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Appendix A: The outline of the method

The techniques used to analyse the data are based on the rigourous application of non-linear, weighted, least-squares methods used simultaneously for all relevant data that contain phase information. Our technique does not use an O-C diagram as an intermediate stage of data processing; O-C diagrams are used only as a visual check on the adequacy of the models. Let us assume that all observed phase curves of a star are adequately described by the unique general model function F(ϑ,a), described here by g_a parameters contained in a parameter vector a, a = (a₁,...,a_j,...,a_ga). In our computation we assume that the form of all the phase variations is constant and that the time variability of the observed quantities are given by a phase function ϑ(t,b), which is a monotonic function of time t. The fractional part of it corresponds to the common phase, the integer part is the so-called epoch (E). We can express the phase function by means of a simple model quantified by g_b parameters b, b = (b₁,...,b_k,...,b_gb). The instantaneous period is given simply by the equality .

For a realistic modelling of phase variations for all data types for a star, we need g_a free parameters for the description of the model function F(ϑ,a) and g_b free parameters for the description of the phase function ϑ(t,b). The computation of the free parameters was iterative under the basic condition that the weighted sum S(a,b) of the quadrates of the difference Δy_i of the observed value y_i and its model prediction is minimal (w_i being the individual weight of the i-th measurement). (A.1)We obtain here g = g_a + g_b equations of g unknown parameters. The weights of individual measurements w_i are inversely proportional to their expected uncertainty. The system is non-linear; we have to determine the parameters iteratively. With a good initial estimate of the parameter vectors a and b, the iterations converge very quickly. Usually we need only several tens of iterations to complete the iteration procedure.

A.1. Virtual O-C diagrams. Evolution of periods

The short-term modulation of the phase function was analysed by means of the residuals of the observed data Δy_i, creating individual values of the phase shifts expressed in days (O–C)_j with adapted individual weight W_j for each observed datum and averages of the phase shifts defined for arbitrarily selected groups of measurements or deflection of the mean period from the instant model period ΔP_k(t_k): (A.2)Computations of and ΔP_k followed after the model parameters were found; consequently they had no influence on the model solution. They were used only to visualise of the solution. Similarly, we can compute virtual “observed” values of the instant period from a group of observations to generate the model curves in our figures.

Relation A.1 can also be used to determine reliably zero-phase times for selected groups of observations O_k. These quantities depend only marginally on the chosen model of the phase function ϑ(t,b). Therefore we can use these O_k in the process of the phase function modelling.

Appendix B: Brief specification of CU Vir data

The data used for the analysis of CU Vir are given in Table B.1. Here we used the following abbreviations: EW – the equivalent width, RV – radial velocity, β – Hβ photometry, UBV – Johnson UBV photometry, uvby – Stroemgren uvby photometry, B_TV_TH_p – Hipparcos photometry, B_eff – the mean longitudinal magnetic induction, spf – magnitudes derived from spectrograms obtained by UV satellites OAO 2 and IUE, and radio – timings of radiopulses.

© ESO, 2011