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Fig. 3.8 The orbit of HR 6469 with e = 0.672. Fig. 4 in Scarfe et al. (1994), courtesy C. D. Scarfe

Fig. 3.8 The orbit of HR 6469 with e = 0.672. Fig. 4 in Scarfe et al. (1994), courtesy C. D. Scarfe

A useful relation can be derived from ( 3.1.23) if i is close or equal to 90c e cosa = 2P(tu - ti - ^. (3.1.24)

Because all quantities on the right-hand side of (3.1.24) can be determined with high accuracy, (3.1.24) can be used to derive a lower limit for e

Another useful approximation connects e sin a to the durations, ©a and ©p, of eclipses at apastron and periastron [cf. Binnendijk (1960, Eq. 385)]:

Note that (3.1.25) and (3.1.26) allow us to derive approximations for e and a separately because 0a and 0p can also be measured directly. Modeling of the surface configurations of eccentric EBs involves astrophysical considerations and computations beyond the orbital calculations. The shapes and surface gravities of the components are phase dependent, whereas stellar volume and bolometric luminosities are nearly independent of phase.11 Strictly speaking, the resulting forces for eccentric orbits cannot be described by a potential because the force field is time dependent and therefore nonconservative. If, nevertheless, models do make use of the potential formalism for eccentric orbits (Wilson 1979), it is under this assumption: If a binary can adjust to equilibrium on a timescale short compared to that on which forces vary, an effective potential [Avni (1976), Wilson (1979)] can be defined locally at each point of the orbit without significant inconsistency. The timescale for re-adjustment is that for free nonradial oscillations , which is normally much shorter than an orbital period.

The purely orbital calculations are connected with the Keplerian problem that considers two point masses moving on ellipses around their center of mass. In addition to the orbital elements we need the true anomaly u, measured from periastron to the star's position in the orbit. The true anomaly u and the eccentric anomaly E are illustrated in Fig. 3.9 and are related by

Fig. 3.9 True anomaly and eccentric anomaly. The figure shows the relative orbit of a body around another one located in the focus F1 of the ellipse. The eccentric anomaly E is computed by solving Kepler's equation. Once E is available the true anomaly u can be computed

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