A&A 458, 259-267 (2006)
DOI: 10.1051/0004-6361:20065423
A. H. Córsico1,2, - L. G. Althaus1,2,
- M. M. Miller Bertolami1,2,3,
1 - Facultad de Ciencias Astronómicas y Geofísicas,
Universidad Nacional de La Plata, Paseo del Bosque S/N, (1900) La
Plata, Argentina
2 - Instituto de Astrofísica La Plata, IALP, CONICET-UNLP
3 - Max-Planck-Institut für Astrophysik, Garching, Germany
Received 12 April 2006 / Accepted 8 July 2006
Abstract
Aims. We reexamine the theoretical instability domain of pulsating PG 1159 stars (GW Vir variables).
Methods. We performed an extensive g-mode stability analysis on PG 1159 evolutionary models with stellar masses ranging from 0.530 to
,
for which the complete evolutionary stages of their progenitors from the ZAMS, through the thermally pulsing AGB and born-again phases to the domain of the PG 1159 stars have been considered.
Results. We found that pulsations in PG 1159 stars are excited by the -mechanism due to partial ionization of carbon and oxygen, and that no composition gradients are needed between the surface layers and the driving region, much in agreement with previous studies. We show, for the first time, the existence of a red edge of the instability strip at high luminosities. We found that all of the GW Vir stars lay within our theoretical instability strip. Our results suggest a qualitative good agreement between the observed and the predicted ranges of unstable periods of individual stars. Finally, we found that generally the seismic masses (derived from the period spacing) of GW Vir stars are somewhat different from the masses suggested by evolutionary tracks coupled with spectroscopy. Improvements in the evolution during the thermally pulsing AGB phase and/or during the core helium burning stage and early AGB could help to alleviate the persisting discrepancies.
Key words: stars: evolution - stars: interiors - stars: oscillations - white dwarfs
Pulsating PG 1159 stars - after the prototype of the spectral class
and the variable type, PG 1159-035 or GW Vir - are very hot
hydrogen-deficient post-Asymptotic Giant Branch (AGB) stars with
surface layers rich in helium (), carbon (
)
and oxygen (
)
(Werner & Herwig 2006) that exhibit
multiperiodic, low degree (
), high radial order (
)
g-mode luminosity variations with periods in the range
from about 300 to 3000 s. Some GW Vir stars are still embedded in
a planetary nebula; they are commonly called PNNVs (Planetary Nebula
Nucleus Variable). PNNV stars are characterized by much higher luminosity
than the "naked'' GW Vir stars (those without nebulae)
.
GW Vir stars are particularly important to infer fundamental properties about pre-white
dwarfs in general, such as the stellar mass and the surface
compositional stratification
(Kawaler & Bradley 1994; Córsico & Althaus 2006).
PG 1159 stars are believed to be the evolutionary connection between post-AGB stars and most of the hydrogen-deficient white dwarfs. These stars are thought to be the result of a born again episode triggered either by a very late helium thermal pulse (VLTP) occurring in a hot white dwarf shortly after hydrogen burning has almost ceased (see Fujimoto 1977; Schönberner 1979 and more recently Althaus et al. 2005) or a late helium thermal pulse (LTP) that takes place during the post-AGB evolution when hydrogen burning is still active (see Blöcker 2001, for references). During a VLTP episode, most of the hydrogen-rich envelope of the star is burnt in the helium-flash convection zone, whilst in a LTP hydrogen-deficient composition is the result of a dilution episode. In both cases, the star returns rapidly to the AGB and finally into the domain of high effective temperature as a hydrogen-deficient, quiescent helium-burning object.
A longstanding problem associated with pulsating PG 1159 stars is
related to the excitation mechanism. The early work by
Starrfield et al. (1983) was successful in finding
the correct destabilizing agent, namely the
-mechanism associated with the partial
ionization of the K-shell electrons of carbon and/or oxygen in the
envelope of models. However, their models required a driving region very
poor in helium in order to be able to excite
pulsations; even very low amounts of helium
could weaken or completely remove the destabilizing effect of carbon
and oxygen (i.e. "helium poisoning'').
The latter requirement led to the conjecture that
a composition gradient would exist to make compatible the
helium-devoid driving regions and the helium-rich photospheric
composition. Even modern detailed calculations still reveal the necessity
of a compositional gradient in the envelopes of models
(Bradley & Dziembowski 1996; Cox 2003). The presence
of a chemical composition gradient is difficult to explain
in view of the fact that PG 1159 stars are still
experiencing mass loss (
yr-1 for PG 1159-035; Koesterke et al. 1998),
a fact that prevents the action of gravitational settling of carbon
and oxygen, and instead, tends to homogenize the envelope of hot
white dwarfs (Unglaub & Bues 2000).
Clearly at odds with the hypothesis of a composition gradient in the PG 1159 envelopes, calculations by Saio (1996), Gautschy (1997), and Quirion et al. (2004) - based on modern opacity OPAL data - demonstrated that g-mode pulsations in the correct ranges of effective temperatures and periods could be easily excited in PG 1159 models having a uniform envelope composition. The most recent study of PG 1159-type pulsations is that of Gautschy et al. (2005) (hereinafter GAS05) based on a full PG 1159 evolutionary sequence starting from the zero-age main sequence (ZAMS) and evolving through the thermally pulsing and VLTP phases (see Althaus et al. 2005). These authors found no need to invoke composition gradients in the PG 1159 envelopes to promote instability.
As important as they are, the vast majority of the studies of pulsation
driving in PG 1159 stars rely on simplified stellar models.
Indeed, the earliest works
employed static envelope models and old opacity data. Even more modern
works, although based on updated opacity data (OPAL), still use a series of static envelope models that do not represent a real evolutionary
sequence, or evolutionary computations based on simplified
descriptions of the evolution of their progenitors. The
only exception is the work of GAS05, which employs equilibrium
PG 1159 models that evolved through the AGB and born-again stages,
beginning from a
zero age main sequence model star.
GAS05 analyzed four model sequences,
with
0.530, 0.55, 0.589 and
,
the
sequence being derived directly from the evolutionary
computations of Althaus et al. (2005). The remaining sequences were
created from the
one by changing the
stellar mass shortly after the end of the born-again episode.
On the basis of full evolutionary PG 1159 models covering the
whole range of observed GW Vir masses, this paper aims to
confirm and extend the results of the stability analysis
by Saio (1996), Gautschy (1997), Quirion et al. (2004),
and GAS05. We analyze the pulsational stability of seven different evolutionary
sequences of PG 1159 models with stellar masses between 0.530
and
.
Here, all of the PG 1159
evolutionary sequences have been derived by considering
the complete evolution of their
progenitors, an aspect that constitutes an improvement over previous
studies. One of such sequences (
)
was presented by Althaus et al. (2005) and analyzed by GAS05, and
the remaining ones were computed by Miller Bertolami &
Althaus (2006), with the exception of the
sequence, which is
presented for the first time in this work.
The pulsational results presented here based on extensive full evolutionary models increase our understanding of the GW Vir stars and place previous studies on a solid basis,
regarding stellar modeling. The paper is organized as follow: in
the next section we briefly describe the input physics of
our evolutionary code and the PG 1159 evolutionary sequences
analyzed. A brief description of our nonadiabatic
pulsation code is presented. In Sect. 3 we describe the
stability analysis and in Sect. 4
we compare our predictions with the observed properties of known
GW Vir stars. In Sect. 5 we summarize
our main results and make some concluding remarks.
The nonadiabatic pulsational analysis presented in this work relies on stellar models that take into account the complete evolution of the PG 1159 progenitor stars. The evolution of such models has been computed with the LPCODE evolutionary code, which is described in Althaus et al. (2005). LPCODE uses OPAL radiative opacities (including carbon- and oxygen-rich mixtures) from the compilation of Iglesias & Rogers (1996), complemented in the low-temperature regime with the molecular opacities of Alexander & Ferguson (1994) (with solar metallicity). Chemical changes are performed via a time-dependent scheme that simultaneously treats nuclear evolution and mixing processes due to convection, salt fingers and overshooting. Convective overshooting is treated as an exponentially decaying diffusive process above and below any convective region.
Specifically, the background of stellar models has been extracted from
the evolutionary calculations recently presented
in Miller Bertolami & Althaus (2006) and Althaus et al. (2005), who
computed the full
evolution of 1, 2.2, 2.7, 3.05, and 3.5 stars.
We include a new sequence of initially 3.75
.
All of the sequences were evolved from the ZAMS
through the thermally pulsing and mass loss phases on the AGB. After
experiencing several thermal pulses, the progenitors depart from the
AGB and evolve towards high effective temperatures. Mass loss during
the departure from the AGB has been arbitrarily fixed
so as to obtain a final helium shell flash
during the early white dwarf cooling phase. After the born-again
episode, the hydrogen-deficient, quiescent helium-burning remnants
evolve at constant luminosity to the domain of PG 1159 stars with
surface chemical composition rich in helium, carbon and oxygen. The
masses of the remnants span the range
.
For the sequence of
two different AGB evolution have been considered, with different
mass loss rates so as to obtain different numbers of thermal pulses
and, eventually, two different remnant masses of 0.530 and
0.542
.
The main characteristics of the sequences
considered in this work are given in Table 1. We list
the initial and final stellar
mass (at the ZAMS and PG 1159 stages, respectively),
and the surface abundance of the main
chemical constituents during the PG 1159 stage. The sequence with
an initial mass of 2.7
is the same as presented by Althaus et al. (2005).
Table 1: Initial and final stellar mass (in solar units), and the final surface chemical abundances by mass (PG 1159 regime) for the evolutionary sequences considered in this work.
Our PG 1159 models are characterized by envelopes with uniform chemical
compositions that extend from the surface downwards well below the
driving region (i.e., the chemical composition at the driving region
is the same as at the stellar surface). Thus, our models are not characterized by
chemical gradients between the driving region and the stellar
surface. Note that our post-VLTP models predict a range in the surface
composition. In particular, the final surface abundance of helium spans the
range 0.28-0.50 by mass,
which is in agreement with the range of
observed helium abundance in most PG 1159s (see Werner & Herwig 2006).
Our sequences with helium abundances larger
than the standard ones observed in PG 1159 stars will allow us to
explore the role of helium in the instability
properties of pulsating PG 1159s. Mass loss episodes after the VLTP have not been considered
in the PG 1159 evolutionary sequences we employed here.
The pulsation stability analysis was performed
with a new finite-difference nonadiabatic pulsation code
based on the adiabatic version described
in Córsico & Althaus (2005, 2006). The nonadiabatic code solves the
full sixth-order complex system of linearized equations and boundary
conditions as given by Unno et al. (1989).
Our code provides the dimensionless complex
eigenvalue ()
and eigenfunctions (
)
as given by Unno et al. (1989). Nonadiabatic
pulsation periods and normalized growth rates are evaluated as
and
,
respectively.
Here,
and
are the real and the
imaginary part, respectively, of the complex eigenfrequency
.
Our code
also computes the differential work
function,
,
and the running work integral, W(r), as
in Lee & Bradley (1993). In this work the
"frozen-in convection'' approximation was assumed because
the flux carried by convection is usually negligible
in PG 1159 stars. Also, the
-mechanism for mode driving
was neglected in the computations because nuclear-burning shells in PG 1159
models usually destabilize very short periods that are not observed
in GW Vir stars (Kawaler et al. 1986; Kawaler 1988;
Gautschy 1997). We employed about 3000 mesh-points to
describe our background stellar models, most of them distributed
in the envelope region where all the pulsation driving and damping
occur. We employed the "Ledoux modified'' treatment to compute the
Brunt-Väisälä frequency (N) (Tassoul et al. 1990).
![]() |
Figure 1:
The normalized growth rate in terms of period (in seconds)
for overstable g-modes corresponding to
two
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We analyze the stability properties of about
2400 stellar models covering a wide range of effective temperatures
(
)
and a range of
stellar masses (
). For
each model we have restricted our study to unstable
g-modes with periods in the range
s, thus comfortably covering the full period spectrum observed in GW Vir stars.
Consistent with other stability studies of GW Vir stars, all unstable g-modes in our PG 1159 models are driven by the
-mechanism associated with the opacity bump due to partial
ionization of K-shell electrons of C
and O
centered at
(Quirion et al. 2004; GAS05).
We start by discussing the stability properties of two template
PG 1159 models. These properties are common
to all PG 1159 models of our complete set of evolutionary sequences.
The normalized growth rate (
)
in terms of pulsation periods (
)
for overstable
modes corresponding to the two selected models are shown in Fig. 1. Model (a) is representative of the high-luminosity, low-gravity pre-white dwarf
regime, and model (b) is typical of the low-luminosity, high-gravity
phase, when the object has already entered their white
dwarf cooling track (see Fig. 3). Note that modes
excited in model (a) have pulsation periods in the range
s, substantially longer than
those excited in model (b) (
s).
For each model,
reaches a maximum
value in the vicinity of the long-period boundary of the instability
domain. In other words, within a given band of unstable modes, the
excitation is markedly stronger for modes characterized by long
periods. This effect is particularly strong in model (b),
the value of the growth rate for the shortest periods being more than seven
orders of magnitude smaller than for the modes with longer periods
.
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Figure 2:
The opacity and opacity derivatives ( left scale),
and the differential work function ( right scale)
for two selected ![]() ![]() |
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Figure 2 shows details of the driving/damping process in model (a) for two selected dipole modes. We restrict the figure to the
envelope region of the model, where the main driving and damping occurs.
Thick continuous curve corresponds to
for an unstable mode with
s and
10-5
(marked as a circumscribed dot in panel (a) of Fig. 1),
while the thick dashed curve depicts the situation for a stable mode with
s,
10-3. Also plotted is the Rosseland opacity,
,
and
its logarithmic derivatives,
and
.
The
region that destabilizes the k= 70 mode (where
)
is
clearly associated with the bump in the opacity centered at
[
], although the maximum driving
for this mode comes from a slightly more internal region (
). Note also that in the driving region the quantity
is increasing outward, in agreement
with the well known necessary condition for mode excitation
(Unno et al. 1989). Since the contributions to driving
at
from 7.5 to 10 largely overcome the damping effects at
,
the mode with k= 70 is globally excited. At variance, the strong damping experienced by the mode with k= 150 (denoted by negative values of
)
makes this mode globally
stable. The situation in the low-luminosity, high-gravity
phase as in model (b) of Fig. 1 is
similar, the only important difference being that the
driving/damping regions are located in considerably
more external layers. This is due to an outward
migration of the opacity profile, induced by the evolution of the star.
Here, we examine the location of the unstable domains on the HR diagram. In Fig. 3 we show the evolutionary tracks for our complete set
of PG 1159 model sequences, where the thick portions of
the curves correspond to models with dipole unstable modes.
A well-defined instability domain, bounded by a red (cool)
edge at high luminosities, and by a blue (hot) edge both at high
and low luminosities, is apparent in the plot.
The blue and red edges for dipole and quadrupole modes
for each sequence are connected by thin
curves as given by standard nonlinear least-squares algorithms.
The instability domains for
and
look
very similar, although the edges for
are
slightly shifted to higher effective temperatures,
and the region of instability is somewhat wider than for
.
Figure 3 should be compared with Fig. 5 of GAS05.
The global agreement between our results and the predictions
of GAS05 is excellent, in particular for the sequence of
- the only sequence in common between those authors and our work.
At variance with GAS05, in this work we have employed PG 1159 models with different masses derived from the complete evolution of the progenitor stars. This has enabled us to extend the pulsational stability analysis to lower effective temperatures in the high-luminosity, low-gravity region. As a result, we have found, for the first time, a reliable high-luminosity, low-gravity red edge of the GW Vir instability strip. The red edge is markedly sensitive to the stellar mass, being hotter for the more massive models.
![]() |
Figure 3:
The HR diagram for our complete set of PG 1159 model sequences
of
0.530, 0.542, 0.565, 0.589, 0.609, 0.664 and
![]() ![]() ![]() ![]() ![]() |
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Our blue edge (both for dipole and quadrupole modes)
cannot be exactly represented by a straight line. It is clearly
seen for dipole modes in Fig. 3.
The departures from a simple linear relation have
their origin in the different surface chemical compositions
with which our models of different stellar masses
reach the domain of the PG 1159 after emerging
from the born-again episode. The
unstable portions of the evolutionary tracks corresponding to models
characterized by a surface helium abundance of
0.28-0.39 by
mass (that is, the sequences with masses of
0.542, 0.565 and
)
extend slightly beyond the linear parameterization of the blue
edge, as compared with the case of the more massive models which have
greater helium abundances in the envelope (
0.50, 0.47 and 0.48 for
stellar masses of
0.609, 0.664 and
,
respectively)
. Increasing the helium abundance in the
driving region, the efficiency of pulsational driving is
reduced (see GAS05). This is consistent with the finding of
Quirion et al. (2004) that decreasing the helium mass fraction in
the driving regions, the blue edge of the instability domain
shifts to higher effective temperatures.
![]() |
Figure 4:
Same as Fig. 3, but for the
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Since spectroscopic calibration of PG 1159 stars gives effective
temperatures and surface gravities, we plot the
evolutionary tracks and the instability domains on the
plane. Figure 4 gives the
complete set of PG 1159 sequences. We emphasize with thick solid lines the stages with overstable
g-modes and
the loci of the blue and red edges for dipole
and quadrupole modes with thin curves as parameterized by
standard nonlinear least-squares
procedures. We postpone to Sect. 4
a complete discussion of the general agreement between our pulsation
models and the observed GW Vir stars.
In the current computations, the dipole unstable domain for each
evolutionary sequence is separated into two regions, one of them
corresponding to the high-luminosity phase (low gravity) and the other
corresponding to low luminosities (high gravity) (see
Figs. 3 and 4). The only exception is the
sequence with
which exhibits an unique unstable
domain that extends uninterruptedly from the high-luminosity phase to
the low-luminosity regime. This is at variance with the results
reported by GAS05, who found that for all of their sequences (including
that with
)
the tip of the evolutionary "knee'' is
pulsationally stable in the case of dipole modes. This discrepancy is
related to the different loci of the tracks on the HR diagram and
on the
plane. The evolutionary
sequences considered in this work reach lower
and
luminosity values for a fixed
than in the ones of GAS05. This reflects the fact that in the present work we have considered
PG 1159 evolutionary sequences derived from the full evolutionary
computations of the progenitor stars.
We now explore the ranges of periods of unstable modes. Figure 5 shows the
instability boundaries of dipole modes on the
plane for the complete set of evolutionary sequences.
Note that generally the periods of unstable modes for each sequence
are clearly grouped in two separated regions, one of them characterized
by long periods and high-radial overtones, corresponding to evolutionary
stages before the knee at high luminosities, and the other one
characterized by short periods (low k values) corresponding to the
hot portion of the white dwarf cooling track (low luminosities). As
mentioned before, the sequence with
shows a unique instability domain. Thus, this sequence shows instability even around the knee. The splitting of the instability domains into two separate regions can be understood in terms of the magnitude of
the thermal timescale
at the driving region. See GAS05 for a demonstration of this.
The high-luminosity domain of instability exhibits a strong mass
dependence, the longest unstable periods being shorter for the
more massive models. In addition, for the less massive models the
longest pulsation period is reached at lower effective temperatures
the more massive models. In all the cases the
long-period limit is attained shortly after the beginning of the
instability domain. The shortest unstable period for each sequence is
also markedly sensitive to the stellar mass. Generally, the minimum
period is smaller for the less massive models. Thus,
the period-width of the instability domain is
larger for the less massive models. As can be seen from
Fig. 5, the instability island on the
plane is almost vertical for the model with
,
but
its slope gradually decreases as we go to sequences with higher
masses.
![]() |
Figure 5:
The instability domains on the
![]() ![]() |
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The low-luminosity domain, on the other hand, shows a moderate
dependence on the stellar mass. The maximum overstable period is
always larger for the less massive models. The minimum overstable
period, however, does not show a clear trend with the value of the
stellar mass. We note that the shortest unstable periods are of
55 s and correspond to the sequence with
.
This short-period limit is substantially lower than that
reported by GAS05, of about 190 s for their sequence of
.
In our computations the shortest overstable periods have very small growth rates, of the order
of
10-9-10- 13, and thus we could consider that these modes
are stable (see panel (b) of Fig. 1). If so, our
short-period limit agrees with the predictions of GAS05.
The morphology of the instability domains for quadrupole modes looks
very similar to the case of dipole modes, the novel feature being a shortening of the overstable periods, in agreement with
GAS05. Indeed, the long-period limits of the high-luminosity
instability domains for quadrupole modes are shortened by about 3000 s
relative to dipole ones. A less severe decreasing of
overstable quadrupole periods is also present
in the low-luminosity domain. As a result, both regions of instability are closer together to such a degree that the high- and low-luminosity instablility domains
of the
0.530, 0.542 and
sequences merge into an uninterrupted region
.
The lowest
short-period limit for the
low-luminosity domain, which corresponds to the sequence with
,
is of about 50 s.
In all of the computations presented in this paper we
neglected chemical diffusion, and thus the stability calculations were
performed on PG 1159 models with a constant chemical composition in the
driving region. In particular, no helium enrichment in the driving region
was allowed, and consequently all of our sequences shown
pulsational instability well beyond the
empirical red edge of the GW Vir stars at low luminosities
(
K; Dreizler & Heber 1998), even
down to the domain of the variable DB white dwarfs.
Table 2: Stellar parameters and pulsation properties of all known pulsating PG 1159 stars. The approximate surface abundances (in % by mass) have been derived from Table 2 of Werner & Herwig (2006) by assuming a composition made of 4He, 12C and 16O, except for HS 2324+3944 (data taken from Table 1 of Werner & Herwig 2006) and for Abell 43 (data taken from Miksa et al. 2002).
In this section we compare our theoretical predictions with the observed properties of GW Vir stars. Currently, 11 pulsating PG 1159 stars are known. In Table 2 we show the main spectroscopic and pulsation data available. Note that there are five GW Vir stars (termed PNNV) that are still embedded in a planetary nebula. The remaining objects lack a surrounding nebula and are commonly called "naked'' GW Vir stars. There are two objects with measurable amounts of hydrogen in their spectra; they are termed pulsating "hybrid-PG 1159''. HE 1429-1209 is a naked GW Vir star but its effective temperature and gravity place it in the region of the HR diagram usually populated by PNNVs.
In Fig. 6 we plot the location of pulsating and
nonpulsating PG 1159 stars, as well as PG 1159 stars that have not been
observed for variability, on the
plane
(data taken from Werner & Herwig 2006). The plot also shows the
evolutionary tracks for our complete set of sequences of PG 1159
models. The blue and red edges for dipole and quadrupole modes are
also shown. Regarding pulsating PG 1159 stars, the agreement between
observations and model predictions is excellent. All
of the GW Vir variables lie inside our predicted instability domains of dipole
and quadrupole modes. However, there are several
non-variables occupying the unstable region. The existence of
non-variable stars within the instability domain could in part be
understood in terms of a variation in surface chemical composition
(and thus in the driving region) from star to star.
For instance, Quirion et al. (2004) found that the helium enrichment in the
driving region is the cause of the nonpulsator
MCT 0130-1937 (with a helium abundance of about 75%)
within the instability strip. Note, however, that PG 1151-029,
Longmore 3, Abell 21 and VV47 (not included in the analysis of Quirion
et al. 2004) are found to have standard helium abundances (see Table 2 of Werner & Herwig 2006) and however are non-variables. On the
other hand, it is remarkable the existence of the pulsating star
NGC 246 with a helium abundance (
)
unusually large among pulsators (see Table 2). These controversial cases remain to be explained.
![]() |
Figure 6:
The distribution of the spectroscopically calibrated variable
and non-variable PG 1159 stars
in the
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Regarding PG 1159-035 (GW Vir) - the prototype of both the variable
and the spectroscopic classes - our analysis predicts that
this star pulsates for a stellar mass of
(see Fig. 6). It is a markedly higher value
than the mass required by GAS05 to excite pulsations in PG 1159-035
(
), and, at the same time, closer
to the long-recognized "seismic'' mass of
of Kawaler & Bradley (1994). So, the employment of PG 1159
models derived from the complete evolution of their progenitor stars
appears to be a key factor to alleviate the discrepancy between the
seismic and the spectroscopic mass of PG 1159-035.
In what follows we focus on the observed period ranges for the
known GW Vir stars. In Fig. 7 we show the
diagram, in which the effective temperatures and the
period ranges are taken from Table 2. For comparison, we have included the theoretical instability boundaries for
.
Also plotted are the ranges of
unstable periods as predicted by Quirion et al. (2004). We are only interested in a qualitative comparison between the
observed and the theoretical ranges of unstable periods. Thus, we do
not attempt a detailed asteroseismic period fitting for
each individual GW Vir star. As can be seen from
Fig. 7, the agreement between theory and observations
is reasonably good for all cases, and for several stars the agreement
is excellent. Note, for instance, that for seven stars (Abell 43, HS 2324+3944, K1-16, NGC 246, HE 1429-1209, PG 2131+066 and PG 0122+200) the observed ranges of periods are completely contained in the theoretical instability domains corresponding to at least one stellar mass. For the case of PG 1159-035, the observed periods are
compatible with the instability domain of the
models, although the shorter detected periods (
530 s) lie
below the theoretical short-period limit. The opposite is true for PG 1707+427, whose longer observed periods (
760 s) lie above the highest long-period limit which corresponds, at the effective temperature of this star, to the
sequence. In the
case of RX J2117+3412, the range of observed periods partially
overlaps with four instability regions, corresponding to the sequences
of
0.5890, 0.609, 0.664 and
,
but no instability
domain accounts for the observed periods shorter than
800 s. Finally, in the case of Longmore 4 the band of observed periods is partially covered by the instability domains of the
0.530, 0.542 and
sequences. Note that the agreement between the
observed and predicted periods of unstable modes is comparable to that
reported by Quirion et al. (2004).
![]() |
Figure 7:
The period ranges of known
variable PG 1159 stars on the
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The predicted instability domains resulting from our analysis nearly reproduce the observational trend that the periods exhibited by pulsating PG 1159 stars decrease with decreasing luminosity (increasing surface gravity) (see O'Brien 2000, for a discussion of such a trend). This is well documented by Figs. 5 and 7. Note that Longmore 4, being a high-luminosity PNNV that shows short periods, is an exception to this trend. Longmore 4 is particularly interesting because it showed a surprising behaviour in its spectral type which suddenly changed from PG 1159 to [WCE] and back to PG 1159. According to Werner et al. (1992), this could be the result of a transient but significant increase in the mass loss rate. According to our computations, Longmore 4 is located very close to the red edge of the instability strip (see Fig. 6); so, this star could have just entered the instability phase.
In addition to the issue of pulsation instability,
valuable information about pulsating PG 1159 stars can be
extracted from the period spacing between consecutive overtones of
fixed value. The period spacing depends primarily on the
stellar mass and is only weakly dependent on the luminosity and
surface compositional stratification (Kawaler & Bradley 1994;
Córsico & Althaus 2006). Thus, this
quantity allows a determination of M* to a very high accuracy.
Here, we consider five GW Vir stars for which detailed
asteroseismic studies have been carried out:
PG 1159-035 (
s; Kawaler
& Bradley 1994), PG 2131+066 (
s; Reed
et al. 2000), PG 1707+427 (
s; Kawaler et al. 2004), PG 0122+200 (
s; O'Brien et al. 2000), and RX J2117+3412 (
s; Vauclair et al. 2002). We compare the observed mean period spacing of
each star with the average of the computed period spacing,
(
),
corresponding to models with an effective temperature as close as
possible to the value of
of the star under
consideration.
We first consider the case of PG 1159-035. According to its location
on the
plane, this star should have a stellar mass of
.
We refer this value as
the "spectroscopic mass'',
(see Fig. 6).
Our stability analysis predicts that a model with this mass at the
effective temperature of PG 1159-035 is pulsationally
unstable. However, this model should have a value of
s, which is in conflict with the observed
mean value of 21.5 s. To have a
value
compatible with the observed period spacing the stellar mass of PG 1159-035 should be
.
Thus, we found a discrepancy of
,
somewhat lower than that reported in the
literature (
,
Dreizler & Heber 1998;
,
Kawaler & Bradley 1994). We note that
a model with
at
140 000 K should be pulsationally stable in the framework of our
stability analysis.
In the case of RX J2117+3412, the evolutionary tracks of
Fig. 6 suggest a stellar mass of about
(see Miller Bertolami & Althaus 2006). A model with this mass and at the
of this star should be
pulsationally unstable. The corresponding value of
should be lower than 16 s, clearly at odds with the measured mean value of 21.5 s. So, in order to have
values comparable to the observed, we should
be forced to consider models with masses lower than
.
Thus, for RX J2117+3412 we found a large disagreement between the
spectroscopic and the seismic mass (
)
and in the opposite direction than for PG 1159-035. Note that models
with such low masses (at the effective
temperature of RX J2117+3412) are outside of the instability domain
(see Fig. 6).
Finally, we have the cases of PG 2131+066, PG 1707+427 and PG 0122+200. According to our evolutionary tracks, the mass of these
stars should be of
for PG 2131+066 and of
for PG 1707+427 and PG 0122+200.
However, in order to have values of
compatible with the observed mean period spacings, the stellar masses
should be substantially larger, of about
for PG 2131+066 (
),
for PG 1707+427 (
), and
for PG 0122+200 (
).
Models with such high stellar masses (at
the effective temperature of the stars under consideration) are
pulsationally unstable (see Fig. 6).
Given the spectroscopic uncertainties in the determination of
,
these solutions still could be compatible with those
derived from our evolutionary tracks and stability analysis.
We conclude that the stellar masses of naked GW Vir stars as
predicted by our evolutionary tracks are generally
lower than those suggested by the period spacing data. On the contrary,
for the PNNV RX J2117+3412 the evolutionary tracks predict a spectroscopic
mass about
higher than the seismic derivation.
Although our full evolutionary PG 1159 models hint at a general agreement
between the spectroscopic and seismic masses of pulsating PG 1159 stars,
persisting discrepancies could still reflect a problem in the stellar
modelling during the pulsing AGB phase of progenitor stars, as noted by
Werner & Herwig (2006).
In this paper we re-examined the pulsational stability
properties of GW Vir stars. We performed
extensive nonadiabatic computations on PG 1159 evolutionary models with
stellar masses ranging from 0.530 to
.
For each
sequence of models, we computed the complete evolutionary stages of
PG 1159 progenitors starting from the Zero Age Main Sequence. Evolution
was pursued through the thermally pulsing AGB and born-again (VLTP)
phases to the domain of the PG 1159 stars. The employment of such full
evolutionary PG 1159 models constitutes a substantial improvement over
previous studies on GW Vir stars regarding the stellar modelling.
Numerous detailed investigations of pulsating PG 1159 stars have been performed on the basis of artificial stellar models. In spite of the fact that significant pulsation damping and driving occur in PG 1159 envelope stars, the employment of such simplified stellar configurations appear not well justified in the case of these stars. This is in contrast to the situation of their more evolved counterparts, the white dwarf stars, for which their thermo-mechanical structure has relaxed to the correct one by the time the pulsational instability domains are reached. The main goal of the present work has been to assess to what degree the conclusions arrived at in previous studies on PG 1159 stars change when realistic stellar configurations are adopted.
Our study confirms the following results, already known from previous studies:
Acknowledgements
We wish to thank our anonymous referee for the constructive comments and suggestions that greatly improved the original version of the paper. This research was supported in part by the PIP 6521 grant from CONICET.