3 Analysis of the power spectra

3.1 Time variations

A comparison of power spectra from the campaigns covering 3 seasons shows that the amplitude varies dramatically for most of the observed modes. This is well illustrated by examining the amplitude evolution of the dominant modes in Table 6. In the discovery data and in the succeeding 1992 WET run, the dominant mode was at 1217 $\mu$Hz, but its amplitude decreased by a factor 3 on a one year timescale. During the 1993 campaign, most of the modes observed one year before had decreased their amplitude by a comparable factor of 3, while a few modes increased their amplitude. Only one mode - at 1315 $\mu$Hz - maintained an almost constant amplitude over the two year period covered by our observations. This mode may be useful for determining $\dot{P}$, which will be discussed in Sect. 4.9. This mode was the dominant mode during the 1993 low amplitude phase of RXJ 2117+3412. In contrast, the power spectrum of the 1994 WET campaign was dominated by a mode at 958 $\mu$Hz, which was below the detection limit during the 1992 WET run and barely present during the 1993 campaign. The amplitude of this mode increased by a factor 6 in one year interval. Such large amplitude variations are a common property of the PNNV and the pulsating PG 1159 stars (Bond et al. 1996; Kawaler 1998). The amplitude variations observed in RXJ 2117+3412 are reminiscent of what has been described for the PNNV NGC 1501 (Bond et al. 1996). In that case, amplitude variations, up to a factor of 2, are sometimes also associated with frequency variations. Some frequency variations are also observed for a few modes in RXJ 2117+3412. Amplitude variations are not uncommon in some cooler DBVs (Vuille et al. 2000) and ZZ Ceti stars (Kleinman et al. 1998).

3.2 Fine structure, multiplets

While the amplitude of the modes changed within the two years interval of the observing campaigns, most of the modes observed more than once were at the same frequencies, within the observational uncertainties. Some interesting exceptions will be discussed below. In the following, we will assume that any non-linear effects that are present only affect the mode amplitudes and have a negligible effect on the mode frequencies. Therefore, we can still rely on linear pulsation theory to compute the frequencies of the observed pulsation modes. Asteroseismology depends on having the maximum number of pulsation modes available for an accurate inference of the internal structure of a star. Given the large amplitude changes present in RXJ 2117+3412 from one season to the next, we had to observe the star with three multisite campaigns in order to detect enough modes to decipher the structure of RXJ 2117+3412. Combining the sets of frequencies derived from these observing campaigns, allows us to significantly increase the number of modes usable for asteroseismological analysis. The 1994 WET data set has the best frequency resolution and coverage. The superior coverage makes the window function the most useful for deciphering the power spectrum. This is seen in Fig. 9 where the window functions from the three multisite campaigns are compared. We start the process of mode identification with the best power spectrum (1994 WET) and then proceed to the other two campaigns. For this reason, Table 6 lists the frequencies identified in the data in the order of worsening spectral window, i.e., 1994 WET, 1993, 1992 WET, which is also the inverse chronological order.

Figure 9: A comparison of the window functions obtained during the three multisite photometric campaigns used in the present paper: from top to bottom, they correspond to the September 1992 WET run, the September 1993 run, and the August 1994 WET run.

A quick look at the frequency list derived from the 1994 WET campaign alone (Table 6, Col. 1), shows a total of 42 significant peaks. A number of them are separated in frequency by about 5 $\mu$Hz. This is uncomfortably close to one half of the one day alias (5.8 $\mu$Hz). If we examine the window function in 1994, there are no peaks present in the range of 5 to 6 $\mu$Hz (as shown in Fig. 9), so we conclude that this frequency separation must be real. We interpret this splitting as due to slow rotation, implying that the star is rotating with a period of about one day, assuming these are $\ell =1$ modes. We follow this guideline to "read'' the frequency distribution and propose a mode identification. However, the 1994 WET frequency list by itself shows only doublets separated by about 5 $\mu$Hz, and no triplets or quintuplets, which would be the clear signatures of $\ell =1$ or $\ell =2$ modes split by rotation. One only sees several cases of two modes separated by about 5 $\mu$Hz. Considering the 1994 WET frequencies together with those derived in the previous campaigns, some of the missing multiplet members show up, which greatly aids our effort to decipher the power spectrum. This game can be difficult as the previous campaigns did not realize as good a coverage as the 1994 WET campaign; this is reflected in their poorer window function (see Fig. 9). The deconvolution of the power spectrum in some rich frequency domains could not be done unambiguously with the 1992 WET or 1993 data alone, and we relied on the 1994 WET data to help resolve ambiguities between the true frequencies and their aliases. In the following discussion, we discuss the features of the combined frequency list, which will be the basis for describing the fine structure used later to determine the rotational splitting and the period spacing.

The lowest frequency significant peak lies at 653.987 $\mu$Hz. This peak is seen in both the 1992 and the 1994 WET runs, but was below the detection limit in 1993. Note that the tentative detection of modes at frequency below 650 $\mu$Hz (Vauclair et al. 1993) is not confirmed by any of the multisite campaigns or by a re-reduction of the discovery data (see Fig. 1). We believe that they were probably the result of inadequate extinction and/or transparency corrections.

 

Table 7: Frequency list from the IAC CCD photometry.

$f~(\mu$Hz)	$\delta$$f~(\mu$Hz)	A (mma)
717.65	0.25	2.78
830.93	0.28	2.55
958.45	0.19	3.62
1179.60	0.32	2.23
1217.40	0.31	1.14
1315.61	0.58	1.23

The next few modes appear as single peaks. The feature seen at 717.714 $\mu$Hz in the 1994 WET data is also present in the power spectrum of the CCD photometry obtained during the 3 consecutive nights subset, but with an amplitude of 2.78 mma, as compared to 1.96 mma for the whole 1994 WET. As the amplitude of the other modes found in both the photomultiplier and the CCD photometry are in quite good agreement (see the discussion below), so we interpret the amplitude discrepancy as the signature of an amplitude change of this mode on a time scale shorter than the WET campaign (15 days). To check this hypothesis further, we break the 1994 WET data set into two parts and recalculate the amplitudes of the modes by a non-linear least-squares fit to each half of the data. We find that during the second half of the WET run, which encompasses the three nights where we acquired the CCD simultaneous photometry, the amplitude of the 717.714 $\mu$Hz exceeds by 44% its value during the first half of the run. This confirms the short time scale variability of that particular mode.

The first apparent fine structure feature is formed by the next two peaks at 789.042 and 793.783 $\mu$Hz present only in the 1994 WET data. They seem to form a doublet with a frequency separation of 4.741 $\mu$Hz, possibly due to rotational splitting. If this were the case, they would be $\ell =1$ modes with $\delta$m=1. However, as we will discuss later, this is not a single, rotationally split mode.

The next mode at 830.708 $\mu$Hz is also present in the CCD data with an amplitude in agreement with that of the whole 1994 WET run. However, this mode is also present in 1992 WET at a frequency shifted by 0.7 $\mu$Hz and with a smaller amplitude. The neighboring two modes at 836.067 and 840.367 $\mu$Hz, seen in the 1994 WET data, form the first true doublet. As will be shown later, they do not form a triplet with the 830.708 $\mu$Hz mode. The 836.067 $\mu$Hz mode is also present in the 1993 data, but with the frequency shifted to 835.000 $\mu$Hz; the frequency shift is significant when compared to the least-squares fit errors. Either we are seeing different modes in the 1994 WET and 1993 campaigns, or the same mode is exhibiting an unexplained (non-secular) frequency shift. The frequency separation, $\Delta f= 4.3$ $\mu$Hz measured in the 1994 WET spectrum suggests that these two peaks are two components of a $\ell =1$ triplet, with $\delta$m=1. As we have no explanation for the observed frequency shift of the 830 and 836 $\mu$Hz peaks, we will use the best determined frequency, i.e., the 1994 WET values which have the smaller least-squares fit errors, in the following analysis.

The next two peaks at 851.483 $\mu$Hz (seen only in the 1993 run) and at 872.337 $\mu$Hz (seen only in the 1994 WET data) are single peaks. More interestingly, the next two peaks seen in the 1992 WET data at 889.587 $\mu$Hz and 894.800 $\mu$Hz form another doublet. This doublet is also seen in the discovery data (Fig. 1), although strong aliasing made unambiguous frequency identification impossible. The frequency separation is 5.213 $\mu$Hz. We interpret this doublet as two components of an $\ell =1$ mode split by slow rotation, with the third component missing.

The two next peaks at 906.378 $\mu$Hz and 921.721 $\mu$Hz, seen only in 1994 WET data, are single peaks. The next three peaks at 940.563, 945.156, and 949.909 $\mu$Hz form the first identified triplet, suggesting an $\ell =1$ mode split by rotation. The WET 1994 data show only the m=+1 and -1 components of the triplet, while the all three components were detected in the 1992 WET run. By contrast, only the 940 $\mu$Hz mode was marginally visible in the 1993 data. Taking the best determined frequency for the m=-1 and m=+1 modes from 1994 WET data and the central m=0 mode frequency from the 1992 WET data, one finds a frequency separation of 4.593 $\mu$Hz from m=-1 to m=0 and 4.753 $\mu$Hz from m=0 to m=+1. Also, the frequency separation between the extreme components of this triplet differs between the two WET data sets by as much as 1.15 $\mu$Hz, which is significant compared to the frequency resolution of the data sets. The triplet was wider during the 1992 WET run. Given that the fine structure splitting of this and other modes changes from season to season, we try wherever possible to base our frequency splittings on the 1994 WET data, since this data set has the best window function.

The next two peaks at 958.533 $\mu$Hz and 963.282 $\mu$Hz form a doublet separated by 4.749 $\mu$Hz. The 958 $\mu$Hz peak also happens to be the largest amplitude mode in the 1994 WET data. While neither peak was detected in the 1992 WET data, they were both present in the 1993 data. The frequency separation suggests that these two peaks are also two adjacent components of a $\ell =1$ triplet. The CCD data also show a mode at 958.45 $\mu$Hz with an amplitude of 3.6 mma both values in excellent agreement with the values in Table 6.

The next two small amplitude peaks at 978.874 $\mu$Hz and 988.726 $\mu$Hz, form a doublet separated by 9.852 $\mu$Hz or 2$\times$4.926 $\mu$Hz. We interpret these peaks as the m=-1, +1 components of a triplet ($\ell =1$) whose missing central (m=0) component should be near 983.8 $\mu$Hz.

The next two peaks at 1005.645 and 1010.541 $\mu$Hz, form another doublet seen only in the 1994 WET data. The frequency separation is 4.896 $\mu$Hz. The doublet is interpreted as two adjacent components ($\delta$m = 1) of an $\ell =1$ mode split by rotation.

The next peak at 1023.594 $\mu$Hz is seen in the three runs, with its largest amplitude occurring in the 1992 WET run. The next two peaks at 1045.690 $\mu$Hz and 1055.703 $\mu$Hz, separated by 10.013 $\mu$Hz or 2$\times$5.006 $\mu$Hz are interpreted as the m=-1 and m=+1 components of a triplet whose m=0 mode is not seen, but should be near 1050.7 $\mu$Hz. The 1046 $\mu$Hz component is seen in all three runs, though only marginally in 1993, while the 1056 $\mu$Hz component was seen only in the 1994 WET data.

There is a final triplet formed by the peaks at 1096.712, 1101.942 and 1107.224 $\mu$Hz. The 1097 $\mu$Hz component is seen in the 1994 WET and the 1993 data (though significantly displaced by 0.65 $\mu$Hz to 1096.060 $\mu$Hz in 1993). The central component at 1101.942 $\mu$Hz is seen in the 1994 WET data, as well as in the 1992 WET data (where it is displaced by 0.73 $\mu$Hz), but it is absent in 1993. The third component at 1107.223 $\mu$Hz is seen in the 1992 WET and in 1993 data, but it is absent in the 1994 WET data. The components of this triplet are nearly symmetrically separated from their central m=0 mode by 5.230 $\mu$Hz and 5.282 $\mu$Hz respectively. We supplement the two 1994 modes with the 1992 WET m=+1 mode, although choosing the 1993 frequency would only change the splitting from 5.282 to 5.358 $\mu$Hz. We note that the data suggest a decreasing frequency splitting for the modes of this triplet from 1992 to 1994. In 1992, the m=0 to +1 splitting is 6.021 $\mu$Hz, while the average splitting in 1993 is 5.620 $\mu$Hz, and it decreases further to 5.230 $\mu$Hz in 1994.

The following mode at 1123.747 $\mu$Hz is a single peak while the next two peaks at 1179.955 and at 1190.578 $\mu$Hz form a doublet with a 10.623 or 2$\times$5.311 $\mu$Hz frequency separation. Only the 1179 $\mu$Hz mode was present in all three data sets. We interpret this doublet as two components of a triplet whose missing m=0 component should be near 1185.3 $\mu$Hz. The 1179 $\mu$Hz mode is present in the CCD data at 1179.60 $\mu$Hz and an amplitude of 2.23 mma; the frequency and amplitude are in good agreement with the values listed in Table 6. Next, one finds a doublet formed by the 1212.490 and the 1217.865 $\mu$Hz modes. The 1217 $\mu$Hz is present in all the data sets and was the largest amplitude mode in the 1992 WET data and the second largest mode in the 1993 data set. This peak is also seen in the CCD data at a frequency of 1217.40 $\mu$Hz, in good agreement with the 1994 WET data, but with an amplitude (1.14 mma) which differs significantly from the amplitude of the whole WET run (1.38 mma). However, in contrast with the case of the 717 $\mu$Hz discussed above, the frequency resolution of the CCD data is not sufficient to separate the two modes at 1212 and 1217 $\mu$Hz. In this case, the amplitude discrepancy reflects the fact that these two modes interfere in the power spectrum of the CCD light curve, while they are resolved in the power spectrum of the whole WET data. With a separation of 5.375  $\mu$Hz, this doublet is two adjacent components of an $\ell =1$ triplet.

Careful scrutiny of the combined frequency list does not reveal any other multiplets. The rest of the modes have single peaks of very low amplitude sparsely distributed in frequency up to 4340 $\mu$Hz.

Looking at the possible linear combinations and harmonics, one finds only a few cases. We searched for all possible quadratic (f₁+ f₂=f₃) and cubic ( $f_{1}+f_{2}\pm f_{3}=f_{4}$) linear combination peaks. A selection of such linear combinations is listed in Table 8. Considering that both quadratic and cubic combination peaks are not very abundant in the power spectrum, and that the largest amplitude modes do not necessarily generate them, we expect peaks from 4th order or higher linear combinations are unlikely. Therefore, all peaks which cannot be explained as 2nd or 3rd order linear combination are most likely true pulsation modes. Among those, the peaks with frequency 1572 $\mu$Hz, 2109 $\mu$Hz, 2133 $\mu$Hz, 2154 $\mu$Hz, 2164 $\mu$Hz and 2174 $\mu$Hz must be true pulsation modes. All remaining peaks above 1550 $\mu$Hz can be explained as 2nd and 3rd order combination peaks and are not independent modes.

 

Table 8: Linear combinations of frequencies in RXJ 2117+3412.

		1994 WET
f1	f2	f3	f4	$\delta f$
($\mu$Hz)	($\mu$Hz)	($\mu$Hz)	($\mu$Hz)	($\mu$Hz)
958.533	988.726		1947.334	0.075	a
666.938	1289.129		1956.008	0.059	b
789.042	1179.955		1968.952	0.045	b
963.282	1005.645		1968.952	0.025	a
1010.541	1179.955	1217.812	3408.257	0.051	a
921.721	1123.747	1362.734	3408.257	0.055	a
653.987	1315.055	1439.198	3408.257	0.017	a
789.042	1179.955	1439.198	3408.257	0.062	b
872.337	1096.712	1439.198	3408.257	0.010	a
958.533	1010.541	1439.198	3408.257	0.015	a
963.282	1005.645	1439.198	3408.257	0.132	b
836.067	1023.684	1548.653	3408.257	0.147	b
1010.541	1217.812	1289.129	3517.490	0.008	a
789.042	1289.129	1439.198	3517.490	0.121	b
921.721	1055.703	1539.991	3517.490	0.075	a
988.726	988.726	1539.991	3517.490	0.047	a
963.282	1005.645	1548.653	3517.490	0.090	b
1212.490	1315.055	1397.385	3924.971	0.041	a
978.874	1397.385	1548.653	3924.971	0.059	a
1055.703	1315.055	830.708	1539.991	0.059	a
1023.684	1397.385	872.337	1548.653	0.079	a
1212.490	1315.055	978.874	1548.653	0.018	a

 

Table 8: continued.

		1993
f1	f2	f3	f4	$\delta f$
($\mu$Hz)	($\mu$Hz)	($\mu$Hz)	($\mu$Hz)	($\mu$Hz)
963.416	1179.761		2143.374	0.197	b
2133.122	2174.884		4308.046	0.040	a
2153.980	2153.980		4308.046	0.086	a
939.838	1245.457	2153.980	4339.147	0.128	a
1179.761	1968.222	963.416	2184.777	0.210	b
1397.242	1968.222	963.416	2402.113	0.065	a

		1992 WET
f1	f2	f3	f4	$\delta f$
($\mu$Hz)	($\mu$Hz)	($\mu$Hz)	($\mu$Hz)	($\mu$Hz)
653.811	1315.181		1968.915	0.077	b
653.811	1315.181	2109.129	4077.942	0.179	b
945.156	1023.594	2109.129	4077.942	0.063	a
1179.893	1315.181	945.156	1549.959	0.041	a

Notes: The table lists selected linear quadratic ( f1 + f2 = f3) and cubic ( $f1 + f2 \pm f3 = f4$) combinations. The difference between the frequency resulting from the combination and the frequency of the observed peak is listed in Col. 5 as $\delta f$. The quality of the agreement is given in Col. 6 as: a) if the frequency mismatch is consistent with zero within formal errors, or b) if consistent with zero within twice formal errors.

Among the modes involved in linear combinations is the mode at 1315 $\mu$Hz, which showed a nearly constant amplitude. Otherwise it would have been a good candidate for a $\dot{P}$ measurement. It appears in one quadratic combination and in at least four higher order combinations.

3.3 Frequency table summary

Among the 63 frequencies listed in Table 6, we find 15 linear combinations, which leaves 48 independent pulsation modes. Among them, we find two complete triplets and eight doublets. We interpret the doublets as triplets with one missing component. Among these doublets, three are interpreted as triplets with the central m=0 component missing.

As no multiplet structures more complex than triplets are found, we conclude that the multiplets recognized in RXJ 2117+3412 are probably $\ell =1$modes split by rotation. The rotational splitting averaged between all multiplets is $\approx$5 $\mu$Hz. If $\ell =2$ modes were present, and if RXJ 2117+3412 is in an asymptotic pulsation regime, we would expect to detect all or part of quintuplets with components separated in frequency by about 8.3 $\mu$Hz. The only peaks listed in Table 6 which could potentially be identified as components of rotationally split $\ell =2$ modes are the 1539.991 $\mu$Hz-1548.653 $\mu$Hz ( $\Delta f=8.662$ $\mu$Hz) and the 1947.334 $\mu$Hz-1956.008 $\mu$Hz ( $\Delta f=8.674$ $\mu$Hz) doublets. However, these peaks can be explained as previously mentioned by quadratic and cubic combinations (Table 8) and we do not consider them to be real modes. We conclude that there is no evidence for $\ell =2$ modes split by rotation in the power spectrum.

Significant amplitude variations are seen in RXJ 2117+3412 as in most of the PNNV and GW Vir stars. They are accompanied by significant frequency variations for the two modes at 830 $\mu$Hz and 836 $\mu$Hz and the two triplets centered on 945 $\mu$Hz and 1101 $\mu$Hz. One can think of at least two explanations for these amplitude variations and frequency shifts: i) changes in the UV flux, as reported by Feibelman (1999), may reflect modifications in the chemical composition and in the structure of the outer layers which, in turn, affect the oscillatory properties of the modes having substantial amplitudes in those regions; ii) non linearities result in both amplitude and frequency variations for selected modes as described by Goupil et al. (1998).