## 3. Periodic changes with the 1 37 period## 3.1. Is there a secularly stable 1 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 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 74, advocated by Clarke (1990). I, therefore, restricted my initial data analyses to the neighbourhood of the 1 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 25 and 1
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
372 (frequency 0.7289 c d
The problem of the determination of an accurate value of the 1
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
Putting aside the early RVs I therefore adopted a period of 1
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.
(The best fit period in the neighbourhood of the period derived by Baade (1984) is 1 366791. It gives a much worse phase curve than the period of 1 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
To check on this suspicion, I used the best-fit period 1 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 1 372 period. Note that if one assumes a strictly sinusoidal variation of the O-C changes, the best-fit period is (5650 400) d, the semi-amplitude of the variation is 0 174 0 017 and the mean O-C amounts to 0 940. 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 37196 0 00004, 1 37192 0 00003 and 1 37162 0 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 With the numerical values derived above, formula (3) predicts 1 3720, 1 3721 and 1 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 1 372 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
I also analyzed
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
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 My tentative conclusion is that one observes a combination of a
slow and probably cyclic small variation of the 1
372 period and of an amplitude variation on a
time scale of 10 ## 3.3. Polarimetric and light changes with the 1 372 periodFig. 5 shows the polarimetric observations by Clarke (1990) plotted vs. phase of the mean 1 372 period from ephemeris (1). Some mild variability may be suspected.
A firm detection of photometric variations with the 1
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 1 372 period © European Southern Observatory (ESO) 1998 Online publication: May 15, 1998 |