General Approach to Modeling Eclipsing Binaries

This chapter provides the basis to compute observables for a given set of EB parameters and a given set of times or phases. Typical observables are light curves, radial velocity curves, polarization curves, and line profiles. In this chapter the focus is on general considerations; no details of implementation are given. Those are found in Chap. 6 for several light curve models.1 EB data analysis leads to a nonlinear least-squares problem in which observed curves are compared with model curves. The presentation is greatly simplified if we take the following formal approach: We formally define an eclipsing binary observable curve, O, as a mathematical object

O := {(tk, Ok) | 1 < k < n}, i.e., as a set of n elements in which each element is a pair, (t, o), where t represents an independent, time-related quantity and o is the corresponding observable. The quantity t used as the independent quantity may either represent time or the photometric phase @ defined in formula (2.1.1).

In the past, at least, in light curve analysis the phase ^ is used. In more recent years, this has changed. If the period and epoch are to be determined, or if apsi-dal motion effects are considered, or polarization data and pulse arrival times are included in the analysis, as demonstrated in Sect. 7.3.1, it is necessary to use the time instead of phase, or in addition to phase; Section 3.8 provides an example of how this is done.

The term light usually is used in the abstract sense in this book and may represent not only the photometric brightness (i.e., the observed radiant power or flux in a particular passband) but any observable,2 such as

• the light at a given wavelength;

• the radial velocity;

1 Sometimes we use the expression light curve model in a general and abstract sense meaning "a model for computing eclipsing binary observables."

2 This terminology (without the formal mathematical approach) has appeared already in Wilson (1994).

J. Kallrath, E.F. Milone, Eclipsing Binary Stars: Modeling and Analysis, Astronomy 75

and Astrophysics Library, DOI 10.1007/978-1-4419-0699-1_3, © Springer Science+Business Media, LLC 2009

• polarization;

• photospheric spectral line profile;

• spectral distributions due to circumstellar flows; and

• any other quantity associated with the phase, but also other quantities independent of phase which we call systemic observables. Given, say, good Hipparcos data, or if the binary happens to be a member of an assemblage (star cluster, galaxy,...) with known distance D, the parallax n is available and can be considered an additional observable.

It may represent

• a measured value of an observable; and

• a value derived from a light curve program (more generally, an "observables generating program " as per Wilson (1994) based on a light curve model ("model,"for short).

Thus an "observable curve" ("observable" for short), may be, for instance

• an observed light curve Oobs;

• a calculated light curve Ocal;

• a wavelength-dependent light curve OA;

• a radial velocity curve Ovel;

• a polarization curve Opo1;

Before 1970 an observed light curve Oobs of an EB was analyzed following rectification procedures which trace back to the early 1900s. However, the underlying physical models were relatively simple and neglected effects which later turned out to be relevant. Photometric and spectroscopic data were analyzed separately and with different methods.

Today's methods permit analysis of photometric, spectroscopic, and other data simultaneously. If the vector x represents all relevant EB parameters, for each phase $ the corresponding observable ocal($, x), or several observables, ocal($, x), of type c can be computed with a light curve model. For a given set of phases, a whole observable curve Ocal(x) or a set of several curves, OJf^x*), can be computed; this problem is denoted as the direct problem. The inverse problem is to determine a set of parameters x* from a set of EB observations by the condition that a set of curves3 Ocal(x+) best fits a set of observed curves O0^. The system parameters x are modified according to an iterative procedure until the deviation between the observed curves and the calculated curves Ocal(x*) becomes minimal in a well-defined sense. The system parameters x*, corresponding to the observed curves O^bs, are ordinarily regarded as the solution of a least-squares problem.

In Chap. 4, the inverse problem is discussed. Obviously, in order to tackle the inverse problem we need to be able to solve the direct problem, i.e., the mapping

3 We show that it is advantageous to fit several light curves or even different types of eclipsing binary observations simultaneously.

x ^ Ocal(x), which is the subject of the present chapter. Each realistic model for computing the observable Ocal(x) for a given set of parameters consists of three major parts:

1. The physics and geometry of orbits and components.

2. Computation of local radiative surface intensity as a function of local gravity, temperature, chemical composition, and direction. The proper formulation of the radiative physics requires the use of accurate model atmospheres.

3. Computation of the integrated flux in the direction of the observer. This computation must take eclipses into account. The inclusion of other effects such as circumstellar matter, i.e., gas streams, disks, attenuating clouds, etc., may be desirable.

3.1 System Geometry and Dynamics

Orbis scientiarum (The circle of the sciences)

The shapes of the stellar surfaces are either explicitly specified a priori (as, e.g., by spheres and ellipsoids) or, in more sovarPhisticated treatments, determined implicitly by a physical model.4 The theoretical bases for the modeling of stellar shape distortions are varied. Particular light curve models emphasize one of the following: They adopt Chandrasekhar's (1933a, b) results on the theory of polytropic gas spheres and centrifugal- and tidal-force perturbations (Wood 1971) or the Roche model (see Sect. 3.1.5 for references). If the underlying forces can be determined completely by a potential function, the stellar photospheres are assumed to be equipotential surfaces. Surfaces of constant density then coincide with surfaces on which the potential energy per unit mass is constant and the local gravity and surface orientation are given by the gradient of the potential. This approach is generally applicable if the stars move in circular orbits. Under some limited assumptions it is also a good approximation for eccentric orbits (see comments on page 102).

3.1.1 Coordinates and Basic Geometrical Quantities

Figures 3.1 and 3.2 illustrate the geometry of the coordinate system used in most of the models presented in this book. We introduce for present and future purposes a generalized right-handed Cartesian coordinate system (x, y, z) with origin in the center of mass of a star. The x-axis points to the center of mass of the other star, the z-axis is normal to the orbital plane,5 and the

4 We refer to light curve models in which the geometry of components is fixed a priori as "geometric models," and to those based on equipotential surfaces as "physical models."

5 The rotation axes for orbital and proper rotation of the stars are assumed to be parallel to the normal of the orbital plane.

Fig. 3.1 Definition of a right-handed Cartesian coordinate system. The origin is in the center of mass of one of the stars. The x-axis points to the center of mass of the other star, the z-axis is normal to the orbital plane, and the y-axis is fixed by the "right-handed" stipulation
Fig. 3.2 Definition of spherical polar coordinates. The angles 0 and 9 denote longitude (zero in the direction toward the companion star, increasing counterclockwise) and colatitude (zero at the "North" pole), respectively

y-axis is fixed by the "right-handed" stipulation. This coordinate system is called C1. Additionally, spherical coordinates (r,e,0) are used, where the unit of r is the relative orbital semi-major axis, a. The radius vector r is represented as

z) Vv V cos e where r is the modulus of the vector r and where X, and v are the direction cosines. The angles 0 and e denote longitude (zero in the direction toward the companion star, with 0 increasing counterclockwise) and colatitude (zero at the

"North" pole), respectively. Next, we introduce the direction cosines (nx, ny, nz) of the surface normal vector n = (nx, ny, nz)T. (3.1.2)

The formulas to compute n are different for various classes of surfaces, such as spheres, ellipsoids, Roche equipotentials, and are provided in the appropriate sections. Once n is known we can compute the angle ft between the radius vector r and the surface normal n as shown in Fig. 3.3 and get r cos ft = er n = - n = Xnx + uny + vnz, r = |r|. (3.1.3) r


Fig. 3.3 Surface normal and line-of-sight. This figure shows the radius vector r, the normal vector n, the line-of-sight vector s, and the angles ft and y observer

Fig. 3.3 Surface normal and line-of-sight. This figure shows the radius vector r, the normal vector n, the line-of-sight vector s, and the angles ft and y

The distance r from a surface point to the center is a function r = r(0, <P\p) of angular position, (0, $), and the parameters p defining the shape of the surface. In these spherical coordinates, as shown in Appendix C.2, the differential volume element dV is given by dV = r2 sin 0 d0 dr, (3.1.4)

and the differential surface element by

cos ft

For discussing eclipse effects it is useful to introduce the plane-of-sky coordinates (xs, ys, zs). The origin of this right-handed coordinate system, P1, is the center of component 1. The traditional sense is such that the xs-axis is positive away from the observer and coincides with his line-of-sight. As shown in Fig. 3.4, the zs-axis is up when the ys-axis shows exactly to the left. If we want to model polarimetry as in Wilson & Liou (1993, p. 672) and want to keep right-handed coordinate systems, it is necessary to have the positive xs-axis pointing toward the observer. To derive appropriate formulas let us consider the transformation in detail. In the traditional sense P1 is related to C1 as follows. At first we rotate C1 counterclockwise around its z-axis by 180° - $, getting an intermediate coordinate system (x', y', z'). This system is rotated counterclockwise around its y'-axis by an angle of 90° - i (see Fig. 3.5). Therefore, according to the rotation matrices described in Appendix C.1 we can relate the coordinates by

Fig. 3.4 Plane-of-sky coordinates I. The figure shows the plane-of-sky and illustrates the orientation of the orbital plane w.r.t. the plane-of-sky

(xs, ys, ZS)T = Ry(i)Rz(180° - $) (x, y, z)T (3.1.6)

which, with6 sin(180°- $) = sin $ and cos(180°- $) = cos $, andcos(90°-i) = sin i and sin(90° - i) = cos i leads to sin i 0 cos i cos $ sin $ 0 ys | = ( 0 1 0 - sin $ cos $ 011 y| (3.1.7)

6 If the photometric phase, $, appears in an additive term involving an angle or as the argument of a trigonometric function, e.g., sin $, the term has to be interpreted as sin $ = sin 9($), where the geometric phase or true phase angle 9($) is evaluated according to (3.1.19) in the circular, or according to (3.1.37) in the eccentric orbit case.

y-axis y-axis

Fig. 3.5 Plane-of-sky coordinates II. This figure relates the plane orthogonal to the plane-of-sky, and the orbital plane

and finally gives x sin i cos 0 + y sin i sin 0 + z cos i y | = | —x sin 0 + y cos 0

As a special case we compute the plane-of-sky distance 5 between the component centers as a function of phase. If d is the distance between the centers at phase 0, the plane-of-sky distance 5 follows by setting x = d, y = z = 0 as

or equivalently7

The plane-of-sky coordinates just introduced are also useful to represent the line-of-sight vector S pointing from the observer to the plane-of-sky. According to our definition of the plane-of-sky coordinate system the observer is located at

and thus in this coordinate system ss is given as ss = S = (+1, 0, 0)T, 5 =|S|.

7 Replace cos2 0 = 1 — sin2 0, simplify, and replace again cos2 i = 1 — sin2 i.

The inverse transformation associated with (3.1.7) is x \ i cos $ - sin $ 0 \ /sin i 0 - cos A / xs y = sin $ cos $ 0 0 1 0 | ( ys z / \ 0 0 1 / \ cos i 0 sin i

or fx \ /cos $ sin i - sin $ - cos $ cos i y I = I sin $ sin i cos $ - sin $ cos i \z) \ cos i 0 sin i

and thus, in the coordinate system C\, s = (sx, sy, sz)T takes the form

' cos $ sin i - sin $ - cos $ cos A / cos $ sin i " s = | sin $ sin i cos $ - sin $ cos i I ss = sin $ sin i I, (3.1.15) cos i 0 sin i J y cos i with direction cosines (sx, sy, sz). The angle y between the line-of-sight s and n follows as (see Fig. 3.3)

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