Binary Stars and Their Light Curves

Astrophysics of Variable Stars ASP Conference Series, Vol. 349, 2006 C. Sterken & C. Aerts Binary Stars and their Light Curves R. E. Wilson Astronomy Department, University of Florida, Gainesville, FL 32611 U.S.A. Maria Mitchell Observatory, Nantucket, MA 02554 Abstract. This guide to persons newly entering the eclipsing binary field begins by comparing eclipsing binary modeling and analysis with the corre- sponding problems for visual and spectroscopic binaries, including issues of data quantity, reliability, and astrophysical information content. Emphasis is on the value of intuition acquired by visualization, by numerical-graphical experiments, and by examination of graphs during the progress of a solution. An important part of a solution is in judgments as to what can realistically be found, so as not to attempt the impossible yet optimally extract the resident information. Intuition is needed to get close and to recognize false solutions (avoid possible local minima). Effects on light curves of ellipsoidal variation, reflection, and surface irregularities are discussed. Parameter combinations to be found from light curves alone, velocity curves alone, and light and velocity curves together are tabulated. Schemes to deal with uniqueness and convergence problems include the Levenberg-Marquardt algorithm, the Method of Multiple Subsets, and Vector Length Reduction. Orbital ephemerides are most often found by eclipse timings, although a relatively new way is by inclusion of ephemeris parameters in solutions of mixed whole light and velocity curves. Apsidal motion may also be found from whole curve solutions. Close binary morphology is reviewed, including its origins, conceptual basis, and relation to binary system evolution. The connection between morphology and solutions can be introduced by solution constraints that reduce parameter lists. Photometric mass ratio foundations are discussed, with comparison of the semi-detached and overcontact cases and of the complete and partial eclipse cases. A few common misconceptions are mentioned and some new directions are briefly indicated. The guide concludes with solution strategies and weighting, including things to avoid in both of those areas. 1. Background and Prospects Eclipsing binary (EB) modeling naturally partitions into several areas, headed by the overall issue of how to find astrophysically useful numbers. A simple count of parameters quickly confirms intuition in regard to the wealth of information in EB data. Contrasted with the short parameter lists of purely spectroscopic, visual, and astrometric binaries, information about EB stars consists of masses, luminosities, dimensions, temperatures, surface brightness distribu- tions, rotations, and distances, while EB orbit information is in dimensions, eccentricities, orientations, ephemerides, and apsidal motions. Uniqueness and convergence difficulties can follow from adjustment of excessive numbers of pa- 71 72 Wilson rameters, so good judgment is needed to decide what realistically can be found from observations. Input to the EB fitting problem consists of light and radial velocity curves, spectra, and various kinds of supporting observations. Conditions such as lobe filling and overfilling connect relative star/orbit dimensions to mass ratios and come under the heading of morphology, with the standard morphological types being detached, semi-detached, overcontact, and double contact. The means are thought experiments, logic, mathematics, and experience. Here we consider how to go from observations to results so as to complete the overall goal of observing, both by intuition and by impersonal solutions. Now is a good time to work on EB analysis, with innovative observational projects such as Gaia, RAVE, and KEPLER promising to break new ground in coverage1 and accuracy. Although surprisingly widespread opinion sees today’s surveys as having made dedicated observing programs of individual objects obso- lete, large scale surveys to date have sacrificed accuracy in favor of coverage and have not been substitutes for dedicated programs. The new projects and space missions promise coverage and accuracy, although not necessarily together. New capabilities on the modeling side are coming at a steady pace, but major devel- opments in efficient processing (automated pipelining) are needed to handle the anticipated flood of data from the coming missions. Automated schemes should preserve reliability so as to avoid false solutions, although that goal is not so easily reached. Anyone who has analyzed ONE EB can appreciate that point - now do a million EBs. 2. From Geometry of Points to a Binary in a Box As automated computation advanced from vacuum tube machines to amazing little desktop and laptop boxes over the past 50 years, EB light curve models advanced from spherical stars with first-order corrections for non-spherical phenomena to thoroughly physical isomorphs of real binaries - a binary in a box that can mimic real binaries with ever increasing generality and fidelity. Incen- tives for such improvements extend beyond hardware capability and include the lure of probing local astrophysics in star systems that are seen only as point sources by the largest telescopes. A broad view of observations and of models for various kinds of binaries (visual, spectroscopic, eclipsing, etc.) that follows below may provide background. 2.1. Binary Star Observations and Eclipsing Binary Models The observational and modeling problems for EB light curves differ greatly from those of visual and most spectroscopic binaries, which usually can be regarded as point masses without significant interaction. The visual and spectroscopic cases have difficulties that require experience, but mainly they are in the observations (insufficient angular resolution, insufficient spectral resolution, line blending) while the models are simple orbiting points. The situation is reversed 1Here coverage is used for brevity and refers jointly to sky coverage, magnitude coverage, and orbital phase coverage. Binary Star Light Curves 73 for EB light curves, whose data are typically clean and periodic, with only minor analogies to the resolution and line blending problems of their spatially resolved or Doppler resolved cousins. Not only are EB light curves relatively accurate and reproducible, but they are much easier to observe than velocity curves, so we have more points per curve (by one to two factors of 10) and we have data for more binaries. There are ape- riodicities and other problems for some EBs, but they are met less often and are on a much smaller scale than those of the other cases. EBs consist of tidally and rotationally deformed stars with local effects associated with spatially extended objects, so astrophysics replaces orbiting points. Local effective temperatures are needed for computation of light curves and other observables (e.g. polariza- tion). The stars heat one another with re-radiation, brightness varies over the surfaces due to variation in local gravity (gravity brightening, also called gravity darkening), and recorded light is a superposition of local radiation curves - not just those of blackbodies but the much more intricate radiation of real stars. Eclipses are a diagnostic phenomenon whose value can scarcely be overstated, as eclipses are indispensible astrophysical probes. 2.2. Learning from Numerical-graphical Experiments Numerical EB experiments bolster intuition for dealing with particular data sets and also provide long term experience. They may involve eclipses, proximity effects, surface irregularities, eccentric orbits, noise sources, and whatever other phenomena might be relevant to a given EB or to general background knowl- edge. The guiding word is visualization, and the idea is to develop a facility for connecting a binary system’s mental image with its light curves. Extraction of astrophysical information from a light curve need not involve computational details, as public programs already exist, but start-up, middle stages, and final results need to be checked intuitively for reasonableness. Ellipsoidal Variation An important proximity effect is the ”ellipsoidal”2 variation caused by tides. Here our mental imagery needs to connect tides with the double-peaked light curve waves they produce. The system is bright when we see the largest projected area, midway between the conjunctions, and dim at the conjunctions. A part of the effect is called geometric ellipticity to distinguish it from the surface brightness effect of photometric ellipticity, which arises because mean surface brightness is higher for a side-on view than for an end-on view. The main cause of photometric ellipticity is gravity brightening, whereby local surface brightness goes up with local gravity. The geometric and photometric effects essentially add to make an overall ellipticity effect, and limb darkening increases the overall amplitude slightly. In a simple, uncomplicated example of non-identical stars, the light curve maxima have the same level but the two minima differ, as the stars are right-left and up-down symmetric but not front- back symmetric. Ellipsoidal variation waveforms are not far from double cosines (i.e. cos 2θ, where θ is phase) although not accurately cosines, largely because of the lack of front-back symmetry. Familiar examples of large ellipsoidal varia- 2Tidally stretched stars are not formally ellipsoidal, but are close to ellipsoids unless they exceed, fill, or nearly fill limiting lobes. 74 Wilson tion are the very abundant W Ursae Majoris

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