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 corresponding 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 distributions, 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. polarization). 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 knowledge. 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 (W UMa) stars, whose ellipsoidal amplitudes are large because tides are large on both stars and also because the stars are not just in contact with their lobes, but overcontact (OC). Eclipse depth estimation can be subjectively difficult in W UMa’s because much or even most of light curve amplitude may be tidal. Amplitudes would be even larger if W UMa’s had the large gravity effect associated with radiative envelopes (Von Zeipel, 1924a,b,c), but their envelopes transfer energy convectively so they have the smaller gravity brightening predicted by Lucy (1967) - about one-third of the radiative effect3. Rare OC ’s of spectral types O, B, and early A have the larger ellipsoidal amplitudes characteristic of radiative envelopes, with V606 Centauri (e.g. Lorenz, Mayer, & Drechsel, 1999) and TU Muscae (e.g. Terrell, et al., 2003) being well observed examples. Figure 1 illustrates the above-described effects for simulated binaries.

10.5 10.5

10.6 10.6

10.7 10.7

10.8 10.8

V magnitude 10.9 V magnitude 10.9

11 11

11.1 11.1

11.2 11.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Phase Phase

Figure 1. Left: Enhanced ellipsoidal variation for a simulated binary that is 24 percent overcontact (higher curve) compared with an otherwise identical binary in marginal contact (lower curve). Right: Ellipsoidal variation for a simulated OC binary with a radiative envelope (lower curve) compared with one with the same geometry but a convective envelope (higher curve).

Reflection The other proximity phenomenon of widespread importance is the reflection effect, whereby some of each star’s radiation is re-processed by the other. Most “reflection” for stars actually is heating with thermal re-radiation, although free electron scattering is important for very hot stars and Rayleigh scattering is important for low mass brown dwarfs and planets. Numerical experiments can forge mental connections between input and output, where input consists of star sizes and temperatures, as well as orbit character and orientation, while light curves are the output. The inward facing reflection caps may cover substantially more or less than a hemisphere in very close binaries because of the small separations, as the relative sizes of source and target stars affect the basic geometry. Consequently an accurate reflection phase function cannot be written once and for all, even for simple examples, as it differs somewhat

3Comments that refer to convective atmospheres in relation to gravity brightening and bolometric albedos have become commonplace in the literature. However the astrophysics of albedos involves large parts of the envelope, not just the paper thin atmosphere, so one should speak of convective envelopes. Binary Star Light Curves 75 from one simple example to another. A rigorous bolometric theory of reflection, including eccentric orbits and multple reflection, is in Wilson (1990). The point source model is a very good approximation to reflection geometry in many realistic circumstances and is exactly correct for spherical sources, except in penumbral zones. However the irradiated star’s geometry requires attention to detail. Reflection in most eccentric binaries is a minor effect, as the large relative radii (R1,2/a) needed for major reflection quickly lead to circularization, so relative radii seldom exceed ≈ 0.2 in eccentric binaries. However V1647 Sgr is an interesting example (Figure 2) with a prominent reflection bump at periastron.

1.1

0.9

0.8

0.7 light in V band

0.6

0.5

0.4 0 0.2 0.4 0.6 0.8 1 Phase

Figure 2. An unusual periastron reflection bump appears around phase 0.30 in simulated V -band data of V1647 Sgr (with artificial noise added). Real light curves of V1647 Sgr (Clausen, Gyldenkerne, & Grønbech, 1977) also show the effect.

Interplay of Ellipticity and Reflection Reflection has the essential form l = A0 + A1 cos θ + A2 cos 2θ for each star separately (l for light), an approximation that serves reasonably well also for ellipticity, but A1 and A2 are positive for reflection whereas they are negative for ellipticity. With a shift of π radians for the phase function of star 2 (assume circular orbit), the cos 2θ terms for ellipticity and reflection are unaffected by the effective whole cycle shift so their difference appears in the light curve due to the intrinsically opposite signs. The π radian shift for star 2 produces a sign flip in its cos θ reflection term, thereby leaving a net difference of cos θ reflection for stars 1 and 2 in the light curve, and ordinarily with negative sign, as reflection is larger for the lower temperature star (conventionally star 2). “Visual” connection of these sign rules with physical and geometrical configurations is a worthwhile exercise. Overall consequences are 76 Wilson that differenced reflection appears in light curves as a cos θ variation and summed reflection appears as a cos 2θ variation, so the cos 2θ wave can affect light curves even for very similar stars. Summed reflection appears as a waveform that is concave-upward around phases 0.25 and 0.75 so as to destructively interfere with the main ellipticity variation. The (negative) cos 2θ part of ellipticity usually overpowers (positive) cos 2θ reflection, but not in all examples. WZ Ophuchi and EI Cephei have net upward concavity between the eclipses due to reflection winning over ellipticity.

Magnetic Star Spots and Other Irregularities Some centuries ago, well before discovery of the first EB, explosions accounted for all known variable stars. Many exploding stars, including novae, dwarf novae, and some of the super- novae, have turned out to be close binaries, and some are EBs. With improved observations came discovery of smaller irregularities, from outbursts involving only part of a star down to the inevitable lack of repetition that attends minor events such as we see on the Sun, as stars are not the idealized objects of our imaginations. Actually, the fidelity with which many EB light curves repeat is remarkable and we should not agonize over slight departures from repetition, but we should be alert for tell-tale clues about interesting behavior. At the scale of a tenth-magnitude or so we have magnetic star spots (like Sun spots) that are incorporated within several EB light curve models such as described in Budding (1977), Yamasaki (1982), Poe & Eaton (1985), Strassmeier (1988), Linnell (1989), Kang & Wilson (1989), and Wilson (1993). Numerous papers have stressed the lack of uniqueness in spot solutions and most star spot models have accordingly been kept simple, for example with circular spots and with the only parameters being latitude, longitude, radius, and temperature, as in the WD model (Wilson & Devinney, 1971; Wilson, 1979; 1990). Development of intuition for spot modeling follows obvious lines - visualize the binary as it turns and place spots where the light curves seem to need them. One should keep in mind that magnetic spots are a phenomenon of convective envelopes, and preferentially deep ones, so surface temperatures should be ≈ 7500K and lower. RS CVn and W UMa type binaries are particularly noted for light curve waves caused by spots. Spectroscopic methods for quantifying spot distributions (e.g. Vogt, 1981; Vogt & Penrod, 1983) suffer less from uniqueness problems than do photometric methods, although photometry reaches fainter limiting magnitudes. Much of the theoretical and observational literature on magnetic spots in stars of various types can be traced by going to the many contributions in Byrne & Mullan (1992).

Ephemerides: Periodic, Quasi-periodic, and Aperiodic Phenomena Eclipses are timing ticks, like pulsar pulses, that keep track of intrinsic and apparent period changes. Today EB observation times are corrected for light travel across Earth’s orbit, but the process could be reversed. Although there are more accurate ways, one or more steady EBs could serve to measure the speed of light via the (Earth orbit) light time effect, analogous to the famed measurement by O. Rømer (1676) based on occultations of Jupiter satellites. If EB eclipse timings accurate within a few minutes had been measured in Rømer’s time, they could have established the correctness of the heliocentric theory well before the first stellar aberration and parallax measurements. Although that thought is Binary Star Light Curves 77 highly conjectural since no EBs were then known, the heliocentric light-time effect probably was measurable from EBs within the technology of the time. Orbit periods can change by a variety of mechanisms such as impulsive or con- tinual angular momentum transfer due to matter exchange, angular momentum changes due to systemic matter losses, and spin-orbit exchanges caused by tides. A third star can exert dynamical effects that are discussed, for example, by Har- rington (1968; 1969), Soderhjelm (1975), and Mayer (1990) and their contained references. Apparent period changes can result from periodic phase excursions of eclipses caused by rotation of eccentric orbits (apsidal motion) and by light-time effects as an EB orbits the barycenter of a multiple system. Apsidal motion is (surprisingly often) incorrectly called precession, even in text books. Of course precession is a conical motion of one plane’s normal with respect to that of another, as in the precession of the equinoxes. It therefore necessarily involves two planes, whereas apsidal motion involves only one. Very small effective phase excursions can result from eclipse distortions caused by spots and possibly from other surface phenomena, where there are no changes in conjunction times but only in perceived conjunction times. Analyses of timing diagrams that plot residuals from trial ephemerides vs. time are traditional means of quantifying periods and period changes. They have usually been called by the un-informative name “O−C diagram” (i.e. “Observed – Computed”) but timing diagram, timing residual diagram, and ephemeris diagram are better names, as a huge variety of astrophysical diagrams display observed minus computed something. Regardless of the name, timing diagrams have great value, as they make use of abundant measures (times of eclipse minima) that come from both professional and amateur astronomers and are not seriously undermined by astrophysical idiosyncracies. They go well back in time for many binaries, with the classically strange ǫ Aurigae and β Lyrae and the pro- totype of EBs, Algol, being prominent examples of coverage over centuries. Tim- ing diagrams are a major help to intuition, as a steady period change appears as a concave upward (P increase) or downward (P decrease) parabola and a discon- tinuous change appears as a sudden change in slope. The Kreiner, Kim, and Nha (2001) atlas and companion web site (www.as.ap.krakow.pl/o-c/index.php3) have timing diagrams and ephemerides of many EBs in very well organized and convenient form.

Ephemerides – a New Way A new route to ephemerides has come along in the last decade. The idea is to find an ephemeris from a general solution of mixed light and velocity curves, entering time rather than phase into the machine. Ephemeris parameters T0, P , and dP/dt become ordinary solution parameters, found together with the other parameters, and the light curve and radial velocity templates are the corresponding solution curves. The history, references, and mathematical basis are in Wilson (2005). Advantages are that radial velocities can fill gaps in timewise coverage, that velocities can serve alone (despite lacking sharp timing ticks) where eclipses are absent, that the timing information in light and velocity curves is combined coherently, and that whole light and velocity curves contribute to accuracy. Weighting is important to success, so a good weighting strategy should be followed (Sec. 6.2). One can always do it both ways (timing diagram and whole curve analysis) if in doubt as to which way gives better or more interesting results. The timewise data distribution will differ 78 Wilson between whole curves and times of minima, so results for the two procedures will never be identical and may sample substantially different parts of the binary’s history. However good agreement has been the rule in cases to date where approximately the same intervals were sampled by minima and whole curves.

3. Close Binary Morphology

3.1. Physical Models and Morphology in Concept The essence of close binary morphology is that critical lobes limit or are closely related to star size, with a limit also on system size. Limiting cases involve accurate coincidence between a star surface and a lobe surface, while the “closely related” cases involve overcontact, with both stars extending modestly beyond their lobes and following a common equipotential. Early morphological ideas and the morphological type names of detached, semi-detached, and contact came from Kuiper (1941) and Kopal (1954). Conceptual and naming history is traced in Wilson (1994a), and the history and relative merits of the contact vis-à- vis overcontact names can be followed in arguments by Rucinski (1997) and Wilson (2001). The limiting lobes that set the scene for the three types were originally defined in the context of synchronous rotation and circular orbits, although there is a logical extension to asynchronous and eccentric cases (Plavec, 1958; Limber, 1963; Avni, 1976; Wilson, 1979). Asynchronous lobes are smaller (super-synchronous) or larger (sub-synchronous) than synchronous lobes. A fourth type of double contact (Wilson, 1979) applies to rare binaries with lobe overflow onto critically rotating super-synchronous components. Close binary star morphology begins with recognition that a star’s surface coincides with one member of an infinite family of equipotentials, which are surfaces of constant potential energy per unit mass. At a given point within a binary system, the gravitational potentials of the two stars and a rotational potential add to the total potential. Limiting lobes in the potential field set a maximum size for each star, beyond which lobe overflow to the companion removes matter on a dynamical time scale. In most situations the effective time scale for mass loss is the thermal time scale of the star’s envelope, as the promptly lost gas must be re-supplied by envelope re-adjustment.

Figure 3. Left: Stars detached from lobes. Right: Lower mass star ﬁlling lobe. Figures by courtesy of D. Terrell. Binary Star Light Curves 79

Figure 4. Stars overﬁlling lobes with a common level surface. Figure by courtesy of D. Terrell.

Figures 3 and 4 show a typical overall configuration for circular orbits and synchronous rotation, with limiting lobes indicated4. The two lobes taken together (with figure-8 shape) constitute the inner contact surface. An equilibrium configuration, with both stars extending beyond the inner contact surface and following a single surface equipotential, is a natural consequence of each star being larger than its lobe so that neither can serve as a sink for excess gas from the other. That configuration is not only possible but very common among the W UMa type binaries and is part of the modern definition of the W UMa class. Also shown in Figure 4 is the outer contact surface, which limits the size of an overcontact binary. Briefly stated, the central ideas related to close binary morphology are that we work in co-rotating frames (identify static configurations in frames that co-rotate with a given component or with the whole system), star surfaces follow equipotentials, effective gravity null points set lobe sizes, and lobe sizes limit star sizes due to lobe overflow. An overcontact binary results if both stars somewhat exceed their lobes. A general definition is that a limiting lobe is bounded by an equipotential for which the effecive gravity is zero on the line of centers at periastron (Wilson, 1979).

3.2. Morphological Origins How a given morphological configuration came about may not be obvious in all binaries, but we have only a modest number of evolutionary avenues. A star can expand with age, eventually reaching an equipotential that includes the effective gravity null point, thus leading to semi-detached lobe filling and matter transfer. Matter loss is through a nozzle at the inner Lagrangian point - a small area where gas pressure is not balanced by gravity. Another possibility is that

4See www.boulder.swri.edu/∼terrell/talks/aavso2001/frame03.html for versions in color. 80 Wilson star separation can shrink due to loss of orbital angular momentum, leading to lobe filling and mass transfer. Magnetic winds are currently understood to be the main agent for angular momentum loss that can lead to the OC condition of a W UMa system (viz. Maceroni & Van’t Veer, 1996). Alternatively, a system can be formed already filling or overfilling lobes, and a subset of very close binaries, such as some of the W UMa’s, may be born that way. More spectacular is common envelope evolution, an ultra-fast process by which cataclysmic variables (CV ) are believed to be formed (Sparks and Stecher, 1974; Paczynski, 1976). The idea, now well accepted, posits a red giant’s main sequence companion entering the giant’s envelope because of the giant’s evolutionary expansion and quickly spiralling inward, liberating the low density envelope on the way by converting orbital energy to thermal and then gravitational energy of the ejected gas (see also Webbink, 1992; Taam & Bodenheimer, 1992; Iben & Livio, 1994). The result, under a restricted set of initial conditions, can be a pre-CV consisting of the former giant’s core, now a white dwarf, and a surviving part of the main sequence star in a tiny orbit. A magnetic wind then further reduces orbit size until the main sequence star contacts its limiting lobe. Subsequent angular momentum loss by the wind, and later by gravitational radiation, drives lobe overflow and the hallmark accretion of a CV.

4. Light and Radial Velocity Solution Essentials

EB light, radial velocity analysis goes back about a century, with historical and intuitive background to be found in Wilson (1994a) and Wilson (1994b), respectively. Light and velocity information is mainly diﬀerent and complementary, although with some overlap in derivable information that allows the parameter list to be reduced via simultaneous solutions, as there is no need for separate light and velocity mass ratios or separate orbital eccentricities, arguments of periastron, or ephemerides. A smaller parameter list not only avoids logical dilemmas (which eccentricity is right?) but can properly balance the two information sources and typically leads to faster and surer convergence.

4.1. Absolute and Relative Orbits: Direct Observational Data Both absolute and relative orbit concepts enter the overall binary star problem, as velocity curves relate to absolute orbits while visual binary data and light curves relate to relative orbits. An absolute orbit is defined with respect to the system center of mass while a relative orbit is defined with respect to the other star, so a binary has two absolute orbits and two relative orbits. All the orbits of a given binary are co-planar and have the same shape (eccentricity), and the absolute and relative orbits of each star have the same orientation. The absolute orbits have size ratio a1/a2 = m2/m1 and add in size to a relative orbit (i.e. a1 + a2 = a). The ordinary complete input data set is a light curve of the whole system and velocity curves of both stars. Better is multi-band light curves and velocity curves of both stars. Velocity curves are almost always under-populated, if they exist at all, as they require more intricate instrumentation and longer exposures. However, light curves require clearer skies if comparison star observations are not effectively simultaneous with the EB observations. Binary Star Light Curves 81

4.2. What Can Be Found from Light Curve and RV Curve Combos? Persons interested in binaries will know the content of Table 1 that broadly summarizes the parameters to be found from combinations of light and velocity curves. Here i is inclination, ω is argument of periastron, a and e are respectively orbital semi-major axis and eccentricity, and m is star mass. Nodal position an- gles are not to be found from these kinds of data, although they can be found for visual and astrometric binary data and from polarization data. Mass ratios can be found from light curves of semi-detached and overcontact binaries where eclipses are complete (i.e. total-annular), but realistically not otherwise 3 3 2 (viz. Terrell & Wilson, 2005). The mass function, m2 sin i/ (m1 + m2) , is subscripted in the table for the case that star 1 was observed, and will be 3 3 2 m1 sin i/ (m1 + m2) if star 2 was observed. Ephemeris parameters (say reference epoch (T0), Period (P ), and dP/dt) can be found for any of the cases, as can the dω/dt of apsidal motion.

Table 1. Derivable Parameters for Data Combinations

Parameter(s) Light One RV Both RV Light Curve Curve Curve Curves +Both RV’s

Orbit Orient. i, ω ω ω i, ω Orbit Ecc. e e e e Orbit Size --- ak sin i a1,2 sin i a1,2 Star Radii R1,2/a ------R1,2 3 3 2 3 3 Star Masses m2/m1 m2 sin i/ (m1 + m2) m1,2 sin i m1, m2

4.3. Limb Darkening Laws Solutions that utilize linear (cosine) limb darkening continue to appear, with some papers stating that a linear law is “good enough”. Thought experiments are unlikely to tell whether that view is realistic, but graphical comparisons of the linear law with model stellar atmosphere intensities answer the question – linear laws fall far short of several non-linear laws in accuracy as shown by Van Hamme (1993). Laws with at least one term beyond the linear terms are needed in modern work.

5. The Morphology - Solution Connection

Not only are equipotential models directly physical, but they implicitly contain essential close binary morphology and can use it to constrain solutions (i.e. ex- ploit physical conditions to eliminate parameters). Except for detached binaries, solutions and morphology go hand in hand. A dimensionless version of the potential can be written as a function of the mass ratio and re-scaled coordinates (Kopal, 1959). A definite dimensionless potential then corresponds to lobe filling for any hypothetical mass ratio. Therefore one parameter can be made to disappear for the lobe-filling case, making the solution mathematically simpler (semi-detached constraint). The semi-detached constraint was made a part of 82 Wilson the WD model in about 1975 and first explicitly explained in Leung & Wilson (1977). A simple way to handle lobe-filling in solutions is to compute the lobe-filling potential from the momentary mass ratio so that surface potential is no longer a free parameter. A light curve computer program can thus be constructed so as to produce a semi-detached morphology if so asked. How can one know whether a star fills its lobe? Evidence can be in the period changes expected for matter transfer by lobe overflow, emission lines of gas in an overflow stream, the overall character of a system in regard to binary star evolution, and light curve experiments that indicate essential agreement between star size and lobe size. Other constraints that derive from physical conditions, are the overcontact constraint (Wilson & Devinney, 1973) where stars 1 and 2 have the same potential, and the double-contact constraint (Wilson, 1979) where both stars accurately fill their lobes, which is physically plausible only if the overflow target rotates super-synchronously. A constraint that derives from the binary-observer configuration is usually called the X-ray binary constraint (Chanan, Middleditch, & Nelson, 1976; Wilson & Wilson, 1976) since all applications (e.g. Wilson & Terrell, 1994; Wilson & Terrell, 1998) have been to X-ray binaries, although suitable configurations without X-rays can be imagined. A requirement is that one of the stars have negligible size compared to the other, thus being eclipsed as a point source. In that case, only one size for the large star can agree with the width of its eclipse of the small star, given definite values for the other parameters. The parameter inter-relation is relatively simple for circular orbits, with the large star’s potential dependent only on orbit inclination, mass ratio, and rotation rate. The eccentric case (Wilson, 1979) is considerably more intricate and, of course, involves more parameters. Solutions constrained by morphology are incorporated in the WD computer model and solution program, where 8 modes of operation correspond to 8 morphological rules, as shown in Table 2. A longer review of EB morphological origins is in Wilson (1994a).

5.1. Photometric Mass Ratios for SD’s and OC ’s Thoughts on measuring mass ratios from light curves go back (in published form) somewhat over a half-century and are based on the same morphological ideas that are built into WD and now into other EB programs (e.g. Prsa & Zwitter, 2005). The logic is direct and can be stated in four points: 1. The lobe conﬁguration depends on mass ratio (big mass goes with big lobe). 2. Star surface coincides with lobe surface in lobe overﬂow situations (semi- detached condition, SD). 3. Star size ratio is (nearly) the lobe size ratio in the overcontact case (OC ). 4. Total-annular eclipses give accurate star sizes, R1/a, R2/a. Morphological physics (items 1, 2, 3) couples with light curve geometry (item 4) to restrict possible mass ratios. Notice that the SD size-mass coupling involves accurate matching of one star size with one lobe size while the OC coupling essentially compares the ratio of star sizes to the ratio of lobe sizes. One must say ‘essentially’ because the star-lobe correspondence for OC ’s is not an accurate match, as the star surfaces are somewhat outside the lobe surfaces. Formally, OC Binary Star Light Curves 83

Table 2. “Modes” in WD Model and Corresponding Constraints

Mode Morphological Condition

-1 Surface potential ﬁxed by eclipse duration (usually X-ray eclipse of white dwarf, neutron star, or black hole)

0 No constraints, not even on luminosities via radiative physics

1 Potential 2 same as potential 1 and T2 = f(T1) so as to have surface brightness continuity on common envelope surface

2 Only constraint is that L2 = f(L1, etc.), in agreement with radiative physics

3 Like mode 1 except that T2 = f(T1) condition is not applied (temperatures can be considerably diﬀerent)

4 Semi-detached with star 1 accurately ﬁlling lobe

5 Semi-detached with star 2 accurately ﬁlling lobe

6 Double contact with both stars accurately filling lobes (non-synchronous) surface shape and size depend on mass ratio and overcontact level (sometimes called fill-out), so the additional parameter detracts slightly from the otherwise simple correspondence between lobe-size and star-size ratios. The SD case involves points 1, 2, and 4 while the OC case involves points 1, 3, and 4 so the basis for photometric mass ratios qptm) differs fundamentally for the two cases. Accordingly the SD and OC qptm ideas are distinct and have distinct histories that are reviewed briefly in Wilson (1994a). Having total-annular (complete) eclipses is a great help, as system geometry then is found far more reliably than for partial eclipses. The qptm concept does not apply to detached binaries (DB), where there is no connection between lobes and star surfaces.

Is SD or OC Better for qptm? Which case plays a more prominent role for accurate qptm, SD or OC ? Several issues enter, some favoring SD’s and others favoring OC ’s. The initial issue is the likelihood of complete eclipses. To have eclipses at all is favored by a large sum of relative radii, (R1 + R2)/a, and completeness is favored by unequal radii. OC ’s obviously win on the ﬁrst count by their deﬁnition, and OC ’s (or at least the W UMa’s) win statistically on the second count, as the range of R2/R1 for W UMa’s is mainly from about 0.1 to 0.7, with examples near unity very rare. Binaries of the classical Algol type are numerous and so can represent the SD’s, and many of those have R2/R1 ≈ 1. The reason is that the low mass lobe-contacting star (call it star 2) is evolved and therefore large for its mass, often being similar in size to the higher mass companion. Of course its actual size is set by the lobe, so small 84 Wilson

Algols (short period, small orbit) tend to have R2/R1 < 1 while large Algols tend to have R2/R1 > 1. Algols with periods in the much observed range of a few days have R2/R1 ≈ 1 and accordingly need nearly edge-on orbits if eclipses are to be complete. However SD’s have an advantage on another issue, in that the star-lobe correspondence is one of direct matching, whereas OC ’s have the complication of a second parameter (overcontact level). A point not widely appreciated is that radius accuracy is lost in Algols that have annular secondary eclipses, which are the typically shallow eclipses of the low temperature lobe fillers. Visualization shows that the sum of the radii is nicely set by the width of either eclipse, so we need another item of radius information, either (R1 −R2)/a or R2/R1, to have R2/a and know the dimensionless lobe size. One might think of measuring the difference (again vizualizing) by the width of totality in the deep eclipse of the detached star, but a reality check shows it to be entangled with the inclination, which cannot be measured independently – basically one does not usually know the chord of star 1 traversed in the eclipse of star 2 accurately enough to be much help. How about measuring R2/R1? That comes from the annular (in this example) eclipse of the contact star and, in the first approximation, is just the square root of the fractional area “cut out” from the large star by the small star, as measured by the eclipse depth. However our example deals with the shallow – often very shallow – secondary eclipse, whose depth may be nearly lost in the noise, so the accuracies of R2/R1 and thereby R2 may suffer, leaving the SD at a disadvantage vis-à-vis a typical OC as a qptm provider. The overall situation can be placed in perspective by noting that qptm estimation extracts information from two radii for OC ’s, one radius for SD’s, and no radii for DB’s, so one expects qptm reliability to decrease in that order. However a wider view considers the immediate and longer range (i.e. big picture) application of a qptm after it is in hand. One immediate application could be to check a radial velocity mass ratio (of course the check goes both ways), while another 3 3 2 is combination of qptm and i with a mass function, m2 sin i/ (m1 + m2) , to find individual masses for a single-lined binary. For the larger picture, note that qptm targets are far from randomly distributed within the astrophysical zoo of binary star evolution, with SD’s mostly represented by the classical Algols and cataclysmic variables (CV ) and OC ’s mostly by the W UMa’s. To cite one area of active interest, consider EBs as distance indicators within the Local Group of Galaxies (e.g. Paczynski, 2000; Guinan, et al., 1998; Wilson, 2004). High lu- minosity is indispensible for that problem, so W UMa’s, all with low to modest luminosities, are not a realistic present option, but some SD’s should be lumi- nous enough, at least for the nearer galaxies. In the areas of stellar structure and evolution, W UMa’s have been the focus of repeated attempts to understand energy transfer within common envelopes for at least several decades. Testing of old and new theoretical ideas is best done with accurate knowledge of masses and other parameters for W UMa’s, for which qptm’s play an important role.

Completely Eclipsing OC’s – the Amplitude-qptm Light Curve Connection A- mong the armaments in light curve intuition’s arsenal, none is more reliable and effective than the amplitude-qptm connection for completely eclipsing OC binaries (Wilson, 1978). It is as if such a light curve carries a printed tag from which one can read the mass ratio. Imagine a pair of spherical stars so Binary Star Light Curves 85 as to avoid effects that are not essential to intuition. Notice that the nearly equal mean surface brightnesses of OC components render the problem entirely geometrical – light curve form is the same for any surface brightness. Now picture an annular eclipse of nearly identical stars – it will be rather deep, with almost half of system light lost. Next do the same for annular eclipse by a vanishingly small star – depth will be vanishingly small, with no chance to increase depth by raising surface brightness, which is uniform for the whole binary. The corresponding total eclipse will have the same depth as the annular one, again because of the uniform system surface brightness. Clearly there will be a monotonic relation between eclipse depth and ratio of radii. Now throw in reflection, limb darkening, and (overcontact) tides. The numbers differ from the spherical case but a steep monotonic relation persists - deep eclipses go with equal radii and shallow eclipses go with unequal radii. Now we come to qptm, for which overcontact equipotentials show that the ratio of mean radii is a unique function of q and overcontact level, which can be inverted to give q for given ratio of radii and overcontact. Of course eclipse depth and light curve amplitude are not the same thing, but a light curve computer model easily deals with that small problem. We now almost have the desired unique relation between light curve amplitude and mass ratio, except that overcontact level plays a role. However, the role is minor and overcontact level is usually known reasonably well from a light curve solution. With a little experience, qptm can be estimated by eye for a completely eclipsing OC binary, and can be derived with very good confidence from light curve modeling.

A Common Misconception Comments to the effect that qptm’s derive from the ‘ellipsoidal’ variation produced by tides are fairly common in the literature, but that notion is thoroughly incorrect. As outlined above, qptm’s derive from radii (R/a) in the SD case, or from ratios of radii (R2/R1) in the OC case, and therefore from eclipses, which allow measurement of radii. Tides have essentially nothing to do with the situation. If doubt should arise on this issue, one can try to find mass ratios from ellipsoidal OC light curves that just miss having eclipses – there will be no success, although ellipsoidal variation will be substantial. Then try to measure mass ratios from SD’s with complete eclipses and very little tidal variation (because the lobe filling star is dim). It will work despite the lack of help from ellipsoidal variation. An easy way to do the experiments is with synthetic light curves that have synthetic noise, as the actual parameters will be known (viz. Terrell & Wilson, 2005).

6. Solution Strategies and Algorithms

6.1. Overview of Light-Velocity Solutions Intuition should be developed beforehand via thought experiments about how parameters affect light curves (viz. eg. Wilson, 1994b). Such intuition allows reliable estimation of what can be found so as to avoid attempts at the impossible. Parameters for which realistic observational evidence is weak or absent need to be assigned astrophysically reasonable values and not adjusted. That advice applies to any non-linear parameter estimation problem, of which the EB problem is only an example. Most attempts at the impossible are due to inex- 86 Wilson perience, although even EB veterans sometimes fail to think things through and try for more than the data can deliver. Convergence failure in the iterations of an impersonal solution algorithm may be a sign that the impossible has indeed been attempted. Specific reasons for a parameter to be unretrievable can be that its variation affects the theoretical output very little or not at all, or that it affects the output in essentially the same way as another adjusted parameter. A general rule is: For estimation of a parameter to be locally unique5, changing the parameter must change the observable record differently than changing any other parameter. “Observable record” refers to the theoretical counterpart of an ideal data set. Obviously, if parameters A and B affect the modeled observables in the same way, one cannot tell whether A or B is responsible for observed behavior. To check on the rule in DC, examine whether any two columns of partial derivatives (∂f/∂p, where f may be light or radial velocity and p is a parameter) are essentially multiples of one another. An easy way is to graph weighted ∂f/∂p vs. phase. If one parameter column has the same form as another, their parameters affect the observables in the same way, the matrix of (weighted) normal equations will be singular, and the solution will fail. WD’s DC program prints the partial derivatives, so graphical checks are easy.

The Solution Process Solution begins in a subjective trial and error stage with graphical experiments that make liberal use of common sense and aim for an approximate fit, with use of whatever information exists (spectra, astrometry, reddening studies, etc.) so as to be astrophysically realistic. The importance of making graphs to monitor progress can scarcely be over-emphasized. A particularly extreme mistake among beginners is to look only at the minima – not graphically but only by comparisons of observed and computed numbers at eclipse bottoms. Not only is much time wasted but the package of experience that could have been gained from graphs is thrown away, unopened. The subjective stage is followed by an objective stage in which an iterative algorithm leads to a weighted Least Squares solution, with standard error estimates. Progress should continue to be monitored with graphs. The documentation monograph for the WD program has some guidelines on strategy that are mainly specific to the DC (Differential Corrections) algorithm. Other algorithms (not discussed here, but each with a large literature) include Steepest Descent, Simplex, Itera- tive Minimization (one parameter at a time), Simulated Annealing, and various Genetic Algorithms. All objective solution algorithms are iterative, as the problem is strongly non-linear in many of the parameters (curvature in parameter space). DC “helpers” include the Levenberg-Marquardt scheme (Levenberg, 1944; Marquardt, 1963), which strikes a compromise between DC and Steepest Descent, and the Method of Multiple Subsets (MMS; Wilson & Biermann, 1976), which reduces correlation difficulties by keeping the correlation matrix relatively small at each iteration. Iterative Minimization can be seen as an extreme case of MMS, where the subsets contain only one parameter. A well known strategy that can stabilize solutions (avoid divergence) for most or perhaps all schemes for solving non-linear systems of equations might be called Vector Length Re- duction (VLR). The VLR idea is to go in the direction indicated by the latest

5Here “locally” refers to parameter space. Binary Star Light Curves 87 iteration but not all the way. All corrections are multiplied by a common factor (presumably less than unity) so as to shorten the solution vector while leaving its direction unchanged. VLR may not seem very eﬃcient in some practical work, as multipliers as low as 0.05 may be needed for success, with consequent increase in numbers of iterations. However it usually does lead to convergence (we get there eventually). The ideas of the preceding paragraph need to be automated if one is to deal with large numbers of light curves eﬃciently, which will be no easy task.

6.2. Weighting Weighting in solutions may be very important or unimportant according to one’s aims, with the safe route being to apply logical and consistent rules routinely (viz. Wilson, 1979 for rules). Weights are particularly crucial in simultaneous solutions that combine two or more kinds of data and should be based on objective criteria, not prejudice. While that advice may seem obvious, publications can be found where velocities were weighted arbitrarily relative to light curves because of a wish to increase their influence. The justification usually offered is that light curve points greatly outnumber velocity points, so the authors thus compensated for the “unfair” advantage on the photometric side – like heavily weighting votes from a minor political party to compensate for their low number. However, the Least Squares criterion, coupled with a good model, properly accounts for both the intrinsic influence of properly weighted points and their quantity. Naturally the worst data should not exert the most influence, and correct weighting prevents that undesirable outcome. With proper weighting, all parts of all data sets have essentially the same weighted variance, so “big scatter, small weight; small scatter, big weight”. Overall weight is the product of the separate weights now to be discussed.

Curve-dependent Weights In a solution of multiple curves (e.g. two velocity curves plus U, B, V, R, I), some curves may be noisier than others, so curve- dependent weights are needed. The DC program of WD deals with the problem as follows: A (human) choice is made of a reference phase range where each curve is approximately level, and the variance in that range (essentially [sum of squared residuals]/[number of points]) is measured. The reference ranges need not be the same for the various curves. The variances are deﬁned for reference ranges rather than for whole curves because noise typically varies with level for light curves, and the level dependence will be accounted for separately (see below). Next enter the square roots of the variances into DC as reference standard deviations, σ. The machine then uses the σ’s to compute curve-dependent weights, w = K/σ2, where K is a multiplier, internal to DC, that places weights in a convenient range for purposes of print format. Multiplication of all weights by a common factor has no eﬀect on a solution, as only relative weights matter.

Level-dependent Weights Light curve noise (scatter) usually increases with light level, either proportionally to level for noise sources such as variable sky transparency and scintillation, or with the square root of level for counting statis- 2 1/2 tics. More generally, with f for ﬂux, σ = ¡C1f + C2f¢ so, with the C’s being empirically estimated constants, the f 2 term dominates for bright stars and the 88 Wilson f term for faint stars. Noise is accordingly largest at maximum light and small- est at minimum light, and since noise, not noise/signal, counts in Least Squares, weights are greatest at minimum light. However, because C1 and C2 depend on several instrumental characteristics and usually are not reliably known, DC assumes for simplicity that only one or the other (or possibly neither) actually operates.

Individual Assigned Weights The third weight category is that of individual point by point assignments. Ordinarily, one should not apply curve-dependent or level-dependent weights and also individual weights, as that would use the same information twice. Individual weights are suitable for situations where two or more noise sources are comparably important and an external program computes the weights. Weights assigned on an unknown (perhaps subjective) basis by authors of a utilized data paper are sometimes entered as individual weights if no other strategy seems appropriate.

6.3. Common Strategic Mistakes Strategic errors in machine solutions that are not specific to a particular computer program include: 1. Failure to make adequate use of graphs (“flying blind”). Graphs that compare fitted curves with observations should be viewed routinely as solution iterations proceed. A glance at a graph may show the cause of a problem when a hundred hours of staring at numbers will not. 2. Following a recipe without thinking about the meanings and values of input numbers. There is much to think about, such as which parameters to adjust, initial values, control integers, parameter subsets to be followed, and parameter increments (called DEL’s in DC ) for computation of partial derivatives, ∂f/∂p, where f is flux and p is a parameter. 3. Failure to examine and actively think about ongoing iterations. 4. Insufficient thought given to weighting. Going back to re-do all the iterations is not fun, so have correct weighting from the beginning. 5. Belief in everything that comes out of the machine. Always ask “does it make sense”? 6. Applying only part of a solution (a solution for n parameters is a coherent entity. Part of it is not). The documentation for WD has a longer and more WD-specific list under the heading Common Difficulties.

7. Acknowledgments

Email discussions with D. Terrell improved parts of the meeting presentation, and D.T. also contributed the ﬁgures illustrating detached, semi-detached. and overcontact morphology to both the meeting review and this paper. The work was supported by U.S. National Science Foundation grant 0307561. Binary Star Light Curves 89

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