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Astron. Astrophys. 339, 822-830 (1998)

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3. Period search

Because the photometric sets provide the largest part of the observational data and usually are more accurate than the spectroscopic data from the point of view of the dispersion, they were given a higher priority in the period analysis.

The light curves of CU Vir have slightly different amplitudes and shapes in different spectral bands. Therefore we first normalized all data by their amplitudes to obtain homogeneous data sets. The same procedure was done with the spectroscopic observations. We then divided our data into two groups: U+u+[FORMULA] (Si II 6347) and B+b+[FORMULA] (Si II 4128-31, 4201). The intensity variations of the Si II 6347 line is closer in shape to the U+u and possibly the y light curves while those of Si II 4128-31 have a similar shape to the B+b light curve. Three different approaches were used for period analysis, discussed below.

3.1. Constant period

First we searched for a constant period over the whole range of the photometric data using Stellingwerf's (1978) method. The best period is 0[FORMULA]52070281[FORMULA]0.00000016 and it is defined mainly by the recent more extensive photometric data sets. Fig. 2a represents a plot of all data with the constant period (left panel). The right panel shows a plot of the deviations from the mean curve. The rms is 0.168. One may easily see that a few data sets do not fit at all with this period. Note, the the error in the period determinations is mainly defined by the entire time interval of the data used, therefore it is smallest for the constant period because the time interval is the largest, more than 40 years.

[FIGURE] Fig. 2a-c1. Photometric variations vs. phases calculated with three different period approaches. a with a constant period 0[FORMULA]52070281; b with a linearily changing period (Sect. 3.1); and c with two periods constant periods P1=0[FORMULA]5206778 and P2=0[FORMULA]52070308. In c, combined B+b (b ) normalized magnitudes are from Table 1, data sets #1, 3, 5, 10, 11 (P1) and data sets #15, 16, 18-22, 25, 26 (P2). Right panels show the deviations from the mean curves.

3.2. Linearily changing period

The search for the linearily changing period was done using the method by Cuypers (1986) realized in Pelt's (1992) package. The best fit to all our data was achieved with the following ephemeris:

[EQUATION]

[EQUATION]

where [FORMULA]=2435178.92 and S=[FORMULA]. This corresponds to P_[FORMULA] days per cycle. A plot of all photometric data with this period is shown in Fig. 2b. Again, the right panel of Fig. 2b shows deviations of the points from the mean curve with the rms being 0.127. The linearily changing period shows practically the same scatter in the final plot for the photometry as does a solution with 2 different periods for all but the # 5 data set, which is definitely shifted by about 0.2 of the period. This shift cannot be explained by any incorrectness of the reduction or normalization procedure because Winzer's data were always included in any period search by previous investigators, and showed a very good phase agreement with all photometric and spectroscopic variations reported before 1985.

Moreover, when we plot the equivalent width data with the linearily changing period (Fig. 3a) we obtain noticible phase shifts between the different data sets.

[FIGURE] Fig. 3a and b. Spectroscopic variations vs. phases calculated with the linearily changing period (a ) and with two constant periods (b) (see Fig. 3). [FORMULA] (Si II 4128-31 and 4201) are plotted; Table 1 data sets #4 and 6 (P1) are represented by filled triangles and open squares, respectively; and data sets #14 and 24 (P2) by open triangles and filled circles, respectively.

3.3. Two periods solution

First we constructed an O-C diagram with the ephemeris,

[EQUATION]

The period was estimated by Pyper (1994) as the one that best fit the photometric data from 1955-1984. Then we measured the phases of the maxima for each data set; they are plotted in Fig. 4a and b. The phases of the maxima and minima were found by a sinusoidal fit to the observational points. It is seen that our data can be fit by two straight lines that intersect near epoch JD=2446000 (1985). Period analyses perfomed separately for two groups of data before and after 2446000 by Stellingwerf's (1978) method resulted in two different periods: 0[FORMULA]5206778[FORMULA]0.00000020 (JD[FORMULA]2446000) and 0[FORMULA]52070308[FORMULA]0.00000019 (JD[FORMULA]2446000). The O-C diagram obtained with two periods is shown in Fig. 4c,d for maxima and minima and for different spectral bands. The U+u and B+b light curves give slightly different phases of minimum, therefore we averaged them and finally obtained the following ephemeris which fit all photometric and spectroscopic observations for 40 years (more than 29000 rotations) of the observations:
JD(U,B light min) = 2435178.6417 +[FORMULA]

[FIGURE] Fig. 4a-d. O-C diagrams for CU Vir with one constant period P1=0[FORMULA]5206778 (a ,b ), and with two constant periods, P1 and P2=0[FORMULA]52070308 (c ,d ). Photometric observations before 2446000 are shown by filled circles, those after 2446000 are shown by open circles. Spectroscopic observations are shown by asterisks; in (a,c) [FORMULA] (Si II 6347) are plotted, in b and d [FORMULA] (Si II 4128-31, 4201) are plotted. In a and b , the dashed lines represent 99% confidence levels. In plots c and d the upper line represents light and spectrum minima, and the lower line represents maxima.

Those who would like to use the maximum as a starting phase can use the following moment JD(B light max)=2435178.9025.

Combined B+b light curves obtained with the above two periods for all photometric observations are plotted on Fig. 2c (left panel). The deviations from the mean curve are shown in the left panel of Fig. 2c. The rms is 0.114. According to the Fisher test the rms-values in all three approaches for the period search are different with 99.3% confidence level. Combined Si II intensity variations are plotted on Fig. 3b ([FORMULA] 4128-31, 4201). Note that we slightly shifted the equivalent widths from different spectroscopic sets by constant values. These shifts may arise from different treatments of the continuum as well as from different registration (CCD, Reticon, photographic plates), and have no influence on the period search procedure. Fig. 5 displays combined curves of the effective magnetic field variations (a); He I 4026 (b); and He I 4471 (c) line intensity variations. Here we did not need any vertical shifts to combine the spectroscopic data. Fig. 6 represents the hydrogen line equivalent width variations (a); variations of the [FORMULA]-index (b), and radial velocity variations (c).

[FIGURE] Fig. 5a-c. Combined effective magnetic field variations (a ); [FORMULA] (He I 4026) (b ); and [FORMULA] (He I 4471) (c ). a Table 1, data sets #8 (P1) and 30 (P2) are represented by open circles and by filled circles, respectively. b Data sets #6 and 9 (P1) and 12 (P2) are represented by filled circles, open circles, and filled triangles, respectively. c Data sets #6 (P1) and 23 (P2) are represented by filled circles and open circles, respectively.

[FIGURE] Fig. 6a-c. Combined hydrogen line variations. a Normalized [FORMULA] of Balmer lines vs. phases. Table 1, data set #4, [FORMULA] (P1) are open circles; and data set #24, [FORMULA] (P2) are filled circles. b [FORMULA]-index vs. phase. Table 1, data sets #10-11 (P1) are filled triangles; and data set #17 (P2) are open circles. c Radial velocity vs. phase. Table 1, data set #2 (P1) are open circles, and data set #24 (P2) are filled circles.

3.4. Does a period change still continue?

Continuing observations with the FCAPT are being made in order to see whether the period of CU Vir continues to change or has stabilized. A period search of the P10 data (data sets #15, 16, 18) using the Scargle (1982) algorithm, yielded a slightly shorter period of 0[FORMULA]5206987. O-C diagrams of the P10 and FCAPT data using this period are plotted in Fig. 7. Investigators of eclipsing binary light curves (e.g., Cherewick & Young 1975) have found that a continually varying period results in an ephemeris

[EQUATION]

where [FORMULA] is the rate of change of the period ( d cyc-1). If the FCAPT data for 1997 are included, least squares quadratic polynomial regressions of the u and b data result in [FORMULA] and [FORMULA], respectively. At present, the residuals of these fits are only slightly better than those for a linear regression on the same data. There is an additional indication of the continually changing period in the O-C diagrams (Fig. 4), where the open circles do not lie on the straight line but rather lie on the parabolic line, which is typical for a changing period. Several more years of observations will be necessary to determine whether the period of CU Vir is still changing.

[FIGURE] Fig. 7a and b. O-C diagrams, calculated with P=0[FORMULA]5206987 for the CU Vir U+u (a ) and B+b (b ) minima from the P10 UBV (Table 1, data sets #15, 16, 18) and FCAPT uvby (data sets #19-22, 25-27) data. In both plots, the P10 data are open circles and the FCAPT data are closed circles. The solid lines are least squares quadratic regressions.

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© European Southern Observatory (ESO) 1998

Online publication: October 22, 1998
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