
observed irradiance variation is indeed solar luminosity change. Thus, the sun may
modulate its energy generating rate in the core in sync with the solar cycle. Simple
calculation show that the time for a photon to diffuse from the core to the surface
is of the order of 105 years. It is then hard to imagine how something happening
near the surface can have any effect on what happened in the core 105 years ago.
Another possibility is that the solar core does not generate more energy during
solar maximum. The increased surface magnetic activities simply increase the total
effective emissivity, and thus increases the total luminosity during solar maximum.
This scenario requires an energy reservoir for the excess energy not radiated away
during solar minimum. This energy cannot be stored as magnetic energy, since most
of the models of solar magnetic cycle (See, for example, the discussions in Foukal,
1990) predict that the magnetic activity is the weakest during solar minimum.
The solar convection zone, with its huge heat capacity, can easily store the 10-3
solar luminosity not radiated away without causing any significant change in the
convection zone. If this excess energy is stored purely in the form of thermal energy,
then the accompanying temperature perturbation should be of the same order of
magnitude, and should be an observable effect. However, we must note that the net

result of the complementary effects on solar luminosity of the sunspots and faculae

15

are not clear yet. There are observations and models arguing that the flux excess
of faculae and deficit of sunspots should balance each other (Chapman, Herzog,
Lawrence, and Shelton 1986, Schatten 1985, Bruning and LaBonte 1983). If this
is true, then we need some other mechanism to explain the solar cycle irradiance

variations.
1.2.3 The Subsurface Magnetic Fields

We believe that the increased surface magnetic activity during solar maximum is
the surface signature of a toroidal magnetic field near the base of the solar convection
zone intensified by the solar differential rotation. The estimated strength of the
magnetic field is of order of a few thousand gauss (Parker, 1987b). MHD theory,
as discussed in Section 1.2.1, predicts that such an intense magnetic field should
block convective flow. Thus, it is important to consider the effect of this intense
magnetic field on the solar luminosity variation. Interestingly, its effect depends
on the location of the field. If it lives in the convection zone, then its effect is
to block the convective flow, thus reducing the efficiency of the convective energy
transport. The solar plasma directly above these field line receives less energy from
the below, thus forming a ”thermal shadow”. On the other hand, if these field live
in the radiative zone, since the magnetohydrostatic equilibrium must be maintained
in and outside the tube, the plasma density inside the magnetic flux tube must be
lower, as in the case of the faculae flux tube. The ”evacuated” flux tube thus has
lower opacity and higher radiative transfer efficiency. The solar plasma above it
will see more flux coming through, and the temperature should be higher than the
plasma at the same level without a magnetic field below. Thus, if the toroidal field
is responsible for the observed 11-year luminosity variations, it must exist near or

below the base of the solar convection zone.

16

The exact location of the toroidal magnetic field is not well determined, both
theoretically and observationally, however. Also, the interactions between the mag-
netic field and the energy transport mechanism are more complicated than the
simplified picture above. Parker, in a series of papers (1987a, 1987b, 1987c), con—
sidered the dynamics of a strong and broad horizontal magnetic field in the con-
vections zone of a star. He argued that the result of the interaction between the
magnetic buoyancy of the intense horizontal magnetic field and the weight of the
thermal shadow formed above the field due to the blocking of the convective flow
is that the weight of the thermal shadows would overcome the magnetic buoyancy
and press the magnetic field down against the bottom of the convection zone. Also,
the inhibition of the convective flow results in the accumulation of heat beneath
the magnetic field. This causes the gas density below the field to decline, and
eventually triggers the Rayleigh-Taylor instability. The Rayleigh-Taylor instability
arises between the interface of two fluids with different density in the presence of
a gravitational field (Chandrasekhar 1970). Consider two vertically adjacent fluids,
with the lower fluid having density p; and the upper layer with density p2, and
the gravitational field is pointing downward. If p2 < m, then the configuration is
stable against perturbations, whereas it is unstable if p2 > p1. If there is surface
tension then the configuration will be stabilized for perturbations with sufficiently
short wavelengths. If a horizontal magnetic field is introduced, the magnetic tension
of the field serves as the surface tension due to the frozen-in field approximation.
Thus, the accumulation of heat beneath the magnetic field will eventually cause the
hot gas to penetrate through the field to raise to the surface, carrying some of the
magnetic field with it. This phenomenon is interpreted as the cause for the onset of
the active regions. Parker’s work provided some support to the model in which the

subsurface magnetic field is the modulation mechanism, since the predicted location

17

of the subsurface magnetic field should increase the energy flux received by the solar
plasma directly above it. The amplitude and phase of this modulation should also

be in phase with the sunspot cycle, consistent with observations.

Spruit (1991) made a mixing-length calculation on the effect of the surface and
subsurface magnetic field on the solar luminosity and radius variations. Two im—
portant time-scales in this calculation are the turbulent diffusive timescale Tm, and
the thermal time scale of the convection zone, Tth- The estimated values are Tm z 1 ‘
yr., and Tu, a: 105 yr. An isotropic turbulent diffusivity was assumed. Because of
the short turbulent diffusion timescale and the large heat capacity of the convection
zone, perturbations originating from the base of the convection zone are smeared
horizontally and lose their identity before reaching the surface. He concluded that,
any temperature perturbations originating from layers deeper than 40000 km be-
low the surface will have unobservable effects on the luminosity and radius, and all
the observable effects result from the surface magnetic field. Thus, this calculation
would rule out the subsurface magnetic field as the cause of the observed irradiance
variations. However, numerical simulations of realistic convection are not consistent

with Spruit’s assumptions (Kuhn 1991).

1.3 This Thesis

It is clear from the above discussions that the question of how, and which solar
magnetic field component is modulating the long-term solar luminosity variations is
not answered by the current theoretical modeling. There is room for improvement
at both theoretical and observational front. We took a step forward by constructing
a specialized instrument, the Solar Photometric Telescope, to obtain precision full—
disk photometric observations of the Sun. New observing techniques, along with

new analysis methods were developed to achieve high photometric sensitivity.

18

The Solar Photometric Telescope (SPT), is designed to achieve both small and
large scale photometric accuracy. We employed a cryogenically cooled CCD array
to obtain better photometric accuracy. The optical system is carefully designed to
minimize noise due to scattered light, internal reflection, and optical imperfection
of the optical components. CCD array observations require careful calibration of
the difference between the gain of each detector element, the so—called ”fiatfielding
problem”. The SPT was designed to allow us to take gain calibration data, the
Hatfield, with exactly the same optical setup used for taking data. This is essential
to achieve the desired photometric accuracy, since the ”true” gain variation includes
both the pixel-to-pixel sensitivity variation of the CCD, and the optical modulation
of the light beam due to optics, e.g., the dust shadows. The daytime sky was
used as our ”flatfield”. Although it provides the best flatfield over other uniform
light sources, it is the most time-consuming step of the whole observing procedure,
taking up 2/ 3 of the observing time. This is a compromise we have to make to
achieve the large scale photometric accuracy. Compared with the SPT, Nishikawa’s
instrument employed video CCDs operating at room temperature. He also used
diffusers to make calibration observation. Our experience showed that this method
of flatfielding introduced large scale error, since the insertion of diffusers changes
the optical configuration of the system. This is the limiting factor of the largescale

photometric accuracy of his instrument.

Although the sun radiates about 70 % of its energy in the visible part of the
spectrum with wavelengths below 1;: (Foukal 1990), the infrared solar spectrum is
still of particular interest to the solar astronomers. The opacity of H‘, the dominant
source of absorption in the solar photosphere (Chandrasekhar and Breen 1946), is

smallest at the infrared J band at 1.6 it. Thus, we have access to the deepest

19

observable layer of the solar photosphere at this wavelength (Vernazza, Avrett, and
Loeser 1976). Although this layer is only 35 km below the layer with T5000, this
is an important 35 km since the solar opacity is changing rapidly in this region.
For example, some photometry showed that the faculae appear dark, not bright
near disk center at 1.6 p (Foukal, Little, and Mooney 1989, Foukal, Little, Graves,
Rabin, and Lynch 1990). Although this result was not confirmed by other observers,
it could bear important information on the structure of the small magnetic flux tube
if proved to be true. Another important property of the infrared spectrum is the
height dependence of the source function, i.e., at longer wavelengths, we observe
higher up in the solar atmosphere. This is due to the fact that the H' dominates
the continuum opacity for light with wavelength longer than 1.6”. Also, the H"
is in Local Thermal Equilibrium (LTE) with the electron gas. Thus, the infrared
spectrum can be used to probe the temperature structure of different layer of the

solar atmosphere.

Infrared observations have numerous advantages as described above; however,
imaging in this part of the spectrum has long been closed to astronomers due to the
lack of adequate infrared array detectors. There were some single element infrared
detectors available, but imaging with a single element detector is difficult. The
long time required to scan an image limits the time resolution and data rate, and
the changing sky condition and the small amount of data limit the photometric
accuracy achievable. Some examples of this kind of observation can be found in

Koutchmy (1978), and Worden (1975).

With recent progress in infrared array technology, large format infrared arrays
became accessible to astronomers. Another new instrument, an infrared camera was

built with a Rockwell TCM1000C infrared array, sensitive to light with wavelength

20

from 1 to 2.5 p. It has a large format of 128 x 128 pixels, with a pixel size of 60 pm.
The full well capacity is of order of 107, and is operated at liquid Nitrogen temper-
ature of 77 K. These properties make it an ideal instrument for solar photometric
observations. To take advantage of this new instrument, we used it to obtain images
of the Sun at the infrared J (1.2 p), H (1.6 p), and K (2.2 p) band. The infrared
data presented in this thesis were obtained at the National Solar Observatory at
Sacramento Peak, New Mexico. This observatory, at 3200 meters above the sea
level, is ideal for infrared observations since it is above most of the water vapor in

the earth’s atmosphere which is responsible for most of the atmospheric absorption

of the infrared light (Jackson, 1985).

One of the new analysis techniques developed is an algorithm to calculate the
flatfield of an array detector with only a set of shifted data frames. This tech-
nique is indispensable when good flatfields are not easily available, like the infrared
observations where sky brightness is fractionally lower than in the visible. New al-
gorithms for finding the coordinates of the solar disk center, averaging images, and
constructing the contrast image of the averaged solar images were also developed.

These will be presented in Chapter 3.
1.4 Related Subjects

The solar cycle irradiance variation is not the only subject that can be studied
with new high accuracy photometric observations. For example, the energy balance
between sunspot and faculae, the facular and network contrast, and the thermal
signature of large scale solar convection like supergranulation and giant cells are

interesting subjects that can be studied.

The photospheric network is known to outline the supergranulation seen in

photospheric velocity maps and the photospheric magnetic network (Frazier 1970,

21

Muller 1983). High resolution observations show that it consists of isolated bright
points, with contrast of 1.2 to 1.4, and sizes of 0.2”, imbedded in intergranular lanes,
and was considered to be the same as the facular bright points (Muller and Keil
1983, and references therein). The observation of N ishikawa and Hirayama (1990),
which showed that the center-to—limb variation of the facular two-color contrast
in the quiet network and active region are the same, supported this finding, but

different views exist.

The center— to-limb variation of the contrast bears the information of the vertical
structure of the small magnetic flux tube considered to be the basic constituent of
network and faculae. However, no reliable photometric measurement of the network
contrast in the continuum is available up to date, due to its smallness. Becker
(1968) obtained z 0.3% contrast with broad band continuum images centered at
4850 A. Frazier (1970) found 1 % contrast at 4850 A. Liu (1974) used time-averaging
techniques on white light. images to obtain images of the network near disk center,
but the contrast was not measured. Infrared observations (Worden 1975) at 1.6 it
showed temperature structure weakly correlated with Chromospheric network, but
the author reported a huge temperature difference of 500 K. Other monochromatic
studies found 0.29 ‘70 at 5265 A (Skumanich, Cmythe, and Fazier 1975), 0.1 % at
5256 A (Foukal and Fowler 1984).

Since the network is found both in the active and quiet sun regions, it will be
instructive to study the differences of the characteristics in these two regions. This
study should yield information on how the network is formed, and its relation to

the supergranulation.

F aculae, when observed in continuum wavelengths with low to medium spatial

resolution, appear as extended bright patches in the active regions near solar limb.

22

They are almost invisible near disk center. As pointed out earlier, the Facular
Bright Points are considered by some to be identical to the Network Bright Points,
but it is not proved, or disapproved observationally. The major difference between
faculae and network has been the spatial appearance of the two components. It
is an important component to the solar irradiance since it is the brightest feature
observable in a continuum image. Nevertheless, the spatial distribution, as well as
the center-to-limb variation of contrast, are still not well determined. The CLV of
facular contrast has been measured in different region of p, in a wide range of wave-
lengths (e.g., Lawrence, Chapmann, and Herzog 1988, Lawrence 1988, Hirayama,
Hamana, and Mizogaki 1985, Libbrecht and Kuhn 1986, Wang and Zirin 1987, and
Nishikawa and Hirayama 1990). Most of these results showed that the facular con-
trast near disk center is vanishingly small, except the result of Foukal and Fowler
which showed negative contrast at 1.6 )1 near disk center. The generally adopted
value today is zero. Near the limb, there is better agreement between observations.
However, at the extreme limb ([1 S 0.2) Libbrecht and Kuhn found that the faculae
contrast peaked near the limb and falls off outward, while Wang and Zirin found

that it continues to rise toward the limb.

The interesting question is, is FBP the same as the NBP? If not, what’s the
difference between the two? If yes, then what is the mechanism that distributes
the NBP in a network-like structure. What is the vertical structure of the facular
magnetic flux tube? There is a lot of work to be done to answer all these questions,
and an accurate measurement of the contrast, and contrast center-to—limb varia-
tions will certainly provide us valuable information toward the resolutions of these

problems.

This thesis reports on our effort to try to answer the questions we have raised

23

so far. We will start by describing the instrument in details in Chapter 2. The
observation and data reduction procedures are discussed in Chapter 3. Results and
interpretations of the observations of photospheric network and faculae contrast are
given in Chapter 4. Chapter 5 discusses subjects related to the total solar irradiance

variation, and summarizes the whole work.

CHAPTER 2

INSTRUMENTATION

2.1 General Considerations

The goal of this research is to detect the brightness variations that are causing
the long-term solar irradiance changes. The limb brightness measurement provides
a good guideline for the instrumental precision required. The observed brightness
excess in the active latitudes range from about almost zero during solar minimum
to about 7 K near solar maximum. Thus it is desirable to have a 1 K sensitivity
to monitor the yearly change of the photospheric temperature. The sensitivity
required to resolve a temperature variation of amplitude 6T from a blackbody at

temperature T at wavelength A is

61_ 7’57- 6T
I eIp(-:&.%)-1T

 

(2-1)

where k is the Boltzmann constant, h is the Planck constant, and c is the speed
of light. At the red band pass of A = 6,500 A, g} = 6.5 x 10‘4 for 6T = 1 K,
and T = 6,000 K. In the infrared at A = 16,000 A, 4} = 3.3 x 10-4 for the same
magnitude temperature variation. The full well capacity of the Texas Instrument
CCD used for this observation is 65,000 photons. The photon noise per pixel per
exposure is 260 photons. Thus, the fractional noise is 4 x 10’3, approximately the

sensitivity desired. The full well capacity of the infrared array is about 50 times

24

25

that of the Texas Instrument CCD, hence the photon noise is not the limiting factor.
However, there are other sources of error, e.g., scattered light, and flatfielding error.

Thus, it is necessary to average a large number of images to minimize these errors.

Another important factor for the success of this observation is the flatfielding
problem. Flatfielding is necessary for any array detector observation. Conventional
flatfielding techniques use either ”dome flat”, or ”diffuser flat”. Our experience
with these methods showed that they are good only to 10’2 level on the large
scale. The ’true flatfield’ should measure both the pixel-to-pixel gain variations
of the detector and the ’gain’ variations due to optical imperfections, like dust
shadows. When a diffuser is used, or when the telescope is turned to the dome, the
illumination pattern of the system is changed, hence the optical configurations used
to register the data is different from that used to register the flatfields. Thus, the
diffuser and dome flatfields usually produce some large scale flatfielding error. The
best flatfield available for daytime observations is the sky. The sky is virtually flat,
except for a small linear gradient. The measured amplitude of this linear gradient is
approximately 10‘5 / pixel, when looking at 30 degrees away from the sun. This can
be fit for and removed aposteriori. The disadvantage of using the sky for fiatfielding
is that the sky brightness is approximately 10’5 to 10‘6 that of the sun, hence the

exposure time to obtain a skyflat would be 105 times longer.

The brightness of the infrared sky is lower than in the visible, due to the
wavelength dependence of the scattering cross-section of photons scattering off air
molecules. Thus, using sky as flatfield for infrared observation requires very long
integration and becomes unpractical. We developed an iterating algorithm which

can calculate the flatfield of an imaging array with a set of shifted data frame.

This algorithm is used to calibrate the infrared data. This algorithm was tested for

 

26

nighttime star cluster observation with the MSU 24” telescope, and for the visible
wavelengths full disk solar observation on the SPT. The flatfielding quality was

found to be comparable to the skyflats in the visible data.

The above considerations call for a system with low scattered light, and high
data rate. It also requires a 106 dynamical range of exposure time. The Solar Pho-
tometric Telescope was designed to meet these criteria. The following paragraphs

describe the system in details.

2.2 The Solar Photometric Telescope

Figure 2.1 shows the system diagram of the SPT. Figure 2.2 and Figure 2.3 are
pictures of the SPT. The heliostat was mounted on a heavy steel platform sitting
on the roof of the Michigan State University Observatory. It consists of a 15 cm
primary quartz mirror mounted on a Meade 8” telescope drive, and a 6 cm secondary
mirror reflects sunlight down through a hole in the roof to the imaging optics. All
the imaging optics and the CCD camera were mounted on a bracket attached to
the heliostat platform. A fast shutter assembly, consists of a electromechanical
leaf shutter and a rotating wheel shutter is mounted on a vertical bench, isolated
from the imaging optics. The imaging system is enclosed in a light-tight box,
and well baffled with tubes inside the box. An IBM-compatible computer is used
to control the shutters, the heliostat, and the CCD camera. Images are stored on
magnetic tape for later analysis. The CCD was upgraded from the Texas Instrument
384 x 594 array to a Ford Aerospace Inc. 1024 x 1024 array during the winter of
1991. The imaging optics, as well as the computer control and data storage system

were modified consequently to take advantage of the higher spatial resolution offered

by the new CCD.

27

   
     
  
     

Electra-Mechanical
Shutter

— Neutral Density Filter
—Primary Lens

Rotating Wheel Shutter

Lens

5 Light Battling Tubes

Lens

 

Exabyte Drive

 

Figure 2.1 System Block Diagram of the Solar Photometric Telescope.

28

 
 
 
   

Secondary Mirror

Primary Miirror

Figure 2.2 The Heliostat of the SPT.

29

   
   
 
   

Primary Lens
Shutter Assembly and
Filter Wheel

 

  
   
  
 

' t

3'; Light Baffling
~ .Tubes

5 Coillmator
_Lens

 

Reimaging Lens
CCD Camera
: Electronics

CCD Dewar

Figure 2.3 The Optical System of the SPT.

30

2.2.1 Optical System

In the first phase of the operation, the imaging optics consists of a f=900 mm
primary lens, forming a full-disk solar image of size 8.3 mm at the focus of a f=166
mm secondary lens. This secondary turns the focused beam from the primary lens
into parallel beam again. A third f=135 mm reimaging lens then forms the solar
image on the CCD with a size of 270 pixel. The secondary and the reimaging lens
combination works as an optical relay. The addition of these optics may appear
redundant, and ideally, we would like to avoid any unnecessary optics to reduce
scattered light. But the benefits of the optical relay described below more than

compensate its drawback, and its use is justifiable.

The diameter of the primary lens is 60 mm. To avoid saturation problem, it was
stopped down to 25.4 mm by an aperture stop. This results in a very slow beam,
with f/D=35.4. A slow beam forms sharp dust shadows on the CCD. They are
difficult to calibrate by flatfielding. The optical relay reduces the f/ D ratio to 28.8,
and makes it easier to calibrate the effects of dust shadow. To avoid vignetting,
the neutral density filter, the interference filters, and the shutters were all placed
as close to the primary lens as possible. This also minimizes the effect of filter
inhomogeneities. But focusing with primary lens becomes impossible, since the
components are so close to each other. The optical relay thus provides a convenient
way to focus the image. Since the final position of the image is independent of
the relative distance between the secondary and reimaging lens, focusing can be
achieved simply by adjusting the secondary to focus it on the image plane of the
primary lens. The optical design was checked with a computer ray-tracing program

to ensure that the system is free of vignetting problem. Figure 2.4 shows the ray-

tracing diagram of the optical system.

 

 

31

Ray Tracing Diagran of The Solar Photonetric Telescope
Lou Resolution Node (6" resolution)

SI1 L2 1.3 CCD

20. C. I

 

-2.0 on

i m.

 

 

LAWT: Annotate Ciriienu Zaxle "die ”die Poen Hagn Randon Ila: Quads 380

Figure 2.4 Ray-Tracing Diagram of the Solar Photometric Telescope in Low Reso-
lution Mode.

32

With the installation of the 10242 array, the image size increases to 350 pixel
due to the smaller pixel size of the new array. To take advantage of the larger
format of this array, the secondary and reimaging lenses were replaced by lenses
with f=81 mm and f=160 mm, respectively. The new image size is 886 pixel, or
2.16”/ pixel. This is well below the diffraction limit of 6.5” of a 25.4 mm telescope
at 6,500 A . To achieve 2”/ pixel resolution, the aperture stop of the primary lens
is removed to yield a resolution of 2.09” at 5,000 A and 2.72” at 6,500 A . This
resolution is barely enough for the green wavelength, but not enough for the red.
Since experiences from nighttime observations showed that the seeing of our site is
typically 3”, we decided to go ahead with the D=60 mm primary lens. Since the
aperture size of the shutter and the interference filters were only 25.4 mm and 50.8
mm, respectively, they have to be increased to avoid vignetting. An alternative is
to place them between the secondary and reimaging lens, where the beam size is
small. The later is adopted since it does not require the acquisition of new filters
and shutter. The f/ D ratio of this configuration is 29.6, not significantly different

from the old configuration. Figure 2.5 shows the ray tracing diagram of this setup.
2.2.2 The Shutter Assembly

The shutter assembly consists of an electromechanical leaf shutter and a rotat-
ing wheel shutter. Figure 2.6 shows a picture of the shutter assembly. The leaf
shutter has an aperture of 25.4 mm. The fastest speed is about 5.5 msec. With
the initial setup with a D23 neutral density filter, this exposure time still leads to
saturation when taking solar images. Also, an integration time of 5 x 105 longer for
taking the skyflat is 45 minute. This is far from acceptable. We achieved a faster
shutter speed by synchronizing the leaf shutter with the shutter wheel, running at

40 revolution per second (RPS) speed. The shutter wheel is a copper disk with

33

Ray Tracing Diagran of the Solar Photonetric Telescope
High Resolution Diode (2" resolution)

 

 

 

 

 

 

68.0 on 2 13.0 on
IATOUT: Annotate Ciriienu Zaxis iidin Odin Posn iiagn Randol- like Quads ESC

Figure 2.5 Ray-tracing Diagram of the Solar Photometric Telescope in High Reso-
lution Mode.

34

two 36 degree wedge cutoff, spaced 180 degree apart. One of the cutoff is masked
by tape, allowing only one exposure for each revolution. The symmetrical cutoff
is necessary to balance the wheel. The shutter is driven by a DC motor, equipped
with an optical encoder. The optical encoder generates an index pulse for each
revolution. The shutter wheel is aligned with the optical encoder such that when
the index pulse is on, the shutter wheel cutoff is aligned with the optical axis, al-
lowing the light from the Heliostat to pass through. Thus, when taking exposure,
the control computer detects the index pulse, delays for 1/4 of the revolution, then
opens the leaf shutter. On the second index pulse, the control computer delays for
another 1/4 revolution, then closes the leaf shutter. The fastest speed this shutter
assembly achieved was 2 msec., limited by the maximum speed of the DC motor.
When taking skyflats, the shutter wheel can be parked at the open position by the
computer, and the integration time is controlled by the leaf shutter. The minimum
shutter speed is limited only by the number of register of the counter, which is 214
second. The large range of possible integration time of this shutter system thus al-
low us to take both the solar images and the skyflats with exactly the same optical

setup. This is critical to achieve high photometric sensitivity.
2.2.3 The Computer Control and Data Acquisition System

The first control computer used was a IBM PC-AT computer. It controls the
shutter wheel speed through a D / A controller. A bidirectional parallel port is used
to control the leaf shutter, the heliostat motion, and is used to monitor the status of
the wheel shutter. The first. mass storage system used was a 9—track tape drive. The
CCD camera system is manufactured by Photometrics Inc., and is also controlled by
the computer. An observing program, written with fortran and assembly language,

integrates the operation of the subsystems. The upgrade to the Ford 10242 array

35

 

6 The Shutter Assembly and the Filter Wheel.

Figure 2

36

presents a 5 fold increase in the amount of data. To handle this large amount of
data, the IBM PC-AT was replaced with a 386 PC, and the 9-track tape drive was

replaced with an Exabyte tape drive.

2.3 The Infrared Array Detector

The Infrared Array system is a Rockwell TCM1000C 128x128 pixels infrared
array which is sensitive to light with wavelength between 1 and 2.5 micron. A filter
wheel mounted inside the dewer carried an H (1.6 micron) and a K (2.2 micron)
filter. Both filters have a band pass of 0.2 micron FWHM. In the infrared, the
brightness of thermal radiation from objects at room temperature sometimes ex-
ceeds faint celestial sources. Thus, a warm filter in front of the infrared array will
radiate significant amount of infrared light to contaminate the observations. Thus,
the filters are mounted inside the dewar and maintained at liquid Nitrogen tem—
perature to minimize their thermal emission. The infrared array was electronically
shuttered so that no mechanical shutter was necessary. A detailed description of the
Infrared Array System is presented in another paper (McPherson, Lin, and Kuhn

1991)

The infrared data were obtained at the National Solar Observatory at Sacra-
mento Peak, New Mexico, with the 6” ceolostat in the Evans Facilities, and with
the Vacuum Tower Telescope (VTT). To obtain spatial resolution comparable to
the SPT, we used a f=600 mm lens to form a 200 pixels diameter partial solar
image on the infrared array. Neutral density filters were used to reduce the amount
of light. Like the visible wavelengths observations, scattered light variation is the
most important source of error. Since we don’t have a permanent blackbox setup,
we tried to minimize the scattered light by covering and baffling the infrared system

with black felt and tubes. Although the covering and baffling is simple and maybe,

37

well, ugly, it served its purpose well. Figure 2.7 shows the experimental setup for

the infrared observation.

38

       

925 3.380

>83 2:22..
_

 

 

 

 

 

_
5...... 8:20:85

3.: 2.55 2a:

2.9.. ESE—n.

3...“. 2.2.8 .8502

Figure 2.7 Experimental Setup for the Infrared Observations.

CHAPTER 3

THE OBSERVATIONS AND DATA REDUCTION

3.1 Observing Procedure

A complete set of imaging array data must include dark frames, flatfields, and
the data frames. For the visible wavelength observations, the skyflats were used
for flatfielding. Thus, the observing sequence usually starts with a set of 5 dark
frames, then 20 solar images, and finally a set of five 4—rninute exposure skyflats.
The 20 solar images were taken continuously, and take approximately 10 minutes
to complete. Thus, a complete dataset in one color takes approximately 45 minutes
to collect. The filters were then switched to the next color, and the observing
sequence repeated. we call a set of data with both colors a time step, since the
solar images within this set will be averaged to make two time-averaged solar images
in green and red wavelength. Occasionally, the sky clouded up during observation
and skyflats were inaccessible. Except when the sky is completely overcasted, a
set of 10 shifted solar images would be collected to use the iterating algorithm
to calculate the flatfield. Since the incident angle of the light beam changes as the
heliostat traces the sun across the sky, the dust shadow pattern on the CCD changeS+

also. Thus, frequent flatfield observations are necessary.

The iterating algorithm was the primary calibrating method for the infrared
observations, and because the array covers only part of the solar disk, a data set

in one color consists of dark measurements, and then a series of shifted images to

39

40

cover the full solar disk. Diffuser flats were also obtained for initial flatfielding for

the iterating algorithm.
3.2 The Visible Wavelength Data

The construction of the SPT was completed in the summer of 1990. It collected
5 days’ data during that summer; however, only 3 days’ data lasted more than fours
hour. The SPT was dismounted for roof work and the CCD camera upgraded with
a 1024 X 1024 CCD in the following fall and winter. The SPT resumed operation
the next summer, and obtained 9 days’ data. Figure 3.1 shows a raw full disk solar
image obtained with the SPT in the green bandpass. Table 1 summarize the data
set of the visible wavelengths. The length of the data set varies from a single time

step to 4 time step which last 6 hours depending on weather condition.

Table 1. Summary of Visible Wavelength Observations

 

Date Red (6,500 A) Green (5,000 A)
July 4, 1990 1 time step’ 1 time step
Aug 4, 1990 3 time steps 3 time steps
Aug 14, 1990 3 time steps 3 time steps
June 4, 1991 1 time step No data

June 5, 1991 3 time steps 1 time step
June 6, 1991 2 time steps 2 time steps
June 8, 1991 2 time steps 2 time steps
Sept 6, 1991 3 time steps No data

Sept 21, 1991
Sept 28, 1991
Sept 29, 1991

3 time steps
4 time steps

2 time steps

3 time steps
4 time steps

2 time steps

 

*Each time step consists of 20 solar images.

41

 

Figure 3.1 Raw Solar Image in the Green Bandpass. This image is obtained on June
4th, 1991. Image size is 448 x 448 pixels, and the solar radius is 170 pixels. A large
sunspot near the limb, as well as several small sunspot groups appear as sharp dark
spots are visible. Also visible are shadows of dust particles on the CCD window
which appear as fuzzy large dark spots. Solar north is oriented approximately 135

degree counterclockwise to the +9 direction.

 

42

3.3 The Infrared Data

The infrared data were obtained at N SO at Sacramento Peak during the Fall
of 1990, and February of 1992. The 6” ceolostat of Evans Facility was used for the
1990 observations, and the Vacuum Tower Telescope (VTT) was used for the 1992
observations. Only 10 images in both H and K band were obtained in the 1990
observation. Nevertheless, this was enough to resolve the low contrast photospheric
network. Several series of scans across the solar disk consisting of 100 images in J
and H band were obtained for the 1992 observations. Figure 3.2 shows a typical

raw image in the H band. Table 2 summarizes the Infrared dataset.

Table 2. Summary of Infrared Wavelength Observations

 

Date J band (1.211) H band (1.611) K band (2.2/1)
Oct. 12, 1990 No data 10 images 10 images
Feb. 25, 1992 2 time steps’ 1 time step No data

Feb. 26, 1992 1 time step 2 time steps No data

 

* Each time step is a 100 images scan covering the full disk.

3.4 Data Reduction

The data reduction starts with the standard procedure of dark subtraction and
flatfielding. Figure 3.3 and 3.4 show two typical flatfields in the green and H band,
respectively. The solar disk centers were then found and recorded. The flatfielded
solar images within a time step can now be rotated, and shifted to correct for the
image rotation and shift due to the heliostat. These images were then rescaled and
averaged, and the fractional residual, or contrast, of the averaged solar image can be
constructed. Figure 3.5 and Figure 3.6 show two flatfielded, time-averaged images

in the green and H band. Figure 3.7 to Figure 3.11 show the contrast images of the

43

 

Figure 3.2 Raw Solar Image in the Infrared H Band. This image is obtained on
February 12th, 1990. Image size is 130 x 132 pixels, and the solar radius is 100
pixels. The two fuzzy dark spots on the upper half of the image are dust shadows.
The finger-like dark strips on the lower half of the image are due to large pixel-to-

pixel gain variations.

44

five wavelengths. Since the contrast images are constructed with the time-averaged
solar images, contributions from temporal brightness structures, like the 5-minute
solar oscillation, are small. Low contrast structures can now be examined more
easily from these contrast images. However, the imaging optics of the SPT distort
the solar images to an elliptical shape, with an oblateness of 7 x 10—3. Thus, the
contrast images in the visible wavelengths showed a large second harmonics term
near the limb, with an amplitude of 10'2. This can be seen easily in the contrast
images. Examination of this brightness structure in different time steps showed that
they have a constant phase, i.e., these brightness structures do not rotate with time.
Thus, it is evident that they are not of solar origin, but are due to the optics. The
amplitude of the distortion can be calculated, and the images corrected before being
averaged. The distortion corrected contrast of the image of Figure 3.8 is shown in
Figure 3.12. It is clear that the large optical distortion error is significantly reduced.

The details of each reduction step are described in the following paragraphs.
3.4.1 Dark Subtraction and Flatflelding

The dark frames and flatfields from each time step are averaged first to make an
average dark, and average flatfield for each time step. In the case that no skyflats
were available, or when reducing the infrared data, the iterating algorithm is used

to calculate an iterated flatfield for that time step.

The principle of the iterating algorithm can be understood by observing a simple
experiment in which 2 array detector images are obtained with the 2nd displaced
with respect to the first image by 1 pixel in the horizontal direction. Assuming that
the source flux is constant in time, then each horizontal pair of pixel between the 2
images are illuminated by the same flux. Thus we can obtain the ratio of the gain

between every neighboring pixel in the horizontal direction. The same experiment

45

 

Figure 3.3 Green Bandpass Flatfield. An average skyflat obtained by summing over
5 skyflats each with 4-minute integration time. The position of the sky is 30 degrees
east to the Sun. The flatfield is normalized by the average intensity of the image.
Display range is 0.9 to 1.1.

46

 

Figure 3.4 Infrared H Band Flatfield. This Flatfield was obtained with the Gain-
Iterating Algorithm developed by Kuhn, Lin, and Loranz (1991). Ten partial-disk

images like the one in Figure 3.2 are used for the calculation. The iteration were

carried to the 10th step. The Infrared array has a large pixel—to—pixel gain variation
(50 ‘76). Display range was 0.9 to 1.1

47

 

Figure 3.5 Green Time-Averaged Solar Image. This image is obtained by summing

over 20 solar images, like the one in Figure 3.1, after dark-subtraction and flatfield
calibration. It is clear that the dust shadows in Figure 3.1 were well calibrated.
Only a 400 x 400 pixels subimage is displayed, with the Sun centered in the image.

Solar north is oriented in the y direction, and west is to the right of the image.

48

New.
A \- ~\‘:\ 33w
w a \
assets.» ..

M
%
a...

\ i t
‘&\"\L
git.» - A 53%, x

, .
.3»

. m

 

Figure 3.6 H Band Time-Averaged Solar Image. This image is constructed by
summing over 100 partial-disk images of a scan obtained in February 26th, 1992.
The radius of the Solar disk is 126 pixels.

49

 

Figure 3.7 Time-Averaged Residual Image in the Green Bandpass Before Distortion
Correction. The two dark patches at the top and bottom of the image are due to

optical distortion. Display range is -0.02 to 0.02.

50

 

Figure 3.8 Time-Averaged Residual Image in the Green Bandpass. Optics distortion
were corrected. The amplitude of the large-scale error are reduced compared with

Figure 3.6. Display range is -0.02 to 0.02.

51

 

Figure 3.9 Time-Averaged Residual Image in the Red Bandpass. Optics distortion

were corrected. Display range is -0.02 to 0.02.

52

 

Figure 3.10 Time-Averaged Residual Image in the Infrared J Band. This is a residual

contrast image of one of the scan obtain in January 26th, 1992. The image was

filtered by a band-pass filter to remove large-scale error due to the error in adjusting
the average intensities from image to image, as well as high frequency noise. Display
range is -0.02 to 0.02.

53

 

Figure 3.11 Time-Averaged Residual Image in the Infrared H Band. This is a

residual contrast image of one of the scan obtain in January 26th, 1992. The image

was filtered by a band-pass filter to remove large—scale error due to the error in
adjusting the average intensities from image to image, as well as high frequency

noise. Display range is -0.02 to 0.02.

54

 

Figure 3.12 Residual Image in the Infrared K Band. This is the residual image

of a single exposure obtained on Oct. 12th, 1990. The arch structures are error

introduced by the iterated flat-fielding algorithm. The amplitude of this error is
z 1 x 10‘s, and can be reduced by averaging more images. The averaged residual
images in Figure 3.10 and Figure 3.11 didn’t show such error. The display range is
-0.02 to 0.02.

55

can be repeated in the vertical direction, and the gain function of the array detector
can be determined up to a scaling constant. However, in practice, it is impossible
to shift the image by exactly one pixel, nor is the source flux constant in time.
Nevertheless, this method can be repeated with different shifts and a least-square
algorithm can be formulated to solve for the gain function g(:i:'). The full discussion
of this algorithm was published in Kuhn, Lin, and Loranz (1991). Here, only the

result is given.

The rth iteration of the logarithm of the gain function, C(x), is given by

G'+1(x) = I\'(x)+;(i)‘—)Z[G'(x—ai+aj)+G'(x—aj+ai)] (3—1)
Ki
with
K(x)=n—— (1)02] D.(x)— D(x—a;+aj)j(]+[D (—x) D.-(x—aj+ai)]

where the index 1‘ is the number of iteration, index i and j is the exposure indices,
a; are the displacement of the ith exposure, and D(x) are the logarithm of the data
frames. Note that the term K(x) is a constant and depends on the data frames

only.

This algorithm was found to converge reasonably fast. The variance between
successive iteration decreases like a power law of the iteration count 1‘. 10 iterations
is usually enough to reduce the variance to 104, and takes approximately 20 min—
utes on a DecStation 3100 computer with 10 400 x 400 data frames. The iterated

gain was found to be as good as the skyflats.

56

3.4.2 Calibration of Solar Disk Center

The alignment of the heliostat was less than perfect, and it produces an im-
age motion of approximately 2”/ min. Also, images from a heliostat rotate with
360° / day rate. Thus, the position of the disk center of each solar image is needed
for rotation correction and averaging. The disk center position is found by a two-
step algorithm. The first step calculates the center—of—mass of the solar disk, using
the average intensity of the full image as a threshold, i.e., all the pixels with intensity
higher than the average intensity are considered inside the solar disk. This method
is subject to error caused by the sky gradient, the presence of sunspots, and any
bright pixels outside the solar disk. Depending on the number of sunspots and bad
pixels, and the amplitudes of the linear gradient, the centering error ranged from a
few tenths of a pixel typically to as much as 3 pixels when the image is degraded by

, e.g., passing cloud or high cirrus. No iterative process was applied to this method.

The second step uses a least-square-minimization algorithm. Since the measured
oblateness of the sun is of the order of 10‘5 (Dicke and Goldenberg 1967), we can
assume that the sun is symmetrical about its center, and can be described by a
radially symmetrical function I (r), where r is the position vector in the disk center
coordinates system. Since the sky is used for flatfielding, the linear gradient in the

skyflats breaks this symmetry, and the image now is described by

Sway): 1(7‘)(1+0$+/3y) (3—2)

We can find the functional form of the image within any coordinate system with

origin at (r'c, yé), by a transformation of the form

57

S'(;1:',y') =S(r—Ar,y—Ay) (3-3)

where A2: and Ag are the relative shift between the two coordinate systems. The
transformation is complicated in general, however, if Ar, Ay,a, and ,3 are small,
we can perform the transformation with a first order expansion of Ar, Ag, (1, and
,3. The sky gradient is of the order of 10"5 per pixel at 30° away from the sun. The
error in the disk center position obtained with the center of mass method is small,
hence, the linear expansion is a good approximation. Thus, we can formulate a
least square fitting algorithm to fit an observed image with the function of equation
(3-3) to the parameters Ar, Ag, 0, and ,3. (Ar, Ay) thus give the deviation of the
center-of-mass center from the true disk center, and (a, 13) give the sky gradient, and
can be removed. The detailed formulation of the algorithm is included in Appendix
C. Test with fake solar images showed that it achieved a precision at the order of

10"3 pixel.
3.4.3 Image Shift and Rotation

The solar images were constantly moving and rotating, due to the missalignment
and the optical nature of the heliostat. If the shift between two images is an integral
number of pixel, then these two images can be summed without any degradation
to the spatial resolution. However, summing images with fractional pixel shift does
degrade the spatial resolution. Image rotation produces shift proportional to the
radius which, when summing, also degrades spatial resolution of pixels near the
limb. Thus, the solar images were shifted to the nearest grid point, and rotated to
have the same orientation before averaging. The shift and rotation can be described

by a transformation of the form

58

I’(F) = [(3') (3 — 4)

where I (F) is the intensity of the image at coordinates F. The coordinate transfor-

mation is

"'1.

=R-F+T (3—5)

where R is a 2 x 2 rotational matrix,

c056 sing
R = (3 - 6)

—-.sin€ c0319

where 6 is the angle of rotation, and T = ((117, dy) is the translation vector. Since we
don’t have infinite resolution, [(17) is really the integration of the function I within
the area of the pixel at F. The new function I '(F) thus must be interpolated from the
neighboring pixels of if. A second order Taylor series is used for the interpolation.
The error introduced by the interpolation is thus of the order of the third order
derivative of the image function. This is usually negligible except for pixels near

sunspots and at the extreme limb.
3.4.4 Image Distortion Correction

The optics distort the solar images to an elliptical shape. The distortion can be
described by a mathematical transformation like equation (3—4), with the coordinate

transformation defined as

59

"’1
II

C.”
:0
'11

(3-7)

S is a 2 x 2 matrix with the form

1 0
S=(" ) (3—8)
0 1

describing the distortion of the coordinates system, and R is a rotation matrix. The
angle 9 is defined as the angle between the x axis and the direction of distortion.
The intensity plot along a fixed radius circle Ir(6) will not be constant, but will

have a nonzero 2nd harmonics component, i.e.,

I,(6l) = L(r)- (13(1) cos(26l) (3 — 9)

where L(r) is the lin‘ib-darkening function. Notice that this is different from the
effect of a linear gradient, which produce a nonzero lst harmonic term. Thus, the
linear gradient and image distortion can be calculated separately. This transforma-
tion can be carried out digitally, like the image rotation and shift operation. It is
also interpolated with a second order Taylor series. To find the oblateness of the
distorted solar images, a symmetrical solar image is constructed with the average
limb-darkening function from the data. This artificial solar image is then numeri-
cally distorted with different oblateness, and the 2nd harmonics amplitudes at each
radius are calculated, and compared with a2(r)da¢a. Figure 3.13 shows the averaged
ag(r)da¢a plotted with the a2 calculated with artificial images with different distor-
tion. It is clear that the distortion with a = 1.007 agrees best with the measured

curve.

60

 

 

0.02 1 r r i I I l .
’grna2.da;i’ NH
’reda2.d§:’ H--i
'fake1a2.da:’
'fake3a2.dai' --
’fake4a2.dat' --
I I
0.015 - i ' .
l
1 1
I
I
.'
l
I
A i
u <
V 0.01 ' - '
N
4

 

0.005 -

 

 

 

 

 

 

O 20 4O 60 80 100 120 140 160 180
radius [pixel]

Figure 3.13 A2(r) Optical Distortion Amplitude. The A2(r) coefficients of the red
and green images are plotted with that from simulated data. The simulated curves
were obtained by distorting a perfectly symmetrical artificial solar image. The
distortion of fake4a2.dat was 1.007, 1.01 for fake1a2.dat, and 1.0025 for fake1a3.dat.
The red A2(r) was displaced by one pixel to distinguish it from the green curve.
A2(r) of green images follows fake4a2.dat nicely. A2(r) of the red images are larger
than the simulated curve for r < 120, due to the larger scattered light noise in the

red images. There were 10 green images and 12 red images used for this calculation.

The error bar is the 30 error bar.

61

Once the amplitude and the phase of the distortion are found, correction can

be made simply by performing a transformation with distortion amplitude 01-1.

3.4.5 Image Averaging

Given the amplitude and the phase of the distortion, the solar images can now
be corrected and averaged. Since the sky transparency may change from time to
time, the average intensities for each image are different, and areas with bad pixels
will have less data for averaging. Thus, every solar images was rescaled to the same
disk center intensity and then added. The summed image is then divided by the

number of images contributing to a given pixel to construct a time-averaged image.
3.4.6 Fractional Residual Image Construction

A fractional residual image is constructed by the following formula:

.11 1m

70?) : D(r)

 

—1 (3—10)

where D(r) is the circularly averaged limb-darkening function, obtained by binning
pixels with radius r from the disk center in 1 pixel intervals. Sunspots, faculae,
and bad pixels were discarded by an iterative 30 cut. Thus, the fractional residual,
or contrast image represent the local deficit, or excess of intensity in units of the
average photospheric intensity. The low contrast structures were greatly enhanced
as is evident in Figure 3.7 and Figure 3.8. Faculae are now clearly visible near the
limb. The other interesting feature is the network-like structure visible over the
whole disk. This structure is also seen in the infrared data, shown in Figure 3.9 to
Figure 3.11. On the large scale, we still see some dark patches appear randomly.
This is more profound in the red wavelength images taken after September 6 of

1991. This is identified as due to the scattered light in the system. Next chapter

62

describes the analysis of the physical characteristics and interpretation of the various

brightness structures.

CHAPTER 4

PROPERTIES OF PHOTOSPHERIC
NETWORK AND FACULAE

4.1 Introduction

Direct imaging reveals brightness structures of all kind, as is evident from the
residual images in Figure 3.8 to Figure 3.12. Solar astronomers named these bright-
ness features by their appearance. The sunspots and faculae are well known, and
can be identified clearly in these images. The low contrast cells, resembling the ap-
pearance of the better known Chromospheric network, is the photospheric network.
The photospheric network has been known for decades, however, direct imaging of
it is difficult because of the small contrast. Most of the observations up-to—date
were obtained at the line core of photospheric absorption lines where the contrast
is much higher. We will show in this chapter that the low contrast cells are indeed

the photospheric network.

To measure the properties of any feature, we have to find all the image pixels as-
sociated with it first. However, it isn’t as straightforward as naming when it comes
to attributing an image pixel to a particular brightness component. For example,
it is easy to identify a sunspot in the image visually, but where do we define the
boundary between the sunspot and the surrounding photosphere? Some investi-
gators use magnetic field measurements to distinguish magnetic and nonmagnetic

features. When only contrast measurements are available, most investigators make

63

64

the attribution by setting a contrast range for each component. This separation is
not clear cut, and is acceptable only if the brightness distribution of all components
are well separated. This, unfortunately, is not true for most of the cases. An ex-
ample is the photospheric network which, we will show later, not only is co—spatial

with the faculae, but their contrast distribution have a large overlapped region also.

Thus, to make an unbiased separation of the image pixels, we compare the
histograms from a 25 x 100 pixels disk-center quiet region and an active region
containing sunspots and faculae on the east limb with the same number of pixels.
The range of p is 0.3 < ,u < 0.5. These histograms are plotted in Figure 4.1. While
the quiet region histogram has a symmetric profile, the active region histogram is
more complicated. It has a population scattered over the negative contrast region,
and an enhanced positive contrast tail. The negative contrast population is clearly
due to the presence of the sunspots. Traditionally, the term faculae refers to the
bright patches in the active regions near solar limb in a white light image. Thus, the
positive contrast tail can be interpreted as due to the presence of the faculae. If we
assume that a quiet region histogram represents the ”normal” contrast distribution
of the quiet photosphere without sunspots and faculae, then we can measure the
contrast distribution of sunspot and faculae by taking the difference between an
active and a quiet region histogram, as shown in Figure 4.2. In this figure, the small
bump with positive contrast represents the contrast distribution of the faculae. The
sunspot appears as the group of image pixels scattered over the negative side of the
histogram. The dip around zero contrast then represents the number of pixels in
the active region that is transformed from a quiet photospheric pixel to sunspots or

faculae pixel.

The quiet photosphere is not featureless, however. It includes the photospheric

m

 

160 I I I I I I

 

 

 

 

 

 

 

 

 

T
'ActiveHistogram’ --
’QuietHistogram' -----
M0” -
120 ~ -
100 - -
z 80 - -
60 - -
40 ~ _ .
20 - t! -
i,
0 I . J..- I J .“1 11... I11 I I k I
—0 12 —0.1 —0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06
contrast

Figure 4.1 Histograms of Contrast From Quiet and Active Regions. The active
region contrast histogram was obtained from a 25 x 100 pixels, or 125” x 500” area

on the east limb with 0.3 < p < 0.5.

66

 

 

 

 

 

 

 

 

 

 

‘10 I I I I I I I I
'Active-Quiet' —

20 r 1

0 ,. ....... , ........ Mun-u..- -1“.-- Lhmluun ._-.. ........ .
~20 r a
-40 r a
-60 I- -
..80 i- .-
-100 l I 1 L L 4 l 1

-0.12 —0.1 -0 08 -0 00 -0.04 -0.02 O 0.02 0.04 0.06

contrast

Figure 4.2 Difference Between Active and Quiet Region Contrast Histogram. The
positive contrast bump is interpreted as the contrast distribution of faculae. The

flat negative contrast component is due to sunspots.

67

network. Although we can visually identify the photospheric network, the separa-
tion of a quiet region into network pixels and quiet photospheric pixels is not possible
in our data. The 6” resolution of the SPT smoothed the network brightness to such
a low value that it does not appear as a distinct feature in the histograms. Thus, we
cannot make an independent measurement of the contrast distribution of the net-
work , and we can only separate the full-disk solar image into a quiet photosphere
component, a faculae component, and a sunspot component. The contrast distri-
butions are important for the measurements of the irradiance contribution, and the
fact that we can’t measure the network contribution to the total solar irradiance
is disappointing since the ”active network” model predicts that the photospheric
network should play an important role in the modulation of the long—term solar
irradiance variations. Nevertheless, we can still measure its properties like the spa-
tial scale, and correlation to Chromospheric network. We can also measure various

properties of the root-mean-square contrast.

Having defined the three brightness components, we can go on to measure their
properties. Before doing that, we will first measure the noise in the system. Then,
we will present the evidence which identifies the cell structures in the residual im-
ages as the photospheric network, and measurements of its properties. The mea-
surements of facular contrast at disk center, and its center-to-limb variations will
be presented next. The global scale structure of the quiet photosphere, and the
irradiance contributions of these three brightness components will be discussed in

the next chapter.

4.2 Noise Estimate

The noise in a photon counting device like a CCD follows Poisson statistics, i.e.,

a = VN, where a is the error of the measurement in unit of number of photons,

68

and N is the number of photons collected by the detector. The normalized noise,
7‘6 2 7117 is a measure of the relative photometric accuracy of the measurement.
Observations with a CCD detector consist of three measurementszthe measurement
of the source (the solar image), the dark current measurement, and the pixel-to-
pixel gain variation measurement (the flatfield). The accuracy of each measurement
is limited by the photon noise. In addition, there exits a fourth term of noise, the
readout noise of the detector. This term is usually small, of order of a few electrons,
and can be neglected. The reduction of CCD data consists of dark subtraction, and
flatfielding, as discussed in chapter III. Since the measurements of the solar flux

and dark current are independent, the noise in the dark—subtracted images is the

quadratic sum of the individual noise terms, i.e.,

 

03 z \/U:920n + 0301‘}: (4 — 1)

When flatfielding, however, the noise in the flatfield and in the dark-corrected image
do not add up quadratically. Denote the dark-subtracted image by S, and the
flatfield by g, each subjected to photon noise as and 09, respectively. Then we can
write 5 = 50 + 03, and g 2 go + 09, where 50 and go are the true signal and gain,

respectively. Thus, the flatfielded image can be expressed as

S s
1:_°_+_3_ (4_2)
90+Ug
S
z—Q‘fas‘fswfl
91 90

The two noise terms, as and So- if, are measured independently. Thus, the total

noise in the dark subtracted and flatfielded image is the quadratic sum of the two,

 

69

U
02 :Ugun+arfark +Sg(-g_g)2 (4—3)
0

The normalized noise then can be determined by

 

g = (fig-rug)? (4—4)

Thus, it is the normalized noise that is summed quadratically when flatfielding.

The average pixel intensity in a dark frame is only 150 adu. This is small when
compared with the typical disk center intensity of 8,000 to 10000 adu in a solar
image. Thus, the noise in the dark frame is usually negligible, and we can write
a, = a”... The photometric accuracy in the skyflats varies as the sky condition
changes. On clear days the sky brightness is lower, and the average pixel intensity
from a skyflat with 20 minute integration time is typically 1/2 to 1/3 that of the
disk center intensity of the sun. The photometric accuracy of the skyflats thus is
about 40 % to 70 ‘70 worse than that of the solar images. In hazy days the average
intensity of a skyflat with the same integration time is usually comparable to the
disk intensity of the solar images. In this case, the normalized noise from the skyflats

and solar images are comparable.

As will be discussed in the following section, the small-scale noise can be mea-
sured by power spectrum analysis. For images obtained with the new Ford CCD,
power spectrum fitting of disk center regions showed an average (normalized) noise
level near 1 x 10"3 / pixel for the average image of each time step. The average
light level of a 20 minutes integration time skyflat is about 104 adu/ pixel. With a

gain of 10 photon/adu, and averaged over 20 solar images, the statistical noise is

70

7 x 10’4 / pixel / time-step. The noise in a solar image is comparable to the skyflat
in the worse case, and better in some cases when the sky was clear. Thus, the com-
bined noise level is 1 x 10'3 / pixel or better, in good agreement with the measured
noise. A 5 hours observation yielding 4 time steps in both color will reduce the

noise level to 5 x 10‘4/pixel/day.

The infrared array has a large gain of 160 photon/adu, and the light level on
the solar disk for each exposure is about 3000 adu. Thus each exposure of the sun
yields noise comparable to the noise of the average of 10 visible solar images. Since
the flatfields were calculated with the gain iterating algorithm, the theoretical noise
in the iterated gain is limited by the noise in the data frames averaged over the
number of data frame used for the calculation. This is small and can be ignored.
However, the largest source of error in the gain iterating algorithm is the uncer-
tainty in adjusting the average signal due to the fluctuation of sky transparency.
This produces bright and dark patches with amplitudes of approximately 1 X 10—3
roughly aligned with the solar limb. Although the amplitude of this error is compa-
rable to the network signal, it can be identified clearly due to the different spatial
appearance. This error in the 1992 dataset was greatly reduced in the averaged

images.

The photon noise is the dominant source of noise on small scale. On large scale,
variation of scattered light is the largest source of error. The visible wavelength
contrast images in Figure 3.8 and Figure 3.9 show some extended dark patches
with amplitude of approximately 4 x 10"3 even after the optical distortion was
corrected. This error was more profound in all the red images taken after Sept. 6,
1991, as shown in Figure 4.3. More careful examination of the images showed that

it has roughly the same pattern, and same orientation, but shifts from frame to

71

In”

11"" “f...

7‘ ”'5 '5. saw. 7. ~-“ 13"?”

 

Figure 4.3 Red Contrast Image with Large Scattered Light Contamination. This
image was obtained on Sept. 6th, 1991. Display range was -0.02 to 0.02.

72

frame. We believe that it is an additive effect due to the internal reflection between
the CCD chip and the window of the CCD dewar. Some large dust particles must
have fallen onto the red interference filter between Sept. 4th, and Sept. 6th. The
increased scattered light in the red band pass increases the amplitude of the noise
in all the red images taken after Sept. 6th. Reflections due to the highly reflective
surfaces of the neutral density filter and the interference filters are unlikely, since
they are placed far away from the CCD, and they are tilted slightly with respect to

the optical axis.
4.3 Properties of The Photospheric Network
4.3.1 Cross-Correlation Between Different Time Steps and Colors

One of the interesting feature in Figure 3.8 and Figure 3.9 is the bright and
dark mottles resembling the appearance of network. To better show the structure
of these mottles, the running average of a 32 pixels diameter mask was calculated,
and subtracted from each residual images. This process removes the low frequency
structure from the image. We then apply a low pass filter by computing the running
average of a 2 pixels radius mask. Figure 4.4 and Figure 4.5 are the filtered image
of Figure 3.8 and Figure 3.9 respectively, and show more clearly the structure of
the mottles. To verify that this structure is indeed of solar origin, cross-correlations
between the two band passes, and between different sets for each band pass were
calculated. Figure 4.6.c, Figure 4..6.f, and Figure 4. 4.6.i show the cross-correlations
of the disk center region between the red and green band pass for the first to third
set. Strong correlation was observed in the 2nd and 3rd sets. The cross-correlation
of the lst set is noisy, but it still shows significant correlation signal. Since there
are approximately 40 minute intervals between successive sets, the cross-correlation

between different datasets should show a solar rotational signal. This is clearly seen

73

from Figure 4.6. We conclude that the dark and bright mottles seen in Figure 3.8

and 3.9 are indeed structures on the solar photosphere.

Figure 3.10 to Figure 3.12 are infrared residual images, filtered like the visible
images. Structures similar to the network in the visible images are clearly visible.
There are also are structures introduced by the iterated flatfielding routine in the
K band image. This error was reduced in the J and H band images by averaging.
This error was due to the uncertainty in adjusting the mean signal variation due to
the fluctuation of sky transparency. Figure 4.7 shows the cross—correlations between
the H and K band images of the 1990 observations near disk center, as well as the
cross-correlations between different exposures of the same band pass. Figure 4.8
shows the cross-correlation between the J and H band images near disk center from
the 1992 data set. Since there were only 10 images covering different part of the
solar disk in each bandpass in the 1990 data set, we cannot average the images to
reduce the residual flatfielding noise. The cross-correlations between the H and K
band images are noisy. Nevertheless, images of the same bandpass show significant
crossscorrelation. In the 1992 data set, there were approximately 100 images for
each scan, so we could average the images to reduce the flatfielding noise, and the

good correlation between the J and H band images is evident in Figure 4.8.
4.3.2 Power Spectrum Analysis

The scale of the network was measured by power spectrum analysis. Power
spectra of four 128 X 128 pixels (890” X 890”) regions at the disk center, filtered
by a high pass filter with cutoff frequency of 0.05 radian/ Mm, were calculated, and
averaged to obtain an average spatial power spectrum for both the red and green
band pass. Three 64 X 64 pixels (606” X 606”) images were used to obtain the average

power spectrum of the H and K band in the infrared. Figure 4.9 shows the average

 

74

 

Figure 4.4 Filtered Green Band Contrast Image Showing Photospheric Network.
This is the same image that is displayed in Figure 3.8. Display range is -0.02 to
0.02.

75

A

.35"
_ w

e
:5“

 

Figure 4.5 Filtered Red Band Contrast Image Showing Photospheric Network. This
is the same image that is displayed in Figure 3.9. Display range is -0.02 to 0.02.

76

 

g 4, h II it

Figure 4.6 Cross-Correlation of Visible Wavelengths Images. Black lines mark zero
shift position. Panel c, f, and i are the cross-correlation of the lst, 2nd, and 3rd
time step between the red and green images at disk center, respectively. Strong
correlation between the two colors is evident. Panel a, d, g and b, e, h are the
cross—correlation of the red and green images between the lst and 2nd, lst and 3rd,
and lst and 4th time step, respectively. The peaks of the time correlations (Panel
a, d, c, and Panel b, e, h)shift to the right due to solar rotation. Each time step is
about 80 minutes apart. The area used to do this calculation is 64 x 64 pixels, or
448” x 448”.

77

 

g , h

Figure 4.7 CrossCorrelation of Infrared Wavelengths Images. Panel a and d: H

band residual images. The time interval between the two images were about 10
minutes. Panel g: Cross-correlation between a and (1. Panel b and e: K band
residual images. Panel h: Cross-correlation between b and 6. Panel c: Cross-

correlation between a and b. Panel f: Cross-correlation between d and e.

78

 

    

Cross-correlation

Figure 4.8 Comparison of Faculae in J and H Band Near Disk Center. The lower
panel is the cross-correlation map of the two images. Sunspots were marked such
that they don’t contribute to the cross-correlation calculation. Large positive peak

at zero shift indicating positive correlation is clearly visible.

79

Photospheric Network Power Spectrum.
0.25 I I I I I I I

’red.avg’ +-
'grn.avg' +--
’irh.avg’ -B-~
’irk.avg’ 96“
’red.fit' — .
’grn.fit’ —
'irh.fit’ —
'irk.fit' —

 

 

 

 

 

 

 

 

 

 

T
E
3 J
11 I, :IF” '- : .ESES‘ Fa-II-‘i-éheees --- -- - ‘ 3:4
0 I- n. . . .'.'..-.,"""-'i -'-""‘"".'I“‘.'..I;i. .'.:'"' ’ I
I I I I I I I l
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
[radian/Mm]

Figure 4.9 Average Power Spectrum of Photospheric Network. A Gaussian profile
was fitted to the shoulder at k=0.2 radian Mm-l. Solid lines show the fit.

80

power spectrum of the red, green, H, and K bandpass data. There are no significant
peaks in the spectra. This is not unexpected, since a peak in the power spectrum
implies the existence of long range order in the structure. Phenomenon such as
waves do have long range order. However, the random distribution of dark and
bright patterns observed in the residual images does not suggest long range order.
The power spectrum of this kind of structure should be flat at low frequencies and
fall off at higher frequencies, e.g. Lorentzian (Keil, Kuhn, Lin, and Reardon 1992).
So, the feature of interest in Figure 4.9 is the shoulder around 0.2 radian / Mm. The
presence of this shoulder in the power spectrum of all 4 band-pass indicates that it

is not an artifact. This is the signal associated with the network structure.

We can construct a simple model to explain the observed power spectrum. Con-
sider a 2D array populated with random numbers with rms amplitude a. If this
array is convolved with a Gaussian function with characteristic length A, i.e., replac-
ing the array element with the local average weighted by the gaussian envelope, the
resulting image will have structures similar to the bright and dark mottles observed.
There will not be long range ordering, since the correlation between any two points
with distance d apart is the strength of the gaussian function at distance d away

from the centroid. The power spectrum of this structure is

PM) = [000 - Go»)? (4 — 5)

where C(k, A) = e:I:p(—A2k2 / 2) is the Fourier transform of the gaussian function

1 r( :32)
v27rAe p 2A2

 

C(r, A) = (4 - 0)

81

Since the gaussian function falls to 0.5 ‘70 of its central value at distance 3A, a good

estimate of the average cell size should be I = 6A.

To calibrate this model, we calculated the power spectrum of solar granulation
images in the infrared H band obtained with the infrared array camera with the
vacuum tower telescope at Sac. Peak. The average power spectrum was fitted to

the formula

P(k) = Aerp(—A2k2) + B (4 — 7)

In this model, the amplitude A is a measure of the amplitude of the contrast of the
structures, B is a measure of the noise in the images, and A is the characteristic
length of the structures. The fitted value of A is A = 213 km, which gives cell size
of 6A 2 I = 1,278 km. This agrees well with the measured granule size of 500 km
to 2,000 km. The fit of the shoulder of red, H, and K band in Figure 4.9, and power
spectrum from green images of Sept. 4th, 1991 gives I = 33,480 :t 2400 km. The
green power spectrum from the 1990 data gives I = 39, 000 km, but we have some

evidence that the green images were poorly focused.
4.3.3 Cross-Correlation With The Chromospheric Network

Having established the size of the network, we calculated the correlation of the
observed network with the chromosphere network. The N30 at Sac. Peak provided
the CaK image of August 2nd, 1990. It was digitized by the Fast Scanning Micro
Densitiometer also at Sac. Peak. Figure 4.10 shows the cross correlation between
the observed network with CaK network in regions near the solar north pole, disk
center, and solar south pole. The chromosphere CaK network has been known

to outline the supergranulation (Simon and Leighton 1964), as the photospheric

82

network does (Muller 1983). Hence, the positive correlation shown in Figure 4.10

comes as no surprise.
4.3.4 Cross-Correlaion With the Supergranulation Velocity Field

It is also interesting to check the correlation between the observed network and
SG velocity field. Assuming the velocity field of the supergranulation is driven by
a convective process, the temperature structure should be hot at the cell center,
and cool around the cell boundary. The divergence of the velocity field should be

correlated with the temperature field. The simplest model suggests that

V-i7=aT (4—8)

where I7 is the 2 dimensional surface velocity field, T is the surface temperature
field, and a is a scaling constant. If we assume that I7 = V45, then we can solve for

05, given the measured temperature field, by solving the Poisson equation for (b,

V23 = 0T (4 — 9)

Thus we can infer a velocity field from 17,” = V45 based only on the measured

temperature field.

Since the velocity field of supergranulation is predominantly horizontal, we
should look for a correlation between the inferred velocity field and the measured
velocity field near the limb. The velocity measurements we use were obtained at
Big Bear Solar Observatory, provided by Dr. M. Woodard. Figure 4.11 shows the
measured velocity field at the north (4.11c) and south (4.11d) limb, the inferred

velocity field of the same regions projected along the direction toward the disk

83

 

g... , h .. . I

Figure 4.10 Cross-Correlation Between Photospheric Network and CaK Network.
Panel a to c: 64 X 64 pixels area of the red residual images near the solar north
pole, disk center, and south pole. Panel (1 to f: CaK images from the same area as
panel a to c. Panel g, h, and i: cross-correlation between panel a and d, b and e,
and c and 1', respectively. Strong correlation between the residual images and the

CaII K images are clearly visible.

84

 

e V f

Figure 4.11 Cross-Correlation Between Supergranulation and Photospheric Net-
work. Panel a and b: Inferred velocity field projected along the line-of—sight from
regions near the solar north and south pole, respectively. Panel c and (I: measured
velocity field from the same regions as a and b. Panel e and f: Cross-correlation
between a and c, and b and (1, respectively. There are signs of negative correlation
peak near zero shift.

85

center (4.11a and 4.11b), and the 2D cross-correlation between the measured, and
inferred velocity field at the north (4.11e) and south (4.11f) limb. The covered area
is about 450” X 450”. This simple model to infer the velocity field doesn’t take into
account surface projection effects. The correlation maps are also noisy. However,
both cross-correlation maps show definite negative peak near zero shift. The dis-
placement of the peaks from the zero shift is probably due to the uncertainty in
selecting the corresponding area from the velocity and temperature measurements,
since there was nearly a 4 hour difference between the observations. Nevertheless,
the negative correlation suggests that if the observed brightness structures are the
thermal signature of a convective process, then the velocity field generated will have
opposite direction with respect to the SG velocity field. Thus, we conclude that
the observed brightness structure with SG scale is not the thermal signature of the
convective flow associated with the supergranulation. This negative correlation can

be expected from the positive correlation with CaK network.

It is clear from the above analysis that this brightness structure with charac-
teristic scale of 32,000 km is the photospheric counterpart of the Chromospheric

network, the photospheric network.
4.3.5 Line Contribution to Network Contrast

The band pass of 100 A includes many absorption lines, which are formed higher
in the atmosphere. The solar spectrum in the red and green bandpass are shown in
Figure 4.12 and Figure 4.13, respectively. Since the spectroheliograms obtained in
the core of the photospheric lines show high intensity modulation in the network,
the broad band photometric measurements are contaminated by the line emission.
The Ha line is included in the wing of the red filter, as shown in Figure 4.12. The

central 1 A of Ho line contributes 0.1 % of the total intensity in the red band

 

 

 

 

 

 

 

 

 

 

 

0.01 I I I i r
’RedWindow’ —
0.009 - -
0 008 - 1
. l
5 0.007 - -
‘8
‘ri
t 0.006 - -
Q)
0
U
c 0.005 - 4
.9
in
2 0.004 6 ~
E
U)
C
E 0.003 » .
P
0.002 - .
0.001 - -
0 l l l l
6350 6400 6450 6500 6550 6600 6650

wavelength [Anstron]

Figure 4.12 Transmission Function of the Red Filter. The transmission function of
the filter (the Gaussian profile) was multiplied by the solar spectrum. The Ha line
at 6563 A is included near the wing.

87

 

0 ° 01 I I I I I
’GreenWindow' —

0.004 r

 

 

 

0.003 r

Transmission Coefficient
C)
C)
0
U1
T

 

 

0.002 '-

 

 

 

 

 

 

 

 

 

 

 

 

 

0.001-

 

 

 

 

 

0 I I I

4850 4900 4950 5000 5050 5100 5150
wavelength [Anstron]

Figure 4.13 Transmission Function of the Green Filter. The transmission function

of the filter (the Gaussian profile) was multiplied by the solar spectrum.

88

pass, since the transmission coefficient of the filter at Ha is about 10 % of the
peak transmission. Thus, a 10 % intensity modulation in the center of Ha in the
network will produce an intensity modulation of 10". We estimated the total
line contributions to the observed intensity by computing the fraction of the line
emission to the total emission within the band-pass. The digitized Kitt Peak solar
atlas was used for this calculation. The lines were identified with a pre-defined
transmission coefficient threshold, ie., all the emission with strength less than the
threshold were identified as line emission. The line contributions vary with the
threshold. For a threshold between 0.9 to 0.3, the line contributions vary from 6.2
% to 0.13 (70 in the red, and 19 (’70 to 0.6 (70 in the green. This result predicts a green
intensity modulation of 3 to 5 times larger than the red. This is inconsistent with
our observations as discussed in the following section. Thus, we do not believe the

line emission is the source of the observed intensity modulation in the network.

4.3.6 Photospheric Network Contrast near Disk Center

The above analysis established that the observed bright and dark mottles seen
in Figure 4.4 and 4.5 are the photospheric network. Of interest are the contrast
at disk center, and the center-to-limb variation of the contrast. The contrast of
the network at disk center is measured by calculating the standard deviation of the
central 450” X 450” area. The amplitude of the contrast is (2.34 i 0.38) x 1073,
(1.83:1:O.51) x 10‘3, (1.0210.21)x 10-3, and (1.11i0.21)x 10-3 for the green, red,
H, and K band, respectively. Figure 4.14 shows the contrast vs. wavelength, with
the contrast of a 5,800° K blackbody with 2.9° K deviation. The green contrast
was calculated from the June 1991 data taken with the new CCD. The 1990 green
images were not used due to the focusing problem. The red contrast calculated

from the August 1990 data and June 1991 data are consistent. It is clear that the

89

brightness signal of the network has approximately thermal wavelength dependence.
Whether this is truly thermal due to some kind of local heating mechanism (e.g.
magnetic heating), or is associated with the ”hot wall” of the thin magnetic flux

tubes is not clear, however.
4.3.7 Center-to-Limb Variation of Network Contrast

Figure 4.15 to Figure 4.18 shows the center-to—limb variation (CLV) of the con-
trast of the north and south active latitudes, and along the north—south axis in
the green, red, H and K band. The regions from north (south) 15 degree to 45
degree were used to calculate the CLV of the north (south) active latitudes, and a
region 30 degree wide, centered around the north-south axis was used for the quiet
region. The CLV of the contrast was obtained by computing the rms amplitude of
the pixels within a radial bin of 1 pixel interval within the selected region. Facular
and sunspot pixels were discarded by repetitive 30 cut. Since the number of pixels
per radial bin is proportional to the radius, it is small near disk center. Thus, the
rms contrast in this region shows some fluctuations. At II = 0.9, or r = 56 pixels,
the radial bins of the active and quiet regions should contain more than 100 pixels.
Thus, the rms amplitude is a good representation of the network contrast. The
CLV of the quiet region contrast shows little variation in the visible wavelength,
and that of the north and south active latitudes increase linearly from near disk
center to H z 0.35, then rises sharply toward the limb. This is expected from the
increase of bright pixels in the form of enhanced network and bright patches in the

active regions.

The CLV of the contrast in the infrared was calculated by averaging the CLV
of images from the 1990 observations. Since each image covers only part of the

solar disk, the covered range in II is different from image to image. The CLV of the

90

dT:2.9K @ T=5,800K
0.007 1 i i I

’dI.thermal' —
'dI.dat’ ”-1

 

0.004

dI/I

 

 

 

 

 

O 1 1 I l

500 1000 1500 2000 2500
wavelength [nm]

 

Figure 4.14 Photospheric Network Contrast Near Disk Center. The solid line is the
theoretical contrast of a temperature perturbation of a blackbody at 5800 K with
perturbation amplitude of 2.9 K. The error represent the 30 scatter of the data.

0.01

91

CLV of the Network contrast, Green

 

0.009 -

I

0.008

I

0.007

0.006

0.005 -

dI/I

I

0.004

0.003

0.002

I

0.001 -

 

 

l

I

l l

I

I

I I

’GrnActiveNorth’ --
’GrnActiveSouth’ ----- ‘
’GrnQuiet' ------

 

0
0.1

0.2

0.3

0.4

0.5 i 0.6
cos(theta)

0.7

0.8 0.9

Figure 4.15 Network Contrast Center-to-Limb Variation in Green Band.

 

92

CLV of the Network contrast, Red

 

0'01 I I I I I I I I

’RedActiveNorth’ --
'RedActiveSouth’ ----- .
’RedQuiet' ------

0.009 -

0.008 '

0.007 -

0.006 '

0.005 *

dI/I

0.004 -

0.003 r

0.002 r

 

0.001 - -

 

 

 

0 l l I 1 l l l I

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
cosltheta)

Figure 4.16 Network Contrast Center-to—Limb Variation in Red Band.

CLV of Network Contrast H band
0-01 I 0 I I I I I I I
l. 1 ’H_ActiveNorth’ 1H
1 'H_ActiveSouth’
0.009 1' l“ IH—Quietl .a.-. -
0.008 - '1,
I 0
0.007 - l 0 -
l 0
0.006 - 1! l -
\ 0
H ‘i‘
2 0.005 - ‘.\ .
'U I
‘I‘ 0
0.004 - -
1
ix \ I) J
0.003 - x, .. , <>
0410\0 ‘
0.002 r 7" I. -
.1. j ‘. :é, IIAI‘II Isl-I 57?: I1 ‘3
0.001 - l l iii"
0 l I I I l I
0 0.1 0.3 0.4 0.5 0.6 0.7
cosltheta)

Figure 4 17 Network Contrast Center—to—Limb Variation 1n Infrared H Band

93

 

 

 

 

 

 

 

 

 

 

 

 

 

94

CLV of Network Contrast, K band
0'01 I I I I I I I I
'K_ActiveNorth’ ro—4
’K_ActiveSouth’ -+-_
’K_Quiet’ +3--

 

dI/I
o

 

 

 

 

 

 

 

 

 

 

 

0 1 .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 l
cosltheta)

Figure 4.18 Network Contrast Center—to-Limb Variation in Infrared K Band.

95

contrast in the H band shows no distinction between the active and quiet region
for 0.4 < y < 1 and a rapid rise for p < 0.4 in both the active and quiet regions.
The CLV of the K band images are similar to that of the H band, except that the
CLV of the quiet regions did not rise to a high value, as the visible data. Since a
filter with diameter of 20 pixels was used to remove the low frequency structures
from the images, the filtering in the regions within 10 pixels from the limb have
less weight. Thus, the CLV of the contrast in these regions, ie., the regions with
p < 0.4 are noisier, as is reflected in the large error bar in this range. In the
range between 0.4 < p < 1 we expect to see a steady increase of contrast in the
active latitudes similar to the visible data. Although the error in this range is
smaller, it is still large compared to the small linear rise, if it does exist. Thus we
conclude that the infrared data are too noisy to give a reliable measurement of the
difference between the CLV of the active and quiet network between 0.4 < p < 1.
Although the CLV near the limb are very noisy, nevertheless, the rapid rise of the
H band CLV is interesting. The H band continuum originates from the deepest
observable layer in the solar photosphere when observed near disk center — 35 km
below 7500 = 1. The K band continuum is formed about 25 km above T500 = 1
(Vernazza, Avrett, and Loeser 1976). This is an interesting region of the solar
atmosphere where the opacity is rapidly changing. Toward the limb, the H band
images integrate larger physical depths due to the smaller opacity. Thus more of
the network structures were included in each limb pixel in the H band than in the
K, and visible wavelengths bands. This may cause the rms contrast to rise faster
in the H band. Whether or not this difference between the CLV of the contrast in
the H band and the other wavelengths represents different physical conditions at

different heights needs further investigation.

96

4.3.8 Temperature Structure of Supergranulation

The supergranulation (SC) is suggested to be the convective flow associated with
the He ionization layer (Simon and Leighton 1964, Leighton 1964) Many authors
(Beckers 1968, Frazier 1970, Worden 1975, Koutchmy 1978, Foukal and Fowler,
1984) attempted to observe the thermal structure associated with this large scale
convective flow, but the results are inconclusive. To date, there are no published
theoretical models, nor numerical simulations of SC predicting the contrast of the
thermal structure of the SG. Our observation suggests that any thermal structure
associated with a convective flow of the SG must be imbedded in the brightness

signal of the photospheric network and have an amplitude less than 3 K.
4.4 Properties of Photospheric Faculae

4.4.1 Spatial Distribution of Faculae

It is clear from the residual images that faculae, the extended bright patches
seen near the limb, is al')sent near disk center. Tracing the faculae as they rotate
into disk center from images taken on consecutive day, like the June 4th to June 8th
series of 1991, showed that the facular contrast decreases rapidly with increasing [1.
Thus, identifying faculae near disk center is difficult. Furthermore, the observation
of Muller and Keil (1983) showed that FBPs form the perimeter of the network cell.
Thus, faculae and network are really indistinguishable, at least near disk center. In
the active regions near the limb, the issue is more complicated. There exist extended
bright patches, as well as enhanced network. We had shown from comparison of
active and quiet regions histograms that faculae increase the number of bright pixels
in an active region. This can be explained either by a higher concentration of the
facular bright points (FBP), or higher FBP contrast, or both. There also exist

bright features of size from 0.5” to 1.5” called facular knots (Spruit and Zwaan

97

1981) or facular granules (Mehltretter 1974, Muller 1977) in the active regions which
may contribute to the increased contrast. To study the spatial structure of bright
patches, the power spectrum of 5 64 x 64 pixels regions from the red residual image,
four from the active region at the northeast, northwest, southeast, and southwest
corner of the solar disk, and one from a disk center quiet region were calculated
and fitted to the formula of equation (4-7). If these bright features (F BP’s, Facular
knots, and F acular granules) appear as isolated bright points scattered over the
active regions, then they should contribute a spectral frequency independent power
to the power spectrum, thus a higher B. On the other hand, if the faculae appear as
large extended bright patches, then they contribute to the low frequency part of the
power spectrum. All the power spectra show structures with the same characteristic
length, and same level of noise B. However, the active region power spectra show
amplitudes A larger than the quiet region power spectra, suggesting that the high

contrast pixels appear in the enhanced active network.

4.4.2 Faculae Contrast Near Disk Center

And Its Center-to-Limb Variation

The center-to-limb variation of facular contrast is of special interest to solar
astronomer, since it bears information about the vertical structure of the thin mag-
netic flux tube, which is the basic constituent of faculae. It is also important for
the proxy estimate of the faculae contribution to the solar irradiance variation.
However, the facular contrast near disk center is not well—defined up to date. Most
authors adopted zero contrast at disk center, but spatially resolved study showed
that individual facular flux tube may have contrast as high as 40 ‘70 (Muller and
Keil, 1983). The most interesting observation might be the measurements of Foukal,

Little, Graves, Rabin, and Lynch (1990), and Foukal, Little, and Mooney (1989)

98

which showed that the faculae contrast is negative in the infrared H band near
disk center. Since our analysis method described in Section 4.1 defines faculae as
the excess number of bright pixels of histogram when compared with average quiet
region histogram, we cannot check if the faculae contrast is indeed negative near
disk center in the infrared H band with this definition. Nevertheless, we can still
check the correlation between our infrared H band and J band images of the same
area taken near disk center. This was shown in figure 4.8. Unit opacity of J band
occurs at about the same height of T5000 2 1, hence if Foukal’s claim is true, we
would see negative correlation between the J and H band images. Although the
two images were taken 90 minutes apart, there exists long-lived features such that
the cross-correlation map showed positive peak at zero shift, suggesting positive
correlation between the two images. Thus our observations suggest that most of
the faculae in the infrared H band near disk center are bright compared with the

surrounding photosphere.

The results of facular contrast distribution observations up-to—date are highly
subjective to bias introduced by the analysis method. As we argue below, these
methods tend to overestimate the facular contrast in most circumstances. Most
authors (Nishikawa and Hirayama 1990, Hirayama, Hamana, and Mizugaki 1985,
Chapman, Herzog, Lawrence, and Shelton 1986) define facular pixel in the images
by choosing pixels with contrast higher than a fixed threshold. This definition is
certain to bias the facular contrast. Lawrence (1988), and Lawrence, Chapman,
and Herzog (1988) found the facular contrast distribution by assuming that an
active region histogram is the superposition of a quiet photosphere component and
a faculae component. In their analysis, two adjacent rectangular areas containing a

few hundred pixels each were selected: one containing some facular pixels, the other

99

only quiet photosphere. The quiet photosphere rectangle were used to construct a
quiet region contrast distribution. This distribution was then fitted to the low-
contrast end of the combined histogram of the two rectangles by adjusting its mean
and height. The difference of the combined histogram and the fitted histogram was
interpreted as the contrast distribution of faculae. The two—component model of
the histogram is a more reasonable assumption, however, the fitting procedure to
find the quiet sun contrast distribution is really redundant, since it should be well
represented by the contrast distribution of the quiet photosphere rectangle. Also the
choice of regions containing easily identifiable facular pixels unavoidably involves
human bias as to what is perceived as facular and runs the risk of neglecting the
dimmer facular pixels. Thus, their analysis also tends to overestimate the facular

contrast.

The method we used to obtain the faculae brightness distribution as described
earlier is unbiased, since a large area including all the possible facular candidates
were used to obtain the contrast distribution. However, the power spectrum analysis
in Section 4.3.2 showed that most of the bright features appear as network. Thus,
the division of it into a network and a faculae component seems groundless. On
the other hand, the intrinsically different shape of the active and quiet regions
contrast histogram does suggest that there are some differences in the properties
between the two regions. Thus, we follow the traditional convention and define
the excess number of bright pixels in the active region as faculae and treat them
separately. What is the difference in the structure of the small flux tube in quiet
regions network and in active region network? We are not in a position to answer
this question here. An observation measuring the distribution of magnetic field

strength in active and quiet regions, and the correlation between the magnetic field

100

strength and continuum contrast will provide more information on this issue.

To calculate the CLV of facular contrast, an average quiet region histogram
nqu,ct(c), where c is the contrast, is obtained by averaging several quiet region
histograms taken from different data set. Notice that Nquict(c) has no p dependence,
since fiat network contrast CLV in quiet region derived in section 4.3.7 inferred that
there are no difference between the histograms of quiet photosphere at disk center
and near solar limb. Histograms N ( y, c) of all the pixels with distance [.1 from solar
disk center is also generated. Since each radial bin would contain different numbers
of pixels, the histograms were normalized by the total number of pixel for each

radial bin. That is,

 

(4 — 10)

The summation over the whole range of contrast gives the total number of pixels in
the given p bin. The average quiet region histogram has to be normalized similarly.
Denote the normalized quiet region histogram by fiqu,et(c), we then calculate the
differences between mm, c) and fi.qu.-et(c). The excess number of bright pixels then

gives the contrast distribution of faculae at that radius, i.e.,

nfac(,u,c) = n(p,c) — fiqu,‘€¢(c),for c 2 0,and'n 2 0 (4 — 11)

The average facular contrast as a function of ,u can then be found by calculating

the average contrast of all the excess bright pixels.

101

00

_ Z} nfac(p,c) - c
Clad/1) = 5"“ <4 — 12)

00

Z Mad/1,0)

c=0

 

We calculated the CLV of the facular contrast of the 1991 visible wavelengths data
listed in Table 3.1 and the 1992 infrared wavelengths data listed in Table 3.2, and
the average C LV is plotted in Figure 4.19. The facular contrast measured by other
authors were plotted also. The 100 images of each scan of the infrared data were
mosaic together to make a full—disk infrared residual image in both the J and H
band. Since there are large-scale scattered light error in the visible images, and the
mosaicking procedure also produce large-scale error due to the uncertainties in ad—
justing overall intensity level between images, the full-disk contrast images were first
filtered by a high-pass filter. The amplitude of these large-scale error are of order
of a few tenth of a percent, this represents a error of the same order of magnitude
in the small—scale contrast, which is acceptable. The CLVS of each full disk image
were then averaged to obtain the average CLVs in Figure 4.19. The infrared CLVs
rise sharply around )1 _—. 0.3. The visible wavelengths CLVs rise around )1 = 0.15.
This may be due to contaminations of filtering error near the limb. This plot shows
that the facular contrast does not vanish even at disk center. Also, we found that
the facular contrast near the limb is smaller than some previously measured val—
ues. This is not a surprising result judging from the peak position of the facular
contrast distribution in Figure 4.2. The difference between our measurements and
the results of the others is clearly due to the lack of good references with which
one can identify facular pixels unambiguously in the other observations. The low
contrast facular pixels were simply been ignored by most observers. For example,

Hirayama, Hamana, and Mizugaki (1985) used Ha filtergrams to help identify fac-

102

ular regions in the continuum images, and the maximum excess intensities of the
identified facular regions were used as representative of the facular contrast. Thus,
they only counted the brightest facular pixels, and it is obvious that they should

derive a much higher facular contrast.

The size of individual F BP is small, and with a spatial resolution of 1” or worse,
each image pixel would contain several F BPS. The spatial resolution of the SPT
changes from 7” in the 1990 dataset to 5” in the 1991 dataset. It is 2” in current
configuration. We did not see any significant spatial resolution dependence in the
observed contrast value. Since the spatial resolutions of the other observations
fall mostly in this range, differences between spatial resolution may not play an

important role here.

One may argue that with our definition of facular contrast distribution, one may
always find image pixels with contrast higher than the average, and one cannot tell
whether the excess contrast is really due to the presence of FBPS in the pixel, or it is
just some brighter nonmagnetic pixels. Also, since the image pixels lost their iden-
tity in the histogram, we cannot make comparison with magnetic measurements to
help identify facular pixels. Thus, one may argue that the nonzero facular contrast
derived may not be accepted with confidence. However, if the brighter pixels are
really due to the brightness variation of nonmagnetic photosphere, the difference
between the histograms should show white-noise like distribution. This is not the
case in the facular contrast contribution in Figure 4.2, which shows a population
with positive contrast. Thus, we are confident that our definition of facular contrast
distribution of equation (:4-2) is of statistical significance. The spatially resolved fac-
ular contrast measurements of Muller and Keil ( 1983) derived a contrast of 0.4 near

disk center. Assuming a 1 % filling factor, we would expect a 4 x 10‘3 contrast

103

facular contrast
0.08 x I I I I

t.,__ 'Green' —
“a I Red I .....
0. C7 _ ' J -band' """" ..
a, z t 'H-band' r-r
., ‘ '1awrence.dat’ 0
"k1.-.“ 'hirayama.dat' +
0.06t E 'hillock.dat’ El 4

 

dI/I

 

 

 

 

Figure 4.19 Center-to-Limb Variation of F aculae Contrast. The diamond curve was
taken from Lawrence, Chapman, and Herzog (1988), the cross curve was ~taken from
Hirayama, Hamana, and Mizugaki (1985), and the square curve was taken from the
hillock model of Schatten, Mayr, Omidvar, and Maier (1986).

104

from spatially unresolved observations. This is in good agreement with our results.

 

CHAPTER 5

THE SOLAR IRRADIANCE VARIATIONS
AND CONCLUSION

5.1 Introduction

The primary goal of this eXperiment is to identify the contribution of the var-
ious brightness structures to the total solar irradiance. The sensitivity of our in-
strument is limited by two factors. First, the poor spatial resolution of the SPT
limits our ability to calculate the irradiance contributions of each brightness compo-
nents independently. \N'e are presently only able to isolate the sunspot and faculae
contributions. Network and quiet photosphere are treated together as the ”aver-
aged photosphere". Secondly, absolute photometry with 10'3 accuracy from ground
based observation is virtually impossible because the transparency of Earth’s atmo-
sphere is continuously changing. Relative photometry, however, can achieve high
degree of accuracy with carefully chosen reference. Assuming that the intensity of
the quiet sun is constant in time, it gives us a convenient reference to calculate the
flux excess or deficit of each brightness component. This is a reasonable assump-
tion on the time-scale of a few months—the duration of the dataset—for the following
reasons. The radiative fiux output of the photosphere is strongly modulated only
by the presence of strong surface magnetic fields, like the dark sunspots and bright
faculae. Preferably, we would like to use the nonmagnetic quiet photospheric in—

tensity as the reference. However, a large quiet region inevitably includes network

105

 

106

bright points, and again, since we can not separate the network pixels from the
nonmagnetic quiet photosphere pixels, we can only use the average quiet region
photospheric intensities for the references. The filling factor of the NBP is approx-
imately 0.01. With a spatial resolution of 6”, the contrast of the network pixels
is only 2 x 10“3 in the visible wavelengths as discussed in Chapter 4. The bias
introduced by the inclusion of network element in the calculation of the average
quiet photosphere intensity is of comparable magnitude. Since it is used as the
reference for the calculation of image pixel contrast, the amplitude of the error in
the contrast is 10‘3 times the ”true” contrast. This is small, and can be neglected.
The limb brightness observations did suggest that the average photosphere would
become brighter toward solar maximum, however, the brightness increase is small,

and the error introduced can also be neglected for the time—scale considered here.

To calculate the flux deficit and excess of the sunspots and faculae, we first cal—
culate the total irradiance of a quiet sun by integrating the average limb-darkening

function over the full disk, i.e.,

L(p)d€dp (5 — 1)

\n:

=0

1
[quiet : /
#:09

where L(;1) is the averaged limb-darkening function. The irradiance excess or deficit

of an image pixel in unit of the total flux of the quiet sun is then given by

= center) (5 — 2)

where 5 is the coordinate vector of the image pixel, and

107

£01)
[quiet

 

1701) = (5 - 3)

is the normalized limb-darkening function which, when integrated over 0 and p as

in equation (5-1), gives unit total quiet sun flux.
5.2 Irradiance Contributions of Faculae and Sunspots

We have obtained the facular contrast distribution n fac(p,c) as a function of
p in Chapter 4. Multiply niac(p,c) by the total number of image pixel at p, the
contrast c, and the normalized average limb—darkening function D(p), and integrate
over [1 and c, then we obtain the full-disk faculae contribution to the solar irradiance

in units of the total solar irradiance of the quiet sun. That is,

/ nfac(#,C)CN(/1)L'(u)dcd# (5 — 4)

where Cfacffl) is the average facular contrast function defined in equation (4-3),
and N fac(/1) is the total number of facular pixels at p. The full-disk sunspot irradi-

ance contributions can be evaluated similarly with the sunspot contrast distribution

n3p0t(#a C),

00

1
I....= / / "spamC)CN(/1~)i'(#)dcd# (5—5)

”:0 c=—oo

108

Napot(u)5spot(u)i'(u)d#

H
.“=\._

Table 3 list the facular and sunspot irradiance contribution of data from 1991
observations. It is found that the total facular contribution is about a few times 10-4
of the total solar irradiance. The total sunspot contribution varies with sunspot
number and positions, ranging from comparable to a few times that of the total

facular contribution in our data.

Table 3. The Sunspots and Facular Irradiance Contributions of 1991 data

 

Date Sunspots( red) Faculae( red) Sunspots( green) F aculae( green)
June 4 —1.2 x 10'3‘ 0.47 x 10‘3 No data
June 5 -1.30 x 10-3 0.73 x 10-3 —1.8 x 10-3 0.67 x 10-3

June 6 —1.7 x 10-3 0.47 x 10-3 —2.43 x 10-3 0.40 x 10-3
June 8 —-3.4 x 10-3 0.82 x 10-3 —3.19 x 10-3 0.97 x 10-3
Sept 6 —1.27 x 10‘3 0.99 x 10‘3 No data

Sept 21 —0.81 x 10-3 0.83 x 10-3 —0.74 x 10-3 0.60 x 10-3
Sept 28 —2.51 x 10-3 0.83 x 10-3 —1.41 x 10-3 0.44 x 10-3
Sept 29 —2.33 x 10"3 0.76 x 10-3 —1.68 x 10-3 0.39 x 10-3

 

‘ In unit of the total quiet sun irradiance.

It is also interesting to examine the integrand of equation (5-4) for images from
consecutive time-steps, since they tell us how the facular and sunspot irradiance
contributions evolve as they rotate across the Solar disk. Figure 5.1 shows the
integrands of images of the June and September 1991 time series. It is clear that
the facular contribution curves do not change appreciably, regardless of the changing

position and area of the brightest facular area. The sunspot contribution curves, on

109

the other hand, show strong dependence on the position and the size of the sunspot

groups, as might be expected.
5.3 The Energy Balance of Active Region

Observations with larger data bases have shown that sunspots and faculae are re-
sponsible for the short-term irradiance variations. The long-term effect of these two
components are less certain. Bruning and LaBonte (1983), using monochromatic
continuum images at the wing of Fe I A5250 line, showed that the total irradiance
contribution from the magnetic features, i.e., sunspots and faculae, is nearly zero
when averaged over a one—year period. Chapmann and Meyer (1986) used proxy
measurements of sunspot and faculae areas of about 40 active regions to show that
the irradiance excess of faculae and the deficit of sunspots roughly balance each
other over the lifetime of active regions. These observations were interpreted as
evidence for detailed balance between the sunspot energy deficit and facular energy
excess, and there were some models (e.g.,Schatten and Mayr 1985, Schatten, Mayr,
Omidvar, and Maier 1986) put forward to explain these results. In these models,
the energy flux blocked by the sunspots is reradiated away by the faculae. How-
ever, it is still a subject of controversy. If these results were correct, they imply
that a third, or more components are responsible for the observed 11-year solar
irradiance variations and there are some unknown mechanisms operating below the
photosphere to channel the blocked energy to the faculae. Also, we must remember
that we only measure the irradiance at Earth. The irradiance excess of faculae and
deficit of sunspots may cancel each other, statements about the balance between
the total energy flux is a dangerous extrapolation. We already know that faculae
are not isotropic radiator. They radiate more in the direction tangent to the solar

surface, since they are brighter near the limb. Thus, although a disk center facular

110

 

 

 

 

 

- - IIIIII-ifd J u a
_ n . I.
u u n .
. . . IIIHII
III}
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I 1... 1 1 1 1 «I. r . . I
g g g g g 9 ~ a...
C C C t t t * nnnnnnnnnnnnnnnnnnnnnnnnnn
a a a O O O — uuuuuuuuuuuuuuuuuuu
f f f p p p u ........... a
568 S S S ~. uu-nnuc
] 1.35 6 8 ~ -------
I I I I J 3.] u .......... I
I I I IIIIIIIIIIIIIIII
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. lllll
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llllllllll
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.IIHHHl|
a II. lllllllll JI
.. IIIIIIIIIII I!
I Iltvllll I
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us . \\\
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\\
. A.
~
c v \
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— .
_ . n
.— u .
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I — . I
. .
. If
... .
I
. I
a [.4].
.. :5
I .. Lhn.. I
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. I ..
._ X, .
u /
. II
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\s. \V
s .. \\ .k
|~u|~ I\IIIIIbFFf~lh'f
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\\\
IIIIIII
“III-I. ..... I
P b - I D -
5 S 0 5 S 5 5 l
0 O 0 0 0 O 0
_ . . _ . . 0
e e e e e e 0
4 2 2 4 6 8 u
_ _ _ _ 0
.

ahwucouv H\H

0.9

0.8

0.4 0,5

0.3

0.1

mu

Figure 5.1 Sunspots and Faculae Irradiance Contributions.

111

contributes little to the total solar irradiance we observe, it still radiates measur-
able amount of energy in directions normal to our line-of-sight. The total energy
radiated by any brightness element at any time is the integration of its radiation
field over the 47r solid angle. To measure the radiation field I (9, ¢), where 9 and ti)
are the polar and azimuthal angles respectively, one would need to place detectors
around the brightness element to get a faithful measurement. This is impossible in
practice. Nevertheless, in the simplest model, the radiation field of any brightness
element can be represented by the center-to—limb variation of its contrast assuming
an azimuthal symmetric radiation field. The total energy output in excess or deficit

AF of a brightness element with respect to the average photosphere is

—1
AF = 2.. / coat-0w (5 — 6)
74:0
where C(p) is the contrast of the brightness element at )1. Thus, the total flux excess

radiated by all the facular elements under consideration is

0
Ffac = / lVfac(/1')AFfacd/f' (5 — 7)
u’=l

l
: 27erac CIaC(/1)E,(H)d#

u=0

where Cfac(/1) is the average contrast function of the faculae, and N [ac is the total
number of facular pixels in the area under consideration. The energy flux deficit of
the sunspots can be calculated similarly using the sunspot contrast function. Note

that equation (5-4) and (5—7) give only the instantaneous irradiance and energy

 

112

output. They should be integrated over the lifetime of the faculae and sunspots to

obtain the total irradiance and energy output.

Our observations of the facular contrast function showed that it does not have
large variation over the time scales considered here. Hence, the use of an average
contrast function for all the faculae can be justified. The sunspot contrast func-
tion, however, varies with the size of the sunspot. Thus, the contrast function of
any sunspot must be measured separately. Another difficulty associated with the
problem of active region energy budget is that we don’t know where to draw a line
for the active region, since we don’t know how far-reaching the sunspots affect the
photosphere. Also, all the features on the sun spend half of their lifetime on the
invisible side of the solar hemisphere due to solar rotation. Thus, there is always
some uncertainty in any observation trying to prove or disprove the energy balance

between sunspot and faculae.

Facing the observational difficulties associated with the problem of active region
energy balance, Kuhn’s (1992) idea that the horizontal energy transport is more
important than the vertical energy transport in supplying the excess energy of
faculae seems to provide a natural explanation for the observed irradiance balance
between faculae and sunspots. Since horizontal energy transport is more effective,
the energy blocked by sunspots is deposited in the surrounding layer, and then
picked up by the faculae if they appear simultaneously. Thus, the total energy
radiated by the sunspots and faculae in an active region is balanced. However, the
sunspot blocked energy has not necessarily been channeled to the faculae. There are
occasions where isolated sunspots appear without accompanying facular regions. In

this case, the blocked energy is re-radiated by the surrounding photosphere.

The above picture is attractive, but how do we go from energy balance to

113

irradiance balance? Judging from the very different form of the integral equations of
irradiance (Equation (5-4)) and total flux (Equation (5-7)), It is hard to understand
how the time-integral of the irradiance and flux integral of sunspot and faculae can
be balanced simultaneously. On the other hand, if we consider the irradiance excess

or deficit of a single magnetic element 61( p) when it is positioned at p on the solar

disk, then

6101) = C(#)L(u) (5-8)

If we trace it across the solar disk, and integrate over time, then the time—integrated
irradiance contribution is exactly the flux integral of equation (5-7). Thus, inte-
grating a large number of faculae and sunspots over a long period of time, as the
observations of Bruning and LaBonte (1983) and Chapmann and Meyer (1986), is
equivalent to measuring the average flux integral of the faculae and sunspot. It is
then not too surprising to find that the time-integrated irradiance contribution of

the two components should balance each other.
5.4 The Average Photosphere

The limb brightness observations of Kuhn Libbrecht, and Dicke (1988) demon-
strated the existence of a global scale brightness structure. They also argued that
the contrast of this structure is not subject to a facular limb brightening function,
but has a n independent contrast. The amplitude of this global scale structure
should be of a few times 10—3, depending on the phase of the solar cycle. However,
in a full disk contrast image, this structure is mingled with the extended facular
regions near the limb. The large scale scattered light noise further hampers our

ability to detect this structure. Since the scattered light noise appears randomly in

 

114

the images, there is no systematic way of removing it. The data base is not large
enough to reduce its effect by direct averaging either. Thus, we applied a biased
averaging method by discarding all the pixels with contrast greater than 0.6 ‘70 or
less than -0.6 %. This threshold is chosen based on the observation that the quiet
region network contrast is z 0.2%, and a 30 cut should exclude the faculae and
sunspots pixels. This also excludes most of the scattered light error in the images.
The averaged contrast images of 1991 visible wavelengths data are shown in Figure
5.2 and Figure 5.3. We did not attempt to do the same averaging for the infrared
wavelength data, since the mosaicking error is of the order of a few times 10—3.
This is large enough to mask the small signal in the infrared. Near the limb, the
green average contrast image shows that some of the additive error leaked through
the 30 filtering, while the red average contrast image appears cleaner. This is not
unexpected, since the amplitude of the error in the green images is smaller and has
a higher probability of leaking through the filter. Despite the difficulty in averaging,
both images show brighter active latitudes compared with the solar equator and the

pole regions. This is consistent with the limb brightness observations.

To check if we can reproduce the limb brightness profile observed by the solar
oblateness telescope, the pixels with the same latitude were averaged to make a
contrast profile as a function of latitude I for both colors. This is assuming that
the contrast of this large-scale structure does not change appreciably as across
the solar disk. The results are shown in Figure 5.4. The equivalent brightness
temperature profile is shown in Figure 5.5. Considering the very different amplitudes
and distribution of the large scale noise in the green and red contrast images, it is
surprising to find the excellent agreement of the two curves between —70° and 70°.

The amplitude in the green is slightly larger, in agreement with the assumption that

115

 

Figure 5.2 The Average Photosphere in Green Bandpass.

116

 

Figure 5.3 The Average Photosphere in Red Bandpass

117

this signal is of thermal origin. The data 70° poleward had only a small number
of pixels for averaging, and the discrepancy between the two colors in these region
are not unexpected. Comparing this plot with the limb brightness temperature of
Kuhn et a1, it shows qualitatively the same feature. We see temperature excess at
both north and south active latitudes. The small dip at the equator also agrees
well. To check if these features are )1 independent, curves averaging over different
range of )1 were calculated, and it showed that the temperature profile of Figure
5.3 is indeed independent of p. The amplitude of the temperature excess at active
latitudes with respect to the polar area is approximately 2 K. This is a large drop
from the 7 K amplitude from 1990. Unfortunately, the limb brightness experiment
was discontinued after the summer observing season of 1990, and no data were
available for comparison with this observation. The ERB data in Figure 1.1 shows
that the irradiance remains near peak value on the first half of 1991, and only shows
some sign of decrease. Thus, the 2 K amplitude does not seem to agree well with the
ERB data. This maybe due to error from the 30 filtering. Nevertheless, the position
of the peak at about 20° south and about 30° north is also consistent with the trend
in the shift of the position of the peak observed in the limb brightness observations.
Thus, we believe that this is the large-scale brightness structure observed by the

limb brightness observations.
5.5 Interpretations

We have successfully resolved the photospheric network, and we can measure
its rms contrast as it varies with solar cycle. However, we still cannot deduce its
irradiance contribution to our data. Thus we still cannot answer the important
question of whether the active network contrast is responsible for the observed

solar cycle irradiance variations. Even if we can reduce the network irradiance

 

 

118

Average Background Contrast of 1991

 

0.0015 I I I I l

I I I
’redavg.dat’ 0
'grnavg.dat' +
0.001 - .
o
0
0.0005 - ~
900

 

 

 

 

 

._ ............... «It ......
0 0 0 ++ o {777%”, Ff
0 + + °flf db 0* W
3 + +722” ° °°°
c + #+ + 0 o
0 ++ {i Q03? 0
0 -0.0005 - + W; ° + ‘
+ #
sh *+
0
0001 - + -
#130 +
a
”5,!
00015 ~ + .
_0.002 I I I J i I I I I
-100 -80 -60 -40 -20 0 20 40 60 80

latitude [degree]

Figure 5.4 Limb Brightness Profile of the Average Photosphere.

100

119

Average Background Temperature of 1991

 

 

 

 

2.5 I I I I I I I I T
5 ’BkT 'o
o i c emp
o
5 o
2 ~ g o .
to°o E 80
§ 9°
‘00 ° E 0 ° 0
00° 0 % 6
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- o o °§% 9 °
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t& = ,
0° 9
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)- % '4
0.5 o
o
o
e
z t
O 1 1 1 1 1 1 L l e 1
-100 -80 -60 -40 -20 0 2O 40 60 80

latitude [degree]

Figure 5.5 Brightness Temperature Profile of the Average Photosphere.

100

 

 

120

contribution, it would still take observations spanning over the whole phase of a
solar cycle before we can make any conclusions. Thus, it seems appropriate to look
into other kinds of observations which might provide useful information to help

resolve the confusion.

The observations of the solar p-mode oscillations provide information about the
interior of the Sun that is inaccessible with photometric observations. One of the
major differences between the active network model of Foukal et al and the hot
active-region photosphere model of Kuhn et al is the thermal structure of the Solar
Convection Zone (SCZ). The active network model speculated that the changing
contrast and area of the active network is the cause of the Solar cycle irradiance
variations, but it did not elaborate on how the changing active network might affect
the thermal structure of the underlying solar atmosphere. The calculations of Spruit
assume the SCZ redistributes the energy not radiated away during Solar minimum
uniformly throughout the SCZ. Thus the thermal structure of the SCZ would not
deviate significantly from a spherically symmetric model. Kuhn et al’s model, on
the other hand, argued that the thermal structure of the solar photosphere along
the active latitudes is modulated by the toroidal field deep below. In a separate
paper Kuhn (1988) argued that a small thermal perturbation originated deep in
the convection zone would not be smoothed out uniformly over the entire solar
convection zone, but would produce a visible temperature perturbation (IT in a
shallow layer below the photosphere. This would produce an observable asphericity
in the thermal structure of the photosphere. Coode and Kuhn (1990) developed
a model in which the brightness temperature dT was related to the local effective
sound speed. They were able to explain consistently the 2nd and 4th order p-mode

splitting coefficients variations observed by various groups over different phase of the

 

 

 

121

solar cycle. Moreover, the measured centroid frequency shifts of low-order p—modes
also agrees well with the calculated shifts. The 2nd and 4th order p-mode splitting
coefficients measure the deviation of the structure of the SCZ from a spherically
symmetric model. Thus, the calculations of Spruit are inconsistent with the p-mode
splitting coefficients observations. The centroid frequency shifts are proportional
to the temperature excess dT. The active network model requires the SCZ to store
the excess energy not radiated away during solar minimum. Thus the SCZ should
be hotter during Solar minimum. The amplitude of the temperature increase of the
SCZ may be too small to be observed, but the sign of the temperature perturbation
dT should predict a centroid frequency shifts with a 180 phase shift with respect to

the shifts predicted by the limb brightness temperature observations.

The p-mode frequency shift and splitting observations also provide information
on the radial structure of the temperature perturbations. Since the penetration
depth of a mode depends on the mode frequency, the radial temperature perturba-
tion would produce frequency dependent frequency shifts. Comparing the BBSO
frequency shifts observations between 1986 and 1988 with model radial tempera-
ture perturbation profile, it was concluded that the radial temperature perturba-
tion must peak a few hundred km below the solar photosphere, and then decreases
toward the photosphere. Exactly how the toroidal field setup the observed tem-
perature perturbation is not easily solvable, nevertheless, Kuhn (1991) was able to
show with numerical simulation qualitatively how a small perturbation at the base
of a convective box can evolves in time to produce the observed radial temperature
perturbation profile. These results strongly support the interpretations of the ob-
served dT as a true thermal perturbation originated deep in the SCZ. The active

network model, on the other hand, seems unlikely to produce the thermal structure

 

 

 

122

observed by the seismology observations.

5.5 Conclusion

We presented multi-wavelength measurements of, (1) the photospheric network
contrast, its center-to—limb variation, and a statistical measurement of its charac-
teristic size; (2) the facular contrast and the facular contribution to the total solar
irradiance; (3) the sunspot contribution to the total solar irradiance; (4) the bright-
ness profile of the average photosphere. From these measurements, we discussed
the various possible physical processes that could lead to the observed active region
irradiance balance, and the solar cycle irradiance variation. These observations also
demonstrated that, with carefully designed instruments, observation techniques,
and data reduction algorithms, it is possible to achieve better than 10"3 / pixel ac-
curacy on a spatial scale ranging from 104 km to the full disk. We feel justified to
say that we have achieved what we set out to do. However, it is not enough, still.
It is clear that more observations spanning the whole phase of the solar cycle are
needed to provide information for a full understanding of the physics at work. Be—
side a longer time series, improved spatial resolution is indispensable to resolve the
network contribution to the irradiance variation. This is one problem that we are
unable to solve due to the limited spatial resolution of our instruments. The first
dataset obtained with the improved optics of the SPT with 2” resolution showed
that it does not provide more new information than what we can get with the
medium resolution optics. We feel that sub-arcsecond resolution is needed. Hope-
fully instruments of upcoming projects will provide enough resolution to solve this

problem.

APPENDICES

 

APPENDIX A

THE GAUGE FREEDOM OF THE
GAIN ITERATING ALGORITHM

In our 1991 paper (Kuhn, Lin, and Loranz 1991) we stated that there is an
ambiguity associated with the solution of the gain iterating algorithm, due to the
gauge invariant property of the formulation under a gauge transformation of the

form

C(X) -* C(X) + 45(X) (A - 1)

where C(x) is the logarithm of the gain function of the imaging array, and ¢(x) is
the gauge of the transformation. We also pointed out that one of the possible form

for 05(x) is

7500:8556 (A—2)

where 5 is a constant vector. In other words, we can only determine a solution of
the algorithm up to a linear constant term in the logarithm of the gain. However,
the question of whether other possible form of ¢(x) exist was not fully investigated.
In this section, we will demonstrate that the solution for ¢(x) is equation (A—2) is

the only possible solution of the algorithm as follows.

123

 

124

Assume that C(x) is the logarithm of a solution of the gain integrating algo-
rithm. Let’s now perform a transformation of the form in equation (A-l), and
substitute C(x) + ¢(x) in the formulation. It is then straightforward to show that

the iterating formula in Chapter 3 (equation (3-1)) becomes

G'+1(x )2 I\(x) +(———)Z[G'(x —a;+aj)+G'(x—aj+a3) (A—3)

(MK) - ¢(x - 3i + 85)) - (¢(X) - ¢(x - 85 + ai))l

where all the symbols are as defined in Section 3.3.4. Thus, the condition that ¢(x)

must satisfy to make the transformation gauge invariant is

2000:) aux—a.+aj))+(¢(x)—¢(x—aj+aa))]=0 (A-4)

Note that this equation must hold for any x within the domain of the array. Define

ai — 33 E dxij, we can rewrite equation (A-4) as

Zuax) — ¢<x-—dx.j)) +(¢><x) +¢fx+dxijlll = 0 (A-s)

In the special case that the shifts are infinitesimal, equation (A-5) reduces to a
Laplace equation. There are many solutions that will satisfy the Laplace equation,
e.g., any harmonic function. However, since equation (A-5) must be satisfied with
arbitrary choice of number of shifted images and size of shift, ¢(x) must be an odd
function around the point under consideration. This is a more restrictive condition

than the Laplace equation, but there are still a large number of functions that satisfy

 

125

this condition. In addition to equation (A-5), there is another criteria that ¢(x)
must satisfy. That is, equation (A-5) must be satisfied on every point in the valid
pixel domain of the imaging array, or, equation (A-5) is translational invariant.
Summarizing, we have two conditions that ¢(x) must satisfy to make the gauge

transformation of equation (A-1) invariant:

¢(x + dx) = -¢(x — dx) (.4 — 6)

¢(x+xo+dx) = —¢(x+xo —dx) (A—7)

where dx is the displacement vector around x between a pair of images, and x0 is
a translation vector. To examine what kind of solutions condition (A-6) and (A-7)
yield, let’s expand 05(x) by a polynomial expansion around some arbitrary point x,

i.e.,

N
07):)..de Em x)dx (A—8)

The condition in equation (A-6) requires that all the even order coefficients, a2.-
where i is arbitrary integer, must vanish. Condition (A-7) further requires that all
the odd coefficients except the first order term vanish. This is due to the fact that
any term in the expansion with 3rd power or higher will introduce terms with even
power, thus violating condition (A-6). Thus, the only possible family of solution

that satisfies both equation (A-6) and equation(A—7) is

 

126

¢(x)=6’.x (A—9)

This proves our observations in the 1991 paper that there can be an ambiguity
in the form of a linear gradient term in the logarithm of the gain function in the
solution of the gain iterating algorithm, and that it is the only possible form of the

ambiguity.

The above derivation is valid for an perfect infinite array. However, notice that
we don’t have an infinite array, and there are always bad pixels. Thus, the square
symmetry of equation (A—5) is not satisfied in these regions. However, as long as the
numbers of bad pixels and boundary pixels are small, the solution of the algorithm
will not be significantly affected, and an perfect infinite array approximation can

be assumed. This result was confirmed by numerical experiments.

 

APPENDIX B

LEAST SQUARE ALGORITHM TO FIND
THE DISK CENTER COORDINATES
OF FULL DISK SOLAR IMAGE

A CCD image can be described by a 3-dimensional function (x, y, and intensity)
[(17) = [(3, y), where F = (.r, y) is a position vector in the array coordiantes system.
A solar image without any distortion can be considered symmetrical about its center,
(.rc, yc). The image function [(77) have radial symmetry if the origin of the coordi-
nates system is translated to (IC, yc), i.e., I(;r —> .7: — :rc, y —+ y -— yc) 2 1(17) 2 I(r).
The solar images have a linear gradient introduced by the use of sky as the Hatfield.

Thus, the image function that describes the sky-calibrated solar image is

Shay) = 1(r.y)g(r.y) (3 - 1)

= I(r)(1 +01: + 5y)

where a and ,3 are the amplitude of the sky gradient in the x and y direction, respec-
tively. Note that the coordinates system used now is the solar center coordinates
system. However, we have only limited accuracy in the knowledge of (see, ya). Given
an estimated disk center position with a small error (32, yé) = ($12,; +Az, yc+Ay), the
image function 5'0" ) that describes the observed image in the coordinates system

of (1:2, yé) is

 

128

S'(x',y') = 5(1: — A17, y — Ag) (B — 2)
=Hr—ALy-Awflx-Aay-Aw

=HHH+IMMxAx+yAyWI+Mx-Afl+flw-AWD

where D(r) = i-yggl. Given the two functions [(7‘) and D(r), we can find the

parameters a, 73. Ar, and Ay by fitting a observed solar image d(7"' ) to the function

5' with least square fitting method. Define

02 = ZIS’U") — deal" (B — 3)

and minimize a“ With respect to a, )3, Ar, and Ag gives 4 equations,

 

 

c102
— = B — 4
do 0 ( )
(102
__ = B _
dd 0 ( 5)
(10'2

d A x Z O (B _ 6)

and

do2

(1 A y — 0 (B — 7)

Assume that a, [3, Ar, and Ag are small, then to the first order of these parameters,

we have a system of linear equations:

 

129

[[1202] A 1 + Zpypz] A y + Z[12(2I — d)D — I(I — 11)]11
+2111“ ‘ZI—d)D]fl=:[a:( d—I)D] (B—8)
Eta/Dz] A x + Z:[y2D2] A y + Z:[$y( (21— d)D]a

+Z[y(21— d)D— Ira—(1)13: Zly<d—I)Dl (B—9)
:[1-(2 (1)—D 1(1— 11)] Ax+Z[:ry(2I— d)D] Ay+:[x212]a
+Z[ry12])3= 2m d [)1 (B — 10)
2mm —d)D]A:r+Z:[y'( (:21- d)D— 1(1— 11)) Ay+Z[1-y12]a
+Zly21213= [m (d— 01 (B— 11)

which can be solved for a, ,3, A23, and Ag easily.

Since we do not know the true center at the beginning, the functions I (7‘) must

be calculated with the estimated center (r'c, yé), i.e.,

I'(r')= / S'(f')d9 (B—12)

1":1’0
We can estimate the error of this calculation by integrating in the coordinates
system of the true center. Discarding terms with 2nd order and higher of A1: and

Ag, we have

 

130

r' = ((1 - M2 + (y — AMI/2 (B — 13)
z [r2 + Ar2 — 2r(cos9 A .7: + sin9 A y)]1/2

zr+(cos9A:r+sin9Ay)

Thus,

I'(r') = / I(r)(1 + arcos9 + firsin9)d9 (B — 14)

r=ro—(cos9AI+sin9A3/)

 

= / [1(r0) + d2”) |,=,0(cos9 A 1: + sin9 A y)](1 + arcos9 + flrsin9)d9
7‘

Note that all the terms with 1st order of Ar, Ay,a, and 9 contain either c039 or
37719, which vainsh when integrating over 9. Thus, the error in I I(1") is of second

order or higher of Ar, Ay,a, and ,9, and I'(r') can be used for I(r).

 

 

 

 

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