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Astron. Astrophys. 354, 589-594 (2000)

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3. Analysis

3.1. Light curves

The light curves from each observing night typically cover one minimum and one maximum, with a few longer stretches in between. An example can be seen in Fig. 2: the quality is generally good, see the top panel. The difference between the two constant stars shows no sign of variability, and the rms-scatter is typically about 4 mmag. Since one of these stars is about 3 magnitudes fainter than V 1162 Ori and the comparison star, we expect the rms-scatter in the light curve of V 1162 Ori to be lower than 4 mmag (see Sect. 3.4 for a discussion of the formal noise level).

[FIGURE] Fig. 2. Examples of light curves, from the night 1998 Jan 20 for V 1162 Ori.

From the light curves it became clear that the amplitude of the light variations in V 1162 Ori increased significantly towards April 1998. From January to March 1998 the amplitude (full light range) in y, which can be compared with the V-amplitude of 0[FORMULA]10 found by Hintz et al. (1998), was 0[FORMULA]11-0[FORMULA]12 (as derived from the individual light curves). The quoted range in amplitude reflects cycle-to-cycle variations as well as uncertainties in the determination of the amplitudes, which are formally derived in Sect. 3.4. In April 1998 the amplitude in y had grown to 0[FORMULA]14. Cycle-to-cycle amplitude variations of this order were also found by Poretti et al. (1990), but in 1998 there appears to have occurred an overall increase in amplitude. In b the amplitude increased from 0[FORMULA]14-0[FORMULA]15 in January-March 1998 to 0[FORMULA]17 in April 1998. In early 1999 the amplitude was a little lower than in April 1998, 0[FORMULA]12-0[FORMULA]13 in y, and 0[FORMULA]15-0[FORMULA]16 in b. However, in March 1999 the amplitude had again increased, to 0[FORMULA]15-0[FORMULA]16 in y and 0[FORMULA]19-0[FORMULA]20 in b. During one night in January 1999 we have observed through the V-filter, and found an amplitude of 0[FORMULA]125, thus similar to the y-amplitude, as one would expect.

In the lower panel of Fig. 2 we show an example of a colour curve. Since b and y frames were obtained sequentially, we constructed the [FORMULA] index from an observed y magnitude and the corresponding interpolated b magnitude.

In both the upper and lower panel of Fig. 2 we have matched the zero-points to the indices determined by Hintz et al. (1998). On the basis of observations of 4-5 standard stars (Jonch-Sorensen 1994) each night during a few photometric nights, we obtained zero-point shifts that yield, within the observational errors, the same y, [FORMULA] indices as found by Hintz et al. (1998).

3.2. [FORMULA] diagrams

We have used the observed times of minimum and maximum light to investigate the pulsational period of V 1162 Ori. The times of extreme light were determined by fitting a third degree polynome to the light curves in the vicinity of each extremum. The times of minimum and maximum light can be seen in Table 3. We have used the count scheme from Table 1 of Hintz et al. (1998). It should be noted that the zero-point of this scheme is not well defined due to the period break in between the observations of Poretti et al. (1990) and Hintz et al. (1998). The first [FORMULA] of Hintz et al. (1998), given in their Table 1 ([FORMULA]) is consistent with the E-numbers of our times of maximum light, but the E-numbers for the times of maximum from Poretti et al. (1990), given in the same table, may be off by one or two cycles.


[TABLE]

Table 3. New times of maximum and minimum light (average of y and b filters, HJD-2,400,000.0). The cycle count scheme is based on Hintz et al. (1998).


After the observations in the beginning of 1998, it was clear that the period determined by Hintz et al. (1998) did not fit our data. The ephemeris given by Eq. 2 in that paper agreed fairly well with our times of maximum light in the beginning of January 1998, but the [FORMULA] value grew towards March and April that year, see Arentoft & Sterken (1999). We searched for the period in the 1998-data, and found that a linear fit could be obtained from the January-March 1998 times of maximum, yielding a period of 0.0786987 days:

[EQUATION]

This fit is represented by the horizontal line in Fig. 3. However, the times of maximum found in April 1998 are not reproduced with the above period, as can also be seen in Fig. 3. The data-points from this run, which are marked by the arrow, lie significantly below the horizontal line. Furthermore, the increased amplitude, discussed in Sect. 3.1, indicates that changes have occurred. Including the data obtained in 1999, it was clear that the period had changed. Instead, a shorter period could be found using a linear fit to the April 1998 and the 1999 times of maximum, demonstrated by the inclined fit in Fig. 3:

[EQUATION]

This finding suggests that V 1162 Ori has undergone two period changes within only a few months, one in late 1997 or early 1998, and one in March/April 1998.

[FIGURE] Fig. 3. [FORMULA] diagram for V 1162 Ori, based on the times of maximum light in both the y and b filter. The leftmost data-points corresponds to observations made in January 1998, the rightmost to observations from March 1999. The points from April 1998 are marked by an arrow. (1) and (2) refer to Eqs. (1) and (2) in the text.

A different approach is to fit the times of maximum light from Hintz et al. (1998) together with our new data. The data from Poretti et al. (1990) are not included because of the period break reported by Hintz et al. (1998). Such a fit gives a value for the period of 0.07869122 days, which is close to the value found by Hintz et al. (1998), and the corresponding [FORMULA] diagram is shown in Fig. 4. This figure suggests that the times of maximum light can be described with only one value of the period, with a quasi-cyclic [FORMULA] variation superimposed. The ephemeris found from this dataset is:

[EQUATION]

The cyclic behavior could be due to a beat between two very closely-spaced frequencies (see Sect. 3.4), or to the presence of a binary companion. In the latter case, the cyclic variations can be caused by light-time effects due to orbital motion, as was seen in some other [FORMULA] Scuti stars, e.g. CY Aqr (Ai-Ying & Jian-Ning 1998) and AD CMi (Jian-Ning & Shi-Yang 1996). The evidence presented in Fig. 4 is not convincing, but a cyclic variation can also not be completely ruled out.

[FIGURE] Fig. 4. [FORMULA] diagram for V 1162 Ori, based on the times of maximum light of data from Hintz et al. (1998) and our y and b measurements, using a period of [FORMULA] (Eq. (3)). The fitted sine has a period of 2.1 years.

By fitting a sine-function to the [FORMULA] values in Fig. 4 we obtain a period of the cyclic variation of about 2 years, and an amplitude of 0.0075 days. Using the well-known formulae for the light-time effect, see e.g. Irwin (1959), we find that such values for the period and amplitude are not unreasonable for a low-mass companion, considering that one component (V 1162 Ori) has a mass of about 2[FORMULA]. It is, however, clear that the [FORMULA] behavior cannot be described by a truly cyclic function alone: in Fig. 4 large deviations from the fit are clearly present, and the [FORMULA] variations must also have another cause than the light-time effect, if present at all. A much shorter period is also a possibility, but we do not find that we presently have enough data for a detailed analysis, and more data is needed over the coming years.

Due to the uncertainty in the count scheme following the period break found by Hintz et al. (1998), we have tried to modify the count scheme to see if all available data can be fitted with one single period, with cyclic variations superimposed. We have found that even then it is not possible to make a linear fit including the data from Poretti et al. (1990), i.e. the period break reported by Hintz et al. (1998) is real and cannot be ascribed to cyclic variations in the [FORMULA] diagram.

In Fig. 5 we have collected the periods and amplitudes which have been determined for V 1162 Ori. From this figure, there seems to be no clear connection between the period and amplitude changes: whereas the 9-year gap between the second and the third point would suggest that the amplitude decreases with increasing period, the remaining data do not support such a correlation. The size of the period changes is of the order of [FORMULA].

[FIGURE] Fig. 5. The evolution of the pulsational period and amplitude (full light range) of V 1162 Ori with time. The error on each point is smaller than the plotted symbols. The first and second points are, respectively, from Lampens (1985) and Poretti et al. (1990), the third point is from Hintz et al. (1998), and the last 2 points are from this work.

3.3. Phased light curves

We have used Eq. (1) to phase the light curves from January 1998, see Fig. 6. The mean magnitude in this plot is again from Hintz et al. (1998), as described in Sect. 3.1. In Fig. 7 we show a colour phase diagram from the same month, using the same period. The [FORMULA] curve given by Hintz et al. (1998) is not a smooth curve, but has a dented maximum. This is to some extent also the case in the second maximum of the colour curve in Fig. 2, while the phased colour curve is smooth.

[FIGURE] Fig. 6. Phased light curve for V 1162 Ori, based on Eq. (1) and data from January 1998.

[FIGURE] Fig. 7. Phased colour curve for V 1162 Ori, based on Eq. (1) and data from January 1998.

It can be seen from Fig. 3 in Poretti et al. (1990) that the minimum in the phase diagram occurs after phase 0.5. This is also the case in our phase diagram shown in Fig. 6, and the light curves are thus not perfectly symmetric. From all our light curves, where we have both a maximum and a minimum, we have found the phase difference between a maximum and the following minimum to be [FORMULA] for the period Jan.-March 1998 (33 points), and [FORMULA] for the period April 1998-March 1999 (46 points).

3.4. Frequency analysis

We have also performed Fourier analysis of the time-series, in order to search for a possible second period. Because of the changes taking place, we had to split the data in two sets, one covering January-March 1998, and one covering April 1998 - March 1999. The amplitude spectra were calculated using the program Period (Breger 1990). The derived amplitude spectrum for the b-measurements of 1998 (January-March) is shown in Fig. 8. The formal resolution of the amplitude spectrum is 0.03 c/d, but the actual precision of the detected frequencies is higher than that. The main pulsation is clearly visible at a frequency of about 12.7 c/d. It is also clear from Fig. 8 that the spectral window function for these observations is poor. In early 1998, the full amplitude of the main pulsation, from the Fourier analysis, was 0[FORMULA]112 in y and 0[FORMULA]138 in b, and in 1999 0[FORMULA]123 in y and 0[FORMULA]152 in b.

[FIGURE] Fig. 8. Amplitude spectrum for V 1162 Ori, based on b-measurements from January, February and March 1998. The amplitude shown here is half the full light range. The insert shows the most significant section of the prewhitened amplitude spectrum (on a larger amplitude scale).

The periods determined from the two datasets (and in both y and b) were, within the errors, the same as the periods found from the [FORMULA] analysis in Sect. 3.2.

After removing the main pulsation from the b time-series, we arrive at the residual spectrum shown as the insert in Fig. 8. Two structures with a highest peak of equal amplitude are now visible in the spectrum. The amplitude of both is about 4.5 mmag, which (at a noise level in the region of about 1.1 mmag) corresponds to a 4 [FORMULA] detection. We do, however, find it doubtful that these peaks are due to real variations in the light curve. The highest peak in the structure around 25 c/d does not correspond to the 2[FORMULA] term, but it does, within the resolution, correspond to a 1 c/d alias. Assuming this structure is caused by the 2[FORMULA] term, this term will then have a half-amplitude of about 4.5 mmag, in agreement with Hintz et al. (1998). We do not find evidence of a second period at 16.48 c/d, as suggested by Hintz et al. (1998), and we can put an upper limit on the full amplitude of a possible second period at about 9 mmag. We must stress, however, that this does not rule out the presence of low-amplitude variations, which can have amplitudes much lower than this.

If the cyclic behavior of the [FORMULA] diagram (Fig. 4) is caused by a beat between two very closely-spaced frequencies, it would be expected that an increase in amplitude for the main pulsation should lead to a decrease in the amplitude of the secondary pulsation. We do not see evidence for such a mechanism, and especially the dramatic decrease in amplitude (50 percent) between the studies of Poretti et al. (1990) and Hintz et al. (1998) can only be explained by a beat phenomenom if the two frequencies are so closely-spaced that they are non-resolved. However, if the period of the beat is 2 years, which we cannot be sure of from the present data, the secondary period would be so close to the primary that they would not be resolved in the observations presented here, covering a time base of about 500 days. A time base of at least 1000 days is in such a case needed to resolve the spectrum. A cyclicly changing period could cause a higher noise level in the amplitude spectrum, as seen in the prewhitened spectrum in Fig. 8. In this case, the Fourier analysis is no longer valid.

The noise level at high frequencies in the prewhitened spectrum is 0.24 mmag.

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© European Southern Observatory (ESO) 2000

Online publication: February 9, 2000
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