A&A 381, 122-150 (2002)
DOI: 10.1051/0004-6361:20011483
G. Vauclair1 - P. Moskalik2 - B. Pfeiffer1 - M. Chevreton3 - N. Dolez1 - B. Serre1 - S. J. Kleinman4,18 - M. Barstow5 - A. E. Sansom5 - J.-E. Solheim21 - J. A. Belmonte6 - S. D. Kawaler23 - S. O. Kepler7 - A. Kanaan7,22 - O. Giovannini7 - D. E. Winget4 - T. K. Watson4 - R. E. Nather4 - J. C. Clemens4,19 -
J. Provencal4,20 - J. S. Dixson4 - K. Yanagida4 - A. Nitta Kleinman4 - M. Montgomery4 - E. W. Klumpe4 - A. Bruvold21 - M. S. O'Brien23,24 - C. J. Hansen25 - A. D. Grauer8 - P. A. Bradley4,9,27 - M. A. Wood26,27 - N. Achilleos10 - S. Y. Jiang11 - J. N. Fu1,11 - T. M. K. Marar12 - B. N. Ashoka12 - E. G. Mei
tas13 - A. V. Chernyshev14 - T. Mazeh15 - E. Leibowitz15 - S. Hemar15 - J. Krzesinski16 -
G. Pajdosz16 - S. Zoa16,17
1 - Université Paul Sabatier, Observatoire Midi-Pyrénées,
CNRS/UMR5572,
14 Av. É. Belin, 31400 Toulouse, France
2 - Copernicus Astronomical Center,
Ul. Bartycka 18, 00-716 Warsaw, Poland
3 - Observatoire de Paris-Meudon, DAEC,
92195 Meudon, France
4 - Department of Astronomy and McDonald Observatory,
Texas University at Austin,
Austin, TX 78712, USA
5 - Department of Physics and Astronomy,
University of Leicester,
University Road, Leicester, LE1 7RH, UK
6 - Instituto de Astrofisica de Canarias,
38200 La Laguna, Tenerife, Spain
7 - Instituto de Física-UFRGS,
Av. B. Goncalves 9500,
91501-900 Porto-Alegre, RS, Brazil
8 - Department of Physics and Astronomy,
University of Arkansas at Little Rock,
Little Rock, AR 72204, USA
9 - Los Alamos National Laboratory,
X-2, MS B-220,
Los Alamos, NM 87545, USA
10 - Siding Spring Observatory,
and Department of Mathematics,
Australian National University,
Canberra, Australia
11 - Beijing Astronomical Observatory,
Chinese Academy of Sciences, 20A,
Datun Road, Beijing 100012, PR China
12 - Indian Space Research Organization,
Airport Road, Vimanapura PO,
Bangalore 560017, India
13 - Institute of Theoretical Physics and Astronomy,
Gostauto 12,
Vilnius 2600, Lithuania
14 - Astronomical Institute, Astronomicheskaya 33,
Tashkent 700052,
Uzbekistan
15 - Wise Observatory,
Tel Aviv University,
Tel Aviv 69978, Israel
16 - Mt. Suhora Observatory,
Cracow Pedagogical University,
Ul. Podchorazych 2, 30-084 Cracow, Poland
17 - Astronomical Observatory, Jagiellonian University,
ul. Orla 171,
30-244 Cracow, Poland
18 - Sloan Digital Sky Survey, Apache Pt. Observatory, PO Box 59,
Sunspot, NM 88349, USA
19 - Department of Physics, University of North Carolina, Chapel Hill,
NC 27599-3255, USA
20 - Department of Physics and Astronomy,
University of Delaware,
Newark, DE 19716, USA
21 - Department of Physics,
University of Tromso,
9037 Tromso, Norway
22 - Departamento de Física, Universidade Federal de Santa Catarina,
CP 476, CEP 88040-900, Florianópolis, Brazil
23 - Department of Physics and Astronomy,
Iowa State University, Ames, IA 50011, USA
24 - Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218, USA
25 - Joint Institute for Laboratory Astrophysics, University of Colorado,
Box 440, Boulder, CO 80309, USA
26 - Department of Physics and Space Sciences and SARA Observatory,
Florida Institute of Technology, Melbourne, FL 32901, USA
27 - Guest Observer, Institute for Astronomy, Honolulu, HI, USA
Received 17 July 2001 / Accepted 15 October 2001
Abstract
The pulsating PG 1159 planetary nebula central star RXJ&nb
sp;2117+3412 has
been observed over three successive seasons of a multisite photometric
campaign. The asteroseismological analysis of the data, based on
the 37 identified
modes among the
48 independent pulsation frequencies detected in the power spectrum, leads to the
derivation of the rotational splitting, the period spacing and the mode
trapping cycle and amplitude, from which a number of fundamental parameters can be
deduced.
The average
rotation period is
days. The trend
for the rotational splitting to decrease with
increasing periods is incompatible with a solid
body rotation. The total mass is 0.56
+0.02-0.04
and the
He-rich envelope mass fraction is in the range
0.013-0.078
M*.
The luminosity derived from asteroseismology is
log(
+0.23-0.32 and the distance
760
+230-235 pc. At such a distance, the linear size of the
planetary nebulae is
pc. The role of mass loss
on the excitation mechanism and its consequence on the amplitude
variations is discussed.
Asteroseismology is a powerful tool to explore the internal structure of stars and test the predictions of stellar evolution. Its application to the late stages of medium mass stellar evolution, i.e., to planetary nebulae nuclei and white dwarfs, has been particularly fruitful owing to the short period oscillations observed in these compact stars, allowing the accumulation of a large number of pulsation cycles over relatively short observation times. The organization of coordinated multisite observing campaigns was a breakthrough in the study of compact multiperiodic pulsators. In this respect, the Whole Earth Telescope network (WET, Nather et al. 1990), performing coordinated photometric campaigns, successfully contributed to this effort.
In these late stages of stellar evolution, stars have several opportunities
to become pulsationally unstable.
The first instability region is encountered during the high
luminosity planetary nebula phase, and nine variable planetary nebulae
nuclei (PNNV) are presently known (Ciardullo & Bond 1996).
The second
instability region is found among the pre-white dwarf stars of the PG 1159
spectral type, which are direct
descendants of a significant fraction of PNN. These stars have passed the turning
point in the H-R diagram, where planetary nebulae nuclei reach their
highest effective temperature and start cooling towards lower temperatures and
luminosities, as they begin contracting towards the white dwarf
cooling sequence. Six such pulsating PG 1159 stars, also known as
GW Vir variable stars, are presently known.
Five are plain PG 1159 type stars that show no hydrogen in their spectra:
PG 1159-035 (McGraw et al. 1979;
Winget et al. 1991),
PG 2131+066 (Bond et al. 1984; Kawaler et al. 1995),
PG 1707+427 (Bond et al. 1984; Fontaine et al. 1991;
Grauer et al. 1992),
PG 0122+200 (Bond & Grauer 1987; Vauclair et al. 1995;
O'Brien et al.
1996, 1998; Vauclair et al. 2001)
and RXJ 2117+3412, the subject of this paper.
The sixth object, HS 2324+3944 (Silvotti
1996; Silvotti et al. 1999),
is a "hybrid'' PG 1159 type star, which has hydrogen in its spectrum.
Both the PNNV and the PG 1159 instability
strips are not "pure'' instability strips, i.e., both non-variable and variable
stars are present in the same temperature and luminosity range. We still do not
understand why stars of the same effective
temperature and surface gravity, have some that pulsate, while others do not. In the
case of the PG 1159 type stars, the only known distinction comes
from spectroscopy: pulsating PG 1159 stars show nitrogen in their spectra while
non-pulsating PG 1159 stars do not (Dreizler 1998; Dreizler & Heber
1998).
However, there is the noticeable and puzzling
exception of PG 1144+005, which shows N in its spectrum at the same level
as the pulsating PG 1159 stars, but was not found to pulsate (Grauer et al. 1987).
Observatory | Location | Telescopes | (m) | |
1992 | 1993 | 1994 | ||
OMP | Pic du Midi, France | 2.0 | ||
Teide | Tenerife, Canary Islands | 1.5 | 0.8 | |
Roque de los Muchachos | La Palma, Canary Islands | 2.5 | 1.0 | |
LNA | Itajuba, Brazil | 1.6 | ||
McDonald | Mount Locke, Texas | 2.0 | 0.9 | 2.0, 2.6 |
Steward Observatory | Mount Bigelow, Arizona | 1.5 | ||
Steward Observatory | Mount Lemmon, Arizona | 1.5 | ||
U. of Hawaii | Mauna Kea, Hawaii | 0.6 | 0.6 | |
Siding Spring | Siding Spring Mt., Australia | 1.0 | ||
Beijing Observatory | Xinglong, China | 2.1 | ||
Vainu Bappu | Kavalur, India | 2.2 | 1.0 | |
Maidanak | Maidanak, Uzbekistan | 1.0 | 1.0 | |
Wise | Mount Ramon, Israel | 1.0 | ||
Suhora | Mount Suhora, Poland | 0.6 | 0.6 | 0.6 |
For completeness, we note that there are
two more white dwarf instability strips.
They are the pulsating helium atmosphere
white dwarfs (8 DBVs known) and the pulsating hydrogen atmosphere white dwarfs
(31
DAVs or ZZ Cetis known).
The DBV instability strip is not a "pure'' instability strip either. The
fraction of non-variable stars found within the instability strip varies
between 25% and
50% depending whether their
atmospheric parameters are derived from pure He model atmospheres or from
model atmospheres allowing for a small admixture of undetectable
hydrogen (Beauchamp et al. 1999).
In contrast with the PNNV, PG 1159 and DBV instability strips, the DAVs
form a "pure'' instability strip, i.e., no stable stars are found within
the domain of the HR diagram (or equivalently in the
-
diagram) where the DAVs are located, once the mass dependence
of the blue edge of the instability strip is properly accounted for (Kepler et al.
2000). The list and the properties of the variable planetary nebulae nuclei,
variable PG 1159 type stars, DBVs and DAVs are summarized in Bradley (2000).
The variable stars in the pre-white dwarf evolutionary stage and on the white dwarf cooling sequence are non-radial gravity mode pulsators. This is unambiguously demonstrated for the two ZZ Cetis, R 548 (Robinson et al. 1982) and G 117-B15A (Kepler 1984). In the framework of the linear pulsation theory, it has been possible to extract fundamental stellar parameters for most of the pulsators in the PG 1159 instability strip: the total mass from the period spacing, the rotational period from the frequency splitting, the depth of the chemical composition transition zone between the helium-rich outer layer and the carbon-oxygen core etc. As a by-product of the asteroseismological analysis, the luminosity and distance can be derived for each star (Winget et al. 1994). In the case of the PNNV, few have been studied with the same scrutiny, because observing them requires CCD photometry campaigns to remove the surrounding nebula. In addition, the PNNV mode amplitudes vary on short time scales (days to weeks), making the mode identification difficult. However, the best studied case, NGC 1501 (Bond et al. 1996) shows many similarities with the GW Vir stars. The evolutionary link between the PNN and the PG 1159 stars is now well established and the stellar parameters deduced from asteroseismology of PNNV and GW Vir stars provide further confirmation.
The discovery that RXJ 2117+3412, an X-ray source detected in the ROSAT
sky survey, is a member of the PG 1159 spectral class (Motch et al. 1993)
is an additional
evidence of an evolutionary link between the PNN and the white dwarfs.
The low surface brightness planetary nebula surrounding RXJ 2117+3412
was discovered
to be the largest planetary nebula known
(Appleton et al. 1993). The nebula has an angular diameter of 13 arcmin, and at an estimated
distance of 1.4 kpc (Motch et al. 1993), its linear extent
should be about 5.3 pc. Furthermore, the complex
structure of the nebula, which shows many thin filaments, is reminiscent of
the structure predicted for the shock produced when a
"superwind'' generated by the hot central star collides with
the material ejected at
the end of the previous AGB phase (Appleton et al. 1993).
|
Telescope | Date | Start Time | Run Length |
(UT) | (UTC) | (s) | ||
|
Suhora 60 cm | 22 September 92 | 01:07:15 | 4155 |
jesem-05 | Maidanak 1 m | 22 September 92 | 16:00:00 | 19930 |
x-8008 | Suhora 60 cm | 22 September 92 | 23:32:30 | 9550 |
int-0014 | Isaak Newton 2.5 m | 24 September 92 | 01:05:40 | 5920 |
ro-021 | Itajuba 1.6 m | 24 September 92 | 01:23:40 | 5825 |
pab-0147 | McDonald 82
![]() |
24 September 92 | 07:15:30 | 7620 |
jesem-06 | Maidanak 1 m | 24 September 92 | 14:49:50 | 22765 |
x-8011 | Suhora 60 cm | 25 September 92 | 01:05:10 | 4685 |
int-0017 | Isaak Newton 2.5 m | 25 September 92 | 20:50:10 | 18695 |
x-8013 | Suhora 60 cm | 25 September 92 | 23:53:20 | 6670 |
pab-0156 | McDonald 82
![]() |
26 September 92 | 01:46:00 | 14135 |
jesem-10 | Maidanak 1 m | 26 September 92 | 15:22:30 | 17945 |
x-8016 | Suhora 60 cm | 27 September 92 | 00:38:00 | 7480 |
int-0019 | Isaak Newton 2.5 m | 27 September 92 | 02:00:30 | 6025 |
pab-0160 | McDonald 82
![]() |
27 September 92 | 08:10:30 | 5260 |
maw-0107 | Mauna Kea 24
![]() |
27 September 92 | 10:35:20 | 5845 |
x-8018 | Suhora 60 cm | 27 September 92 | 21:26:45 | 17025 |
int-0022 | Isaak Newton 2.5 m | 28 September 92 | 01:27:40 | 7765 |
pab-0163 | McDonald 82
![]() |
28 September 92 | 06:54:30 | 9915 |
x-8019 | Suhora 60 cm | 28 September 92 | 18:27:35 | 2095 |
pab-0166 | McDonald 82
![]() |
29 September 92 | 07:39:00 | 6975 |
ro-023 | Itajuba 1.6 m | 29 September 92 | 22:15:50 | 16245 |
pab-0168 | McDonald 82
![]() |
30 September 92 | 01:40:30 | 28170 |
maw-0111 | Mauna Kea 24
![]() |
30 September 92 | 05:30:00 | 24085 |
sjk-0208 | Siding Spring 40
![]() |
30 September 92 | 12:02:00 | 8240 |
ro-025 | Itajuba 1.6 m | 30 September 92 | 23:51:40 | 8410 |
pab-0171 | McDonald 82
![]() |
1 October 92 | 01:50:00 | 27345 |
maw-0114 | Mauna Kea 24
![]() |
1 October 92 | 08:36:20 | 14360 |
x-8020 | Suhora 60 cm | 1 October 92 | 20:29:00 | 9780 |
pab-0173 | McDonald 82
![]() |
2 October 92 | 01:50:00 | 26910 |
maw-0118 | Mauna Kea 24
![]() |
3 October 92 | 06:02:20 | 15230 |
|
Telescope | Date | Start Time | Run Length |
(UT) | (UTC) | (s) | ||
|
TCS 1.5 m | 14 September 93 | 21:26:00 | 15420 |
rx-0915 | TCS 1.5 m | 15 September 93 | 20:46:00 | 20670 |
suh-0001 | Suhora 60 cm | 15 September 93 | 21:19:40 | 5260 |
a-402 | Mt. Bigelow 61
![]() |
16 September 93 | 03:51:00 | 19310 |
k93-0214 | Kavalur 90
![]() |
16 September 93 | 18:14:00 | 8325 |
rx-0916 | TCS 1.5 m | 16 September 93 | 21:29:00 | 14660 |
rx-0917 | TCS 1.5 m | 17 September 93 | 20:13:00 | 21690 |
a-404 | Mt. Bigelow 61
![]() |
18 September 93 | 02:36:00 | 21430 |
k93-0215 | Kavalur 90
![]() |
18 September 93 | 14:16:10 | 18390 |
suh-0002 | Suhora 60 cm | 18 September 93 | 18:47:00 | 5890 |
rx-0918 | TCS 1.5 m | 18 September 93 | 20:19:00 | 21910 |
a-405 | Mt. Bigelow 61
![]() |
19 September 93 | 02:46:00 | 22500 |
suh-0003 | Suhora 60 cm | 19 September 93 | 18:16:00 | 30210 |
rx-0919 | TCS 1.5 m | 19 September 93 | 20:48:00 | 19100 |
a-407 | Mt. Bigelow 61
![]() |
20 September 93 | 02:39:00 | 28930 |
ra-288 | McDonald 36
![]() |
20 September 93 | 03:34:40 | 18120 |
suh-0004 | Suhora 60 cm | 20 September 93 | 19:17:30 | 23685 |
rx-0920 | TCS 1.5 m | 20 September 93 | 20:27:00 | 20390 |
a-408 | Mt. Bigelow 61
![]() |
21 September 93 | 02:29:00 | 28460 |
a-409 | Mt. Lemmon 60
![]() |
22 September 93 | 02:43:00 | 27070 |
suh-0005 | Suhora 60 cm | 22 September 93 | 19:16:00 | 21825 |
rx-0922 | TCS 1.5 m | 22 September 93 | 20:37:00 | 19530 |
a-410 | Mt. Lemmon 60
![]() |
23 September 93 | 02:37:00 | 27300 |
suh-0006 | Suhora 60 cm | 23 September 93 | 18:33:00 | 20515 |
rx-0923a | TCS 1.5 m | 23 September 93 | 20:21:00 | 4620 |
rx-0923b | TCS 1.5 m | 23 September 93 | 22:51:00 | 4720 |
The subsequent analysis of a HST high resolution spectrum of
RXJ 2117+3412, using NLTE model atmosphere, indicates that it is
the hottest known PG 1159 type star with
K,
+0.3-0.2, and abundance ratios typical of other PG 1159 stars:
/23.8/6.2 (by numbers) (Werner et al. 1996;
Rauch & Werner 1997). The HST spectrum also
shows evidence of ongoing mass loss from the central star. The mass loss
is confirmed by more recent observations; it is estimated to be of the order of
=
from C IV line (Koesterke
et al.
1998), or
from O VI line
(Koesterke &
Werner 1998), with a terminal velocity of 3500 km s-1.
Because of the association of a planetary nebula with a PG 1159-type central star,
and because it is
close to the point in the HR diagram where high luminosity PNN turn to
lower effective temperature and luminosity to join the white dwarf
cooling sequence (Dreizler & Heber 1998, see their Fig. 8),
RXJ 2117+3412 is presently the best example of a PNN on its way to the
white dwarf sequence.
Shortly after RXJ 2117+3412 was announced as a new PG 1159 type star, photometric observations were performed to determine whether it is a pulsator. Watson (1992) and Vauclair et al. (1993) independently discovered that RXJ 2117+3412 is pulsating. This opened the opportunity to investigate the internal structure and evolutionary status of this unique object.
This paper presents the results of an asteroseismological study of RXJ 2117+3412. The observational campaigns, which cover the 1992, 1993 and 1994 seasons are described in Sect. 2. Section 3 gives the analysis of the power spectra. The various stellar parameters derived for RXJ 2117+3412 from this analysis are discussed in Sect. 4. Section 5 summarizes the results and suggests some ideas for future work.
The observations described in this paper result from multisite rapid photometry campaigns organized on three consecutive seasons in 1992, 1993 and 1994. The participating sites are listed in Table 1.
The data have been obtained with 2-channel or 3-channel photometers all equipped with blue sensitive photomultipliers (Hamamatsu R647-04 or similar) and used without a filter (white light). These instruments fulfill the specifications and requirements as prescribed by Kleinman et al. (1996). The sampling time was either 5 s or 10 s. In the former case, the data were coadded to 10 s afterwards. For 2-channel photometers, the observing procedure consists of simultaneously monitoring the target star in one channel and a comparison star in the second channel. The sky background is measured at random time intervals in both channels. For 3-channel photometers, the sky background is continuously monitored by the third channel, with the target and comparison stars placed in the other two channels.
After the announcement that RXJ 2117+3412 was a PG 1159 type star
(Werner 1993; Motch et al. 1993), the star
was immediately tested for photometric variability. It was found to be variable
by Watson (1992) and
Vauclair et al. (1993) independently. The data set
obtained at the 2.5 m NOT is described in Vauclair et al. (1993).
It was obtained with the Chevreton three-channel photometer - a
short description is
given in Vauclair et al. (1989).
The data consist
of 28 hr of time-series photometry accumulated during 4 consecutive nights,
and allowed to extract 27 peaks in the
power spectrum. The largest amplitude mode was found at 1217.8 Hz
(821 s period)
with a 4.6 mma amplitude, after re-reduction of the data.
The frequency resolution of these single-site
discovery data was only 2.7
Hz.
Figure 1 shows the power spectrum of the light curve, re-reduced for the
present paper.
![]() |
Figure 2: Normalized light curve of RX J2117+3412 during the 1992 WET (XCOV8) campaign. The modulation intensity is plotted as a function of time (UT). Each panel corresponds to one day. |
A WET campaign had been planned for September 1992, shortly after
the discovery of the variability of RXJ 2117+3412. The star was its
third priority target. This campaign obtained 78 hr
of non-redundant data. This WET campaign
will be referred to as 1992 WET (or also XCOV8) in the following discussion.
The observing sites involved are listed in Table 1.
The total duration of
the campaign was 10.7 days, with a corresponding frequency resolution
in the Fourier transform of 1.1 Hz.
The coverage of the
1992 WET for RXJ 2117+3412 was 35%, a rather satisfactory coverage for a third
priority target. The observation log is given in Table 2, and
Fig. 2 shows the normalized light curve of the 1992 WET data.
A multisite campaign was organized, independently of the WET network, one year
after the 1992 WET campaign.
105 hr of fast
photometry were obtained during 9.1 days. The coverage was 48%
and the frequency resolution achieved 1.3 Hz.
The sites involved are listed in Table 1 and the observation log is
given in Table 3.
Surprisingly, the average amplitude of the pulsations observed during this campaign was much smaller than one year earlier. The normalized light curve of this campaign is shown in Fig. 3. Note that the vertical scale of Fig. 3 is the same as in Fig. 2.
![]() |
Figure 3: Normalized light curve of RXJ 2117+3412 during the September 1993 campaign. The modulation intensity is plotted as a function of time (UT). Each panel corresponds to one day. |
![]() |
Figure 4: Normalized light curve of RXJ 2117+3412 during the 1994 WET (XCOV 11) campaign. The modulation intensity is plotted as a function of time (UT). Each panel corresponds to one day. |
RXJ 2117+3412 was the first priority target of the 1994 WET campaign. The observing sites involved are listed in Table 1 and the observation log is given in Table 4.
The campaign
had a total duration of 15.0 days, of which only 13.8 days are used in the
forthcoming reduction, implying a frequency resolution of
0.8 Hz
in the power spectrum.
175 hr
of non-redundant data were obtained, leading to a coverage of 49%. The light
curve, shown in Fig. 4, looks quite different from those of
the two previous campaigns.
Here, the largest amplitude mode is at a frequency of
958.5
Hz (1043 s period).
This campaign is referred to as 1994 WET (or also XCOV11) in the following text.
In addition to the data listed in Table 4, which were obtained with photomultiplier-based photometers, some CCD photometry data have been acquired at the Teide Observatory IAC 0.80 m telescope during part on the 1994 WET on three consecutive nights: 1994 August 9-11. Images of the RXJ 2117+3412 field were taken every 250 s, on average, with an exposure time of 150 s. Because of the different sampling time, the CCD data are reduced separately and are not included in the calculation of the power spectrum. They are useful for a comparison of the CCD photometry with photomultiplier photometry. Table 5 is the log of the CCD photometry observations.
|
Telescope | Date | Start Time | Run Length |
(UT) | (UTC) | (s) | ||
|
Maidanak 1 m | 31 July 94 | 18:34:40 | 9750 |
emcav-04 | Maidanak 1 m | 1 August 94 | 16:56:40 | 10170 |
suh-0015 | Suhora 60 cm | 1 August 94 | 23:36:50 | 6010 |
gv-0414 | TBL 2 m | 2 August 94 | 00:54:00 | 8530 |
emcav-05 | Maidanak 1 m | 2 August 94 | 16:18:20 | 25010 |
suh-0016 | Suhora 60 cm | 2 August 94 | 22:20:30 | 12600 |
gv-0416 | TBL 2 m | 2 August 94 | 22:36:00 | 16300 |
sjk-0374 | JKT 1 m | 3 August 94 | 01:49:30 | 2990 |
sjk-0375 | JKT 1 m | 3 August 94 | 03:03:30 | 10310 |
emcav-06 | Maidanak 1 m | 3 August 94 | 16:14:00 | 24240 |
sjk-0376 | JKT 1 m | 3 August 94 | 21:11:30 | 22900 |
gv-0418 | TBL 2 m | 3 August 94 | 22:34:00 | 17340 |
sjk-0377 | JKT 1 m | 4 August 94 | 03:37:00 | 7640 |
pab-0179 | Mauna Kea 24
![]() |
4 August 94 | 09:29:10 | 19260 |
emcav-07 | Maidanak 1 m | 4 August 94 | 16:39:40 | 23820 |
suh-0017 | Suhora 60 cm | 4 August 94 | 20:16:30 | 18690 |
sjk-0378 | JKT 1 m | 4 August 94 | 21:33:00 | 17700 |
gv-0420 | TBL 2 m | 5 August 94 | 00:10:00 | 11570 |
sjk-0379 | JKT 1 m | 5 August 94 | 02:30:30 | 12290 |
pab-0182 | Mauna Kea 24
![]() |
5 August 94 | 07:12:00 | 27060 |
emcav-08 | Maidanak 1 m | 5 August 94 | 17:35:30 | 3530 |
emcav-10 | Maidanak 1 m | 5 August 94 | 18:38:30 | 5700 |
gv-0422 | TBL 2 m | 5 August 94 | 20:55:00 | 22930 |
suh-0018 | Suhora 60 cm | 5 August 94 | 21:07:00 | 16250 |
pab-0183 | Mauna Kea 24
![]() |
6 August 94 | 06:22:30 | 24330 |
pab-0184 | Mauna Kea 24
![]() |
6 August 94 | 13:15:30 | 5730 |
sjk-0380 | JKT 1 m | 6 August 94 | 21:07:30 | 6730 |
gv-0424 | TBL 2 m | 6 August 94 | 22:47:00 | 7400 |
sjk-0381 | JKT 1 m | 6 August 94 | 23:00:30 | 5060 |
|
Telescope | Date | Start Time | Run Length |
(UT) | (UTC) | (s) | ||
|
Mauna Kea 24
![]() |
7 August 94 | 06:15:00 | 30920 |
k44-0259 | Kavalur 40
![]() |
7 August 94 | 14:08:00 | 3220 |
k44-0260 | Kavalur 40
![]() |
7 August 94 | 15:17:40 | 23390 |
emcav-11 | Maidanak 1 m | 7 August 94 | 16:04:10 | 25870 |
sjk-0382 | JKT 1 m | 7 August 94 | 21:36:30 | 23050 |
gv-0445 | TBL 2 m | 7 August 94 | 22:40:00 | 16700 |
ra-340 | McDonald 82
![]() |
8 August 94 | 06:01:30 | 15200 |
pab-0186 | Mauna Kea 24
![]() |
8 August 94 | 06:30:10 | 23930 |
pab-0187 | Mauna Kea 24
![]() |
8 August 94 | 13:42:00 | 1770 |
emcav-12 | Maidanak 1 m | 8 August 94 | 16:01:40 | 26060 |
suh-0019 | Suhora 60 cm | 8 August 94 | 19:45:30 | 1940 |
sjk-0383 | JKT 1 m | 8 August 94 | 21:03:30 | 32000 |
gv-0426 | TBL 2 m | 8 August 94 | 22:07:00 | 18500 |
ra-341 | McDonald 82
![]() |
9 August 94 | 02:53:10 | 21410 |
pab-0188 | Mauna Kea 24
![]() |
9 August 94 | 06:38:10 | 29840 |
sh-0000 | Wise 40
![]() |
9 August 94 | 18:40:40 | 14690 |
suh-0020 | Suhora 60 cm | 9 August 94 | 23:26:00 | 9810 |
ra-342 | McDonald 82
![]() |
10 August 94 | 02:54:50 | 30450 |
pab-0189 | Mauna Kea 24
![]() |
10 August 94 | 06:16:30 | 31140 |
sh-0003 | Wise 40
![]() |
10 August 94 | 18:34:10 | 3650 |
sh-0004 | Wise 40
![]() |
10 August 94 | 20:02:00 | 2240 |
sh-0006 | Wise 40
![]() |
10 August 94 | 21:48:20 | 14860 |
ra-343 | McDonald 82
![]() |
11 August 94 | 05:30:20 | 21010 |
pab-0190 | Mauna Kea 24
![]() |
11 August 94 | 06:14:30 | 31300 |
sh-0007 | Wise 40
![]() |
11 August 94 | 18:13:30 | 24950 |
ra-344 | McDonald 82
![]() |
12 August 94 | 02:48:40 | 30740 |
ra-345 | McDonald 107
![]() |
13 August 94 | 05:05:50 | 6310 |
pab-0191 | Mauna Kea 24
![]() |
13 August 94 | 07:18:00 | 13490 |
ra-346 | McDonald 107
![]() |
13 August 94 | 09:00:50 | 8160 |
ra-347 | McDonald 107
![]() |
14 August 94 | 03:43:30 | 11760 |
ra-348 | McDonald 107
![]() |
14 August 94 | 07:04:10 | 2330 |
ra-349 | McDonald 107
![]() |
14 August 94 | 08:12:30 | 11620 |
gv-0472 | Xinglong 2.16 m | 14 August 94 | 12:31:00 | 25850 |
gv-0474 | Xinglong 2.16 m | 15 August 94 | 12:44:10 | 25890 |
The photomultiplier photometer data have been reduced in a now standard way (Nather et al. 1990; Kepler 1993). In both 2- and 3-channel photometers, the sky background is measured at the beginning and at the end of each run in all channels. This is used to determine the sensitivity ratios of the channels. In 3-channel photometric data, the sky background is monitored continuously in one channel, allowing for point by point subtraction of the sky background from the target and comparison star channels, after application of the proper sensitivity ratios. For 2-channel data, the sky background is normally measured at irregular intervals in both channels. The sky background is then constructed by polynomial interpolation. Each star channel is then corrected for extinction and normalized. When conditions show evidence for transparency variations, the normalized target star channel counts are divided by the smoothed comparison star channel counts. Subtracting unity from the resulting time series gives the time series on which the barycentric correction to the time base is applied.
Each of the observing campaigns has been reduced shortly after the
observations. For the purpose of the present paper however,
all the data have been
re-reduced in an homogeneous way. A few runs have been rejected where
the noise level was too high, which was usually due to clouds or instrumental problems.
In case of overlapping
data, we kept the best signal/noise ratio run in our analysis.
The power spectrum of each time series is obtained by a Fourier
Transform.
A non-linear least-squares fitting routine (which fits the
frequencies, amplitudes and phases of sine waves to the time series),
followed by prewhitening, is used to extract the significant modes from
the power spectra. The discovery
data obtained in August 1992 at the NOT, were also re-reduced for comparison with
the original reduction, although these data were not used in the present paper
because they are single site
data with too poor a frequency resolution.
The power spectrum of these data is
shown in Fig. 1. The power spectra of the time series obtained during
the 1992 WET, the 1993 multisite campaign and the 1994 WET runs are shown in
Figs. 5-7 respectively.
Figure 8 illustrates the prewhitening sequence on a portion of the 1994 WET power
spectrum.
|
Start time | Run length* |
(UT) | (UTC) | (s) |
|
22:33:42 | 22250 |
August 10, 1994 | 22:57:12 | 24500 |
August 11, 1994 | 23:09:39 | 24750 |
Power is seen without ambiguity in the range 650 Hz-4340
Hz.
Most of the peaks with significant power are found in the restricted
range 650
Hz-1600
Hz.
For each observing season,
Table 6 lists the frequencies f, with their uncertainties
f (in
Hz)
and the
amplitudes A (in mma) of the peaks considered significant in the power spectra.
These values are derived by the non-linear least squares fit.
To decide whether a peak in a power
spectrum is significant on an objective basis, the following rules were
applied: a False Alarm Probability (FAP) (Kepler 1993) was estimated on the
1000
Hz
frequency range embedding most of the significant power.
All peaks with a
10-3 were considered
as significant. Several peaks with
FAP >10-3were also included in the list
(marked with colons in Table 6), but only if: i) they fit the
pattern of
rotationally split multiplets,
or ii) they fit
the period spacing (
)
distribution, or iii) they have
frequency equal to, or close enough to, the frequency of a significant peak
observed in other seasons.
CCD photometry obtained at the IAC 0.80 m telescope has been
reduced independently. The images were taken without a filter. The basic
reductions (bias subtraction and flatfield corrections) were made by use
of the IRAF
package. The photometric reductions were done using the MOMF
package (Kjeldsen & Frandsen 1992). The resulting time series was analyzed
by a non-linear least-squares fit. Frequencies extracted from this data
set are listed in Table 7.
![]() |
Figure 5: Power spectrum of the 1992 WET (XCOV 8) light curve. The units are the same as in Fig. 1. Note the different vertical scale on each panel. The window function is shown at the same frequency scale in the insert. The prominent sidelobes in the window function correspond to the 1 and 2 day aliases. |
![]() |
Figure 6: Power spectrum of the 1993 light curve. The units are the same as in Fig. 1. Note the different vertical scale on each panel. The window function is shown at the same frequency scale in the insert. The prominent sidelobes in the window function correspond to the 1 day alias. |
![]() |
Figure 7: Power spectrum of the 1994 WET (XCOV 11) light curve. The units are the same as in Fig. 1. Note the different vertical scale on each panel. The window function is shown at the same frequency scale in the insert. Note the small amplitude of the 1 day alias sidelobes. |
![]() |
Figure 8:
Illustration of the prewhitening sequence. This figure is an enlarged
part of the power spectrum of the 1994 WET shown in Fig. 7, restricted to the frequency
range 750-1250 ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
|
1994 | 1993 | 1992 | |||||
WET | WET | |||||||
f | ![]() |
A | f | ![]() |
A | f | ![]() |
A |
653.987 | 0.018 | 0.90 | 653.811 | 0.029 | 1.46 | |||
655.556 | 0.031 | 1.33 | ||||||
666.938 | 0.028 | 0.57: | ||||||
706.260 | 0.039 | 0.98: | ||||||
717.714 |
0.008 | 1.96 | ||||||
789.042 |
0.022 | 0.71 | ||||||
793.783 |
0.019 | 0.84 | ||||||
830.708 |
0.006 | 2.68 | 831.412 | 0.022 | 1.78 | |||
836.067 |
0.022 | 0.74 | 835.000 | 0.040 | 0.66 | |||
840.367 |
0.040 | 0.43: | ||||||
851.483 | 0.041 | 0.64 | ||||||
872.337 | 0.029 | 0.55 | ||||||
889.587 | 0.033 | 1.23 | ||||||
894.800 | 0.019 | 2.16 | ||||||
906.378 |
0.016 | 1.03 | ||||||
921.721 |
0.031 | 0.53 | ||||||
940.563 |
0.020 | 0.82 | 939.838 | 0.063 | 0.43: | 939.948 | 0.051 | 0.89 |
945.156 | 0.040 | 1.05 | ||||||
949.909 |
0.022 | 0.83 | 950.445 | 0.022 | 2.07 | |||
958.533 |
0.005 | 3.64 | 957.959 | 0.042 | 0.65 | |||
963.282 | 0.025 | 0.67 | 963.416 | 0.031 | 0.86 | |||
978.874 |
0.032 | 0.52 | ||||||
988.726 |
0.038 | 0.44: | ||||||
1005.645 |
0.033 | 0.49 | ||||||
1010.541 |
0.032 | 0.52 | ||||||
1023.684 |
0.037 | 0.45 | 1023.292 | 0.029 | 0.90 | 1023.594 | 0.020 | 2.03 |
1045.944 |
0.050 | 0.35: | 1044.904 | 0.072 | 0.36: | 1045.690 | 0.045 | 0.90 |
1055.703 |
0.042 | 0.41 | ||||||
1096.712 |
0.016 | 1.05 | 1096.060 | 0.055 | 0.52 | |||
1101.942 |
0.039 | 0.41 | 1101.203 | 0.065 | 0.67: | |||
1107.300 | 0.063 | 0.46 | 1107.224 | 0.027 | 1.61 | |||
1123.747 |
0.049 | 0.32: | ||||||
1179.955 |
0.007 | 2.16 | 1179.761 | 0.061 | 0.43 | 1179.893 | 0.029 | 1.43 |
1190.578 | 0.054 | 0.79: |
|
1994 | 1993 | 1992 | |||||
WET | WET | |||||||
f | ![]() |
A | f | ![]() |
A | f | ![]() |
A |
|
0.016 | 0.95 | 1212.419 | 0.047 | 0.58 | |||
1217.812 |
0.012 | 1.38 | 1217.886 | 0.022 | 1.21 | 1217.865 | 0.010 | 4.07 |
1245.457 | 0.059 | 0.46 | ||||||
1289.129 |
0.035 | 0.45 | 1289.136 | 0.060 | 0.45 | 1289.160 | 0.015 | 2.63 |
1315.055 |
0.012 | 1.24 | 1315.032 | 0.021 | 1.26 | 1315.181 | 0.026 | 1.55 |
1362.734 |
0.056 | 0.29: | 1362.495 | 0.074 | 0.36 | |||
1397.385 |
0.057 | 0.28: | 1397.242 | 0.061 | 0.45 | |||
1439.198 |
0.025 | 0.62 | ||||||
1539.991 |
0.055 | 0.29: | ||||||
1548.653 |
0.043 | 0.37 | ||||||
1549.959 | 0.050 | 0.75 | ||||||
1572.012 | 0.094 | 0.27 | ||||||
1947.334 |
0.068 | 0.26: | ||||||
1956.008 |
0.027 | 0.58 | 1956.785 | 0.103 | 0.29 | |||
1968.952 |
0.023 | 0.67 | 1968.222 | 0.085 | 0.35 | 1968.915 | 0.047 | 0.84 |
2109.129 | 0.049 | 0.76 | ||||||
2133.259 |
0.064 | 0.25: | 2133.122 | 0.055 | 0.47 | |||
2143.374 | 0.124 | 0.21: | ||||||
2153.980 | 0.079 | 0.33 | ||||||
2164.116 | 0.122 | 0.21: | ||||||
2174.884 | 0.076 | 0.35 | ||||||
2184.777 | 0.110 | 0.23: | ||||||
2402.113 | 0.098 | 0.26 | ||||||
3408.257 | 0.046 | 0.35 | ||||||
3517.490 |
0.049 | 0.30 | ||||||
3924.971 | 0.062 | 0.27 | ||||||
4077.942 | 0.100 | 0.41 | ||||||
4308.046 | 0.074 | 0.35 | ||||||
4339.147 | 0.132 | 0.20: |
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 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
Hz - maintained an almost constant amplitude over the two year period covered by
our observations.
This mode
may be useful for determining
,
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
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).
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.
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 Hz. This is uncomfortably
close to one half of the one day alias (5.8
Hz).
If we examine the window function in 1994, there are no peaks present in the range of 5 to 6
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
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
Hz, and no triplets or quintuplets,
which would be the clear signatures of
or
modes split by rotation.
One only sees several cases of
two modes separated by about 5
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 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
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.
|
![]() ![]() |
A (mma) |
|
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 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
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 Hz present only in the
1994 WET data. They seem to form a doublet with a frequency separation
of 4.741
Hz, possibly due to rotational
splitting. If this were the case, they would be
modes
with
m=1. However, as we
will discuss later, this is not a single, rotationally split mode.
The next mode at 830.708 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
Hz and with
a smaller amplitude.
The neighboring two modes at 836.067 and 840.367
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
Hz mode.
The 836.067
Hz mode is also present in the 1993 data, but with the frequency shifted to
835.000
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,
Hz measured in the
1994 WET spectrum
suggests that these two peaks are two components of a
triplet,
with
m=1. As we have no explanation for the observed frequency shift of the 830 and
836
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 Hz (seen only in the 1993 run) and at
872.337
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
Hz and
894.800
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
Hz.
We interpret this doublet as two components of an
mode split
by slow rotation, with the third component missing.
The two next peaks at 906.378 Hz and 921.721
Hz, seen only in 1994
WET data, are single peaks.
The next three peaks at 940.563, 945.156, and 949.909
Hz
form the first identified triplet,
suggesting an
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
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
Hz from m=-1 to m=0 and 4.753
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
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 Hz and 963.282
Hz form a doublet
separated by 4.749
Hz.
The 958
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
triplet.
The CCD data also show a mode at 958.45
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 Hz and
988.726
Hz, form a doublet separated by 9.852
Hz or
2
4.926
Hz.
We interpret these peaks as the m=-1, +1 components of a triplet (
)
whose missing central (m=0) component should be near 983.8
Hz.
The next two peaks at 1005.645 and 1010.541 Hz, form another
doublet seen only in the 1994 WET data. The frequency separation is 4.896
Hz.
The doublet is interpreted as two adjacent components (
m = 1) of
an
mode split by rotation.
The next peak at 1023.594 Hz is seen in the three runs, with its largest
amplitude occurring in the 1992 WET run.
The next two peaks at 1045.690
Hz and 1055.703
Hz, separated by
10.013
Hz or 2
5.006
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
Hz. The 1046
Hz
component is seen in all three runs, though only marginally in 1993,
while the 1056
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 Hz.
The 1097
Hz component is seen in the 1994 WET and the 1993 data
(though significantly displaced by 0.65
Hz to
1096.060
Hz in 1993). The central component at 1101.942
Hz is
seen in the 1994 WET data,
as well as in the 1992 WET data (where it is displaced by 0.73
Hz),
but it is absent in 1993. The third
component at 1107.223
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
Hz and 5.282
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
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
Hz, while the average
splitting in 1993 is 5.620
Hz, and it decreases further to 5.230
Hz in 1994.
The following mode at 1123.747 Hz is a single peak while the next two peaks
at 1179.955 and at 1190.578
Hz form a doublet with a 10.623 or 2
5.311
Hz
frequency separation.
Only the 1179
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
Hz.
The 1179
Hz mode is present in the CCD data at 1179.60
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
Hz
modes. The 1217
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
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
Hz discussed above,
the frequency resolution of the CCD data is not sufficient to separate the two
modes at 1212 and 1217
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
Hz, this doublet is two adjacent components of an
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 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
(
)
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
Hz,
2109
Hz, 2133
Hz, 2154
Hz, 2164
Hz and 2174
Hz must
be true pulsation modes. All remaining peaks above 1550
Hz can be
explained as 2nd and 3rd order combination peaks and are not independent modes.
1994 WET | |||||
f1 | f2 | f3 | f4 | ![]() |
|
(![]() |
(![]() |
(![]() |
(![]() |
(![]() |
|
958.533 | 988.726 | 1947.334 | 0.075 | a | |
|
1289.129 | 1956.008 | 0.059 | b | |
|
1179.955 | 1968.952 | 0.045 | b | |
963.282 | 1005.645 | 1968.952 | 0.025 | a | |
|
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 |
|
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 |
|
1315.055 | 1397.385 | 3924.971 | 0.041 | a |
978.874 | 1397.385 | 1548.653 | 3924.971 | 0.059 | a |
|
1315.055 | 830.708 | 1539.991 | 0.059 | a |
|
1397.385 | 872.337 | 1548.653 | 0.079 | a |
1212.490 | 1315.055 | 978.874 | 1548.653 | 0.018 | a |
1993 | |||||
|
f2 | f3 | f4 | ![]() |
|
(![]() |
(![]() |
(![]() |
(![]() |
(![]() |
|
963.416 | 1179.761 | 2143.374 | 0.197 | b | |
|
2174.884 | 4308.046 | 0.040 | a | |
2153.980 | 2153.980 | 4308.046 | 0.086 | a | |
|
1245.457 | 2153.980 | 4339.147 | 0.128 | a |
|
1968.222 | 963.416 | 2184.777 | 0.210 | b |
|
1968.222 | 963.416 | 2402.113 | 0.065 | a |
1992 WET | |||||
f1 | f2 | f3 | f4 | ![]() |
|
(![]() |
(![]() |
(![]() |
(![]() |
(![]() |
|
653.811 | 1315.181 | 1968.915 | 0.077 | b | |
|
1315.181 | 2109.129 | 4077.942 | 0.179 | b |
945.156 | 1023.594 | 2109.129 | 4077.942 | 0.063 | a |
|
1315.181 | 945.156 | 1549.959 | 0.041 | a |
Among the modes involved in linear combinations is
the mode at 1315 Hz, which
showed a nearly constant amplitude. Otherwise it would have been a good
candidate for a
measurement. It appears in one quadratic
combination and in at least four higher order combinations.
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 modes split by rotation. The rotational splitting averaged between all
multiplets is
5
Hz.
If
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
Hz. The only peaks listed in Table 6 which could potentially
be identified as components of rotationally split
modes are
the 1539.991
Hz-1548.653
Hz (
Hz) and the
1947.334
Hz-1956.008
Hz (
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
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 Hz
and 836
Hz and the two triplets centered on 945
Hz and 1101
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).
For the mode identification, we proceed by iteration as follows.
We assume that all the modes seen in RXJ 2117+3412 have the same value of .
This assumption relies on the equidistant period spacing, discussed above
(Sect. 4.1), and on the similar rotational splitting found in the multiplets.
It is not possible to give an absolute identification for the k order
of the modes. Only differential k can be asserted. The reference mode
is chosen at 945.156
Hz, the central m=0 component
of a triplet, whose korder is some unknown k0. We will attempt now to identify the
relative order
,
as well as the azimuthal number m for as many modes as possible.
Note that the
convention chosen for the sign of
m is different from the one used in classical textbooks: here
m=-1 is associated with the low frequency component of a triplet
(retrograde mode) while m=+1 corresponds to the high frequency component (prograde
mode). This sign convention is the same as in Unno et al. (1989) and Winget et al.
(1991, 1994).
The identification procedure starts by considering modes for which the
m=0 frequency is well determined, either from the two complete triplets,
(945
Hz, 1101
Hz),
or from the three doublets whose components are separated by twice the
rotational splitting (978/988
Hz, 1045/1055
Hz and 1179/1190
Hz).
In this latter case the m=0 frequency is
obtained by averaging the two components, assuming that the triplets are
symmetric. These five m=0 modes give a unique determination of the period spacing,
either by applying the inverse variance test (O'Donoghue 1994),
which yields a period spacing of
s (HWHM),
or a linear unweighted least-squares fit which yields
s.
The inverse variance procedure also yields the values of the relative radial order of the modes,
;
they are listed in the fourth column of Table 9.
We are not able to assign an absolute k value, because there are no models of
suitable quality at that high a luminosity in the literature.
Starting from this point,
new m=0 modes are identified in a step by step procedure,
using linear
interpolation/extrapolation.
We allow for up to 2 s departures from the linear trend to
account for possible mode trapping effects. Later, we will show that this is
consistent with the actual trapping cycle, whose maximum amplitude
is 1.3 s.
An illustration of the mode identification procedure is given in Fig. 11 for the case
of the
and -4 modes.
From the straight line fit to the first five modes, we find that the m=0
mode for
should appear at
s or in the
frequency interval
1003.8 to 1007.9
Hz.
This estimate allows a unique identification of the m=0 component within the
doublet at 1005/1010
Hz as the 1005.645
Hz mode.
We can now include this mode into linear regression and
continue the procedure.
We then estimate the frequency range of the m=0 mode for
,
which is 1025.9 to 1030.2
Hz. It is clear that the mode
observed at 1023.594
Hz must be an m=-1 component of the triplet. The
m=0 component is not seen, but its frequency can be estimated from the
rotational splitting. We include the new m=0 frequency into the linear regression
and continue the procedure until we have determined the relative radial order (
)
and azimuthal order (m) for as many of the observed modes as is possible.
At each intermediate step, a linear fit
to m=0 modes is redetermined.
We notice that the rotational splitting changes with period
(see Fig. 13), so we must interpolate the value for the
,
-8 modes. For
smaller than
-11, the behaviour of rotational splitting is not known, and we assume a constant splitting
of
Hz for these modes.
While this is an arbitrary assumption, it has a negligible
effect on the trapping cycle parameters (Sect. 4.3) that we determine below.
We stop the mode identification at
frequencies lower than 780
Hz, because below this limit the spacing between adjacent mmodes and between different overtone modes become comparable and we can no longer tell them apart.
Also, we refrained from
determining periods of unobserved m=0 modes
for
larger than 0,
because the variation of rotational frequency splitting
with
is not well established here because there are not enough multiplets detected at these
low frequencies.
In addition, at low frequencies,
any error in the interpolated/extrapolated value of
would translate in relatively large error of the
mode's period and affect our mode trapping results.
Our procedure shows that some modes which could have been interpreted
as multiplets by considering only their frequency difference in Table 6 cannot be so.
That is the case for the modes at 789
Hz and 793
Hz, which cannot
correspond to the same
value, and for the modes at
830, 836 and 840
Hz, which must be a combination of a single mode and
a doublet of the successive k mode. Our proposed mode identification is
summarized in Table 9.
|
Frequency (![]() |
m | ![]() |
|
789.042 | +1 | +10 |
|
793.783 | -1 | +9 |
|
830.708 | +1 | +7 |
|
836.067 | -1 | |
1189.956 | 840.367: | 0 | +6 |
|
851.483 | -1 | +5 |
|
872.337 | 0 | +4 |
|
889.587 | 0 | +3 |
1117.568 | 894.800 | +1 | |
|
906.378 | 0 | +2 |
|
921.721 | -1 | +1 |
|
940.563 | -1 | |
1058.026 | 945.156 | 0 | 0 |
1052.732 | 949.909 | +1 | |
|
958.533 | -1 | |
1038.118 | 963.282 | 0 | -1 |
|
978.874 | -1 | |
(1016.467) | (983.800) | 0 | -2 |
1011.403 | 988.726: | +1 | |
|
1005.645 | 0 | -3 |
989.569 | 1010.541 | +1 | |
|
1023.594 | -1 | |
(972.247) | (1028.545) | 0 | -4 |
|
1045.690 | -1 | |
(951.750) | (1050.697) | 0 | -5 |
947.236 | 1055.703 | +1 | |
|
1096.712 | -1 | |
907.489 | 1101.942 | 0 | -7 |
903.160 | 1107.224 | +1 |
|
Frequency (![]() |
m | ![]() |
|
1123.747: | -1 | |
(885.736) | (1129.004) | 0 | -8 |
|
1179.955 | -1 | |
(843.692) | (1185.267) | 0 | -10 |
839.928 | 1190.578: | +1 | |
|
1212.490 | -1 | |
821.145 | 1217.812 | 0 | -11 |
|
1245.457 | -1 | |
(799.495) | (1250.789) | 0 | -12 |
|
(1283.828) | 0 | -13 |
775.699 | 1289.160 | +1 | |
|
1315.055 | -1 | |
(757.354) | (1320.387) | 0 | -14 |
|
1362.495 | 0 | -15 |
|
1397.242 | -1 | |
(712.975) | (1402.574) | 0 | -16 |
|
1439.198 | -1 | |
(692.267) | (1444.530) | 0 | -17 |
Note that there is no correlation between mode trapping and mode amplitude,
where trapped modes are those defining the minima in Fig. 12. There
are 6 such minima. The corresponding modes have periods ()
of
733.97 s (-15), 799.49 s (-12) and/or 821.15 s (-11), 885.74 s (-8)
and/or 907.49 s
(-7), 972.25 s (-4), 1058.03 s (0) and 1124.11 s (+3). Looking at
the trapped mode amplitudes during different runs (Table 6),
we see that trapped modes can have either low or high amplitudes.
Mode trapping and amplitude were also found to be uncorrelated in
PG 1159-035 (Winget et al. 1991) and in the DBV GD 358
(Winget et al. 1994).
Clearly, the amplitude
of a mode is not
simply governed by its linear growth rate.
The period spacing found here for RXJ 2117+3412 is remarkably similar to
the ones found in other PNNV and GW Vir stars:
s in
NGC 1501 (Bond et al. 1996), 21.5 s in PG 1159-035 (Winget et al.
1991),
21.6 s in PG 2131+066 (Kawaler et al. 1995) and 21.1 s in
PG 0122+200,
(Vauclair et al. 2001).
We do not understand what mechanism is forcing these pulsators - that have
different masses and luminosities - to display the same
period spacing. O'Brien (2000) suggests that an interplay between the
driving zone depth and the maximum allowed pulsation period, as a
function of
and the total mass, can explain the tendency for
higher mass pre-white dwarfs to pulsate at cooler
than lower mass
ones. If higher mass GW Vir stars pulsate at cooler temperature than low mass ones,
then the average period spacings could be similar for all of the GW Vir stars.
However, it is still unclear how this preserves the almost constant
observed over a factor of almost 1000 in luminosity.
Clearly, the "numerology'' is telling us something about the nature of
the GW Vir and PNNV stars, but we do not yet understand it.
where
takes a simple form in the asymptotic limit of high
order gravity modes, which applies to pre-white dwarfs, as discussed by
Winget et al. (1991). In that case
as shown by Brickhill (1975).
In the above expression for the frequencies, solid body rotation is
assumed. Any differential rotation present in the star would result in
an additional term
to
which depends on k, and so is different for different modes.
Assuming all the modes to be
modes, we derive an average rotation
period for RXJ 2117+3412, using the relation
One should keep in mind that the frequency
separation within multiplets may deviate from uniformity for different reasons.
There are at least four physical processes that could affect the
frequencies: i) non-linearities resulting from resonant coupling between
components of multiplets, ii) mode trapping, iii) structural changes
in the wave propagation cavities, and iv) magnetic field.
In case i), slight changes in the frequency of multiplets components are expected
even in
the case of modest non-linearity in the pulsations. The non-linearities result
in both amplitude and frequency variations for selected modes as described in
Goupil et al. (1998). In their application to the case of the DBV GD358,
however, they find that the frequency splitting is changed by the non-linear effects
by no more than 2%.
In case ii),
mode
trapping also introduces small variations in the frequency shift due to
rotational splitting (Kawaler et al. 1999).
Since the radial structure of the mode
is affected by trapping, so is the rotational kernel. If rotation is non-uniform,
it will affect the splitting constant.
In their discussion of the rotational
splitting in PG 1159-035, Kawaler et al. (1999) show that the
effect of mode trapping may change the frequency separation of the rotationally
split components by 2.5% in the period range 400
s.
They also show that the rotation rate inside PG 1159-035 decreases with increasing radius.
The variations could be much larger would the gradient of the rotation curve be
steeper.
As a consequence, any variations in the frequency splitting measured between
multiplets may be due to a combination of at least
these four effects and deriving any information on a potential
differential rotation profile is a speculative task.
Figure 13 shows the
variation of the rotational splitting as a function of
period for all multiplets observed in RXJ 2117+3412. Only secure detections are considered
(marginal detections are rejected). Rotational splittings derived from
1994 WET data are preferred because these data have the best window function. For data
from either the 1992 WET or the 1993 data set, splittings showing the smallest
formal error were preferred.
Figure 13 shows a clear trend of decreasing rotational splitting
with increasing period. Only one mode does not fit the trend: it is the
1124.117 s mode, seen only during the 1992 WET.
Several explanations can account for this discrepancy. We note from Table 6 that
for the few cases where we have frequency splitting information for the
same mode in different years, that the 1992 WET data have the largest splitting
for unknown reasons. We suspect that the systematically larger frequency splittings
in the 1992 data affect
the 1124.117 s mode, although we cannot rule out the possibility that there is an
incompletely corrected 2d alias present.
The observed
rotational splitting changes by a factor of 1.14 in the period
interval 821 s to 1058 s, covering 11 k orders. The factor is 1.24 if the longest
period mode (at 1189 s) is included, covering 17 k orders.
This variation of
is 13-14 times greater than the
non-differentially rotating case considered by Kawaler et al.
(1999) who find
% change in the rotational
splitting between 400 s and 800 s, covering 20 k orders, in their model
of PG 1159-035.
The conservative conclusion is that the rotational
splitting in RXJ 2117+3412 is inconsistent with solid-body rotation.
Kawaler et al. (1999) show
that a rotation law that decreases or increases outwards may have
similar signatures in a
-Period diagram. They also show that mode
trapping affects the rotational splitting and that it is the phase shift
between the trapping seen in the period spacing (
-Period
diagram) and the one seen in the rotational splitting (
-Period
diagram) which contains the pertinent information on the rotation velocity
law. We do not see any such trapping cycle in the
-Period
diagram
of RXJ 2117+3412, which is in agreement with the weak trapping indicated by the small
amplitude of the trapping cycle, and this precludes any further statement on
the internal rotation profile of RXJ 2117+3412.
The observed trend of the rotational splitting in RXJ 2117+3412 is
surprisingly smooth. One would have expected a rather complex
internal rotation law,
if one considers that the star is i) still contracting towards the white
dwarf cooling sequence with a short time scale (
)
and ii) is loosing mass at a rate of a
(Koesterke et al. 1998;
Koesterke & Werner 1998).
In the absence of any consistent physical interpretation of the
rotational splitting variation, we can only conclude that the
average rotation period is 1.1 d, and that the frequency splitting
is not consistent with solid body rotation.
The average rotation period for RXJ 2117+3412 is within the range of values
derived from asteroseismology for other pre-white
dwarfs. The PNNV NGC 1501 has a rotation period of 1.17 d (Bond et al.
1996).
Among other pulsating PG 1159 stars having rotation periods derived from
rotational splitting
PG 1159-035 has a period of 1.38 d (Winget et al. 1991),
PG 2131+066: 5.07 h (Kawaler et al. 1995) and PG 0122+200: 1.61 d
(O'Brien et al.
1996,
1998; Vauclair et al. 2001).
This trend persists with the cooler DBV and DAV white dwarfs, which have rotation
periods ranging from 9 to 58 hours (Bradley 2001).
Spruit (1998) argues that such rotation periods around one day for white dwarfs
can be expected if some small non-axisymmetries occur in the mass loss process
along the AGB evolutionary phase. In the case of RXJ 2117+3412, which is still losing
mass in its present pre-white dwarf phase (Werner et al. 1996;
Koesterke et al.
1998; Koesterke & Werner 1998), the complex structure
of its low surface brightness planetary nebula suggests such a
non-axisymmetrical mass loss.
The existence of a magnetic field would also lift the degeneracy of the
modes by splitting a mode of degree
into
+1 components. Since the
cumulative power spectrum of RXJ 2117+3412 does show some triplets,
the fine structure must at least be dominated by rotational splitting as
the magnetic field alone would only produce doublets for
modes.
However, if a weak magnetic field is superimposed on the rotation, its
effect would be to shift each component relative to its non magnetic
frequency, with the shift in frequency proportional to
m2B2,
where B is the strength of the magnetic field. Both the
components are equally shifted by the magnetic field to higher
frequencies. The m=0 component is also shifted to higher frequency by
a smaller amount (see Unno et al. 1989; Jones et al.
1989). As a result, a frequency asymmetry in the triplets
could be the signature of such magnetic field.
Unfortunately, there are only two true triplets
in the power spectrum of
RXJ 2117+3412 to search for such an asymmetry.
Considering these two triplets, one does find that the differences between
their prograde and
retrograde mode frequency splitting is within the formal uncertainties in
the frequency measurement with
Hz. The corresponding
upper limit of the magnetic field, obtained by scaling the results of
Jones et al. (1989) for
modes (their Fig. 1) is of the
order of
G. As this is taken from the calculations for a pure
carbon white dwarf model by Jones et al. (1989), it can only be an approximate
value when scaled to RXJ 2117+3412.
Mode trapping is interpreted as the signature of chemical
stratification in the star. Such a stratification is induced by the
previous history of nucleosynthesis within the star and the
gravitational settling combined with diffusion acting in a strong gravitational field.
The effect of mode trapping on the frequency of the pulsation modes has been
studied in detail for the pre-white dwarf pulsators (Kawaler & Bradley
1994), although for luminosities lower than that of RXJ 2117+3412.
The trapping cycle observed in RXJ 21117+3412,
folded by the
trapping phase, is shown in Fig. 14. It does not show evidence of a
double peaked structure; this absence suggests that we detect probably only one
chemical composition transition zone between the He-rich outer layers
and the C/O core. The amplitude of the trapping cycle (A)
depends on the gradient of the mean molecular weight through the
transition zone and on the thickness of the He-rich layer,
while the period of the trapping cycle depends mainly on the thickness of the
He-rich outer layer.
The thickness of the He-rich outer layer in
RXJ 2117+3412 could be precisely determined only
through the calculation of realistic
models, which are not yet available. The best we can do now is to
use the results published by Kawaler & Bradley (1994)
and extrapolate them to the range of
parameters of RXJ 2117+3412. The extrapolation may not be too bad, since at least
the average period spacing is only weakly dependent on luminosity.
The trapping period depends on the thickness
of the He-rich envelope at fixed
,
as shown in Fig. 3 of Kawaler & Bradley.
As can be inferred from this figure, the logarithm of the outer layer
fractional mass
is related to the trapping period through
a very tight linear relation. Similarly,
at fixed mass of the He-rich outer layer, the trapping period depends
on log (
),
as shown in their Fig. 4. Again, the trapping period as a
function of log (
)
is accurately fitted by a linear relation.
We combine these two relations to construct an
interpolation formula representing the models of Kawaler & Bradley:
![]() |
Figure 14:
Folded trapping cycle. The residuals of the period distribution (same as in Fig. 12)
but plotted vs. trapping cycle phase. The mode
![]() |
The above equation can be now rewritten as
Despite similar trapping periods, we derive vastly different envelope thickness.
This is entirely due to the difference in
between RXJ 2117+3412
and PG 1159-035. A comparison of the trapping amplitudes of PG 1159-035
and RXJ 2117+3412 shows that the
amplitude of RXJ 2117+3412 is only half that of PG 1159-035.
The smaller trapping amplitude of RXJ 2117+3412 could arise from two effects (see Kawaler & Bradley
1994, their Fig. 3). First, the trapping amplitude decreases with increasing
He-rich layer mass fraction and second, it also decreases towards longer
periods. Both effects are the result of the peak amplitude portion of the eigenfunction moving away
from the composition gradient, which decreases the resonance effect of mode trapping.
The observed periods of PG 1159-035 are in the range of
430 s to 840 s, while in RXJ 2117+3412 they are in the range of 690 s to 1190 s.
The presently available evolutionary models for the transition between
the planetary nebulae nuclei and the white dwarfs are not
suitable for interpreting
RXJ 2117+3412. While the 0.7
evolutionary
sequence of Wood & Faulkner (1986) fits the location of
RXJ 2117+3412 in the
-
diagram, the pure He surface
composition of their models does not adequatly represent the observed
abundances of RXJ 2117+3412. The more recent calculations by
Gautschy(1997), while using a composition more compatible
with the surface abundances of PG 1159 stars, do not fit the high
luminosity and temperature of RXJ 2117+3412, except by considering
stellar models with masses in excess of 0.7
,
which
disagrees with the mass derived from asteroseismology (see below). None
of these models takes into account the chemical stratification induced
by diffusion in the presence of mass loss. As a consequence, one can hardly
use them to calculate
theoretical period spacings which one could use for asteroseismological mass
determination.
In Vauclair et al. (1993), the mass estimate
was based on an interpolation formula which did
not take into account the luminosity dependence of the period spacing
(Winget et al. 1991). This formula, used to estimate the mass of PG 1159-035
is probably not appropriate for RXJ 2117+3412, which is two orders of
magnitude more luminous than PG 1159-035. Kawaler & Bradley (1994)
calculated the period spacings,
,
for a grid of pre-white dwarf models of
various masses,
including the luminosity dependence (their Fig. 2). They derive an
interpolation formula that is valid for luminosities
between
.
Extrapolating this formula to the luminosity of RXJ 2117+3412
log(
is risky. Rather
than using their interpolation formula, we extrapolate
directly from their Fig. 2 by spline functions.
Using the period spacing derived
in Sect. 4.3,
s, and assuming that this period spacing is
valid for
asymptotic
modes, we obtain
s.
At the luminosity of RXJ 2117+3412 (see next section),
such a period spacing corresponds to a model of
0.56
.
The formula of Winget et al.
(1991) gives a mass of 0.58
.
The
value obtained from Kawaler & Bradley's interpolation formula would be
0.52
,
and the range
is a rough
estimate of the uncertainty for this preliminary mass determination.
We should stress that the uncertainty we quote is entirely due to the fact that we
have to extrapolate the existing theoretical calculations
and does not include any observational uncertainty in
or
.
A much more precise
mass estimate (as good as for PG 1159-035) must await models
covering the parameter range of RXJ 2117+3412.
For now, our best value for the mass of RXJ 2117+3412 is
.
If we interpret the observed period spacing as corresponding to
modes, the derived
mass would be
0.32
.
Such a low mass would be in conflict with the spectroscopically determined log g= 6.0.
It would imply that the progenitor of RXJ 2117+3412 was in a binary system, for which we do
not see evidence, and that the star would have a helium core as a result of previous mass
transfer in the binary system.
Higher
values would imply even lower mass estimates for RXJ 2117+3412,
which would make the conflict even more severe.
On these grounds, we
conclude that the modes observed in RXJ 2117+3412 must be
modes. This is
in agreement with the
value implied by the fine structure found in the power spectrum.
Knowing the total mass of the star from asteroseismology on one hand and the surface gravity and effective temperature from spectroscopy on the other hand, it is straightforward to derive the luminosity and the distance of the star.
The best fit model atmosphere for RXJ 2117+3412 indicates a surface gravity
log g=6.0
+0.3-0.2 (Rauch & Werner 1997)
and an effective temperature
(Werner et al. 1996).
The radius derived from the mass (
)
and gravity is:
Adopting this new distance estimate, and its associated uncertainty, the linear
extent of the planetary nebula is:
This estimate of the evolutionary time scale would imply a rate of
period change of the order of
(
)
for the
low He-rich envelope mass case (high He-rich envelope mass)
for periods around 1000 s. This is comparable to the
measured
for the 516 s mode in PG 1159-035 (Costa et al. 1999).
Such a high rate of period change would probably be detectable in
RXJ 2117+3412 if one could find an isolated mode with a stable
amplitude.
As shown in Fig. 12, mode trapping in RXJ 2117+3412 offers the
potential to measure
for both trapped and nontrapped modes.
In the case of the trapped modes the resultant
would be dominated
by the evolutionary time scale of the outer layers, where the structure reflects the
competition between contraction and mass loss. The
of the nontrapped
modes would be dominated by the core cooling time scale.
Unfortunatly, it will be difficult to get an unambiguous
measurement for any mode of RXJ 2117+3412,
because of the rich pulsation spectrum (which requires multisite data)
and because of the large amplitude variations of the modes.
The mode at 1315.05 Hz, whose amplitude remained almost constant in
the data sets presented here, unfortunately takes part in a
linear combination
with the 653.987
Hz to form the 1968.952
Hz
(plus some other higher order combinations). So it can not be
used as a clean mode to measure
.
As this mode may be suspected to
take part in a mode coupling, its frequency variation may
not reflect global stellar evolution, but would rather reflect some
properties of the mode coupling. It is worth mentioning however that,
while both the 653
Hz and 1968
Hz peaks
suffered large
amplitude variations in the period covered by the data, as seen in Table 6,
the 1315
Hz amplitude varies by only 20% during the same time. If those
three modes were coupled, it is difficult to understand how the
amplitude of two
components could vary so much while the third remains almost constant.
So long as we cannot determine whether the 1315
Hz mode is an eigenmode or coupled
to other modes, it would be hazardous to infer any
physical meaning from a
measurement for that particular mode. One must
conservatively conclude that
with the data presently available, there is no stable enough mode
in a clean enough part of the power spectrum which we could
identify as a potential
candidate for a
measurement.
But given the presently poor physical constraints on evolution in this part of the H-R
diagram, attempting to measure a rate of period change is a worthy challenge.
The rapid evolution of RXJ 2117+3412 shows up in the mass loss.
RXJ 2117+3412 has a measured mass loss rate of
,
which has interesting implications for
the region where pulsation driving takes place.
The excitation mechanism (
mechanism due to carbon
and oxygen partial ionization) operates at the depth of
K, which
lies in the outer
10-8 M* mass fraction of the star
(Bradley & Dziembowski 1996).
The observed mass loss rate implies that the material in the driving region is renewed
on a time scale considerably shorter than the evolutionary time scale
(
50 days!). As a consequence, the chemical composition
of the driving region could change on this time scale if there is compositional
stratification of the outer layers. This outflow of mass through the driving region
may affect the efficiency of the excitation mechanism, especially if the
mass loss rate is time dependent. The amplitude variations observed in
the star, including variations on time scale as short as that
exhibited by the mode at 717
Hz during the 1994 WET campaign,
could be related to the effect of the mass loss on the
effectiveness of the driving in this region.
The non-radial g-mode instability in pre-white dwarf stars is triggered
by the
and
mechanism induced by the partial ionization of carbon and
oxygen at
K, as first suggested by Starrfield et al.
(1983, 1984) and confirmed by the subsequent analysis
of Stanghellini et al. (1991). More recently, instability
studies using models computed with the new He/C/O OPAL opacities (Iglesias &
Rogers 1993) show a better agreement with the observed
blue edge of the instability strip and put some constraints
on the composition of the
driving region (Bradley & Dziembowski 1996; Saio 1996;
Gautschy 1997). However, Kawaler et al. (1986)
had also anticipated that during the PN and pre-white dwarf evolutionary
phases, the possibility of a remnant He-burning shell that could drive g-modes by
the
-mechanism. As the He burning necessarily occurs at the
bottom of the He-rich outer layers,
the periods of these unstable g-modes are in
the range of 70 s-200 s, corresponding to low k orders for
modes.
Saio (1996) and Gautschy (1997) also find
g-modes triggered by the
-mechanism in some of their models,
with typical periods between
110 s and
150 s. However,
such short period g-modes excited by
-mechanism have not
been found in the surveys of PNN conducted by Grauer et al.
(1987) and by Hine & Nather (1987).
At the high frequency end of the list given in Table 6, one finds some peaks
which could be candidates for such -mechanism driven modes, since
their periods are in the range 230-290 s. But, as discussed in Sect. 3.2,
all the peaks with frequency above 2180
Hz (
s) are
the result of
linear combinations of lower frequency modes; they are not independent
modes. The highest frequency independent mode has a frequency of 2174.884
Hz (period of 459.8 s). A careful scrutiny of the power spectrum at
even higher frequencies (up to 12000
Hz), where the highest
noise peaks are at a 0.20 mma level, does not reveal any significant peak. We
conclude that our data show no evidence for low-k order mode driven
by the
-mechanism.
We performed an asteroseismological analysis of the pulsating
PG 1159-type
planetary nebula central star/pre-white dwarf
RXJ 2117+3412 based on three multisite campaigns. Because of its
observed amplitude variations, a property shared with most of
the variable
planetary nebulae nuclei and pre-white dwarf stars, three campaigns
were necessary for us to decipher the complex light curve and identify the
pulsation modes. The cumulative power spectrum leads to the detection of
48 independent modes. Among them, we detected two
triplets and eight doublets, which have an average frequency splitting of
5
Hz. We identify these modes as
g-modes split by rotation.
From the analysis of the period distribution, we assign relative overtone numbers and
m values to all the modes
observed between 780 Hz and 1450
Hz.
The mean rotational splitting of
Hz gives an
average rotation period of
d. The
rotational frequency splitting decreases with increasing period, but there
is no signature of mode trapping on the rotational
splitting, so we cannot infer the internal rotation law.
We can only conclude that the observed trend is not compatible with a
solid body rotation law.
The lack of significant asymmetry between the prograde and retrograde
components in the triplets indicates that the magnetic field
strength is probably small, with
Gauss.
From the period spacing (
s) we derive a
total mass of
.
The mode trapping indicates
that the He-rich outer layer mass should be at
least 0.013 M* and could be as large as 0.078 M*.
The asteroseismological mass and the spectroscopic atmospheric
parameters (log g,
)
allow us to derive a luminosity of
.
The distance of RXJ 2117+3412, taking into account the interstellar
absorption on the line of sight, is
760
+230-235 pc. At this distance, the linear size of the
surrounding planetary nebula is
pc, confirming its status as
the largest known planetary nebula.
Speculating on the probable evolutionary link between a star like
RXJ 2117+3412 and PG 1159-035, we infer that it should take at least
yr (or
yr) to
RXJ 2117+3412 to shed its excess He-rich envelope through mass
loss, if its He-rich envelope mass fraction is
0.013 M*(if 0.078 M*). This rapid evolutionary time scale
leads to a predicted
between
and
for periods of about 1000 s.
Unfortunately, interpreting any
measurement of
for
the modes identified in RXJ 2117+3412 presents a severe challenge:
most of the modes suffer amplitude variability,
the frequency spectrum will likely require multisite data to properly resolve the modes,
and the rapid evolution means that we may require several observing runs each season to maintain
an accurate cycle count.
Through its pulsations, however, RXJ 2117+3412
may provide a clue to understanding the complex physical
processes that compete during this rapid evolutionary phase. Considering
the asteroseismological data presented here together with the evidence
from spectroscopy and photometry presented elsewhere, it is tempting to
speculate that time dependent mass-loss plays a dominant role in what we see
in RXJ 2117+3412. For example, the present chemical composition of the He-rich
outer layers
should be the product of the interplay between diffusion
and mass loss
(see the recent calculations by Unglaub & Bues 1998,
2000).
Changes in the mass-loss rate should translate into a change in the heavy elements
distribution within the envelope and the photosphere.
These changes could in turn affect the opacity and the efficiency of the driving
mechanism. The structure of the outer layers should be modified
accordingly, producing changes in the spectrum and in the UV flux, since
most of the UV opacity is due to the heavy elements.
As the abundances in the excitation region and the whole structure of
the mode propagation cavities are affected by such variations, one would expect
both the frequencies and amplitudes of the g-modes to vary. The
presently available data provide some support for such a scenario, since a
high UV flux was observed in 1993
(Feibelman 1999),
while the star showed low amplitude oscillations. The
subsequent decrease in the UV flux would then be explained by the
restoration of the appropriate heavy element abundance producing the
opacity in the UV and the associated -mechanism efficiency. The
missing piece of evidence is whether the mass-loss rate was really
varying simultaneously. A fully consistent study of this complex
problem would be to carry out: long-term observations of the
spectroscopic and photometric variations in the UV; search for a time
dependence in the mass-loss rate; and obtain simultaneous asteroseismological
observations to determine how the g-mode frequencies and amplitudes respond
to mass-loss and UV flux variations.
Acknowledgements
The WET gratefully acknowledges support from the National Science Foundation (US) through Grant AST-9876655 to Iowa State University. GV and PM acknowledge support from the French/Polish "Jumelage'' programme. GV and MC acknowledge support from the CNRS(France)/CSIC(Spain) exchange programme and from the CNRS(France)/Chinese Academy of Sciences "PICS'' programme. MAB was supported by PPARC, UK through an Advanced Research Fellowship. XCOV8 WET run was partly supported by the EU-HCM grant CHRX-CT94-0434. During the XCOV11 WET run, EGM was attacked and badly injuried as he was observing at the Maidanak Observatory. Despite his bad experience, EGM thanks the local astronomers for a 6km mountain-running to the military base, the pilots of the military helicopter for the quick delivery to the hospital, and the doctors of the Kitab hospital who rescued him. ADG thanks the Director and Staff of the Steward Observatory for telescope time and technical support with the telescopes on Mt. Bigelow and Mt. Lemmon. JNF thanks the K.C. Wong Education Foundation, Hong-Kong, China and the French Ministery for National Education for financial support. Astronomical research at the Wise Observatory is supported by the Basic Research Foundation of the Israeli Academy of Sciences. PM, JK, GP and SZ are supported in part by KBN grants No 2-P03D-015-08 and 2-P03D-014-14 in Poland.