Eclipsing binary stars in open clusters John K. Taylor M.Sci. (Hons.) St. Andrews Doctor of Philosophy School of Chemistry and Physics, University of Keele. March 2006 iii Abstract The study of detached eclipsing binaries allows accurate absolute masses, radii and luminosities to be measured for two stars of the same chemical composition, distance and age. These data can be used to test theoretical stellar models, investigate the properties of peculiar stars, and calculate its distance using empirical methods. De- tached eclipsing binaries in open clusters provide a more powerful test of theoretical models, which must simultaneously match the properties of the eclipsing system and the cluster. The distance and metal abundance of the cluster can be found without the problems of main sequence fitting. Absolute dimensions have been found for V615 Per and V618 Per, which are eclipsing members of h Persei. The fractional metal abundance of the cluster is Z ≈ 0.01, in disagreement with literature assumptions of a solar chemical composi- tion. Accurate absolute dimensions have been measured for V453 Cygni, a member of NGC 6871. The current generation of theoretical stellar models can match these properties, as well as the central concentration of mass of the primary star as derived from a study of the apsidal motion of the system. Absolute dimensions have been determined for HD 23642, a member of the Pleiades. This has allowed an investigation into the usefulness of different methods to find the distances to eclipsing binaries. A new method has been introduced, based on calibrations between surface brightness and effective temperature, and used to find a distance of 139 ± 4 pc. This value is in good agreement with other Pleiades distance measurements but does not agree with the controversial Hipparcos parallax distance. The metallic-lined eclipsing binary WW Aur has been studied using extensive new spectroscopy and published light curves. The masses and radii have been found to accuracies of 0.6% using completely empirical methods. The predictions of theoretical models can only match the properties of WW Aur by adopting Z = 0.060 ± 0.005. iv Acknowledgements I am grateful to Pierre Maxted for being an excellent supervisor and to Barry Smalley for being exceptionally useful. Thanks are also due to others who have collaborated with me on this work: Shay Zucker, Paul Etzel and Antonio Claret. Data have been made available by Ulisse Munari, Philip Dufton, Danny Lennon and Kim Venn. Useful discussions have been undertaken with Jens Viggo Clausen, Liza van Zyl, Steve Smartt, Ansgar Reiners, Roger Diethelm, Ron Hilditch, David Holmgren, Rob Jeffries, Nye Evans, Onno Pols, Jørgen Christensen-Dalsgaard, Frank Grundahl, Hans Bruntt and Sylvain Turcotte (in no particular order). Overly frank discussions have also been conducted with Ulisse Munari. v Contents Abstract ....................................... iii Acknowledgements ................................ iv 1 Detached eclipsing binary stars ...................... 1 1.1 Stars . 1 1.1.1 Stellar characteristics . 4 1.1.1.1 Stellar interferometry . 4 1.1.1.2 The effective temperature scale . 4 1.1.1.3 Stellar chemical compositions . 4 1.1.1.4 Bolometric corrections . 5 1.1.1.5 Surface brightness relations . 7 1.1.2 Limb darkening . 11 1.1.2.1 Limb darkening laws . 11 1.1.2.2 Limb darkening and eclipsing binaries . 14 1.1.3 Gravity darkening . 15 1.2 Stellar evolution . 16 1.2.1 The evolution of single stars . 16 1.2.1.1 Main sequence evolution . 17 1.2.1.2 Evolution of low-mass stars . 18 1.2.1.3 Evolution of intermediate-mass stars . 18 1.2.1.4 Evolution of massive stars . 19 1.3 Modelling of stars . 19 1.3.1 Details of some of the physical phenomena included in theoretical stellar evolutionary models . 21 1.3.1.1 Equation of state . 21 1.3.1.2 Opacity . 21 1.3.1.3 Energy transport . 22 1.3.1.4 Convective core overshooting . 22 1.3.1.5 Convective efficiency . 25 1.3.1.6 The effect of diffusion on stellar evolution . 27 1.3.2 Available theoretical stellar evolutionary models . 29 1.3.2.1 Granada theoretical models . 29 1.3.2.2 Geneva theoretical models . 29 1.3.2.3 Padova theoretical models . 30 1.3.2.4 Cambridge theoretical models . 30 1.3.3 Comments on the currently available theoretical models . 31 1.4 Spectral characteristics of stars . 31 1.4.1 Spectral lines . 31 vi 1.4.2 Stellar model atmospheres . 33 1.4.2.1 The current status of stellar model atmospheres . 34 1.4.2.2 Convection in model atmospheres . 34 1.4.2.3 The future of stellar model atmospheres . 35 1.4.3 Calculation of theoretical stellar spectra . 36 1.4.3.1 Microturbulence velocity . 37 1.4.3.2 The uclsyn spectral synthesis code . 38 1.4.4 Spectral peculiarity . 38 1.4.4.1 Metallic-lined stars . 39 1.5 Multiple stars . 41 1.5.1 Binary star systems . 42 1.5.2 Eclipsing binary systems . 43 1.6 Detached eclipsing binary star systems . 44 1.6.1 Comparison with theoretical stellar models and atmospheres . 49 1.6.1.1 The methods of comparison . 50 1.6.1.2 Further work . 52 1.6.1.3 The difference between stars in binary systems and sin- gle stars . 53 1.6.2 The metal and helium abundances of nearby stars . 54 1.6.3 Detached eclipsing binaries as standard candles . 55 1.6.3.1 Distance determination using bolometric corrections . 56 1.6.3.2 Distances from surface brightness calibrations . 58 1.6.3.3 Distance determination by modelling of the stellar spec- tral energy distributions . 59 1.6.3.4 Recent results for the distance to eclipsing binaries . 60 1.6.4 Detached eclipsing binaries in stellar systems . 61 1.6.4.1 Results on detached eclipsing binaries in clusters . 62 1.7 Tidal effects . 64 1.7.1 Orbital circularization and rotational synchronization . 64 1.7.1.1 The theory of Zahn . 65 1.7.1.2 The theory of Tassoul & Tassoul . 68 1.7.1.3 Comparison with observations . 69 1.7.2 Apsidal motion . 72 1.7.2.1 Relativistic apsidal motion . 73 1.7.2.2 Comparison with theoretical models . 75 1.7.2.3 Comparison between observed density concentrations and theoretical models . 76 1.8 Open clusters . 77 2 Analysis of detached eclipsing binaries ................. 80 2.1 Observing detached eclipsing binaries . 80 vii 2.1.0.4 Photometry of dEBs . 80 2.1.0.5 Spectroscopy of dEBs . 81 2.2 Determination of spectroscopic orbits . 81 2.2.1 Equations of spectroscopic orbits . 81 2.2.2 The fundamental concept of radial velocity . 83 2.2.3 Radial velocity determination from observed spectra . 84 2.2.3.1 Radial velocities from individual spectral lines . 85 2.2.3.2 Radial velocities from one-dimensional cross-correlation 90 2.2.3.3 Radial velocities from two-dimensional cross-correlation 91 2.2.3.4 Radial velocities from spectral disentangling . 94 2.2.4 Determination of spectroscopic orbits from observations . 95 2.2.4.1 sbop – Spectroscopic Binary Orbit Program . 98 2.2.5 Determination of rotational velocity from observations . 99 2.3 Photometry . 100 2.3.1 Photometric systems . 100 2.3.1.1 Broad-band photometric systems . 101 2.3.1.2 Broad-band photometric calibrations . 103 2.3.1.3 Str¨omgrenphotometry . 104 2.3.1.4 Str¨omgrenphotometric calibrations . 106 2.4 Light curve analysis of detached eclipsing binary stars . 109 2.4.1 Models for the simulation of eclipsing binary light curves . 110 2.4.1.1 ebop – Eclipsing Binary Orbit Program . 111 2.4.1.2 The Wilson-Devinney (wd) code . 114 2.4.1.3 Comparison between light curve codes . 117 2.4.1.4 Other light curve fitting codes . 118 2.4.1.5 Least-squares fitting algorithms . 118 2.4.2 Solving light curves . 120 2.4.2.1 Calculation of the orbital ephemeris . 122 2.4.2.2 Initial conditions . 123 2.4.2.3 Parameter determinacy and correlations . 128 2.4.2.4 Final parameter values . 129 2.4.3 Uncertainties in the parameters . 130 2.4.3.1 The problem . 130 2.4.3.2 The solutions . 132 3 V615 Per and V618 Per in h Persei .................... 134 3.1 V615 Per and V618 Per . 134 3.1.1 h Persei and χ Persei . 136 3.2 Observations . 139 3.2.1 Spectroscopy . 139 3.2.2 Photometry . 140 viii 3.3 Period determination . 144 3.3.1 V615 Per . 144 3.3.2 V618 Per . 145 3.4 Spectral disentangling . 148 3.5 Spectral synthesis . 150 3.6 Spectroscopic orbits . 151 3.6.1 V615 Per . 151 3.6.2 V618 Per . 156 3.6.3 The radial velocity of h Persei . 157 3.7 Light curve analysis . 157 3.7.1 jktebop .............................. 157 3.7.2 V615 Per . 158 3.7.3 V618 Per . 161 3.8 Absolute dimensions and comparison with stellar models . 164 3.8.1 Stellar and orbital rotation . 164 3.8.2 Stellar model fits . 167 3.9 Discussion . 167 4 V453 Cyg in the open cluster NGC 6871 ................ 170 4.1 V453 Cyg . 170 4.1.1 NGC 6871 . 174 4.2 Observations . 174 4.3 Period determination and apsidal motion . 179 4.4 Spectral synthesis . 180 4.5 Spectroscopic orbits . ..