Ephemerides Derived from Whole Light Curves and Radial Velocity Curves

In most publications on EBs, we find that ephemerides (epoch, period, and period changes) are derived from the times of minima. This usually requires a long-time coverage of the binary, which is a useful basis to establish the period and period changes. However, if the EB has just been discovered, the times of minima are only a fraction of the data available when complete light curves and radial velocity curves have been observed. An alternative method has been proposed by Wilson (2005) and by Van Hamme & Wilson (2007). It does not depend on traditional timing diagrams but derives an ephemeris from whole light and radial velocity curves. The time coverage is improved as there may be epochs with only light curves and other epochs with only radial velocity curves. Wilson, in Elias et al. (1997), had used this technique already in an analysis of AX Monocerotis, a (K giant, Be giant) binary that lacks eclipses, with radial velocities from two epochs being the only reliable means to an ephemeris. Because "time markers" such as light curve eclipses are absent in radial velocity curves, an algorithm was needed for treating whole curves, and the idea for AXMon was to work within a full binary star observables program (in this case, WD), rigorously compute phases from time, and let WD take care of all sophistications of the synthesized velocity curves. The analysis involves the orbit ephemeris parameters T0 (reference epoch), P0 (period at reference epoch), P = dP/dt (period time derivative), and dddt (orbit rotation, i.e., apsidal motion). For P = 0 and thus constant P0 = P, phase and time are connected by

where A0 denotes the length A0 = 0 - 00 of a phase interval [00,0], and At the corresponding time interval. Thus we can compute phase as a function of time by t - T0

For constant, but nonzero P = 0 we need to derive5 a relation between between phase, 0(t ), and time for nonconstant period. We start with


Using the integral relation

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