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.

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.

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.

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.

   
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.

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.

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.

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.

   
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.

Figure 6: The new probable frequency (f6) detected from the combined data set. The position of the frequency is marked with a square.

 

 

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/f_n), 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.

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.

   
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.

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.

Figure 9: Variation in phase of f1 (top) and of 2f1 (bottom). A sinewave with the period deduced from the amplitude variations is superimposed.

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. A_f1 is $0.0015\pm0.0004$, thus formally significant, but the fit does not appear fully convincing to us. Furthermore, as A_2f1 is expected to scale with A_f1² (see Garrido & Rodriguez 1996 and references therein) such a variation is not surprising; A_2f1/A_f1² does not correlate with A_f1 (not shown). In any case, the variation appears uncorrelated with the trend of $\phi_{21}$ vs. A_f1. 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.

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).

  
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.

Figure 11: Phase and amplitude of f2 and f3 as function of time.

   
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 M_V=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.

Figure 12: Regions of interest for V1162 Ori, pointing to f1 being radial. See text for discussion.

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.