Issue 
A&A
Volume 534, October 2011



Article Number  L5  
Number of page(s)  5  
Section  Letters  
DOI  https://doi.org/10.1051/00046361/201117784  
Published online  06 October 2011 
Online material
Appendix A: The outline of the method
The techniques used to analyse the data are based on the rigourous application of nonlinear, weighted, leastsquares methods used simultaneously for all relevant data that contain phase information. Our technique does not use an OC diagram as an intermediate stage of data processing; OC 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 = (a_{1},...,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 socalled epoch (E). We can express the phase function by means of a simple model quantified by g_{b} parameters b, b = (b_{1},...,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 ith 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 nonlinear; 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 OC diagrams. Evolution of periods
The shortterm 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 zerophase 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_{T}V_{T}H_{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
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