Light Curves of W Ursae Majoris Systems
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The Light Curves of W Ursae Majoris Systems by Ian Stuart Rudnick A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1972 UNIVERSITY OF FLORIDA 3 1262 08666 451 2 To my wife, Andrea , ACKNOWLEDGMENTS The author sincerely expresses his appreciation to his conmiittee chairman and advisor, Dr. Frank BradshawJ Wood for his comments and suggestions, which greatly aided the completion of this work. The author wishes to thank Dr. R. E. Wilson of the University of South Florida for provide ing one of the synthetic light curves and for serving on the author's committee. Thanks are also due to Drs. K-Y Chen and R. C. Isler for serving on the author's committee. The author expresses his gratitude to Dr. S. M. Rucinski for providing the other synthetic light curve. The author wishes to thank Drs. J. E. Merrill and J. K. Gleim for their many helpful discussions. Thanks are also due to R. M. Williamon and T. F. Collins for their help in ob- taining some of the data and for many enlightening conver- sations. W. W. Richardson deserves highest commendation for his untiring work on the drawings. The author extends his thanks to the Department of Physics and Astronomy for providing financial support in the form of graduate assistantships, and to the Graduate School for support in the form of a Graduate School Fellow- ship* The author is indeed grateful to his parents and to his wife's parents for their encouragement. The author's wife deserves more than appreciation for her patience, encouragement, and hard work during five years of school life. Her devotion and understanding helped as nothing else could. TABLE OF CONTENTS Page ACKNOWLEDGMENTS . Hi LIST OF TABLES . vii LIST OF FIGURES . x ABSTRACT xii CHAPTER I INTRODUCTION 1 The Russell Model 2 CHAPTER II FIRST SYNTHETIC LIGHT CURVE 6 Rectification. 6 Solutions from Graphical Rectification 22 Solutions from Least Squares Rectification .... 44 Orbital Elements 55 Figures of the Components 58 Comparison of Solution with Input Parameters ... 59 CHAPTER III SECOND SYNTHETIC LIGHT CURVE. 64 Rectification , 69 Solutions ......-.„ 75 Orbital Elements and Figures of the Components . 78 Comparison of Solutions with Input Parameters. 78 CHAPTER IV THE SYSTEM OF 44i BOOTIS 87 History 87 Visual Binary 89 Spectroscopic Binary 91 Eclipsing Variable 92 Instrumentation. 93 Observations 96 Rieductioncdf Data 98 Times of Minimxam Light and the Period. 126 Variation in the period caused by motion in a visual binary system . 126 Page A recent period change 132 Light Variations. 133 Rectification . 144 Orbital Elements 146 CHAPTER V SUMMARY AND CONCLUSIONS 153 LIST OF REFERENCES 156 BIOGRAPHICAL SKETCH 161 1 LIST OF TABLES Page I Observations of First Synthetic Light Curve,; .... 7 II Graphical Rectification Coefficients for First Synthetic Light Curve 13 III Least Squares Rectification Coefficients for First Synthetic Light Curve 15 IV Rectified First Synthetic Light Curve Using Graphi- cal Rectification Coefficients 16 V Rectified First Synthetic Light Curve Using Least Squares Rectification Coefficients .... 17 VI Solutions with Different Values of Limb Darkening. 25 VII Solution for x = 0.8, 1 - £P^= 0.18, 1 - Z^^^= 0.16, o o ' = = k 0.45, p^ -0.94, a°^ = 0.9943, a^^= 0.9698. 26 VIII Solution for x = 1.0, 1 - £P^= 0.18, 1 - il^®^= 0.16, o o ' k =0.45, p^ = -0.90, a°^= 0.9905, a^^= 0.9375 ... 27 IX Solutions for x = 0.6, 1 - Z^^/9P^= 0.2142, -l^^) + o o (1 o (1 - 1°^) =0.34 33 X Solution for x = 0.8, 1 - l^^= 0.18, 1 - 1^^^= 0.16, o o ' k = 0.50, p = -0.6608, a°^= 0.89, a^^= 0.7915 ... 34 XI Solution for x = 0.8, 1 - Z^^= 0.18, 1 - 1^^^= 0.16, o o ' k = 0.55, p^ =-0.4798, a°^= 0.78, a^^= 0.6659. ... 35 XII Solution for x = 1.0, 1 - £P^= 0.18, 1 - £^®^= o o 0.16, k = 0.60, p = -0.3225, a°^= 0.68, a^^= 0.5332 36 ^ o o o ... VI Page OP "tT" XIII Shape Curves for 0^ = 50°, X = 3.955, x = oc 4.605;- X = 0.8 and = 45?9, x = 3.476, ^^ = 4.047, X = 0.8 37 XIV Depth Curve for x = 0.8, 1 - £^ = 0.16, 1 - 1^^= 0.18 41 XV Solution for x = 0.8, 1 - i^^= 0.16, 1 -!i^^^= oc 0.145, pr-oc, k = 0.65, p = -0.1625, a^ = 0.55, a^^= 0.4548 47 o jpr_ „ ,, , „sec XVI Solution for x = 0.8, 1 - 5,^ = 0.16, 1 - H^ 0.145, pr-tr, k = 0.65, p = -0.1271, a°^= 0.525, a^''= 0.4314 ..48 o XVII Solution for X = 0.8, 1 - 2p^= 0.16, 1 - Z^^^= 0.145, k = 1.0, p = 0.1540, a = 0.305 49 ^o o XVIII 0-C's from Solution for k = 1, p = 0.1540, a = 0.305 51 o XIX Solution for x = 0.8, 1 - 1^^= 0.165, 1 - ^q^*^= 0.145, k = 0.70, p = -1.429, a°^= 1.0, aj^ = ^o o o 1.067, T= 0.558590, L^ = 0.588 ^^ XX Orbital Elements for Solutions of First Synthetic Light Curve 57 XXI Figures of the Components 60 XXII Observations of Second Synthetic Light Curve • • • 67 XXIII Rectification Coefficients for Second Synthetic Light Curve 70 XXIV Rectified Second Synthetic Light Curve -j^. OC tr XXV Solution for x = 0.4, k = 0.545, a =1.0, a„ = ' o o 1.016, p =-1.10,T = 0.3126, 1 - 2.g^= 0.265, 1 -i^^^= 0.233 -76 Vlll , Page oc XXVI Solution for x = 0.4, k = 0.65, a^ = 1.0, a^^= 1.039, p = -1.538,T = 0.447537, 1 - il?^= o o ' o 0.265, 1 - 1^^^= 0.233, L-,= 0.234 79 XXVII Orbital Elements for Solutions of Second Synthetic Light Curve 80 XXVIII Figures of the Components 81 XXIX Input Parameters for Second Synthetic Light Curve 83 XXX Comparison Stars . 97 XXXI Observations of 44i Bootis 99 XXXII Standard Stars . 112 XXXIII Recent Times of Minimum Light 127 XXXIV Times of Minimiam Light with Corrections for Motion in a Visual Binary Orbit ,...,i.,.,. 131 XXXV Solutions and Orbital Elements Mth x = .6 1 _ jiP^= 0.23, 1 - £^®°= 0.13, = 41?0, and o o ' e m /m^ = 0.50 150 LIST OF FIGURES Page 1. Observations of First Synthetic Light Curve 8 2. Graphical Rectification Plots for First Synthetic Light Curve 12 3. Rectified First Synthetic Light Curve Using Graphical Rectification Coefficients . 19 4. Rectified First Synthetic Light Curve Using Least Squares Rectification Coefficients 21 5. Solution for X = 0.8, 1 - Sp^= 0.18, 1 -l^^^= 0.16, o o k = 0.45, p^ =-0.94, a°^= 0.9943, a^^= 0.9698. ... 29 6. Solution for x =1.0, 1 - £P^= 0.18, 1 - il^®°= 0.16, ' o o ' k = 0.45, p =-0.90, 0.°^= 0.9905, a^^= 0.9375. ... 31 7. Depth Curve and Shape Curves 39 8. Superpositions of Primary and Secondary Eclipses . 43 9. 0-G's from Solution for k = 1, p = 0.1540, a = 0.305 °. 52 o 10. Observations of Second Synthetic Light Curve .... 66 11. Rectified Second Synthetic Light Curve 74 12. Visual Binary Orbit of 44i Bootis 90 13. Comparison Star Extinction for July 1, 1970 110 14. Second-order Extinction for Albireo 114 15. First-order Yellow Extinction for Albireo 117 16. First-order Blue Extinction for Albireo 119 Page 17. First-order Ultra-Violet Extinction for Albireo . 121 18. UBV Transformation Coefficients 123 19. Variation in the Period of 44i Bootis B Caused by Motion in a Visual Binary System 130 20. 0-C's from Pbhl's Light Elements 134 21. 0-C's from New Light Elements 135 22. Yellow Light Curve of 44i Bootis 136 23. Blue Light Curve of 44i Bootis 137 24. The Light Curves of Different Authors 139 25. The Light Curves of Different Authors 140 26. Deformities of the Light Curve of 44i Bootis on Two Nights 143 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The Light Curves of W Ursae Majoris Systems by Ian Stuart Rudnick June, 1972 Chairman: Frank Bradshaw Wood Major Department: Astronomy Two synthetic light curves computed from theoretical astrophysical models of W Ursae Majoris systems are discussed. Solutions of these light curves, based on the geometrical Russell model and the Russell-Merrill method of solution of the light curves of eclipsing binaries, are presented. The relatively shallow minima caused by the partial eclipses of the first synthetic light curve lead to a problem of indeterminacy in the solution. The "observed" points in the shoulders of both synthetic light curves fall below the theoretical light curves predicted by the solutions. The addition of third light in the solution of the two synthetic light curves improves the fit of the solutions to the light curves; however, there is no sound basis for adding this third light. The orbital elements predicted by the Russell- Merrill solutions of the two synthetic light curves are not at all close to the orbital elements used to generate these light curves from the theoretical astrophysical models. In particular the Russell-Merrill solution underestimates the sizes of the components. It is concluded that the Russell model is not compatible with the theoretical astrophysical models used to generate the synthetic light curves.