A&A 439, L31-L34 (2005)
DOI: 10.1051/0004-6361:200500154
A. H. Córsico1,
- L. G. Althaus1,2,
1 - Facultad de Ciencias Astronómicas y Geofísicas,
Universidad Nacional de La Plata, Paseo del Bosque, s/n, 1900 La
Plata, Argentina
2 -
Departament de Física Aplicada,
Universitat Politècnica de Catalunya, Av. del Canal Olímpic,
s/n, 08860 Castelldefels, Spain
Received 18 May 2005 / Accepted 8 July 2005
Abstract
The present letter is aimed at exploring the influence of
overshooting during the central helium burning in pre-white dwarf
progenitors on the pulsational properties of PG 1159 stars. To this
end we follow the complete evolution an intermediate-mass white dwarf
progenitor from the zero age main sequence through the thermally
pulsing and born-again phases to the domain of the PG 1159 stars. Our
results suggest that the presence of mode-trapping features in the
period spacings of these hot pulsating stars could result from
structure in the carbon-oxygen core. We find in particular that in
order to get enough core structure consistent with observational
demands, the occurrence of overshoot episodes during the central
helium burning is needed. This conclusion is valid for thick
helium envelopes like those predicted by our detailed evolutionary
calculations. If the envelope thickness were substantially smaller,
then the occurrence of core overshooting would be more difficult to
disentangle from the effects related to the envelope transition
zones.
Key words: stars: evolution - stars: interiors - stars: white dwarfs - stars: oscillations - stars: convection
Convective overshooting is a longstanding problem in the theory of stellar structure and evolution. It is well known on theoretical grounds that during many stages in their lives stars experience overshoot episodes, that is partial mixing beyond the formally convective boundaries as predicted by the Schwarzschild criterium of convective stability (Zahn 1991; Canuto 1992; Freytag et al. 1996; see also Renzini 1987). In particular, core overshooting taking place during central burning is an important issue because it has significant effects on the stellar structure and evolution. Over the years, considerable observational effort has been devoted to demonstrating the occurrence of core overshooting. Indeed, confrontation of stellar models with a wide variety of observational data suggests that convective overshoot takes place in real stars (see Stothers & Chin 1992; Alongi et al. 1993; Kozhurina-Platais et al. 1997; Herwig et al. 1997; von Hippel & Gilmore 2000, among others).
However, most of the evidence about the occurrence of core overshooting relies primarily on observational data from the very outer layers of stars from where radiation emerges. A more promising and direct way of placing constraints on the physical processes occurring in the very deep interior of stars is by means of the study of their pulsational properties. Pulsating white dwarf stars are particularly important in this regard. In fact, white dwarfs constitute the end product of stellar evolution for the vast majority of stars, and the study of their oscillation spectrum through asteroseismological techniques has become a powerful tool for probing the otherwise inaccessible inner regions of these stars (Bradley 1998; Metcalfe et al. 2002; Metcalfe 2003). White dwarf asteroseismology has also opened the door to peer into the physical processes that lead to the formation of these stars. In particular, Straniero et al. (2003) have raised the issue of using pulsating white dwarfs to constrain the efficiency of extra mixing episodes in the core of the white dwarf progenitors.
This letter is aimed at specifically assessing the feasibility of employing pulsating PG 1159 stars to demonstrate the occurrence of core overshoot during the core helium burning phase. Pulsating PG 1159 stars (or variable GW Virginis) are very hot hydrogen-deficient post-AGB stars with surface layers rich in helium, carbon and oxygen that exhibit g-mode luminosity variations. These stars are thought to have experienced a very late helium-shell flash during their early cooling phase after hydrogen burning has almost ceased - a born-again episode; see Fujimoto (1977), Schönberner (1979). As shown by Kawaler & Bradley (1994), variable PG 1159 stars are particularly important to infer fundamental properties about pre-white dwarfs in general, such as the location of chemical interfaces and envelope masses. Specifically, we present an adiabatic pulsation study based on detailed stellar models, the evolution of which has been followed from the zero-age main sequence through the thermally pulsing phase and born-again episode to the PG 1159 state. This fact allows us to obtain PG 1159 models with a physically sound internal structure consistent with the predictions of the theory of stellar evolution.
The adiabatic pulsational analysis presented in this work are based on
evolutionary stellar models that take into account the complete
evolution of the progenitor star (see Althaus et al. 2005, for
details). Specifically, the evolution of an initially
stellar model from the zero-age main sequence has been followed all
the way from the stages of hydrogen and helium burning in the core up
to the tip of the AGB where helium thermal pulses occur. After
experiencing 10 thermal pulses, the progenitor departs from the AGB and evolves toward high effective temperatures, where a final thermal
pulse takes place soon after the early white dwarf cooling phase is
reached - a very late thermal pulse and the ensuing born-again
episode, see Blöcker (2001) for a review. During this episode, most
of the residual hydrogen envelope is engulfed by the helium-flash
convection zone and completely burnt. After the occurrence of a
double-loop in the Hertzsprung-Russell diagram, the now
hydrogen-deficient, quiescent helium-burning 0.5895-
remnant evolves at constant luminosity to the domain of PG 1159 stars
with a surface chemical composition rich in helium, carbon and oxygen:
(4He, 12C, 16O) = (0.306, 0.376, 0.228). This is in
good agreement with surface abundance patterns observed in pulsating
PG 1159 stars (Dreizler & Heber 1998; Werner 2001). Also, the surface
nitrogen abundance (about 0.01 by mass) predicted by our models is in
line with that detected in pulsating PG 1159 stars (see Dreizler &
Heber 1998). We mention that abundances changes are described by
means of a time-dependent scheme that simultaneously treats nuclear
evolution and mixing processes due to convection and overshooting. A
treatment of this kind is particularly necessary during the extremely
short-lived phase of the born-again episode, for which the assumption
of instantaneous mixing is inadequate. Overshooting is treated as an
exponentially decaying diffusion process and has been considered
during all evolutionary phases. In particular, overshooting occurring
toward the end of core helium burning phases yields a sharp variation
of the chemical composition in the carbon/oxygen core. Radiative
opacities are those of OPAL (including carbon- and oxygen-rich
compositions, Iglesias & Rogers 1996), complemented, at low
temperatures, with the molecular opacities from Alexander & Ferguson
(1994).
![]() |
Figure 1:
The run of the squared Brunt-Väisälä frequency in terms
of the mass coordinate, corresponding to a 0.5895-![]() ![]() ![]() ![]() |
Open with DEXTER |
As for pulsational calculations, we have computed adiabatic g-mode
periods by employing an updated version of the pulsation code
described in Córsico et al. (2001). We limited our calculations
to the degrees
because the periods observed in pulsating PG 1159 stars
have been identifiedwith
.
To get values of periods as
precise as possible, we have employed about
2700-3000 mesh-points to
describe our background stellar models. The prescription we follow to
assess the run of the Brunt-Väisälä frequency (N) is the
so-called "Ledoux Modified'' treatment (see Tassoul et al. 1990),
appropriately generalized to include the effects of having several
nuclear species with varying abundance in a given region. In this
numerical treatment the contribution to N from any change in
composition is almost completely contained in the Ledoux term B;
this fact renders the method particularly useful to infer the relative
weight that each chemical transition region have on the mode-trapping
properties of the model.
![]() |
Figure 2:
The forward period spacing
![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
We begin by examining Fig. 1 in which a representative
spatial run of the Brunt-Väisälä frequency of a PG 1159 model is
displayed. The model is characterized by a stellar mass of 0.5895 ,
an effective temperature of
139 000 K and a
luminosity of
.
In addition, the plot shows
the internal chemical stratification of the model for the main nuclear
species (upper region of the plot), and for illustrative purposes the
profile of the Ledoux term B (inset). The figure emphasizes the
role of the chemical interfaces on the shape of the
Brunt-Väisälä frequency. At the core region there are several
peaks at
resulting from steep variations in
the inner oxygen/carbon profile. The stepped shape of the carbon and
oxygen abundance distribution within the core is typical for
situations in which extra mixing episodes beyond the fully convective
core during central helium burning are allowed (Straniero et al. 2003).
In particular, the sharp variation around
induced by mechanical overshoot could be a potential source of mode
trapping in the core region. The bump in N at
is other possible source of mode trapping, in this case associated
with modes trapped in the outer layers. This feature is caused by the
chemical transition of helium, carbon and oxygen resulting from
nuclear processing in prior evolutionary stages.
The influence of the chemical composition gradients on the pulsation
pattern is clearly shown in panel (a) of Fig. 2, in which the
forward period
spacing (
)
is plotted in terms of the periods (
)
for
the same model analyzed in Fig. 1. The plot shows very
rapid variations in
everywhere in the period spectrum,
with trapping amplitude (measured as the interval in
between a period spacing maximum and the adjacent minimum) up to about
6 s and a trapping cycle (measured as the interval in
between
the period spacing minima) of
70 s. The rather complex
period-spacing diagram shown by Fig. 2 is typical of models
characterized by several chemical interfaces. In order to disentangle
the effect of each chemical composition gradient on mode trapping, we
follow the procedure of Charpinet et al. (2000). Specifically,
we minimize - although no completely eliminate
- the effects of a given chemical interface simply by forcing the
Ledoux term B to be zero in the specific region of the star in which
such interface is located. In this way, the resulting mode trapping
will be only due to the remainder chemical interfaces. Specifically,
we have recomputed the entire g-mode period spectrum assuming (1)
B= 0 at the region of the O/C/He chemical interface (
), and (2) B= 0 at the region of O/C chemical interface
(
)
(see inset of Fig. 1). The
results are shown in panels (b) and (c) of Fig. 2,
respectively. By
comparing the different cases illustrated, an important conclusion
emerges from this figure: the chemical transition region at
is responsible for the non-uniformities in
only for
s (panel c), whereas the chemical composition gradients
in the core region (
)
cause the mode-trapping
structure in the rest of the period spectrum (panel b).
From the above discussion, it is clear that the mode-trapping
features predicted by full evolutionary PG 1159 models are dominated by
the core chemical structure left by prior overshoot episodes. This is
particularly true for the range of periods observed in GW Vir
stars. On its hand, the more external chemical transition has a minor
influence, except in the regime of short periods. This finding is
clearly at odds with previous results reported by Kawaler & Bradley
(1994). Indeed, these authors have found that the mode-trapping
properties of their PG 1159 models are fixed mainly by the outer O/C/He transition region, to such a degree that they have been able to employ
mode-trapping signatures as a sensitive locator of this transition
region. We note that our PG 1159 stellar models differ considerably
from those employed by Kawaler & Bradley (1994), particularly
concerning the details of the treatment of the evolutionary stages
that lead to the formation of PG 1159 stars. Of particular interest is
the presence of a much less pronounced chemical transition in the C/O core of the Kawaler & Bradley (1994) models, as compared with the
rather abrupt overshoot-induced chemical gradients at
in our full PG 1159 evolutionary models. In addition, we
note that our evolutionary treatment predict thick helium envelopes.
Thus, trapping structure predicted by our models is due mostly to the
core chemical gradients. If we artificially minimize the effect of
these gradients we immediately recover the results of Kawaler &
Bradley (1994) for the case of thick helium envelopes.
In addition to the issue of core overshooting, other relevant
point regarding the core chemical stratification of our models are
the adopted reaction rates during the central helium burning phase.
Following a suggestion of an anonymous referee, we have explored the
possibility of a smaller rate of the 12C
O
nuclear reaction (see Kunz et al. 2002). We found
that with a lower rate of that reaction, the resulting central
abundances of oxygen and carbon are quite similar, partially
smoothing the chemical steps in the core. However, appreciable
structure remains that is still able to give clear pulsational
signals associated to the occurrence of prior core overshooting.
![]() |
Figure 3:
The forward period spacing
![]() ![]() ![]() ![]() |
Open with DEXTER |
In Fig. 3 we compare the predictions of our calculations with the pulsational spectrum of PG 1159-035, the prototype of GW Vir stars. We do not intend to perform here a detailed asteroseismological fit to this star. Instead, we want to show that the structure associated with the observed period spacing of PG 1159-035 could be reflecting the presence of chemical gradients in its inner core, a fact that is borne out by Fig. 3. In fact, we find that according to the predictions of the theory of post-AGB evolution for the chemical stratification for the pulsating PG 1159 stars, the occurrence of core overshoot episodes during central helium burning is required to be consistent with seismological demands of these stars.
We judge that the period-spacing distribution exhibited by PG 1159 stars bears the signature of core overshoot episodes and that improved observations would eventually turn these pulsating pre-white dwarfs into a powerful tool for shedding new lights on the mixing and central burning processes occurred in the white dwarf progenitors of intermediate masses. However, it is not unconceivable that stellar winds during the PG 1159 stage could reduce the helium content considerably, with the consequent result that the trapping features induced by core overshooting could not be disentangled from that caused by the O/C/He transition zone.
Acknowledgements
This research was partially supported by the Instituto de Astrofísica La Plata. L.G.A. acknowledges the Spanish MCYT for a Ramón y Cajal Fellowship.