for constant P. Except for the new formulae (5.2.21) and (5.2.22) to compute phase, radial velocities and light curve flux may be computed in the usual way from an k
5 This derivation is reproduced from R. E. Wilson's personal notes of Dec. 1995 and Feb 1997.
EB model. Terms in Tj, P0, dP/dt, and dw/dt were added to the DC equation of condition in the WD model so, with f the general symbol for an observable quantity (here radial velocities or flux), the equation becomes f— fc = i57^ + f 8 P + f5(d P /dt) + f5(dw/dt} + ± f ■ (5.2.24)
Subscripts o and c are for "observed" and "computed," and the 8pk values are corrections to input parameters pk in the iterative scheme. The partial derivatives in To, Po, dP/dt, and dw/dt can each be expressed as the product of a numerical derivative and an analytic derivative. For instance, dfc/d T0 is given by how the basic observables depend on phase, dfc/dip, and an analytic derivative that quantifies how phase depends on a given parameter (e.g., dp/dT0 ). Accordingly, each of the four derivatives df^ _ dfc. dp = f dP
includes a numerical factor, dfc/dp, that needs to be computed only once (not four times) per data point, and also an analytic factor. The analytic factors require negligible computing time and the numerical factor is the same for all four parameters, so four derivatives can be generated for the computational price of one. To compute, for instance, dp/dT0, we provide the inverse relation to (5.2.21)
0 dP/dt and its Taylor series expansion
Exploiting the analytic derivatives eliminates the need to specify appropriate increments for T0, P0, dP/dt, or dw/dt in computing the four derivatives dfc/dT0, etc.; we only need to think about one increment in
Traditional timing plots and the ephemeris solutions described here are complementary in that timing diagrams are naturally visual and intuitive, while "multiple whole-curve" solutions potentially access a larger body of information. Ideally, one should include the times of minima as another set of observables into the light curve analysis exploiting the ideas outlined in Sect. 5.1.1.
Was this article helpful?