1 Introduction

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).

 

Table 1: Participating sites.

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 $\approx$25% and $\approx$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 $\log g$- $\log T_{{\rm eff}}$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).

 

Table 2: Journal of observations: 1992 WET (XCOV8).

Run Name	Telescope	Date	Start Time	Run Length
		(UT)	(UTC)	(s)
x-8006	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 $^{\prime\prime}$	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 $^{\prime\prime}$	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 $^{\prime\prime}$	27 September 92	08:10:30	5260
maw-0107	Mauna Kea 24 $^{\prime\prime}$	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 $^{\prime\prime}$	28 September 92	06:54:30	9915
x-8019	Suhora 60 cm	28 September 92	18:27:35	2095
pab-0166	McDonald 82 $^{\prime\prime}$	29 September 92	07:39:00	6975
ro-023	Itajuba 1.6 m	29 September 92	22:15:50	16245
pab-0168	McDonald 82 $^{\prime\prime}$	30 September 92	01:40:30	28170
maw-0111	Mauna Kea 24 $^{\prime\prime}$	30 September 92	05:30:00	24085
sjk-0208	Siding Spring 40 $^{\prime\prime}$	30 September 92	12:02:00	8240
ro-025	Itajuba 1.6 m	30 September 92	23:51:40	8410
pab-0171	McDonald 82 $^{\prime\prime}$	1 October 92	01:50:00	27345
maw-0114	Mauna Kea 24 $^{\prime\prime}$	1 October 92	08:36:20	14360
x-8020	Suhora 60 cm	1 October 92	20:29:00	9780
pab-0173	McDonald 82 $^{\prime\prime}$	2 October 92	01:50:00	26910
maw-0118	Mauna Kea 24 $^{\prime\prime}$	3 October 92	06:02:20	15230

 

Table 3: Journal of observations (1993).

Run Name	Telescope	Date	Start Time	Run Length
		(UT)	(UTC)	(s)
rx-0914	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 $^{\prime\prime}$	16 September 93	03:51:00	19310
k93-0214	Kavalur 90 $^{\prime\prime}$	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 $^{\prime\prime}$	18 September 93	02:36:00	21430
k93-0215	Kavalur 90 $^{\prime\prime}$	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 $^{\prime\prime}$	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 $^{\prime\prime}$	20 September 93	02:39:00	28930
ra-288	McDonald 36 $^{\prime\prime}$	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 $^{\prime\prime}$	21 September 93	02:29:00	28460
a-409	Mt. Lemmon 60 $^{\prime\prime}$	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 $^{\prime\prime}$	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 $T_{{\rm eff}}= 170\,000$ K, $\log g= 6.0$ ^+0.3_-0.2, and abundance ratios typical of other PG 1159 stars: $\rm He/C/O=47.5$/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 $\dot{M}$= $10^{-7}~M_{\odot}\,{{\rm yr}}^{-1}$ from C IV line (Koesterke et al. 1998), or $\dot{M}= 4\times 10^{-8}~M_{\odot} \,{{\rm yr}}^{-1}$ 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.