A&A 374, 1056-1070 (2001)
DOI: 10.1051/0004-6361:20010794

V1162 Ori: A multiperiodic $\delta$ Scuti star with variable period and amplitude[*],[*]

T. Arentoft1 - C. Sterken1,[*] - G. Handler2 - L. M. Freyhammer1,3 - A. Bruch4 - P. Niarchos5 - K. Gazeas5 - V. Manimanis5 - P. Van Cauteren6 - E. Poretti7 - D. W. Dawson8,9 - Z. L. Liu10 - A. Y. Zhou10 - B. T. Du10 - R. R. Shobbrook11 - R. Garrido12 - R. Fried13 - M .C. Akan14 - C. Ibanoglu14 - S. Evren14 - G. Tas14 - D. Johnson8 - C. Blake15 - D .W. Kurtz 16,17,18

1 - University of Brussels (VUB), Pleinlaan 2, 1050 Brussels, Belgium
2 - South African Astronomical Observatory, PO Box 9, Observatory 7935, South Africa
3 - Royal Observatory of Belgium, Ringlaan 3, 1180 Brussels, Belgium
4 - Laboratório Nacional de Astrofísica, CP 21, 37500-000 Itajubá - MG, Brazil
5 - Department of Astrophysics, Astronomy and Mechanics, University of Athens, 157 84 Zografos, Athens, Greece
6 - Beersel Hills Observatory, Belgium
7 - Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate, Italy
8 - Department of Astronomy, San Diego State University, San Diego, California, USA
9 - Department of Physics and Astronomy, Western Connecticut State University, Danbury, Connecticut 06810, USA
10 - Beijing Astronomical Observatory, Chinese Academy of Sciences, Beijing 100012, PR China
11 - Research School of Astronomy and Astrophysics, Australian National University, Weston Creek PO, ACT 2611, Australia
12 - Instituto de Astrofisica de Andalucia, CSIC, Apdo. 3004, 18080 Granada, Spain
13 - Braeside Observatory, Flagstaff, Arizona, USA
14 - Ege University Observatory, Bornova 35100, Izmir, Turkey
15 - Astrophysical Sciences Department, Princeton University, Princeton, New Jersey 08544, USA
16 - Centre for Astrophysics, University of Central Lancashire, Preston PR1 2HE, UK
17 - Department of Astronomy, University of Cape Town, Rondebosch 7701, South Africa
18 - Laboratoire d'Astrophysique, Observatoire Midi-Pyrénées, 31400 Toulouse, France

Received 26 April 2001 / Accepted 31 May 2001

We present the results of multisite observations of the $\delta$ Scuti star V1162 Ori. The observations were done in the period October 1999 - May 2000, when 18 telescopes at 15 observatories were used to collect 253 light extrema during a total of 290 hours of time-series observations. The purpose of the observations was to investigate amplitude and period variability previously observed in this star, and to search for low-amplitude frequencies. We detect, apart from the main frequency and its two first harmonics, four additional frequencies in the light curves, all with low amplitudes (1-3 mmag). Combining the present data set with data obtained in 1998-99 at ESO confirms the new frequencies and reveals the probable presence of yet another pulsational frequency. All five low-amplitude frequencies are statistically significant in the data, but at least one of them (f5) suffers from uncertainty due to aliasing. Using colour photometry we find evidence for a radial main frequency (f1), while most or all low-amplitude frequencies are likely non-radial. We show that the main frequency of V1162 Ori has variable amplitude and period/phase, the latter is also displayed in the O-C diagram from light extrema. The amplitude variability in our data is cyclic with a period of 282 d and a range of nearly 20 mmag, but earlier amplitude values quoted in the literature cannot be explained by this cyclic variation. O-C analysis including data from the literature show that the period of V1162 Ori displays a linear period change as well as sudden or cyclic variations on a time scale similar to that of the amplitude variations.

Key words: stars: variables: $\delta$ Scuti - stars: individual: V1162 Orionis - techniques: photometric - methods: data analysis

1 Introduction

The $\delta$ Scuti stars are pulsating A-F stars situated on or just above the main sequence. They display a large range in pulsational amplitude, from the mmag level observed in the low-amplitude, multiperiodic $\delta$ Scuti stars up to almost one magnitude found in some of the high-amplitude $\delta$ Scuti stars (HADS). The HADS generally have amplitudes exceeding 0 $.\!\!^{\rm m}$3 and slow rotational velocities ($v\sin i$ below 30 kms-1). V1162 Ori is often considered a HADS, although it does not qualify as such due to its full amplitude of only 0 $.\!\!^{\rm m}$1-0 $.\!\!^{\rm m}$2 and its high projected rotational velocity ($v\sin i$ of 46 kms-1, Solano & Fernley 1997). It is an intermediate amplitude, up to now monoperiodic Pop I $\delta$ Scuti star with a frequency of 12.7082 d-1: Hintz et al. (1998) claimed a secondary frequency near 16.5 d-1, but this was later shown to arise from a variable comparison star (Lampens & Van Cauteren 2000). We will, however, show that V1162 Ori is not monoperiodic and that it is indeed positioned in the narrow HADS instability strip given by McNamara (2000). V1162 Ori has in the past shown very large amplitude changes, ranging from half peak-to-peak values of 98 mmag observed by Poretti et al. (1990) to 50 mmag observed by Hintz et al. (1998), who also detected a period break using O-C analysis of times of maximum light. Later changes observed by Arentoft & Sterken (2000, hereinafter Paper I) could be due to period breaks or cyclic period changes, and also these authors detected amplitude variations. As a result it was decided to organise a multisite campaign on V1162 Ori, spanning a full observing season. The aims were to investigate the time scales of the changes, how or if the amplitude and period/phase variations are related and if possible to gain information on the underlying physical processes causing them. Although amplitude and period variations are common phenomena in $\delta$ Scuti stars, the causes are far from understood (see e.g. Breger & Pamyatnykh 1998; Breger 2000a). Even the involved time scales are very different from star to star: 4 CVn, for example, shows amplitude variability on time scales of years (Breger 2000b), whereas XX Pyx displays period and amplitude variability on time scales as short as 20 d (Handler et al. 2000). Breger (2000a) discusses the possibility that amplitude variability can be related to multiperiodicity, as the monoperiodic HADS appear to have more stable amplitudes (e.g. Rodriguez 1999) than the multiperiodic $\delta$ Scuti stars of low and possibly also high amplitude.

The philosophy of the present multisite campaign is different from normal campaigns on $\delta$ Scuti stars: the aim was to collect as many extrema as possible over the observing season (8 months). Thus, the participating teams observed V1162 Ori whenever they had sufficient time to spare to cover an extremum. These observations, which often covered short light curve sections - sometimes only 20 min - had the purpose of following the evolution of the main pulsational period, and were complemented with dedicated time-series observations from several sites, also distributed over the long time span. The latter allow us to monitor amplitude changes as well as to search for low-amplitude frequencies - however without the usual multisite advantage of suppressed side-lobes in the amplitude spectra.

Finding low-amplitude frequencies is very important for understanding changes in the light curve: low-amplitude frequencies can interfere with the main mode and cause e.g. amplitude variations through beating or give rise to cycle-to-cycle variations. Furthermore, detection of additional pulsation frequencies would yield tighter constraints on stellar models.

{\includegraphics[width=12cm]{1425f1n.eps} }\end{figure} Figure 1: Examples of light curves obtained in two of the cases of overlapping data. Dots are data from SAAO, open circles data from Athens University Observatory. We also show the difference between the datapoints from Athens and interpolated values of the SAAO datapoints (triangles, shifted by 120 mmag).
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2 The data

Data were obtained with 18 different telescopes at 15 sites in 12 countries, utilising both CCDs and PMTs. For the CCD observations, care was taken to avoid having the very bright and close-by star $\upsilon$ Ori on the frames by placing V1162 Ori near the edge of the field of view. For PMT observations the disturbing effect of a close (10 $\hbox{$^{\prime\prime}$ }$), 3 $.\!\!^{\rm m}$1 fainter neighbouring star on the observational noise was minimised by including the star in the aperture. The observations were, with few exceptions, done through the Johnson V filter. The list of participating observers and sites is given in Table 1, where we also give the number of extrema and hours of data collected with each telescope.

The bulk of the data was reduced by the individual observing teams, and several different reduction procedures were therefore applied. It is beyond the scope of the present paper to describe them all, we will just add that the applied procedures follow general and established methods for reduction of CCD and PMT data. Differential magnitudes were, both for the CCD and PMT data, measured with respect to the star GSC 4478-0019, which is slightly brighter and situated only 3$^\prime$ from V1162 Ori. Times of observations were recorded as mid-exposure and converted to Heliocentric Julian Date.


Table 1: List of sites participating in the campaign. Telescope diameters are given in meters.
Observatory Location Observer (#extrema) Telescope Detector #hours
SAAO S. Africa T. Arentoft (44), L. Freyhammer (30), G. Handler (16) 1.00 CCD 77.0
SAAO S. Africa G. Handler (6) 0.75 PMT 4.9
SAAO S. Africa G. Handler (6) 0.50 PMT 3.1
Athens University Greece P. Niarchos, K. Gazeas, V. Manimanis (26) 0.40 CCD 33.2
Kryonerion Greece P. Niarchos, K. Gazeas, V. Manimanis (17) 1.22 CCD 22.9
LNA Brazil A. Bruch (26) 0.60 CCD 33.5
Xinglong China Z. L. Liu, A. Y. Zhou, B. T. Du (16) 0.85 CCD 18.9
Beersel Hills Belgium P. Van Cauteren (14) 0.40 CCD 20.6
Ege University Turkey C. Akan, C. Ibanoglu, S. Evren, G. Tas (14) 0.48 PMT 25.5
San Pedro Martir Mexico E. Poretti (3) 1.50 CCD 3.8
Merate Italy E. Poretti (7) 0.50 PMT 10.8
Mt. Laguna USA D. W. Dawson (9), D. Johnson (2) 0.50 PMT 14.5
Siding Spring Australia R. R. Shobbrook (5) 0.61 PMT 6.2
Sierra Nevada Spain R. Garrido (4) 0.90 PMT 4.9
Braeside USA R. Fried (4) 0.40 CCD 4.6
ESO Chile C. Sterken (3) 1.54 CCD 4.3
Lick USA C. Blake (1) 1.00 CCD 1.4
Total   (253)     290.1

Using BV photometry obtained at SAAO and by photometry obtained at ESO (PaperI), we determined the relative colours of V1162 Ori, the comparison star and a check star, GSC 4778-0025, which will be used to investigate the stability of the comparison star. The V, B-V values for V1162 Ori were fixed to those of Poretti et al. (1990, V=9.89, B-V=0.31), and b-y to that of Hintz et al. (1998, b-y=0.187). We found that the comparison star, GSC 4778-0019 (V=9.73, B-V=1.55, b-y=0.91) is very red, and differs significantly in colour from V1162 Ori. The check star, GSC 4778-0025 (V=12.58, B-V=0.80, b-y=0.47), is somewhat fainter than V1162 Ori and the comparison star.

Photometric light curves of very different length and quality were collected at the many sites during the campaign. As noted above, some light curves cover only single maxima or minima, while others cover several hours and cycles. The observations were not coordinated, but in a few cases did observations overlap, unfortunately only shortly and in poor weather at one or both sites. In Fig. 1 we show examples of overlapping data from two nights during which the weather was non-photometric at one of the two sites. The agreement between the overlapping data is fairly good. We show the difference between the data from the two sites (triangles); the scatter is high in the first night (7.4 mmag rms) and at a more acceptable level during the second night (4.8 mmag rms). In the left-hand panel there is also a systematic trend present in the difference. Such trends can be due to differences in filter passbands or, because the comparison star is very red, extinction. Our best PMT data have rms-scatter within a night of just below 2 mmag, whereas our best CCD data have rms scatter of about 2.5 mmag.

3 Frequency analysis

The frequency analysis was carried out using the excellent Fourier analysis tool Period98 (Sperl 1998). Amplitudes are in the following given as half the peak-to-peak value, and as criterion for detection of a pulsational frequency we require the corresponding peak in the amplitude spectrum to have an amplitude of at least 4 times the average noise, determined after prewhitening, in the frequency domain where it is found (Breger et al. 1993). This requirement can be lowered to 3.5 for combination frequencies as they occur at known positions (Breger et al. 1999).

3.1 Low-frequency analysis and stability of the comparison star

The stability of the comparison star, GSC 4778-0019, was investigated using CCD observations also including the check star, GSC 4778-0025, on the frames. Of the campaign data we used for this purpose the extensive time-series data obtained at SAAO during 15 nights. We also used the CCD data obtained at ESO.

Low-frequency variations are clearly present in the SAAO V1162 Ori data, giving rise to peaks in the amplitude spectrum of up to 8 mmag in the frequency range 0-2 d-1, as shown in the upper panel of Fig. 2. The variations are also directly visible as shifts in the nightly zeropoints, especially after subtracting the main pulsation frequency at 12.7082 d-1. The zeropoints of V1162 Ori minus the comparison star were compared with those of the comparison star minus the check star. The sizes of the night-to-night changes in the former have values not systematically different from the changes in the latter, but with opposite signs, as is seen in the middle panel of Fig. 2. The nightly changes thus originate from the comparison star.

{1425f2n.eps} }\end{figure} Figure 2: Amplitude spectra in the low-frequency domain of V1162 Ori (SAAO data, top). The lower panels show a comparison between the zeropoints in the differential light curves of V1162 Ori  minus comparison (filled circles), comparison minus check (open circles) and V1162 Ori  minus check (squares). The error bars are 3 times the error on each nightly mean value.
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The cause of the variability in GSC 4478-0019 is unclear. It can be variable on a time scale of days or, as the star is very red, the night-to-night changes could be due to extinction effects. However, as the zeropoint shifts are very similar relative to two stars of different colour, V1162 Ori and the check star, the variations cannot be ascribed to extinction. From the ESO data, we find the shifts to have the same size in the b and y filters. We calculated the amplitude spectra of the ESO 1998, ESO 1999 and new SAAO comparison minus check star data separately and searched for re-occuring peaks, but did not find any. The variations of the comparison star seem nonperiodic, or are not stable from year to year.

The difference between V1162 Ori and the check star shows a much smaller degree of variation, although some datapoints in the lower panel of Fig. 2 deviate from the zero mean. However, the effects are small and could be caused by extinction. The corresponding amplitude spectrum, which is also shown in Fig. 2, has little power at low frequency. There are some $\sim$2 mmag peaks present near 0.9 d-1 (and 1 d-1 aliases), but similar peaks are not present in the corresponding ESO data. We do therefore not find evidence for the presence of low-frequency variations in V1162 Ori.

We are mainly interested in the absence of signal in the frequency range 5-50 d-1 in the comparison star data, and using the SAAO CCD measurements of the comparison star relative to the check star, there are no outstanding peaks in this part of the amplitude spectrum. The amplitude spectrum, with a 4$\sigma $ significance curve superimposed, is shown in Fig. 3. All peaks above 5 d-1 in frequency are statistically insignificant and can be considered noise peaks. If we combine the ESO and SAAO comparison star data, and correct for the changes in nightly zeropoint, the comparison star shows no periodic variability up to 0.7 mmag below 15 d-1, and 0.5 mmag above (peak values). This is shown as the insert in Fig. 3. The amplitude spectrum was calculated up to 300 d-1 and no high-frequency periodic components were detected either. The individual light curves have typically a rms-scatter of about 4 mmag. The residuals after correcting for nightly zeropoint variations are, for both the ESO and SAAO data (and the combination of the two) normally distributed, and can be represented by Gaussians with values of $\sigma $ of the expected 4 mmag.

{1425f3n.eps} }\end{figure} Figure 3: Check for periodic variations of the comparison star in the frequency range where variations in V1162 Ori are found. This diagram is based on the difference between the comparison star and the check star in the SAAO data. The solid line is a 4$\sigma $ significance curve, found from local noise levels in the amplitude spectrum. The insert displays the amplitude spectrum of the ESO and SAAO data sets combined.
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3.2 Frequency analysis of V1162 Ori

All data for which we could determine times of maximum or minimum light are included in the O-C analysis in Sect. 4. For Fourier analysis of V1162 Ori we selected, from the campaign data, long data strings covering more than one cycle and having well-defined zeropoints. Furthermore, only data obtained through the V-filter and of a sufficiently high quality were included.

Obvious bad points were removed from the data based on visual inspection of the light curves. In total, the data set for the Fourier analysis consists of 5388 datapoints, covering a time base of 134 days with an effective length of 139 hours of photometry obtained during 37 nights. 108 datapoints were rejected as being bad.

The data selected for Fourier analysis were then low-frequency filtered. This was done by zeropoint correcting data from the individual nights, and removing slow trends by fitting 3rd degree polynomials to residuals from a provisional frequency solution and subtracting them from the original data. The filtering removes signals at low frequencies, up to about 5 d-1. It was checked on the main pulsation at 12.7082 d-1 that frequencies in this area were not affected by the filtering: the amplitude and relative sizes of the side-lobes remained constant. This is expected as the frequency regions where peaks occur are well separated. However, the procedure has only marginal effect on the noise levels in the frequency regions under investigation here (5-50 d-1), but we perform the filtering as we will later subdivide the data into smaller segments which will be more susceptible to effects of 1/f-noise.

V1162 Ori is known to display amplitude and period/phase variability on a relatively short time scale (Hintz et al. 1998; PaperI). We have to keep in mind the possibility that such changes occur within the time span of our data set, and if so, our analysis should take this into account. Such variability can lead to spurious peaks and/or increased noise levels in the residual amplitude spectrum (see e.g. Handler et al. 2000). Part of the latter could also be caused by filter passbands mismatches.

Using the filtered campaign data we first performed a regular frequency analysis of V1162 Ori, not allowing for phase and amplitude variability. This was done to get an idea of the frequency content of the light curves before we include the earlier ESO data and allow the phase and amplitude of the main frequency to vary.

3.2.1 The campaign data

The amplitude spectrum of V1162 Ori is dominated by the main periodicity at 12.7082 d-1 (f1), but after prewhitening with this frequency, we detect the first two harmonics of f1, and four additional frequencies on a statistically significant level. The successive (simultaneous) prewhitening of the amplitude spectrum is demonstrated in Fig. 4 and discussed below. The detected frequencies are marked with a square in each of the panels (a-f). The upper panel shows the original amplitude spectrum, and due to the high amplitude of f1 it represents the spectral window function as well.

After removing f1, the dominant set of peaks belongs to 2f1, and 3f1 is visible near 38 d-1 (panel b). Removing also the harmonics reveals several additional peaks in the residual amplitude spectrum (c). The highest of those occurs at 12.94 d-1, i.e. close to, but clearly resolved from, f1. However, to test the reality of this peak we subdivided the data in two nearly equal parts, and found it to be present in both, showing that this peak is not an artifact of f1. Furthermore, the resolving power in each of the two subsets is, with time bases of 70 days, about 0.02 d-1 (Loumos & Deeming 1978), ten times higher than the separation between the two peaks in question. The 12.94 d-1 peak is also found in the SAAO data alone (and subsets thereof) and is thus not a spurious effect of merging data from several sites. Gradual prewhitening by including the residual frequency of highest amplitude in the frequency solution (which is optimised after each additional frequency) allows us to detect the four low-amplitude frequencies. We label them f2-f5 and give the values of the frequencies, amplitudes and S/N in Table 3. The tabulated values, however, are the solution from the combined ESO and campaign data set and will be discussed below.

The choice of f5 is not obvious (Fig. 4f), as three peaks have equal amplitude. We selected the central peak at 15.99 d-1, but this peak may be an alias and not the true frequency. In the campaign data, the S/N is only 4.3, but it will be confirmed when we include the ESO data, as will a peak at 27.77 d-1 (g, marked position). This represents the first detection of multiple frequencies in V1162 Ori.

{1425f4n.eps} }
\end{figure} Figure 4: Amplitude spectrum of V1162 Ori with successive, simultaneous prewhitening of the detected frequencies (see text), using time-series data obtained during the campaign. Note that the low-amplitude frequencies, especially f5 suffer from uncertainty due to aliasing.
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We prewhitened the light curves for the 7 significant frequencies and calculated statistical weights, following Frandsen et al. (2001), from the residuals, to see if applying such weights could improve the noise levels. We also tried to decorrelate SAAO data residuals for effects of seeing, sky-background levels and relative position on the CCD chip, using the methods described by Frandsen et al. (1996, or see Arentoft et al. 2001). As neither of these two methods proved to have any effect on the noise in the investigated frequency region, we did not apply them to the data used in the further analysis. After subtracting the 7-frequency solution there are indications of additional peaks or increased noise levels in the 10-15 d-1 range of the amplitude spectrum, possibly originating at least partly from phase and amplitude variations of f1. Dividing the data in smaller subsets suggested amplitude and phase variability of f1 - the amplitude changes between 60 and 73 mmag during the campaign, and there is a drift in phase of about 30$^{\circ}$. We will therefore take into account ($A,\phi $)-variations of f1 in the subsequent analysis. The reason why the residual amplitude spectrum displays relatively weak signal compared to the size of the amplitude variations is that, as will be shown below, the vast majority of the campaign data have nearly constant (high) amplitude - data with low amplitude constitute only a small fraction of the data set.

{1425f5n.eps} }
\end{figure} Figure 5: The upper panel displays the evolution in phase of f1 (12.708264 d-1) determined from the four ways of dividing the data in subsets: a)- d) in Table 2. For error bars, see Sect. 3.3. The different parts of our data set are given above the figure. Dots are from a), triangles from b), open circles from c) and crosses from d). The lower panel gives the residual amplitude spectrum for each of the cases a)- d) after removing f1and its harmonics, taking ($A,\phi $)-variations into account (see text). The position of f1 and 2f1 has been marked in the amplitude spectra. The 2.2 mmag peak close to f1 in c) is not a remnance of f1 but a close-by noise peak.
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3.2.2 Including the 1998-1999 ESO data

At this point we also include the data obtained previously at ESO (PaperI) in the analysis, to seek confirmation of the newly detected low-amplitude frequencies in an independent data set, to expand our time base, and to use the combined data set to search for additional pulsation frequencies. The ESO data comprise about 111 hours of both b and y-photometry collected over 7 observing runs at ESO in early 1998 and 1999, and we add the y data to the 139 hours of V-photometry from the campaign data set. The pulsation amplitude was found to be similar in the V and y filters in PaperI, but varied from about 60 to more than 75 mmag in y on a time scale of months. Furthermore, period changes were also present; including those data in the analysis requires that we take ($A,\phi $)-variations into account. We also include an additional 10 hours (335 datapoints) of mainly short light curves obtained during the campaign, which were not included in the Fourier analysis above. They will increase the time resolution in the investigation of ($A,\phi $)-variations.

To be able to take ($A,\phi $)-variations of f1 into account, we need to subdivide the dataset into smaller subsets to allow fitting of f1 and 2f1 within each subset. The subsets should, when possible, have a time base sufficiently long for the individual frequencies to be resolved, but not so long that possible variations are undersampled. In short, our final results must not depend on the choice of subsets. We tested four different ways of subdividing the data: (a) treating each night individually, (b) using 18 subsets of 2-8 nights of data (2 nights only in cases of isolated data), (c) 13 subsets of slightly longer time base, or (d) 8 subsets combined of 2-3 of the subsets in (b). We will refer to the labels a-d below.

The four sets are outlined in Table 2[*]. In each case we performed a preliminary frequency analysis allowing for ($A,\phi $)-variations. We fixed f1 to the optimal frequency, 12.708264 d-1, determined from a fit to the combined data set, and left the amplitude and phase of f1 and 2f1 as free parameters within each subset. The residual amplitude spectra after subtracting f1 and 2f1 are displayed in Fig. 5, lower half. Subdivision (a) leads to overfitting of the data, which is seen as a suppression in the noise level around f1 and 2f1 - this is an artifact of fitting with too many degrees of freedom. (b) gives reasonable results, but there is a small dip in the noise level at f1, and the amplitude of the close-by peak at 12.94 d-1 is slightly lower than in (c) and (d), which gives similar results for the amplitudes of the low-amplitude modes. The noise level in the region 10-20 d-1 of the residual spectra is slightly lower in case (d), also after prewhitening with the low-amplitude frequencies detected above. However, (d) has the disadvantage of the ($A,\phi $)-variations being poorly sampled, and for investigating such variations we will use subdivisions (b) and (c) to obtain higher temporal resolution.

In searching for low-amplitude frequencies we are interested in as low a noise as possible, and subdivision (d) will therefore be used for this analysis. The results of the frequency analysis should not differ between (c) and (d), and we can use (c) as control, thus minimising the risk of detecting spurious peaks. In the four cases we can determine the evolution of the phase of f1 in time as seen in Fig. 5, upper panel, which has the same shape regardless of choice of subset sizes. This figure shows that ($A,\phi $)-variations of f1 must be taken into account in the analysis. f1 is by far the dominant frequency, which is why even fitting within the individual nights gives reasonable values for the phases, despite poor frequency resolution.

If ($A,\phi $)-variations are disregarded when subtracting f1 there remains a residual signal in the amplitude spectrum near f1 (at 12.71197 d-1) of 13 mmag and near 2f1, at (f1 +12.71197 d-1), of 2 mmag. The peak close to f1 may be a real peak, or a result of amplitude and phase variability of f1. In the latter case is a peak at the sum-frequency also expected, as 2f1 will be modulated in the same way as f1. Such close (real) peaks would cause amplitude and phase variability through beating, and to test their reality, we included them in the frequency solution instead of allowing f1 to vary. This resulted in a 30% higher noise level, indicating that they are indeed not real frequencies. Another way of testing their reality is, following Handler et al. (2000), to compare the amplitude ratio of f1 and 2f1 to their close-by peaks. These ratios should differ if the frequencies are real and the pulse shapes of the individual signals should not vary. The ratios differ, but our data may not be sufficient to allow a reliable determination. Consequently, we leave these peaks close to f1 and 2f1 out of our frequency solution, but will return to them in the discussion.

3.2.3 The combined ESO and campaign data set

The combined ESO y and campaign data set consists of 7552 datapoints spanning a time base of 815 d. We successively subtracted the frequencies detected in Sect. 3.2.1, allowing for ($A,\phi $)-variations of f1 and its harmonics using subdivision (d). It was verified that these frequencies were present in the combined data set as well, with f2 and f3 as the dominant peaks in the spectrum after subtracting f1. The residual spectrum after subtracting f1 and its harmonics can be seen in Fig. 5, bottom panel. The alias ambiguity in the determination of f5 is not cleared by the combined data set.

We then searched for additional significant peaks. The peak at 27.77 d-1 is statistically significant in the combined amplitude spectrum, as tabulated in Table 3. This peak has the same amplitude, when using subdivision (c), but its presence may be uncertain due to the very low amplitude. Its likely presence is displayed in Fig. 6. No further frequencies were found, and we note that f1 is the lowest frequency of the detected modes.

{1425f6.eps} }\end{figure} Figure 6: The new probable frequency (f6) detected from the combined data set. The position of the frequency is marked with a square.
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Table 3: Table of detected frequencies in V1162 Ori , from the combined data (campaign and ESO 1998-1999). The amplitudes of f1 and its harmonics are average values, as we have allowed for amplitude and phase variations. Overall residual scatter after subtracting this frequency solution is 6.2 mmag. The last column gives the frequency ratio (f1/fn), discussed in Sect. 3.4.
ID Frequency Amplitude S/N f1/fn
  (d-1) (mmag)    
f1 12.7082 66.6 151 1.000
2f1 25.4164 6.6 27 -
3f1 38.1246 1.3 8 -
f2 12.9412 3.2 7 0.982
f3 19.1701 3.0 9 0.663
f4 21.7186 2.4 8 0.585
f5 15.9901 2.1 5.5 0.795
f6 27.7744 1.1 5 0.458

The ESO light curves covered only 1-3 hours per night, resulting in a poor spectral window function. After removing f1 and 2f1, allowing for ($A,\phi $)-variations, the residual amplitude spectrum of the ESO data set alone can be seen in Fig. 7. The above detected low-amplitude frequencies are also present in the ESO data set, and are thus confirmed as they are found in two independent data sets.

{1425f7.eps} }\end{figure} Figure 7: Residual amplitude spectrum of the ESO 1998-99 data, after prewhitening with f1 and 2f1. The position of the low-amplitude frequencies detected from the combined data set are indicated. The insert shows also the low-frequency part of the amplitude spectrum.
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3.3 Amplitude and phase variations

We will now seek to characterise in detail the ($A,\phi $)-variations taking place in f1, and furthermore search for variability of the low-amplitude frequencies. The procedures described in this section are largely based on the methods used by Handler et al. (2000) in their analysis of XX Pyx. We keep in mind that these authors, using a larger data set, did not consider phase variability of modes with amplitudes lower than 1.5 mmag, as the phases of those modes were not sufficiently constrained.

Before we start the analysis we will address the question of error bars on the amplitude and phase values. We calculated these following Montgomery & O'Donoghue (1999). However, the formal error bars are unrealistically small, as was also discussed by Handler et al. (2000) who found the error calculations to be underestimated by a factor of two or more. In our data we would expect residual noise levels in the amplitude spectrum of less than 0.1 mmag (assuming white noise). The residual noise levels were 0.4 mmag at 15 d-1 and 0.16 mmag at 38 d-1. In the following we therefore multiply the formal errors by a factor of two to obtain more realistic, but possibly still underestimated, error estimates.

3.3.1 $\mathsfsl{f_1}$

We used the combined data set prewhitened for the low-amplitude frequencies to investigate ($A,\phi $)-variations of f1. To subtract the low-amplitude frequencies we first subtracted f1 and its harmonics, allowing for ($A,\phi $)-variations using subdivision (d). This led to a time string whose amplitude spectrum displayed only noise at f1, 2f1 and 3f1. Having removed the influence of f1, f2 - f6 were then fitted to the residuals, creating a synthetic time string which was subtracted from the original data. It was checked that this method removed the low-amplitude frequencies well.

Because we used a data set prewhitened for the low-amplitude modes, and as a result of Fig. 5, upper panel (showing that the choice of subset sizes does not influence the shape of the phase variations), we used the 18 subsets of subdivision (b) for the investigation of ($A,\phi $)-variations of f1, as we are interested in a high time resolution.

We show the evolution in amplitude of f1 in Fig. 8. Very large variations are present, and they appear cyclic. We have superimposed a sinewave with a period of $282\pm6$ d, which seems to describe the amplitude variations of both f1 and 2f1 well, although the scatter in the bottom panel is high and the agreement with the fit only suggestive. Especially the data from March 1999 (at 460 d) show a very high amplitude value of 2f1. The amplitudes of the sinewaves are 9.85 mmag for f1 and 1.7 mmag for 2f1, with average values of 64.36 and 5.98 mmag, respectively. The parameters of the fit were determined by least-squares fitting to the (only) 18 f1 data points, and residual scatter is 1.47 mmag for f1 and 0.90 mmag for 2f1, lower than the scatter in the individual data subsets.

Figure 8 also shows that a large fraction of the campaign data have nearly constant amplitude (over 70 mmag), as mentioned in Sect. 3.2.1. There is a larger scatter around the last maxima in the upper panel of Fig. 8 (f1), where the fit deviates up to 2 mmag from the datapoints. This may be due to the presence of additional effects, and the reason it is seen being the larger amount of data available. Another explanation may be that the error bars still underestimate the real scatter. To test the reality we determined the amplitude with smaller and larger subsets, but the same shape remained. Regardless, the suggested cyclic variation cannot explain the amplitude variations observed prior to the ESO data, as Lampens (1985) found an amplitude of 92 mmag, Poretti et al. (1990) derived one of 98 mmag, and Hintz et al. (1998) found amplitudes of 72 and 50 mmag from two different data sets - three of these four measurements are thus outside the amplitude range of the upper panel of Fig. 8. We note that the shape of the amplitude variations does not change when using the original, non-prewhitened data instead. Given the cyclic shape of the amplitude variations it is not surprising that a peak is present in the amplitude spectrum very close to f1 (Sect. 3.2.2). The beat period of f1 and the close peak is about 270 d, consistent with the time scale of the cyclic variation in amplitude. The shape of the phase variations of f1 in Fig. 9 is the same as in Fig. 5, showing that the low-amplitude modes do not influence the phase determinations - they have been prewhitened in Fig. 9 but not in Fig. 5. Phase changes are clearly present, both in f1 and in 2f1. The shape of the phase changes appears parabolic, or, as the maximum is very broad, possibly piecewise linear. This will be discussed in detail in Sect. 4.

The amplitude and phase variations are not directly correlated, which is especially seen from the high amplitude of the datapoints from March 1999. We have in Fig. 9 superimposed the suggested sinewave from the amplitude variations, but with different values for phase and amplitude. There is reasonable agreement with the curve for the phases of the ESO data (before HJD 2451300), but not for the phases of the campaign data. The descending branch seen in the latter is less steep than expected from the fit, and the overall shape is clearly not purely sinusoidal. Thus simple beating between two close frequencies alone cannot describe the observed variations.

{1425f8.eps} }\end{figure} Figure 8: Variation in amplitude of f1 (top) and of 2f1 (bottom). The variations appear sinusoidal, and are present in both f1 and 2f1. Error bars are discussed in the text. The dashed curves are sinewaves found from an optimised fit to the amplitude values of f1 with scaled amplitude for 2f1.
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{1425f9.eps} }\end{figure} Figure 9: Variation in phase of f1 (top) and of 2f1 (bottom). A sinewave with the period deduced from the amplitude variations is superimposed.
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As the ($A,\phi $)-variations are clearly present in the ESO data, which were all taken using the same instrumental setup, the variations in amplitude are not caused by spurious effects of data merging or from filter passband mismatches between individual sites.

There are several mechanisms which could cause variations in the light curve shape of a pulsating variable, e.g. a beat phenomenon, where the pulse shape will vary according to the beat phase (Poretti 2000). Consequently we calculated, within the subsets, the phase difference ($\phi_{21}$) and the amplitude ratio (R21) of f1 and 2f1. The results are shown in Fig. 10 as a function of time (upper panels) and of the amplitude of f1 (lower panels). The straight line in the bottom panel is a weighted linear fit to the data.

The figure shows that the phase difference between f1 and 2f1 remains constant both as a function of time and f1-amplitude, whereas the amplitude ratio (R21) may grow with larger amplitude of f1. The slope of the fit to R21 vs. Af1 is $0.0015\pm0.0004$, thus formally significant, but the fit does not appear fully convincing to us. Furthermore, as A2f1 is expected to scale with Af12 (see Garrido & Rodriguez 1996 and references therein) such a variation is not surprising; A2f1/Af12 does not correlate with Af1 (not shown). In any case, the variation appears uncorrelated with the trend of $\phi_{21}$ vs. Af1. This suggests that the pulse shape of f1 remains nearly constant during the amplitude variation, supporting an intrinsic amplitude variation rather than a beat phenomenon.

{1425f10.eps} }\end{figure} Figure 10: Effects of the amplitude and phase variations on the pulse shape, as a function of time (top) and amplitude of f1 (bottom). Whereas the phase difference ( $\Delta \phi = 2\phi _{f_{1}}-\phi _{2f_{1}}$) is constant, this appears not to be the case for the amplitude ratio ( A2f1/Af1, see text).
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3.3.2 Low-amplitude frequencies

We used the combined data set prewhitened for f1 and harmonics to investigate possible variability of the low-amplitude modes. The five low-amplitude frequencies were optimised to the complete data set and fixed. The amplitudes and phases were then optimised while allowing one frequency at a time to have variable amplitude and phase. This gave, for each frequency, a set of amplitudes and phases as a function of time. For each frequency we then created "single-mode data sets", as in Handler et al. (2000), by subtracting from the light curves all the other frequencies but the one under investigation. This was done using a (n-1) simultaneous fit to the data.

We tried different ways of subdividing the data (b,c), but only for f2 and f3, the strongest of the low-amplitude signals, were meaningful results obtained - i.e. only for these two frequencies were the results independent of the method used. The results are displayed in Fig. 11. The scatter in this plot is quite high, and although trends or deviations from point to point in some cases are present, Fig. 11 does not show convincing evidence for ($A,\phi $)-variations of the low-amplitude modes. The amplitude modulation present in f1 does not seem to be present in the low-amplitude modes, only the first amplitude values (from the 1998 ESO data) have a shape similar to that of f1.

{1425f11.eps} }\end{figure} Figure 11: Phase and amplitude of f2 and f3 as function of time.
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3.4 Colour photometry

From the ESO data we determined the average difference between the times of minimum and maximum light in the b and y filters. This difference ( $T_{{\rm ext},b}-T_{{\rm ext},y}$) amounts to $-0.00010\pm0.00007$ d, or a phase shift between b and y of +0 $.\!\!^\circ$5$\,\pm\,$0 $.\!\!^\circ$3. From the light curves themselves we find a phase shift for f1 of +0 $.\!\!^\circ$75$\,\pm\,$0 $.\!\!^\circ$4 between b and y, with the error estimate again scaled by a factor of two.

Using $uvby\beta$ photometry and physical parameters from Hintz et al. (1998) we verified that V1162 Ori is well placed in the HADS instability strip (McNamara 2000). Using the Moon & Dworetsky (1985) code we found a $T_{\rm {eff}}=7400$ K and MV=1.89, in agreement with Hintz et al. (1998).

We then determined from our own data a f1 phase difference $\phi_{b-y}-\phi_{y}=+5^{\circ}\pm2^{\circ}$, and an amplitude ratio $A_{b-y}/A_{y}=0.23\pm0.02$. The values agree well between two subsets of the data (1998 and 1999, separately). The colour data are not sufficiently abundant to allow meaningful determination of phase shifts for the low-amplitude frequencies.

In Fig. 12 we compare these values for f1 with theoretical predictions (Garrido et al. 1990; Garrido 2000). Model atmospheres were calculated assuming Pop I, $T_{\rm {eff}}=7400$ K, $\log\,g=3.96$ (Hintz et al. 1998) and $\alpha=1.25$. For Qwe used the calibration given in Breger & Pamyatnykh (1998), and found Q=0.029 d assuming a mass of 1.8 $M_{\odot}$ (Hintz et al. 1998) and a bolometric correction of -0 $.\!\!^{\rm m}$1. The positive phase shift indicates a radial f1. The deviation from the predicted amplitude ratio is likely due to the present accuracy of the model atmospheres (see e.g. Garrido 2000), which are furthermore highly temperature dependent.

{1425f12.eps} }\end{figure} Figure 12: Regions of interest for V1162 Ori, pointing to f1 being radial. See text for discussion.
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The error associated with Q is too large to distinguish between the fundamental mode (Q=0.033) and the first overtone (Q=0.026). However, the dominant frequency in HADS is expected to be the fundamental (e.g. McNamara 2000).

The last column of Table 3 gives the frequency ratios relative to f1. For $\delta$ Scuti stars a ratio of fundamental to first overtone of 0.77-0.78 is expected (see e.g. Petersen & Christensen-Dalsgaard 1996), which is not found for any of the low-amplitude frequencies. Furthermore, they do not show a regular frequency spacing with f1, and most of them are very likely non-radial. Especially f2 is too close to f1 for both of them to be radial.

4 O-C analysis

Following Sterken et al. (1987), the times of maximum and minimum light were determined by fitting 3rd degree polynomials to the extrema in the light curves. During the campaign, eleven extrema were measured at two sites simultaneously, which offers another order-of-magnitude estimate of the precision in the timings. The simultaneous measurements deviated mutually with a mean of 0.0009 d (0.0005 d median), and they provide a realistic uncertainty estimate of the timings. The extrema collected during the campaign are presented in Table 4, the earlier ESO-extrema are published in PaperI.

{1425f13.eps} }\end{figure} Figure 13: O-C diagrams for times of maximum and minimum light in the combined data. The superimposed solutions are piecewise constant periods (upper panel) and a sinewave (middle panel). The latter is repeated with binned data (on a larger scale) in the lower panel. The error bars show the errors on the mean values of the bins. P0 is 0.07868895 d, corresponding to the frequency of f1 found in Sect. 3.2.2.
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The O-C diagrams are displayed in Fig. 13, which includes times of maximum as well as minimum light. A constant O-C shift between the maxima and minima of $0.0027\pm0.0004$ d (caused by the non-sinusoidal shape of the light curve of f1) was found and corrected for. This corresponds to a positive shift of the minima in pulsational phase of f1 of $0.03\pm0.01$ cycles, similar to what was found in PaperI. This asymmetry is a general feature of HADS (McNamara 2000). To check whether deviations from regularity in the light curve shape, (e.g. from a changing pulse shape) could influence our results, we also performed the analysis on the maxima and minima separately. The results were found to agree.

The shapes in the O-C diagrams correspond to the shape of the phase variations in Fig. 9, as one would expect - the two diagrams are equivalent. In Fig. 13 we show two possible fits to the data, piecewise linear segments (period breaks), and an optimised sinewave with a period very similar to that of the amplitude variations. The best of these solutions is the piecewise linear fit which leaves residual O-C scatter of 0.0016 d, while the sinewave leaves a scatter of 0.0020 d. This is not unexpected as the piecewise linear fits have more degrees of freedom. In the lower panel of Fig. 13, the data have been binned in smaller segments. The binned O-C values show overall deviations from a sinewave fit larger than the error bars. The campaign datapoints are systematically below the fitted curve at maximum, and above at minimum.

{1425f14.eps} }\end{figure} Figure 14: O-C diagrams for our times of maximum and minimum light combined with the times of maximum published by Hintz et al. (1998). Upper panel: large scale variations can be described by a parabola. Middle panel: subtracting the parabolic shape leaves residuals which, except for the first datapoints, may possibly be described by a sinewave with a period of 285 d and an O-C amplitude of 0.0029 d. In the lower panel have the data been binned (shown on a larger scale). P0 is 0.07868910 d.
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Our data only cover a relatively short time base in terms of O-C patterns, and may sample only a small part of a large scale structure. We therefore include the extrema published by Hintz et al. (1998) in the analysis, as shown in Fig. 14, upper panel. The overall shape of this diagram is parabolic, but with additional effects present. A parabolic fit with a linear period change rate of $-5\times10^{-9}$ ss-1 (calculated as described by e.g. Sterken 2000) is subtracted in the middle and lower panels. The residuals can now to some extent be described by a sinewave, but several points still deviate by several $\sigma $, and they may be better described by piecewise linear segments. The same is the case for the phase values in Fig. 9: the residuals after correcting for a linear phase change still deviates from a sinusoidal shape (not shown). The superimposed sinewave has a period of $285\pm3$ d, in agreement with the time scale of the amplitude variations ( $282\,\pm\,6$ d).

If a cyclic variation is present in the O-C diagram, the time scale is thus the same as for the amplitude modulation. The fact that these match is an argument that the variations in the O-C diagram are indeed cyclic - except if the phenomenon causing the amplitude variations also causes the main period to change abruptly on the same time scale. In any case, the first datapoints in Fig. 14 are not described by a cyclicly changing period (unless the overall shape in the upper panel is not parabolic, but rather sinusoidal with a very long period), and neither are the data from Poretti et al. (1990): a model of a slow linear change with a cyclic component superimposed cannot fit the data completely. For the binned data, subtracting four piecewise linear segments (in Fig. 14: $E<10\,000$, $10\,000<E<15\,000$, $15\,000<E<21\,000$ and $E>21\,000$) leaves a residual scatter of 0.0007 d. Subtracting a sinewave leaves one of 0.0010 d, and in both cases the residuals vs. epoch leaves no trend.

Subtracting the sinewave before performing the parabolic fit (and excluding the first datapoints) yields a linear period change rate of $-5.1\times10^{-9}$ ss-1, or a value of $(1/P)({\rm d}p/{\rm d}t) \sim$ $-2.4\times10^{-5}$ yr-1. This is a very high rate of change, and only expected for pre-main sequence or very evolved stars (Breger & Pamyatnykh 1998) - and with opposite sign of what is expected for the majority of $\delta$ Scuti stars.

The residual scatter in the non-binned O-C diagram, is, after subtracting parabolic and cyclic variations (or piecewise linear segments) 0.0018 d. This is a factor of two larger than what is expected from our measurements of the precision on the individual timings (0.0009 d). As the cause of this could be the multiperiodicity of V1162 Ori, we re-determined the extrema from the data set prewhitened with the low-amplitude frequencies, but found the same scatter to be present. This type of enhanced scatter may be an intrinsic feature in HADS (Szeidl 2000).

Table 4: New times of maximum and minimum light (HJD-2450000). The error on the time is estimated to be 0.0009 d. The cycle count scheme is based on Hintz et al. (1998).
$T_{\max}$ E $T_{\max}$ E $T_{\max}$ E $T_{\min}$ E $T_{\min}$ E $T_{\min}$ E

55279 1528.4862 56142 1573.4161 56713 1460.6223 55279 1539.7801 56285 1606.3490 57131
1464.5925 55330 1528.5654 56143 1573.4942 56714 1466.9130 55359 1546.2318 56367 1607.2947 57143
1465.5370 55342 1528.5654 56143 1574.2797 56724 1466.9959 55360 1548.4375 56395 1608.3136 57156
1466.7965 55358 1528.6426 56144 1574.3571 56725 1469.5985 55393 1549.3013 56406 1611.3046 57194
1466.8718 55359 1528.7222 56145 1575.3043 56737 1471.0086 55411 1549.3796 56407 1620.9812 57317
1466.9523 55360 1528.7995 56146 1575.3812 56738 1481.3219 55542 1553.3919 56458 1631.9209 57456
1469.0000 55386 1530.3744 56166 1575.4601 56739 1483.2822 55567 1562.2839 56571 1638.2145 57536
1469.6322 55394 1530.4536 56167 1576.3249 56750 1483.3650 55568 1562.3629 56572 1661.4271 57831
1471.0436 55412 1530.5302 56168 1576.4041 56751 1483.6802 55572 1562.4420 56573    
1480.3290 55530 1531.3188 56178 1577.0364 56759 1496.5817 55736 1562.5207 56574    
1481.1957 55541 1531.3950 56179 1577.2705 56762 1497.2898 55745 1564.4883 56599    
1481.2765 55542 1532.3416 56191 1577.3507 56763 1499.9645 55779 1566.3755 56623    
1483.2425 55567 1532.4205 56192 1577.3509 56763 1500.2799 55783 1569.3649 56661    
1483.3202 55568 1532.4990 56193 1578.2935 56775 1500.3640 55784 1570.3080 56673    
1483.4008 55569 1532.5803 56194 1582.3065 56826 1500.9890 55792 1570.4677 56675    
1483.6330 55572 1533.2851 56203 1582.3826 56827 1501.6138 55800 1571.2517 56685    
1483.7140 55573 1533.3672 56204 1583.2500 56838 1505.9471 55855 1571.3345 56686    
1485.5243 55596 1533.4444 56205 1583.4087 56840 1509.6402 55902 1571.3366 56686    
1488.5147 55634 1533.5217 56206 1584.2723 56851 1509.7258 55903 1571.4150 56687    
1496.6174 55737 1534.4690 56218 1584.3515 56852 1512.3995 55937 1571.8035 56692    
1497.2472 55745 1536.3564 56242 1584.4336 56853 1512.5590 55939 1572.2784 56698    
1498.5080 55761 1537.2988 56254 1591.5902 56944 1515.5460 55977 1572.2779 56698    
1499.9237 55779 1539.8154 56286 1592.3795 56954 1515.6238 55978 1572.3569 56699    
1500.3148 55784 1546.2699 56368 1593.3995 56967 1515.7048 55979 1572.3568 56699    
1501.5761 55800 1549.3378 56407 1595.2901 56991 1516.4114 55988 1572.4336 56700    
1502.4431 55811 1549.4178 56408 1605.2797 57118 1516.4909 55989 1572.5145 56701    
1502.5203 55812 1551.3022 56432 1605.3595 57119 1517.4364 56001 1573.3016 56711    
1504.5661 55838 1552.3268 56445 1606.3039 57131 1519.8706 56032 1573.3771 56712    
1505.9004 55855 1554.2949 56470 1607.2497 57143 1524.8303 56095 1573.4592 56713    
1507.5560 55876 1562.3208 56572 1607.3288 57144 1528.3726 56140 1574.3229 56724    
1509.6023 55902 1562.3221 56572 1608.2734 57156 1528.4519 56141 1575.3465 56737    
1509.6795 55903 1562.3978 56573 1610.3186 57182 1528.5296 56142 1575.3465 56737    
1509.7570 55904 1562.4779 56574 1611.2627 57194 1528.6095 56143 1575.4250 56738    
1511.4895 55926 1563.5002 56587 1612.2810 57207 1528.6857 56144 1575.5020 56739    
1511.5713 55927 1564.5231 56600 1617.3184 57271 1528.7664 56145 1576.2886 56749    
1512.3591 55937 1566.3330 56623 1620.9394 57317 1530.4165 56166 1577.0770 56759    
1512.4337 55938 1568.5370 56651 1632.2740 57461 1530.4950 56167 1577.2348 56761    
1512.5917 55940 1569.3225 56661 1635.2620 57499 1530.5765 56168 1577.3137 56762    
1514.7970 55968 1569.4807 56663 1637.2295 57524 1531.3606 56178 1577.3166 56762    
1515.5830 55978 1570.3457 56674 1638.2490 57537 1531.4411 56179 1582.4268 56827    
1515.6585 55979 1570.5031 56676 1647.2206 57651 1532.3842 56191 1583.2940 56838    
1515.7390 55980 1571.2901 56686 1650.2122 57689 1532.4642 56192 1583.3729 56839    
1516.4483 55989 1571.2899 56686 1659.4158 57806 1532.5429 56193 1584.3165 56851    
1517.3906 56001 1571.3679 56687 1660.4388 57819 1533.3318 56203 1584.3953 56852    
1517.4713 56002 1571.3694 56687 1662.4069 57844 1533.4075 56204 1591.5550 56943    
1518.4955 56015 1572.2361 56698     1533.4864 56205 1593.3676 56966    
1519.9111 56033 1572.3133 56699     1534.4335 56217 1595.3274 56991    
1524.8671 56096 1572.3928 56700     1534.5092 56218 1605.2476 57117    
1528.3279 56140 1572.4706 56701     1535.6093 56232 1605.3247 57118    
1528.4067 56141 1573.3361 56712     1536.3184 56241 1605.3263 57118    

5 Discussion and conclusions

Several very interesting phenomena are present in the light curves of V1162 Ori, including multiperiodicity and cyclic amplitude variability. The O-C analysis reveals the presence of a linear period change whose size is too large to be reconciled with evolutionary changes as given by Breger & Pamyatnykh (1998). However, these authors have collected available information on observed period changes in $\delta$ Scuti stars and find that the observed values, which are distributed nearly evenly between increasing and decreasing periods, disagree with stellar evolution calculations. They conclude that the observed linear period changes are not caused by evolutionary effects, but rather by long-period binarity or nonlinear mode interactions.

On top of the linear change are O-C variations which can be explained by frequent period changes or a cyclic variation combined with sudden changes. In both cases is the time scale of the period variation similar to that of the amplitude modulation, making a common origin probable.

Cyclic variations in the O-C diagram are explained either by the light time effect caused by the motion of the pulsating star in a binary system, or by beating of two (or more) very closely-spaced frequencies. These possibilities were put forward for V1162 Ori already in PaperI. In Sect. 3.2.2 we noticed a peak in the amplitude spectrum very close to f1, but considering the nearly cyclic amplitude and phase variations present in the data, such a close peak would always be expected, as discussed in Sect. 3.3.1.

Including this close peak in the frequency solution still leads to an increased noise level in the residual amplitude spectrum of 15% even if we take the linear variation in period of f1 into account. Furthermore, with a pure beat of f1 with a single close frequency, we cannot explain the deviations from a sinusoidal shape in the O-C diagram corrected for the linear period change (Fig. 14, middle and lower panels). A linear change combined with beating between two close frequencies is therefore, based on the present data, not a likely (or at least not the only) explanation of the observed changes.

In the case of binarity there is more room for deviations from strict periodicity. We get from the light time effect (following Irwin (1959) and assuming a circular orbit), that a binary configuration realistically models the possible cyclic part of the O-C variations. The separation between the two components would be small, of the order of 1 AU. We have no good interpretation of the amplitude variations in case of binarity: as f1 is probably a radial mode, a changing aspect angle would not give rise to amplitude variations. One could speculate that in such a relatively close system the amplitude variations could be caused by tidal deformation of V1162 Ori, but in this case one would expect all the detected frequencies to behave in a similar way, which is not the case. We note that such a binary system is not expected to give rise to detectable low-frequency variability in the light curves, even for a mass ratio of 1.

In conclusion, we have shown that V1162 Ori, previously considered monoperiodic, is a multiperiodic $\delta$ Scuti star. Apart from the main mode f1 (with harmonics), we find 5 additional frequencies in the amplitude spectrum. All these have very low amplitude and are most likely non-radial. V1162 Ori displays period and (cyclic) amplitude variability. The main pulsational period shows a linear period change with a change rate of $(1/P)({\rm d}p/{\rm d}t) \sim$ $-2.4\times10^{-5}$ yr-1. In addition, the O-C diagram reveals the presence of additional changes on the same time scale as the amplitude modulation.

The goals of the present campaign have partly been reached: we have obtained a much better knowledge of the changes present in the light curves of V1162 Ori - but we have not been able to determine their cause. Further photometric observations are clearly needed, and so is an extensive spectroscopic investigation spanning several years to establish or reject a possible binary nature of this star.

TA and CS acknowledge financial support from the Belgian Fund for Scientific Research (FWO). This project was supported by the Flemish Ministry for Foreign Policy, European Affairs, Science and Technology, under contract BIL 98/11/52, and the National Research Foundation of South Africa. LMF acknowledges support from IUAP P4/05 financed by Belgian DWTC/SSTC. AB acknowledges support through a grant of the Conselho Nacional de Desenvolvimento Científico e Tecnológico (grant No. 301029). PVC is grateful to the Royal Observatory of Belgium for putting at his disposal material acquired by project G.0265.97 of the Fund for Scientific Research (FWO) - Flanders (Belgium); sincere thanks go to Dr. J. Cuypers. Our research has made use of the SIMBAD database operated at CDS, Strasbourg, France, and the NASA Astrophysics Data System.


Copyright ESO 2001