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 synchronous rotation, the ellipticity of star 1 is assumed to be representable by equations of the kind a1 — b1 3m2 3

r1 2m 1

r1 2m 1

where a, b = aj 1 - n2, and c = aj 1 - Z2 are the radii (a) in the line through the stellar centers, (b) in the direction perpendicular to this line but in the orbital plane, and (c) in the polar direction, respectively. Note that b and c are the radii "seen" during eclipses. The quantity K depends on the central condensation and lies in the range 0.0018-0.018. The surface brightness1 in a given passband is assumed to obey

1 Note that J is the usual symbol for mean intensity in radiative transfer theory. Russell used J

instead of I for ordinary intensity.

where J0 is the mid-disk surface brightness, x is the limb-darkening coefficient, g is the (local) gravitational acceleration, g is the average of g over the surface, and y is given by

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