Modelling ellipsoidal variables with known distances can lead to exact determination of the masses of both components, even in the absence of eclipses. We present such modelling using light and radial velocity curves of ellipsoidal red giant binaries in the LMC, where they are also known as sequence E stars. Stars were selected as likely eccentric systems on the basis of light curve shape alone. We have confirmed their eccentric nature and obtained system parameters using the Wilson-Devinney code. Most stars in our sample exhibit unequal light maxima as well as minima, a phenomenon not observed in sequence E variables with circular orbits. We find evidence that the shape of the red giant changes throughout the orbit due to the high eccentricity and the varying influence of the companion. Brief intervals of pulsation are apparent in two of the red giants. We determine pulsation modes and comment on their placement in the period-luminosity plane. Defining the parameters of these systems paves the way for modelling to determine by what mechanism eccentricity is maintained in evolved binaries.
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