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Astron. Astrophys. 331, 171-178 (1998)

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3. Results

3.1. Frequency analysis

Analysis of frequencies of our data was carried out using the Discrete Fourier Transform method, as described in López de Coca et al. (1984). The v filter was the first analysed because both amplitude and intensity are, commonly in [FORMULA]  Sct variables, larger in this band than in all the other three. Firstly, we have analysed the C2-C1 differences. When the Fourier analysis is applied to C2-C1, we obtain C2 and C1 not showing any sign of variability within about 0.m 0006, in the range from 0 to 30 cd-1. The standard deviation of C2-C1 was found to be of 0.m 0026. In our case, the significance level, as defined by Breger et al. (1996), is of about 0.m 0010. That is, the noise level as the average amplitude in an oversampled amplitude spectrum (as defined in Handler et al. (1996)) is of about 2.m 25*10-4 in the same range of frequencies.

When the Fourier analysis is applied to 20 CVn, the periodograms showed a principal peak at [FORMULA] =8.2168 cd-1 in very good agreement with the period P=0.d 1217 derived by earlier authors. Fig. 2 shows the spectral window and power spectra of 20 CVn in the v filter before and after prewhitening for the above derived frequency. The line in the bottom panel of Fig. 2 means the significance level as described by Breger et al. (1996). As can be seen, after prewhitening for the frequency [FORMULA], the resulting periodograms did not show any trace of another peak, suggesting the monoperiodic nature of this star within 0.m 0005. Moreover, no signal for the second harmonic of the frequency [FORMULA] is present in the periodogram. This is due to the very small amplitude of the light curve. Thus, the light curve is a perfect sinusoid. As we can see in Fig. 2, our resulting power spectrum did not show any peak with an amplitude signal-to-noise (S/N) ratio larger than 4 in the range where other peaks may be suspected to be found in this star. The same analysis was performed for the other three uby filters and the results were consistent in the sense that the frequency found was always the same within 0.0001 cd-1.

[FIGURE] Fig. 2. Power spectra of 20 CVn in the v filter before and after prewhitening the frequency [FORMULA] =8.2168 cd-1 to our data

Frequency analysis was also carried out on the different data sets available from earlier authors, especially on the three best ones, i.e. to the 1969 B and V data of Shaw (1976) and the 1980 V data of Peña & González-Bedolla (1981). In all the cases we find similar results to those found with our data, that is, when the main frequency is prewhitened the resulting periodograms do not show any trace of another peak. Nevertheless, in these cases, the noise level is much higher than in our data. Therefore, we conclude that there are no remaining periodicities in the light variation of 20 CVn.

In our case, the standard deviations, as determined by residuals from the solution, were of 0.m 0048, 0.m 0019, 0.m 0017 and 0.m 0018 for the u, v, b and y =V bands, respectively. In Fig. 1 we plot the resulting Fourier fitting, in the filter v, with the observed light curves. We can see that the synthetic light curve satisfactorily reproduces the data. The results of the Fourier analysis are listed in Table 1 with the amplitudes, phases, mean values and residuals calculated for the four uvby filters together with the b-y and c1 indices. The initial time is HJD=2449736.d 5738 which corresponds to our former observational datum.


Table 1. Results from Fourier analysis

3.2. O-C analysis

Assuming 20 CVn as a monoperiodic pulsator, the classical O-C method can be used. Twelve times of light maxima were obtained from the new data by using the method described in Rodríguez et al. (1990), where each light maximum was derived as an average over the three vby bands. The u band was not considered for the averages since the time of maximum for this filter is shifted with respect to the other three by about 0.015 cycles (see Table 1), in agreement with other [FORMULA]  Sct type pulsators (Rodríguez et al. 1995band references therein). In addition, ten times of maxima have been derived from data of Shaw (1976) and Chun et al. (1983) as averages over the BV bands of the Johnson photometric system, another fifteen from Wehlau et al. (1966), Peña & González-Bedolla (1981) and Bossi et al. (1983) in the V band and another one from Nishimura et al. (1983) in the B band. As we can see, the times of maxima are obtained in different filters B, V and vby but it is not necessary to correct the different times of maxima to refer all to the same band. These corrections are very small (of about one ten-thousandth of a day) and do not change the final results (Rodríguez et al. 1993). However, the two maxima obtained from Wehlau's et al. (1966) data (HJD=2439278.d 722 and 2439293.d 688) were not used to determine the ephemeris of 20 CVn. These observations were obtained in nearly the same epoch than those from Shaw (1976) but the instants of these maxima are in disagreement with the ephemeris obtained using different combinations of data sets.

With the aim of using the largest number of maxima as possible to determine the ephemeris of 20 CVn, we made use of the radial velocity data available from the bibliography to calculate the corresponding maxima and to transform to light maxima. For this purpose, we calculate the phase shift between the instants of maximum in radial velocity and light using the spectroscopic data from Mathias & Aerts (1996) and our own photometric data. Note, that these two sets of observations were collected at the same epoch. In this way, we obtain that light maximum occurs after that corresponding to radial velocity by 0.d 0443 ([FORMULA] 0.0010)=  [FORMULA]  ([FORMULA] 3). In addition, we can determine the phase shift between the maxima in temperature and radius, assuming the curve in temperature as the curve in (b-y). In this case we obtain Tmax (Te)-Tmax (R)=- [FORMULA] ([FORMULA] 5) in the sense that radius maximum occurs later. Please, note that there is a mistake in the times of the radial velocity curve Fig. 1 from Mathias & Aerts (1996). The true values were kindly supplied by Mathias (1997, priv. comm.) and a value of HJD=2449707.d 6599 was derived for the instant of radial velocity maximum. Using this method, four new times of light maxima were obtained from radial velocity data of Nishimura et al. (1983), Yang & Walker (1986) and Mathias & Aerts (1996).

Consequently, forty times of maxima (from 1969 to 1997), listed in the second column on Table 2, were used to determine the ephemeris of the pulsation of 20 CVn by means of the classical O-C method. At first, we applied this method to the maxima obtained in the epoch 1980-1983. T0 =2444376.d 673 (first light maximum derived from Peña & González-Bedolla 1981) was adopted as initial epoch and P0 =0.d 12170 (from our Fourier analysis) as the initial period. A least squares fit of a linear ephemeris leads to the following elements: T1 =2444376.d 6765 ([FORMULA] 0.0011) and P1 =0.d 1217027 ([FORMULA] 0.0000002). This value seems to be good and the residuals appear to be randomly distributed around zero. For a second stage, we applied the O-C method to all the forty maxima listed in Table 2 using as initial ephemeris the above solution T1 and P1. Now, the resulting elements of the new linear ephemeris are: T2 =2444376.d 6644 ([FORMULA] 0.0024) and P2 =0.d 12170234 ([FORMULA] 0.00000008). The resulting cycles Ei and residuals (O-C)l are listed in the third and fourth columns of Table 2. The standard deviation of the fit is of 0.d 0146. This value is too large and, from Table 2, it can also be seen that the residuals (O-C)l are not random distributed around zero. This leads us to make a reinterpretation of all the maxima. A least squares fit of a quadratic ephemeris as Tmax =T3 +P3 E+AE2 gives the following coefficients: T3 =2444376.d 6772 ([FORMULA] 0.0007), P3 =0.d 12170268 ([FORMULA] 0.00000002) and A=-19.d 3([FORMULA] 0.7)10-12. Now, the standard deviation of the fit is of 0.d 0031 in good agreement with the error bars in the determination of the earlier maxima. Moreover, the new residuals (O-C)q, listed in the fifth column of Table 2, appear to be randomly distributed around zero showing that the residuals (O-C)l fit well to the parabola.


Table 2. Times of maxima of 20 CVn. The sources are: 1) Shaw 1976; 2) Peña & González-Bedolla 1981; 3) Bossi et al. 1983; 4) Chun et al. 1983; 5) Yang & Walker 1986; 6) Nishimura et al. 1983; 7) Mathias & Aerts 1996; 8) present work

This result means that the pulsation period of 20 CVn is decreasing at a rate of dP/dt=-11.6([FORMULA] 0.4)10-8  dy-1 or dP/Pdt=-95([FORMULA] 3)10-8  y-1. However, an observed negative period change is in contradiction with the theoretical positive period changes expected from stellar evolutionary tracks inside the [FORMULA]  Sct region in the H-R diagram, assuming that other physical reasons for period changes can be excluded (Rodríguez et al. 1995a). In particular, for 20 CVn, a star near to the overall contraction phase (using an evolutionary model with core overshooting), the period change rates expected from evolutionary tracks at the two stages considered in Sect. 3.4 (i.e., main sequence and post-main sequence stages using the evolution models of Schaller et al. 1992) were calculated. The results indicate increasing periods with values, for dP/Pdt, of 0.39x10-8  y-1 and 52.4x10-8  y-1, respectively. However, we know that negative period changes are observationally shown for a number of evolved Population I high amplitude [FORMULA]  Sct stars. Nevertheless, the rate of decreasing period found for 20 CVn is also too high, as compared with those observed rates, in about one order of magnitude. Moreover, as we mentioned above, the two maxima derived from Wehlau's et al. (1966) data do not fit well to our resulting ephemeris. Then our result must be used with caution and more data are necessary during the next years in order to get a definitive conclusion.

3.3. Amplitude variations

Long term amplitude variations seem to be common in low amplitude [FORMULA]  Sct stars. This is different to that suspected in the high amplitude ones. For any of the known high amplitude [FORMULA]  Sct stars, long term amplitude variability had not been established at the time. In this sense, we have analysed the different data sets available for 20 CVn from the bibliography. Unfortunately, only the two early data sets from Shaw (1976) in BV and Peña & González-Bedolla (1981) in V are reliable for this subject. Table 3 lists the amplitudes from these data sets together to the one obtained from our data. In all the cases, these amplitude determinations were carried out making the Fourier fitting with only one term for the frequency 8.2168 cd-1 obtained in Sect. 3.1. For the sake of homogeneity, Johnson's B amplitude, from Shaw (1976) data, has been transformed to V equivalent amplitude using our uvby data and assuming that the b and v measurements can be averaged to approximate the variations in B. Then, a factor of 0.709 ([FORMULA] 0.011) can be applied to transform B to V amplitudes. From Table 2 it seems that there are not significant variations in the luminosity amplitude of 20 CVn. If there is some real change from one session to another, this variation would be of only a very low percentage of the total amplitude. Nevertheless, only three sessions (years 1969, 1980 and 1995) are available for this analysis. Hence, more data sets are necessary to obtain a definitive conclusion.


Table 3. Amplitudes, as determined by means of the Fourier analysis (i.e., semiamplitudes), on different data sets. The sources are: 1) Shaw 1976, 2) Peña & González-Bedolla 1981, 3) present work

3.4. Photometry

Fig. 3 shows the light and colour index variations of 20 CVn along the pulsational cycle assuming the quadratic ephemeris derived in Sect. 3.2. As can be seen, the curves in V and b-y are phased. However, the maximum in the c1 index curve occurs about 0.05 cycles later than the maximum in b-y, in good agreement with other [FORMULA]  Sct pulsators (Garrido & Rodríguez 1990), due to the temperature and gravity variations. The m1 index curve seems to be constant, as was expected, from the low luminosity amplitude of this star.

[FIGURE] Fig. 3. Light curve and colour index variations of 20 CVn along the pulsation cycle

In order to discuss the pulsational characteristics of this star it is useful to know its physical parameters. The following Strömgren values of V=4.m 73, b-y=0.m 180, m1 =0.m 231, c1 =0.913 and [FORMULA] =2.m 778 (Rodríguez et al. 1994) were assumed for the variable. The intrinsic indices were derived using the reference lines of Philip & Egret (1980) with the appropriate corrections in gravity and metallicity (Crawford 1975a, b; Philip et al. 1976). Thus, a colour excess of 0.m 022, -0.m 007 and 0.m 005 were found for b-y, m1 and c1. Then, deviations from the ZAMS's values of [FORMULA] m1 =-0.m 043 and [FORMULA] c1 =0.m 172 are obtained. This value for [FORMULA] m1 indicates that 20 CVn is overabundant in metals. In this case, it has been shown that the c1 index is not a good luminosity indicator (Kurtz 1979): the luminosity determined directly from [FORMULA] c1 index is underestimated. It is similar to that shown in Fig. 1 of Rodríguez et al. (1994) where some Am [FORMULA]  Sct stars seem to lie outside of the cool border of the [FORMULA]  Sct region. Fortunately, Guthrie (1987) derived a relation to correct [FORMULA] c1 in order to be useful as luminosity indicator. This relation depends on metallicity and rotational velocity. In our case, the rotational velocity term can be neglected because the value of vsini for 20 CVn is very small (15 Km/s, Rodríguez et al. 1994). Then, its contribution is less than one thousandth of a magnitude.

With these corrections and using the relation by Crawford (1975b) for luminosity, Code et al. (1976) for bolometric correction and the grids by Lester et al. (1986) with [Me/H]=0.0 for temperature and gravity we obtain the following values of Mbol =1.m 01, Te =7540 K and log g=3.54. In addition, a value of [Me/H]=0.54 is obtained using the Smalley's (1993) calibration for metal abundances. From a comparison with the ([FORMULA] m [FORMULA], [FORMULA]) grids by Rodríguez et al. (1991), a m1 index variation of less than 0.m 001, reversed with respect to the V light curve, must be expected over the full cycle of pulsation of 20 CVn. This expected variation is too small to be detected from the m1 index curve as it can be seen in Fig. 3.

The values found for Te, log g and [Me/H] are in good agreement with those compiled in the Cayrel de Strobel's et al. (1992) catalogue from earlier authors. This is also true for [Me/H]=0.48 obtained by Hauck et al. (1985). However, our determinations for Te and log g seem to be higher than those of 7200 K and 3.0 obtained by these authors from spectrograms. In order to decide this point, we have reanalysed our data. As mentioned in Sect. 2, a few H[FORMULA] measurements were collected in this work. Then, the standard magnitude differences of 20 CVn minus C1=HD 115271 were transformed to apparent magnitudes of the variable assuming the following values for C1: V=5.m 79, b-y=0.m 109, m1 =0.m 205, c1 =0.m 943 and [FORMULA] =2.m 856 (Hauck & Mermilliod 1990). Then, we obtain the following Strömgren indices for 20 CVn: V=4.m 73, b-y=0.m 168, m1 =0.m 231, c1 =0.912 and [FORMULA] =2.m 798 in very good agreement with our previous determination. Only there are some differences in b-y and [FORMULA] but they indicate a hotter and, hence, less luminous star. Following the same procedure mentioned above, we obtain: Mbol =1.m 33, Te =7760 K, log g=3.73 and [Me/H]=0.47. Then, we assume the mean values of 1.m 17, 7650 K, 3.64 and 0.51 for these parameters, with typical errors of 0.m 3, 150 K, 0.1 and 0.1, respectively (Lester et al. 1986; Breger 1990; Smalley 1993), in good agreement with the values derived above.

Moreover, using the relation by Petersen & Jorgensen (1972), a value of Q=0.d 028 ([FORMULA] 0.005) (Breger 1990) is found for the pulsation constant. This indicates that 20 CVn is pulsating in the fundamental mode or first overtone assuming radial pulsation. Finally, it is possible to gain some insight into the mass and age of this star using the evolutionary tracks of Schaller et al. (1992) for Z=0.020. This way, a mass of 2.16([FORMULA] 0.1) M [FORMULA] and age of 1.0([FORMULA] 0.2) 109  years can be obtained in a post-main sequence stage of evolution. When a main-sequence stage is considered for 20 CVn, a mass of 2.33 M [FORMULA] and age of 0.7 109  years are found.

3.5. Pulsation mode identification

Due to only one frequency being found in the light curve of 20 CVn, methods based on period ratios or frequency differences are excluded in order to identify the pulsation mode of this star. However, the methods based on the phase shifts and amplitude ratios between the observed light and colour variations, in uvby photometry (Garrido et al. 1990) or in UBV photometry (Watson 1988), can be used to determine the nature of radial or nonradial pulsation in this star and to identify the mode in which 20 CVn is oscillating. This point is of interest because this star has also been observed by other authors (Mathias & Aerts 1996) at high resolution spectroscopy and found as a nonradial pulsator, possibly a tesseral l=3 or a sectorial l=2 mode with [FORMULA] m [FORMULA] =2, using the moment method.

Table 4 lists the phase shifts, in degrees, and amplitude ratios between the different bands and colour indices, as resulting from Table 1. We can see that the phase shifts of both v and b filters with respect to the y band are nearly zero, allowing only small positive or negative values. This is also true for the pair (b-y,y). These effects suggest radial pulsation for this star by comparing with the "amplitude ratios versus phase shifts" diagrams of Garrido et al. (1990). However, the figures given by these authors refer to a [FORMULA]  Sct model with Te =8000 K [or 7850 K in the (b-y,y) diagram], log g=4.0 and Q=0.d 030. With the aim of deciding this point we have calculated the regions of interest for a model with the appropriate parameters (Te =7650 K, log g=3.64, Q=0.d 028) for this star. Figs. 4a, b show the predictions of this model for l=0, 1, 2, 3 and the (v,y) and (b-y,y) pairs. These graphs again suggest radial pulsation for 20 CVn, although the amplitude ratios seem to be slightly large especially for [FORMULA] v/ [FORMULA] y. However, we consider this disagreement to be minor, since the amplitude ratios (but not the phase shifts) are sensitive to the adopted atmospheric parameters (Breger 1997a,b). In fact, our results lead to the same conclusions when we take into account the error bars in Te, log g and Q. In addition, radial pulsation is also suggested from phase shifts in UBV photometry (Watson 1988) using the data set of Shaw (1976). Only these observations are available in the bibliography with measurements in both B and V filters. We obtain [FORMULA] B - [FORMULA] V = [FORMULA] 8 ([FORMULA] 5.7), [FORMULA] [FORMULA] - [FORMULA] V = [FORMULA] 8 ([FORMULA] 8.6), [FORMULA] B/ [FORMULA] V=1.44 ([FORMULA] 0.16) and [FORMULA] (B-V)/ [FORMULA] V=0.44 ([FORMULA] 0.12). In this case, the error bars are larger but the results agree well with our conclusions in uvby photometry.

[FIGURE] Fig. 4a and b. a Observed phase shift and amplitude ratio between the v and y filters for 20 CVn. The predictions of the model, for Te =7650 K, log g=3.64 and Q=0.d 028, are also shown for l=0,1,2,3. b Same as a for b-y and y


Table 4. Observed phase shifts and amplitude ratios

The question now arises when interpreting the high resolution spectroscopy results of Mathias & Aerts (1996). They find that 20 CVn is a multiperiodic variable pulsating in a nonradial mode with l=3 or 2. Their data seem to be of very high quality but, we feel they collected a too small amount of data to get a definitive conclusion. In fact, they have observations of only one pulsational cycle. Nevertheless, nonradial pulsation in 20 CVn was also suspected by Smith (1982) and Yang & Walker (1986). In the former case, Smith (1982) used the classical method which consists of a comparison between the observed spectrum and a theoretically computed one on a trial-and-error basis. This author found that on profile fitting alone the radial pulsation solution can not be ruled out for the primary mode of 20 CVn. Then, his nonradial suggestion is based in two other points: 1) multiperiodic behaviour and 2) low value of 2K/ [FORMULA] mv for the main pulsation. This last point is also the one pointed out by Yang & Walker (1986). With respect to the first point we now know that this star is a monoperiodic pulsator, at least from a photometric point of view. In relation to the second point, Smith (1982) found a value of 2K/ [FORMULA] mv =42 Km s-1 mag-1 for 20 CVn assuming 2K=1.3 Km s-1 and [FORMULA] mv =0.m 03 (full light amplitude in the V band). However, we now know that [FORMULA] mv =0.m 0207 (mean value from Table 3). Moreover, from earlier authors, the value of 2K ranges from 1.2 (Yang & Walker 1986) to 1.7 (Nishimura et al. 1983). Then, the ratio 2K/ [FORMULA] mv ranges from 58 to 82 Km s-1 mag-1. These values are not a strong indication of nonradial pulsation. Similar values of 2K/ [FORMULA] mv =84 Km s-1 mag-1 were also found by Smith (1982) for the radial primary mode of [FORMULA]  Sct and [FORMULA]  Del. Furthermore, to use the 2K/ [FORMULA] mv ratio to discriminate safely between radial and nonradial pulsation is subject of controversy (see e.g., Riboni et al. 1994and references therein). In summary, we feel that radial pulsation is the most reliable identification for the pulsational nature of 20 CVn.

However, taking into account the spectroscopic results of Smith (1982) and Mathias & Aerts (1996) it might be that the spectroscopic and photometric behaviour of this star is different. In this sense, and in order to accurately identify the mode(s) and to decide on the mono/multiperiodic behaviour of 20 CVn, it would be very interesting to perform a long campaign to observe this star collecting simultaneously high resolution spectroscopy and multicolor photometry.

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Online publication: February 4, 1998