## 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 rectification process into those of spherical stars. The geometrical model for distorted stars is a tri-axial ellipsoid. The physics is limited to Planck-ian radiation, linear limb-darkening law (3.2.23), gravity brightening, and a simple treatment of the reflection effect. In principle, only systems that conform to the "rectifiable model" are to be treated. In practice, it is not always obvious whether a given binary conforms, except, of course, that some systems may not yield solutions.

The basic Russell model was implemented not by a computer program but by manual techniques, augmented in the 1950s and 1960s by computation of "intermediate" or final orbit least-squares techniques [see, e.g., Irwin (1962)]. Manual methods continued to be used until recently (although at a declining rate since the 1970s) because of simplicity, and where computing power was unavailable, for convenience. The value of the basic technique is that we can obtain and test a solution, with a relatively straightforward procedure, and with the help of either a set of tables or nomograms. In the precomputer era, this method was the most elaborated and powerful technique available to EB astronomers. Nowadays, the use of tables and nomograms is decreasing. Another disadvantage of the Russell-Merrill technique is that it allows for only limited augmentation of its astrophysical content. Why do we describe it so extensively, then, here, and in Appendix D.1? Even though the basic method is less and less favored by light curve modelers, it still may be in use, and it was the method of choice for most of the twentieth century. In the early 1970s, there were computerized versions [Jurkevich (1970), Proctor & Linnell (1972)] of the Russell-Merrill approach available but they are no longer maintained. Although the programs EBOP and WINK cannot be considered as successor codes of the Russell-Merrill approach, the early versions of these codes benefited from ideas in the Russell-Merrill model. In order to be able to evaluate the results of light curve analyses carried out with this and similar methods, it may help modelers to be familiar with both its concepts and practices, even if most of those solutions have been redone, or at least retried.

The Russell-Merrill method uses the geometric phase introduced in Sect. 3.1.2. One of its basic ingredients is the function a = a(5, k, x1, x2), defining the light lost at any phase compared to that lost at internal tangency (i.e., at second or third contact). As shown in Fig. 6.1, the quantity a depends on the quantities introduced in Sect. 3.1.3: The projected separation of centers 5, the ratio of radii k = rs/rg, and the assumed limb-darkening coefficients xg and xs .1 The light curve is assumed to be in units of normalized flux, such that2 I = Lg + Ls = 1 at maximum light, which is assumed to be flat. For total eclipses, it is possible to derive the ratio of the components' luminosities from the light levels during total phases. For given Lg the light loss of an eclipsed star is then found in a table containing a set of empirical a values for observations of a given minimum. Equation (3.1.10) would then provide, for different values during eclipse, a set of equations of conditions for the radius of the eclipsed star and the inclination. The following discussion is an abbreviated version of Russell & Merrill's (1952) description of the procedure.

In general, the light observed at any phase may be written as loc = 1 - Ls tocaoc = 1 - Lsaoc (6.2.1)

during an occultation eclipse, and ltr = 1 - Lg Tatr (6.2.2)

1 Russell and Merrill refer to the components as "greater" and "smaller" stars, abbreviated g and s, respectively.

2 Note that this Russell notation equates light and luminosity and is thus an inconsistency in theory. This problem did not cause trouble in the old models because I is proportional to L for spherical stars.

Fig. 6.1 The relation among 8/rg, k = rs/rg, and a. The geometrical depth p = (8 — rg)/rs represents the degree of overlap of the disks. The hatched area a is the same independent of the particular star covered; adapted from figure 134 in Binnendijk (1960, p. 266). Note that the meaning of a here is slightly different from the a used in the Russell-Merrill model, where it is defined as a ratio of light lost at any phase to that lost at internal tangency. Thus the hatched area is Ta, if expressed as a unitless fraction of the eclipsed star's disk area

Fig. 6.1 The relation among 8/rg, k = rs/rg, and a. The geometrical depth p = (8 — rg)/rs represents the degree of overlap of the disks. The hatched area a is the same independent of the particular star covered; adapted from figure 134 in Binnendijk (1960, p. 266). Note that the meaning of a here is slightly different from the a used in the Russell-Merrill model, where it is defined as a ratio of light lost at any phase to that lost at internal tangency. Thus the hatched area is Ta, if expressed as a unitless fraction of the eclipsed star's disk area during a transit eclipse. Here, t is the ratio of the light lost at internal tangency to that of the entire eclipsed star:

1 lint Lg

For an occultation, toc =-= 1, whence the right-hand side of (6.2.1):

To study complete eclipses, Russell and Merrill defined a conveniently normalized photometric measure of projected separation, f, a function of k, a, and the limb-darkening coefficient, x :

sin2 0 - sin2 01 _ ¿2 - S2 sin2 01 - sin2 02 = S2 - ¿2

where 01 corresponds to one fixed a on a branch of the minimum and 02 to another; Russell & Merrill's (1952) prescription was to look up 01 at a1 = 0.6, and 02 at a2 = 0.9, with hand-drawn curves through the data of the minimum for interpolation. Defining and computing quantities A = sin2 61, B = sin2 01 — sin2 02, (6.2.5) may be written as sin2 0 = A + B f, f = ~~B—"' (6,2,6)

For a range of values of a, 0 is read from the plot, and f is computed. For each value of a, a value of f is found, so that a table can be constructed. The ratio of radii, k, can be found by interpolation in tables for foc and for ftr, appropriate for occultation and transit eclipses, respectively. It may not be known in advance whether a given eclipse is an occultation or a transit, and both sets of tables can be consulted. If total, the occultation eclipse is flat-bottomed and in the case of EA light curves tend to be the deeper eclipse, whereas the bottom of the transit eclipse is usually gently curved because of the limb darkening of the larger star.

If the external contact (first or fourth contact) phase can be identified, then from (3.1.10), r2

sin2 0ext = A + Bf (x, k, a = 0) = -2- (1 + k)2 - cot2 i (6.2.7)

sin i and, from the phase of internal contact, r 2

sin2 0int = A + Bf (x, k,a = 1) = -^(1 - k)2 - cot2 i, (6.2.8)

sin2 i and from these two equations, both rg and i can be found. The equations are independent of whether the eclipse is a transit or an occultation, but we must consult different tables for the two cases.

In the partial eclipse case, both the shape and depth must be evaluated; for complete eclipses information from either the shapes or depths is sufficient in principle (of course, in this case also it is safer to use both kinds of information). A new variable n is defined in terms of the relative light lost in the minimum, a, and that at mid-eclipse, a0, so that a 1 -l n = - = ---. (6.2.9)

This is the measured relative light loss fraction. The predicted fraction will be different for occultations and transits; both hypotheses must be tried in any solution unless good spectral information is available. Often one or the other can easily be excluded. A function x is defined such that sin2 0 (n)

The procedure is that n is calculated for a range of values of l on both descending and ascending branches (often folded by reflection about an axis through the mid-minimum point). Initially, neither a0 nor a is known. The corresponding phase, sin2 e - A

0, is then read from the light curve, and x is computed. For a given x, a tabular relationship between k and a0 can be produced from x tables. Several such curves, generated by selecting a new n, computing the corresponding x, and returning to the tables to find the new set of values of (k, a0), will intersect in a region defining the "best" values of k and a0. The tables for both xtr and xoc must be tested because the relationships will be different for the occultation and transit cases. These provide the "shape relations" for partial eclipses.

The "depth relations" provide indispensable information about the solution in partial eclipse cases. To discuss these, Russell introduced the q function

1 - ltr atr Lg Tatr + ltr - 1 q := T— = T^goc = 1 ,oc > (6-2-11)

Ls where the last equivalence results from the condition that Lg + Ls = 1, the equivalence that

q and substitution of equations (6.2.1) and (6.2.2). The evaluation of q thus depends on the availability of data from both eclipses. Again, both hypotheses about the eclipses must be tested: First, one of the eclipses is assumed to be an occultation; the computation is done, the same eclipse is then assumed to be a transit, and the computation redone. Russell and Merrill recommended as a first assumption that the occultation is the deeper eclipse. After q is computed, a0 is computed from equation (6.2.12), and the table of k(xg, xs, aoc, q) is used to establish a table of relations between k and a0. The elements are then computed, and from these a theoretical light curve is constructed and plotted against the observations. At this point, if the observations were good enough, improvements could be made.

The analysis depends on the assumption that the stars are well represented by the spherical model. Therefore, in order to use the Russell-Merrill method, the light curve first must be transformed into what it would be if the stars were spherical. This process is called rectification, and in principle can be applied only to "rectifiable cases." These rectifiable stars are similar prolate ellipsoids with limb-and gravity-darkening, and "gray-body" radiators (i.e., not perfectly efficient black-bodies). The contributions due to "ellipticity" (called elsewhere the "oblateness effect"), gravity and limb-darkening, and "reflection" are computed theoretically and applied semi-empirically, since they will depend on the surface brightnesses, sizes, and separations of the components. The rotation of a prolate ellipsoid could cause a sinusoidal variation of the surface area, and thus of the brightness. If stars were true ellipsoids, such a correction would be exact (with additional corrections for expected limb darkening and gravity darkening). However, stars are not ellipsoids. Russell and Merrill recognized that the ellipsoidal assumption was only an approximation to the true shape of a binary star component. It was a question of the adequacy of the approximation weighed against the tedium of exact calculation - a formidable and daunting task in the early 1950s, and a difficult one, in any case, until the 1970s.

Rectification begins with a Fourier analysis of the light curve outside of eclipse. The representation is of the following kind:

where n was usually taken as 2 although some practitioners used as many significant terms as necessary to fit the data satisfactorily. Any term other than A0, A1, and A2 was considered "unexplained" (if the minima are centered on phases 0 ;and 0.5 - otherwise such terms are generated even if there are no asymmetries in the light curve) and had to be dealt with in an arbitrary way. The Arterm, which peaks at 0 = 0° and reaches minimum at 180°, and the A2-term, which peaks at 0° and 180° and reaches minimum at 90° and 270°, are the contributors to the reflection effect. The cos 20 term may be at least partially due to what has been called the "oblateness effect," the effect of tidal distortion on the light curves. The presence of a B1-term indicates a difference between the maxima of the light curve - the O'Connell effect [see page 6, footnote 37 on page 135, and Davidge & Milone (1984)], - and higher-order B-terms indicate other light curve perturbations. Russell and Merrill recognized that it was "against human nature" to defer light curve analysis data solely because of unexplained asymmetries, and so, to permit analysis, they provided a general prescription for full rectification (Russell & Merrill 1952, p. 53).

Further details of the Russell-Merrill method can be found in Appendix D.1. Here we merely cite one example of the utility of the method applied to an apparently intractable - but rectifiable - problem.

As an example of the Russell-Merrill approach, we illustrate the method with the plot of the depth and shape relations for the secondary (occultation) eclipse of the system RT Lacertae (see Fig. 6.2). Although this is not an ideal system for application of the Russell-Merrill technique (in fact its many complications make it a truly difficult challenge for any light curve analysis program), nonetheless this technique at least permits a solution, and so RT Lac provides a suitable example of the usefulness, even today, of the Russell-Merrill technique. Precisely because of the complications in this binary, the Russell-Merrill method succeeds at least as well as other methods, and thereby demonstrates the continued usefulness of the method. The binary is a 5d07-period system, with a sinusoidal wave of period ~ 10 years. The primary minimum is more negative (suggesting higher temperature for the eclipsing star) in both (B-V) and (U-B) color indices. Ha, Ca II H and K, and far-ultraviolet spectroscopy reveal signatures of gas streams associated with Algol systems and thus mass exchange [cf. Huenemoerder (1985); Huenemoerder & Barden (1986a, b)]. Infrared photometry indicates the presence of a phase-dependent

RT LAŒRTAE - SELECTED Si°= FUNCTIONS FROM SECONDARY MINIMUM

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