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.