The Eclipsing Binary Orbit Program EBOP

Etzel's (1981) Fortran program EBOP is based on the Nelson & 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 & Merrill (1952). However, as EBOP computes light curves directly, it is much more flexible and it provides options to implement more physics. It is not necessary to rectify for proximity effects. Nevertheless, the model is purposely tied to light curve defined geometrical parameters rather than "astrophysical" (Etzel 1993).

EBOP makes use of spheroidal stars moving in circular or eccentric orbits; here we explain only the circular case. It uses the linear limb-darkening law (3.2.23). The eclipsed area, a, and the light loss during an eclipse is integrated semi-analytically using some basic formulas for circular disks, rings, and sectors. The stellar disk of the covered star is partitioned into concentric rings of radius r sin # and width Ay = r cos #, where # is the angular distance to the disk center. Integration over the entire disk f I (# )ds' 1 fn/2 fn/2

yields the averaged flux F. The accuracy of this "eclipse function" depends on the width, Ay, of the rings and the accuracy of the computations involved. Choosing Ay = 5° yields a relative error of 10-4 which is usually sufficient. The advantage of this semi-analytic integration procedure is its efficiency. It is faster and more accurate than the standard procedure based on elliptic integrals or purely numerical procedures (two-dimensional GauB quadrature or direct summation over a stellar grid). The major parameters to compute light curves of spherical stars with EBOP are

• relative surface brightness at the disk center of the secondary component, Js;

• relative radius rp of the primary component;

• ratio k = rs/rp of the radii rs and rp of the secondary and primary components, respectively (note that these radius definitions differ from those of Russell-Merrill);

• limb-darkening coefficients xp, xs;

and the associated parameters:

• Eccentric orbit characterization (e cos Q, e sin Q);

• ephemeris phase correction A0;

• normalization of the light curve mq; and

• size of the integration rings Ay.

4 Here we discuss only the program's application to spherical stars. For the extensions including slightly deformed components modeled as ellipsoids, the reader is referred to Etzel (1981); the implementation is based on the evaluation of oblateness as described in Binnendijk (1974) and requires also the mass ratio.

It should be stressed that in EBOP the central surface brightness, Js, is used and not the stellar temperature. This is inherited from the Russell-Merrill method. Note that Js is the relative surface brightness at the disk center of star 2 because the value of Jp is defined as unity. In the NDE spheroidal star model, Js is directly connected to the ratio of depths of minima, whereas temperature has only an indirect influence on the light curve - one that is related to the physical model. The advantage of Js over effective temperatures or the ratio of bolometric luminosities is that, to a large degree, it can be determined empirically from the light curve whereas stellar temperatures are related less directly to the light curve, through the many assumptions of the radiative model. Therefore, for checking as well as comparative purposes, computation of this quantity in other light curve models is desirable.

Based on the linear limb-darkening law (3.2.23) in EBOP, the unnormalized luminosities5 of each spherical component follow as ls = n Jsr:

lp n Jprp

according to (E.29.16). The luminosity ratio l± = k2 js 1 Xs/3 (6.2.16)

depends only on the ratio of radii, ratio of surface brightness, and a correction term for the limb darkening. The relative luminosities ls + lp ls + lp in normalized units are used in the absence of third light. If third light has to be considered, EBOP uses a modified definition for the luminosities and imposes the normalization

EBOP does not support direct modeling of the proximity effects caused by distortion of the components and the reflection effect. However, terms for some perturbations, such as oblateness, are considered (Etzel, 1981, pp. 114-115); they are basically derived from Binnendijk (1960).

The perturbation terms describing the reflection effect are based on the assumption of a point source (the illuminating star) that illuminates the facing stellar hemisphere of the other star. Quantitatively, this is described by the simple bolometric phase law found in Russell & Merrill (1952, p. 44) and Binnendijk (1960, p. 119):

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