2.Observations and Data Reduction

Our data consist of three sets of electronic spectra obtained between June 1992 and April 2003 in both the coudé and Cassegrain foci of the Ondřejov Perek 2-m telescope between HJD 48813.4316 and 52754.8733.

The main features of the spectrograms and a description of the detectors used are summarized in ^{Saad et al. (2004)}, where the details about the main reduction procedures are also discussed. Table 1 lists the journal of the spectroscopic observations of the system.

2.1.Radial Velocity Measurements

The RV measurements were performed interactively using the computer program SPEFO (developed by the late Dr.Jiří Horn) by matching the original and mirrored line profiles. The RV analysis is based mainly on the Hα line, where RVs are measured in several parts of the line (namely in the steep parts of the line wings, the central absorption, the absorption bump, and the violet and red peaks displacements) and in strong lines of other elements in its vicinity such as He I 6678 Å and Si II 6347, 6371 Å. In addition, for HEROS (i.e. the wide range) spectra, RVs for Balmer lines up to H7, and other metallic lines such as Mg II 4481 Å and He I 4471 Å were determined. In the latter cases, RVs for these lines were measured for the outermost parts of the absorption wings, neglecting the details in the line core. Measurement of RVs in κDra was not an easy task due to the low amplitude of the variations (only a few km/s) and the line-profile distortion.

The measured RVs for different lines are listed in Table 2, where the file ID, Heliocentric Julian Date of observations (HJD), and the RVs in km/s for each line are listed.

2.2 V/R Measurements

As we have already discussed in detail in ^{Saad et al. (2004)}, the optical and near infrared spectra of κDra in the region 3450-7850Å are characterized by the presence of emission in lower Balmer lines (Hα, Hβ, Hγ), in some Fe II lines, and in the oxygen triplet line OI 7772, 7774, 7775Å. All the above mentioned lines display double-peaked profiles. The V/R ratio is obtained for these lines whenever it is measurable. The violet (V) and red (R) peak intensities of Hα are variable on a long-timescale (see Figure 4a, b in ^{Saad et al. 2004}). The intensity ratio V/R (concerning the normalized continuum) was found to be variable with time. However, the first inspection of its time plot reveals that it varies independently of other line parameters (strength and intensity), which show cyclic long-term variations on a time scale of decades. In some phases, it was quite difficult to identify the double-peak structure of the Hα line or one of its peaks. For both Hβ and Hγ lines, V/R variations were obtained over the last three years of the observations, since older Reticon spectra do not cover corresponding spectral regions. Their V/R variations generally behave similarly to those of Hα. V/R variations of Hβ show stronger R phases than V for the same epochs and its value is longer below unity, (V/R > 1) in addition to its lower amplitude scale. Due to the weak emission feature of the Hγ line, V/R variations are not significant and the values are distributed almost around unity. Figure 1 illustrates temporal V/R variations for the Hα, Hβ and Hγ lines. Moreover, V/R variations were found for some other photospheric lines affected by emission, namely for the iron lines Fe II 7462, 7712 Å, and for the triplet O I 7772-5 Å. Their relative V and R intensities are very weak, 4-6 % of the continuum level for the Fe II 7712Å and O I 7772-5Å lines, and 1-3 % for the Fe II 7462Å line. During our observations, the Fe II 7462Å line has phases dominated by relatively strong R peaks, in contradiction to the FeII 7712Å line, where the ratio V/R is almost around unity. For the O I 7772-5Å line, V/R displays consequent slight changes from values >1 to < 1.

Fig. 1 Simultaneous time distribution of the V/R ratio for Hα, Hβ, and Hγ lines. 

2.3.Period Analysis

Preliminary analysis of the RV variations was important to select a suitable set of lines for further evaluation of the orbital elements. The period search was carried out separately for several line data sets (basically those which cover a long time interval of observations), where two independent numerical period searching routines were used. One is based on the phase dispersion minimization (PDM) technique (^{Stellingwerf 1978}), the other is a computer program PERIOD04 (^{Breger 1990}). For , starting values for the frequencies need to be determined by some other technique such as Fourier (which is available in the program itself) or using PDM. Within limits given by the alias structure of the observations, PERIOD can improve the frequency values by minimizing the residuals of a sinusoidal fit to the data. A search for periodicity in the interval 1-160 d has been performed for three sets of RVs, namely RVs of the Hα emission wings, of the metallic lines Si II 6347, 6371Å, and of the He I 6678Å line. The analysis confirmed the 61^d.5549 period for two sets of lines: He I 6678Å and Si II 6347Å (which ^{Juza et al. 1991} reported for the orbital period) with amplitudes of 5.9km/s and 8.9km/s, respectively. For the Hα data set, we detected two frequencies, at 0.9987cd^-1 and 0.0162cd^-1 (the orbital one). The upper and middle panels of Figure 2 represent a periodogram (frequency vs. power) of the original data; the right insets in each panel illustrate the spectral window function around the detected frequency, while the left insets represent the residuals pre-whitened for the detected frequency for the He II 6678Å and Si II 6347Å lines, respectively.

Fig. 2 Each main panel displays the periodogram of the orginal data, the other left and right panels represent the spectral window for the detected frequency and the residual after prewhitening with the detected frequency, respectively. The arrows refer to the peak of the predicted frequency. The panels represent period search results. Upper panel: for He I 6678 Å, Middle panel: for Si II 6347 Å and Lower panel: for Hα line. 

In Figure 2, the lower panel is similar to the upper and middle ones, but for the Hα data set. The pre-whitened spectra for a higher frequency of 0.9987cd^-1 show the disappearance of the frequencies and that of the orbital period as well. Figure 3 displays the corresponding phase diagrams for both data sets. We adopted a similar procedure for searching the period of the V/R ratio of Hα. The frequency of 0.0162cd^-1 (P = 61^d.64) was detected for the V/R variations (which is close to the orbital frequency), the corresponding phase variations for this frequency are illustrated in Figure 4. Phase diagrams of the V/R variations for Hα Hβ and Hγ lines display consistent behavior with each other.

Fig. 3 Corresponding phase diagrams for RVs of 6347Å and 6678Å lines folded with the period 61^d5549. 

Fig. 4. V/R variations of 𝐻𝛼 folded with the 61^d.64 period. 

Figure 5 represents the phase diagrams folded with P= 61^d.64 for the Hα emission wings (upper panels) and Hα absorption center (lower panels).

Fig. 5 Phase diagrams folded for (P= 61^d.5549) Upper panel: Hα emission wings, Lower panel: Hα absorption center. 

4.1.RVs Variations

RVs of the red and violet emission peaks show long-term variations in terms of years. Figure 6a displays its time-distribution. These long-term variations were found for the strength and intensity of the whole line (more details in ^{Saad et al. 2004}). A signature of the 61^d.5549 orbital period has been detected after pre-whitening it for the long-term variations period. Figure 6b and Figure 6c illustrate the periodograms of their RVs after pre-whitening for long-term variations, with that (8044 d) cycle obtained by ^{Saad et al. (2004)} from brightness, line intensity, and equivalent width variations. The arrow indicates the peaks of the detected frequencies, which are very close to that of the orbital period. Therefore we can say that the violet and red peaks in their displacement follow two different movements of the systems, one of them is related to the circumstellar matter surrounding the system (which is the dominant one as it is clear from a time-distribution diagram), while the other expresses the orbital motion of the binary system. In this respect, we have to note that we failed to find similar behavior in other parameters with dominant long-term variations (intensity of red and violet peaks, and the line strength).

Fig.6.  Violet and red peaks variations of Hα. (a): the long-term distribution of the RVs of V and R peaks, (b):the periodogram of the RVs residuals of the V peak after prewhitening for long term variations, (c): similar to (b) but for the R peak.The arrow indicates the peaks of the predicted frequencies. 

4.2.V/R Variations

Violet-to-red peak intensity ratio (V/R) variations are one of the most striking features of the emission lines, which give a measure of the line asymmetry. Usually, the V/R ratio behaves quasi-periodic on a timescale of years. Be stars may change from a stable V/R =1 ratio to V/R variability and back (^{Porter & Rivinius 2003}). ^{Hanuschik et al. (1996)} supposed that the majority of Be stars have emission lines with stable and equal V and R peaks, although one-third of them show a cyclic variation of V/R.

On average, periods of V/R variations are usually about seven years (^{Mennickent & Vogt 1991}) and connected with long-term variations. In some particular cases, they display different successive phases of activity, as reported for 59Cyg (^{Harmanec et al. 2002}) or ζ Tau (^{Hubert et al. 1982}). For several other stars, V/R variations are quasi-periodic on a long-timescale, e.g., γ Cas with a period of 9150 d, and κCMa with a period of 8052 d (^{Mennickent & Vogt 1991}). We found erratic or quasi-periodic variations, as reported also for β CMi (^{Mennickent & Vogt 1991}). In most cases with V/R variations, long-term variations are found to be superimposed on short-term ones, which are sometimes misinterpreted as irregular variations. In the particular case of 59Cyg, two significant periods have been obtained. A short one, of 28.1971 d, which follows the orbital period of the binary system, and a long one of 722 d, which is connected with the formation of the new Be envelope (for more details see ^{Harmanec et al. 2002}).

κDra is one of a few Be stars, which are known for the synchronization of the V/R variations with that of the orbital motion and its behavior independent of its long-term variations. Other similar systems include 4 Her, where the V/R variations follow its orbital period of 46^d.1921 period (^{Koubský et al. 1997}), 88 Her, which varies with a 86^d.7221 period (^{Doazan et al. 1985}), and Per with a period of 126^d.696 (^{Poeckert 1981}). They are known as systems with V/R phase-locked with their orbital period. Concerning Hα, the V/R asymmetry of κDra, ^{Arsenijevic et al. (1994)} noted that, according to the polarization results, the internal layers of the envelope have axial symmetry, but the outer Hα emitting region, affected by the presence of the companion, is probably associated with nonaxisymmetric external layers. In different panels of Figure 7 we give a series of simultaneous profiles of Hα, Hβ, and Hγ showing variations of the V/Rat different phases. Various hypotheses and models have been devoted to explain long-term V/R variations. ^{Struve (1931)} suggested an elliptical-ring model (a geometrical one), which was elaborated by ^{McLaughlin (1961)}. Further modifications were done by ^{Huang (1973}, ¹⁹⁷⁵⁾, who attributed the V/R variations to the apsidal motion of an elliptical ring, in which emitting atoms revolve around the star according to Kepler’s law. A binary model proposed by ^{Kriz & Harmanec (1975)} is another view, where the companion star deforms the disk into an elliptical one through tidal interactions.

Fig. 7 Variations of the double-emission peaks for different phases during one period. Left panel: for Hα, Middle panel: for Hβ, and Right panel: for Hγ. 

^{Kato (1983)} and ^{Okazaki (1991)} explained the long-term V/R variations by one-armed global oscillations, where a thin non-self gravitating Keplerian disk is distorted by a density wave. Let us apply this model to κDra. The emission structure originates in the envelope, while the binary star is sitting somewhere inside. Therefore, it affects the innermost parts of the surrounding structure. This part of the disk rotates most rapidly. Due to the Roche lobe geometry, there is a high asymmetry in the outer parts of the disk, which therefore reflects in an asymmetry between the red and blue components (i.e V/R variations), alternating around the orbit. In κDra, binarity seems to play the most important role in the line asymmetry (^{Panoglou et al. 2018}), which explains the V/R phase-locked variations with the 61^d.5549 period.

7.2.Rapid Variability

For 𝜅Dra, rapid variability was reported by ^{Hubert & Hubert (1979)} and ^{Andrillat & Fehrenbach (1982)}. Three decades ago lpv of 𝜅Dra was studied by ^{Hill et al. (1991)}. They detected rapid variations (on a timescale of hours) with a period of 0.545 days in the width and asymmetry of the absorption lines He I 4471 Å, Mg II 4481 Å, and He I 4388 Å, and attributed these variations to nonradial pulsations. However, ^{Juza et al. (1991)} showed that the detected periods of rapid variations are all aliases of one physical period and that they are related to the rotation of the star.

In this respect, we derived more constrained limits of the rotational period of the primary (0^d.95 -1^d.91) in comparison to (0^d.5-2^d.0) reported by ^{Juza et al. (1991)}, based on the revised values of the Be star radius 6.4±0.5R_e and projected rotational velocity vsini = 170 km/s (^{Saad et al. 2004}) for different inclination angles, from 30^o to 90^o, and using explicitly the relation

Prot=50.633R/R⨀vsin⁡i-1sin⁡i

For the orbital inclination i = 30^o, the rotational period is 0.95 3± 0.07 days. The error accounts only for the uncertainty in the determined radius. An ultra-rapid variation on a time ≈2min in the Hα profile structure detected on 14/15 Feb. 1993 (from 36 co-added and averaged profiles) is reported by Anandarao et al. (1993). The lpv manifest themselves mainly through the line profile asymmetry, line width, and sometimes give rise to the moving (absorption or emission) sub-features.

For different regions of the spectra, our RV measurements were examined for rapid variability. We searched periodicity using the two above mentioned independent numerical period searching routines. As mentioned earlier (in § 2.3) for Hα, the periodogram for the RV measurements of the outermost parts of the line (wings), the lower panel of Figure 2 gives a higher frequency at 0.9987cd^-1, while the other frequency is centered at the orbital period.

After the pre-whitening process other frequencies peaks completely disappear, which supports the suggestion that this rapid variation in the RV (around one day) is not a real one, and it is probably associated with the orbital frequency. For some absorption lines (mainly for He I 6678 Å and ) we searched for short period variations. After removing the orbital variation from the data (pre-whitened) of RV measurements of the mentioned two lines, the residuals were searched for short term variations. Two different short periods with lower powers are found for each set of data, 0^d.171198 (at an amplitude 3.775 km/s) and 0^d.416805(at amplitude 2.9956 km/s) for HeI 6678 Å and Si II 6347 Å respectively.

The various sets of RVs available from all lines (366 points which were previously used to obtain the final solution with FOTEL), after pre-whitening for the orbital period, were searched for short term variations. The detected period of 0.^d 39526 with low power (3.321kms^-1) may belong to the noise level.

Finally, we have used the measurements of equivalent width (EW) and line central intensity (I_c) of Hα to search for the short term variability. We searched for a period in both data sets in the range between 0.1 - 2 d first using the program PERIOD. Figure 11 represents a periodogram of the original data, a spectral window, and a periodogram of the pre-whitened spectra. Two main frequencies (ordered by power) at: 1.00020cd^-1 and at 2.00343cd^-1 have 75% and 65% of confidence level (as shown in the spectral window), respectively. Using the PDM technique for the same data sets and the range of periods gave the highest peak at f = 1.00020cd^-1, similar to that obtained using PERIOD. Pre-whitening the original spectra with an f of 1.00020cd^-1, removed the one at 2.00343cd^-1; however, an f of 2.0025cd^-1 at lower amplitude is still found (third panel of Figure 11).

Fig 11 Periodogram of the original data for the EW of Hα, spectral window, and the periodograms through the process of successive prewhitening for the detected short-term period. 

Since the obtained frequency is around 1 day, this raises the question of whether a one-day periodicity may be caused by the cadence of the observations. To solve this problem we applied the randomizing technique suggested by ^{Eaton et al. (1995)}. We created the randomizing sample for our data, then we calculated the power again for the periodicity in the same frequency range; the same results were obtained. Comparable highest peaks are found in the normalized power of the original data and those obtained from a randomized data set.

7.2.Line Profile Variation

The profile variations (lbv) of the He I 6678 Å absorption line were investigated by inspection of the residuals (subtraction of the individual spectra from the average one). Two sets of closely time-spaced observations of this line were obtained. Eleven spectra were obtained in the course of two successive nights, on 23/24 April 1994, between epochs 49466.3311-4946.5566(≈29h) using a Reticon detector in the coudé spectrograph. Another ten spectra were obtained during a 40-min period of the night of April 24 (at JD 52754.8454 -52754.8733) using the coudé spectrograph and the CCD detector. The lower panel of Figure 12 displays the residuals of the individual spectra obtained by subtraction from the average one, and the time evolution of the corresponding profiles of individual lines, respectively. Each set of observations related to different phases of line strength (the first set obtained in 1994 corresponds to phases of higher line strength in comparison to that obtained in 2003) as was already known from the study of long-term variations of the line. The deviation from the mean profile is clear in some spectra. The upper panel of Figure 12 shows the line wings affected by the emission. The residuals of the individual spectra from the averaged one show fluctuations from one exposure to the other. However, no clear sequence of variation can be detected. No night-to-night changes similar to those found in the Be star FX Lib (as reported by ^{Guo, 1994}) were found in the line profile of κDra.

Fig. 12 Upper panel: Line profiles of 6678Å observed on 24 April 2003 and 23/24 April 1994. Each profile is fitted with the average one (dashed lines). The number at the right hand side is the midexposure HJD. Lower panel: The residuals of individual spectra subtracted from the averaged one.