Eclipsing Variables - What They can Tell Us and What We can do with Them The aim of the present book will be to provide an introduction to the inter- pretation of the observed light changes of eclipsing binary stars and their analysis for the elements of the respective systems. Whenever we study the properties of any celestial body - be it a planet or a star - all information we wish to gain can reach us through two different channels: their gravitational attraction, and their light. Gravitational interaction between our Earth and its celestial neighbours is, however, measurable only at distances of the order of the dimensions of our solar system; and the only means of communication with the realm of the stars are their nimble-footed photons reaching us - with appropriate time-lag - across the intervening gaps of space. As long as a star is single and emits constant light, it does not constitute a very revealing source of information.
A spectrometry of its light can disclose, to be sure, the temperature (colour, or ionization) of the star's semi-transparent outer layers, their chemical composition, and prevalent pressure (through Stark effect) or magnetic field (Zeeman effect), it can disclose even some information about its absolute luminosity or rate of spin. It cannot, however, tell us anything about what we should like to know most - namely, the mass or size (i.e., density) of the respective configuration; its absolute dimensions, or its internal structure.
Eclipsing Variables-What They can Tell Us and What We can do with Them.- I. The Code of the Language.- I. Light Changes of Eclipsing Binary Systems: Spherical Stars.- 1. Equations of the Problem.- 2. Loss of Light During Eclipses.- A. Evaluation of the Integrals.- 3. Loss of Light as Aperture Cross-Correlation.- A. Cross-Correlation: a Definition.- B. Hankel Transforms.- C. Evaluation of the Hankel Transforms.- 4. Fourier Transform of the Light Changes.- A. Evaluation of Fourier Transforms.- B. Synthesis of the Fourier Transform.- Bibliographical Notes.- II. Light Changes of Close Eclipsing Systems.- 1. Light Changes of Distorted Stars Outside Eclipses.- A. Discussion of the Results.- 2. Light Changes of Distorted Systems Within Eclipses.- A. Theoretical Light Curves.- Bibliographical Notes.- III. Special Functions of the Theory of Light Curves.- 1. Associated Alpha-Functions and Related Integrals.- A. Literal Evaluation.- B. Discussion of the Results.- 2. Recursion Relations.- A. Recursion Relations for the ?nm-functions.- B. Recursion Relations for the Boundary Integrals.- 3. Differential Properties.- A. Differential Properties of the ?-functions.- B. Differential and Recursion Properties of the Boundary Corrections: the J-integrals.- C. Differential Properties of Hankel Transforms.- Bibliographical Notes.- II. Decoding of the Light Changes.- IV. Analysis of the Light Changes in the Time Domain.- 1. Direct Methods: Historical.- 2. Iterative Methods: Total and Annular Eclipses.- A. Formation and Weighting of the Equations of Condition.- B. Differential Corrections for ? and U.- C. Combination of Both Minima.- D. Evaluation of the Elements.- 3. Iterative Methods: Partial Eclipses.- A. Combination of the Alternate Minima.- B. Generalized Equations of Condition: Differential Corrections.- C. Evaluation of the Elements.- 4. Effects of Orbital Eccentricity.- A. Central Eclipses.- B. Inclined Orbits.- Bibliographical Notes.- V. Analysis of the Light Changes in the Frequency-Domain: Spherical Stars.- 1. Introduction: Uniformly Bright Stars.- A. Total (Annular) Eclipses.- B. Partial Eclipses.- C. Relation with Fourier Series.- 2. General Case: Limb-Darkened Stars.- A. Evaluation of the Moments A2m of Eclipses.- B. Alternative Evaluation of the Moments A2m.- C. Differential Properties of the Moments A2m.- D. Atmospheric Eclipses.- 3. Solution for the Elements.- A. Optimization of Solutions.- B. Combination of Alternate Minima.- C. Eccentric Orbits.- Bibliographical Notes.- VI. Analysis of the Light Changes in the Frequency-Domain: Distorted Stars.- 1. Modulation of the Light Curves.- A. Particular Cases.- 2. Photometric Perturbations.- A. Particular Case: Total Eclipses.- B. Summary of Procedure.- Bibliographical Notes.- VII. Error Analysis.- 1. Error Analysis in the Time-Domain.- A. Errors of Auxiliary Constants.- B. Other Combinations of Elements; Limb-Darkening.- 2. Differential Corrections.- A. Equations of the Problem.- B. Combination of Both Minima.- 3. Error Analysis in the Frequency-Domain.- A. Error Analysis by Fourier Series.- B. Relation between the Fourier Transform and Moments of the Light Curves.- C. Recursion Relations between Different Moments of the Light Curves.- D. Solution for the Moments and Their Uncertainty.- Bibliographical Notes.- Appendices.- Appendix II. Explicit Forms of Associated ?-Functions of Even Orders and Indices.- Appendix III. Definition of Hypergeometric Series.- References.- Name Index.