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Astron. Astrophys. 334, 558-570 (1998)

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3. Periodic changes with the 1 [FORMULA] 37 period

3.1. Is there a secularly stable 1 [FORMULA] 37 period?

First, I analyzed the available RVs. For all least-square fits discussed below, I formally used the program FOTEL (Hadrava 1990, 1995a) which is designed for orbital (and light-curve) solutions of spectroscopic (eclipsing) binaries, uses the simplex method and calculates realistic errors based on the covariance matrix. I verified that Baade's RVs can indeed be best reconciled with periods close to 1 [FORMULA] 37. Other periods, including both 1-d aliases, seem to give much worse phase diagrams. On the other hand, the scatter is not significantly reduced for the double-wave periods near 2 [FORMULA] 74, advocated by Clarke (1990). I, therefore, restricted my initial data analyses to the neighbourhood of the 1 [FORMULA] 37 period.

Fig. 1 shows Stellingwerf's (1978) PDM periodograms (structured into 5 bins with 2 `covers') for the individual RV data sets for periods between 1 [FORMULA] 25 and 1 [FORMULA] 54. It is seen that the resolution in frequency is inevitably low for the short data strings while the longer strings of data suffer from aliasing problems. Yet, it seems clear that the only period common to all periodograms is a period close to 1 [FORMULA] 372 (frequency 0.7289 c d-1), i.e. a one-year alias of the 1 [FORMULA] 36673 period derived by Baade (1984). It is a bit curious that the RVs which best show the 1 [FORMULA] 372 periodicity - without too much aliasing - are the Kitt Peak RVs published by Abt & Levy (1978) who concluded from them that the RV of [FORMULA]  CMa is constant. (During preliminary analyses, I was obtaining a different period from the Reticon RVs published by Baade 1984. Upon a closer examination, I came to the conclusion that the correct HJDs of the three October 1982 Reticon spectra must be higher by 1 day than what is given in Table 1a of Baade 1984. Upon my request, Dr. Baade very kindly checked his original records and confirmed that it is indeed so. After this correction of dates (already applied in Table 1 and Fig. 1), also the RVs from Baade's 1982-83 Reticon observations seem to be best reconciled with a period of 1 [FORMULA] 372.)

[FIGURE] Fig. 1. Stellingwerf's PDM periodograms of [FORMULA]  CMa for various RV data subsets

The problem of the determination of an accurate value of the 1 [FORMULA] 37 period is not trivial, however. This is not only because of the heterogeneity of the data but also due to the fact that the amplitude and mean RV of the apparent RV changes varies not only with the dispersion and resolution of the spectrograms (as already pointed out by Baade) but probably also from physical reasons. This is best illustrated by Fig. 2 where the RV curves, based on two consecutive observing runs by Baade with the same instrument, are compared. The semi-amplitudes of the two curves are (30.61 [FORMULA] 0.34) km s-1 and (14.5 [FORMULA] 1.3) km s-1, respectively, the rms of the fit per 1 observation being 3.1 km s-1 in both cases. As Baade remarked, the Balmer emission has decreased between these two epochs. Therefore, the change of the amplitude of the 1 [FORMULA] 37 RV curve can indicate either that the variation of the line cores is affected by the strength of the circumstellar matter or that it is controlled by more than one period close to 1 [FORMULA] 4.

[FIGURE] Fig. 2. Radial-velocity curves of the He I triplet lines of [FORMULA]  CMa (P = 1 [FORMULA] 372) from two consecutive observing runs by Baade with the same instrument. The upper one is based on data secured between HJD 2443884 and ...895, the lower one on data taken between HJD 2444951 and ...955. A large difference in the amplitudes of the two curves is clearly seen

Putting aside the early RVs I therefore adopted a period of 1 [FORMULA] 3718 which resulted from trial period searches over all more recent RVs, and calculated local sinusoidal fits to individual subsets of data covering no more than 400 d (for Abt and Levy's data) and much less for all other data sets obtained since HJD 2440607. 1 This way, the particularly chosen exact value of the 1 [FORMULA] 37 period played a negligibly small role in the determination of locally derived semi-amplitudes and systemic velocities of individual RV subsets. Using these locally derived values, I then transformed all RV data (HJD [FORMULA] 2440607) into an interval [FORMULA] and subjected this homogenized data set to a period analysis. Clearly the best period was indicated in the neighbourhood of 1 [FORMULA] 3719. A sinusoidal fit led to the following linear ephemeris:


(The best fit period in the neighbourhood of the period derived by Baade (1984) is 1 [FORMULA] 366791. It gives a much worse phase curve than the period of 1 [FORMULA] 371906.)

3.2. Cyclic changes of a single period or an interference of several short periods?

A phase diagram of the transformed RVs for ephemeris (1) is shown in the upper panel of Fig. 3 where the individual data subsets are denoted by different symbols. Comparing this plot to Fig. 2, one can see an increased scatter in the phase diagram based on all RVs. (Note, however, that the scatter in the transformed RVs is a bit confusing since, say, a deviation of 0.1 from the mean curve represents 1.5 km s-1 for a subset with a semi-amplitude of 15 km s-1 but 3.5 km s-1 for another one, with a semi-amplitude of 35 km s-1.) On a closer inspection, one clearly sees that the individual subsets are slightly shifted in phase with respect to each other. This indicates that the period is changing with time.

[FIGURE] Fig. 3. A RV curve of the KPNO, La Silla and Calar Alto observations, transformed locally into an interval [FORMULA]. Upper panel: RVs plotted vs. phase of the linear ephemeris (1). Bottom panel: RVs re-plotted vs. phase of a periodically varying 1 [FORMULA] 372 period (see the text for details). Individual data subsets localized in time and denoted by various symbols in both plots refer to the following epochs (in HJD-2400000): C: 40607.5-40608.8; D1: 42033.0-42452.8; D2: 42724.9-42887.6; E1: 43091.8-43096.9; E2: 43499.6-43510.6; E3: 43884.5-43895.8; F: 44951.7-44955.9; G1: 45245.9-45247.9; G2: 45351.8-45357.7

To check on this suspicion, I used the best-fit period 1 [FORMULA] 371906 to re-calculate local epochs for the individual data subsets. Then I derived the O-C deviations of these epochs from the linear ephemeris (1). Their plot vs. time (see the upper panel of Fig. 6a-d) shows that the change is smooth and indicative of a slow and possibly periodic change of the 1 [FORMULA] 372 period. Note that if one assumes a strictly sinusoidal variation of the O-C changes, the best-fit period is (5650 [FORMULA] 400) d, the semi-amplitude of the variation is 0 [FORMULA] 174 [FORMULA] 0 [FORMULA] 017 and the mean O-C amounts to 0 [FORMULA] 940. This latter value implies that the epoch of maximum RV of ephemeris (1) for the mean period should be corrected to HJD 2442805.802. One should adopt this result as a tentative one, however, since the O-C variations - even if they vary strictly periodically - need not follow either an exact sinusoid or the period found from a few data points only.

As a test of internal consistency of the procedure, I analyzed three subsets of more recent RVs, each of them spanning more than 400 d, namely (in HJD-240000): 40607 - 42452, 42724 - 43895, and 44951 - 45357. The resulting local periods of their sinusoidal fits were 1 [FORMULA] 37196 [FORMULA] 0 [FORMULA] 00004, 1 [FORMULA] 37192 [FORMULA] 0 [FORMULA] 00003 and 1 [FORMULA] 37162 [FORMULA] 0 [FORMULA] 00006, respectively. It is possible to compare them to those from the sinusoidal model used: If one assumes that the O-C varies sinusoidally, then


where T is the time of observation, [FORMULA] and [FORMULA] are the period and time of maximum of the short and long variation, respectively, A is the semi-amplitude of the long variation and E is a generalized epoch, i.e. cycle and phase of the observation. Since [FORMULA], one can substitute, with a high accuracy [FORMULA] into the cosine term in eq.(  2). For the instantaneous period one then obtains


With the numerical values derived above, formula (3) predicts 1 [FORMULA] 3720, 1 [FORMULA] 3721 and 1 [FORMULA] 3716, in a reasonable agreement with the above results of the local fits if one realizes inevitable inaccuracies of the tentative model function used.

Finally, I derived the phases of the 1 [FORMULA] 372 variation, assuming its slow sinusoidal change, from eq. (2) and re-plotted all normalized RVs in the bottom panel of Fig. 3. Note that the remaining larger deviations all come from photographic spectra of moderate dispersions.

Next, I considered the possibility of multiperiodic changes. I first searched for periodicity the O-C residuals from the normalized RV curve for a constant period of ephemeris (1). I used Breger's program PERIOD (Breger 1990), searching over the whole range of frequencies up to 20 c d-1. I indeed found a periodic variation with a period close to the 1 [FORMULA] 3719 period, namely 1 [FORMULA] 348548 [FORMULA] 0 [FORMULA] 000026. The phase curve for this period is shown in Fig. 4. However, the fit of normalized RVs with these two periods gives a larger rms error per 1 observation than the fit with one periodically variable period.

[FIGURE] Fig. 4. The O-C from the 1.372-d RV curve (normalized RVs) plotted vs. phase of the 1 [FORMULA] 348548 period. A calculated epoch of the maximum residual RV, HJD 2442805.817, is used as phase zero

I also analyzed the original RVs for multiperiodicity to see whether the amplitude change could be due to interference of several frequencies. In every step, I fitted the original data with all the frequencies found and analyzed the new residuals from the fit again. The results of this analysis are summarized in Table 4. The rms errors of individual original RVs range from 3 to 6 km s-1 which is comparable to the rms from the multiperiodic fit. I repeated this analysis for two data subsets, covering the first, and the last about 3000 d, respectively. The first two periods with largest amplitudes (in brackets) were 1 [FORMULA] 37196 (22.6) and 1 [FORMULA] 36933 (7.7) in the former, and 1 [FORMULA] 37189 (24.0) and 1 [FORMULA] 34631 (12.7) in the latter subset. Further frequencies with smaller amplitudes, found in each subset, differ totally from each other and from those of Table 4. The rms error of the fit for the second subset (containing solely Baade's data) was only slowly decreasing with more frequencies added. I verified, however, that the first two periods of Table 4 also fit the first data subset quite well. Since these two frequencies are clearly essential to modelling of the amplitude changes, this may in turn indicate that there is some regularity in the amplitude changes of the RV curves. At the same time, it is obvious that the instrumental effects of the very different spectral resolutions used also affects the actually measured amplitude of each RV curve from heterogeneous sources. I calculated RV amplitudes for about 1-d long subsets of the longest set of homogeneous RV observations (HJDs 2443884-895) to find that any variation of the RV amplitude must occur on a time scale much longer than 10 d. On the other hand, Fig. 6a-d shows that a significant amplitude change took part within 300 d. Note that the beat period between the first two periods of Table 4, 72 [FORMULA] 274, is indeed within these limits. I also created artificial data using the first two periods of Table 4 and calculated local RV fits exactly as for the real data, for the 1 [FORMULA] 371906 period fixed. This gave an approximate (far from ideal) reproduction of the run of the O-C deviations for the local epochs, a rather fair reproduction of all local amplitudes, and a poor reproduction of local mean velocities. Moreover, the artificial local RV curves showed significant systematic variations over the intervals covered by real data subsets as well as significant deviations from the sinusoidal shape - in contrast to real observations.


Table 4. Results of a multiperiodic fit to the original more recent RVs of [FORMULA]  CMa (the rms errors of the last 2 digits of the periods are given in brackets; the rms per 1 observation of the fit is 4.28 km s-1)

At this stage I decided to carry out a numerical test. Using the sinusoidal fit to the O-C changes (cf. the upper panel of Fig. 6a-d), I constructed a model function describing a sinusoidal variation with the mean period of 1 [FORMULA] 371906 which periodically varies with a period of 5644 d and generated an artificial data set for HJDs identical to those of real RV observations. Then I subjected the model function to a Fourier analysis over the range of frequencies from 0.0001 to 2 c d-1. This analysis indeed led to the detection of several periods in the neighbourhood of the 1 [FORMULA] 3719 period, the second largest amplitude being detected for a period of 1 [FORMULA] 354, reminiscent of what I obtained from the Fourier analysis of the real RVs - both normalized and original ones (note that my model is close to, but not identical to the real data since it is based on pure sinusoids with no scatter added). I also verified that if the normalized RVs were pre-whitened for the periodically variable period, no other periods close to 1 [FORMULA] 3 - 1 [FORMULA] 4 were detected in the residuals. Some remaining power was found in the frequency range of about 0.01 to 0.1 c d-1 but I was unable to find any clear periodicity in these residuals.

My tentative conclusion is that one observes a combination of a slow and probably cyclic small variation of the 1 [FORMULA] 372 period and of an amplitude variation on a time scale of 100 -101  d which may be related to changes in the Be envelope, as already suggested by Baade and as may be suspected from Fig. 6a-d below. New systematic spectral observations are clearly needed to check on the possible true periodicity of the amplitude changes.

3.3. Polarimetric and light changes with the 1 [FORMULA] 372 period

Fig. 5 shows the polarimetric observations by Clarke (1990) plotted vs. phase of the mean 1 [FORMULA] 372 period from ephemeris (1). Some mild variability may be suspected.

[FIGURE] Fig. 5. A phase plot of the Stokes parameter u with respect to intrinsic stellar axes (from Clarke 1990) for the 1 [FORMULA] 37 period. Phases were calculated using ephemeris (1)

A firm detection of photometric variations with the 1 [FORMULA] 372 period is seriously hampered by the presence of light variations on at least two different longer time scales which are discussed below. I carried out various trials only to find out that the 1 [FORMULA] 372 period is not convincingly present in accurate photometric data sets after their proper prewhitening for changes on longer time scales (see below).

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© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998