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Astron. Astrophys. 320, 497-499 (1997)

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2. Observations and period analysis

Additional photometry was carried out in December 1993, again in the uvby system, with the SAT telescope of La Silla. 14 points were obtained over an interval of one month. The homogenized data are given in Table 1.


Table 1. Differential photometry of HD 208217. The comparison star is HD 207964=HR 8352. The data are homogenized, so the average level is zero in each band and each photometric system.

The period search is performed by looking for the best fit of a mathematical model to the observational data. The fitting function is a cosine wave and its first harmonic:


where m is the magnitude, P the fundamental period, I the total number of harmonics, t the time and [FORMULA] the origin of time. Subscript s designates the distinct photometric data sets which, in all generality, may differ because of minor instrumental changes (only affecting the zero points), the use of different comparison stars, or the absence of the latter. The least-squares fit and the uncertainties of the estimated parameters are computed by the general method described by Press et al (1992).

The use of this period-searching algorithm has two advantages. Firstly, Eq. (1) is known to be a very good approximation for most well-studied Ap stars; higher-order harmonics, if at all present, are of small amplitudes (see, e.g., Mathys & Manfroid 1985, Manfroid & Renson 1994). Secondly, the zero-point differences between the Danish and SAT photometric systems are automatically adjusted by the procedure for each trial period. This is of little concern for large data sets with homogeneous phase coverage, where the zero-point shifts can be calculated from simple averages. With only a small observational material at our disposal, it becomes a big asset since the zero points of peculiar stars cannot be accurately estimated from regular standard-star transformations.

Because of the 12-year gap separating both sets of observations, an unambiguous determination of the period proves to be impossible, with dozens of equally likely candidates spread over a rather broad interval centered at 8:d6 (see Fig. 1 for the periodogram relative to the b data set).

[FIGURE] Fig. 1. Periodogram of the Strömgren b data; in this, and the next two figures, the quantity in ordinate is the reduced [FORMULA] of the fit by a cosine wave and its first harmonic

A few additional measurements of the star have been secured in September 1993 at the ESO 1 m telescope in a special narrow-band filter having the same central wavelength as Strömgren b. No comparison star was measured and the overall accuracy of the "all-sky" measurements cannot be compared to that achieved in the other data sets. Nevertheless they proved to be valuable in constraining the range of possible periods to about half a dozen candidates (see Fig. 2).

[FIGURE] Fig. 2. Periodogram of the Strömgren b data together with the narrow-band data

Lifting the remaining ambiguity was made possible by the analysis of the variations of the mean magnetic field modulus (Mathys et al. 1996). 31 magnetic mesurements were secured over two years. The resulting periodogram obtained with the same method as for the photometric data is shown on Fig. 3.

[FIGURE] Fig. 3. Periodogram of the magnetic data


Table 2. Parameters of the least-squares fits for the photometric and magnetic variations of HD 208217. The error on each parameter is indicated in parentheses. [FORMULA] is the scatter around the least-squares fit. r is the total range of the analytical model. Units are magnitude and Gauss.

By combining the photometric and magnetic results, the most likely period appears to be [FORMULA]. The next candidate is [FORMULA] which, although less probable, cannot be totally excluded.

The uvby lightcurves obtained with the newly determined period are shown in Fig. 4. The coefficients of the cosine-wave decomposition are given in Table 1. The origin of time is [FORMULA].

[FIGURE] Fig. 4. Lightcurves in uvby (bottom to top) plotted with the period 8:d44475; ticks on the ordinates are separated by 0.01 mag; open triangles represent the Danish telescope 1981 data; filled squares are the SAT 1993 observations

HD 208217 is characterized by strongly anharmonic light curves. The amplitude of the first harmonic is practically equal to that of the fundamental wave in each band. A search for higher-order harmonics hints at a marginally significant second-order wave in the v band only (and in the indices [FORMULA] and [FORMULA]). Third-order harmonics are totally negligible.

Although their ratio stays more or less constant, the amplitudes [FORMULA] and [FORMULA] change dramatically with wavelength, with a sharp maximum in the v band and a reversal between the b and y bands. The phases of the various components are more consistent and the double-wave pattern is recognizable in every band.

The phase diagram of the mean magnetic field modulus, with a two-wave fit superimposed, is shown on Fig. 5. The coefficients of the fit are given in the last line of Table 1. While the presence of a first harmonic in the magnetic variations is now well established, the observational errors do not allow to conclude on possible higher-order terms.

[FIGURE] Fig. 5. Phase diagram of the mean magnetic field modulus with the period 8:d44475
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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998