Structure and Energy Transport of the Solar Convection Zone A

Structure and Energy Transport of the Solar Convection Zone

A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAI'I IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN ASTRONOMY

December 2004

By James D. Armstrong

Dissertation Committee:

Jeffery R. Kuhn, Chairperson Joshua E. Barnes Rolf-Peter Kudritzki Jing Li Haosheng Lin Michelle Teng © Copyright December 2004 by James Armstrong All Rights Reserved

iii Acknowledgements

The Ph.D. process is not a path that is taken alone. I greatly appreciate the support of my committee. In particular, Jeff Kuhn has been a friend as well as a mentor during this time. The author would also like to thank Frank Moss of the University of Missouri St.

Louis. His advice has been quite helpful in making difficult decisions. Mark Rast, Haosheng Lin, and others at the HAO have assisted in obtaining data for this work. Jesper Schou provided the helioseismic rotation data. Jorgen Christiensen-Salsgaard provided the solar model. This work has been supported by NASA and the SOHOjMDI project (grant number NAG5-3077). Finally, the author would like to thank Makani for many interesting discussions.

iv Abstract

The solar irradiance cycle has been observed for over 30 years. This cycle has been shown to correlate with the solar magnetic cycle. Understanding the solar irradiance cycle can have broad impact on our society. The measured change in solar irradiance over the solar cycle, on order of0.1%is small, but a decrease of this size, ifmaintained over several solar cycles, would be sufficient to cause a global ice age on the earth. By understanding the changes that cause the solar cycle one might be able to determine if longer period solar irradiance cycles exist.

One possible cause of the irradiance cycle is that magnetic fields at the solar surface create "holes)} in the solar surface allowing more energy to be radiated. This class of models assumes that energy radiation from the un-perturbed quiet sun remains constant over the solar cycle, and that the solar surface features are effectively coupled to the entire convection zone. Energy excess or deficits from solar features are drawn from the entire solar convection zone, which reradiates the excess or deficit over the thermal time scale of the entire convection zone.

A second proposed mechanism is that structural changes in the solar convection zone enhance energy transport during the solar maximum. The assumption is that magnetic fields generated near the base of the solar convection zone surface, bringing with them an increased energy flux. This excess energy is radiated by the solar surface and observed as the solar irradiance cycle.

v Observations and simple physical modes are presented here. It will be shown that

surface features alone do not cause the solar irradiance cycle. Another component is necessary. Strong evidence is also given that surface features only redistribute the

surface energy flux in a spatial and angular fashion, but do not contribute to the solar

irradiance cycle.

vi Table of Contents

Acknowledgements iv

Abstract ... v

List of Tables x

List of Figures xi

Chapter 1: Introduction 1

1.1 Prologue.... 1

1.2 Observational History . 2

1.3 Magnetic Cycle .... 6

1.4 The Solar Irradiance Cycle . 12

1.5 Proxy Models ...... 16

1.6 Generation and Initial Rise Of Magnetic Fields 19

Chapter 2: On Proxy Models for Solar Irradiance 31

2.1 Introduction . 31

2.2 Simulated Irradiance Data 32 2.3 Results.. 38 2.4 Discussion 39 Chapter 3: Structure Of the Solar Convection Zone 42 3.1 Abstract...... 44

vii 3.2 Introduction . 45

3.3 Non-Orthogonal Oblateness Calculations 46 3.4 Vector Harmonic Solution ... 48

3.5 Solar Rotation and Model Data 50 3.6 Results and Discussion 52 3.7 Conclusion..... 60 3.8 Acknowledgements 61

3.9 Discussion Of The Paper . 61

Chapter 4: Small Magnetic Flux Tubes - Facular Regions . 63

4.1 Introduction .... 63

4.2 Data and Analysis 69

4.3 Comparison To Models. 75

4.4 Conclusions ...... 82

Chapter 5: Sunspot Bright Rings 85 5.1 Introduction. 85 5.2 Observations. 90

5.3 A Simple Model . 97 5.4 Numerical Simulations 102

5.5 Results...... 105

5.5.1 Temperature Profiles 105 5.5.2 Reradiation Of Energy 109

5.5.3 Facular Contamination 110

5.6 Scattered Light 115

5.7 Discussion... 120 Chapter 6: Discussion and Conclusions 124

6.1 discussion . 124

viii 6.2 Conclusions ...... 132

ix List of Tables

3.1 Shape Coefficients and Gravitational Oblateness ...... 55

3.2 Shape Coefficients and VSH-Eq. (6) Sol. differences for Quadratic Rotation Law ...... 57 3.3 Shape Coefficients with Surface Rotation Matched Model 59

3.4 Shape Coefficients with Localized Radial and Angular Rotation Perturbations ...... 60

5.1 Fraction of Energy Reradiated by Sunspot Bright Rings. 110 5.2 Scattered Light Corrections ...... 119

x List of Figures

1.1 Maunder (1922) This figure shows the observed latitude of sunspots

over several cycles...... '...... 5

1.2 The stretching of magnetic fields (a-c) can be thought of as adding a toroidal (d-f) field to the original poloidal field. 9

1.3 a) A section of the solar disk is shown here for reference and

comparison. b) Sunspots are considered to consist of uniformly dark

areas, uniformly bright areas, and unchanged quiet sun. By measuring

the area of the dark and bright regions the change in the solar

irradiance is then calculated. 17

1.4 The difference between the solar rotation rate at the equator and 60

degrees longitude (wo""oo - WO=60o ). The position of the tachocline is

0 shown by the dotted line. In the inset the solar rotation rates at 0 ,

30 0 and 600 is shown...... 21

1.5 The entropy per unit mass up to an overall constant of integration is

shown here. The solar convective zone has the highest per unit mass

energy. Elsewhere the per unit mass entropy increases. The decrease

near the solar surface is interpreted as evidence of the solar wind, and

known concerns over helioseismic inversions near the solar surface. 29

xi 2.1 The contributions to the simulated irradiance data from sunspots and faculae are shown here. The lack of a periodic signal confirms that sunspots and faculae do not contribute to the irradiance cycle. .. .. 35

2.2 The modeled irradiance data is shown here. a) Shows the contribution from faculae. b) Shows the contribution from sunspots. c) Shows the total modeled irradiance plus the 120-day running average...... 37

2.3 The 120 day average of the simulated irradiance data is shown with a solid line. The portion of the simulated solar cycle, which is not

generated by faculae or sunspots is shown with the dot dash line. The residual resulting from a least squared fit of the simulated sunspots

and faculae to the simulated irradiance is shown with a dashed line. The absence of any periodic signal in this residual is an indication the proxy models would not detect underlying irradiance cycles...... 40

3.1 Graphs of the vector harmonic components of acceleration vs. radius.

The scaled A2,1,O, A4,3,O, and A4,5,O are plotted with dash-dot, dashed

and solid lines, respectively. The A2,1,O, ~,3,O and A4,5,O terms have been scaled by -1, 10, and 50 to fit them on the same vertical scale.. 51

3.2 The variation of d2 with radius. The solid line shows the density oblateness computed using the VSH approach, and the dashed line shows the corresponding polynomical expansion...... 54

3.3 (a) Graph of d2 and 82 versus radius. The density and pressure surfaces are represented by the solid and dashed lines, respectively. (b) The

variation of d4 (solid line) and 84 (dashed line) with radius...... 56

xii 4.1 The simple hot wall model accurately describes the limb brightening

profile of facular regions. The model assumes that a flux tube is a simple cylindrical depression in the solar surface. The walls are hot. The walls heat the floor. Larger faculae have cold floors since the walls

are unable to heat the larger area. Small faculae have warm floors... 68

4.2 Red and Blue residual images are measures of the solar surface temeperature. The points each represent an individual pixel from

PSPT data. The line is the theoretical computation of blue residual from the red residual based on a blackbody spectral distribution.. .. 70

4.3 The mean and standard deviation of CalIK residuals are shown here. Solid lines show means, while dashed lines show standard deviations.

Lines with *'s include only pixels with a CalIK residual ofgreater than

0.1, while lines without symbols include all pixels with a CaIIK residual

of greater than -0.05. The more sensitive cut of 0.1 shows some trend

as a function of J1.. However, since the trend is similar in both the standard deviation and the mean, it is believed to be an artifact of incomplete limb darkening removal. 73

4.4 CaIlK is shown to be an effective tracer of magnetic flux density. CalIK

is shown as a function of magnetic flux density, for varying values of

J1.. Near disk center the magnetic field is seen to have a "v" shape with

a very small region where the slope is nearly flat and thus CaIIK is

unresponsive to increasing magnetic flux density. Near the limb, The functional form is more accurately described as a "u" shape...... 74

xiii 4.5 The dependence of line of sight magnetic flux density is shown here. Each trace represents a bin of CalIK residual with a width of 0.02, ranging from a CalIK residual of 0 to 0.16. For small values of CalIK

residual the line of sight magnetic flux density is nearly independent

on /1. For increasing CalIK residual the magnetic field gains a stronger dependence on the viewing angle...... 76

4.6 The line of sight dependence of blue continuum is show here, as a function of GalIK residual. Each trace represents a bin of GalIK of

width of 0.02. As GalIK residual increases, facular limb brightening increases. Thus near the limb, features with stronger magnetic field

appear brighter when compared to features with a weaker magnetic

field, or areas with no magnetic field. Near disk center, features with strong magnetic field appear darker than regions with little or no magnetic field...... 77

4.7 The energy excess or deficit a..s a fraction of quiet sun luminosity is

shown as a function of CaIIK. Weak GalIK is associated with small

flux tubes while stronger GalIK residuals indcate larger flux tubes. Small flux tubes are associated with a local increase in brightness. As small flux tubes become larger they are associated with larger local

increases in brightness. In large flux tubes, the "floor" of the flux tube is no longer be heated by the walls. Eventually a point is reached where the cold floor dominates the redistribution of energy, and the

flux tube decreases the local surface brightness. The solid line was

computed using data taken near solar maximum. The dashed line was constructed using data taken 18 months earlier to approximate solar

minimum . 79

xiv 5.1 The extrapolation routine is demonstrated here. The scatter plot shows the median values of the residual for the center region of the sunspot NOAA 8525, in red. The solid line is the resulting extrapolation. The same extrapolation procedure was used for all annuli, however it should be noted that the functional relationship for regions of the bright ring tended to be noisier than the above graph...... 95

5.2 The temperatue pretubation profiles are shown for 4 sunspots, NOAA 8263 (Upper-Left), 8640 (Upper-Right), 8525(Lower-Left) and 8706(Lower-Right). Solid lines show the temperature perturbation computed from the PSPT red data, dashed lines PSPT blue, and the dotted lines are from the MDI. As discussed in the text, the MDI profiles have been shifted by a constant. 96 5.3 The schematic for the model discussed in section 5.3 shown here. A sunspot blocks energy by inhibiting convection. The blocked energy can be considered as a source which is split between two paths to the ground. 101

5.4 The thermal conductivities and heat capacity used in the diffusion models are shown here. The mixing length conductivity is marked with "+". The vertical conductivity extrapolated from Kuhn & Georgobiani (2000) is marked with "*". The horizontal conductivity extrapolated from Kuhn & Georgobiani (2000) is marked with diamonds. The heat capacity is marked with triangles...... 106

5.5 The modeled temperature profiles using mixing length conductivities are show here for 5 depths. 107

5.6 The modeled temperature profiles using the numerical conductivities of Kuhn & Georgobiani (2000) are show here for 5 depths...... 108

xv 5.7 Facular regions in sunspot bright rings (shown with a dashed line) are slightly fainter than their non-bright ring counterparts (shown with a solid line)...... 114

5.8 Point spread functions are computed by deconvolution ofthe image and

the limb darkening function. Six point spread functions were computed for the blue images. The mean and one standard error from these are

shown...... 117

xvi Chapter 1 Introduction

1.1 Prologue

Suspense is a good thing for a novel. It is a bad thing for a dissertation. I will begin by (hopefully) dispelling much of the suspense. The main finding of this work is that surface magnetic features are not the only cause of the solar irradiance cycle. Part of the irradiance is associated with changes in the quiet sun.

In the first chapter I will introduce several concepts to the reader. These concepts are the basic building blocks to understanding the solar irradiance cycle and the relationship to the magnetic cycle.

In the second chapter I present a plausibility argument. Proxy models have been demonstrated to very accurately model the solar irradiance based on the size of observed surface magnetic features. Due to this success I have been asked how proxy models could so accurately describe the solar irradiance cycle yet still conflict with implications of energy transport. The argument was constructed as an answer to that question. The point is that surface magnetic features are correlated to other mechanisms that drive the solar irradiance cycle - correlation does not imply causality. Additionally I will present the signature of this correlation.

1 In the third chapter I present a study of the solar oblateness (quarupole moment). The solar quadruple moment may be computed by knowing the interior solar rotation rate. Over the course of the solar cycle is believed that rotational shear near the base of the solar convection zone is converted into the magnetic field of the solar magnetic cycle. Observation of the solar oblateness during different stages of the solar cycle can give us important clues about the mechanism that drives the observed solar irradiance cycle.

In chapter four I present a study of solar facular regions. There are two concepts that should be taken away from this chapter. First that the primary effect of facular regions is to redirect energy radiated from the sun in a spatial and angular fashion. Second, proxy models overestimate even the local energy budgets for facular regions. As discussed in chapter two, overestimating the energy budget for facular regions is one of the hallmark signs that proxy models do not include one ofthe physical causes of the solar cycle. Instead the contribution to the irradiance cycle from these physical causes is attributed to facular regions.

In chapter five I present observations and numerical models of sunspot bright rings. As in chapter four, I will be looking at the energy budgets of sunspots and the bright rings associated with sunspots. This discussion will go beyond simply measuring energy budgets. Comparison of the observations to numerical models puts constraints on energy transport in the solar convection zone.

1.2 Observational History

In order to understand the role, and many of the important conclusions about the structure and energy transport mechanisms in the solar convection zone, it is

2 important to begin with a review of what we know about the solar cycle, and the associated irradiance cycle.

Many of the known characteristics of the solar cycle can be understood with a discussion of Maunder's "Butterfly Diagram". This was originally published in 1913. A later study was also published by Maunder (1922) which included more data, and is shown here as figure 1.1. The figure shows the position of observed sunspots on the solar disk. The vertical axis shows the latitude where the sunspot was observed. Time is tracked on the horizontal axis.

The first conclusion that can be drawn from figure 1.1 is the presence of a cyclic nature in the appearance of sunspots. Maunder (1913) explained that this solar cycle has a period of approximately 11 years. From figure 1.1 it can be seen that during the early stages of a solar cycle sunspots predominantly appear at latitudes near 20° north/south latitude. As the solar cycle progress, the preferred location shifts toward the solar equator, but does not actually reach the equator. It is generally accepted that the appearance of a sunspot at the solar equator is rare, and though there are only a few sunspots, which are reported to be at the solar equator, even this may be an over estimate due to the uncertainty of the location of the solar equator.

The location of sunspots is not the only property of the solar cycle that was noted at the time Maunder (1922) was written. It was also apparent that the number of sunspots, which occurred in a given time frame, varied during the course of the solar cycle. Also the fraction of the solar disk, which is covered by sunspots, varies over the solar cycle. This can also be seen in figure 1.1. By taking a ruler and sliding it across the page it can be noted that significantly fewer sunspots appear just before the beginning of the "wings" or just after the "head" of the butterfly. In this figure it can be most clearly seen in the years 1899, 1900, and 1901. This is that hallmark of

3 the sunspot cycle. The number of sunspots increases to a maximum, passes through a minimum and returns to a maximum with a period of approximately 11 years.

Prior to the observation of the locations of sunspots it was believed that sunspots might have a cause external to the sun. It was popular to speculate about what might cause the observed ll-year cycle in appearance of sunspots. Due to the migrations of latitude in which sunspots were observed, it was realized that it was more likely that the underlying cause of the sunspots could be found in the solar interior.

It was observed that the sunspots were vortical structures Hale (1908b). These vortical structures were compared to tornadoes on the surface of the earth. As was known at the time, the circulation of terrestrial tornadoes in the northern hemisphere was left-handed, while on the southern hemisphere they are right-handed. The speculation was that vortices on the sun might also follow such a rule. This lead to the belief that charged particles circulating in this vortical structure, might create a magnetic field (Hale 1908a).

These beliefs lead to observations of the magnetic field in sunspots. In 1919, Hale et al. published their landmark paper on these observations. It was found that sunspot.."l did contain a magnetic field and that in sunspot groups the lead sunspot has a different polarity than the following sunspots. It was also presented that the polarity of the field was reversed in the southern hemisphere, when compared to the northern hemisphere. The study of the magnetic fields comprised a portion of two solar cycles. This enabled the discovery that the magnetic field reversed from one solar cycle to the next. Thus if lead sunspots in the northern hemisphere have a polarity referred to as "+" during a given cycle, then in the following magnetic solar cycle, lead sunspots in the northern hemisphere would have a "-" polarity.

4 MOI4THL.Y NOllCES OF R.A.S.

Figure 1.1 Maunder (1922) This figure shows the observed latitude of sunspots over several cycles.

5 1.3 Magnetic Cycle

To facilitate this discussion on magnetic fields, I wish to define some terms for later use. The first is a poloidal magnetic field. A poloidal magnetic field is the field configuration that naturally results from a magnetic dipole. Field lines are thought of as coming out of the north pole of the magnet, stretching around and returning to the south pole of the magnet. A magnetic field of this configuration is generally symmetric about the defining line of the dipole, and thus it can be well described by its dipole strength. In the case of the sun the defining line of the dipole is aligned with the rotational axis of the sun. This requires that the field lines from a poloidal field are orthogonal to the rotational velocity fields in the sun.

Second, a toroidal field can be thought of as an annulus or annular distribution of magnetic flux. In the case of the sun, the primary axis of a toroidal field is aligned with the rotational axis of the sun, and thus the rotational velocity fields are parallel with the field lines from a toroidal field. Since there is no assumption of symmetry in the case of a toroidal field, a distribution of field strength with respect to latitude is required to describe a toroidal field.

Rotational shear is the derivative of the angular rotation rate. Radial rotational shear refers to the derivative with respect to radius. Angular or latitudinal shear refers to the derivative with respect to the pole, or equivalently with respect to latitude.

It will also be helpful to draw a rough picture ofthe solar structure to keep in mind.

The nuclear fusion that powers the sun occurs in the core. This extends to roughly

O.25Rev . Radiative transfer is the primary mechanism for energy transport in the core of the sun, and out to 0.7Rev. The region from the solar core to the 0.7Rev is known at the radiative zone. Above the radiative zone the primary mechanism of energy transport is by the bulk motions, i.e. convection. This is referred to as the convective

6 zone. This region extends up to the solar surface where the atmosphere becomes transparent and radiative transport again dominates energy transport. The standard analysis of convective stability would indicate that the base of the convection zone is stable against convection. Instead the momentum of the convective down flows drives convection. This overshoot region is referred to as the tachocline, and is a region of particular interest, due to the strong radial rotational shears, transition from radiative to convective energy transport, and it is presumed that this is the site of generation of magnetic fields that create the solar cycle (c.f. Caligari et al. 1995).

While the core and radiative zone of the sun rotate roughly as a solid body, the convection zone does not. The bulk motions of convection cause a transport of angular momentum into and outward in the sun. The result of this transport of angular momentum is that the equator of the sun rotates with a period of roughly

27 days. At 30° north/south latitude the solar surface rotates with a slightly longer period of roughly 28 days.

It is believed that the magnetic field of the solar cycle is generated at the base of the solar convection zone, slightly into the radiative zone. In the solar convection zone it is unlikely that the seed fields could resist the shear forces of overturning convection cells. This would result in the shredding of the seed magnetic field. This shredding might result in large localized fields, but probably would not result in a strong global field that is believed to be necessary to drive the solar cycle. Deeper in the radiative zone the differential rotation becomes quite small which would not allow for the development of the magnetic field.

To understand the necessity of rotational shear in the generation of the magnetic field of the solar cycle, it is often instructive to think of the magnetic fields as threads of elastic, which pass through the solar plasma. The solar plasma, for most intents and purposes, can be treated as a perfect electric conductor. This implies that the

7 magnetic fields are frozen into the plasma. One often compares the pressure of the plasma to the pressure of the magnetic field. f3 is the ratio of the gas pressure to the magnetic pressure, i.e.

(1.1 )

Where p is the pressure, and B is the magnetic field strength. When f3 is greater than 1 then the gas pressure dominates the dynamics of the plasma. When f3 is less than 1 the magnetic fields dominate the dynamics of the plasma.

A seed poloidal magnetic field (see figure 1.2 a.) begins the solar cycle (c.f. Foukal

1990; Choudhuri et aI. 1995). At the base of the convection zone this field is then stretched by the latitudinal rotational shear. (See figure 1.2 b,c) The resulting field can be seen as a combination of the original poloidal field, plus a toroidal field (see figure 1.2d-f and compare to a-c). We can conclude that a toroidal field is generated from a poloidal field and the velocity shear.

The mechanism that is currently believed to be responsible for transporting the magnetic field from the base of the convection zone to the solar surface was originally proposed by Jensen (1955) and Parker (1955). Magnetic fields exert a force orthogonal to the field lines, and a tension along field lines. The pressure exerted by the magnetized plasma is equal to the sum of the magnetic and the gas pressure. If the magnetized plasma is in thermal and pressure equilibrium with its surroundings, then the plasma will be less dense than the surrounding material. This will cause the magnetized plasma to buoy up.

Along the field lines this results in a lower pressure than the external pressure. This, however, does not result in the initial intuitive conflict; One might expect the pressure at the end of the field lines to cause an inward flow of plasma, which would

8 d) e) f)

/ \ / I \ I \ I I

\ I \ 0 I No Toroidal Field

Schematic of Magnetic Field Generation

Figure 1.2 The stretching of magnetic fields (a-c) can be thought of as adding a toroidal (d-f) field to the original poloidal field.

9 result in further expansion of the magnetic field and eventual dispersion. Field lines are closed, and thus there is no end for the plasma to rush in. One model that researchers often use is that of a loop of magnetic field, which is held from collapsing by buoyant pressure. A simple mental analogy is that of a rubber band stretched over a billiard ball. (This is shown as the toroidal fields of figure 1.2.)

In the turbulence of the solar interior one portion of the magnetized plasma will rise faster than another. This portion of the plasma has what one might call a "first mover advantage". Since there is less pressure along the field lines, fluid will flow from the high point in the magnetic field into the lower points. This creates an anchoring effect in neighboring points, and an increase in buoyancy in the "first mover" section.

This results in a loop rising to the solar surface. This process is often referred to as

Parker instability.

As this loop rises, the coriolis force will twist the loop around the direction of it rise. It has been long known that sunspot groups tend to form in a line. In 1919, Joy examined the angles created by the lines of sunspot groups and the lines of latitude.

It was found that the angle increased from 3° at a latitude of 5° north/south to 11° at a latitude of 32°. This relationship is often referred to as Joy's law (Hale et al.

1919). A comparison can be made between Joy's law and models of rising flux tubes.

By creating models with different field strengths D'Silva & Choudhuri (1993) found that in order to match Joy's law a field at the base of the convection zone on order of

105 Gauss was necessary. This is in agreement with the estimates for the necessary field strength, which would be required in order to contain the observed flux of the solar cycle in the overshoot region at the base of the convection zone Caligari et al.

(1995).

When the magnetic flux reaches the solar surface, it inhibits convection. This significantly reduces the energy transport. This decrease in energy transport allows

10 the plasma to cool. The cooled magnetized plasma is seen as a sunspot. This can be thought of in terms of the first law of thermodynamics. The temperature determines the amount of energy radiated. Energy input is restricted by the magnetic field inhibiting energy flow. To conserve energy the solar surface cools, and thus radiates less energy. This continues until the input and radiated energy flow balance.

Continuing with the discussion of the solar cycle, leading sunspots tend to migrate toward the equator. There the magnetic field is met with magnetic field from the leading spots in the opposing hemisphere. Since leading sunspots in the northern and southern hemisphere have opposite polarity, the magnetic fields cancel.

Following sunspots are observed to migrate toward the pole. We can consider the effect of trailing sunspots on the global magnetic field by imagining one of the magnetic field lines. Assuming a polarity the thread passes into the solar interior at the pole. The thread wraps around the radiative zone in the direction of the solar rotation. It then passes upward through the convection zone forming the following spot, and back into the solar interior, forming the leading sunspot. Since the magnetic "thread" is seen as going "into" the solar interior at the pole, and "out" of the solar interior where following sunspots are formed, it can be understood that following sunspots have an opposite sign from that of the global magnetic field at the poles.

As the magnetic field from following sunspots migrate pole-ward they are thought to cancel the field. As more magnetic field from following sunspots reach the pole, this field builds up as a new global magnetic field, with opposite sign. It has been estimated that this neutralization process does not need to be very efficient, requiring only 1% of the total magnetic flux (Foukal 1990).

It is debated how the resulting poloidal field is reintroduced into the solar interior to begin the next cycle. The two most popular methods are negative buoyancy and meridional flows. Negative buoyancy occurs as the magnetized plasma cools the

11 requirement of pressure equilibrium increases the density. Thus as the plasma in a magnetic flux tube cools, and the gas pressure decreases, the external pressure causes the flux tube to contract increasing the density. This increase could be significant enough to overcome the decrease in density caused by the pressure support from the magnetic field. The more dense magnetized plasma sinks down into the solar interior. One of the major advantages to this solution is flux retraction. Flux retraction allows for a recycling of magnetic flux. This would decrease the strength of the magnetic field in the overshoot region that would be required to supply the observed magnetic flux for a solar cycle. One of the major concerns with generation of 105 gauss fields is that they exceed equipartition values - the energy density available from differential rotation would be lower than the energy density of the magnetic field (Parker 1987).

Meridional flows are relatively steady state flows, similar to terrestrial trade winds, in the convection zone of the sun. Near the equator there is an upward flow. At the solar surface this flow is diverted pole-ward, and is seen as a surface flow with a speed of round 7 mls (Dikpati & Choudhuri 1994). Near the poles the flow is diverted inward where it reaches down to the surface of the radiative zone. At the surface of the radiative zone, the flow proceeds to the equator. This offers a second alternative transport mechanism for surface magnetic fields to be reintroduced into the solar interior.

1.4 The Solar Irradiance Cycle

To begin with consider the difference between the solar irradiance and the solar luminosity. Irradiance is the amount of energy radiated by the sun in a particulm direction (Le. energy flux per sterradian). In the case of the sun we typically think in terms of the amount of energy we receive at the Earth. The distance between

12 the Earth and the sun varies by about L7% over a year. If we think of the solar irradiance in terms of power per unit area at the earth, then the solar irradiance varies by more than 3% over one orbit. However, the average distance between the earth and the sun (taken over a one year period) is constant. We can measure the solar irradiance in terms of watts per square meter. This is equivalent to measuring the amount of energy radiated per sterradian. The solar luminosity equals the total amount of energy the sun radiates in all directions. Luminosity is the integral of the irradiance.

L= JJdn (1.2)

Where L is the luminosity and I is the irradiance.

It is often thought that the irradiance of the sun does not depend on the viewing angle. To a large extent this is true. It will be seen later in this paper that this assumption does not hold. Small variations in the solar irradiance on the order of0.1% of the time averaged solar irradiance will be considered. When large sunspot groups appear on the solar surface the asymmetry that they cause in the solar irradiance can be larger than this by a factor of four or more (Chapman 1987).

It can be argued that the first evidence of the solar irradiance cycle was made by Douglas (1919). Douglas was studying the rings in yellow pine trees of Northern

Arizona, in relation to the climate and solar activity. He noted several irregularities in the ring structures, which corresponded to the time of the Maunder minimum, a period from approximately 1650 extending into the early 1700's where sunspot activity observed to nearly stop. This can be considered to be a prolonged solar minimum. Douglas noted that the peculiarities in the ring structures could be explained by unusual weather.

13 The atmosphere of the Earth makes it difficult, if not impossible to take accurate determinations of the soJar irradiance at the level better than 1%. Clouds, temperature variations, dust and turbulence all act to obfuscate the sensitive measurements. To overcome these obstacles, over the last 30 years an ongoing study of the solar irradiance has been undertaken with several spacecraft missions.

To make the measurements cavity radiometers are typically used. A simple picture of a cavity radiometer is that of a black. body cavity operating in reverse. A small opening is created leading to a larger cavity. The interior of the cavity is coated with a material designed to absorb as broad a band of light as possible. (One might think of this as black. paint.) Light enters through the small opening and is almost completely absorbed by the material. An accurate measurement of the temperature resulting from the incident light indicates the magnitude of the irradiance. Frohlich

& Lean (1998) compiled the data from five space borne radiometers experiments: The Hickey-Frieden radiometer aboard NIMBUS-7, ACRIM I aboard the Solar Maximum Mission, the Solar Monitor on ERBS, ACRIM II on board the Upper Atmosphere Research Satellite, and VIRGO on board the Solar and Helioseismic Satellite (SOHO). The compilation removed errors from absolute and relative scale differences in the instruments. Absolute scale differences between instruments are the differences in the total measured luminosity, and are typically dependant on uncertainties in measuring the area of the opening in the cavity. Relative scale uncertainties refer to different magnitudes of change that is measured by two instruments. Instrumental drift (discussed below) was also removed in the study. The result is a record of the solar irradiance, which covered a 30-year period.

The existence of a solar irradiance cycle was confirmed from this record. This solar irradiance cycle is an increase of the average solar irradiance (taken over a suitably long time, typically 120 days) of 0.1%when the surface magnetic fields are observed to

14 be at their strongest (solar maximum). During the periods where sunspots and other magnetic features are observed to be at a minimum, the solar irradiance decreases, from the ll-year average, by a similar 0.1% (Frohlich 2002). The record also shows large variation over short timescales. The variation can be as large as 0.4% on the timescale of a week (Chapman 1987). This is due to the presence of large sunspot groups, faculae, or plage.

One problem, which arises in taking these measurements, is that over time exposure to the harsh environment of space causes degradation of the sensors.

Spacecraft, which are positioned high enough above the Earth's atmosphere and magnetic field to avoid their interference with the measurement, lose the protection provided by the atmosphere and magnetic fields. Intense radiation from the sun alters the physical and chemical properties of the coatings of these instruments. This degradation of the coatings does not affect day-to-day me&Sllrements. The overall error does take effect after several years, and is seen as an instfUIIl-e:q.tai drift;. To remove the instrumental drift an assumption is made that the solar in.. &di~~~ f~twns to the same levels each solar cycle. The problem with this solution is th#t pj.e solar irradiance might have an underlying cycle, which is longer than the known ll-year cycle. If we imagine that there is a 100-year solar cycle, thf'lfl. thtE\ wOllld appear as instrumental drift and would be eliminated from the measurements.

Studies of star clusters and stellar models indicate that the sun has increased its luminosity by 30% over its lifetime (Schwarzschild et al. 19i:i7). Measurements have determined that there 111so exists an ll-year solar cycle. The question still remains whether the sun changes on more intermediate time scales of hundreds to thousands " of years. To understand this change it is necessary to un4~rstand the underlying cause of the short-term (I1-year) changes in the solar irradiance,

15 1.5 Proxy Models

As accurate measurements of the solar irradiance became available, several authors

(c.f. Lawrence et al. 1985; Lean et al. 1998; Fligge et al. 2000; Frohlich 2002; Preminger et al. 2002, etc.) began to model the solar irradiance based on observed surface features of the sun. I will refer to this class ofsolar irradiance models as Proxy Models, as they use the proxy of surface feature observations to predict the solar irradiance.

The purpose of these models is to provide an estimate of the solar irradiance when direct measurements are not available.

Proxy Models can be reduced to three simple assumptions: (1) Sunspots are dark, and thus represent a deficit in the solar irradiance; (2) Faculae are bright, and thus they represent an increase in the solar irradiance; (3) The quiet sun, i.e. the area of the solar surface which is free from significant magnetic disturbances, remains unchanged (See figure 1.3). The quiet sun represents no increase or decrease in the solar irradiance, even over periods of time comparable to the solar cycle.

Since sunspots are dark we can calculate their effect on the solar irradiance by measuring the area of the solar surface, which is covered by sunspots. If we know how dark the "average" sunspot is then we can multiply the area of the sun covered by sunspots by the average deficit they produce per unit area of the solar disk. This will give the amount by which one would expect the solar irradiance to be decreased due to the presence of the observed sunspots.

Faculae are bright and may be treated in a similar manner. The area is measured by observations, which, like sunspots, can be made from ground-based instruments.

The area of the solar disk, which is covered by faculae, is multiplied by a constant to give the increase in solar irradiance, which one expects to be observed.

16 a

b

Schematic of Proxy Models

Figure 1.3 a) A section of the solar disk is shown here for reference and comparison. b) Sunspots are considered to consist of uniformly dark areas, uniformly bright areas, and unchanged quiet sun. By measuring the area of the dark and bright regions the change in the solar irradiance is then calculated.

17 The constants by which one multiplies the area of faculae and sunspots in order to predict the solar irradiance is computed by a least-squared fit of the area of sunspots and faculae (each as independent variables) to the observed irradiance. It might interest the reader why one would go to the trouble of measuring the irradiance and then fitting it to other observations only to predict the original measurement. The reason is that by creating an accurate means of predicting the solar irradiance using ground based observations, the solar irradiance may be determined prior to the space borne observations which are now available.

More advanced proxy models use more than a single constant to convert the area of surface features to irradiance contributions. Typically, the position of the magnetic surface feature on the solar disk is also included. As will be discussed in Chapter 4, faculae appear brighter near the solar limb. Near the center of the solar disk they often disappear, or become dark relative to the surrounding quiet sun. Thus, models assume that the conversion factor is a function of JL, the cosine of the angle between the line of sight and the normal to the solar surface (c.f. Ortiz et al. 2002).

With the inclusion of variations for JL a multi-parameter model is constructed which can accurately account for most of the solar irradiance variation. In particular

Frohlich (2002) reports that 95% of the solar irradiance variability may be explained by use of his method. Preminger et al. (2002) cite that 96% of the solar variability can be accounted for. Both of the papers report that the residual 5 or 4% respectively, of unexplained solar variability shows no trend of a solar cycle. The implied conclusion is that surface features alone are proven to be the cause of the solar irradiance cycle.

With the seemingly remarkable predictive power of proxy models it is tempting to assume that they can be applied to accurately predict the solar irradiance for as far back as we have accurate measurements of the surface features on which they depend.

This assumption is only true if an extrapolation of the model to different solar cycles

18 holds. It is equally as tempting to assume that since no solar cycle is evidenced in the residuals from these models, we can assume that surface features represent the physical cause of the solar irradiance cycle. This is a claim that will be challenged in the next chapter.

1.6 Generation and Initial Rise OfMagnetic Fields

Part of the connection between the irradiance cycle and the magnetic cycle that should be considered is the connection between energy transport and the generation of the magnetic fields. In this section I will discuss some of the concepts that link the energy transport to the toroidal fields as they are being generated. This will not be an exhaustive or complete study. Instead it will offer some insights to the problem as well as a possible foundation for future work.

I will begin this discussion by considering some of the factors have lead to the belief that magnetic fields are generated at or near the tachocline. This will lead to the discussion of transport of the magnetic field to the solar surface. The natural question will be asked, "What are the conditions which cause magnetic fields to be transported to the solar surface?" I will introduce into the discussion two alternative predictions about the transport of energy to the surface.

The magnetic fields observed as the solar magnetic cycle are generated in the solar interior. The equation that governs this generation is:

aB 1 - = \7 x (v x B) + - \72 B (1.3) at ~ where ~ is the electrical conductivity, which is extremely high.

19 The first term on the right hand side of equation 1.3 describes the generation of the fields. It implies that a velocity shear in the plasma is required to generate the toroidal field from the poloidal field. The second condition is that the velocity be orthogonal to the magnetic fields and the shear be directed along the field lines. If we consider the pictorial case shown in figure 1.2, and consider a magnetic field element to be a strand of rubber band which is carried with the fluid, then magnetic field is generated when the plasma flow at one point is larger or smaller than the plasma flow at a different point along the field line. There are two forms of a velocity shear present in the sun. First is the convective motion found in the solar convection zone.

The second is shears that occur in the rotational velocity.

The turbulent motion of the convection zone is unlikely to develop the large-scale magnetic fields that we seek to describe. The turbulent velocities would form magnetic loops on the scale of the solar granulation. We might see this by considering a thread of the magnetic field tracing the motion of the fluid as it turns over. If the magnetic field were allowed to reconnect it would form a horizontal loop of magnetic field. Even if a loop of this type were to form, it would be unsuitable as an explanation of the magnetic field. The problem exists that there is no preferred orientation for the loop, except that it is a vertical loop. In three dimensions the loop has two axes that define the plane on which it lies. The second axis could be oriented north-south, east-west or any orientation in between. The associated sunspots, which develop from these loops, would not have the observed orientation. A magnetic field, which is generated by convective motion, would more likely result in a tangle of shredded magnetic fields which would quickly dissipate due to second term on the right hand side of equation

1.3.

The alternative is that the magnetic fields are generated by shear in the rotational velocities of the plasma. This shear is shown in figure 1.4. As can be seen in the figure

20 Rotational Shear

150 ,---, N Solar Rotation Rate I c L...... J 500 0 ,--, Q) N -+--' 450 ::r:: 0 c 100 cr: L.-..J C 400 60 L- a -+--' 0 0 350 ill -+--' s= a 0.4 0.6 0.8 1.0 (jJ cr: 50 radius [Solar Radius] 0 c 0 --l---' 0 --l---' 0 0 0:::

- 50 L-..---'--_-----'--_---'----_'-----'--_-----'--_---'----_'-----'-_-----'----_---'------1 0.4 0.6 0.8 1.0 radius [Solar Radius]

Figure 1.4 The difference between the solar rotation rate at the equator and 60 degrees

longitude (wo=oo - WO=60 o ). The position of the tachocline is shown by the dotted line. In the inset the solar rotation rates at 0°, 30° and 60° is shown.

the strongest shear in the convection zone, however velocities in convective flows can

reach the local sound speed (c.f. Stein & Nordlund 1998). The convective flows will

exhibit shears many times larger than the shears in rotational flows. The convective

flows would dominate the generation of these magnetic fields, effectively shredding

any global magnetic field generated by the rotational shear.

At this point I would like to introduce a gedanken experiment. When we consider

magnetic fields to be represented by field lines it is often confusing. Field lines that are

21 stretched by velocity shears are strengthened. One might think that while the length of the magnetic field lines is increased the field strength is not. Consider the field lines depicted in figure 1.2. Each field line represents fixed quanta of magnetic flux. Field strength is the density of flux, and is thus represented by the density of lines. We can then estimate the field strength by measuring the separation between the field lines.

When a sheer is applied to the system (depicted in 1.2 a-c), the distance between field lines decreases. Thus the magnetic flux, or magnetic field strength increases. This

(correctly) implies that the total field strength (i.e. magnetic flux density) increases.

Even though the number of field lines plotted in the figure remains constant there is a higher density of field lines in band c compared to a. Thus the total magnetic flux represented by these conditions is higher.

We can create a rough estimate of the field strength by taking the difference in the rotational frequency near the pole (60° latitude), and the rotational frequency near the equator (00 latitude). This represents the shear associated with the rotation. We then multiply by the radius to produce a velocity. The square of this velocity is multiplied by ! the density to generate an energy density associate with the rotational velocity sheer. This energy density is then equated to the energy density of the magnetic field, giving an upper limit on the strength of the magnetic field that we can expect to generate, i.e.:

(1.4)

This results in a maximum field strength of 8200 G. This is roughly in agreement with the estimate of Parker (1987), but is insufficient to provide strong enough magnetic fields to agree with the observation of Joy's law, or provide sufficient magnetic flux for the solar cycle (D'Silva & Choudhuri 1993). Even in the convection

22 zone, where the latitudinal velocity shear is the largest, the difference n these velocities is insufficient to generate the estimated 105 Gauss required is in the solar convection zone, which has already been eliminated as a favorable site for generation.

We now wish to understand how the magnetic field near the surface of the radiative zone affects the energy transport. We understand that the magnetic field is generated below the convection zone. Energy transport in this region is dominated by radiative transport. Thus:

(1.5) where F is the energy flux, a is the Stefan-Boltzman constant, c is the speed of light,

T is the temperature, /1, is the wavelength averaged per unit mass opacity, P is the density, and R is the distance from the center of the sun.

The presence of the magnetic field provides for a pressure support. This pressure support will result in a decrease in density:

(1.6)

This can be simplified by assuming an exponential density P Po exp z/h, and h = kT//-lg

B2 Pe - Pi = 81rgh (1. 7) where, Pe and Pi are the external and internal densities, respectively.

If the system is allowed to reach equilibrium, the energy flowing in from an arbitrarily large depth remains constant, since we assume that the energy generation rate at the core remains fixed. Thus we assume that the energy flux through the

23 region remains constant. The opacity per unit mass is unlikely to change. Thus from equation 1.5 the quantity ~ a:. remains constant. Thus the temperature at the top of the magnetized plasma would increase by an amount:

(1.8)

p where ~ is the fractional change in density, i.e. ~ 8 , and I is the thickness of the = p layer of magnetized plasma.

Thus the temperature at the top of a magnetized region will be higher than the unperturbed system. When the magnetic field rises, in principle we should expect that it contributes an excess entropy' to the surface energy budget. This increase in entropy manifests itself as a net increase in the total solar irradiance. The question that I now wish to address, is what are the conditions for the magnetic field to rise?

There are two forces at play. This first is magnetic tension. The second is a buoyancy force. To understand the interplay we will consider a single loop of magnetic flux that passes around the sun at a constant latitude and depth (such as one of the toroidal flux loops shown in figure 1.2f,g.). The buoyancy force arises from the decrease in density that is a result of the magnetic pressure.

The magnetic field is fixed in plasma, and thus pressure from the field is included in the total pressure in the region of plasma, and results in a decrease in density as described in equation 1.6. The lower density results in a buoyant force that is proportional to the energy density of the magnetic field:

(1.9)

where Fb is the buoyant force per unit length and J-L is average mass per particle

24 The tension force of the magnetic field is then assumed to resist the buoyant force. We compare the buoyant force for the single magnetic loop located in the solar equatorial plane to the tension force that arises in this model. In this case the force from tension acting to resist the rise of the magnetic ring is the magnetic tension force times the derivative of the circumference with respect to radius:

(1.10)

where Ft is the inward component of the tension force, and s = 21fR is the circumference. Equating the force of tension acting to resist the rise of the loop to the buoyant force, and simplification gives:

2kT r=-- (1.11) pg this can be further reduced to:

r = 2h (1.12)

This is analogous to equation 11-5 in Foukal (1990) which states that the condition for a magnetic strand to become buoyantly unstable is that the length of the magnetic strand must be larger than twice the pressure scale height. At the tachocline the local density scale height is about 8% of the solar radius. Thus the magnetic tension is insufficient to prevent the magnetic field from buoying up.

If we consider the full three-dimensional treatment of the problem there is an additional concern. We can consider the magnetic field to be like a rubber band that is stretched over a smooth sphere. The magnetic tension causes an inward force on the magnetized plasma. Buoyant force restricts the loop from sinking into the solar

25 interior, but leaves an unresolved component of force toward the nearest pole. Thus the magnetic field slips toward the solar pole.

At this point we turn to the treatment of the problem by Moreno-Insertis et al.

(1992). Equations 2.4-2.6 are:

(1.13)

F¢ = -2poi;; (1.14)

Fz = - !:i.pgcosfJ (1.15) where Fw, F¢, Fz are the components of force in the cylindrical coordinates, w, 4> and z respectively, !:i.p is the difference between the internal plasma density and the external plasma density, 0 is the rotation rate of the reference frame and Oe is the rotation rate of the external plasma.

These equations may be solved to find the conditions under which a magnetic flux ring remains at a constant depth. For this to occur !:i.p must equal zero. The condition for a magnetic flux ring to resist buoyant force is that the density in the magnetized plasma must be equal to the density outside the magnetized plasma. This requires that the temperature in the magnetized plasma be lower than the surrounding plasma.

We should consider this result. The buoyant force is directed from the center of the sun to toward the solar surface, while the magnetic tension is directed inward toward the z axis. Thus if we balance the buoyant force, there is a residual force directed toward the z axis. This implies that a magnetic flux ring of this type will quickly slip toward the pole of the sun. An estimate of the time for the magnetic field

26 2 to slip to the pole may be constructed. Setting !:1p, ~ and (0; - 0 ) in equation 1.13 to zero:

(1.16)

The acceleration is force per unit area divided by density, and the time can be equated to the square root of the distance divided by the acceleration. This gives the estimated time:

(1.17)

3 5 Taking p = 0.2 g/cm , w = 0.7R0 and B = 10 C it is found that a magnetic loop can be expected to slip to the pole in about 9 days.

Both of the above results are disconcerting. First the drifting of the magnetic flux toward the pole would result imply that sunspots are seen at higher latitudes than where they are observed. Second there must be a cooling mechanism for the magnetized plasma, which would allow it to become colder than the surrounding material.

The resolution to the first problem is that as the loop of magnetic field slips toward the pole, angular momentum is conserved. This increases the velocity with which the plasma rotates, relative to the surrounding plasma. Equilibrium can be attained when the centrifugal acceleration is equal to the inwardly directed force arising from the tension in the magnetic field.

By balancing the pole ward drift we have eliminated the magnetic tension from the problem. We are again left with balancing the buoyant force. The magnetized plasma will rise. Due to the balance achieved between tension and centrifugal force the plasma will rise along cylinders. The solution to the problem of the rising magnetic

27 filed can only be resolved if the density inside the magnetized plasma is equal to the density outside the magnetized plasma (Moreno-Insertis et al. 1992; MacGregor &

Cassinelli 2003). Due to the pressure support provided by the magnetic field, this can only be achieved if the temperature of the magnetized plasma is lower than the surrounding plasma.

2 fj.T = !!:..- B (1.18) pk 87r

Consider a parcel of plasma in a layer ofthe sun with a sub-adiabatic temperature gradient. If we move this parcel of plasma upward the external pressure acting on the parcel decreases. This is due to the fact that there is less plasma above the parcel to exert a downward force. The decrease in pressure results in the parcel of plasma expanding to reach equilibrium with its surroundings. If it is assumed that there is no heat exchange from the surroundings then the temperature in the plasma will decrease. The question is whether the plasma will cool faster than it surroundings.

So far this treatment is similar to the standard convection stability analysis found most introductory astronomy textbooks (c.f. Carroll & Ostlie 1996). To relate to the earlier entropy arguments, a different approach will be taken here. The quantity that is of interest is the entropy per unit mass. First we consider the effect of the expansion on this quantity. The second law of thermodynamics states:

ds = Tdq (1.19)

Since we are considering a parcel of plasma which does not exchange heat with its environment, we must conclude that the entropy per unit mass does not change as the parcel of plasma rises. Equation 5 from Fan et al. (1994) was used with the helioseismological model provided by Jesper Schou to construct figure 1.5. We can

28 Entropy Per Unit Moss

1

I <

: 1.00x 10 1 1 r- 1- L Q)

>, Q 10 2 9.50x10 r- -+--' c W

10 9.00x10 1 1 4.Sx1 0 10S.OX1 0 10S.Sx1 0 106.0X1 0 106.sx1 0 107.0x1 0 10 Radius

Figure 1.5 The entropy per unit mass up to an overall constant of integration is shown here. The solar convective zone has the highest per unit mass energy. Elsewhere the per unit mass entropy increases. The decrease near the solar surface is interpreted as evidence of the solar wind, and known concerns over helioseismic inversions near the solar surface.

see that in the convection zone extending from about.7 T0 to the solar surface the

entropy per unit mass is constant. This is what would be expected from a convective

region, i.e. that the rising (falling) plasma expands (contracts) nearly adiabaticly.

Also seen in figure 1.5 is the fact that the convection zone has higher entropy per

unit mass than any other region in the sun. This is not unexpected. The convection

zone extends to the solar surface. Below the solar convection zone the entropy per

unit mass increases outward from the center of the sun. Ifthis were not the case, then

a parcel of plasma that were displaced slightly upward would have more entropy per

29 unit mass than the surrounding material. This implies that the temperature in this parcel of plasma would be higher than the surrounding plasma. To reach pressure equilibrium the plasma would necessarily have a lower density. The lower density would imply that the parcel of plasma would continue to rise.

The conditions, which have just been described, are exactly the conditions for convective instability. Thus another way to analyze stability against convection would be to compute the radial derivative of entropy per unit mass. Ifthis quantity decreases with increasing radius, then the region is convectively unstable. Convection would set in and a flat profile would soon develop. Understanding figure 1.5 allows us a possible resolution to the storage of magnetic fields in the solar interior.

While this will resolve the poleward drift there is concern. First the magnetic field is not purely torroidal. The loops are a combination of polloidal and torroidal.

The loops of magnetic field do not actually close on themselves in the way they have been described. Thus the spinning up effect arising from the conservation of angular momentum is not strictly an accurate representation.

It is possible that a more complete treatment using either a Hamiltonian or

Legrangian approach will resolve the conflict. However, since this section is to be viewed as discussion for further work, a complete treatment is beyond the scope presented here. The assumption will thus be made that the approach provides meaningful insight to a complex problem.

30 Chapter 2

On Proxy Models for Solar Irradiance

2.1 Introduction

Preminger et a1. (2002) presents a multi-color proxy model, which they find can account for most of the observed irradiance variability. They report that when their model is compared to the original irradiance data, they find an a regression coefficient

R = 0.96. Thus faculae and sunspots can predict 96% of the irradiance variations during the timeframe examined by the study. Frohlich (2002) presents his proxy models and reports that he is able to "explain more than 95%" of the solar irradiance variability. With this success it is tempting to assume that the problem has been solved and that the irradiance cycle can be attributed to surface features. It is more realistic to realize that surface features are correlated with irradiance variations.

In this chapter I will investigate the validity of the assumption that proxy models offer proof that the solar irradiance cycle is caused by surface features by constructing a simple model of the irradiance. From this model I will construct a proxy model to describe the irradiance cycle and will examine its ability to distinguish between an underlying irradiance cycle, and one that is caused purely by surface features.

This model should not be misconstrued as proof that proxy models are not making

31 accurate predictions. It is instead a plausibility argument that models of this type could miss fundamental causes of the changes in the solar irradiance, which are not related to surface features. I will focus the discussion and comparison on a toy model designed to test the proxy models such as the ones of Preminger's and Frohlich's.

These proxy models are particularly well constructed, and demonstrated to predict a high percentage of the irradiance variations. This should not be viewed as a criticism of their approach, but rather a compliment, as the best models chosen for comparison.

2.2 Simulated Irradiance Data

I begin by making several assumptions. There are several choices that may be made in creating the input irradiance data. The choices made here are designed to create a more rigorous test, and to create an irradiance cycle, which has a similar visual appearance to the actual data.

The first assumption is that there is an underlying irradiance cycle not caused by the presence of sunspots or faculae. This underlying solar cycle might be caused by any number of physical processes, however in particular structural changes caused by magnetic fields near the surface of the solar convection zone (c.f. Kuhn et al.

1988; Kuhn & Armstrong 2002) are particularly present of mind. The basis of this assumption is to test whether proxy models could detect an irradiance cycle that is not caused by surface features. I will construct the proxy model and test to see how much, if any, of this original signal is evidenced in the residuals from the model. The portion of the solar irradiance which is not physically caused by surface features will be modeled as a sinusoidal function with a period of 11 years, which is in reasonable agreement with observations (Frohlich & Lean 1998).

32 Second, I assume that the appearance of sunspots and faculae is correlated with the assumed underlying irradiance variation. This assumption is in agreement with the observed correlation of the solar irradiance cycle. By making this assumption,

I will examine how much of the underlying solar cycle is incorrectly attributed to surface features.

Third, I assume that faculae and sunspots have an equal but opposite effect on the solar irradiance variation. This assumption is motivated by the discussion of

Chapman (1987) and references therein. Additionally, by making this assumption the underlying irradiance variations should be easier to detect, as the faculae and sunspots will effectively cancel the effect of each other over the time scale of one solar cycle. The interested reader is encouraged to test the effects of this assumption.

Finally, it is assumed that the surface features, i.e. faeulae and sunspots, and their contributions can be accurately measured, up to an overall multiplicative constant.

Comparing the area of surface features with the solar irradiance gives the overall constant. This reflects the assumption that indicators used in proxy models can accurately measured. This should be compared the underlying assumption made in creating proxy models that the area of the observed features on the solar disk can be accurately measured, and that this area is sufficient for determination of the solar irradiance.

The model is constructed by computing the irradiance variations on a daily basis.

The choice of generating data on a daily basis is arbitrary. Oher cadences my be investigated by an intereste reader, however the purpose of this chapter is to construct a plausibility argument not a definitive model of proxy models. As stated in the first assumption the underlying irradiance cycle, Su is assumed to have a sinusoidal form, with a period of 11 years. The amplitude of this sinusoid is chosen to be 0.1%. To

33 this a sunspot signal and a facular signal were added (see figure 2.1. The sunspot and facular signal are discussed below.

The facular signal was created by generating uniformly distributed random numbers, from 0 to 1, for each day. The uniform distribution was chosen to match the appearance ofdaily irradiance measurements. A normal distribution was investigated, but was found to contribute large spikes that did not agree with the visual appearance of measurements. The choice of using daily cadence was arbitrary, but should not affect the validity of the plausibility argument. The random numbers were multiplied by 0.0012 plus the underlying solar cycle irradiance. This reflects the assumption that the appearance of faculae is correlated with the underlying irradiance cycle.

It does not reflect causality in any way. The addition of 0.0012 to the underlying solar cycle allows the facular contribution to be positive definite, and allow for small contributions even at solar minimum. This function is then normalized to one, by dividing by the daily average. The resulting function is then multiplied by an overall constant, which I will refer to as the magnetic feature amplitude, Am.

Sunspots tend to be longer lived, and are assumed to survive for approximately one disk crossing. Individual sunspots also can have a larger effect on the solar irradiance, which results in large downward spikes. The signal is generated by computing normally distributed numbers for each day of the model. As mentioned above the normal distribution allows for the presence of large spikes in the data. This is in agreement with the effects of large sunspot groups crossing the solar disk. The random numbers were then multiplied by 0.0012 plus the underlying solar cycle. To this a Gaussian smoothing was applied. The full width at half max was chosen to be

14 days. This value is larger than one might choose for the disk crossing time for an individual sunspot, but should be a reasonable approximation when considering large sunspot groups. The resulting sunspot irradiance contributions were normalized to an

34 Variation From Sunspots and Faculae a> u 0.2 c 0 u 0 L L 0.1 L -0 0 (f) c 0.0 a> CJ"l c 0 ..r:: U -0.1 +-' c a> u L a> 0.... -0.2 a 2 4 6 8 10 12 Time [Years]

Contributions From Fa.cula.e and Sunspots

Figur 2.1 Th ontribu ions to h simulat d irradiane data from 'un pot and fa ula ar h wn here. The la f a p riodie signal nfinn tha sunspo s and fa ul do not con ribut to th irradianc cl.

35 average of one-per-day, as above, and multiplied by the magnetic feature amplitude,

Am·

The resulting irradiance signal is then:

(2.1) where S is the solar irradiance, Sq is the quiet sun contribution to the solar irradiance cycle, Sf is the normalized contribution from faculae and S8 is the normalized contribution from sunspots.

The choice of the magnetic feature amplitude is somewhat arbitrary. It was found that increasing the value of this the magnetic feature amplitude improves the fit; a conservative value was taken for this reason. It was observed that actual irradiance observations that the lower bound of the variations during solar maximum nearly coincided with the upper envelope at solar minimum. This is shown as a line between o and -0.1 in figure 2.2. This feature can be approximately matched by the choice of

6 X 10-4 for the magnetic feature amplitude.

The resulting facular and sunspot contributions can be seen in figures 2.2a and

2.2b. The resulting irradiance variation can be seen in figure 2.2c. Also shown in figure 2.2c is the 12G-day running average. This figure should be compared to figure

2 from Frohlich (2002). While there is some visual disagreement between actual and modeled observations, such as a more ragged upper edge, and the absence of large downward spikes in the model, the model should suffice for the purposes of this plausibility argument. The amplitude of the observed solar irradiance cycle is also slightly larger than the input value. Taking the difference between the maximum and minimum of the 120-day average gives a value of approximately 0.21 %. This is likely due to large modeled facular activity creating a temporally localized spike near

36 Modeled Facular Contribution 0.3 a 0.2 0.1 0.0

Modeled Sunspot Contribution 0.1 0.0 -0.1 -0.2

Modeled Solar Irradiance (f) 0.3 c .-c 0.2 (l) (J'l 0.1 c 0 0.0 ..c U -0.1 ~ -0.2 0 2 4 6 8 10 12 Time [Years]

Modeled olar Irradian Variations

Figure 2.2 The modeled irradiance data is shown here. a) Shows the contribution from faculae. b) Shows the contribution from sunspots. c) Shows the total modeled irradiance plus the 120-day running average.

37 solar maximum. While it is possible to scale the overall model such that the peak to trough was exactly equal to 0.2% this is unnecessary for the quantities, such a.."l the correlation coefficient R, which will be examined here.

To verify that there is no irradiance cycle that comes from the sunspots and faculae, the sum of these two contributions, along with the 120 day running average of this sum, has been shown in figure 2.1. The lack of a periodic signal demonstrates that the facular and sunspot contribution effectively cancel (which follows by construction). The irradiance cycle in the system is due to the assumed irradiance cycle, not contributions from surface features.

2.3 Results

A multi-parameter linear regression was performed with the modeled irradiance as the dependant variable, and the facular and sunspot contributions as the input variables.

This is an approach analogous to the one presented in Preminger et al. (2002) and

Frohlich (2002).

Several runs were performed and the correlation-coefficient, R, was found to be around 0.94 to 0.95. We consider how this compares to a proxy model, which has a regression coefficient R of 0.96. In fact it is quite unfavorable for the proxy modeL

The proxy model demonstrated here has only two dependant variables, while the model of Frohlich has three. The addition of extra variables to a regression will

2 2 never decrease R , and will often increase R (c.f. Pyndyck & Rubinfeld 1991). By introduction of a single variable, which consists of nothing more than uncorrelated, normally-generated numbers R for this model is increased to 0.96. This is nearly as rigorous a fit as Frolich, who cites an R of 0.97.

38 The implied result is that the irradiance cycle has been absorbed somehow into the observed features. Examining the difference between the input irradiance and the irradiance computed from the linear regression allows a visual confirmation. This residual is shown in figure 2.3. Also shown is the original irradiance cycle signal for comparison. It can be seen that most of the irradiance signal is accounted for by the model - a model that does not take into account the true source of the modeled solar cycle.

To understand how this can occur we examine the regression coefficients. It should be clear that the regression-coefficients should both be 1.0. The regression-coefficients for the sunspots were found to be between -0.24, and -0.3 for several simulations. The regression-coefficients for the facular contribution were found to be between 1.12 and

1.22 for several simulations. Since the facular and sunspot contributions are both correlated with the underlying solar cycle, overestimating the contribution of faculae, and underestimating the contribution of sunspots results in the best fit.

2.4 Discussion

Proxy models assume that the irradiance from the quiet sun does not change over a solar cycle, and that the solar cycle is caused entirely by observed magnetic features. It is then found that the irradiance cycle can be predicted by multiplying the area of sunspots and faculae by coefficients that are believed to be the average decrease or increase these areas contribute to the solar irradiance cycle. The simple model presented in this chapter shows that proxy models can successfully predict the irradiance cycle, even if the assumption that the quiet sun irradiance is invariant is not a valid assumption.

39 Residual Solar Cycle (]) u 0.15 c .-0 v 0 0.10 L L

L 0 - 0.05 0 (f) '\ '"\J'vA, I 1"\ I \ '. /\ c 0.00 \I J\ I J J\ '\ "''. 1\-"'- ~-----""""'-'---...... ,.,.-/ (]) J \ \!V " OJ C ~ 0 -0.05 ....c u ,j

-+--' L,. c -0.10 '- (J) u L (J) 0.- -0.15 0 2 4 6 8 10 12 Time [Years]

Simulated Irradiance Data and Proxy Residual

Figure 2.3 The 120 day average of the simulated irradiance data is shown with a solid line. The portion of the simulated solar cycle, which is not generated by faculae or sunspots is shown with the dot dash line. The residual resulting from a least squared fit of the simulated sunspots and faculae to the simulated irradiance is shown with a dashed line. The absence of any periodic signal in this residual is an indication the proxy models would not detect underlying irradiance cycles.

40 This simple model shows that while proxy models are able to provide accurate estimates of the solar irradiance, they may be missing a fundamental cause these cycles, such as those proposed by Kuhn et al. (1988); Kuhn & Armstrong (2002). An important feature to compare is the estimated contribution of facular regions to their energy budget. If the predicted contribution is excessive it will indicate that there is factor contributing to the solar irradiance cycle that is not included in these models.

To resolve this a careful analysis of facular regions and sunspots is needed.

41 Chapter 3

Structure Of the Solar Convection Zone

Here I include a paper which was published in the Astronomical Journal. The paper presents a calculation of the solar oblateness (quadrupole moment). By examining solar oblateness over a solar cycle it might be possible to detect generation of magnetic fields near the base of the convection zone. I will begin with a brief explanation of the relevance of the solar oblateness to the solar irradiance cycle.

The paper will follow this. The section will be concluded with a few estimates to understand the limitations on the predictive power of this type of work.

To understand the implications of the solar oblateness we begin with a derivation of Van Ziepel's theorem. A similar derivation may also be found in Dicke (1970).

Consider a non-rotating, free-floating mass of gas. Assume that the mass of this system is sufficient to hold it together. The condition of hydrostatic equilibrium is:

vp= -pv¢ (3.1)

Where p is the pressure, p is the density, and ¢ is the gravitational potential. By taking the curl of this equation we find that:

vpxv¢=o (3.2)

42 Thus the gradient of the density is parallel or anti-parallel to the gradient of the potential. Simple consideration (e.g. equation 3.1) shows that it is anti-parallel. Thus surfaces of equipotential are coincident with surfaces of equal density. Uniqueness theorem then requires that this solution to be the only solution. The same solution could be obtained by invoking isotropy. The center of the mass is chosen as the center of the coordinate frame of reference. A lack of a preferred direction in space implies that the system is symmetric with respect to any rotation about the origin. This approach has the added value of providing two additional results at no extra cost.

Since there is nothing to break the symmetry in the case of temperature and pressure, isotherms and isobars are also coincident with the spherical surface of equipotential.

To transform the problem to the case of rotating coordinates, we return to the approach in 3.1. All that is needed is to transform the potential to the effective potential. The result in this case is that surfaces of equal effective potential are coincident with surfaces of equal density. It should be noted that rotation could violate the assumptions here. This derivation only applies to rigid body rotation.

Remaining in the rotating coordinate system, we seek similar solutions for temperature and pressure. Ifwe take the cross product of the gradient ofthe potential with equation 3.1

V'¢ x V'P = -pV'¢ x V'¢ (3.3)

V'¢xV'P=O (3.4)

Thus isobars will also be coincident with surfaces of constant density, and effective potential. Finally we use the ideal gas law:

PV = Nkt (3.5)

43 or equivalently:

T = j.Lp (3.6) pk

Ifwe follow a surface of constant pressure, then we also follow a surface of constant density. The right hand side of equation 3.6 is invariant along a surface of constant pressure, and thus surfaces of constant temperature are likewise coincident with surfaces of constant potential, density, and pressure.

The next point to consider is the effect of magnetic fields on this derivation. As was discussed in chapter 1, the presence of magnetic fields alters the density and conductivity of the plasma in which it is embedded. This leads to the conclusion that understanding of the oblateness of the sun, and in particular the changes of this oblateness over a solar cycle, can give us important clues about the mechanism, which drives the observed solar irradiance cycle.

The following paper was published in The Astrophysical Journal in November of

1999, Volume 525 pages 533-538 (Armstrong & Kuhn 1999).

3.1 Abstract

Accurate measurements of the solar oblateness have recently been obtained from the

MDI/SoHO satellite experiment. The new data are sufficiently accurate to measure non-negligible multipole shape terms of higher order than the oblateness. Here we extend earlier solar limb shape calculations and compare the new data with the helioseismic evidence for a complex internal solar rotation profile. We find that the quadrupole (1=2 referred to as the oblateness) and hexadecapole (1=4) shape terms are marginally inconsistent with the solar rotation data.

44 3.2 Introduction

The precise shape of the Sun has been actively debated since Dicke and Goldenberg's

(1974) oblateness measurements. Although later observations were unable to confirm a large equatorial bulge (cf. Lydon & Sofia 1995; Kuhn et al. 1998, - KBSS) , many of the solar shape measurements have been interpreted as evidence that the Sun's oblateness may vary during the course of the solar cycle (cf. Rosch et al. 1996) . Lydon

& Sofia (1995) used a high altitude balloon experiment, while KBSS used satellite measurements from SoHO/MDI, to obtain consistent, and temporally constant, solar oblateness determinations.

Thus far, only the balloon and satellite experiments from above the Earth's atmosphere have achieved the required sensitivity to measure the expected small, higher order, solar limb shape. It is interesting that the Lydon and Sofia and KBSS solar hexadecapole shape measurements are not consistent. Perhaps this is a result of global solar changes due to the solar cycle, as the measurements were obtained at different phases of the magnetic cycle. Whatever the cause of this difference, the physical implications of the hexadecapole data for the solar interior rotation have yet to be fully analyzed.

The photospheric oblateness is, in principle, sensitive to the interior rotation rate

(d. Dicke 1970; Goldreich & Schubert 1968; Ulrich & Hawkins 1981; Paterno et al.

1996; Lydon & Sofia 1995) although, in practice, the observations have been too uncertain to provide important constraints on the interior rotation. In contrast, the solar rotation between a few thousand kilometers below the surface, down to about half a solar radius, has been measured helioseismically (d. Kosovichev et al.

1997; Schou et al. 1998). It appears to exhibit shear zones immediately below the photosphere and at the base of the convection zone, while there is apparently only

45 a small radial and angular rotation gradient beneath the convection zone. The helioseismic measurements provide only weak constraints on the solar rotation in the deep interior and are vague near the photosphere and at the poles (Schou et al.

1998; Kosovichev et al. 1997; Tomczyk et al. 1995; Thompson et al. 1996). It is likely that improved solar oblateness, and higher order solar shape multipole measurements, could yield useful constraints on the Sun's interior structure.

Previous calculations of the solar distortion were based on a quadratic expansion of the rotation in terms of fL, the cosine of the colatitude (Ulrich & Hawkins 1981).

To avoid truncation errors inherent to this non-orthogonal fL expansion, and to more accurately account for a complex interior rotation profile, we have computed the solar shape from an orthogonal Vector Spherical Harmonic expansion of the global interior acceleration terms. We note that because the solar rotation is not constant on cylindrical surfaces, the visual oblateness is not proportional to the gravitational potential oblateness adjusted by a rotational "effective potential" contribution. The interpretation of the surface shape is also complicated by the fact that surfaces of constant density, pressure, and potential do not exactly coincide. Here we will directly compare shape measurements to self-consistent calculations derived from our solution for the gravitational potential.

3.3 Non-Orthogonal Oblateness Calculations

(Ulrich & Hawkins 1981) compute the solar oblateness by extending an approach described by Goldreich & Schubert (1968). They used the Howard et al. (1980) photospheric rotation data and an expansion of the form (where Wo and W2 are empirically fitted coefficients)

2 W = Wo + W2fL . (3.7)

46 The solar shape follows from the equation of hydrostatic support,

(3.8)

Where P, p,

(3.9)

Here lowest order (spherically symmetric) quantities are indicated with a subscript 0, while perturbed (non-spherically symmetric) quantities are indicated with a prime.

We assume that the rotational acceleration is small compared to the gravitational term and that the perturbed density, pressure, and potential can be expanded in

Legendre polynomials of the form:

00 rJ(r, JL) = L pz(r)Pz (JL) (3.10) 1=1

Similar Legendre expansions III terms of the perturbed pressure PI and the perturbed potential 0):

__1_ (J2

Using eqns. (3.7-3.11) we recover eqns. (13) and (14) of Ulrich & Hawkins (1981), with the correction that -4/21 should be replaced by -8/21 in their eq. [13],

47 We assume that

To the extent that the interior solar rotation is well represented by eq. 3.7 this expansion would be adequate, although the last few years of helioseismic measurements have shown that there is significant structure in the rotation, particularly near the base of the convection zone (d. Kosovichev 1996; Tomczyk et aI.

1995) and in the outer layers near the photosphere (Wittmann 1996). In the light of these measurements, and because of improvements in the accuracy of the shape measurements, it is useful to improve and extend the oblateness calculations.

3.4 Vector Harmonic Solution

We start from the perturbed equation of hydrostatic support (eq. 3.9). We again assume that the pressure, density, and gravitational potential can be expanded as a series of Spherical Harmonics (actually Legendre polynomials, since we assume axial symmetry) We then expand the vector terms in Vector Spherical Harmonics (VSH ­

Edmonds 1957). With summation over l implied, one obtains:

48 (~-~) ~ (~ ~) ~ -J 2ll ++11 dr r PI 1,1+1,0 + J2l +l 1 dr + r PlYi,I-l,O =

dcPo [ (Ttl ~ rz-- ] PI dr V2l+lYi,I+l,O - V2l+lYi,I-l,O

(Ttl d l ~ rz-(d l+1) ~ ] + Po [V2l+T(dr - -;)(h 1,IH,O - V2l+1 dr + -r- (h 1,1-1,0

l+1 ~ (~ ;:;'\ - --POW X w X rJ (3.14) /¥J47f Multiplying this equation by ~,l±l,O and integrating over all solid angles gives:

d l ) dcPo ( d l ) 2l + 1 ---.,... -- PI = Pl- + Po - -- cPl - Po Al 1+1 0 (3.15) ( dr r dr dr r J47f (l + 1) ,, and

d l+1) d¢o (d l+1) 2l+1 PI Pl-+PO cPl+Po--All- (3.16) ( -+dr --r = dr -+--dr r yI4;l ,,10 where

211" [11" Aj,l,m == [ ~:l,m(e, cP) . wX (w X T) sin(B)dBd¢ (3.17) . 0 . 0

49 These equations may be solved for PI giving

2 r [dPo (~ + rpo A rpo A) _2l+1 A ] PI = GM dr cPI + dr £±2)r (V47rl 1,1-1,0 + J47r (l + 1) 1,1+1,0 V47rl Po 1,1-1,0 (3.18)

This can be supplemented with Poisson's equation (3.11), and solved for ~r~l to give:

2 47rr [dPo ( d l + rpo rpo ) _ 2l + 1 A ] +M(r) dr cPI + dr + -r-1) (J47rl AI,I-1,0 + J47r (l + 1) AI,I+1,0 J47rl Po 1,1-1,0 (3.19)

Here M (r) is the enclosed mass within a spherical shell of radius r. This ordinary 2nd order differential equation was solved using a Hamming predictor-corrector method.

The boundary conditions were met using a Newton-Raphson iteration procedure, with the assumption that the boundary at the surface depended upon the constant of proportionality for the boundary at r = 0, as in the non-orthogonal approach.

Once the gravitational moment cPI has been determined, the corresponding multipole density can be calculated using equation (3.18) or Poisson's equation (3.11).

The pressure multipole can also be calculated from the difference of equations (3.15) and (3.16):

=pA.+__Por A +-r====Por A (3.20) PI 0'1'1 J47rl 1,1-1,0 J47r(l + 1) 1,1+1,0

3.5 Solar Rotation and Model Data

We use solar rotation rates which were determined from MDI observations (Scherrer et al. 1995). Rotation rates are found through inversion of the helioseismic frequency splittings. This technique yields the only significant empirical constraint on the

50 2.0X10- 11

1.5X10- 11

/ 11 / 1.0X10- _J--~ I I 5.0x10- 12 I E I --; I - / <{ 0 /

-5.0x10- 12 A 450 -1.0X10- 11

-1.5x10- 11 0.0 0.2 0.4 0.6 0.8 1.0 Radius

Figure 3.1 Graphs of the vector harmonic components of acceleration vs. radius.

The scaled A2,1,O, A4,3,O, and A4,5,O are plotted with dash-dot, dashed and solid lines, respectively. The A2,1,O, A4,3,O and A4,5,O terms have been scaled by -1, 10, and 50 to fit them on the same vertical scale.

interior rotation, although it should be noted that the polar region and depths below

DARCo, are not well sampled by the acoustic inversions. Thus the helioseismic rotation

inferences are of limited precision in these regions (Schou et al. 1998; Kosovichev

et al. 1997; Thompson et al. 1996; Kosovichev 1996). Figure 3.1 shows the 3

lowest order non-zero expansion coefficients computed from eq. (3.17). Note that

A2,3,O = JIA 4,3,O'

The transition from rigid body rotation below the convection zone « 0.7RG ) to rotation which is nearly constant along radial contours is described here by the

sharply increasing magnitude of A4,3,O near O. 7~. The local maxima in the last two

51 acceleration coefficients occur near the shear zones in the upper and lower boundaries of the convection zone. This has been noted for example, by Tomczyk et al. (1995) in the rotation rate profile they describe.

A VSH expansion of the acceleration also has the virtue of converging rapidly.

Fig. 3.1 shows that successive higher order terms decrease by nearly an order of magnitude over most of the volume of the Sun.

We are using a solar model generated by Christensen-Dalsgaard (1992). As we discuss below there are numerical and solar model sensitivities that we have not fully explored which originate from the near-surface layers. To minimize this sensitivity we define our reference solar surface at a radius of 0.995R0 .

3.6 Results and Discussion

Contours of temperature, density, or pressure should be nearly coincident near the photosphere. Differential rotation, magnetic fields and turbulent pressure are the largest local acceleration sources which violate von Zeipel's theorem (d. Dicke 1970).

Since these asphericities are relatively small in the Sun, we may describe the radius of, for example, density contours with:

R(JL) Ip=constant= Ro + L dz(Ro)p,. (JL) (3.21) z where Pz is a Legendre Polynomial of degree land Ro is the mean contour radius.

Here dl = - pz/dP

52 potential defined by J1 = Kl where K = R(')/GM(') at the solar surface. All shape coefficients here and below are expressed in units of the solar radius.

By solving equation (3.19) for the potential coefficients 2 and 4, the shape of density and pressure surfaces can be determined from eqns. (3.18), (3.20), and (3.21).

Figure 3.2 compares the density oblateness, d2 , obtained from eq. (3.7), analogous to the Ulrich and Hawkins approach, to the VSH technique. The two solutions are in reasonable agreement until r ~ O.6R('). After about r ~ 0.7R('), which is the base of the convection zone and the region where the rotation begins to depart from the form of eq. (3.7), the solutions differ appreciably. Above r ~ 0.97R(') there is significant disagreement between the two approaches due to the fact that the rotation there is not well described byeq. 3.7.

While the fractional deviation between the two computed density oblateness calculations is small (~15% ) it can be misinterpreted to imply a large error in the gravitational oblateness, J2 . We note that the visual oblateness, or fractional pole-to-equator solar radius variation, is often described by

(3.22)

where Ds is an ill-defined "characteristic" surface rotation rate Paterno et al. (cf.

1996). This result assumes that rotation is constant on cylinders. Since the second term of the RHS of eq. (3.22) is much larger than the first, a small change in the oblateness could have a relatively large effect on the implied J2 • As we show below eq.

(3.22) gives a poor description ofthe relationship between the shape of solar density or pressure surfaces and the potential oblateness, J2 , because of the complex solar rotation profile.

53 "\ \

\ N \ U " "\ " ~, , \ \ ) \ \ /

- 6 x 10 - 6 ~~------'------'---'-----L-----L-----'------'------'------L--'------'------'------'----'------'------'-----'----' 0.0 0.2 0.4 0.6 0.8 1.0 Radius

Figure 3.2 The variation of d2 with radius. The solid line shows the density oblateness computed using the VSH approach, and the dashed line shows the corresponding polynomical expansion.

54 Table 3.1. Shape Coefficients and Gravitational Oblateness

Coefficient l=2 l=4

6 6 dl -5.87 X 10- 0.616 X 10- 6 6 S1 -5.84 X 10- 0.593 X 10- q -5.27 ± 0.38 x 10-6 1.3 ± 0.51 x 10-6 6 9 JI -0.222 X 10- 3.84 X 10-

Density and pressure surfaces (defined by dl and SI) also differ the most at the base of the convection zone and surface. Figure 3.3 shows the deviation between l = 2 and l = 4 density and pressure components. Surface boundary condition errors appear in the calculations above r = 0.998R0 because of the vanishing density terms in eq. (3.18). To avoid these errors we evaluated our tabulated results (below) at r = 0.995R0 . Table 3.1 summarizes the density, pressure, and potential contour shape coefficients from the VSH expansion. For comparison, the visual oblateness and hexadecapole (C2 and C4 ) derived from MDI measurements (KBSS) are also listed.

These calculations used the interior rotation data from Fig. 3.1.

One measure of the uncertainty in the calculated shape coefficients follows from a comparison of the VSH calculations to the solution from rotation on cylinders eqns.

(3.12) and (3.13) using a rotation profile of the form of eq. (3.7). Coefficients wo(r) and w2(r) were obtained by fitting equation (3.7) to the higher resolution helioseismic interior rotation data. This quadratic form was then used to define a depth dependent rotation profile that was used in eqns. (3.12), (3.13) and (3.19) (VSH). Table 3.2 shows the derived shape coefficients from the VSH approach and the difference between these and the non-orthogonal calculation values. We note that Pijpers (1998) has also computed J2 (with a very different formalism) from the MDI rotation data to

55 Figure 3.3 (a) Graph of d2 and 82 versus radius. The density and pressure surfaces are represented by the solid and dashed lines, respectively. (b) The variation of d4 (solid line) and 84 (dashed line) with radius.

56 Table 3.2. Shape Coefficients and VSH-Eq. (6) Sol. differences for Quadratic Rotation Law

Coefficient [= 2 [=4

-5.09 - 0.1 x 10-6 5.50 - 0.7 x 10-7 -4.84 + 0.07 x 10-6 5.13 - 0.9 x 10-7 -0.226 - 0.002 x 10-6 3.34 - 0.4 x 10-9

6 obtain -0.223 x 10- - in good agreement with our J2 calculation in Table 3.1 (after accounting for a difference in sign convention).

The third row of Table 3.1 shows the measured quadrupole (C2) and hexadecapole

(C4) solar limb shape from MDI/SoHO data (KBSS). These results were obtained by measuring small displacements in the solar limb darkening function- as such, the derived Cl coefficients should be comparable to the isodensity surface shape described by dz (and since the difference between density and pressure shape surfaces is smaller than the measurement errors, we will ignore this interesting distinction).

The coefficient differences indicated in Table 3.2 must result from numerical errors, since the VSH and eq. (3.12) calculations used identical quadratic rotation laws.

Thus, these differences represent a crude estimate of the systematic numerical errors.

It is notable that both the [ = 2 and [ = 4 "errors" are significantly smaller than the

7 statistical measurement errors. Our J2 numerical error, 0.02 x 10- , easily brackets the difference between the Table 3.1 J2 calculation and Pijper's. The agreement between Pijper's integral over his semianalytic J2 kernal and our differential solution for the potential function lends confidence to both calculation techniques.

Unfortunately Jz is not directly observable- it is the surface shape distortion which we measure and the helioseismic rotation data imply a surface oblateness which is

6 0.6 X 10- larger in magnitude than the quadratic rotation shape implications and

57 over 1.5 standard deviations (measurement errors) larger than the observations. The hexadecapole shape distortion implied by the helioseismic rotation profile is also too small by over 1.5 standard deviations in comparison to the measurements. Evidently the shape measurements are marginally inconsistent with the rotation data.

Equation (3.22) is often invoked to connect the surface rotation and oblateness to the potential distortion, J2 . To realistically account for the data in Table 3.7 Ds must correspond to a rotation period longer than 30 days, and the interior rotation would have to be constant on cylinders - in direct conflict with the helioseismic results.

The helioseismic data imply a surface rotation rate that is in conflict with the

Doppler photospheric rotation data. Thus, we explored rotation models that smoothly match the surface and interior measuremeents. This was done by interpolating the observed surface rotation (Howard et al. 1980)

2.82 - O.33,i - 0.53/-l4 (3.23)

(in units wadis) to match the helioseismic interior rotation on a spherical surface of radius Re. A grid of rotation data was computed from a linear interpolation in the radial direction from Rc (using the helioseismic data) to the Doppler data at the surface. Table 3.3 summarizes the derived density and potential coefficients. Even interpolating from 0.95R(') (where the helioseismic results are probably reliable) up to the surface does not reduce the oblateness enough to obtain formal one (J consistency with the observations. The change in the hexadecapole is also insufficient, and is in the direction to increase the l = 4 discrepancy. It is clear that the observed surface rotation cannot reconcile the helioseismic rotation data with the shape measurements.

The effect of solar core rotation on the surface shape has also been considered by

Paterno et al. (1996). We used their core rotation model (their eq. 12) with central

58 Table 3.3. Shape Coefficients with Surface Rotation Matched Model

0.95 -5.79 x 10-6 -0.222 X 10-6 5.84 X 10-7 3.84 X 10-9 9 0.99 -5.87 x 10-6 -0.222 X 10-6 5.59 X 10-7 3.84 X 10-

rotation rates as high as six times the standard model and found that the density

6 oblateness, d2l increases by only 0.16 x 10- above the results in Table 3.1. When the core rotation was 1/10 of the standard model d2 decreased in magnitude by only

6 0.002 X 10- . It is evident that extreme changes in the core rotation would be needed to significantly affect the surface oblateness.

Finally, we have also explored the effects of radial and latitudinal rotation rate perturbations on the derived surface density shape. Table 3.4 shows how the oblateness and hexadecapole shape vary in response to a 10% increase in the rotation rate localized in radius or colatitude. TheBe coefficients were computed by scaling the rotation by a Gaussian function of the form

(3.24)

or for latitudinal changes by

()-() w'(r, ()) = w(r, ()) x (1 + 0.1 exp(-7C ?) x O.lr/R (3.25) 0.1 0

Equation (3.25) is applied to the region 0 :S () :S ~. The rotation rate for ~ :S () :S 1f is then constructed by reflection through the equator.

The results in Table 3.4 suggest that changes in the rotation rate by about 10% in the midlatitudes in both the northern and southern hemispheres could generate a

59 Table 3.4. Shape Coefficients with Localized Radial and Angular Rotation Perturbations

6 7 Rc Oc(deg) d2 (x1O- ) d4 (x1O- )

0.5 -5.91 6.17 0.7 -6.24 6.53 0.9 -6.97 7.32 0 -5.94 5.55 30 -6.42 4.08 60 -6.72 8.80 90 -6.10 7.92

hexadecapole shape which is consistent with the observations. Other perturbations

(i.e. reductions) in the rotation profile of the same order would be needed to reduce the oblateness to match the observations. Without additional constraints there appears to be no unique solution for the interior rotation that recovers both the oblateness and hexadecapole shape.

3.7 Conclusion

Measurements of the quadrupole and hexadecapole limb shape are mildly inconsistent with current solar rotation models. The discrepancy does not appear to be resolved by the known mismatch between the helioseismic surface and Doppler surface rotation measurements. It is unlikely that the core rotation uncertainty will remove the difference. Progress on this problem depends on improved measurements of the limb shape, and a better understanding of the 2-dimensional solar interior rotation in the convection zone.

60 3.8 Acknowledgements

We are grateful to NASA and the SoHO/MDI project (grant number NAG5-3077).

This work was also supported by the NASA SRT program through a grant to JRK.

We thank Jesper Schou for providing the helioseismic rotation data and Jorgen

Christensen-Dalsgaard for the solar model data.

3.9 Discussion Of The Paper

Rozelot et al. (2003) has referred to oblateness observations as an observational window into the sun. While accurate measurements, combined with careful computations can provide insight into the solar interior, care must be taken. We now can consider some limitations on what may be inferred from accurate measurements of the solar oblateness.

Setting aside stated concern, we consider that in equation 3.22 we use the

6 calculated value of -2.2xlO- for J2 . If we chose n to correspond to the rotation rate of 27 days, then we find that the contribution of the second term on the right

7 hand side is 2.36 x 10- which is comparable to the contribution from the J2 term.

Thus if we if have an accurate measure of the surface rotation rate and the surface oblateness we would, in principle, know the gravitational moment of the sun.

The presence of magnetic fields at the base of the solar convection zone changes the assumptions that were made in deriving Van Ziepel's theorem. Magnetic fields have two primary effects which might be detectable with solar oblateness observations.

First they create density perturbations. Second they alter the rotational profile in the solar interior.

61 Magnetic fields contribute to the pressure support of the plasma in which they are embedded. Thus if the magnetized plasma is in thermal equilibrium with the surrounding plasma, then pressure equilibrium will require that the magnetized plasma will have a lower density than plasma free from magnetic fields. Since sunspots primarily appear at mid-latitudes on the solar surface we have a strong argument that the magnetic fields that are generated at the base of the solar convection zone will also primarily be located in these mid latitudes. This is in agreement with simple consideration of the process that generates the magnetic fields. Rotational shear is necessary to enhance the seed poloidal fields by generation of toroidal fields.

Symmetry tells us that rotational shear at the solar equator should near zero. Since the magnetic fields are generated in the mid-latitudes the expected effect of the magnetic fields, due to pressure perturbations would be to alter the hexadecapole moment of the solar surface

A larger and thus more detectable effect would be the interaction ofmagnetic fields with the solar rotation. As magnetic fields are generated by shear in the solar rotation rate, energy is converted from kinetic to magnetic. The generation of magnetic fields serves as a breaking mechanism to slow the faster rotating solar equator. Presumably this effect will be most apparent in the quadropole moment of the sun.

By studying the variation of the solar moments we can gain an insight into the generation and strength of magnetic fields as they are being generated at the base of the solar convection zone. This can lead to a greater understanding of the solar cycle.

62 Chapter 4

Small Magnetic Flux Tubes - Facular

Regions

4.1 Introduction

In this section I will discuss some of the properties of small magnetic features on the solar surface. While the term "magnetic" flux tube is not strictly interchangeable with faculae I will often refer to flux tubes as faculae. In reality magnetic flux tubes form a continuum of features which range from flux tubes to micropores and pores.

One might even view sunspots as the largest member of this family.

Large regions of diffuse magnetic field have been observed at the solar surface. (cf.

Berger & Title 1996) These regions of magnetic field are known to be made of small

(650 km or less) bundles of intense magnetic field (on the order of 1-2 kG, (c.f. Howard

& Stenflo 1972; Stenflo 1973; Spruit & Roberts 1983; Lin 1995) protruding through the solar surface. Berger & Title (1996) liken these magnetic fields existing in the convective down-flows to high surface tension immiscible oil drops on a corrugated surface. The oil drops flow into the troughs of the convective down-flows found in the inter-granular lanes. There they are confined by the gas pressure and collected by the

63 inward flow to a downward flow. As convective cells form and dissipate the magnetic fields are dominated by the merging surface flows, joining becoming deformed and breaking apart.

Since the surface of the sun is very accurately described as a sphere (deviating by

5 less than 1 part in 10 ), and it rotates with a period of roughly 27 days at the equator we can observe flux tubes from nearly 21r sterradians. (Observation at and near the limb of the disk becomes more difficult as the flux tube is foreshortened.) Near the disk center they are seen from overhead while near the limb they would, in principle be seen from the side.

The viewing angle can introduce a complication in companng models to observations. Many magnetic field observations, such as the ones taken by SOHO, are line of sight magnetic field measurements; only the projection of the magnetic field onto the line of sight is measured. A geometric configuration must be assumed to infer the total magnetic flux density. Typically it is assumed that the magnetic field at the solar surface is normal to the surface. This effectively assumes that the magnetic field has a configuration like a pencil sticking out of the surface of a soccer ball, however, it is known that the magnetic fields become splayed at or near the solar surface.

Observationally, groups of small magnetic flux tubes are seen collectively as facular regions. Individual flux tubes are not resolved. Instead we observe the effects of the flux tube diluted by nearby convective cells and inter-granular lanes. Thus a large facular region is actually a collection of magnetic flux tubes, which cannot be resolved by current magnetic field observations.

The lack of resolution introduces other complications when comparing observations to models. Magnetic field measurements determine the total magnetic flux. Measurements are unable to distinguish between one relatively large flux tube

64 and two smaller flux tubes in close proximity if the large flux tube has the same magnetic flux as the two smaller ones. A similar dilution arises when observing the brightness. A single small very bright point would be indistinguishable from a weaker more diffuse bright point (d. Lin & Rimmele 1999).

A final complication arises. If we consider different sized flux tubes to be part of a large family of solar surface features, then we must include sunspots. The Hale polarity law indicated that leading sunspots have one polarity and that the magnetic field stretches to the following spots. This might imply that flux tubes with one polarity would be inclined toward the east, while flux tubes with the opposite polarity would be inclined toward the west. This trend would then be reversed in the opposite hemisphere. Thus flux tubes that are seen at disk center are not actually seen end­ on. The flux tube is in fact inclined to the normal to the solar surface. This effect was discovered and discussed by Topka et al. (1992). They found that the degree of inclination was approximately 10°.

The contemporary picture of an individual magnetic flux tube is the hot floor model. The magnetic pressure creates a region of decreased density at the solar surface. The decreased density leads to a decrease in opacity and thus we effectively

"see" further into the solar interior. We can then create a simple yet surprisingly predictive picture of a flux tube by considering a simple depression of the solar surface.

(The depth of the depression is referred to as the Wilson depression, Zw.)

We consider the solar surface to be a discrete surface instead of the true continuum surface that exists. This is justified by consideration of the Eddingtion­

Barbier approximation. The Eddington-Barbier approximation assumes that a linear relationship holds between the opacity, (7), and the source function, (SA)' By making this assumption we find that the emerging flux is equivalent to a discrete surface at one optical depth, (7 ~ 1). The approximation can be further justified by realizing

65 that the optical depression (i.e. the depression of the optical surface of the sun) is roughly equal to the Wilson depression. Thus the surface of the sun approximately traces a surface of constant density.

Spruit (1976) constructed a complete model of faculae, including a full radiative transfer treatment. His complete treatment resulted in a simplified model. The iconic picture of a flux tube is a cylindrical depression in the solar surface is the result of this study.

The important result of this study is that while the magnetic field extends vertically below the floor of the flux tube, it does not extend beyond the walls of the tube itself. Thus convection is not inhibited and a substantially higher heat transport can be supported. This leaves the walls of the flux tube hotter than the surrounding photosphere. In small flux tubes, the walls can heat the floor of the flux tube. Thus the temperature of the floor of small flux tubes can be maintained. This implies that there is a continuum of possible sizes of flux tubes in which the floors of small flux tubes is hotter, and therefore brighter than the surrounding unperturbed photosphere, and larger flux tubes in which the floor does not receive sufficient energy from the walls of the flux tube to prevent it from cooling and becoming darker.

The distinction between large and dark flux tubes would manifest itself in the observations. Large flux tubes would appear dark at disk center while smaller flux tubes would appear slightly brighter, or would disappear near disk center. This effect is evidenced in the literature with some authors claiming that faculae, or flux tubes are slightly brighter at disk center (c.f. Hirayama et al. 1985; Lawrence 1988), while other authors find that flux tubes are dark at disk center (c.f. Topka et al. 1992;

Lawrence et al. 1993).

At other positions on the solar disk flux tubes would have a different appearance

(see figure 4.1). Toward the limb of the sun the floor of the flux tube disappears

66 behind one of the walls of the flux tube. At this, and larger, viewing angles, only the far wall of the flux tube can be seen. The angle at which this occurs depends on the size of the flux tube. In the case of a small flux tube the floor is only visible from nearly overhead. In larger flux tubes the floor is seen at larger angles. This should be visible in the data. Large flux tubes will have a greater range of contrasts than smaller flux tubes, which will appear to have a weaker dependence on viewing angles.

Topka et al. (1997) tested the idea of the bright wall model on unresolved images of flux tubes. They constructed a model of the contrast of magnetic flux tubes by assuming flux tube diameters from 350-650 km in diameter. The number density of these flux tubes was taken to be a log linear relationship with the diameter of the flux tubes. An ensemble average of the predicted brightness was then calculated for several viewing angles. This ensemble average was compared to observations of solar faculae. They found that agreement between observations and the hot wall model was excellent. They compared the model to observations on continuum images obtained, and found that a Wilson depression of 100 km was consistent with the data, independent of flux density. This is in agreement with the concept that the magnetic flux density in these features has a constant value and is diluted by limited resolution.

I examined data from the Precision Solar Photometric Telescope (PSPT) in conjunction with nearly simultaneous magnetograms taken by the Michaelson Doppler

Imager (MDI) on board the Solar and Relioseismic Observatory (SORO) to gain a greater understanding of the relationship between smaller magnetic fields and contribution to the solar irradiance. From this study I was able to confirm some of the considerations discussed above.

67

Cok:j Fbor Warm Floor

Figure 4.1 The simple hot wall model accurately describes the limb brightening profile of facular regions. The model assumes that a flux tube is a simple cylindrical depression in the solar surface. The walls are hot. The walls heat the floor. Larger faculae have cold floors since the walls are unable to heat the larger area. Small faculae have warm floors.

6 4.2 Data and Analysis

The MDI provides 1024 X 1024 pixel full disk line of sight magnetic fields of the sun. Thus a single pixel corresponds to approximately 2". (I" corresponds to 725

Km at the sun.) Magnetograms are taken with a cadence of 96 minutes, allowing for nearly simultaneous comparison to the PSPT images. For a complete description of the SOHO instrumentation the reader is referred to Scherrer et al. (1995)

The PSPT images are seeing limited, with a pixel scale 1". The PSPT takes full disk images using a 2048 X2048 CCD camera. Five images are summed resulting in images with statistical photometric errors of less than 0.1% per pixel. The images are taken in three narrow wavelength bands, red (606.7 - 607.2 nrn), blue (409.3 - 409.6 nm) and CalIK (393.1 - 393.4 nm). The red band was chosen to avoid any spectral lines, while the blue band minimizes line contamination. Thus this photometry can reliably provide a measure of the local surface temperature (See figure 4.2), while the

CaIIK photometry is a known tracer of photospheric magnetic field.

Limb darkening was removed by the PSPT standard analysis (c,f. Rast et al.

2001), using the following formula:

R = I - Iz(p,) (4.1) 10 where I is the local intensity, II(p,) is the quiet sun estimate, p, is the cosine of the angle between line of sight and normal to the surface (often referred to as the viewing angle), and 10 is an estimate of the total quiet sun intensity, effectively the integral

6 of the irradiance over the solar disk, divided by 10 .

CaIIK is believed to act as a tracer of total magnetic field. Typical studies of facular contrast have used line of sight magnetic field measurements. To convert the line of sight magnetic fields to total magnetic flux density an assumption is made

69 0.05

E - c 0.00 c-

-0.05 c-

-0.10- - u OJ -0.1 5 - - 0::::: -0.20 -0.30 -0.20 -0.10 -0.00 0.10 Blue - 409.6nm

Red residual vs. Blue Residual

Figure 4.2 Red and Blue residual images are measures of the solar surface temeperature. The points each represent an individual pixel from PSPT data. The line is the theoretical computation of blue residual from the red residual based on a blackbody spectral distribution.

70 about the geometry of the magnetic field. Typically, the line of sight is divided by the cosine of the viewing angle, J.l. This is the assumption that the magnetic field is normal to the solar surface.

To test the assumption that CaIIK measures the total magnetic field, as opposed to line of sight magnetic field, the distribution of CaIIK features was examined. If we consider the distribution of surface magnetic features as a function of the density of magnetic flux, this distribution should not vary with longitude. We can then detect any projection effects by examining the distribution of features as a flillction of the viewing angle, or equivalently J.l.

The mean and standard deviation of CaIIK residual for pixels lying between approximately 20° and 40° latitude in both the northern and southern hemispheres was computed as a function of J.l, combining data from 4 PSPT images, all taken from September and October of 1999. The selection of latitudes was chosen on the observation that these are the bands where the majority of solar activity occurs.

Two tests were performed. First we examined all pixels with a CaIIK residual greater than -0.05. The lower cutoff was applied to filter out sunspot regions. For the size of the data set considered sunspots would be found in only a few locations.

The dark sunspots would skew the data. In the second test, only pixels with a CaIIK residual of greater than 0.1 were included in the computation. By including only pixels with a CaIIK residual of greater than 0.1, the sensitivity of the test was found to increase. Effectively we can think of computing the mean and standard deviation of pixels with a CaIIK residual of greater than 0.1 as measuring the length of the distribution beyond 0.1, and the initial fall off rate (respectively). When all data points are included the sensitivity is lower since the portion of the distribution which corresponds to thermal and stochastic fluctuations are included. Elimination of the thermal variations increases sensitivity to the magnetic features.

71 The results of these tests are shown in figure 4.3. The first test, which included all pixels with a CalIK residual of greater than -0.05, shows no variation with respect to /L. The second test shows slight variation with /L. This confirms the increase in sensitivity, which was obtained by limiting the range of CalIK values, however this trend is small (of order less than 0.01). It was concluded that for the purposes of this study, effectively no trend exists. This indicates that CalIK does trace the total magnetic field, and concerns for effects such as limb darkening have been effectively corrected for in the data.

Since the CalIK residual data has a higher resolution than the magnetic field data, the CalIK data was degraded to match the resolution of the magnetic field data. This was done by under-sampling the CalIK residual data. The data as then binned by magnetic flux density and /L, and the filtered mean of the CalIK residual was taken.

The filtered mean is simply the mean of the middle 95% of the data. The results of this construction is shown in figure 4.4.

CalIK residual can be seen to be an effective tracer of the line of sight magnetic flux density. Near the limb, the relation has the appearance of a broad "U" shape. Near disk center the relationship is tighter, forming more of a 'V" shape. The difference in the functional relationship between disk center and the limb is due to foreshortening of the line of sight magnetic field measurements combined with observation of the solar magnetic canopy.

Examining the magnetic flux density, for constant values of CalIK as a function of /L allows us to examine the assumption that total magnetic flux density may be constructed by dividing the line ofsight magnetic field by /L. The relationship between line of sight magnetic field and /L is shown in figure 4.5. For small CalIK residuals the magnetic field remains roughly constant. For larger values of CalIK residual, and consequently stronger magnetic fields the magnetic field shows an increasing linear

72 Measures of ColiK Distribution 0.10

0.08

o :J U 0.06 (J) Q) cr: ~ 0.04 o U

0.02

o.00 L--..'--'--'------''------''------'L----.'-----'-----'-----'-----'--'--'---'--'--'--'---'--'-----' 0.0 0.2 0.4 0.6 0.8 1.0

Figure 4.3 The mean and standard deviation of CaIIK residuals are shown here. Solid lines show means, while dashed lines show standard deviations. Lines with *'s include only pixels with a CaIIK residual of greater than 0.1, while lines without symbols include all pixels with a CaIIK residual of greater than -0.05. The more sensitive cut of 0.1 shows some trend as a function of J-l. However, since the trend is similar in both the standard deviation and the mean, it is believed to be an artifact of incomplete limb darkening removal.

73 Line of Sight Magnetic Field Line of Sight Magnetic Field 0.15 0.15 p..=025 p..=0.35 0 :J u 0.10 0.10 'Vi / QJ \ Q:' ~ a 0.05 0.05 ~I U

I 0.00 " ..J 0.00 -200 -100 0 100 200 -200 -100 0 100 200

0.15 0.15 p=0.45 ~"- p=0.55 I" \ --" "- \- ~ ( 0 ,v,, I~\ hoi "- :J to' u 0.10 v .... ~ 0.10 -II 'Vi /'1 QJ 6"/ a:: Y: 0 0.05 0.05 u

0.00 0.00 -200 -100 0 100 200 -200 -100 0 100 200

0.15 0.15 ~ I p=0.65 /, \ p=0.75 /,r \I II ,1:- -1"' t ./ "'-, \ ;~..- / ," "'~ :;'" 0 ? :J ! ~ ~ u O. 10 0.10 , Ul " QJ n:: '"1\ Y: 1 0 0.05 0.05 u

0.00 0.00 -200 -100 0 100 200 -200 -100 0 100 200 0.15 0.15 ~::: p=0.85 p=0.94 .... Itr1 ~" '" _'" ~ A -0' 0 -0/ :J " ~ u 0.10 "" 0.10 :: 'iii "" QJ " n:: Y: 0 0.05 0.05 u

0.00 ~~~~~~=~~~~~...... J 0.00 ~~~~~~~G-..-~~~~~ -200 -100 0 100 200 -200 -100 0 100 200 Line of Sight Mognetic Field Line of Sight Mognetic Field

Figure 4.4 CaIIK is shown to be an effective tracer of magnetic flux density. CaIIK is shown as a function of magnetic flux density, for varying values of p. Near disk center the magnetic field is seen to have a "v" shape with a very small region where the slope is nearly flat and thus CaIIK is unresponsive to increasing magnetic flux density. Near the limb, The functional form is more accurately described as a "u" shape. 74 dependence on JL. By examining this relationship it can be seen that the magnetic field is not simply foreshortened by JL.

Since it has been shown that the total magnetic flux density is measured by the

GaIIK residual, this shows that the standard geometric assumption is incorrect. The magnetic fields are in fact divergent near the surface.

While it is a disadvantage of using GaIIK that the GaIIK residual is not linearly related to the magnetic field, the fact that CalIK traces total magnetic flux density is an advantage, which allows for a qualitative study of surface magnetic features. In particular, a better association of the features, with a given magnetic flux density, to their corresponding continuum brightness can be made. Thus we are able to compare similar features at various viewing angles. Blue continuum residual as a function of the GaIIK residual, and JL is shown in figure 4.6.

4.3 Comparison To Models

This first conclusion that can be drawn is that strong magnetic regions do appear dark at disk center, while weaker magnetic regions appear bright. Since it is believed that the density of magnetic flux is roughly constant, the magnetic flux is dependant only on the area of flux tubes:

F=BxA (4.2) where F is the magnetic flux, B is the magnetic flux density, and A is the area, we can understand that a stronger magnetic field measurement i'l-ctllally indicates more flux tubes or a larger single flux tube. The measurements average the flux density over the entire pixel. The observation that features with a high obered magnetic flux

75 Projection of Line of Sight Magnetic Field 200

-0 Q) LL u 150 -+--' Q) C CJl o :2: 100

'+- o 50 Q) C ------.J ------

Ol---'------'------'------'------'------'---'-----.J_L---'------'------I---'------'------'----'_L---..-'---~ 0.0 0.2 0.4 0.6 0.8 1.0

Figure 4.5 The dependence of line of sight magnetic flux density is shown here. Each trace represents a bin of CaIIK residual with a width of 0.02, ranging from a CaIIK residual of 0 to 0.16. For small values of CaIIK residual the line of sight magnetic flux density is nearly independent on f.-l. For increasing CalIK residual the magnetic field gains a stronger dependence on the viewing angle.

76 Blue Residual vs. p 0.030

0.020

0 :=J .-U (f) Q) 0:::" 0.010

Q) :=J m- 0.000

- 0 .0 10 '-----'-----'------'--'-_-'------,--I.------'------'------J._-'---'------"----'------''------'-----.J 0.2 0.4 0.6 0.8 1.0 fL

Figure 4.6 The line of sight dependence of blue continuum is show here, as a function of CalIK residual. Each trace represents a bin of CalIK of width of 0.02. As CalIK residual increases, facular limb brightening increases. Thus near the limb, features with stronger magnetic field appear brighter when compared to features with a weaker magnetic field, or areas with no magnetic field. Near disk center, features with strong magnetic field appear darker than regions with little or no magnetic field.

77 density are dark at disk center supports the idea that there is a preponderance of large magnetic features being observed.

If one considers figure 4.6 to be a set of lines with an increasing CaIIK index, another qualitative feature of facular regions can be seen. This set of lines intersects near p,=0.9. As the magnetic field - as indicated by CalIK - increases, the lines rotate, becoming steeper. Effectively, the primary action of magnetic fields at the solar surface is to redistribute the emerging energy in an angular fashion. The stronger the magnetic field the stronger the anisotropy in the radiated energy.

The secondary effect is to increase the energy, which is radiated locally. By integrating the irradiance over 21f the total energy budget from region equivalent to one pixel at disk center can be constructed. This is shown in figure 4.7. This energy budget must be taken with caution. Spruit (1976) states that the bright wall model does not account for a faint dark ring, which would surround the flux tube. He continues by stating that the deficit energy from this dark ring might be sufficient to cancel the excess energy radiated by the flux tube in the bright wall model. A second speculation of this was made by Deinzer et al. (1984).

To understand this we should more carefully consider the energy flow indicated by the bright wall model. For a large flux tube the floor of the model is colder than the surrounding photosphere. Thus the energy flow to the solar surface is disrupted, and less energy is traveling directly upward.

In contrast the walls of the hot wall model provide for an increase in energy flow from the surface, and in smaller flux tubes, it is the walls that heat the floor of the flux tube. This indicates that the excess energy is diverted from an upward flow to the surrounding solar surface. This diverted energy becomes a deficit to the upward flow in surrounding regions. These surrounding regions then become cooler as they receive less heating from below.

78 Integrated Blue vs. CaK Residual 0.006

o ::J 0.004 V UJ \ ;\ /' ill /' \ ; \ /' \ 0:::: /' \ \I \ (]) 0.002 \; \ ::J / / \ OJ / \ /' \ V /' ill 0.000 -+-J o L 0' ill -+-J C -0.002

- 0 .00 4 '------'---..L--'-----'-~'----'---~'______'_---'------''______'____'______'____'___L__.1____'___L__.1____'____'______'____'____'______'____'____'______' 0.0 0.1 0.2 0.3 CaK Residual

Figure 4.7 The energy excess or deficit as a fraction of quiet sun luminosity is shown as a function of CalIK. Weak CalIK is associated with small flux tubes while stronger CaIIK residuals indcate larger flux tubes. Small flux tubes are associated with a local increase in brightness. As small flux tubes become larger they are associated with larger local increases in brightness. In large flux tubes, the "floor" of the flux tube is no longer be heated by the walls. Eventually a point is reached where the cold floor dominates the redistribution of energy, and the flux tube decreases the local surface brightness. The solid line was computed using data taken near solar maximum. The dashed line was constructed using data taken 18 months earlier to approximate solar minimum.

79 If this is the correct picture then facular regions would not contribute to the time averaged solar irradiance. As the magnetic fields are carried across the solar disk they would represent no time averaged increase (or only a slight increase) in the energy received at the earth. Instead facular regions act to redistribute the solar radiation in a spatial and angular fashion. The angular redistribution is evidenced as limb brightening of facular regions. The redistribution of energy in a spatial fashion would be evidenced in a faint dark ring surrounding faculae, which is currently below detection levels. This spatial redistribution of energy can be thought as nothing more than a preferred channel for the energy to escape the solar surface.

At this point we may make a comparison to the proxy model of Preminger et al.

(2002). It should first be noted that the CalIK filter which was used to take the images used by Preminger et al. (2002) has a width of 1nm, compared to the 0.3nm width of the filter used on the PSPT. This implies that some leeway must be given for differences in the data. Additionally argument may be made that since the curve shown in figure 4.7 is the full 3D energy budget it might not compare directly to the irradiance. This effect is actually small, but more leeway can be conceded.

Preminger et al. (2002) gives a regression constant of 0.11 to 0.19 for CalIK for various fits to the solar irradiance. This constant can be thought of as a conversion factor between CalIK residual for facular regions, and full spectrum continuum residual. Using the equation which was used to construct figure 4.2 the blue continuum can be related to the full spectrum continuum. This implies dividing the conversion factor by 1.5 which arises from assumption of a black body spectrum.

The peak of the curve shown in figure 4.7 reaches a value of 6 x 10-3 at a CalIK residual of 0.16. Using the conversion factor from Preminger et al. the implied blue

2 2 residual would range from 1.1 x 10- to 2.0 X 10- , or a factor of nearly 2 to 3 times higher than the observed blue residual. This type of overestimate was discussed in

80 chapter 2. It is a strong indicator that there is a component of the solar cycle that is not accounted for, but is correlated to observed variables.

Much of this discussion, so far, had ignored an important feature of magnetic flux tubes. That feature is that these surface magnetic fields are found in the cooler down- flows of the inter-granular lanes. A simplified picture can be constructed. On can consider a flux tube to be a thread of magnetic field, which penetrates into the solar interior. Magnetic field lines are closed. Thus a down-flow of ionized plasma cannot persist inside the magnetic thread. If the inward flowing material were sufficiently de-ionized to allow for a significant down-flow into the thread, then the pressure to hold the magnetic field in pressure equilibrium would not be coupled to the magnetic field. The magnetic field would disperse. Thus we rule against a significant down- flow inside of the magnetic thread, and conclude that a sheath of cooler downwardly flowing material must surround the magnetic thread.

The question that should be considered is whether this sheath of cooler down-flow is of sufficient dimension to become optically thick, in which case the energy flowing in from the walls of the faculae comes from the downwardly flowing material, or if the sheath is optically thin and the energy is diverted from the warmer material flowing toward the solar surface. It is tempting to say that since the magnetic thread lies in the cooler down-flow the energy must come from the warmer up flows, since the walls are, by definition, hot. This is a correct argument, which can be seen by considering the rate at which temperature increases with depth into the solar interior.

One can relate the temperature scale height to the pressure scale height in the following way:

1 (4.3)

81 where 7T is the temperature scale height, 'Y is the adiabatic gamma, and 7p is the pressure scale height.

We assume that the depth of the Wilson depression to be 100 Km, the pressure scale height to be 140 Km, and the adiabatic gamma to be 5/3, which corresponds to an ideal gas, and is approximately correct for the solar surface. Thus the depth of the Wilson depression would correspond to 0.29 temperature scale heights, and equation 4.3 predicts that the temperature of the down-flow at the bottom of the

Wilson depression would be 1.3 times the temperature at the surface. At the surface, the down-flows have a temperature ofonly 40% of the mean solar surface temperature.

Thus at the bottom of the Wilson depression the temperature would only be 53% of the average temperature of the solar surface. Thus by the time the down ward flowing plasma reached the bottom of the Wilson depression it would still be colder than the surrounding photosphere.

The coldest features shown in figure 4.6 have a blue residual of approximately

-0.03. This corresponds to a temperature that is half a percent lower than the mean surface temperature. The resolution here is two fold. First, we should consider that the flux tube is at an angle of approximately 70°. (This is due in part to the inclusion of regions that are not exactly at disk center, and due in part to the 10° tilt of flux tubes forward of backward. Second flux tubes are unresolved. The measurement is diluted by the surrounding solar surface.

4.4 Conclusions

It has been shown that CaIIK is a tracer of magnetic flux density. Though the relationship between line of sight measurements and CalIK residual is dependant on both the viewing angle, JL and a non-linear relationship to the magnetic flux density,

82 much of the confusion appears due to the geometry of the magnetic field. Since CaIIK residual traces the total magnetic flux density, the geometry of this magnetic field causes much of the complexity in the relationship between the line of sight magnetic field and CaIIK residual.

Using the property that CalIK residual acts as an indicator of the total magnetic field it was shown that strong magnetic features, which are presumably made up of large magnetic structures, appear dark at disk center. Weaker magnetic field observations show small flux tubes that appear bright at disk center.

Facular regions were also shown to break the isotropy of the solar irradiance. They tend to radiate more energy away from the normal to the solar surface than does the unperturbed solar surface. Stronger magnetic fields create a stronger redistribution of this energy.

It was also shown that proxy models, in particular the proxy model of Preminger et al. (2002) have a tendency to greatly overestimate the maximum contribution of facular regions. As discussed in chapter 2 overestimates of the contribution of surface features indicate a missing but correlated cause of the solar irradiance cycle. This implies that there is a missing component to the proxy model that is correlated to the presence of faculae on the solar surface. This missing component would still be linked to the presence of magnetic fields at the solar surface. It is suggestive that models such of those of Kuhn et al. (1988); Kuhn & Armstrong (2002), but further study would be necessary.

This overestimate is actually a lower bound. Facular regions are found in the cold down-flows of inter-granular lanes. Simple estimates show that even with the adiabatic compression of plasma as it flows into the solar surface, these inflows will remain colder than the average solar surface while they are still visible. This implies that the sheath of down flowing material that surrounds the strands of magnetic

83 field must be optically thin. The excess energy that is radiated from facular regions is diverted from neighboring up flows. The local energy excess radiated in facular regions is effectively shunted from reaching the neighboring solar surface.

84 Chapter 5

Sunspot Bright Rings

5.1 Introduction

It has been known for two decades that the solar irradiance varies by 0.1% over the solar cycle (Willson et al. 1981). Observations of this change continues in the literature today (cf. Frohlich 2000) although there is consensus that the irradiance variations must somehow be driven by the solar magnetic fields. What is not known is whether these changes occur as a superficial consequence of the solar cycle variation of the magnetic fields at the photosphere or whether the irradiance change and its spatial signature at the photosphere retains important information about the deeper rotational shear layer at the base of the convection zone where magnetic fields accumulate during the cycle.

Explanations for this irradiance variation generally fall into two categories. The first suggests that the magnetic flux, as it emerges from the solar interior, creates radiative "holes" in the upper atmosphere of the sun. In this model these holes allow more energy to escape from the solar surface. The holes draw upon the large thermal reservoir of the entire convection zone so that any local entropy deficits are miniscule.

Sunspots may be thought of as blockers to the outgoing radiative flux. In this view

85 the energy which is blocked by the sunspot is simply distributed throughout the solar convection zone, to be radiated away over the thermal timescale ( 105 years) of the entire convection zone (Beck & Chapman 1993; Lawrence et al. 1985; Lean et al. 1998;

Fligge et al. 2000; Solanki & Fligge 2002).

The alternative model argues that the perturbed energy transport properties of the convection zone are not analogous to a very highly conductive medium, and that the flux perturbations due to magnetic fields anisotropically and weakly couple to the bulk convection zone over short time scales. In this case the presence of a magnetic field can alter the local temperature structure of the convection zone. It is these perturbations along with entropy changes carried by the magnetized plasma from the shear layers at the base of the convection zone which are responsible for the 0.1% irradiance variations over the course of the solar cycle (Kuhn et al. 1988; Li & Sofia

2001).

A key observation that distinguishes these models is the existence of bright rings, possibly caused by the reradiation of the blocked energy flux in the vicinity of sunspots. The difficulty ofobserving sunspot bright rings can be seen in the literature.

The first evidence pointing to sunspot bright rings was Hale & Ellerman (1906), who observed bright rings in Ro.. The phenomenon was later observed by Royds (1925) who observed in Ha , and Das & Ramanathan (1953). Hale took this as an indication of a ring, encircling the sunspot, of increased temperature. While there are many factors which can cause excess emission in the hydrogen lines, his speculation turned out to be correct.

To establish that bright rings are regions of enhanced temperature they must be observed with continuum images. According to Zwaan (1968), this was first attempted by Waldmeier, in 1939. While the details are missing, Zwaan (1968) reports that Waldmeier finds that 1% of the energy which is blocked by a sunspot,

86 can be reradiated by rings. It is assumed that the remaining 99% of the blocked energy is distributed throughout the solar convection to be radiated over the time scale of the solar convection zone.

Bright rings were again observed in the continuum during the 1980's by Hirayama

& Okamoto (1981), using photographic plates. Fowler, Foukal, & Duval (1983), used a 512 channel diode array. Hirayama & Okamoto (1981) observed 19 spots, and found bright rings with a peak amplitude of 0.3-0.8% ( 11-46°K). They noted their observations were consistent with 100% of the blocked energy being reradiated, but concluded that on average 30% was reradiated. Fowler et al. (1983) observed 10 spots, and found that only 6 of them had bright rings with amplitudes greater than their detection limits. They concluded that bright rings had a peak amplitude of 0.1-0.3%

( 6-17°K), and their data indicate that an average of about 6% (about a factor of 5 less) of the blocked energy is reradiated, when a bright ring was detected.

The most recent investigation of bright rings was by Rast et al. (1999) and Rast et al. (2001). They reported that bright rings tended to extend out to about 2 sunspot radii from the center ofthe sunspot, and have peak temperature perturbations of lOOK

( 0.2%), which is in reasonable agreement with the Fowler et al. (1983) observations.

They also found that larger sunspots which have larger bright rings have rings with brighter peak amplitudes. They note that with assumption regarding the angular distribution of energy, the bright ring temperature perturbations out to two sunspot radii, seem to account for 10% of the blocked energy.

It is difficult to quantify the systematic errors, but the peak temperature amplitudes of bright ring observations since 1981 appear to be consistent. Yet the important physical quantity is the net energy budget for a facular or sunspot region.

The important connection that is to be made here is between the solar magnetic cycle and the irradiance cycle. Thus to discriminate between models we need to know if

87 energy which is blocked by the sunspot is reradiated on short time scales, or if the energy is stored and reradiated over much longer time scales such as the the thermal timescale of the entire convection zone. This is where the observations differ. The amplitude of the sunspot bright ring plays an important role in the energy budget, but so does the size of the bright ring, effectively providing more area to radiate the energy. Fowler et al. (1983) found that bright rings extend out to about 2 sunspot radii. Hirayama & Okamoto (1981) found a bright ring which extended out many times the radius ofthe sunspot. Rast et al. (2001) and Rast et al. (1999) found that the bright rings extended out to two sunspot radii.

There are several subtleties to be considered In order to deduce the energy reradiated by sunspot bright rings. For example, do faculae contained in a sunspot bright ring represent a net increase in the energy budget of the bright ring? If faculae are effectively coupled to the large-scale convection zone then the energy they radiate would increase the energy radiated by the bright ring. On the other hand if the excess energy that is radiated by facular regions comes from the more immediate surroundings, then facular regions do not represent an increase in the energy budget of the bright ring that is separate from the bright ring phenomenon.

It is also important to carefully measure the geometrical extent of the bright ring, and the angular redistribution of the radiation. Clearly any accounting of the net energy budget of a sunspot region must be integrated over the region, both spatially and in angle. With a single observational vantage point we must depend on solar rotation to see the region over several angles. With the assumption of slow temporal evolution, we can estimate the total luminosity perturbation associated with the sunspot.

Finally we must worry about the photometric zero-level correction. The zero-level correction arises from the difficulty of precisely determining the quiet sun irradiance.

88 What we wish to determine is the difference in the energy budget between the undisturbed photosphere and the bright ring or sunspot. A slightly higher or lower estimate in the measured brightness of the quiet sun is effectively reradiated over the entire bright ring. If the bright ring is extensive then a small correction can account for a large fraction of the energy. This point was discussed by Hirayama & Okamoto (1981), when they noted that changing the zero-level of intensity of their observations by 0.03% ( 2°K) can make their observation consistent with all of the blocked energy being reradiated by the bright ring. The effect is two-fold. First the geometric extent of the bright ring is increased by lowering the zero-level, while the sunspot radius remains effectively constant. Second, an incorrect zero-level adds a non-negligible flux to the energy budget.

A careful comparison of these observations with the models is contradictory. The peak amplitudes ofthe model bright rings have amplitudes of0.1%or less (d. Foukal, Fowler, & Livshitz 1983; Nye, Bruning, & Labonte 1988, and references therein.) Whereas the observations were all brighter than this. The problem becomes more acute when we consider that these results are derived from models where the sunspot depth is un-physically shallow (104 km, for Foukal et al. 1983). Additionally, there is the concern that the models treat sunspots as perfect blockers of energy. This provides more energy to drive the sunspot bright rings to larger amplitudes. Since the temperature perturbations are small we can scale our solution to correct for inefficiency in the sunspot blocker. These effects are explained in section 5.3.

The physically important comparison between models and observations is the fraction of energy that is reradiated by the bright ring. This however is also discrepant. Nye et al' (1988) found that less than 5% of the energy is reradiated,

Foukal et al. (1983) found about 2.5%, Spruit (1982) found 5%, Rast et al. (1999)

89 found 10% and Hirayama & Okamoto (1981) found that 30% was reradiated, but that the data was consistent with all of the blocked energy being reradiated.

5.2 Observations

In order to address some of the difficult observational problems (such as the zero point) we consider photometric data from two nearly simultaneous, but independent sources. The first source is the Precision Photometric Telescope, located on Mauna

Loa in Hawaii, and the second is the Michaelson Doppler Imager (MDI) on board the Solar and Helioseismic Observatory (SOHO).

The data from the PSPT have already been described in chapter 4. For the purposes of this study we modified the data slightly. We found that the choice of the scaling factor in limb darkening removal was awkward. To correct for this we multiplied by 10 and divided by the interpolated limb darkening function at p, = 1. (This is analogous to equations (9) and (10) from Beck & Chapman 1993, .» One can think of temperatures that are calculated using residuals from this equation as

"effective temperatures". This reflects that the temperature is estimated on the basis of total radiated energy. For this approach the form of the residual is:

R = 1- h(p,) (5.1) lz(p = 1)

This change in the data results in division by a factor of approximately 2 when comparing to the data presented in chapter 4.

The MDI provides 1024 by 1024 full disk images of the sun with a cadence of approximately 90 minutes. The white light images are taken with a narrow band filter with a wavelength of xxA. The MDI was not designed to provide photometric

90 images with the same precision as the PSPT. They are used as a way to verify the strengths of the bright ring signals found with the PSPT data.

We performed the limb darkening removal for the MDI data by a..'lSuming a circularly symmetric limb profile, which we determined by binning the solar disk into annuli of equal area. We removed pixels which were more than two standard deviations from the median from our sample, and used the mean of the remaining pixels. This limb profile was interpolated to every pixel of the solar disk, and removed as in equation 5.1.

The sunspots that we analyzed were chosen for simple geometry, which makes analysis simple by allowing for use of cylindrical symmetry, and to minimize confusion over facular contamination. The criteria were:

1) The sunspot should be simple. Sunspots which were not nearly circular, or were compound were excluded.

2) Sunspots were chosen to be Visually free from regions of facular networks in the

CalIK images.

3) Data must be available for the majority of one disk crossing, with no more than one consecutive day which is missing from the data set. (Le. we did not consider sunspots where there was a gap in observations of two days or more.) A full disk crossing is necessary to more accurately determine the functional relationships of the bright ring and sunspot contrast.

From these criteria we found four sets of observations: NOAA 8263 during July 2

- 10, 1998; NOAA 8640 during July 22 - 31 1999; NOAA 8525 during April 30 - May

9, 1999; and NOAA 8706 during September 20 - 30, 1999. We were unable to obtain MDI data for NOAA 8263 as this coincided with the unscheduled down time for the

SORO satellite during June to September, 1998.

91 The first step of our analysis was to construct "disk center" images. A disk center image is a 400 by 400 pixel sampling of the full disk image which shows the distances between features on the solar surface as they would appear if the central feature of the image were at the disk center. IT we define the viewing angle as the angle between the line of sight and the normal to the surface then we may think of this as a 400 by 400 pixel image of the sunspot as it would appear if it were observed from a position with a viewing angle of zero.

To construct the disk center images we assume a Cartesian coordinate system such that the i coordinate was positive toward the observer along the line of sight. Positive fJ was chosen to coincide with north on the solar disk, and positive x was chosen as solar east. we can then assume that the sun is a sphere with radius 1, centered on the origin. With these assumptions, we calculated the 3D coordinates of a region which would be observed as a 400 by 400 pixel region centered on the center of the full disk image of the sun.

We then rotated this pixel map so that it was centered upon the sunspot, by using Euler angle rotations.

cost/; cos sin t/; cos t/; sin sin t/; sin t/; sin (}

X' = - sin 1/J cos 4> - cos () sin 4> cos t/; - sin t/; sin 4> + cos () cos 4> cos t/; cos 1/J sin () x sin () sin 4> - sin (} cos 4> cos() (5.2)

Where we find t/;, 0, and 4> from the sunspot's x and y coordinates.

(5.3)

92 'ljJ = tan-1 (xspot) (5.4) Yspot (5.5)

Where rx and ry are the measured radii of the image associated with the x and y directions.

After rotating the pixel maps we sampled the original images at the pixel which was located nearest to the rotated coordinates. This results in an over sampling of

1. the image by a factor of about p, The resulting images are our "disk center images". We constructed the median residual for individual annuli as a function of J.l and r, where r is the distance, along the solar surface, from the center of the sunspot. These values were used to construct a functional form of the per pixel flux, R(J.l, r), which was then integrated to calculate the total radiated energy per pixel.

L(1') = JR(J.l, r)ftdft (5.6)

L is a measure of the total energy radiated from the region (i.e. up to an overall multiplicative constant), R(p) is the median residual. Here we have chosen the coordinate system such that the viewing angle is equivalent to the polar angle for the integration.

Sunspots generally do not pass directly through disk center, and often PSPT observations near the solar limb become unreliable (typically for p < 0.4) due to the over sampling that occurs when making the disk center images (i.e. foreshortening) and the instability of the limb darkening function near the limb. The functional form of R(p) is incomplete, and some e:>...1;rapolation must he done. We have assumed in this case that the contrast has reached maximum/minimum within the values of J.l which we are able to measure. We assumed a constant profile for values of p which

93 we are unable to observe. (See figure 5.1) While we do not measure the contrast near the extreme limb, the net contribution of these data to our estimate of the total energy budget will not effect our conclusions. To estimate the error introduced by these uncertainties, integration of the curve shown in 5.1 was preformed, and compared the result to the result which we would have obtained if we were to instead extrapolate the data by performing a linear least squared fit to the measurements. The difference is less than 15% of the computed value. The approach of assuming that the contra...'Its have reached an extreme within the observational limits of Jl is the conservative estimate when calculation the fraction of energy reradiated by a sunspot bright ring. Any error will tend to overestimate the amount of energy blocked by the sunspot, and underestimate the energy radiated by the sunspot bright ring. This implies that any error would sere to create a lower-limit estimation.

The resulting surface brightnesses are converted into temperatures by substituting the black body equation: 2hGJJ>..5 B>.(T) = ehc/>.kT _ 1 (5.7) into the residual equation(4.1), and linearizing with respect to the temperature.

We observed that some of the resulting temperature perturbation profiles did not approach zero, even far away from the center of the associated sunspot. We believe this is due to imperfections in the limb darkening removal. To correct for this error we used an approach similar to aperture photometry. After constructing the disk center images, we subtracted the median residual of outlying pixels from the image. The outlying pixels are those pixels in the 400 X 400 disk center images that are further than 200 pixels from the image center. The resulting temperature profiles are shown in figure 5.2.

94 -0.10

-0.15 * 0 * * ::J V (f) Q) a:::: -0.20 v Q) * ~ a:::: -0.25 * *

-0.30 0.0 0.2 0.4 0.6 0.8 1.0 J1,

Red residual vs. J.l

Figure 5.1 The extrapolation routine is demonstrated here. The scatter plot shows the median values of the residual for the center region of the sunspot NOAA 8525, in red. The solid line is the resulting extrapolation. The same extrapolation procedure was used for all annuli, however it should be noted that the functional relationship for regions of the bright ring tended to be noisier than the above graph.

95 15 5 4 10 f- f- 3 '0 '0 2 5

1 :"', '," a a " ',f .,', a 50 100 150 a 50 100 150 Radius [Mm] Radius [Mm]

8 8

6 6

f- f- 4 4 '0 '0

,') 2 2 a a a 50 100 150 a 50 100 150 Radius [Mm] Radius [Mm]

Temperature Perturbations vs. Distance from Sunspot Center

Figure 5.2 The temperatue pretubation profiles are shown for 4 sunspots, NOAA 8263 (Upper-Left), 8640 (Upper-Right), 8525(Lower-Left) and 8706(Lower-Right). Solid lines show the temperature perturbation computed from the PSPT red data, dashed lines PSPT blue, and the dotted lines are from the MDI. As discussed in the text, the MDI profiles have been shifted by a constant.

96 To calculate the fraction of energy which is re-radiated by the sunspot bright ring, we followed the same procedure for constructing disk center images. We then took the mean of residuals from 0 to Tspot and from Tspot to rring, where Tspot is the sunspot radius. We consider the sunspot radius to be the point at which the temperature profile ceases to be cooler than the quiet sun. rring is the assumed outer radius of the bright ring. Since the actual extent of the sunspot bright ring was unknown, we computed the fraction of energy reradiated for several values of r ring' The residual R(f.-l) was otherwise constructed and integrated in the same way as the annuli for the temperature profiles. An example of this can be seen in figure 5.1. From the resulting average fluxes we construct the fraction of energy which is reradiated by the bright ring by dividing the mean bright ring flux by the mean spot flux deficit, and multiplying by the ratio of the areas.

5.3 A Simple Model

To understand the results of the numerical models it is helpful to first construct a simple model that may be analytically solved. We consider the case of a one dimensional diffusion model of the sun's convection zone, in which we have introduced a source. We assume that an effective conductivity, K scales with the density, as mixing length predicts:

K = alvpCp (5.8)

Where a is a dimensionless factor of order unity, v is the convective velocity, p is the density, and Cp is the per unit mass heat capacity (c.f. Spruit 1977)

97 We take z increasing with depth. Since the density increases exponentially with z, we expect the density term to dominate equation 5.8. We approximate K( z) as:

z/h Kzz = ,..oe (5.9)

Where h is the density scale height.

The full three dimensional diffusion equation is:

(5.10)

Where Cp is the per unit mass heat capacity, K is the conductivity tensor, and T is the temperature.

Ifwe assume a diagonal form of ,.. and consider the dominant vertical temperature gradient, aT/oz, then the one dimensional diffusion equation reduces to:

(5.11)

We wish to ask the question, "If we introduce a heat source, at some depth, z', what fraction of the heat escapes from the surface, as opposed to being conducted downward to be "lost" in the interior?

With the following assumptions this problem offers an alalytic solution which is quite instructive:

1) At infinite depth the temperature perturbation is zero

2) At the surface the temperature perturbation tends to zero due to the radiative boundary condition.

3) The system is in equilibrium.

In equilibrium, equation 5.11 becomes:

98 (5.12)

We find that the form of temperature perturbation is:

rC1+ -Z/T T+ = ---e + (Cr 1+/)K:o (5.13) fi:O

T _ = -TC1+ e-Z / T (5.14) fi:O

Where +/- denote above/below the source.

The energy flux of the system is:

(5.15)

Conservation of flux tells us that:

(5.16)

From this we find that the fraction of energy which escapes to the surface is an exponential:

F+--- e-z'/h (5.17)

Where z' is the depth of the introduced source. It should also be noted that the solution is governed by a linear differential equation, thus the magnitude of the energy input has no effect on the fraction of energy which is reradiated. This allows us to compare the effect of physical sunspots, which do not block 100% of the emerging flux with models which do, as discussed in section 5.1. A sunspot does not block 100%

99 of the energy incident on it from below. If we want to compare a modeled sunspot which blocks 100% of the incident energy to a realistic sunspot, we can multiply the temperature solution by the fraction of energy which is blocked by the sunspot.

One important point to learn from this simple example is that in equilibrium the magnitude of the conductivity has no effect on the fraction of energy which escapes the surface. It is only the structure of the conductivity that determines the fraction of energy which will be reradiated on local equilibrium time scales. The magnitude of the conductivity does, however, play an important role since, combined with the heat capacity, it sets the time scale of the features which we see:

(5.18)

Where I is the length scale of the system.

This calculation can be thought of in terms of the simple resistor model depicted in figure 5.3. The system effectively acts as a current splitter. The resistance in each branch can then be defined as:

R=J~dZ (5.19) K zz

Where the integral is taken over the path to the ground.

Following the mathematics through from this starting point will yield the same results as equation 5.17. When considering this model it is important to remember that the current which is being split between the the ground at the surface and the ground at depth is superimposed on a current that is flowing from depth toward the solar surface.

100 - Ground at solar - :surface

Ground depth _.....-- In solar interior Figure 5.3 The schematic for the model discussed in section 5.3 shown here. A sunspot blocks energy by inhibiting convection. The blocked energy can be considered as a source which is split between two paths to the ground.

101 5.4 Numerical Simulations

The motivation for section 5.3 can be seen as we develop a two or three dimensional perturbation solution to the diffusion equation. Starting with equation 5.10 we assume that the system has reached equilibrium.

dU ~ ~ - = 0 = (\7) . (~\7To) (5.20) dt

We then consider a perturbation in the conductivity tensor.

(5.21)

This will drive a perturbation in the temperature in the following way:

(5.22)

The first term on the right hand side of equation 5.22 is simply the steady state equation which is zero by definition. For the remaining two terms on the right hand side, we recall that unperturbed. quantities are constant along contours of constant depth. Their only dependence for the model of the sun is in the z direction. The third term in equation 5.22 is analogous to the total diffusion equation. The second term represents a source or sink. The result of this source term is a perturbative thermal source that spreads out from the base of the sunspot. We can consider the perturbative flux from this source as the current which is split between the ground at the solar surface and the ground at depth which was discussed in section 5.3. We generally consider localized perturbations in the conductivity that return to zero within a finite distance, the net contribution from the analogous source is zero,

102 analogous to a dipole. Since the unperturbed temperature is independent of the horizontal position, all of the dipoles are oriented in the vertical direction.

The model is then driven by dipole pairs of sources and sinks. To compare to the model of section 5.3 if we introduce a perturbation at some depth z', and allow perturbation to return to zero only far from the surface of the sun, then we have the source or sink model of section 5.3.

The advantage of subtracting the steady state solution is that the resulting equation is more stable to numerical analysis. In these models we are comparing the system with perturbed conductivities to the system without perturbations to the conductivity. The quantity of interest is the difference between the two, i.e. we wish to study the change in the system caused by the perturbation. The difference in temperatures between the perturbed system and the unperturbed system is small on the scale of the unperturbed system. These small differences can quickly be lost to numerical noise. By removing the steady state solution we model only the perturbation to the system, and eliminate the numerical noise that could otherwise arise from looking for small changes in relatively large numbers.

We constructed numerical models of sunspots for comparison to the observations.

The models were a two dimensional simulation of a cylindrical plug placed in the solar convection zone. Beginning with the diffusion equation (5.22) we assumed cylindrical symmetry. This allows us to reduce to the two dimensional diffusion equation, in cylindrical coordinates:

(5.23)

Where fl,rr and fl,zz are the conductivity in the radial and vertical directions, respectively.

103 From this we subtract the steady state solution, as described above. For the sunspot model, we assumed that the perturbed conductivity returns to zero at an infinite distance. This decision is arbitrary but we are concerned with the existence of the bright ring. The solution inside the boundary of the sunspot is isolated by the boundary conditions, as described below, and is not of interest.

We assumed a cylindrical sunspot "blocker" in which the conductivity is reduced to zero. No energy was allowed to flow into the simulated sunspot. At the surface of our simulated volume, we applied the blackbody condition as the boundary condition:

(5.24)

Where (J" is the Stephan-Boltzman constant, and E is the emmisivity, which was chosen to match the known photospheric temperature (5777 K) to the radiative flux

10 ::::.~) density at the solar surface. (F0 = 6.27 x 10.0

The extreme boundaries of our volume, Le. at rmax and Zmax we set the the temperature of the last point in the simulated volume at each iteration according to the following formula

(5.25)

Where To is the temperature of the grid point at the boundary, T1 is the temperature at the next grid point, etc. This boundary condition correctly predicts the physically reasonable exponential in the vertical direction. This is the equilibrium for a one dimensional model, with a source function, and an exponential conductivity (See section 5.3). In the horizontal direction, this correctly models the equilibrium case where the flux falls as ~. A ~ profile would be expected as a consequence of energy

104 in the case of a source in a 2 dimensions. (This produces a logarithmic temperature profile.)

We used two variations of the conductivity in these models, a mixing length conductivity and a numerical conductivity. The estimate of conductivity for mixing length theory was derived by Spruit (1977), and is assumed to be isotropic. The numerical model of Kuhn & Georgobiani (2000) was used to construct an anisotropic conductivity. The magnitude of the horizontal and vertical conductivity was taken from the average of the ratio of the mixing length conductivity and the respective numerical conductivity. The values calculated by Kuhn & Georgobiani (2000) were used to a depth of approximately 3 Mm, which is where the scaled MLT and the numerical conductivities were the same. The resulting conductivities are shown in figure 5.4.

5.5 Results

5.5.1 Temperature Profiles

We constructed the temperature profiles from the data, as discussed in section 5.2, and displayed in figure 5.2. We also constructed the temperature profiles from the conductivity models discussed in section 5.4. (See figures 5.5 and 5.6.) The modeled temperature profiles all have a sharp increase in temperature near the boundary, with the temperature maximum located just outside of the sunspot. The temperature falls off with the distance from the center of the sunspot, and can be approximated as a ~ profile, where r is the distance from the center of the sunspot.

Even after the baseline subtraction was performed, the MDI profiles remained negative. This is most likely due to small, spatial errors in limb darkening removal.

105 ,---, ,---, 'I 'I E ~ 20 12 u 'I 'I E ~ -I u 18 10 (j) 'I (j) (j)

(j) OJ L OJ 16 8 (1) L I...-...J (1) L-....J ~ 0- ~ ¥ U '---.../ 14 6 '---.../ >, >, -+-' -+-' > u -+-' 12 4 0 U 0- ::J 0 V U c 10 2 -+-' 0 0 u (1) ::r:: C)) 8 0 0 OJ --.l 0 0 5 10 15 20 --.l Depth [Mm]

Thermal Conductivity and Heat Capacity

Figure 5.4 The thermal conductivities and heat capacity used in the diffusion models are shown here. The mixing length conductivity is marked with "+". The vertical conductivity extrapolated from Kuhn & Georgobiani (2000) is marked with "*". The horizontal conductivity extrapolated from Kuhn & Georgobiani (2000) is marked with diamonds. The heat capacity is marked with triangles.

106 20

c - Depth 1 Mm, 1.8% reradiated 0

~ 0 15 Depth 2 Mm, 1.3% reradiated -.0 L Q) ~ Depth 3 Mm, 0.9% reradiated L Q) 0- Depth 4 Mm, 0.7% reradiated

Q) 10 L Depth 5 Mm, 0.6% reradiated ::::J ~ 0 L Q) 0- E S Q) f--- oF=~~~.-,-d 1.0 1.5 2.0 2.5 3.0 Distance From Sunspot Center/Sunspot Radius

MLT Model Temperature Perturbation Profiles

Figure 5.5 The modeled temperature profiles using mixing length conductivities are show here for 5 depths.

107 50 Depth 1 Mm, 0.42% reradiated c Depth 2 Mm, 0.38% reradiated 0 40 -+--' Depth 3 Mm, 0.19% reradiated 0 ...0 Depth 4 Mm, 0.14% reradiated "- (1) Depth 5 Mm, 0.10% reradiated -+--' "- (1) 30 0.-

(1) "- ::J -+--' 20 0 "- (1) D- E (1) 10 I- o LL....L.L.l-L--==:===~~~~~...... J..~~ 1.00 1.10 1.20 1.30 1.40 1.50 Distance From Sunspot CenterjSunspot Radius

Numerical Model Temperature Perturbation Profiles

Figure 5.6 The modeled temperature profiles using the numerical conductivities of Kuhn & Georgobiani (2000) are show here for 5 depths.

108 To correct for this we have shifted the MDI data upward by a few percent. After performing this shift many of even the small features seen in the PSPT data are reproduced in the MDI data. It should be noted that all three estimates of temperature are in reasonable agreement aside from the baseline shift for the MDI data.

The temperature profiles computed from the conductivity models can be seen in figures 5.5 and 5.6, along with the computed fraction of energy which would be reradiated by the bright ring. As can be seen in the graph, even though the MLT model does not quite reach zero within the simulated volume, this should not greatly affect the total energy budget of the spot, since the temperature has fallen by an order of magnitude from one sunspot radius to three sunspot radii.

The numerical models show some of the features seen in the observed temperature profiles. In particular the shape of the observed temperature profiles generally reach a maximum near the boundary between the sunspot and the bright ring. The observed temperature profiles also show a fall off that is similar to the modeled bright rings.

The most notable difference comes from the magnitude of the peak in the temperature profiles. In all cases the computed temperature perturbation is smaller than observed.

This is important since the models have assumptions, as discussed above, which would naively be expected to overestimate the size of this temperature perturbation.

5.5.2 Reradiation Of Energy

The fraction of blocked energy reradiated by the sunspot bright ring is listed in table

5.1. As can be seen the bolometric fraction of energy computed from red and blue data is in reasonable agreement for the bright rings around NOAA 8263, 8525, and

8706. The bright ring around NOAA 8640 shows about twice as much energy is

109 Table 5.l. Fraction of Energy Reradiated by Sunspot Bright Rings

Sunspot % reradiated in Sunspot Color Radius (Mm) 1 sunspot radius 2 sunspot radii 3 sunspot radii

8263 red 19 20 30 32 8263 blue 23 34 35 8640 red 16 4 8 14 8640 blue 7 15 27 8525 red 19 11 31 59 8525 blue 12 38 70 8706 red 16 14 46 67 8706 blue 15 39 71

Note. -- Column 1 lists the sunspot NOAA number. Column 2 gives the PSPT color the estimates were derived from. Column 3 lists the calculated sunspot radius. The radius was computed using both colors. Thus it is only listed under red entries. Column 4-6 give the percentage of 'blocked' energy which is reradiated by the sunspot bright ring.

reradiated in blue than in red. All sunspot bright rings radiate significantly more energy than is predicted by the diffusion models. (See figures 5.5 and 5.6).

To compare the model of section 5.3, we assume that the sunspot extends to a depth of 5Mm. This value is chosen to be in agreement with the observations of Zhao et al. (2001). We computed the surface solar scale height from values found in Cox

(2000). The value was found to be 140 Km. The resulting fraction of energy which

15 is expected to be reradiated at the solar surface effectively zero « 10- ).

5.5.3 Facular Contamination

The difference between the observed and the predicted fraction of energy reradiated by the sunspot bright ring might be explained by the direct effect of surface magnetic

110 fields. Magnetic fields which are seen at the solar surface provide pressure support to the plasma in which they are embedded. In order to maintain pressure equilibrium, the density of that plasma must be lower than the surrounding plasma. This structural change leads to a decrease in the opacity, and ultimately this will lead to changes in the way energy passes through the region. This change in energy transport has been observed to alter the angular distribution of radiated energy, i.e. facular regions appear brighter near the solar limb, and have a decreased contrast near disk center, which can cause them to appear darker than the unchanged quiet sun (Spruit 1977).

While the angular redistribution of energy is well established, it has also been suggested (cf. Lawrence et al. 1985; Lean et a1.1998; Fligge et al. 2000; Solanki

& Fligge 2002; etc.) that faeular regions locally increase the radiated energy. The excess energy is drawn from the large thermal reserve of the entire convection zone, and thus a faeular region is a net increase in the total energy radiated by the sun.

An alternative view is that their primary effect is to redistribute the radiated energy.

In this picture any excess energy that is radiated from a facular region is drawn from the surrounding solar surface. This results in a slight temperature drop in the surroundings. Thus faculae act only to redistribute the radiated energy.

By nature sunspot bright rings contain magnetic fields. As a sunspot ages the magnetic field diffuses away from the sunspot. This may in effect contaminate the bright ring. While it is important to correct for any excess energy that might be reradiated by a region of magnetic activity, the energy reradiated by such a region should be considered as part of the sunspot bright ring environment.

As a first order correction for this facular contamination, we removed pixels that were bright in CaIIK. Neighboring pixels, i.e. pixels within 2 pixels of pixels with a CaIIK excess of greater than 0.1, were also removed. The neighboring pixels were

111 removed due to concerns over slight errors in the pixel registry. If the continuum images are shifted by one pixel or more, then removing only the single pixel associated with a bright CalIK pixel would not remove the contribution from small flux tubes.

This is important, since the size of the images varies by about 3% when comparing the inferred radius of the sun in red to the inferred radius in CalIK. This is due to slight differences in the focal length of the telescope at different wavelengths. Thus by including neighboring pixels we are confident that the radiative excess associated with the observed CaIIK excess is removed from the analysis.

To determine the threshold of CalIK for filtering as described above we examined the angular redistribution of radiative flux from facular regions. In chapter 4 to confirm that CalIK features would result in the same observed residual intensity viewed from different angles, we computed the median and standard deviation of the

CalIK distribution as a function of the viewing angle. It was found that CalIK was a tracer of total magnetic field strength. This gives the advantage that by using CalIK rather than line of sight magnetic field measurements, it is possible to compare the appearance of similar features in different locations on the solar disk

In figure 4.6 the angular dependence of blue residual is shown as a function of the associated CaIIK residuaL In selecting a cutoff of CaIIK to remove we wish to eliminated the regions which contain magnetic features. CalIK residual is not only dependant on the magnetic field strength, but on the temperature of the region.

Temperature perturbations do not alter the angular distribution of flux. On the other hand, magnetic features are expected and have been shown to redistribute radiant energy. It has also been shown that as CalIK residual increases the angular redistribution of energy increases (i.e. the slopes of the curves shown in figure 4.6).

The correct cutoff is where the angular redistribution becomes significant. We found that this occurs when the CalIK is greater than 0.1.

112 Removing pixels with a CaIIK residual of greater than 0.1 from the bright ring computation, it was found that the rings surrounding sunspots were not bright, but were instead slightly darker than the surrounding photosphere (i.e. "bright" rings that are not bright). Within 2 sunspot radii of the edge of the sunspot the fraction of energy reradiated by the sunspot ring around NOAA 8263 is -5.0%, around NOAA

8640 -3.2%, around NOAA -1.2% and around NOAA -2.0% of the energy blocked by the sunspot is reradiated. The negative sign here signifies that the rings are darker than the quiet sun. When considered individually, these values are small, possibly within expected errors in the computations. However, since all of the rings are observed to be dark after removing the CaIIK-bright and neighboring pixels, the combined observations indicate that this is unlikely to be in error.

The result that sunspot bright rings are found to be dark is surprising. The models imply that bright rings should exist at some level, even if they are undetectable. It does not explain how we could find dark rings around sunspots.

To understand why the rings surrounding sunspots appear to be dark, we compared the facular regions found in the sunspot bright rings with facular regions that were not associated with sunspots. We began by constructing a linear least squares fit of the blue residual vs. f-l for different bins of CaIIK strength. Effectively this is a linear least squares fit to the limb brightening profile for facular regions with a given CaIIK residual. These extrapolations of the radiation field can be easily integrated to compute the total energy radiated from a "typical" facular region with a given CaIIK intensity. The results of this integration are shown in figure 5.7. We find that facular regions, which are bright in CaIIK, found in sunspot bright rings are actually bolometrically fainter than facular regions, with similar CalIK brightnesses, found in regions unassociated with sunspot activity.

113 Foculor Energy 0.030

/' (f) /' (f) 0.025 / Q) / 0 / X / /' W /' 0.020 /' >, /' /' OJ /' L /' Q) /' /' c 0.015 /' W /' /' 0 /' c 0 0.010 /' -+-' /' 0 /' 0 L -" l..L 0.005 -"

0.000 0.10 0.20 0.30 0.40 CallK Residual

Figure 5.7 Facular regions in sunspot bright rings (shown with a dashed line) are slightly fainter than their non-bright ring counterparts (shown with a solid line).

114 5.6 Scattered Light

The explanation for the observation that sunspot bright rings were dark, and also that the faculae found in bright rings are darker than similar features which are unassociated with sunspots might come from scattered light. This was mentioned previously in the literature as a possible explanation for sunspot bright rings (cf. Das

& Ramanathan 1953). In previous studies, however, since the rings were observed to be bright the concern was dismissed, as the scattering of light would have made the bright rings appear darker. Thus correcting for them was not an explanation for the bright ring phenomena.

One might consider the observation a bright region neighboring a dark region.

Scattered light then would represent a smoothing or blurring of the image. This smoothing will make the bright region appear darker in the image, while the dark image will appear brighter. This can be considered in terms of scattering "darkness" from the dark region into the bright region.

A sunspot then can be thought of as a relatively dark region, while the sunspot bright ring can be thought of as the bright region. Light is scattered from the sunspot into the neighboring bright ring.

To correct for this effect the point spread function (PSF) of the telescope was constructed. In construction of the residual images, limb darkening profiles are constructed. The limb darkening profiles can be thought of the circularly averaged brightness profiles of the solar disk.

These limb darkening profiles contain the information needed to reconstruct the

PSF of the telescope; i.e. they have a sharp boundary at the solar limb, while the actual images have a "smoothed" boundary. The image recorded by the telescope system is the convolution ofthe original image and PSF. Convolution is multiplication

115 in Fourier space. Thus to compute the PSF one can take the Fourier transform of the actual image, and one which was taken by the telescope. The Fourier transform of the telescope image is divided by the Fourier transform of the actual image. The

PSF is the inverse Fourier transform of the quotient.

PSF = F- 1 (F (It)) (5.26) F (fa)

Where F denotes Fourier transform, It denotes the image taken by the telescope, and fa denotes the actual mage which would have been taken by a perfect telescope system.

Since perfect images are not available the calculated limb darkening profile was used. The resulting PSF was then applied to the several images to serve as a test of the approach. It was found that several artifacts remained in the images, in particular a band like structure was seen in the corrected images. This was believed to be a result of insufficient signal to noise to compute an accurate point spread function at large distances from the center.

To reduce the noise the resulting PSF was circularly averaged. This is an approach analogous, but not as sophisticated as the one presented by Toner et al. (1997). An example of the resulting PSF can be found in figure 5.8.

Instead of applying the PSF to the images taken by the telescope, the PSF was applied to the residual images. By directly computing the effect of the PSF on the residual images there is no need to recreate the residual images after correction for the PSF. The validity of this approach may be seen by considering that conjugation is a linear mathematical function.

(5.27)

116 Point Spread Function 0.030 \ \ Point Spread Function \ 0.025 \ \ \ 0.020 \ \

\ '. . \ 1.L \ . .\ 0> if) 0.015 \ . 0.- \\ () -----.J \ \ \ \ \ '. 0.010 . \ \ . ., \ . 5 10 15 20 ". '., 0.005 Distance [pixels]

O. 000 ~----,--,----l.---'------'------L--.L-,------,-----~~~~--2:-::§:--~::.--~--~~ o 2 4 6 8 10 Distance [pixels]

Figure 5.8 Point spread functions are computed by deconvolution of the image and the limb darkening function. Six point spread functions were computed for the blue images. The mean and one standard error from these are shown.

117 Where It is the limb darkening computed from the actual image, and 10 is an arbitrary constant, in this case chosen to correspond to the interpolated value of the limb darkening function at disk center.

At this point it was found that correcting the original imaged for the circularly averaged PSF was numerically unstable. By taking the average a large number of zeros were introduced into the Fourier transform of the PSF. However since the effect of the PSF on the bright ring calculations, not the corrected images were sought, another approach was used.

As long as it is known how much light is scattered from the bright ring into the sunspot the bright ring can be corrected by hand. By computing the "forward problem" the amount of light scattered out of the bright ring can be found. The forward problem is the application of the PSF to the actual images. This is the equivalent of the PSF being applied to the actual images twice, one by the telescope and once after the image is taken. The new images are then analyzed in the same manner as the original images.

To understand the approach that was used it is simpler to think of scattering darkness rather than scattering light. The amount of darkness in the sunspot can be seen to be the negative quantity that results from integrating the residual over the sunspot. It was then assumed that a constant fraction of this darkness was scattered into the sunspot bright ring each time the PSF was applied to the image.

This assumption was found to accurately account for 99% of the scattered light.

The results are summarized in 5.2. It seems suggestive that x, the fraction of light scattered into the sunspot, is roughly constant for all sunspots. This however is due to the fact that the chosen sunspots were similar.

After correction for the PSF the sunspot bright rings were found to be consistent with zero when CaIIK features were removed. No bright or dark ring was found

118 Table 5.2. Scattered Light Corrections

n Spot Ring % Reradiated X

NOAA 8263 0 -0.161 7.81 x 10-3 39 .949 1 -0.153 6.54 x 10-3 34 2 -0.145 5.33 x 10-3 29 NOAA 8640 0 -0.127 3.00 x 10-3 19 .948 1 -0.120 2.31 x 10-3 15 2 -0.114 1.65 x 10-3 12 NOAA 8525 0 -0.160 7.56 x 10-3 38 0.966 1 -0.155 7.22 x 10-3 37 2 -0.149 6.90 x 10-3 37 NOAA 8525 0 -0.126 7.86 x 10-3 50 .943 1 -0.119 6.81 x 10-3 46 2 -0.112 5.82 x 10-3 42

Note. - Corrections for scattered light are computed by calculating the amount of light scattered into the sunspot after the PSF is applied. n is the number of times the PSF has been applied, with n = 0 corresponding to an image with no scattered light. X is the amount of light scattered into the sunspot as a fraction of the deficit associated with the sunspot.

119 around NOAA 8263, 8640 and 8706. A slight bright ring may have been detected around NOAA 8525.

After correction for the scattered light the facular regions in bright rings remained as a concern. The overall correction was small (i20%) in comparison to the difference between the two populations. The other source for this correction is the baseline correction that was applied to force the temperature perturbations of bright rings to approach zero at large distances from the sunspot.

If we are to assume that the zero correction is the difference between the two populations it should be removed. This would imply that sunspot bright rings radiate approximately 10% of the blocked energy. This is highly uncertain, as each correction has an associated uncertainty that is indeterminate. At this point the best thing to say is that data is suggestive of a thermal bright ring, which reradiated about 10% of the energy blocked by the sunspot.

5.7 Discussion

As described in section 5.1, a major concern with this analysis is whether facular contamination will effect the energy budget. On the one hand surface magnetic features might simply redistribute the energy budget. Effectively this would yield a higher radiative flux in one area that is compensated by a cooler region surrounding the flux tube. (In this view, an analogous faint dark ring surrounds the flux tube.)

On the other hand a facular region might represent a hole, which allows for energy to leak out of the solar surface at a higher rate. After removing facular regions it was found that the remaining "bright ring" was in fact dark. It was also found that facular regions contained in sunspot bright rings are in fact fainter than similar regions that

120 are unassociated with sunspots. This led to the belief that a significant amount of scattered light might be affecting the measurements.

The point spread function was computed by de-convolving images and the associated limb darkening profile. The point spread function was then circularly averaged to lower the associated noise level. After constructing several point spread functions they were applied to the residual images again. While this did not directly correct for scattered light, it allowed for an understanding of how scattered light would affect the measurement of bright rings. By carefully tracking the scattered light it is possible to compute the effect on the bright ring calculation. This was done and the effect was small (on order of 2%). The correction was large enough to correct for the dark bright rings, but was too small to account for difference in the brightness of facular regions in the bright ring and outside the bright ring.

Ifthe difference between facular regions is to be taken seriously it would imply that sunspot bright ring radiate about 10% of the energy blocked by the sunspot within twice the radius of the sunspot from the sunspot edge. A detection of this type is a clear contradiction to the assumption of diffusive conduct vity. The bright ring would radiate far more energy than is implied by these calculations. This indication is far from certain and a larger sample of sunspots might reduce uncertainties.

The other explanation for the difference between facular regions found in a bright ring and those found outside the bright ring would be that the faculae in bright rings are in a higher density environment. The amount of magnetic flux contained in sunspot bright rings is higher than that oftypical regions unassociated with sunspots.

Effectively the magnetic field diffuses away from the sunspot, as it ages, contributing to the facular contamination. There are two effects that might cause an observable difference in high-density facular regions. First, it is more likely to observe several small faculae in the same region. This would create the appearance of a large singular

121 flux tube. Second, flux tubes might sit in the hypothesized "dark ring" that surrounds flux tubes.

It is unlikely that the blending of two distinct flux tubes into a 'single" flux tubes will explain the observed result. Consider the case of two flux tubes, each with an associated CaIIK residual of 0.2. Using figure 5.7 as a guide we can estimate that the blue residual associated with each individual flux tube would have an associated fractional energy excess of 0.013. If both flux tubes were imaged on a single pixel then the resulting associated fractional energy excess would be the sum of their individual residuals, 0.026. The associated fractional energy excess for a region with a CalIK residual of 0.4 is 0.027. For practical purposes the energy excesses are the same, and not an explanation. There is the additional argument that the brighter faculae, which should be more affected by this effect, are similar in both sunspot bright rings and regions unassociated with sunspot bright rings. The latter is the stronger of the two arguments.

The second possible explanation to the observed differences in faculae is that that faculae do have bight rings. In a high-density environment each facular regions rests in the dark ring of neighboring faculae. This reduces the detected increase in energy radiated by facular regions. Faculae can be thought of as a preferred route for energy to reach the solar surface.

Consider the situation where a neighborhood of the solar surface that radiates a constant energy flux, regardless of the presence or absence of faculae. If flux tubes are effectively "shunts" then in a neighborhood with a high density of faculae, each faculae will shunt less energy to the solar surface, when compared to similar faculae in a low density solar neighborhood. This is in agreement with what we see. Regions with a high level of faculae contain faculae which radiate less energy than similar faculae in regions with a low density of faculae.

122 The question of whether or not to include faculae in computations of the fraction of energy reradiated by sunspot bright rings is less clear. If they are included then at least 40% of the energy blocked by the sunspot is reradiated in the sunspot bright ring. This would be a clear violation of the assumption that a diffusion process can describe energy transport in the convection zone.

On the other hand if faculae represent a contamination that should be removed, then sunspot bright rings are consistent with the assumption that energy transport in the solar convection zone can be described by diffusion. In section 5.3 it was found that the fraction of energy reradiated was independent of the overall scale factor.

This implies that faculae can be treated as energy sinks (an overall minus sign, when comparing to the energy source of sunspots). We then ask how much of the energy deficit is "reradiated" in a dark ring surrounding the bright faculae. Faculae are estimated to have a depth of 100 Km; sunspots have an estimated depth of 5 Mm, a factor of 50 times greater. Using equation 5.17 we find the virtually all of the energy deficit from a facular region would be "reradiated" in the dark ring that surrounds the flux tube.

This constitutes a proof by contradiction that the assumption that the diffusion process can describe energy transport in the convection zone is false a false assumption. If sunspot bright rings are modeled by diffusion then faculae must represent an increase in the energy budget of the neighborhood of the solar surface.

If the diffusion equation governs energy transport in the solar convection zone then faculae do not represent an increase in the energy radiated by the neighborhood of the solar surface. At some level, either large or small energy transport in the solar convection zone must fail.

123 Chapter 6

Discussion and Conclusions

6.1 discussion

The observation of solar irradiance changes is an interesting problem in solar physics.

In the short term - i.e. over the time scale of a few days to two weeks (the time for a feature to cross the solar disk) there are changes in the solar irradiance as large as

0.4%. On the time scale of the ll-year period of the solar cycle, there is an observed

0.1 % change in the solar irradiance. Effectively what we would like to answer is:

"What drives the observed solar irradiance cycle?"

There are two explanations of the solar irradiance cycle. First, there are the proxy models. These attribute the solar irradiance cycle to observed surface features.

Second, there is the theory that the solar irradiance cycle is caused by structural changes in the solar interior that alter energy transport to the solar surface.

It has been observed that sunspots are dark and faculae are bright. Thus sunspots might represent a decrease in the solar irradiance and faculae represent an increase in the solar irradiance. By accounting for the area of sunspots and faculae it is assumed that one can compute the solar irradiance. There are two underlying assumptions:

124 coupling of surface features to the solar convection zone, and invariability of the quiet sun.

It is assumed that these surface features are thermally coupled to the entire convection zone. The excess energy that is associated with faculae is not drawn from the surrounding solar surface, but instead is drawn from the convection zone as a whole. This results in only a minute change in the mean temperature of the solar convection zone as a whole. This small change may be ignored for an accurate description of the solar irradiance. Similarly, the energy that is "blocked" by dark surface features is deposited in the convection zone as a whole.

It is also assumed that the quiet sun (regions of the solar surface that are free from faeulae and sunspots) does not change over the course of a solar cycle. Observed surface features are assumed to explain all of the changes in the solar irradiance. The quiet sun is effectively coupled to the convection zone. Here thermal inertial is too large to allow change on the timescales of the solar cycle.

Proxy models have been successfully used to describe the solar irradiance cycle.

Various implementations can predict 95% of the observed solar irradiance variations

Over the course of several cycles. We can ask whether this success in describing the solar irradiance cycle demonstrates that the models explain the physical cause.

There are several concerns with the use of proxy models. Proxy models have been constructed with irradiance data taken over the last 30 years. During this time there have been no abnormal solar cycles. (An example of an "abnormal" solar cycle is the

Maunder minimum.) If one applies the results from proxy models to unusual solar cycles then one constructs an extrapolation of the model to these conditions. This is dangerous since extrapolations rarely produce accurate results. The irradiance data also represent a source of error. The data is constructed assuming no trend longer

125 than the solar cycle. If there is a trend resulting from a 100 or 500 year cycle, then this trend would be attributed to instrumental drift and removed.

The second explanation of the solar irradiance cycle is that internal structural changes alter energy transport to the solar surface. With this explanation it is assumed that magnetic fields are generated near the base of the convection zone. The presence of magnetic fields provides pressure support that alters the local temperature structure. If the magnetic fields are held in place, the temperature at the top of the magnetized plasma is increased. As the magnetic fields rise to the solar surface, extra entropy is brought to the solar surface. This extra entropy results in the solar cycle.

If the magnetized plasma is not held in place, but is allowed to rise under buoyant force, then the magnetized plasma will cool. When the relatively cooler plasma rises to the solar surface, it represents an entropy deficit. This would result in an anti­ correlated contribution to the solar irradiance cycle that other mechanism would need to account for, in addition to the observed solar irradiance cycle.

Analyzing the shape of the solar limb gives an indication of changes in the solar interior. As the magnetic fields are generated at the base of the convection zone, rotational shear is converted into magnetic field. The internal rotation rates give rise to the surface solar oblateness. By observing changes in the solar oblateness over a solar cycle or more, changes in the solar structure associated with the solar cycle may be detected.

The oblateness calculations presented were constructed using a helioseismic model constructed from data collected during a solar minimum. This was found to agree with concurrent solar oblateness measurements. Later measurements (c.f. Rozelot et al. 2003) did not agree.

Since proxy models are so successful in explaining the solar irradiance cycle, it is necessary to understand how it might be possible for them to describe the solar

126 irradiance cycle without accounting for the physical mechanisms driving this effect. In chapter 2 a plausibility argument was presented. The basic assumption of this model is that surface features do not cause the solar irradiance cycle. Faculae and sunspots made equal but opposite contributions to the solar irradiance cycle. By making this assumption the underlying cause of the solar cycle should be more difficult to "hide" among the signal of the sunspots and faculae.

The plausibility argument showed that the success of proxy models could be reproduced, even with the assumption that surface features did not contribute to the solar irradiance cycle. In this case proxy models succeed by overestimating contributions from facular regions. Thus if it is found that proxy models overestimate the contribution of facular regions to the energy budget of the solar surface, then it is a clear indication that one or more of the physical causes of the irradiance cycle has not been accounted for in the proxy model.

Facular regions were studied for their contributions to the solar irradiance cycle.

Hot wall models were presented as a starting point for this study. The hot wall model describes flux tubes as a cylindrical step function depression in the solar surface. The walls of the depression are hotter than the surrounding solar surface. Energy flows in from the surrounding plasma making the walls of the model "hot". Energy inflow from below the depression is suppressed. Primarily the walls heat the floor of the depression. In this model the floors of large flux tubes would appear cold while the floors of small flux tubes appear warm.

Qualitatively observations were in agreement with the hot wall model. All faculae were seen to brighten near the solar limb. Faculae, which were associated with lower magnetic flux density and were thus associated with small flux tubes, were found to be bright near the disk center, and Facular regions with high magnetic flux density which are assumed to contain larger flux tubes were dark near disk center.

127 Energy budgets of facular regions showed a local increase. This local increase in energy flux could be due to an increase in energy released from the solar surface, or could indicate that flux tubes are preferred routes for energy to escape the solar surface. If a flux tube represents a preferred route for energy to escape the solar surface, then faculae are not one of the causes of the solar irradiance cycle.

The local energy budgets were then compared to estimates derived from solar proxy models. It was found that proxy models overestimated the energy budgets.

This is an indication that proxy models do not account for one or more of the physical mechanisms causing the solar cycle. The physical mechanisms that are unaccounted for are positively correlated with the appearance of surface magnetic fields giving the appearance that the magnetic fields are making a larger contribution to the solar irradiance cycle.

Combining the observations with the physical environment of flux tubes gives insight into the facular energy budgets. Surface flows collect surface magnetic flux, depositing it in down flows. The mantic flux tubes, are expected to be surrounded by the cool down-flows. On the one hand this is supported by the fact that the "floor" of larger flux tubes that is in approximate agreement with the temperature of down­ flow at the estimated depth of the floor. On the other hand, the "walls" of flux tubes are hotter than the mean solar surface. This contradicts reasonable estimates of the temperature structure of the down-flows, implying that flux tubes draw energy from neighboring up-flows. Effectively flux tubes divert energy from reaching one region of the solar surface by redirecting the energy to another. This would be analogous to the "dark rings" predicted by Spruit (c.f. 1976); Deinzer et al. (c.f. 1984).

If the energy associated from flux tubes came from the down-flows, it would represent an increase of energy flux from the solar surface. Instead it is likely that the energy excess in flux tubes comes from neighboring up-flows. The plasma in these

128 up-flows is partially cooled before reaching the surface. Thus the primary effect of facular regions is to redistribute energy radiated in an angular and spatial fashion.

An analogous effect to the hypothesized "dark rings" that surround flux tubes is sunspot bright rings. Sunspot bright rings are where part of the energy, which is blocked by the sunspot, is reradiated by the neighboring solar surface. By comparing observations of sunspot bright rings with theoretical predictions is it possible to understand the mechanisms that transport energy to the solar surface.

Since the solar convection zone is highly conductive, it is often claimed that surface features are thermally coupled to the entire convection zone. A surface feature that is bright is presumed to draw the excess energy that it radiates from the entire convection zone. A surface feature that is dark, is assumed to be similarly coupled to the entire convection zone, and the energy that is blocked from reaching the solar surface is instead distributed over the entire convection zone. The excess is then reradiated or recovered by decreasing the solar luminosity over the time scale of the solar convection zone, on order of 105 years.

By constructing a simple 1 dimensional simple resistor model of the solar convection zone, it was shown that the magnitude ofthe thermal conductivity does not determines the fraction of blocked energy that is reradiated. Similarly the fraction of energy that is drawn from the entire convection zone is not determined by the conductivity. Instead the magnitude of the conductivity combined with the heat capacity determines the time scale to achieve thermal equilibrium.

A more complete 2 dimensional diffusion model was constructed. Conductivity estimates from both mixing length theory and least squared fits to convection simulations were used to calculate the fraction of energy that a sunspot bright ring reradiates. For realistic sunspot depths ( 5Mm), the diffusion models predict that sunspot bright rings reradiate less than 1% of the energy blocked by sunspots.

129 By examining sunspots for one disk crossing, total energy budgets can be constructed. Comparing the energy budgets from observations with predictions from models gives insight into energy transport in the solar convection zone. It was found that sunspot bright rings radiated at least 40% of the energy blocked by the sunspot, far more than is predicted by models. However, the excess energy radiated by bright rings is primarily associated with magnetic features, comparable to faculae.

The question of whether faculae, and their surrounding dark ring make a net contribution to the solar energy budget remains unresolved. While magnetic features are likely to appear in most sunspots, if facular regions make a net contribution to the energy budget, then they represent a source of contamination when comparing to models. By removing facular regions the bright rings were found to have no detectable bright rings. This is in agreement with diffusion models.

The facular regions in sunspot bright rings were noted to radiate less energy, locally, than comparable facular regions that were unassociated with sunspots. This is an indication that flux tubes act as a preferred channel for energy to reach the solar surface. Flux tube density in sunspot bright rings is higher than in regions of the solar surface that are unassociated with sunspots. Flux tubes in bright rings sit in the dark rings of neighboring flux tubes. The neighboring flux tubes divert energy away from each other. This is an indication that flux tubes divert much of the associate energy excess from the surrounding solar surface.

Even if facular regions do contribute to the energy budget, diffusion models remain problematic. If diffusion is to explain the energy transport on the scale of both sunspots and flux tubes then flux tube do not represent an increase in the solar surface brightness. Thus the energy associated with the flux tubes in bright rings does not increase the energy budget of the bright ring. This implies that the bright rings would be comparably bright with or without the flux tubes, and the presence

130 of flux tubes can be ignored. This implies that the thermal bright rings radiate far more energy that is predicted by diffusion. At some scale diffusion cannot describe the energy transport.

It is not surprising that energy transport in the solar convection zone is not well described by diffusion. Diffusion would require that the convective motions approximate Kolmogorov turbulence. Instead convection is found to consist of large­ scale coherent flows. Convection is driven from the solar surface by down-flows which span the depth of the convection zone, and reach speeds on order of the local sound speed (Stein & Nordlund 1998).

The argument thus far has considered mainly time scales of about two weeks.

Energy budgets that have been constructed consider an average of the features over the duration of a complete disk crossing. While it is possible that faculae and sunspots do not contribute to the solar irradiance cycle it is unlikely that they are not the cause of short-term (1 week or less) irradiance variations; these variations can be as large as 0.4%.

At this point one should consider the effect of sunspots and faculae on the short term, Le. a few days vs. the long term, Le. two months or more. Since a large percentage of the blocked energy reemerges in the sunspot bright ring, it must be possible to understand how sunspots can have large short-term effects on the solar irradiance, while still having little effect in the long tenn. First, it should be noted that the energy radiated by sunspot bright rings is associated with flux tubes. Both flux tubes and sunspots are darkest at disk center. If we assume that the sunspot and bright ring system contributes nothing to the long term solar irradiance, then it would still be observed that while the system is at disk center the system appears fainter than on average for the system, which was assumed to be equal to the quiet sun. Thus the sunspot bright ring system is darker then the quite sun when viewed

131 at disk center. When the system is near the solar limb, it would similarly be brighter than the average quiet sun, increasing the solar irradiance.

The question of the physical mechanism that drives the solar irradiance cycle can now be addressed. The physical process must be correlated with the appearance of magnetic fields on the solar surface. The simple answer is that magnetic fields drag excess entropy from the base of the solar convection zone. While this solution shows promise more work must be done to understand this process and turn it into a predictive model.

6.2 Conclusions

It has been shown that diffusion models do not describe energy transport in the solar convection zone. Energy is transported by large-scale bulk motions over large distances. For small features such as flux tubes the effects of radiative transfer also plays a meaningful role. This can be seen by examination of flux tubes at disk center where the "floor" is heated by the "walls".

Faculae and sunspots primarily redistribute energy that is radiated at the solar surface in an angular and spatial fashion. Surface features cannot be thought of as coupled to the entire convection zone. In particular facular regions divert energy from neighboring up-flows; this gives facular regions the appearance that they are bright.

Instead of increasing the solar irradiance, they are preferred channels for energy to escape the solar surface. The result is that facular regions primarily act to redistribute energy radiated from the solar surface in an angular and spatial fashion.

Sunspots have large bright rings that radiate 40% or more of the sunspot's energy deficit. This reradiated energy is associated with magnetic fields in the bright rings.

Even if the sunspot bright ring system were to contribute nothing to the solar

132 irradiance cycle, they can still reproduce irradiance variations on the time scales of a week or less. These variations in the solar irradiance are caused primarily by the angular redistribution of radiated energy, i.e. limb brightening of flux tubes in the bright ring, and a similar brightening of the sunspot near the solar limb.

While proxy models do describe the irradiance variations, they do not explain the physical mechanisms that cause the variations. Proxy models overestimate the energy budget of facular regions, effectively hiding the physical cause in faculae.

Surface features do not explain all of the solar irradiance cycle. The remainder of the irradiance cycle must be driven from the quiet sun, and it is likely to be driven by changes in the energy transport in the solar convection zone.

133 References

Armstrong, J. & Kuhn, J. R. 1999, ApJ, 525, 533

Beck, J. G. & Chapman, G. A. 1993, Sol. Phys., 146,49

Berger, T. E. & Title, A. M. 1996, ApJ, 463, 365

Caligari, P., Moreno-Insertis, F., & Schussler, M. 1995, ApJ, 441, 886

Carroll, B. W. & Ostlie, D. A. 1996, An introduction to modern astrophysics

(Reading, MA: Addison-Wesley, -c1996)

Chapman, G. A. 1987, ARA&A, 25,633

Choudhuri, A. R., Schussler, M., & Dikpati, M. 1995, A&A, 303, L29+

Christensen-Dalsgaard, J. 1992, ApJ, 385, 354

Cox, A. N. 2000, Allen's astrophysical quantities (Allen's astrophysical quantities, 4th ed. Publisher: New York: AlP Press; Springer, 2000. Editedy by Arthur

N. Cox. ISBN: 0387987460)

Das, A. K. & Ramanathan, A. S. 1953, Zeitschrift fur Astrophysics, 32, 91

Deinzer, W., Hensler, G., Schussler, M., & Weisshaar, E. 1984, A&A, 139,435

134 Dicke, R. H. 1970, ApJ, 159, 1

Dikpati, M. & Choudhuri, A. R. 1994, A&A, 291, 975

Douglas, A. E. 1919, A&A, 32, 233

D'Silva, S. & Choudhuri, A. R. 1993, A&A, 272, 621

Edmonds, A. R. 1957, Angular Momentum in Quantum Mechanics (Princeton, Princeton Univ. Press)

Fan, Y, Fisher, G. H., & McClymont, A. N. 1994, ApJ, 436, 907

Fligge, M., Solanki, S. K., & Unruh, Y C. 2000, A&A, 353, 380

Foukal , P. 1990, Solar astrophysics (New York, Wiley-Interscience, 1990,492 p.)

Foukal, P., Fowler, L. A., & Livshits, M. 1983, ApJ, 267, 863

Fowler, L. A., Foukal, P., & Duvall, T. 1983, Sol. Phys., 84, 33

Frohlich, C. 2000, Space Science Reviews, 94, 15

-.2002, Advances in Space Research, 29, 1409

Frohlich, C. & Lean, J. 1998, Geophys. Res. Lett., 25, 4377

Goldreich, P. & Schubert, G. 1968, ApJ, 154, 1005

Hale, G. E. 1908a, ApJ, 28, 315

-. 1908b,ApJ, 28, 100

Hale, G. E. & Ellerman, F. 1906, ApJ, 23, 54

Hale, G. E., Ellerman, F., Nicholson, S. B., & Joy, A. H. 1919, ApJ, 49, 153

135 Hirayama, T., Hamana, S., & Mizugaki, K 1985, Sol. Phys., 99,43

Hirayama, T. & Okamoto, T. 1981, Sol. Phys., 73, 37

Howard, R, Boyden, J. E., & Labonte, B. J. 1980, Sol. Phys., 66, 167

Howard, R. & Stenflo, J. O. 1972, Sol. Phys., 22, 402

Jensen, E. 1955, Annales d'Astrophysique, 18, 127

Kosovichev, A. G. 1996, ApJ, 469, L61 +

Kosovichev, A. G., Schou, J., Scherrer, P. H., Bogart, R S., Bush, R 1., Hoeksema,

J. T., Aloise, J., Bacon, L., Burnette, A., de Forest, C., Giles, P. M., Leibrand, K.,

Nigam, R, Rubin, M., Scott, K, Williams, S. D., Basu, S., Christensen-Dalsgaard,

J., Dappen, W., Rhodes, E. J., Duvall, T. L., Howe, R, Thompson, M. J., Gough,

D.O., Sekii, T., Toomre, J., Tarbell, T. D., Title, A. M., Mathur, D., Morrison,

M., Saba, J. L. R, Wolfson, C. J., Zayer, 1., & Milford, P. N. 1997, Sol. Phys., 170,

43

Kuhn, J. R & Armstrong, J. D. 2002, Bulletin of the American Astronomical Society,

34,791

Kuhn, J. R, Bush, R 1., Scherrer, P., & Scheick, X. 1998, Nature, 392, 155

Kuhn, J. R & Georgobiani, D. 2000, Space Science Reviews, 94, 161

Kuhn, J. R, Libbrecht, KG., & Dicke, R H. 1988, Science, 242,908

Lawrence, J. K 1988, Sol. Phys., 116, 17

Lawrence, J. K, Chapman, G. A., Herzog, A. D., & Shelton, J. C. 1985, ApJ, 292,

297

136 Lawrence, J. K, Topka, K P., & Jones, H. P. 1993, J. Geophys. Res., 98, 18911

Lean, J. 1., Cook, J., Marquette, W., & Johannesson, A. 1998, ApJ, 492, 390

Li, L. H. & Sofia, S. 2001, ApJ, 549, 1204

Lin, H. 1995, ApJ, 446, 421

Lin, H. & Rimmele, T. 1999, ApJ, 514,448

Lydon, T. J. & Sofia, S. 1995, ApJS, 101,357

MacGregor, K B. & Cassinelli, J. P. 2003, ApJ, 586, 480

Maunder, E. W. 1913, MNRAS, 74, 112

-. 1922, MNRAS, 82, 534

Moreno-Insertis, F., Schuessler, M., & Ferriz-Mas, A. 1992, A&A, 264, 686

Nye, A., Bruning, D., & Labonte, B. J. 1988, Sol. Phys., 115,251

Ortiz, A., Solanki, S. K, Domingo, V., Fligge, M., & Sanahuja, B. 2002, A&A, 388,

1036

Parker, E. N. 1955, ApJ, 121,491

-. 1987,ApJ, 312, 868

Paterno, L., Sofia, S., & di Mauro, M. P. 1996, A&A, 314, 940

Pijpers, F. P. 1998, MNRAS, 297, L76

Preminger, D. G., Walton, S. R., & Chapman, G. A. 2002, Journal of Geophysical Research (Space Physics), 6

137 Pyndyck, R S. & Rubinfeld, D. L. 1991, Econometric Models and Economic Forecasts

(New York, Irwin McGraw-Hill, 1991,654 p.)

Rast, M. P., Fox, P. A., Lin, H., Lites, B. W., Meisner, R W., & White, O. R 1999,

Nature, 401, 678

Rast, M. P., Meisner, R W., Lites, B. W., Fox, P. A., & White, O. R 2001, ApJ,

557, 864

Rosch, J., Rozelot, J. P., Deslandes, H., & Desnoux, V. 1996, Sol. Phys., 165, 1

Royds, T. 1925, MNRAS, 85, 464

Rozelot, J., Lefebvre, S., & Desnoux, V. 2003, Sol. Phys., 217,39

Scherrer, P. H., Bogart, R S., Bush, R I., Hoeksema, J. T., Kosovichev, A. G., Schou, J., Rosenberg, W., Springer, L., Tarbell, T. D., Title, A., Wolfson, C. J.,

Zayer, I., & MDI Engineering Team. 1995, Sol. Phys., 162, 129

Schou, J., Antia, H. M., Basu, S., Bogart, R S., Bush, R I., Chitre, S. M.,

Christensen-Dalsgaard, J., di Mauro, M. P., DZiembowski, W. A., Eff-Darwich,

A., Gough, D.O., Haber, D. A., Hoeksema, J. T., Howe, R, Korzennik, S. G.,

Kosovichev, A. G., Larsen, R M., Pijpers, F. P., Scherrer, P. H., Sekii, T., Tarbell,

T. D., Title, A. M., Thompson, M. J., & Toomre, J. 1998, ApJ, 505, 390

Schwarzschild, M., Howard, R, & Harm, R 1957, ApJ, 125,233

Solanki, S. K. & Fligge, M. 2002, Advances in Space Research, 29, 1933

Spruit, H. C. 1976, Sol. Phys., 50,269

-. 1977, Sol. Phys., 55, 3

138 -. 1982, A&A, 108, 356

Spruit, H. C. & Roberts, R 1983, Nature, 304, 401

Stein, R F. & Nordlund, A. 1998, ApJ, 499, 914

Stenflo, J. O. 1973, Sol. Phys., 32, 41

Thompson, M. J., Toomre, J., Anderson, E., Antia, H. M., Berthomieu, G., Burtonclay, D., Chitre, S. M., Christensen-Dalsgaard, J., Corbard, T., Derosa, M., Genovese, C. R., Gough, D.O., Haber, D. A., Harvey, J. W., Hill, F., Howe, R, Korzennik, S. G., Kosovichev, A. G., Leibacher, J. W., Pijpers, F. P., Provost, J., Rhodes, E. J., Schou, J., Sekii, T., Stark, P. R, & Wilson, P. 1996, Science, 272, 1300

Tomczyk, S., Schou, J., & Thompson, M. J. 1995, ApJ, 448, L57+

Toner, C. G., Jefferies, S. M., & Duvall, T. L. 1997, ApJ, 478, 817

Topka, K. P., Tarbell, T. D., & Title, A. M. 1992, ApJ, 396, 351

-. 1997, ApJ, 484,479

Ulrich, R. K. & Hawkins, G. W. 1981, ApJ, 246,985

Willson, R C., Gulkis, S., Janssen, M., Hudson, H. S., & Chapman, G. A. 1981, Science, 211, 700

Wittmann, A. D. 1996, Sol. Phys., 167,441

Zhao, J., Kosovichev, A. G., & Duvall, T. L. 2001, ApJ, 557, 384

Zwaan, C. 1968, ARA&A, 6, 135

139