The EC stars ± X. A multi-site campaign on the sdbv star PG

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1 Mon. Not. R. Astron. Soc. 303, 525±534 (1999) The EC stars ± X. A multi-site campaign on the sdbv star PG D. Kilkenny, 1 C. Koen, 1;2 D. O'Donoghue, 1 F. van Wyk, 1 K. A. Larson, 3 R. Shobbrook, 4 D. J. Sullivan, 5 M. R. Burleigh, 6 P. D. Dobbie 6 and S. D. Kawaler 7 1 South African Astronomical Observatory, PO Box 9, Observatory 7935, South Africa 2 Department of Astronomy, University of Texas, Austin, TX 78712, USA 3 Department of Physics, Applied Physics and Astronomy, Rensselaer Polytechnic Institute, Troy, NY , USA 4 Chatterton Astronomy Department, School of Physics, University of Sydney, NSW 2006, Australia 5 School of Chemical and Physical Sciences, Victoria University, Box 600, Wellington, New Zealand 6 Department of Physics and Astronomy, University of Leicester, Leicester LE1 7RH 7 Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA Accepted 1998 October 23. Received 1998 October 5; in original form 1998 March 31 ABSTRACT Results are reported from a multi-site photometric campaign on PG , a member of the recently discovered class of pulsating hot subdwarfs, or EC stars. The main part of the campaign covered two weeks and produced,180 h of photoelectric photometry from the ve sites involved. Periodogram analysis shows that the light curve is dominated by ve frequencies in the range 1.89±2.74 mhz (periods 529±365 s) with the main frequency at mhz ( s), though there appear to be more than 50 frequencies identi able down to a semi-amplitude of (in fractional intensity) in the range 1.74±4.84 mhz (573± 206 s). Compared with other pulsating sdb stars, the low gravity (log g ˆ 5:25) and long periods indicate that the star has evolved away from the core helium-burning horizontal branch. A preliminary model of a post-horizontal-branch star with appropriate parameters yields many pulsation periods in the range 300±600 s, but detailed mode identi cation is not possible at present. Key words: stars: individual: PG ± stars: oscillations ± stars: variables: other. 1 INTRODUCTION A recent quartet of papers announced the discovery at the South African Astronomical Observatory (SAAO) of a new class of rapidly pulsating star. In Paper I (Kilkenny et al. 1997), results from `high-speed' photoelectric photometry showed that the sdb star EC has two pulsation periods near 144 and 134 s with semi-amplitudes of,0.01 mag. Koen et al. (1997±Paper II) presented results for PB 8783 which appeared to have at least six pulsation periods in the range 120±135 s, again with semi-amplitudes of,0.01 mag or less. In Paper III (Stobie et al. 1997), it was shown that EC pulsates with three periods (139± 152 s) and that sdb models with log g ˆ 6 and T eff ˆ K could have radial- and non-radial pulsation modes with frequencies near to those observed. Because some of the observed frequencies are very close together, it seemed likely that non-radial modes must be involved. Paper IV (O'Donoghue et al. 1997) showed that EC also has three pulsation frequencies (137±159 s), and used low- and medium-dispersion spectrograms to derive temperatures and gravities for the four pulsators ± with results near to log g ˆ 6 and T eff ˆ K in each case. At this stage, it appeared that all of the known sdb pulsators ± or EC stars ± were in binary systems. Analysis of UBVRIJHK and uvby photometry and spectrophotometric uxes indicated that each star had a companion of type F or G on (or near) the main sequence (see Paper IV for details). Since the initial discovery, large-scale searches for further sdb pulsators have been carried out from the SAAO and McDonald Observatory in Texas. To date, several new pulsators have been discovered (from a sample of well over 400 sdb stars tested for variability) and some of these are of particular interest. O'Donoghue et al. (1998b ± Paper VI) have shown that PG , a star from the Palomar±Green survey (Green, Schmidt & Liebert 1986), has at least nine frequencies in the range 104± 162 s [an independent discovery of this pulsating star by BilleÂres et al. (1997) nds at least ve frequencies]. There is, however, no evidence of a companion to PG in the spectroscopic or photometric data. An upper limit of about M0 was placed on the spectral type of any undetected companion: a clear difference from the rst four discoveries. Koen et al. (1998 ± Paper VII) reported the detection of pulsations in the sdb star PG ; in this star, the peak-to-peak variations are up to,0.25 mag and it was clear that many frequencies (probably > 30) are present. PG also shows q 1999 RAS

2 526 D. Kilkenny et al. no sign of being binary; any companion must be later than about M2. In addition, the temperature and gravity appear somewhat lower than in the other stars (T eff ˆ K and log g ˆ 5:25 6 0:1) and the main pulsation period is considerably longer (, 480 s). Finally, Kilkenny et al. (1998 ± Paper VIII) presented results for PG 1336±018 which is a pulsating sdb star in a very short-period (, 0.1 d) eclipsing binary system, remarkably similar to HW Vir (Menzies & Marang 1986; Wood, Zhang & Robinson 1993). A possible driving mechanism for the observed pulsations has been suggested by Charpinet et al. (1997) who nd that local enhancement of the iron abundance can occur as a result of diffusive equlibrium between gravitational settling and radiative levitation. The locally increased iron abundance creates an `opacity bump', which then drives pulsation via the k-mechanism. They nd that radial and non-radial pulsations can be driven for < T eff < K in representative models with M ˆ 0:48 M ( and log g ˆ 5:8. Because multi-mode pulsations are potentially very valuable as diagnostics of the structure of sdb stars, an obvious project is intensive multi-site photometric monitoring of selected stars to separate and identify the frequencies present. Such a campaign has already been carried out for PB 8783 (O'Donoghue et al. 1998a ± Paper V) with the result that at least 12 and possibly 15 frequencies were found which could be qualitatively identi ed with low-order radial and non-radial pulsation modes in a model with appropriate T eff and log g. In this Paper we report on a similar campaign carried out on the sdb pulsator PG This is a prime candidate for such a campaign, as it has a rich frequency spectrum (paper VII) and a large amplitude, and is relatively bright (V, 12.8 mag). 2 OBSERVATIONS `High-speed' photometry of PG was obtained during 1997 April 28 to May 12 using the following telescopes: the 1-m at the Sutherland site of the SAAO; the 1-m at Mt John University Observatory (MJUO) in New Zealand; the Jacobus Kapteyn Telescope (1-m) at the Observatorio de Roque de los Muchachos on La Palma (LaP) in the Canary Islands; the 0.9-m telescope at McDonald Observatory (McD) in Texas, USA; and the 0.6-m telescope of Mt Stromlo & Siding Spring Observatory (MSSSO) in Australia. A three-channel photometer (target star, comparison star and sky) was used at the LaP site, two-channel photometers (target and comparison stars) at Mt John and McDonald, and singlechannel photometers at SAAO and MSSSO. All sites used 10-s integrations of un ltered light (`white light'). Since all photometers were equipped with blue-sensitive photomultipliers, the system response in each case was similar to a Johnson B in effective wavelength but with a much broader bandwidth. At the sites using single-channel and two-channel photometers, programme star observations were interrupted at appropriate intervals, typically, 20±30 min, to make sky measurements. Altogether, about 180 h of useful data were obtained, with a fractional coverage, or `duty cycle', of,43 per cent. In addition to the main multi-site campaign, some high-speed photometry was obtained with the SAAO 0.5-m telescope and single-channel photometer. The latter uses a red-sensitive (GaAs) photomultiplier, so observations were made through a solid CuSO 4 lter which blocks out light redwards of about 6000 AÊ and thus produces a response similar to the un ltered blue-sensitive tubes. More than a week on this system was dedicated to PG before the main campaign; it was planned to continue with 0.5-m Table high-speed photometry of PG Run Date Start Start Run Site Tel (UT) (BJD) (h) (m) d/m fvw062 08/4 22:16: SAAO 0.5 fvw064 10/4 22:08: SAAO 0.5 fvw066 13/4 22:40: SAAO 0.5 fvw067 14/4 22:37: SAAO 0.5 fvw068 17/4 21:54: SAAO 0.5 fvw069 18/4 22:00: SAAO 0.5 fvw078 27/4 21:09: SAAO 0.5 fvw079 28/4 20:33: SAAO 0.5 tex002 29/4 05:26: McD 0.9 kal001 29/4 12:59: MSSSO 0.6 dmk043 29/4 21:53: SAAO 1.0 tex003 30/4 05:17: McD 0.9 kal002 30/4 12:11: MSSSO 0.9 tex004 01/5 05:59: McD 0.9 kal003 01/5 12:14: MSSSO 0.6 dmk046 01/5 21:21: SAAO 1.0 tex006 02/5 04:30: McD 0.9 kal004 02/4 12:07: MSSSO 0.6 tex008 03/5 04:31: McD 0.9 kal005 03/5 13:27: MSSSO 0.6 dmk049 03/5 21:21: SAAO 1.0 tex009 04/5 07:59: McD 0.9 dmk052 04/5 21:08: SAAO 1.0 ds001 05/5 10:23: MJUO 1.0 kal007 05/5 13:34: MSSSO 0.6 dmk053 05/5 22:00: SAAO 1.0 tex010 06/5 05:48: McD 0.9 ds003 06/5 12:21: MJUO 1.0 dod002 06/5 21:16: SAAO 1.0 jkt001 07/5 04:03: LaP 1.0 ds004 07/5 11:12: MJUO 1.0 ds005 07/5 13:00: MJUO 1.0 dod003 07/5 21:25: SAAO 1.0 jkt002 07/5 22:41: LaP 1.0 kal010 08/5 11:53: MSSSO 0.6 dod004 08/5 22:07: SAAO 1.0 jkt003 08/5 23:11: LaP 1.0 kal011 09/5 12:03: MSSSO 0.6 dod005 09/5 20:50: SAAO 1.0 kal012 10/5 11:24: MSSSO 0.6 jkt005 10/5 22:18: LaP 1.0 dod006 11/5 21:18: SAAO 1.0 jkt006 12/5 02:34: LaP 1.0 observations for some time after the campaign, to establish a longer baseline when combined with the main campaign, but the plan was thwarted by poor weather. None the less, a further 37.6 h of data were obtained. Table 1 gives a listing of all the individual runs contributing to the campaign and Fig. 1 shows the duty cycle. The actual data are plotted in the gure and, though the scale is very compressed, it is still possible to discern frequency beating on time-scales near an hour and half a day. The last 0.5-m run (fvw079) is almost continuous with the rst of the `multi-site' runs and is included in Fig. 1 and the analysis of the campaign core. 3 DATA REDUCTION The data reduction was carried out on each individual run using

3 A multi-site campaign on PG Figure 1. Coverage diagram for the main part of the campaign. Each panel represents one day and the panels read left-to-right and top-to bottom, with the Julian day numbers indicated on the extreme right (JD ). Ordinate carets are separated by 0.1 mi (fractional intensity units; see Section 3). The actual data are plotted, though on a very compressed scale. Even so, it is possible to see beating on time scales of about an hour and, 0.5 d (e.g. fourth panel from the top). standard techniques. For the three-channel data, the smoothed sky counts were subtracted from the stellar data on a point-by-point basis; for the other data, the less frequent sky measures were linearly interpolated and subtracted from each star measure. The sky counts were generally suf ciently small compared with star counts and measures of the sky background were suf ciently frequent that this procedure was perfectly adequate. For each run, the mean extinction coef cient that best ` attened' the data was determined and applied. In addition, since different sites, instrument sensitivities and telescope apertures result in different mean light curve levels, each light curve was normalized by its own mean level, yielding data in fractional intensity units, di=i. Because of the inconvenience of the term `fractional intensity units', Winget et al. (1994) have named this unit the `modulation intensity' or mi. One thousandth of an mi is then a `millimodulation intensity' (mmi) which is related by a scaling factor, , to the more conventional millimagnitude (mmag) unit; the latter is, however, logarithmic in the measured light intensity whereas the mmi is linear. We shall adopt the mi (or mmi) throughout the analysis and note by way of example that 10 mmi implies that the star has changed by 1 per cent with respect to its mean brightness level. The ve sites involved in this project fall into three longitude groups ± MJUO/MSSSO, SAAO/LaP and McDonald ± separated by roughly 1208 or,8 h of time. Since PG is approximately equatorial, it was expected that very little overlap would occur between these three groups (because telescope limits would tend to restrict observing time to a maximum of about 8 h at each site). Unfortunately, vagaries of the weather prevented much overlap at all, even from sites at similar longitude (see Table 1). Figure 2. Sample light curves: parts of two overlapping runs on PG from 1997 May 8, dod004 (crosses) and jkt003 (dots). Ordinate carets are separated by 0.1 mi and abscissa carets by 0.01 d. The data run leftto-right and top-to-bottom. The top panel is JD to , the second panel to , and so on. The jkt003 data are displaced downwards in the gure by mi for clarity. Note the very clear frequency beating.

4 528 D. Kilkenny et al. some preliminary idea of the problem, Fig. 3 shows the periodogram for the Fig. 1 data (top panel); the periodogram for the data with ve frequencies removed (second panel); and periodograms with 10 and 20 frequencies removed (third and fourth panels, respectively). Where we show a periodogram in this paper, it is always a semi-amplitude (mi or mmi) versus frequency plot. It will be shown as our tale unfolds that the rst ve (largest amplitude) frequencies are by far the dominant frequencies. Their semiamplitudes are between and mi; the next frequency has a semi-amplitude of only mi. It is clear from the bottom panel in Fig. 3 that even after the removal of 20 frequencies, there is still signi cant power (above the general noise level) at 20 or more frequencies. Intercomparison of the various panels in Fig. 3 shows several good examples of the closeness of some of the frequencies [compare, for example, the top panel with the second: the three main frequencies near 2.1 mhz (2.076, and mhz) have been removed from the data in the second panel, yet there is still signi cant power at a frequency of mhz]. Since the existence of many modes of pulsation gives the possibility of investigating the internal structure of stars (see, for example, Winget et al. 1991), the rst aim of the multi-site campaign was to resolve and measure accurately as many frequencies as possible. This is important, even if the appropriate models do not yet exist. Indeed, it is hoped that the data obtained in this campaign will drive the production of the necessary models. A second (though not necessarily secondary) aim is to look for secular frequency changes which could be related to evolutionary changes. This would probably take several seasons, but such changes have already been observed in the DO pulsator PG (Winget et al. 1991). Both of these aims require a careful analysis of the data. Figure 3. Periodograms (semi-amplitude/frequency) for the main campaign (Fig. 1 data). The top panel is the periodogram for the campaign data; the second panel is that for the data pre-whitened by the rst ve frequencies (i.e. the ve frequencies with the largest amplitudes); the third panel is that for the data pre-whitened by the rst 10 frequencies; and the bottom panel is that for the data pre-whitened by the rst 20 frequencies. Note the changes in the scale of the ordinate axes. Fig. 2 shows most of two runs, from SAAO and LaP, which did overlap considerably. The agreement between the two sets of reduced, normalized data is encouraging. One run has been displaced by mi to demonstrate the detailed agreement. Fig. 2 also shows the beating in the light curve quite well and illustrates the internal consistency of each run. These data are typical, in the sense that they were selected only because of their overlap, not because of unusually high quality. 4 FREQUENCY ANALYSIS The frequency analyses described in this section were all carried out using periodogram analysis following the method of Deeming (1975) as modi ed by Kurtz (1985). It was expected from the results presented in Paper VII that the multi-site campaign on PG would reveal a very rich frequency spectrum. A number of preliminary attempts to remove successively the highest amplitude frequencies from the data (or `pre-whiten' the data) showed that this expectation would be realized. It was clear, from examination of sections of the frequency spectrum, that in some frequency ranges there were several close frequencies and careful separation would be necessary. To give 4.1 Main campaign: data divided As a rst step, the data represented in Fig. 1 were divided into two roughly equal parts of about 7 d each and periodograms were calculated for each data set in the range 1 < f < 6 mhz. (There was no evidence for signi cant power at frequencies 6 < f < 25 mhz; below 1 mhz, any power cannot be distinguished from low-frequency sky variations.) The largest amplitude peak was removed from each set by a least-squares procedure (Deeming 1968) which calculated the best- tting amplitude and phase for the given frequency. The next highest peak was calculated and the two frequencies were then removed simultaneously using a Taylor expansion non-linear least-squares technique. This procedure was iterated with the rst n frequencies being removed with simultaneous non-linear tting and the (n 1)th frequency then being found from the periodogram of the residual data. The two halves of the data were analysed independently in this way and then the results were compared. 22 of the rst 25 frequencies found were present in both data sets, although not in exactly the same order (of amplitude). With the removal of 22 frequencies, the semi-amplitudes of any remaining frequencies were,0.001 mi: a level at which a small difference in amplitude could mean a large shift in ranking, or order of removal. Consequently, the 22 frequencies were removed simultaneously and then the next 30 frequencies were extracted singly. Intercomparison between the two halves of the data was made and any coincident frequencies were removed simultaneously (together with the original 22 frequencies). This procedure was repeated until no further coincidences were found, by which time 35 frequencies had been extracted. Fig. 4 shows the window functions for the two halves of the data considered separately and for the whole data set (as

5 A multi-site campaign on PG Table 2. Frequencies, semi-amplitudes and periods derived from the two halves of the data. f a f b D f s a s b P a P b mhz mhz mhz mi mi sec sec Figure 4. Window functions for the two halves of the full data set ( rst half± top panel; second half±middle panel) and for the full data set represented in Fig. 1 (bottom panel). The functions are normalized to unity at zero frequency. illustrated in Fig. 1). This gure indicates that failure to nd coincident frequencies is not due to missing `matches' because of aliasing. The rst part of Table 2 lists, for the two halves of the data (`a' and `b'), the 35 frequencies ( f a, f b ), the differences D f ˆ f a f b, semi-amplitudes (s a, s b ) and periods (P a, P b ) derived in the nonlinear tting described earlier. The frequencies are listed in the order in which they were extracted from the rst half of the data (`a') although they are not strictly in order of decreasing amplitude because slightly different results are obtained when a number of frequencies are removed sequentially, compared with the case when the same frequencies are removed simultaneously. Fig. 5 shows the periodograms for the two halves of the data after the removal of the same 35 frequencies from each set. It is apparent that the noise increases towards lower frequencies in both periodograms. From the noise level at frequencies higher than 6 mhz (not included in Fig. 5) and the general trend of the noise level apparent in that gure, we claim, conservatively, that peaks exceeding 0.5 mmi at frequencies higher than 1.8 mhz are real. There is no clear evidence in Fig. 5 for real residual peaks at frequencies below 1.8 mhz. It does appear that there are a number of peaks well above the noise level and, more importantly, that appear in both periodograms. However, the next 30 frequencies, extracted one at a time, show no signi cant coincidences. Further frequency searches were carried out in the ranges where there seemed to be closely ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± corresponding peaks in both data sets, namely 1.8±3 and 3.8± 5 mhz. Four more frequencies were found to correspond well and are listed in the second part of Table 2, although these have rather low amplitudes. The baselines for the `half' data sets are roughly 7 d or T, s. The corresponding resolution for complete separation of two frequencies in such a data set is,1:5=t (Loumos & Deeming 1978) or 2.4 mhz for the half data sets. The quantity 1/T can be regarded as a reasonable limit: frequencies more closely spaced than this would be `unresolvable', so we could not expect to separate any frequencies more closely spaced than about 1.6 mhz. By extension, if frequencies extracted from the two halves of the data are closer than 1/T, we can say they are formally the same. For most of the frequencies listed in Table 2, the differences between the results from the two half data sets are less than or comparable to 1/4T (0.4 mhz). Only one pair is different by more than 1/2T and evidence is presented later in this section that the feature at mhz is actually three frequencies, unresolved in the half data sets.

6 530 D. Kilkenny et al. Figure 5. Periodograms (semi-amplitude/frequency) for the rst half of the Fig. 1 data (top panel) and the second half (bottom panel) after removal of the rst 35 frequencies listed in Table 2. From the general trend of the noise level, including noise at frequencies higher than 6 mhz, we claim that peaks exceeding 0.5 mmi at frequencies higher than 1.8 mhz are real (see Section 4.1). 4.2 Main campaign: data united Taking the 35 frequencies from Table 2 as a starting point, the next step was to analyse the total data set for the main part of the campaign, i.e. the data represented in Fig. 1. As before, frequencies were removed by simultaneous, non-linear tting procedures. The top three panels of Fig. 6 show the periodograms for the whole data set, for the data after removal of the rst 25 frequencies in Table 2, and after removal of the rst 35 frequencies in Table 2. It is clear that even after the removal of 35 frequencies, there is still a considerable amount of signal in the data. Furthermore, some of these frequencies have relatively large amplitudes (semi-amplitudes, 0.001±0.002 mi). The process of removing further frequencies was carried out as described earlier until a further 20 frequencies had been removed. The last few frequencies removed had semi-amplitudes of 0.6 mmi, which is still about twice the noise level. The bottom panel in Fig. 6 shows the periodogram of the data with 55 frequencies removed. There is still evidence for some nonrandom structure, but at a very low level (< 0:5 mmi). Table 3 lists the 55 frequencies found in the total data set; the frequencies in parentheses are the frequencies that are not `veri ed' by being obviously present in the two halves of the data. The frequencies are ranked (column 1) in order of decreasing amplitude Figure 6. Periodograms (semi-amplitude/frequency) for the main campaign (Fig. 1 data). The top panel is the periodogram for the campaign data; the second panel that for the data pre-whitened by the rst 25 frequencies in Table 2; the third panel that for the data pre-whitened by the rst 35 frequencies in Table 2; and the bottom panel that for the data pre-whitened by the 55 frequencies in Table 3. Note the changes in the scale of the ordinate axes. and, in what follows, the frequencies are identi ed as f n, where n is the column 1 rank. It is apparent from Fig. 6 and Table 3 that the light curve is dominated by the strongest frequency, f 1 ( mhz with a semi-amplitude of mi) and the next four frequencies, f 2 to f 5 ( , , and mhz, with semi-amplitudes, mi). The other frequencies all have semi-amplitudes < mi. In Paper VII, it was noted that the frequency near mhz was almost exactly twice the strongest frequency ( mhz) and that at least three cases existed where a frequency was close (< 1 mhz) to the sum of two other established frequencies. Examination of the results in Table 3 shows that all but one of the high frequencies (greater than 3.9 mhz) can be matched in this way. The nal two columns of the table show the sums of frequencies that will match a particular high frequency and the difference between the extracted frequency and the sum of the two lower frequencies. The worst difference is 0.12 mhz and the average is 0.05 mhz. All combinations of the four strongest amplitudes are present, but for f 5 only the combinations f 1 f 5 and f 3 f 5 are identi ed. There is the immediate question of why we should nd frequencies in the total data set that are not obviously in both halves. Are these in fact real pulsation modes, or just artefacts? Of course, the latter is always a possibility, especially at such low amplitudes, but

7 A multi-site campaign on PG Table 3. Frequencies, semi-amplitudes and periods for the main campaign (Fig. 1 data). Frequencies in parentheses are those not found in both halves of the main campaign data. Rank P f s freq D n s mhz mi sums mhz ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) f 1 f 5 ± f 8 f f 3 f f 1 f 13 ± f 1 ± f 1 f 3 ± f 1 f f 2 f f 1 f 4 ± f 3 f ( ) f 2 f 4 ±0.07 there are reasons why the extra frequencies should be considered to be real. First, there is the resolution, improved by a factor of 2 by doubling the baseline of the data. As an example, two of the extra frequencies, f 13 and f 42, found near f 8 are within 0.8 and 0.6 mhz of the frequency of f 8 and would have been unresolved in the earlier analysis. Note that f 13 has a substantial amplitude; it is clearly many times the noise level (see the third panel in Fig. 6). In Table 2, the difference D f and the difference in the amplitudes, s a and s b, are attributable to the fact that the frequencies near mhz are not resolved in the two halves of the data. Comparison of Tables 2 and 3 reveals other examples of frequencies (in Table 2) that have poor agreement between the two half data sets but are resolvable into more than one frequency by the full data (Table 3). Secondly, it is quite possible that at least some of the frequencies have variable amplitudes; this has been observed in several of the stars reported previously, where amplitude variability has been seen on timescales of weeks and from year to year (e.g. Papers III, IV, VI, VII). In this case, small variable-amplitude frequencies could be missed in either half of the data. Thirdly, the amplitudes of the frequency components extracted have all been at least twice the apparent noise level in the periodograms. Fourthly, and nally, a number of the extracted frequencies with smallest amplitude (0.6 mmi) are very close to the sums of the largest-amplitude frequencies (see Table 3 and the discussion later in this section), which gives con dence that these are real. In spite of these arguments, the fact remains that frequency components detected only once (in the whole data set), especially weak `satellites' of a nearby strong peak, may arise from amplitude/phase modulation of a single component. This possibility should be remembered when interpreting these results. Even though there is little evidence to suggest normal-mode frequencies greater than 5 mhz, the rst 20 frequencies in the range 3.5±10 mhz were extracted from the residual data (with 55 frequencies removed) to see if any further sums could be identi ed. Several other frequencies were found which could be matched at better than 0.1 mhz to low-frequency pairs (e.g. f 4 f 7, f 5 f 6 ) but the amplitudes are so small, 0.3 to 0.5 mmi, that it would require greater optimism than the current authors possess to claim them to be real. The only harmonic frequency that has been identi ed is 2f 1 ; if other harmonics exist (2f 2 ;...;3f 1 ;..., etc.) their amplitudes are less than 0.3 mmi. The fact that the matches between the high frequencies and sums of two lower frequencies are better than, 0.1 mhz for the lowest amplitude (0.7 mmi) frequencies gives con dence that these are real and also indicates that the errors in extracting the frequencies are generally much better than 1 mhz. 4.3 Main campaign plus 0.5-m data As a nal step, the main campaign data were combined with the earlier data from the SAAO 0.5-m telescope (see Table 1) and reanalysed with the procedures described above. The addition of the 0.5-m data increased the quantity of observations by,21 per cent. For the combined data, an attempt was made to remove the 55 frequencies listed in Table 3 by simultaneous, non-linear tting (not truly simultaneous, as the frequencies were removed in three groups to avoid problems with very close frequencies). After that, the residual data had the next 25 highest peaks removed singly. In this process, 50 of the 55 frequencies were recovered in the `simultaneous' tting and a further three by the single-frequency tting. The two frequencies not recovered were f 36 and f 51 ; both are low-amplitude (< 0:001 mi) frequencies, and both are listed in parentheses in Table 3, indicating that they were not found independently in the two halves of the campaign data. The reality of these frequencies must be doubted. The mean difference between the 53 frequencies recovered and the Table 3 results is

8 532 D. Kilkenny et al. Table 5. Comparison of Table 3 results with Paper VII. n f(t3) s.amp. f (VII) s.amp. Dn Note mhz mmi mhz mmi mhz Figure 7. Periodogram (semi-amplitude/frequency) for the main campaign plus 0.5-m data after simultaneous removal of 50 frequencies (see Section 4.3). 0:03 6 0:12 mhz. For comparison, the increased baseline (,33 d) should give a resolution (1/T) of about 0.4 mhz. It is not surprising that the extended data should give such closely similar results to the data from the main campaign alone, since the latter will clearly carry great weight in the analysis. What is somewhat surprising is that when the 53 `known' frequencies are removed, the single-frequency removal procedure still nds several frequencies with semi-amplitudes greater than mi and one greater than mi, i.e. between f 10 and f 11 in amplitude. Fig. 7 shows the periodogram after removal of 50 frequencies [i.e. it includes the three frequencies (f 38, f 41 and f 48 ) found in the singlefrequency tting] and Table 4 lists the newly found frequencies (with semi-amplitude mi or greater). It is noticeable that some of these frequencies are close to some of the largest amplitude variations already found; they might be artefacts of the solution process. However, it is signi cant that in both instances where the baseline of the data has been increased (adding the two halves of the main campaign, and adding the extra Table 4. Frequencies with semi-amplitude of mag or more found in the residual data (main campaign plus 0.5-m data) after removal of 53 frequencies (see Section 4.3). f s.amp. Note mhz mi near f 1 (0.24 mhz) near f 1 (0.10 mhz) near f 6 (0.25 mhz) near f 1 (0.58 mhz) near f 4 (0.23 mhz) ± ± ± ± ± ± ± ± ±0.89 f 3 f ±0.24 f 1 f ± f f 1 f ± ± f 2 f ± m data to the main campaign) this has apparently resulted in more frequencies being found (see the discussion at the start of Section 6). 5 COMPARISON WITH PAPER VII In Table 5, the results from the current analysis are compared with the results from the discovery paper (Koen et al ± Paper VII). We have searched for the obvious correspondences, but also for correspondence at frequency differences near mhz, the one cycle per day aliases. Examination of the window functions from the data in Paper VII ( g. 3 in that paper) indicates that it is quite possible that one (or more) cycle per day aliasing could have occurred in the frequency analysis. As can be seen from Table 5, we nd 14 of the strongest 25 frequencies in common, plus another ve weaker frequencies. If we extend the search to the two cycle per day aliases (Dn ˆ 23:15mHz) we nd another four correspondences. The only way we can get a match to f 9 is to go to the three cycle per day aliases (Dn ˆ 34:72 mhz); the match in both frequency and amplitude is good. Note that there are ambiguities: three matches (one rather unlikely) can be found for f 2, for example. Given the poorer sampling and smaller amount of data in Paper VII, it is encouraging that so many matches can be made between the two sets of data. A further complication is that we know from observations of other EC stars that substantial amplitude changes can occur on a variety of time-scales (e.g. Papers III, IV, VI and VII), even to the extent that relatively strong frequencies can almost disappear over days or weeks. If this is happening in PG , then it is not surprising that we cannot match all or even most of the frequencies found in Paper VII. In this context, note that the amplitude of the main frequency, f 1, appears to have decreased by a factor of more than 2 between the Paper VII data, obtained in

9 A multi-site campaign on PG , and the current data from a year later. In contradistinction, f 5 has increased in amplitude by more than a factor of 2 and the other strong frequencies have changed little, if at all. If we accept that the four frequencies found close to f 1 are all real (see Tables 3 and 4) and if we assume that these were all added constructively to f 1 in the 1996 data and destructively in the 1997 data, then the effect falls far short (by a factor of about 3) of what is required to explain the difference between the two seasons. This reinforces the suggestion, made at the end of Section 4.3, either that we still have not resolved all frequencies ± even for the strongest frequency, f 1 ± or that some other process, such as selective amplitude modulation, is affecting the results. Given the effort, organizational, observational and analytical, which has been invested in this project, it is somewhat dispiriting to feel so distant from a `complete' solution. It is, however, dif cult to see how a substantially better job could be done without a signi cantly longer campaign. In the current ethos of small telescope neglect and closure, prospects for such campaigns are becoming poorer. 6 INTERPRETATION AND MODELS The frequencies in Tables 2±4 are, formally, those needed to reproduce the observations. The observations were, however, gathered with the intention of deciphering the pulsation frequencies of the star. It is important to bear in mind the implied distinction and to consider the question `to what extent can the tabulated frequencies be considered to represent the pulsation frequencies?'. By the term `pulsation frequencies' we mean the frequencies of the normal modes of oscillation of the star. It is clear that the data set needs to span a time suf cient to resolve all the normal modes. If this is not the case, then in place of, say, two frequencies corresponding to two normal modes, the analysis will produce a single frequency. Of equal concern is the prospect that, for whatever reason (perhaps non-adiabatic, non-linear effects), a normal mode of oscillation might have grown and decayed during the course of the campaign. The analysis will then yield two very closely spaced frequencies instead of the single, variable-amplitude frequency. We believe that the 39 frequencies listed in Table 2 are reliably identi able as normal modes of oscillation. They occur in both halves of the main campaign data and it is unlikely that the same set of frequencies would occur in independent subsets if some of these frequencies were only present to account for intrinsic amplitude variability. Unfortunately, little can be said about whether the additional frequencies in Tables 3 and 4 can be identi ed with normal modes, as they have been detected only once. PG is (so far) unique amongst the EC stars in that almost all of the pulsation periods exceed 300 s. Model calculations indicate that these longer periods are a result of the star being more evolved than the other EC stars and, with so many periods present, the advanced evolutionary status is necessary to explain them. Less evolved horizontal branch (HB) models have low-order g modes in this period range, but the density of modes is much lower than the observed period spectrum of PG More to the point, the position of PG in the Hertzsprung± Russell (HR) diagram indicates its more highly evolved status. To have log g ˆ 5:25 and T eff ˆ K requires that the star be well evolved off the core helium-burning HB. As a rst step towards modelling this star, we compare the observed pulsation periods with those of a post-hb model with parameters appropriate for PG This model has a metallicity of Z ˆ 0:01 and a mass of M (, and was evolved off the Table 6. Pulsation periods from the model (0.474 M (, Z ˆ 0:01) described in Section 6. Trapped modes are indicated by asterisks. Radial: n Period(s) log KE Non-radial:, Period(s) log KE C rot * * * * * * zero-age horizontal branch (ZAHB), with an initial helium core mass of M (. The model was extracted from an extensive grid of HB models computed for a forthcoming Paper exploring the pulsation properties of these stars (Kawaler, in preparation). Model periods in the range of the observed periods are listed in Table 6. Note that the periods are very densely packed. Because of its highly evolved (and centrally concentrated) state, the model shows modes of mixed character; g modes in the deep interior and p modes in the envelope. This dense mode structure is consistent with the large number of modes present in PG , even without considering possible rotational splitting. These are clearly very complex stars in terms of their pulsation parameters. Certain modes in the model are `trapped' modes [see for example Kawaler & Bradley (1994) for a discussion of trapped modes in GW Vir stars] and if we consider only these trapped modes, the spectrum becomes simpli ed. The models suggest that mode trapping by the hydrogen/ helium transition is what `selects' the large-amplitude modes and so trapped modes might correspond to the large-amplitude modes observed in PG At this stage, however, it is clearly not possible to identify individual modes in the star with those of this preliminary model. Table 6 lists periods from the model, along with the rotational splitting coef cient. Trapped modes (identi ed by local minima in the mode kinetic energy) are indicated by asterisks. Fig. 8 shows a schematic comparison of the observed and model frequencies. The general agreement in the range is good. Finally, we note that the evolution of HB stars in the HR diagram

10 534 D. Kilkenny et al. referee, Dr Don Winget, for useful comments. CK gratefully acknowledges the hospitality of the University of Texas as well as nancial support from the South African Foundation for Research Development (FRD) and the Whole Earth Telescope (WET) group at the University of Texas. KAL acknowledges receipt of a fellowship from the US Department of Education. The Jacobus Kapteyn Telescope is operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de AstrofõÂsica de Canarias. REFERENCES Figure 8. Comparison of the model pulsation frequencies from Table 6 with the observed frequencies from Table 3 (excluding the non-linear interactions ± the high-frequency sums of two lower frequencies, and the harmonic 2 f 1 ). The top line represents the periods of the observed pulsations with the ve largest amplitudes shown as extended lines (two, f 2 and f 3, are almost coincident near 475 s). The bottom line represents the model frequencies with radial modes shown as extended lines and `trapped' non-radial modes marked with asterisks. is slow in the vicinity of the ZAHB, and accelerates as the star exhausts its core helium. As a result of this, there should not be many `long-period' EC stars similar to PG because of the speed with which the stars evolve through this part of the HR diagram. The shorter-period EC stars are much closer to the ZAHB (if the models are to be believed at all), evolve much more slowly and have convergent evolutionary tracks for different masses. The current (small-number) statistics bear this out: there is only one `PG 1605' star, but there are an order of magnitude more `2±3 min' EC stars. Additionally, if PG is expected to be evolving more rapidly than the other EC stars, it is the best candidate to search for (evolutionary) changes in the pulsation frequencies over the next few seasons. The models indicate that the modes with periods near 480 s should have dp/dt, 1: ss 1 with a corresponding time-scale for period change of about 10 7 yr (Kawaler 1998). ACKNOWLEDGMENTS We thank the Directors and/or time allocation committees of the SAAO, McDonald, Mt Stromlo and Siding Spring, Mt John and La Palma Observatories for the generous allocations of telescope time which permitted this work to be carried out. We are grateful to the BilleÂres M., Fontaine G., Brassard P., Charpinet S., Liebert J., Saffer R., Vauclair G., 1997, ApJ, 487, L81 Charpinet S., Fontaine G., Brassard P., Chayer P., Rogers F. J., Iglesias C. A., Dorman B., 1997, ApJ, 483, L123 Deeming T. J., 1968, Vistas Astron, 10, 125 Deeming T. J., 1975, Ap&SS, 36, 137 Green R. F., Schmidt M., Liebert J., 1986, ApJS, 61, 305 Kawaler S. D., 1998, in Solheim, J.-E. ed., ASP Conf. Ser., The 11th European White Dwarf Workshop (in press). Kawaler S. D., Bradley P. A., 1994, ApJ, 427, 415 Kilkenny D., Koen C., O'Donoghue D., Stobie R. S., 1997, MNRAS, 285, 640 (Paper I) Kilkenny D., O'Donoghue D., Koen C., Lynas-Gray A. E., van Wyk F., 1998, MNRAS, 296, 329 (Paper VIII) Koen C., Kilkenny D., O'Donoghue D., van Wyk F., Stobie R. S., 1997, MNRAS, 285, 645 (Paper II) Koen C., O'Donoghue D., Kilkenny D., Lynas-Gray A. E., Marang F., van Wyk F., 1998, MNRAS, 296, 317 (Paper VII) Kurtz D. W., 1985, MNRAS, 213, 773 Loumos G. L., Deeming T. J., 1978, Ap&SS, 56, 285 Menzies J. W., Marang F., 1986, in Proc IAU Symp. 118, Hearnshaw J. B., Cottrell P. L., eds, Instrumentation and Programmes for Small Telescopes. Reidel, Dordrecht, p. 305 O'Donoghue D., Lynas-Gray A. E., Kilkenny D., Stobie R. S., Koen C., 1997, MNRAS, 285, 657 (Paper IV) O'Donoghue D. et al., 1998a, MNRAS, 296, 296 (Paper V) O'Donoghue D., Koen C., Lynas-Gray A. E., Kilkenny D., van Wyk F., 1998b, MNRAS, 296, 306 (Paper VI) Stobie R. S., Kawaler S. D., Kilkenny D., O'Donoghue D., Koen C., 1997, MNRAS, 285, 651 (Paper III) Winget et al., 1991, ApJ, 378, 326 Winget et al., 1994, ApJ, 430, 839 Wood J. H., Zhang E.-H., Robinson E. L., 1993, MNRAS, 261, 103 This paper has been typeset from a T E X=L A T E X le prepared by the author.