P

11 See comments on page 102.

The eccentric anomaly E is related to the mean anomaly M through Kepler's equation

and thus also to time, t, because M depends on time. Let us now see how M is related to time or photometric phase. Per definition the mean anomaly, M, is the difference between a given orbital phase, $, and periastron phase, $per,

In eccentric orbit scenarios, some attention has to be paid to model the star correctly with regard to apsidal motion and to the computation of M for a given phase $. A consistent way12 is to start with the true anomaly, uc, of conjunction measured from the ascending node uc = 90°- o. (3.1.30)

Computing the eccentric anomaly, Ec, by (3.1.27) and applying (3.1.28) to compute the mean anomaly, Mc, allows us to compute the phase $per of periastron passage relative to conjunction according to (3.1.29),

From that we derive the phase, $c, of conjunction relative to the adopted zero point of phase, $s:

Note that $s is constant and does not depend on o. $c again gives us the phase, $0

of periastron passage relative to the adopted zero point of phase; the 0.75 term accounts for ( being measured from the ascending node (270° from conjunction). Now, eliminating $ by (2.1.1 ) we are in a position to compute the mean anomaly for a given phase, $,

M = 360° ($ - $p0er) = 360° ('—E J - ( - 270°. (3.1.34)

12 The Wilson-Devinney program (Wilson, 1979) uses this approach.

Once M is known, Kepler's equation (3.1.28) is solved (see Appendix C.4) for E, from which u is derived according to (3.1.27). With known true anomaly u, the star positions in inertial rectangular barycentric coordinates (within the orbital plane) are given by q q

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