Issue |
A&A
Volume 499, Number 1, May III 2009
|
|
---|---|---|
Page(s) | 257 - 266 | |
Section | Stellar structure and evolution | |
DOI | https://doi.org/10.1051/0004-6361/200810727 | |
Published online | 30 March 2009 |
Asteroseismology of hot pre-white dwarf stars: the case of the DOV stars PG 2131+066 and PG 1707+427, and the PNNV star NGC 1501
A. H. Córsico1,2,3 - L. G. Althaus1,2,3 - M. M. Miller Bertolami1,2,4 - E. García-Berro5,6
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,
Argentina
3 -
Member of the Carrera del Investigador Científico y
Tecnológico, CONICET,
Argentina
4 -
Fellow of CONICET, Argentina
5 -
Departament de Física Aplicada,
Escola Politècnica Superior de Castelldefels,
Universitat Politècnica de Catalunya, Av. del Canal Olímpic, s/n,
08860 Castelldefels, Spain
6 -
Institute for Space Studies of Catalonia,
c/Gran Capità 2-4, Edif. Nexus 104,
08034 Barcelona, Spain
Received 31 July 2008 / Accepted 26 March 2009
Abstract
Aims. We present an asteroseismological study on the two high-gravity pulsating PG 1159 (GW Vir or DOV) stars, PG 2131+066 and PG 1707+427, and on the pulsating [WCE] star NGC 1501. All of these stars have been intensively scrutinized through multi-site observations, so they have well resolved pulsation spectra.
Methods. We compute adiabatic g-mode pulsation periods on PG 1159 evolutionary models with stellar masses ranging from 0.530 to
.
These models take into account the complete evolution of progenitor stars, through the thermally pulsing AGB phase, and born-again episode. We constrain the stellar mass of PG 2131+066, PG 1707+427, and NGC 1501 by comparing the observed period spacing with the asymptotic period spacing and with the average of the computed period spacings. We also employ the individual observed periods in search of representative seismological models for each star.
Results. We derive a stellar mass of
for PG 2131+066,
for PG 1707+427, and
for NGC 1501 from a comparison between the observed period spacings and the computed asymptotic period spacings, and a stellar mass of
for PG 2131+066,
for PG 1707+427, and
for NGC 1501 by comparing the observed period spacings with the average of the computed period spacings. We also find, on the basis of a period-fit procedure, asteroseismological models representatives of PG 2131+066 and PG 1707+427. These best-fit models are able to reproduce the observed period patterns of these stars with an average of the period differences of
s and
s, respectively. The best-fit model for PG 2131+066 has an effective temperature
K, a stellar mass
,
a surface gravity
,
a stellar luminosity and radius of
and
,
respectively, and a He-rich envelope thickness of
.
We derive a seismic distance
pc and a parallax
mas. The best-fit model for PG 1707+427, on the other hand, has
K,
,
,
,
,
and
,
and the seismic distance and parallax are
pc and
mas. Finally, we have been unable to find an unambiguous best-fit model for NGC 1501 on the basis of a period-fit procedure.
Conclusions. This work closes our short series of asteroseismological studies on pulsating pre-white dwarf stars. Our results demonstrate the usefulness of asteroseismology for probing the internal structure and evolutionary status of pre-white dwarf stars. In particular, asteroseismology is able to determine stellar masses of PG 1159 stars with an accuracy comparable or even better than spectroscopy.
Key words: stars: evolution - stars: interiors - stars: oscillations - white dwarfs
1 Introduction
Pulsating PG 1159 stars (also called GW Vir or DOV stars) 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 luminosity variations with periods ranging from 5 to 50 min, attributable to non-radial pulsation g-modes (see Winget & Kepler 2008 and Fontaine & Brassard 2008, for a recent reviews). PG 1159 stars are thought to be the evolutionary link between Wolf-Rayet type central stars of planetary nebulae and most of the hydrogen-deficient white dwarfs (Althaus et al. 2005). It is generally accepted that these stars have their origin in a born-again episode induced by a post-AGB helium thermal pulse - see Iben et al. (1983), Herwig et al. (1999), Lawlor & MacDonald (2003), and Althaus et al. (2005) for recent references.
Recently, considerable observational effort has been invested to study
pulsating PG 1159 stars. Particularly noteworthy are the works of
Vauclair et al. (2002) on RX J2117.1+3412, Fu et al. (2007) on PG 0122+200, and Costa et al. (2008) and Costa & Kepler (2008) on PG 1159-035. These stars have been
monitored through long-term observations carried out with the Whole
Earth Telescope (Nather et al. 1990). On the theoretical front,
recent important progress in the numerical modeling of PG 1159 stars
(Althaus et al. 2005; Miller Bertolami & Althaus 2006, 2007a) has
paved the way for unprecedented asteroseismological inferences for the
mentioned stars (Córsico & Althaus 2006; Córsico et al. 2007a,b, 2008). The new generation of PG 1159 evolutionary models
of Miller Bertolami & Althaus (2006) is derived from the complete
evolutionary history of progenitor stars with different stellar masses
and an elaborate treatment of the mixing and extra-mixing processes
during the core helium burning and born-again phases. The success of
these models at explaining the spread in surface chemical composition
observed in PG 1159 stars (Miller Bertolami & Althaus 2006), the short
born-again times of V4334 Sgr (Miller Bertolami & Althaus 2007b), and
the location of the GW Vir instability strip in the
plane (Córsico et al. 2006) renders reliability to the
inferences drawn from individual pulsating PG 1159 stars.
Besides the mentioned three well-studied pulsating PG 1159 stars, there exist two other variable stars of this class that have been also intensively scrutinized through the multi-site observations of the WET: PG 2131+066 and PG 1707+427. In addition, there is a variable central star of planetary nebula (PNNV), NGC 1501, which has been the subject of a nearly continuous photometric coverage from a global observing campaign by Bond et al. (1996). We briefly summarize the properties of these stars below.
PG 2131+066 was discovered as a variable star by Bond et al. (1984) with
periods of about 414 and 386 s, along with some other
periodicities. On the basis of an augmented set of periods from WET
data, Kawaler et al. (1995) obtained a precise mass determination of
,
a luminosity of
,
and a
seismological distance from the Earth of d= 470 pc. Spectroscopic
constraints of Dreizler & Heber (1998), on the other hand, gave
,
K,
,
and
for PG 2131+066. By using this updated determination
of the effective temperature, Reed et al. (2000) refined the
procedure of Kawaler et al. (1995) and found
,
and d= 668 pc.
PG 1707+427 was discovered to be a pulsator by Bond et al. (1984). Dreizler
& Heber (1998) obtained
K,
,
and
then
and
were inferred
from their spectroscopic study. Recently, Kawaler et al. (2004)
reported the analysis of multi-site observations of PG 1707+427 obtained with
WET. Preliminary seismic analysis by using 7
independent
modes with periods between 334 and 910 s
suggest an asteroseismological mass and
luminosity of
and
,
respectively.
NGC 1501 was classified as a [WCE] star, an early low-mass Wolf
Rayet-type PNNV with spectra dominated by strong helium and carbon
emission lines (Werner & Herwig 2006). The effective temperature
and gravity of this star are
K and
(Werner & Herwig 2006). The variable nature of NGC 1501 was discovered by Bond & Ciardullo (1993). The star shows ten
periodicities ranging from 5200 s down to 1154 s, although the largest
amplitude pulsations occur between 1154 s and 2000 s. Based on
period-spacing data, Bond et al. (1996) found a stellar mass of
for NGC 1501.
In this work we complete our small survey of asteroseismological inferences on pulsating PG 1159 stars - see Córsico et al. (2007a,b, 2008) for the previous studies of this series - by performing a detailed study of the GW Vir stars PG 2131+066 and PG 1707+427, and the [WCE] star NGC 1501. We employ the same stellar models and numerical tools as in our previous works. In particular, we go beyond the mere use of the observed period-spacing data by performing, in addition, detailed period-to-period fits on the pulsation spectrum of these stars. In our approach, we take full advantage of the state-of-the-art PG 1159 evolutionary models developed by Miller Bertolami & Althaus (2006). The paper is organized as follows. In Sect. 2 we briefly describe our PG 1159 evolutionary models. In Sect. 3 we derive the stellar mass of PG 2131+066, PG 1707+427, and NGC 1501 by using the observed period-spacing data alone. In Sect. 4 we infer structural parameters of these stars by employing the individual observed periods. In this section we derive asteroseismological models representative of PG 2131+066 and PG 1707+427 (4.1), and discuss their main structural and pulsational characteristics (4.2). In Sect. 5 we compare the results of the present paper with those of the asteroseismological study of Córsico & Althaus (2006, hereinafter CA06). Finally, in Sect. 6 we summarize our main results and make some concluding remarks.
2 Evolutionary models and numerical tools
The pulsation analysis presented in this work relies on a new
generation of stellar models that take into account the complete
evolution of PG 1159 progenitor stars. Specifically, the stellar models
were extracted from the evolutionary calculations recently presented
by Althaus et al. (2005), Miller Bertolami & Althaus (2006), and
Córsico et al. (2006), who computed the complete evolution of model
star sequences with initial masses on the ZAMS ranging from 1 to
.
All of the post-AGB evolutionary sequences were
computed using the LPCODE evolutionary code (Althaus et al. 2005) and were followed through the very late thermal pulse (VLTP)
and the resulting born-again episode that gives rise to the
H-deficient, He-, C-, and O-rich composition characteristic of PG 1159
stars. The masses of the resulting remnants are 0.530, 0.542, 0.556,
0.565, 0.589, 0.609, 0.664, and
.
For details about
the input physics and evolutionary code used, and the numerical
simulations performed to obtain the PG 1159 evolutionary sequences
employed here, we refer the interested reader to the works by Althaus
et al. (2005) and Miller Bertolami & Althaus (2006, 2007a).
We computed
g-mode adiabatic pulsation periods with the
same numerical code and methods we employed in our previous works, see
Córsico & Althaus (2006) for details. In addition, we performed
nonadiabatic computations with the help of the code employed in
Córsico et al. (2006) to evaluate the pulsational stability of the
asteroseismological models presented in Sect. 4. We analyzed about 3000
PG 1159 models covering a wide range of effective temperatures (
)
and luminosities (
), and a range of stellar masses (
).
Table 1: Stellar masses for all of the intensively studied pulsating PG 1159 stars, including also one pulsating [WCE] star. All masses are in solar units.
3 Mass determination from the observed period spacing
![]() |
Figure 1: The dipole asymptotic period spacing in terms of the effective temperature. Numbers along each curve denote the stellar mass (in solar units). Dashed (solid) lines correspond to evolutionary stages before (after) the turning point at the maxima effective temperature of each track. Also shown are the locations of PG 2131+066, PG 1707+427, and NGC 1501. |
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In this section we constrain the stellar mass of PG 2131+066, PG 1707+427, and NGC 1501 by comparing the asymptotic period spacing and the average of the computed period spacings with the observed period spacing. These approaches take full advantage of the fact that the period spacing of PG 1159 pulsators depends primarily on the stellar mass, and the dependence on the luminosity and the He-rich envelope mass fraction is negligible (Kawaler & Bradley 1994; Córsico & Althaus 2006). Most of the published asteroseismological studies on PG 1159 stars rely on the asymptotic period spacing to infer the total mass of GW Vir pulsators, the notable exception being the works by Reed et al. (2000) for PG 2131+066, Córsico et al. (2007a) for RX J2117.1+3412, Córsico et al. (2007b) for PG 0122+200, and Kawaler & Bradley (1994) and Córsico et al. (2008) for PG 1159-035. To assess the total mass of NGC 1501 we have considered the high-luminosity regime of the evolutionary sequences, while for PG 2131+066 and PG 1707+427 we have focused on the stages following the ``evolutionary knee'' for the PG 1159 stars, i.e. the low-luminosity regime.
3.1 First method: comparing the observed period
spacing (
)
with the asymptotic period
spacing (
)
Figure 1 displays the asymptotic period spacing for
modes as a function of the effective temperature for different
stellar masses. Also shown in this diagram is the location of PG 2131+066,
with
kK (Dreizler & Heber 1998), and
s (Reed et al. 2000), PG 1707+427, with
kK (Dreizler & Heber 1998), and
s (Kawaler et al. 2004), and
NGC 1501, with
kK (Werner & Herwig 2006), and
s (Bond et al. 1996). The
asymptotic period spacing is computed as
,
where
![]() |
(1) |
and N is the Brunt-Väisälä frequency (Tassoul et al. 1990). From the comparison between the observed





The method employed here is computationally inexpensive and widely used because it does not involve pulsational calculations. However, we must emphasize that the derivation of the stellar mass using the asymptotic period spacing may not be entirely reliable in pulsating PG 1159 stars that pulsate with modes characterized by low and intermediate radial orders (see Althaus et al. 2007). This is particularly true for PG 2131+066 and PG 1707+427. This shortcoming of the method is due to that the asymptotic predictions are strictly valid in the limit of very high radial order (long periods) and for chemically homogeneous stellar models, while PG 1159 stars are supposed to be chemically stratified and characterized by strong chemical gradients built up during the progenitor star life. A more realistic approach to infer the stellar mass of PG 1159 stars is presented below.
3.2 Second method: comparing the observed period
spacing (
)
with the average of the computed
period spacings (
)
The average of the computed period spacings is assessed as
,
where the ``forward'' period spacing is defined as
(k being the radial order) and N is the number of
computed periods laying in the range of the observed periods. For PG 2131+066,
s, according to Kawaler et al. (1995); for PG 1707+427,
s, according to Kawaler et al. (2004); and for
NGC 1501,
s, according to Bond et al. (1996).
![]() |
Figure 2: Same as Fig. 1, but for the average of the computed period spacings. For PG 2131+066 and PG 1707+427 only the stages after the ``evolutionary knee'' have been plotted. As in Fig. 1, we infer the mass of NGC 1501 by considering the stages before the evolutionary knee (dashed lines). |
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This method is more reliable for the estimation of the stellar mass
of PG 1159 stars than that described above because, provided that the
average of the computed period spacings is evaluated at the
appropriate range of periods, the approach is appropriate for the
regimes of short, intermediate and long periods (i.e., )
as
well. When the average of the computed period spacings is taken over
a range of periods characterized by high k values, then the
predictions of the present method become closer to those of the
asymptotic period spacing approach. On the other hand, the present
method requires of detailed period computations, at variance with the
method described in the above section. In addition, we note that both
methods for assessing the stellar mass rely on the spectroscopic
effective temperature, and the results are unavoidably affected by its
associated uncertainty.
In Fig. 2 we show the run of average of the computed
period spacings ()
for PG 2131+066, PG 1707+427, and NGC 1501 in terms of the
effective temperature for all of our PG 1159 evolutionary sequences.
The run of
depends on the range of
periods on which the average of the computed period spacing is done.
Note that the lines shown in Fig. 2 are very jagged and
jumped. This is because that, for a given star, the average of the
computed period spacings is evaluated for a fixed period interval, and
not for a fixed k-interval
.
By adopting the effective temperature of PG 2131+066, PG 1707+427, and NGC 1501 as given
by spectroscopy we found a stellar mass of
,
,
and
,
respectively. Our results
are shown in the third column of Table 1. These
values are
(for PG 2131+066) and
(for PG 1707+427) smaller than
those derived through the asymptotic period spacing, showing once
again that the asymptotic approach overestimates the stellar mass of
PG 1159 stars that, like PG 2131+066 and PG 1707+427, exhibit short and intermediate
pulsation periods (see Althaus et al. 2008a). On the contrary, there
is a very small discrepancy (in the opposite direction) of
0.9% for the case of NGC 1501, showing that in the long-period regime the
results for the stellar mass obtained using the asymptotic period
spacing and the average of the computed period spacings nicely agree
each other. A similar situation is found in the case of RX J2117.1+3412 (Córsico et al. 2007a).
3.3 Third method: using an approximate formula
To compare with previous works, we make an additional estimation
of the stellar mass of PG 2131+066, PG 1707+427, and NGC 1501 using the approximate
expression for the overall structure parameter
derived by
Kawaler & Bradley (1994):
where qy is the He-rich envelope mass fraction. This expression is derived by considering the dependence of the asymptotic period spacing on the total mass, stellar luminosity, and thickness of the He-rich outer envelope for a large grid of ``quasi evolutionary'' PG 1159 models in the luminosity range

Due to the very weak dependence of
on qy, we
arbitrarily fix it to a value of 10-2. Since the luminosity is
not known at the outset, we can compute it as
,
where
R*2= G M*/g. We use the values of g and
inferred through spectroscopy. Assuming that
is
known from the observed period spacing (
), we obtain an estimation of the
stellar mass from Eq. (2). Our results are shown in the
fourth column of Table 1. For PG 2131+066 and PG 1707+427 the
stellar masses obtained in this way are in very good agreement with
our values derived from the asymptotic period spacing (first row in
Table 1). This is not an unexpected result because,
as mentioned, the expression of Kawaler & Bradley (1994) is also
based on the asymptotic period spacing. The slight differences found
(below
2.5%) could be due to differences in the modeling
of PG 1159 stars. For NGC 1501, instead, there is a substantial difference
(
7%) between the prediction of this formula and our value
inferred from the asymptotic period spacing. This could be due to the
inadequacy of the formula of Kawaler & Bradley (1994) for the
high-luminosity regime characterizing the evolutionary status of NGC 1501.
Table 2:
Observed and theoretical ()
periods of the best-fit
model for PG 2131+066 (
,
K,
). Periods are in
seconds and rates of period change (theoretical) are in units
of 10-12 s/s. The observed periods are taken from
Kawaler et al. (1995).
4 Constraints from the individual observed periods
In this approach we seek pulsation models that best match the individual pulsation periods of PG 2131+066, PG 1707+427, and NGC 1501. For the three
stars, we assume that all of the observed periods correspond to
modes because the observed period spacings and the frequency
splittings by rotation are consistent with
(Kawaler et al. 1995, 2004; Bond et al. 1996). To measure the goodness
of the match between the theoretical pulsation periods (
)
and the observed individual periods (
), we follow
the same
procedure as in our previous works - see, e.g,
Córsico et al. (2007a) and Townsley et al. (2004).
Specifically, we employ the quality function defined as
![]() |
(3) |
where n is the number of observed periods. The observed periods are shown in the first column of Tables 2 and 3 for PG 2131+066 and PG 1707+427, and in Table 7 of Bond et al. (1996) for NGC 1501.
Next, we briefly explain the procedure we follow to found a model
representative of a target star. For a given model (characterized by a
given
)
corresponding to an evolutionary sequence of
stellar mass M*, we consider the first observed period of the list,
namely
,
and compute the successive squared differences
,
where the radial order
k varies from 1 to a given maximum value,
,
which
corresponds to
a theoretical period far longer that the maximum period observed in
the star. Next, we retain the minor squared difference, and then we
repeat the procedure but this time considering the second observed
period, namely
.
After the minor squared difference
associated with this observed period is stored, we proceed with the
next observed period, and so until the minor squared difference
associated with the last observed period
is
stored. The next step is to calculate the sum of these differences and
then obtain the value of
for the model under
consideration. It is worth
mentioning that
with this algorithm, the value of
does not
depend on the particular order in which the observed periods are fitted.
The complete algorithm is
repeated for all of the models of the sequence, and then,
a curve of
versus
is obtained for the complete sequence. This
procedure is carried out for all of our sequences. For each star of
interest, the PG 1159 model that shows the lowest value of
is
adopted as the ``best-fit model''.
Table 3:
Same as Table 2, but for the best-fit model for
PG 1707+427 (
,
K,
). The observed periods are
taken from Kawaler et al. (2004).
![]() |
Figure 3:
The inverse of the quality function of the period fit in
terms of the effective temperature (see text for details).
The vertical grey strip indicates the spectroscopic
|
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4.1 The search for the best-fit models
We evaluate the function
for stellar
masses of
0.530, 0.542, 0.556, 0.565, 0.589, 0.609, 0.664, and
.
For the effective temperature we employed a much
finer grid (
K). The quantity
in terms of the effective temperature for different
stellar masses is shown in Fig. 3 for PG 2131+066 (upper panel),
PG 1707+427 (middle panel), and NGC 1501 (lower panel), together with the
corresponding spectroscopic effective temperatures. We prefer to
show in our plots the quantity
instead
in
order to emphasize the location of models providing good agreements
between observed and theoretical periods. As mentioned,
the goodness of the match between the observed and theoretical
periods is reflected by the value of
.
The lower the value of
,
the better the period match. We will consider - admittedly
somewhat arbitrarily - that a peak in the quality function with
(that is,
)
is a good match between the theoretical and the observed periods.
For PG 2131+066 we find one strong maximum of
for a model
with
and
kK. Such a
pronounced maximum in the inverse of
implies an excellent
agreement between the theoretical and observed periods. Another much
less pronounced maxima, albeit at effective temperatures closer to the
spectroscopic estimation for PG 2131+066, are encountered for
at
kK and
at
kK. However, because the agreement
between observed and theoretical periods for these models are
substantially poorer than for the one with
,
we
adopt this last model as the best-fit asteroseismological model. A
detailed comparison of the observed
periods in PG 2131+066 with the theoretical periods of the best-fit model is provided in
Table 2. The high quality of our period fit is
quantitatively reflected by the average of the absolute period
differences
![]() |
(4) |
where

![]() |
(5) |
Note that




For PG 1707+427 we find one strong maximum of
for a model
with
and
kK, an
effective temperature compatible with the spectroscopic determination.
Another somewhat less pronounced maximum is found for a model with
and
kK. Despite
the fact that this model has an effective temperature very close to
the spectroscopic one, we choose the model with
as the best-fit model for PG 1707+427, because the period fit is
characterized by a better quality. Table 3 shows a
comparison between observed
and computed periods of
the best-fit model. We found in this case
s and
s.
The situation for NGC 1501 is markedly different than for PG 2131+066 and PG 1707+427.
Indeed, as clearly shown in the lower panel of Fig. 3, for
this star the
function exhibits numerous local maxima
at several values of the effective temperature and the stellar mass
(
,
0.565, 0.589, 0.664, and 0.741) that have
roughly the same amplitudes, making virtually impossible to isolate a
clear and unambiguous seismological solution. Thus, for NGC 1501 we are
unable to find a best-fit seismological model. This could be, in
part, due to the fact that the periods detected in NGC 1501 can be
associated to eigenmodes with radial orders k quite different from
each other (with a mean spacing of
), in
such a way that it is easy to find numerous models (characterized by
strongly different
and M*) that reproduce to a some
extent the observed period spectrum of NGC 1501. It could also be that
the impossibility to find a best fit model would be reflecting a
different evolutionary history for NGC 1501 than that assumed in this
work for our PG 1159 sequences.
The fourth column in Tables 2 and 3 shows the
rates of period change associated with the fitted modes for PG 2131+066 and
PG 1707+427, respectively. Our calculations predict that all of the pulsation
periods increase with time (
), in accordance with
the decrease of the Brunt-Väisälä frequency in the core of the
models induced by cooling. At the effective temperatures of PG 2131+066 and
PG 1707+427, cooling has the largest effect on
,
while
gravitational contraction, which should result in a decrease of
periods with time, becomes negligible and no longer affects the
pulsation periods, except for the case of modes trapped in the
envelope. Until now, the only secure measurements of
in
pre-white dwarf stars are those of PG 1159-035, the prototype of the class, by
Costa et al. (1999) and more recently by Costa & Kepler (2008). In
this last paper the authors obtained a mix of positive and negative
values of
,
indicating that for that star gravitational
contraction is still important. In principle, the needed time interval
that the observational data should cover in order to reach a
measurement of a
in PG 1159 stars like PG 2131+066 and PG 1707+427 is of
about ten years. Unfortunately, no future observations in the short
term that could allow a determination of
for these stars
and thus to check the predictions of our models are foreseen.
Finally, the last column in Tables 2 and 3 gives
information about the pulsational stability/instability nature of the
modes associated with the periods fitted to the observed ones. Full
nonadiabatic calculations employing the pulsation code described in
Córsico et al. (2006) indicate that, for the case of PG 2131+066, all the
fitted modes except one (that with period 507.9 s) are predicted to be
unstable. For the case of PG 1707+427, our nonadiabatic computations are able
to explain the existence of periodicities in the range
s only, while they fail to predict pulsational
instability for the observed modes with periods at 726, 746, and 909 s.
4.2 Characteristics of the best-fit models for PG 2131+066 and PG 1707+427
The main features of our best-fit model for PG 2131+066 are summarized in
Table 4, where we also provide the parameters of the star
extracted from other published studies. Specifically, the
second column corresponds to spectroscopic results from Werner & Herwig
(2006), whereas the third and fourth columns present results
from the pulsation studies of Kawaler et al. (1995) and Reed
et al. (2000), and from the asteroseismological model of this
work, respectively. The number in parenthesis is the
spectroscopic estimation of the stellar mass employing
the evolutionary tracks of Miller Bertolami & Althaus (2006).
In the present work, errors in
and
are estimated from the width
of the maximum in the function
with respect
and
,
respectively. The error in the stellar mass
comes from the grid resolution in M*. Errors in the rest of the
quantities are derived from these values. The effective temperature
of our best-fit model (
K) is somewhat higher
than - but still compatible with - the spectroscopic value (
K). On the other hand, the total mass of
the best-fit model (
)
is in agreement with the
value derived from the average of the computed period spacings (
), but at odds (
6% smaller) with
that inferred from the asymptotic period spacing (
)
(see Table 1). Also, the M* value of
our best-fit model is substantially larger than the spectroscopic mass
of
derived by Miller Bertolami & Althaus (2006),
but very similar to
according to Werner & Herwig (2006). A
discrepancy between the asteroseismological and the spectroscopic
values of M* is generally
encountered among PG 1159 pulsators - see Córsico et al. (2006,
2007a,b). Until now, the asteroseismological mass of PG 2131+066 has been about
larger (
)
than the
spectroscopic mass if we consider the early estimation for the
seismological mass quoted by Reed et al. (2000) and the derivation
of Miller Bertolami & Althaus (2006) for the spectroscopic
mass
.
In light of the
best-fit model derived in this paper, this discrepancy is slightly
reduced to about
(
).
Finally, our best-fit model for PG 2131+066 is somewhat more luminous and
less compact than what is suggested by the results of Reed et al.
(2000).
Table 4: The main characteristics of PG 2131+066.
The main properties of our best-fit model for PG 1707+427 are shown in Table 5. The
second column corresponds to spectroscopic results from Werner & Herwig
(2006), whereas the third and fourth columns present results
from the pulsation study of Kawaler et al. (2004) and from the
asteroseismological model of this work, respectively.
As for the case of PG 2131+066, the effective temperature of our
best-fit model for PG 1707+427 is slightly larger than the spectroscopic
measurement, but even in good agreement with it. Regarding the stellar
mass, our best-fit model has
,
which is in
agreement with the value derived from the average of the computed
period spacings (
), but at odds (
9% lower) with that inferred from the asymptotic period spacing
(
)
(see Table 1). On the other
hand, we note that M* for the best-fit model is in excellent
agreement with the spectroscopic derivation of Miller Bertolami &
Althaus (2006) (
versus
), but
is substantially lower than the
spectroscopic value quoted by Werner & Herwig (2006)
(
). Until now, the asteroseismological mass
of PG 1707+427 has been more than
larger (
)
than the spectroscopic mass if we adopt for the
seismological mass the value found by Kawaler et al. (2004) and
the derivation of Miller Bertolami & Althaus (2006) for the
spectroscopic mass. In light
of our best-fit model, this discrepancy is strongly reduced to about
(
). Finally, our
best-fit model for PG 1707+427 is slightly more luminous than what is
suggested by Kawaler et al. (2004).
Table 5: Same as Table 4, but for PG 1707+427.
4.3 The asteroseismological distance and parallax of PG 2131+066 and PG 1707+427
As in our previous works - see, e.g., Córsico et al. (2007a) -
we employ the luminosity of our best-fit models to infer the seismic
distance to PG 2131+066 and PG 1707+427. Following Kawaler et al. (1995), we adopt
for both stars (Werner et al. 1991). We
account for the interstellar absorption,
,
using the
interstellar extinction model of Chen et al. (1998). With all these
ingredients the seismic distance, d, can be easily computed using
the apparent magnitudes, which are
for PG 2131+066 and
for PG 1707+427 (Bond et al. 1984). We obtain a distance
pc and an interstellar extinction
for PG 2131+066 and
pc and
for PG 1707+427.
Our estimation of the distance to PG 2131+066 is 15% larger
than that derived by CA06 (
pc). This is because our
asteroseismological model is somewhat more luminous than that of CA06
(
versus
). On
the other hand, our distance is 20-25% larger than that obtained
by Reed et al. (2000) (
pc) on the basis of their own
asteroseismological analysis. This difference can be understood on the
basis that Reed et al. (2000) uses a luminosity of
,
somewhat lower than that of our
best-fit model for PG 2131+066, of
.
On the
other hand, our asteroseismological distance for PG 2131+066 is about 1.2 times longer than that quoted by Reed et al. (2000) of
680 pc
obtained on the basis of spectrum fitting, although both estimations
are compatible at the
level.
For PG 1707+427, our asteroseismological distance is in agreement with that
quoted by CA06, of 697 pc. Werner et al. (1991) obtain a
distance to PG 1707+427 of
1300 pc, substantially larger than our
estimation, but still within the quoted error bars. The different
value of Werner et al. (1991) is due to that they use a luminosity of
,
substantially higher than the luminosity
of our best fit model (
).
5 Comparison with the results of CA06
Following the recommendations of an anonymous referee, we include
in this section a detailed comparison between the PG 1159 models and
the asteroseismological results of the present paper and those of the
previous study by CA06. These authors performed an asteroseismological
analysis of four GW Vir stars (PG 0122+200, PG 1159-035, PG 2131+066, and PG 1707+427) on the basis of
a set of twelve PG 1159 evolutionary sequences with different stellar
masses (
)
artificially derived from the full evolutionary sequence of
computed by Althaus et al. (2005). That sequence is one of
the sequences we use in the present paper. Specifically, the
sequences of CA06 were constructed using LPCODE by appropriately
scaling the stellar mass of the
sequence before the
models reach the low-luminosity, high-gravity stage of the GW Vir
domain. Although this procedure leads to a series of unphysical
stellar models for which the helium-burning luminosity is not
consistent with the thermo-mechanical structure, the transitory stage
vanishes shortly before the star reaches the evolutionary ``knee'' in
the HR diagram (see Fig. 2 of CA06). As a consequence, those PG 1159
models were not suitable for the high-luminosity, low-gravity regime
corresponding, for instance, to RX J2117.1+3412, NGC 1501, K 1-16, HE 1429-1209, etc.
Because the sequences of CA06 with different stellar masses were
created starting from a single sequence with
,
the central abundances of C and O and the spatial extension of the C-O
core are not completely consistent with the value of the stellar mass,
except in the case of the models of the
sequence
itself. For instance, a model of CA06 with
has
a C-O core that is somewhat smaller and the central abundance of O is
substantially higher than what would be expected if the complete
evolution of the progenitor star were performed, as it is the case for
the models employed in the present paper. Figure 2 of Miller Bertolami
& Althaus (2006) clearly illustrate that, when the complete evolution
of the PG 1159 progenitor stars is taken into account, different
stellar masses are associated with different central abundances of C
and O, and different sizes of the C-O core. Specifically, the more
massive the models, the lower (higher) the central abundance of O (C)
and the larger the C-O core. In summary, for a given value of M*,
and
for stages after the evolutionary knee, the only
structural/physical difference between the PG 1159 models employed in
the present work and those of CA06 is related to the size of the C-O
core and the central abundances of O and C.
![]() |
Figure 4:
Comparison between some PG 1159 evolutionary tracks employed in
the present work (thick lines) and those of CA06 (thin lines).
The values are in solar masses. Note that the track
corresponding to the sequence of
|
Open with DEXTER |
In Fig. 4 we compare some evolutionary tracks (
M*= 0.530,
0.542, 0.589,
)
of the PG 1159 sequences employed in
the present work with the corresponding evolutionary tracks of CA06
(
M*= 0.53, 0.54, 0.59,
). A careful inspection of
this figure reveals that both sets of tracks generally differ, but
when the models reach the beginning of their white dwarf stage (
), they turn be in very close agreement. This
agreement is reached earlier in the case of sequences with stellar
masses close to
,
the value of the sequence from
which the remainder sequences of CA06 were generated. Interestingly
enough, the regime in which the tracks of CA06 are in agreement with
those employed in the present paper embraces the location of PG 2131+066 and
PG 1707+427, which have
and
,
respectively. Because of this, it is expected that the global
pulsation properties (i.e., asymptotic period spacing, average of the
computed period spacing) of both sets of models in that regime should
nearly agree, and consequently the asteroseismological inferences on
PG 2131+066 and PG 1707+427 based on these two different sets of models should not
be substantially distinct.
In Table 6 we present a comparison between the stellar
masses inferred in CA06 (their Table 1) and in the present study. Note
that, not surprisingly, the stellar masses derived in CA06 are in
excellent agreement with the values obtained in the present work,
irrespective of the particular method employed, being the differences
in all of the cases below .
This is a clear indication that the
asteroseismological results of this paper and that of CA06 for PG 2131+066 and PG 1707+427 are not seriously affected by the differences between both
sets of models. This adds credibility and robustness to the
asteroseismological results of the present study. This conclusion
should change for the case of PG 1159 pulsators with stellar masses too
departed from the value
and/or located at
earlier evolutionary stages.
Table 6:
Comparison between the stellar mass values (in
)
obtained in CA06 and in the present work.
6 Summary and conclusions
In this paper we presented an asteroseismological study of the high-gravity, low-luminosity pulsating PG 1159 stars PG 2131+066 and PG 1707+427 and of the high-luminosity PNNV [WCE] star NGC 1501. This is the fourth article of a short series of studies aimed at exploring the internal structure and evolutionary status of pulsating PG 1159 stars which have been intensively observed through multi-site campaigns. Our analysis is based on the full PG 1159 evolutionary models of Althaus et al. (2005), Miller Bertolami & Althaus (2006) and Córsico et al. (2006). These models represent a solid basis to analyze the evolutionary and pulsational properties of pre-white dwarf stars like PG 2131+066, PG 1707+427, and NGC 1501.
We first used the observed period-spacing data to obtain estimations of the stellar mass of PG 2131+066, PG 1707+427, and NGC 1501. The results are summarized in Table 1, where we provide a summary of results from the present and the previous works by us, and also from other pulsation and spectroscopic studies. We obtained three mass values for each star: the first one by comparing the observed period spacing with the asymptotic period spacing of our models (an inexpensive and widely used approach that does not involve pulsational calculations); the second one by comparing the observed period spacings with the average of the computed period spacing (an approach that requires of detailed period computations); and the third one on the basis of the approximate formula of Kawaler & Bradley (1994), which is based on the behavior of the asymptotic period spacing of a large grid of quasi-evolutionary PG 1159 models. The first and the third approaches are almost equivalent, and lead to similar, somewhat overestimated values of the stellar mass for PG 2131+066 and PG 1707+427. The second approach, clearly more realistic, conducts to smaller values of M* (and closer to the spectroscopic inferences) in the case of PG 2131+066 and PG 1707+427, and virtually the same M* value than that obtained from the first approach in the case of NGC 1501.
In the second part of our work, we sought for the models that best
reproduce the individual observed periods of each star. The period
fits were made on a grid of PG 1159 models with a quite fine resolution
in effective temperature (
K) although
admittedly coarse in stellar mass (
). We found asteroseismological models only for PG 2131+066 and PG 1707+427. For NGC 1501 we were unable to find a clear and unambiguous
seismological solution due to the existence of numerous and equivalent
minima characterizing the quality function employed in the period-fit
procedure. The pulsational properties of the ``best-fit'' models for
PG 2131+066 and PG 1707+427 are summarized in Tables 2 and 3,
respectively. In particular, we predict the values of the rates of
period change to be positive and in the range
s/s. Unfortunately, we have been unable to check the reality
of this prediction because the lack of any measurement of
for PG 2131+066 and PG 1707+427 for the moment. The structural characteristics of
these best-fit models are shown in Tables 4 and 5. In particular, the seismological masses are closer to
the spectroscopic ones in light of our best-fit models, as we found in
our previous works. In these tables we also show the seismological
distances and parallaxes of PG 2131+066 and PG 1707+427. We found a reasonable
agreement between our results and those of Kawaler et al. (1995), Reed et al. (2000), Kawaler et al. (2004), and CA06. We stress that
almost all differences between our results (Sects. 4.2 and 4.3) and those of earlier works are within the quoted errors.
In summary, in this work we have been able to estimate the stellar
mass of PG 2131+066, PG 1707+427, and NGC 1501 on the basis of the period-spacing
information alone. We have also been successful in finding
asteroseismological models for PG 2131+066 and PG 1707+427 from period-to-period
comparisons. In particular, the
and
of the
best-fit models are in very good agreement with the spectroscopic
measurements. Unfortunately, we fail to found an asteroseismological
model for NGC 1501. In principle, this shortcoming of our study could be
indicating some inadequacy inherent to the stellar modeling. Another
possible alternative could be the fact that the period spectrum of
NGC 1501 includes periodicities associated with g-modes with radial
orders very spaced from each other, in such a way that our
procedure of period-fit is inefficient to isolate a clear and
unambiguous asteroseismological solution. On the other hand, it would
be kept in mind that while both PG 2131+066 and PG 1707+427 are classified as PG 1159
stars, NGC 1501 is a [WCE] star. Although both classes are suspected to
form an evolutionary sequence, this possibility is still under debate
(Crowther 2008; Todt et al. 2008) and it could not be the case.
Therefore, the failure of our models to fit the period spectrum of
NGC 1501 might be indicative that this star has a very different
evolutionary history than PG 2131+066 and PG 1707+427.
We have also included a comparison between the models and results
of the present work and those of the study by CA06. The models
employed in the present work are the result of the complete evolution
of the progenitor stars, and as a result, they are characterized by
central chemical abundances and a size of the C-O core which are
consistent with the value of the stellar mass. This is not the case
for the models employed by CA06, which were artificially derived from
the full evolutionary sequence of
computed by
Althaus et al. (2005). In spite of these differences, our
asteroseismological results are in excellent agreement with those of
CA06. This adds credibility and robustness to the results of the
present study.
As the main conclusions of the present work, we can mention:
- -
- The full evolutionary models of PG 1159 stars employed in the present work lead to asteroseismological results for PG 2131+066 and PG 1707+427 that do not differ substantially from those predicted by CA06 on the basis of evolutionary sequences generated artificially. We note, however, that this agreement between both sets of computations is valid for PG 1159 stars located at the low-luminosity, high-gravity regime after the stars have passed the evolutionary knee in the HR diagram. It should be kept in mind that this conclusion should change for the case of stars located at earlier stages of evolution.
- -
- At present, the PG 1159 evolutionary models used in this work - and in the previous studies of this series - remain the only suitable for asteroseismological inferences on stars that are located at the high-luminosity, low-gravity regime before the evolutionary knee, such as RX J2117.1+3412 and NGC 1501.
- -
- The detailed fitting of the individual periods
(Sect. 4) gives somewhat different masses than analysis
based on asymptotic period spacing (methods 1 and 3 of Sect. 3), but in very good agreement with
the values of M* derived from the average of the computed period
spacings (method 2 of Sect. 3). Thus, method 2 is a
very appropriate way to estimating stellar masses, and detailed period
fits do not significantly improve the mass determinations. We note,
however, that the period-fit approach yields an asteroseismological
model from which one can infer, in addition to M*, the luminosity,
radius, gravity, and distance of the target star. In addition, the
period-fit approach does not require - in principle - external
constraints such as the spectroscopic values of
and g, i.e., the method works ``by letting the pulsation modes speak for themselves'' (see Metcalfe 2005, for an interesting discussion about this).
- -
- The nonadiabatic stability analysis does not at the moment predict instability for all of the fitted modes. This means that, in the frame of the linear nonadiabatic pulsation theory, some pulsation modes detected in PG 2131+066 and PG 1707+427 should not be excited. It is not clear at this stage the origin of this discrepancy. Maybe it could be attributed to the extreme sensitivity of the stability analysis of PG 1159 stars to the exact amounts of the main atmospheric constituents (see Quirion et al. 2004, for details).
- -
- The main conclusion of this series of papers - the
present work and Córsico et al. (2007a,b, 2008) - is that for
most well-observed pulsating PG 1159 stars (RX J2117.1+3412, PG 0122+200, PG 1159-035, PG 2131+066, and PG 1707+427) it is possible to found a stellar model (the asteroseismological
model) with M* and
near the spectroscopic measurements to a high internal accuracy. The next step is of course an assessment of the question if the asteroseismological models can provide more accurate masses for these objects. The scatter in the masses derived from the different asteroseismological methods (see Table 1) suggests that it may not be the case. In fact, when all asteroseismological methods are considered, the uncertainty in the determination of the mass amounts to
, comparable to the spectroscopic one (
, Werner et al. 2008). However, it is worth noting that, when results based on asymptotic period spacing (an approach that is not correct for the high-gravity regime of PG 1159 stars; see Althaus et al. 2008a) are not taken into account, the scattering in the derived masses is of only
.
Acknowledgements
Part of this work was supported by AGENCIA through the Programa de Modernización Tecnológica BID 1728/OC-AR, by the PIP 6521 grant from CONICET, by the MEC grant AYA2008-04211-C02-01, by the European Union FEDER funds, and by the AGAUR. This research has made use of NASA's Astrophysics Data System.
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Footnotes
- ...-interval
- As the star evolves towards
higher (lower) effective temperatures, the periods generally decrease
(increase) with time. At a given
, there are Ncomputed periods laying in the chosen period interval. Later when the model has evolved enough (heated or cooled) it is possible that the accumulated period drift nearly matches the period separation between adjacent modes (
). In these circumstances, the number of periods laying in the chosen (fixed) period interval is
, and
exhibits a little jump.
- ...
mass
- We elect the value of the spectroscopic mass of PG 2131+066 inferred by Miller Bertolami & Althaus (2006) for this comparison because they use the same post-born again PG 1159 evolutionary models we employ here in the determination of the asteroseismological mass, and because the spectroscopic masses quoted by Werner & Herwig (2006) are based on old post-AGB tracks.
All Tables
Table 1: Stellar masses for all of the intensively studied pulsating PG 1159 stars, including also one pulsating [WCE] star. All masses are in solar units.
Table 2:
Observed and theoretical ()
periods of the best-fit
model for PG 2131+066 (
,
K,
). Periods are in
seconds and rates of period change (theoretical) are in units
of 10-12 s/s. The observed periods are taken from
Kawaler et al. (1995).
Table 3:
Same as Table 2, but for the best-fit model for
PG 1707+427 (
,
K,
). The observed periods are
taken from Kawaler et al. (2004).
Table 4: The main characteristics of PG 2131+066.
Table 5: Same as Table 4, but for PG 1707+427.
Table 6:
Comparison between the stellar mass values (in
)
obtained in CA06 and in the present work.
All Figures
![]() |
Figure 1: The dipole asymptotic period spacing in terms of the effective temperature. Numbers along each curve denote the stellar mass (in solar units). Dashed (solid) lines correspond to evolutionary stages before (after) the turning point at the maxima effective temperature of each track. Also shown are the locations of PG 2131+066, PG 1707+427, and NGC 1501. |
Open with DEXTER | |
In the text |
![]() |
Figure 2: Same as Fig. 1, but for the average of the computed period spacings. For PG 2131+066 and PG 1707+427 only the stages after the ``evolutionary knee'' have been plotted. As in Fig. 1, we infer the mass of NGC 1501 by considering the stages before the evolutionary knee (dashed lines). |
Open with DEXTER | |
In the text |
![]() |
Figure 3:
The inverse of the quality function of the period fit in
terms of the effective temperature (see text for details).
The vertical grey strip indicates the spectroscopic
|
Open with DEXTER | |
In the text |
![]() |
Figure 4:
Comparison between some PG 1159 evolutionary tracks employed in
the present work (thick lines) and those of CA06 (thin lines).
The values are in solar masses. Note that the track
corresponding to the sequence of
|
Open with DEXTER | |
In the text |
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