4 Application to three stars

We apply our method to three binaries that contain an intrinsically variable star fulfilling the condition of having an orbital period much longer than the pulsational period(s). We chose the examples in such a way as to illustrate that the inclusion of the weights can lead to negligible, noticeable, and crucial improvements of the determination of the orbital parameters. The applications show that our method leads to solutions that are at least as accurate as those found without assigning weights.

The code that we used for the determination of the best orbital parameters was first published by Bertiau & Grobben (1969). It is based on the Lehmann-Filhés method and allows that weights are given to each of the data points. The period can be fixed as well as variable.

4.1 $\varepsilon\,$Per

  

Figure 1: Top panel: the orbital radial velocity of $\varepsilon\,$Per found by means of the data taken for De Cat et al. (in preparation). The error bars represent the standard errors. Lower panel: the orbit as determined by De Cat et al. by considering all the data of the star

The star $\varepsilon\,$Per is the primary of an eccentric binary with a period of approximately two weeks. The star is also known for its strong and complicated line-profile variations (Smith et al. 1987; Gies & Kullavanijaya 1988) on a time-scale of hours. It is not yet clear how many and what kind of pulsation modes are active in the star, but high-degree modes are surely involved.

De Cat et al. (in preparation) have recently obtained an extensive set of high-resolution profiles that span the complete orbital period with the aim to study the orbital motion, the pulsation of the primary, and also the coupling between the two. We refer to their study for further details but use their radial-velocity data of the $\lambda\lambda\,4553\,$Å line of $\varepsilon\,$Per to determine the orbital periods with our method.

The data consist of 11 nights for which the pulsational behaviour is well-covered and 3 additional nights during which the weather conditions were poor and only a few spectra were obtained. We determined the standard error (s.e.) of these latter spectra by means of the method outlined above, i.e. by application of formula (12). To achieve this, each of the radial-velocity curves for the fully covered nights was fitted with a sine after having determined the best frequency per night for such a fit with a period-search algorithm (PDM, Stellingwerf 1978). From these fits we derived the average radial velocity of that night with its s.e. and the amplitude. The latter is used in the determination of the s.e. of the data of the uncovered nights.

 

Table 3: Orbital parameters for $\varepsilon\,$Per found by the traditional approach and based on our method that includes the assignment of weights. The data are taken from De Cat et al. (in preparation)

The orbital solution that we find is in very good agreement with the solution found by De Cat et al. (see Fig.1 and Table3). In fact, all orbital parameters agree within their s.e. We refer to De Cat et al. for a full description of the analysis of the complete data set.

This example shows that adding points of poorly covered nights to well-determined nightly averages over a pulsation cycle leads to the same accuracy of the orbital parameters if one assigns a proper weight to the scattered data points. In the example of $\varepsilon\,$Per there is no problem to determine the solution without assigning weights because the data nicely cover the complete orbital cycle and a couple of additional nights provide little extra information. When the latter is not the case our method becomes particularly helpful (see the last example).

4.2 $\kappa\,$Sco

The $\beta\,$Cep star $\kappa\,$Sco was monitored as part of a systematic spectroscopic study of pulsating stars in multiple systems (see De Mey 1997; Aerts et al. 1998b). It consists of a $\beta\,$Cep-type primary and a yet unknown secondary. This object has a pulsational behaviour that is similar to the one of the multiple $\beta\,$Cep star $\lambda\,$Sco and was observed during the same observing runs that were devoted to both stars (see De Mey et al. 1997, for the results on $\lambda\,$Sco). The data set consists of 9 fully covered nights and 78 scattered data points. We refer to De Mey (1997) for a full description of the data set.

Two pulsational frequencies are known in the literature for $\kappa\,$Sco and we fitted the fully covered nights with a sine fit for the main frequency 5.004 c/d (Lomb & Shobbrook 1975). We do not find different results than the ones listed below by considering the second frequency given by Lomb & Shobbrook, nor when we first search for the best frequency per night and make a fit with this frequency.

De Mey (1997) determined the orbital period from the complete radial-velocity data set with the PDM and CLEAN methods and found 195.77 days. She used this fixed period to search for the other orbital parameters. We refer to her work for the results.

 

Table 4: Orbital parameters for $\kappa\,$Sco found by the traditional approach and based on our method that includes the assignment of weights

At first, we used the same data set as De Mey (1997) to search for the orbital parameters in the traditional way without fixing the orbital period. This leads to a period of 195.87$~\pm~$0.01 days and slightly different orbital parameters as found by De Mey (1997). We refer to the second column of Table4 for the values of the other orbital parameters. The rms of this solution is 2.38 km s^-1 while the orbit given by De Mey (1997) corresponds to an rms of 2.44 km s^-1. We therefore take our solution given in Table4 and compare it with the results obtained with the approach outlined in Sect. 2. The results of this comparison are given in Table4 and are shown in Fig. 2. It can be noted from the figure that the solution found by means of our new approach gives a slightly better fit to the data and results in orbital parameters with substantially smaller standard errors.

  

Figure 2: The orbital radial velocity of the $\beta\,$Cep star $\kappa\,$Sco. The data are taken from De Mey (1997). The symbol $\circ$ denotes the scattered data points while $\bigtriangledown$ stands for the average radial velocity in the case of a fully covered night. The full line denotes the orbital solution determined by considering all the radial-velocity data without taking into account weights while the dotted line is the orbital solution found by our proposed method

For this example, the introduction of weights is less crucial for the point estimates than for the standard errors. The reason is that we were able to extend our initial data set with many scattered follow-up data that were obtained to determine the orbit. If the number of scattered points is sufficiently high and well-spread, they will not so much change the average, but may affect precision estimation. Moreover, this star has an orbital velocity amplitude much larger than the amplitude of the pulsation velocity. The last example shows that, when these two conditions do not hold, it can be essential to include weights in a proper way.

4.3 $\beta\,$Cru

 

Table 5: Orbital parameters for $\beta\,$Cru based on our method that includes the assignment of weights

Finally, the $\beta\,$Cep star $\beta\,$Cru is considered. The line-profile variations of this star were recently studied very thoroughly by Aerts et al. (1998a), who were for the first time able to determine the orbital parameters of this binary. Aerts et al. have gathered over 1000 spectra for $\beta\,$Cru, but their time-spread is very poor from a point of view of determining the orbital parameters (only a few nights with each some 50 - 250 spectra were obtained).

The application of our method to the data presented by Aerts et al. (1998a) illustrates that assigning weights as we propose here can make the difference between succeeding and failing to determine the orbital parameters in the case that the time spread of the data is very limited with respect to the orbital period. Indeed, Aerts et al. (1998a) failed to find a suitable orbital solution for $\beta\,$Cru before the application of our method. The reason is that the few follow-up measurements that were gathered with the specific aim to study the orbital motion were almost neglected in the calculation of the orbit compared to the more than 1000 spectra obtained for the study of the pulsational behaviour. Moreover, $\beta\,$Cru is an example in which the amplitude of the orbital motion is comparable to the one of the pulsational velocity. In such a situation, it is crucial to substitute fully covered nights by a single data point and add this measurement to the follow-up data, each of them with a proper weight. In fact, the development of our method originates from the purpose to determine an orbital solution for the complicated case of $\beta\,$Cru from the data presented by Aerts et al. (1998a). We list the solution for the orbit of $\beta\,$Cru in Table5 and refer to Aerts et al. (1998a) for further details of this application.