1 Introduction

The determination of the orbital parameters of the components of a binary has been the subject of many investigations. It is not our intention to treat these existing methods, but rather to focus on one particular case, namely the determination of the orbital parameters of a binary that consists of at least one intrinsically variable component. All existing methods developed so far do not take into account the fact that intrinsic variability, such as pulsation, can significantly influence the radial velocities. In cases where the pulsation results in a peak-to-peak amplitude which is comparable to, or a large fraction of, the orbital amplitude, accurate orbital parameters are hard to determine. In such a case, the intrinsic variability cycle needs to be fully covered with observations in order to be able to determine the average radial velocity due to the binary motion. Having a full cover of the variability cycle is, however, a condition that is often not fulfilled.

The usual criterion of having found the best orbit when the rms is minimal breaks down in the case that one of the components exhibits intrinsic variability, especially when the intrinsic and extrinsic variations have comparable amplitudes. In this paper, we develop a method to treat cases for which single radial-velocity measurements are combined with radial-velocity data that do cover a complete pulsation cycle.

Our work originates from a systematic observational study of line-profile variability in pulsating B-type stars of which more and more targets turn out to be a member of a binary system (Aerts et al. 1998b). These variable stars have pulsation periods between a few hours and a few days and are often multiperiodic. This intrinsic variability is combined with extrinsic variability due to a companion. Orbital periods range from a few days up to years. Our main aim of observing these stars is to disentangle the pulsational behaviour by studying the line-profile variations caused by the pulsation(s) in full detail. Since the study of line-profile variability demands rather large telescopes with accurate detectors, observing runs typically extend over only a few days or weeks at best. The (often unknown) binary nature of the objects can prevent a determination of the pulsational parameters. The usual strategy then is to complete the data set that was gathered to study the pulsations with single measurements that are scattered in time and that should allow a determination of the orbital parameters. Once the latter are known, the effect of the binary motion can be subtracted from the data to start a study of the pulsational behaviour.

Well-known examples of pulsating stars that have been monitored extensively with the aim to derive the pulsational parameters and that were known or turn out to be members of a binary are $\sigma\,$Sco (Levee 1952; Mathias et al. 1991), $\varepsilon\,$Per (Smith et al. 1987; Gies & Kullavanijaya 1988), $\lambda\,$Sco (Lomb & Shobbrook 1975; De Mey et al. 1997), $\delta\,$Sco (Telting & Schrijvers 1997), $\alpha\,$Vir (Smith 1985a,b), $\theta^2\,$Tau (Kennelly et al. 1996); $\theta\,$Tuc (De Mey et al. 1998). For most of these cases, the pulsational analysis was or still is limited due to the lack of accurate orbital parameters. On the other hand, more and more binaries turn out to have a variable component, while the orbital parameters were derived assuming that both components are constant stars. Examples of the latter situation are found in the case of $\eta\,$Ori (Waelkens & Lampens 1988; De Mey et al. 1996), and are presented for V436Per by Harmanec et al. (1997) and for $\beta\,$ScoA by Holmgren et al. (1997). The latter two stars are targets of the so-called SEFONO project introduced by Harmanec et al. (1997). This project concerns a search for forced oscillations in close binaries by means of a search for line-profile variability in one of the components. It is to be expected that more pulsating stars in well-known binaries will soon be encountered in connection with this project. Indeed, in many of the target stars line-profile variability is confirmed or suspected (Harmanec, private communication). The question then is how these variable line profiles affect the determination of the orbital parameters, since the latter have been derived before the intrinsic variability was known.

In this paper, we focus on the determination of the orbital parameters in the case that one part of the radial-velocity data set consists of numerous observations that are concentrated on a few intrinsic variability cycles, while the other part of the data are single observations taken at random during subsequent variability cycles. The aim is to give each radial-velocity measurement a weight according to its ability to predict the radial velocity due to the binary motion. It is clear that the data of a fully covered pulsation cycle result in a much better predictor of the binary radial velocity than the scattered data points. We show how one can combine a single radial-velocity measurement with the fully covered cycles to give a better prediction of the radial velocity due to the binary motion. In Sect. 2, we describe the statistical methodology to derive the predictions of the binary radial-velocity and its standard error. A simulation study is performed to study the accuracy of the method. This is described in Sect. 3. The standard errors of the predictors of the binary radial velocity are then used as weights in existing methods to derive the orbital parameters. Application to the $\beta\,$Cep stars $\beta\,$Cru and $\kappa\,$Sco and to the star $\varepsilon\,$Per are given in Sect. 4. Finally, we end with some concluding remarks in Sect. 5.