## Info

Where q M2 M1 is the mass ratio of the binary, and the radius vector, d, within the elliptic orbit, is given by d d (0) .-t-t-- - 1 e cos E. (3.1.36) Finally, we can relate u to the (true) longitude in orbit and also to the (true) phase angle v u o, 0 0 (0) u + o 90 . (3.1.37) So, finally, we coupled the orbital phase 0 and the geometrical phase, 0, through the mean anomaly, M, and Kepler's equation. In the circular case we just had the simple relation (3.1.19). De mortuis nihil nisi bonum (Of...

## Limb Darkening U1 And U2 Coefficients

With the effective limb-darkening factor, D, D -2 1(1 - x)siny cos y dyd< + x sin y cos2 yl dy d< , we eventually get (assuming a unit disk, i.e., R 1) F DI0, D n (1 - D . (3.2.31) The linear limb-darkening law is a one-parameter law. It is only a very rough representation of the actual emergent intensity. Accuracy is increased if we consider two-parameter, nonlinear limb-darkening laws. These laws and their coefficients are derived from stellar atmosphere models see Van Hamme (1993) and...

## Geometry and Coordinate Systems

N ifforni n Ys > lsxPucn lYa Suvarai (The geometrical identity has great meaning) This part of the appendix contains some additional material on coordinate systems and geometry. Consider two right-handed Cartesian coordinate systems with the same origin. Let the second system with coordinates (x', y', z') be generated by a counterclockwise rotation of the first one with coordinates (x, y, z) around its z-axis by an angle a. This situation is demonstrated in Fig. C.1. Then a point (x, y, z) in...

## The Russell Merrill Model

Locus poenitentiae (Opportunity for repentence) D.1 Ellipticity Correction in the Russell-Merrill Model This material gives further details on the Russell-Merrill model already discussed in Sect. 6.2.1. The expression for the ellipticity correction was developed as follows (Russell & Merrill 1952, p. 43). The basic model for the unrectified system consists of two similar triaxial ellipsoids with equal limb-darkening and gravity (or, sometimes, gravity brightening) coefficients. For...

## Precession and Apsidal Motion

Unfortunately the term precession often is used unadvisedly not only in the binary star literature but also in other areas. For example, many textbooks and public documentaries speak of the precession of planet Mercury's orbit in regard to a well-known prediction of General Relativity Theory (GRT). However, the described GRT phenomenon is orbit rotation, not precession. Orbit rotation (apsidal motion) is rotation within the orbit's own plane, so only one plane is involved, whereas precession...

## Radial Velocities

High resolution is essential for spectral analysis, whereas low resolution may be quite sufficient for spectral classification. To be useful in binary star modeling, radial velocities as displayed in Fig. 2.5 must have a high level of precision and accuracy. Red shifts for distant galaxies are very large, for example, and there may be a paucity of (identified) lines in the visible region, so precisions of tens and even hundreds of km s may not be considered too low. In binary star or pulsating...

## Astronomy And Astrophysics Library

Eckart A. Maeder V. Trimble W. B. Burton A. Coustenis E. K. Grebel B. Leibundgut The Stars By E. L. Schatzman and F. Praderie Modern Astrometry 2nd Edition By J. Kovalevsky The Physics and Dynamics of Planetary Nebulae By G. A. Gurzadyan Galaxies and Cosmology By F. Combes, P. Boisse, A. Mazure and A. Blanchard Observational Astrophysics 2nd Edition By P. Lena, F. Lebrun and F. Mignard Physics of Planetary Rings Celestial Mechanics of Continuous Media By A....

## Astrophysical Problems Solved by Light Curve Methods

Several important astrophysical problems have been solved with major help from light curve solution methods, e.g., the Algol Paradox cf. Pustylnik (2005) for a historical review , the structure of W UMa stars, bolometric albedos of convective envelopes, and undersized subgiants cf. Wilson (1994) . Progress in understanding intriguing binaries such as e Aurigae20 and j3 Lyrae has been made by including gas streams and disks in light curve modeling (see Sect. 3.4.4.1). The improvement of light...

## Adrq q r q v

For and in the coordinate frame of component 2 are used to compute the stellar surfaces and surface normal vectors. That in turn leads to new values of 5*(0, r) and so on. As computed by Drechsel et al. (1995), in extreme cases such as the one shown in Fig. 3.14 increasing radiation pressure can force the secondary to switch from inner to outer contact configuration. So besides changing the stellar shapes the system configuration can be changed completely due to the shift of the positions of...

## L O

Detached systems have Q > Q1 and fR < 1, lobe-filling components have fR 1, and over-contact systems are described by 1 < fR < 2. Similar to fi and f2, for computing f1R and f2R we have to apply the coordinate transformations described above. Another definition being used in Binary Maker 3.0 is BM Q Q - 1 , Q > Q1 which is properly normalized for detached systems between 1< fBM < 0 as well. For circular orbits and synchronous rotation the Roche potential approach led to the...

## The Russell Merrill Model and Technique

In limine (On the threshold at the beginning) The basic assumption in this model is that stars are spherical or, in successor versions, ellipsoidal in shape. The original exposition and the notation for the Russell-Merrill model are given in a series of papers by Russell (1912a, b) and by Russell & Shapley (1912a, b). The technique is best described in Russell & Merrill (1952). Light curves of stars which show evidence of tidal distortion and reflection are transformed by the...

## Subroutines of the Wilson Devinney Program

Finis coronat opus (The end crowns the work) The WD program is a practical expression of the WD model and there have been many more publications about the model than about the program. Even the best programs ordinarily have much shorter lifetimes than good ideas, so the model has been kept conceptually separate from its software implementation. Thus papers on the model essentially never mention the names of subroutines or main programs - only papers specifically about the program e.g., Wilson...

## J

VJ 4 I I ( I r2dr ) sin 0d0dp. (6.3.21) In (6.3.21) the symmetry of the Roche geometry is reflected by the factor 4. For detached and semi-detached systems, the function r(0, y) is well defined for all 0 and y. The volumes computed according to (6.3.21) are dimensionless. Multiplication by R3 yields the physical volume. For particular cases other means, such as harmonic, may be used also. As described in Appendix E.30, relation (6.3.21) also can be used to compute the associated Roche potential...

## Ij pd p sin f

The characteristic function has been replaced by appropriate boundaries or integration limits for longitude (p and colatitude 0. < ps (0 ) is the starting longitude and (pn (0 ) is the ending longitude on a given colatitude 0. 0 and 0u represent the lower and upper limits for colatitude. The limits depend on phase 0 and the geometrical parameters. According to Leibniz's rule, the derivatives are now given by Since no analytic expression exists for derivatives...

## Analytic Techniques and Numerical Analysis

The EB field is sometimes advanced on this. Selecting appropriate increments for computing numerical derivatives is an issue. A crucial part of the numerical analysis is also the use of proper weights and the computation of standard errors of the estimated parameters. The interpretation of the standard errors requires great care if parameters are strongly correlated. Utilia et delectabilia (Useful and delightful) This section is intended to guide the reader to recommended books or articles on...

## Phenomenological Classification of Eclipsing Binary Light Curves

Examples of prototypical light curves are shown in Fig. 1.1. They correspond to the classical categories, discussed above, of Algol, ft Lyrae, and W UMa light curves, also known as EA, EB, and EW light curves, respectively. Fig. 1.1 (continued) Classes of light curves. (a) Shows a synthetic Algol-type light curve (V band). It has been produced using the parameter file algolv.bmd from the Binary Maker 2.0 examples collection (Bradstreet, 1993). (b) Shows a synthetic ft Lyrae-type light curve (V...

## Third Body Effects on Light and Radial Velocity Curves

EBs are sometimes members of multiple star system cf. Mayer (2005) and references therein . Among 728 multiple systems with 3-7 components contained in an extended 6 1999 edition of the Tokovinin (1997) catalogue of physical multiple stars, there are 83 eclipsing binaries. In the simplest case a third body orbits with the binary around the system barycenter. Observational evidence of third bodies comes from spectroscopy (disturbances of the radial velocity curve), or from analysis of times of...

## References

M., Torres, G., Latham, D. W., Sozzetti, A., Mandushev, G., Belmonte, J. A. , Charbonneau, D . , Deeg, H. J. , Dunham, E. W. , O'Donovan, F. T., & Stefanik, R. P. 2004, TrES-1 The Transiting Planet of a Bright K0 V Star, ApJ Letters 613, L153-L156 Basri, G. 2000, Observations of Brown Dwarfs, Annual Review of Astronomy and Astrophysics 38, 485-519 Benedict, G. F., McArthur, B. E., Forveille, T., Delfosse, X., Nelan, E., Butler, R. P., Spiesman, W., Marcy, G., Goldman,...

## Star Spots and Other Phenomena of Active Regions

As is observed on our own Sun (Fig. 3.22 ), stars can have spots. Stellar surface imaging by microlensing cf. Sasselov (1998a, b) shows directly that spots are present on other stars as well. A star spot is a region with higher or lower temperature than the surrounding photosphere, and thus it modifies the local flux. By way of physical analogy to the Sun, we should expect magnetic spots to result from convection in the outer envelope and differential rotation. Accompanying phenomena include...

## Morphological Classification of Eclipsing Binaries

The dynamic forces controlling the stellar mass distributions involve the effects of rotation, tides, and noncircular orbits. For an introductory-level discussion of all these effects, see Wilson (1974). Fortunately, tidal forces produce circular orbits and synchronous rotation in many interacting binaries. A detailed and excellent analysis of the tidal evolution in close binary systems is provided by Hut (1981). The orbital period of a synchronous rotator in a circular14 orbit is the same as...

## Hills Model

The program LIGHT2 Hill (1979), Hill & Rucinski (1993) is the result of mating Hill's previous modeling program called LIGHT, which combined the Roche model with Wood's (1971, 1972) GauB-Legendre quadrature scheme, and Rucin-ski's WUMA3 (see Sect. 6.3.5) model which was derived from Lucy (1968). It achieves an accurate representation of the system brightness while dealing with horizons and eclipses. The LIGHT2 program has the following characteristics blackbody semi-empirical hybrid of...

## Nightfall

NIGHTFALL by Wichmann (2002) is a freely available amateur code2 for modeling eclipsing binary stars. It supports a large range of binary star configurations, including over-contact (common envelope) systems, eccentric (noncircular) orbits, mutual irradiance of both stars (reflection effect), surface spots and asynchronous rotation (stars rotating slower or faster than the orbital period), and the possible existence of a third star in the system. It allows the user to produce animated views of...

## Gravity Brightening

Hydrostatic equilibrium is equivalent to constant density and pressure on equipo-tential surfaces. If we assume that density p, temperature 26 T, and pressure p are related to each other by an equation of state, e.g., the ideal gas law, 26 Note that this temperature is the local thermodynamic temperature which differs conceptually from the effective temperature defined as a function of bolometric flux. p RpT, R 8.31451 J mol-1K-1, (3.2.9) with the universal gas constant R, then on...

## Determining Individual Temperatures

One of the main difficulties of modeling EBs is the accurate determination of the individual temperatures. Frequent practice in the literature is to assume the temperature of one star, or better to obtain it from spectra or color indices in an a priori step as described on page 199, after which the other star's temperature follows by fitting the light curve model to the data. In this two-step approach, the accuracy of the estimated temperature depends either on the how well the individual...

## The Wilson Devinney Program Extensions and Applications

Since the Wilson-Devinney program is the most widely used of all the light curve modeling tools, it is appropriate to describe its features, capabilities, and continuing development in some detail. The WD program itself has seen continual improvements, and the current version briefly summarized in Chap. 6 with its powerful features provides the opportunity to extract a maximum of information from a variety of observational data. As a side-effect, publications on the WD model and on the WD...

## Light Curve Software with Graphical User Interface and Visualization

Ad unguem to a fingernail exactly nicely done In this chapter we summarize the approach and current contributions to this area by a number of authors. Here, however, direction rather than specific packages must be emphasized because this subfield is rapidly changing. Graphics and visual support1 include the plotting of light curves, graphing of the fit and residuals, providing projected views of the components, and sometimes the distribution of such physical quantities as surface brightness,...

## Star Planet Systems and Eclipsing Binary Models

In EB models or programs we need to characterize planets by those parameters usually used to describe stars. The fundamental parameters are mass, radius, and temperature. A star-planet or other low-luminosity object system, with transits and radial velocities for the star only, is analog to a single-lined spectroscopic and detached EB. The orbital period, P, can be obtained from either radial velocities or light curves of the system and is usually the most precisely determined quantity. The...

## Passband Luminosities Phoebe

Where the second column applies to main sequence stars V , sub-giants IV , giants III , and bright giants II , the third one I to supergiants. To make use of this scheme, CIs from other passbands, e.g., Johnson V and Cousins I, need to be trans formed into B - V in order to use Flower 1996 , e.g., by exploiting Caldwell et al. 1993 who provide different color index dependencies on B - V .As these calibrations only serve to obtain an initial value of TB t , there is no need to worry about...

## Phoebe

PHOEBE PHysics Of Eclipsing BinariEs by Prsa amp Zwitter 2005b is a modeling package for eclipsing binary stars, built on top of WD program Wilson amp Devinney 1971, Wilson 1979 . The introductory paper by Prsa amp Zwitter 2005b overviews most important scientific extensions incorporating observational spectra of eclipsing binaries into the solution-seeking process, extracting individual temperatures from observed color indices, main sequence constraining and proper treatment of the reddening ,...

## N

Aiv xk Axk Wvdv xk Axk 0, i 1, , n. 4.3.4 Taylor-series expansion up to first-order derivatives gives A xk AxTG W d xk AT xk Axk 0, or with the Hessian matrix G interpret this as an M n, n matrix at each data point indexed by v 5 Many photometric light curve data have been analyzed with the Wilson-Devinney program Wilson amp Devinney, 1971 who were the first to use the method of Differential Corrections with a physical light curve model. Gij dv x , G e M n, n, N . 4.3.5 Multiplication and...

## The Eclipsing Binary Orbit Program EBOP

Etzel's 1981 Fortran program EBOP is based on the Nelson amp Davis 1972 spheroidal model called the NDE model. It is an efficient software for the analysis of detached binary systems with minimal shape distortion due to proximity effects. 4 It is not appropriate for modeling significantly deformed components. The NDE model and its assumptions are close to those in the rectification model by Russell amp Merrill 1952 . However, as EBOP computes light curves directly, it is much more flexible and...

## Dynamics and Orbits

Points of the stellar surface are considered to belong to an equipotential surface. The mathematics of such level surfaces is similar to that of the zero velocity curves in the restricted three-body problem cf. Szebehely 1967 , in which a particle of negligible mass is subject to gravitational forces of two massive orbiting bodies. Within that framework two cases are distinguished circular orbits and elliptic or eccentric8 orbits. We treat them separately because the circular and the eccentric...

## D

If the distance is measured in parsecs and the parallax is measured in arc-seconds, the constant is kn 1. To couple the parallax to the binary model it is more convenient to measure the distance in units of the semi-major axis a. First, we include the parallax both as an observable and also as an adjustable parameter. Second, instead of the normalized light or flux l 0 usually used in light curve analysis, the flux lD 0 in absolute physical dimensions energy time wavelength unit receiver area...

## Mathematical Nomenclature and Symbols Physical Units

What's in a name Shakespeare Romeo and Juliet, ii, 2 A few general rules are observed vectors are marked as bold characters, e.g., x, n, or r. The product a b of two vectors a, b e IRn is always understood as the scalar product aTb Xn 1 aibi. Matrices are indicated with sans serif font, e.g., A. The list below gives our mathematical symbols and operators. IRn the n-dimensional vector space of real column vectors with n components V gradient operator V Vx gt gt dr applied to a scalar-valued...

## Viq

Reasonably good matches with the observedprofiles of several Algol-type binaries. He also used the more extensive radiative transfer code of Ko amp Kallman 1994 which treats several atomic species, but further work must be done to improve the efficiency of the calculations. Terrell amp Wilson 1993 computed the disk matter distribution and motion. Figure 3.25 shows their result for SX Cassiopeiae disk images at 0.32, 1.11, 3.18, and 7.96 orbital revolutions. Based on these calculations, their...