A&A 372, 233-240 (2001)
DOI: 10.1051/0004-6361:20010485
M. Suran1 - M. Goupil2 - A. Baglin3 - Y. Lebreton2 - C. Catala4
1 - Astronomical Institute of Romanian Academy
2 -
DASGAL, UMR CNRS 8633, Observatoire de Paris-Meudon, France
3 -
DESPA, UMR CNRS 8632, Observatoire de Paris-Meudon, France
4 -
LAT, UMR CNRS 5572, Observatoire de Midi-Pyrénées, France
Received 22 January 2001 / Accepted 30 March 2001
Abstract
Pulsational properties of
1.8
stellar models
covering the latest stages of contraction toward the main
sequence up to early hydrogen burning phases
are investigated by means of linear nonadiabatic analyses.
Results confirm that pre-main sequence stars (pms) which cross the classical
instability strip on their way toward the main sequence are
pulsationally unstable with respect to the
classical opacity mechanisms.
For both pms and main sequence types of models in the lower part of the instability
strip, the unstable frequency range is found to be roughly the
same. Some non-radial unstable modes are very
sensitive to the deep internal structure of the star.
It is shown that discrimination between
pms and main sequence stages
is possible using differences in their oscillation frequency distributions
in the low frequency range.
Key words: stars: oscillations, stars: pre-main sequence
In the vicinity of the main sequence, the HR diagram is confusing as stars of similar global properties but different stages of evolution lie at the same position. In some cases the young pre-main sequence stars are recognized through specific characteristics, for instance the presence of nebulosity or high degree of activity. But, this is not always the case. An alternative is to take advantage of seismological information whenever possible. We therefore carry out a comparative study of the pulsational behavior of pre-main sequence (hereafter pms) and main sequence (hereafter postZams) stars of intermediate mass, in the classical instability strip.
Our study is restricted to the latest stages of the pre-main sequence contraction when traces of the formation phases have already disappeared, so that a pms model can be built with the quasi-static approximation.
The outer layers of pms and postZams stars having almost the same effective temperature and gravity are very similar. As these layers drive the pulsation in this range of temperature, it is reasonable to expect that pms stars in the instability strip are also destabilized by classical opacity mechanisms and that the same type of modes as for the postZams stars are excited. This idea has been confirmed by Suran (1998) and in a more detailed work but for pure radial modes of very young and cool stars by Marconi & Palla (1998). This supports vibrational instability as an explanation for the observed periodic variations detected for a few stars which are suspected to be in the pre-main sequence stage (Kurtz & Marang 1994; Catala et al. 1997; Kurtz 1999).
Differences, on the other hand, exist between the two stages: whereas pms has some relics of its gravitational contraction phase, the main sequence nuclear burning modifies the inner core structure, and postZams models develop chemical inhomogeneities. In a preliminary work, Suran (1998) has stressed some consequences of these differences in the structures of the frequency distributions. The present paper further investigates and compares the pulsational properties of these two phases for non-radial oscillations. It is organised as follows: stellar modelling and its physical inputs are described in Sect. 2, where comparable pms and postZams models are selected. Oscillation frequencies associated with nonradial modes for both types of objects are compared in Sects. 3 and 4, and emphasis is put on the structure of the frequency spectra in both stages. Finally in Sect. 5, we discuss the possibility to infer the evolutionary stage, i.e. the age of a variable object in this region of the HR diagram, through the pattern of its frequency spectrum.
In the early stages of pms evolution, various rapid- dynamical and thermal-processes occur which modify the internal structure of the protostar. By the time the protostar reaches the classical instability strip on its way toward the main sequence, these complex phenomena have long disappeared and the evolution has considerably slowed down. The usual quasi-static approximation then holds and is assumed here. Rotation is not considered and the star is modelled in a spherically symmetric equilibrium state.
Equilibrium models have been computed with CESAM (Morel 1997), an evolutionary code with a numerical accuracy of first order in time and third order in space. The models are built with the fast version of the code i.e. 300 mesh points in mass; pulsation calculations on the other hand use an extended 2000 mesh grid, calculated from the static model by means of the basis of the spline solution.
The EFF equation of state (Eggleton et al. 1973)
is used as it is sufficient for our purpose
in this mass range.
Opacities are provided by the Livermore Library
(Iglesias & Rogers 1996), complemented at
low temperatures (
K) by
Alexander & Ferguson 's (1994) tables.
The nuclear network contains the following species
1H, 3He, 4He, 12C, 13C,
14N, 15N, 16O, 17O.
The main nuclear reactions of pp+CNO cycles are included
with the species 2H, 7Li, 7Be set at equilibrium.
Nuclear reaction rates
are taken from Caughlan & Fowler (1988); weak screening is assumed.
The isotopic helium ratio is fixed at
and
the mixture of heavy elements is solar
(Grevesse & Noels 1993).
In the convection zones, the temperature gradient is
computed according to the standard mixing-length theory, with
the mixing-length defined as
,
where
is the pressure scale height. Assuming a constant
value of
along the main sequence (Fernandes et al. 1998), we have
chosen
,
the value obtained for a calibrated Sun
computed with the same physical assumptions.
The atmospheric layers are treated with the Eddington's grey approximation.
We consider the lower part of the instability strip where one
encounters the group of
Scuti stars defined here as Pop I stars
in a postZams stage.
The associated mass range approximately is 1.6-2
.
A mass of 1.80
with a
standard chemical composition for Population I (X=0.71, Y=0.27, Z= 0.02)
is therefore chosen as typical for the present work.
Figure 1 shows the evolutionary track running from the latest pms stages to the
early postZams ones for this 1.80
stellar model.
Peculiar stages are indicated and their characteristics listed in Table 1.
Local maxima of the luminosity along the pms track
correspond to the onset of rapid nuclear burning stages:
ignition of the p-p reaction occurs close to models M6 and M7;
the C12-N14 transformation (close to models M12, M13 and M14)
gives rise to core convection.
As the CNO
cycle contribution becomes increasingly important and eventually dominant,
the convective core slightly expands till the stage of
M10. Once the M14 stage is reached,
the star definitely settles into the
CNO cycle burning phase.
The zero age main sequence (Zams) model is defined as
the model with the minimum luminosity just
before evolution starts back toward the red. At this stage, M15 in Table 1,
the gravitational energy (
)
represents less than 1% of the total generated energy (
)
and the central initial hydrogen content has been only slightly
modified (see Table 1).
This study does not include the earliest stages, hence
it cannot provide exact
values of ages. However, evolutionary time scales for phases
after the ignition of
the central hydrogen burning phase are reliable.
We have therefore arbitrarily chosen the stage M1 as the initial epoch.
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Figure 1:
Evolutionary sequence of a 1.80
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log T | log L | R | age | ![]() |
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|
M1 | 3.808 | 1.064 | 2.751 | .0 | 0.7143 | 0.000 |
M2 | 3.844 | 1.133 | 2.527 | .4 | 0.7143 | 0.000 |
M3 | 3.863 | 1.160 | 2.389 | .6 | 0.7142 | 0.000 |
M4 | 3.872 | 1.170 | 2.319 | .7 | 0.7142 | 0.068 |
M5 | 3.880 | 1.176 | 2.251 | .8 | 0.7142 | 0.124 |
M6 | 3.887 | 1.177 | 2.184 | .9 | 0.7142 | 0.166 |
M7 | 3.895 | 1.160 | 2.056 | 1.1 | 0.7142 | 0.221 |
M8 | 3.897 | 1.129 | 1.970 | 1.25 | 0.7142 | 0.243 |
M9 | 3.890 | 1.030 | 1.816 | 1.6 | 0.7141 | 0.258 |
M10 | 3.875 | 0.924 | 1.721 | 2.4 | 0.7139 | 0.239 |
M11 | 3.897 | 1.003 | 1.702 | 3.8 | 0.7136 | 0.211 |
M12 | 3.912 | 1.038 | 1.657 | 4.4 | 0.7135 | 0.186 |
M13 | 3.919 | 1.050 | 1.621 | 4.8 | 0.7134 | 0.186 |
M14 | 3.923 | 1.047 | 1.586 | 5.2 | 0.7134 | 0.201 |
M15 | 3.922 | 1.006 | 1.521 | 11. | 0.7120 | 0.205 |
M16 | 3.920 | 1.044 | 1.608 | 300. | 0.6383 | 0.204 |
M17 | 3.918 | 1.058 | 1.649 | 400. | 0.6097 | 0.202 |
M18 | 3.908 | 1.102 | 1.813 | 700. | 0.5127 | 0.197 |
M19 | 3.897 | 1.131 | 1.978 | 900. | 0.4355 | 0.191 |
M20 | 3.893 | 1.139 | 2.029 | 950. | 0.4142 | 0.190 |
M21 | 3.889 | 1.146 | 2.086 | 1000. | 0.3920 | 0.188 |
M22 | 3.876 | 1.162 | 2.248 | 1120. | 0.3340 | 0.182 |
M23 | 3.872 | 1.167 | 2.312 | 1160. | 0.3132 | 0.180 |
M24 | 3.866 | 1.172 | 2.383 | 1200 | 0.2913 | 0.178 |
M25 | 3.851 | 1.183 | 2.594 | 1300. | 0.2320 | 0.170 |
As mentionned earlier, a source of ambiguity comes from
the fact that pms and postZams stars have
similar surface properties and therefore
lie at the same position in a HR diagram.
For instance, our
sequence in Fig. 1 shows that
the three pms models M4, M8, M13 can be confused with
the three postZams models M23, M19, M16 respectively, on
the basis of their sole location in the HR diagram (Table 1).
The question we address here is
whether small differences in the structures of the pms and
postZams models can induce significant
changes in their oscillation spectra.
The second issue is to determine whether these
frequency differences can be used to discriminate between the two evolutionary
stages.
Accordingly, our procedure consists of comparing seismological properties of pms and postZams models having the same position in the HR diagram and the same mass (herafter "associated models'').
As expected, the outer layers of associated models are very similar: density and temperature profiles and therefore the opacity profile are almost identical, we therefore refrain from showing them.
On the other hand, the central regions significantly differ.
A pms model is still slightly contracting
and remains chemically homogeneous.
In contrast, a postZams model has a
core which traces its nuclear history, changes
its chemical composition and density, and builds regions with gradients.
Our youngest models
correspond to the phase of the beginning of the p-p chain, with a very small expanding
convective core. During the main sequence phase, as the star ages,
hydrogen is depleted in the central region
and the fully mixed convective core receeds with time starting at
the Zams stage (approximately M15).
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Figure 2:
Behavior of the Brünt-Vaissälä frequency N
in normalized unit (see Eq. (1)) as a
function of radius (normalized to the total radius) for models
along the 1.80
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Signatures of these modifications are
clearly seen on the
profile of the Brünt Vaissälä frequency, N, as a
function of radius (Fig. 2). The sharp transition between zero values of N
within the convective core and positive values
in the overlaying radiative layers delimits the edge of the convective
core. As it can be seen in Fig. 2, during the pms evolution, the outward
progression of the edge of the expanding convective core
can be followed with the correlated outward progression of the
inner edge of N. The inner N maximum also moves
outward and decreases. When the model
evolves beyond the Zams on the main sequence,
the inner edge and maximum of the Brünt-Vaissälä
frequency receed inwards, while a narrower but
higher peak associated with the
gradient region
forms. It can be expected from Fig. 2 that modes with normalized frequencies
(Eq. (1) below) in the range 4-10 are the most sensitive to the size of the
convective core and consequently to the evolutionary status (pms or postZams)
of the star.
At every stage considered here, the time scale of evolution is much
longer than the pulsation period.
For instance our
sequence indicates time intervals of about
0.1-1 Myr between specific stages of evolution represented
in Fig. 1. Periods of oscillation of
Scuti stars typically range
between 1h-3h, reaching roughly 20 h for unstable g-modes.
Growth rates are of the order
of 100-500 yr, still much smaller than evolutionary time scales.
The oscillation
amplitudes are expected to grow or decay on a much faster (Kelvin-Helmoltz) time scale
than the (nuclear) evolutionary
time-scale, so that the amplitudes have settled onto their
saturating value (limit cycle) long before evolution can have any disturbing
influence.
These considerations authorize investigations of the pulsations
of pms stars by means of usual linear nonadiabatic analyses of static models
at a given stage of evolution (Cox 1980; Unno et al. 1989).
We treat the nonadiabatic effects in the formulation of
Unno et al. (1989), using Suran's oscillation code.
The oscillation eigenfrequencies (rad/s)
are denoted as
.
We will also use
frequencies in
Hz,
,
and normalized frequencies,
defined as
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(1) |
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(2) |
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(3) |
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(4) |
Interaction between pulsation and convection is neglected. This is certainly not justified for the cool stars but the effect is less important for stars close to the Zams. Neglecting this interaction equivalently affects pms and postZams models at the same effective temperature close to the Zams. Hence this cannot significantly affect the main conclusions of the present comparative study.
Figure 3 displays the evolution of the normalized frequencies, ,
for a 1.8
model evolving from stage 1 (pms)
to stage 25 (postZams) along the evolutionary track shown in Fig. 1.
The plot is restricted to
modes. These modes are usually thought to be
the easiest modes to detect, as their visibility
coefficients remain large after integration over the whole stellar disk.
The radial order interval is chosen such as to cover the range of
vibrationally unstable modes.
Evolution is represented in abscissae by a quantity which is
defined so as to change monotonously
throughout the pms and main sequence stages:
is the difference
between the radius of the model, R,
and the radius of a reference model,
,
which for
obvious reasons is chosen to be the Zams model (M15 model of Table 1).
Time goes from left to right. For pms models,
and for postZams models
.
Normalized
frequencies evolve with age quasi-symetrically
with respect to the Zams stage. Frequencies of radial modes and nonradial
pure p-modes
with given
remain nearly constant when evolving from late pms to
early postZams stages because the models change nearly homologously.
On the other hand, g-modes, as they are sensitive to the structure of the inner convective layers, do not behave homologously. One can see in Fig. 3 changes of the frequencies of the g-modes which arise from the "breathing'' of the core, (which moves back and forth) when a new nuclear reaction starts, at models M4 and M12: frequencies of g-modes increase with age for postZams models because the inner maximum of the Brünt-Väissala frequency increases with age and its sharp inner edge moves outward. In a symmetric way with respect to the Zams model, the frequencies of g-modes decrease when the pms model evolves toward the Zams stage because the position of the inner maximum of N moves inward with time.
In addition, the well known phenomenon of
avoided crossing (Unno et al. 1989) occurs
during the latest postZams
stages. The development of a
gradient at the edge of
the convective core of the postZams model
causes the development of
a sharp peak of the Brünt-Vaissälä frequency. Then
nonradial low frequency modes undergo
an avoided crossing which shifts the
frequency with respect to its value in the
absence of avoided crossing.
Only those modes which are sensitive to the edge of the convective core
have a different
behavior before and after the Zams stage.
These interacting modes exchange their physical nature. Their frequencies
penetrate into the p mode frequency domain, breaking
the regularity seen at high frequencies.
The progression of this phenomenon into the p-mode domain
can be followed from mode with
(
)
up to n=8 (
)
for
modes.
This leads to different and recognizable patterns in a
frequency spectrum, which can be considered as the signature of
different structures of the core and can
therefore tell whether the star is a pms or a main sequence one.
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Figure 3:
Normalized frequencies of a 1.8 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 4:
Growth rate ![]() ![]() ![]() |
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Finally, frequency calculations show the existence of strange modes i.e. surface modes trapped in very superficial outer layers (Unno et al. 1989; Buchler et al. 1997; Saio et al. 1998). Whether these modes are excited or not is probably strongly dependent on the convection-pulsation interaction not included here. As these surface modes are not essential for our purpose, they were disregarded in Fig. 3.
The linear nonadiabatic calculations
for modes with
for models of Table 1 show that the pms
models lying in the classical instability strip are vibrationally
unstable. The unstable modes are the same than those for
classical
Scuti stars, namely low radial order
g- and p-modes.
Figures 4 shows the behavior of the parameter
(Eq. (2)) with increasing
normalized frequencies
for
modes
for the M8 pms and the M19 main sequence models. The
instability strength as measured with
is independent of the degree
.
The
curves for pms and main sequence associated models
are almost identical. For these models, modes with radial order in the range
n=1-7 are found to be unstable.
The work integrands are found to be quite similar for associated
pms and postZams models. Computation of the work integrals
shows that, for both pms and
postZams models, the modes are driven by opacity
mechanisms which efficiently operate in the H and He
ionisation zones. The
mechanism contributes only very weakly
in the deep interior even in the youngest models.
Frequency differences for a given mode
between models M4 and M23
reach values as large as 50 Hz in real units.
This is emphasized in Fig. 6. Global parameters are
identical, therefore radial modes with same radial order
have about the same frequencies. The external layers are
similar, therefore the excitation range is about the same.
In this frequency range,
modes are the most affected by the structure of the
deep layers and are the main agents for differences in power spectra.
Pms and postZams models at the same location in the HR
diagram have similar mean densities,
hence one expects the large separation
to be the same for "associated'' models.
This is confirmed in Fig. 7 where the large
separations for
modes (for low radial order modes)
are displayed for associated models
and for the Zams
model. The large separation is indeed the same respectively for M4 and M23;
for M8 and M19; for M13 and M16. The large separation scales as
(GM/R3)1/2. Given the mass,
the large separation increases with age for
pms models (decreasing radii) and decreases with age
for postZams models (increasing radii); it is largest for the Zams
model which has the smallest radius.
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Figure 5:
Frequency distribution for
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Figure 6:
Theoretical power spectra
in the low frequency range where modes are excited. Top: postZams M23
model. Bottom: pms M4 model.
Modes with same ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 7:
Large separations
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The development of a
gradient at the edge of the
convective core can be seen on the frequency
large separation for nonradial modes in Fig. 8.
The large separation for
modes for associated
pms and postZams models
is expressed in normalized units
and plotted against the normalized
frequency
.
The existence of a mode in avoided crossing
generates a large departure of the frequency difference
from its mean value at the frequency of this mode.
The frequency at which a given
mixed mode occurs is correlated with the maximum value of the inner peak of
the Brünt-Vaissälä frequency arising from a larger
gradient.
Such a mixed mode is detected in postZams model M20, M22, M23 and M24 in
Fig. 8.
While the main sequence model ages, the progression of the
mixed mode toward higher frequencies
is clearly seen. The frequency
of the extra mode with the highest frequency
gives an indication of the age of
the postZams star.
On the other hand, pms models show a
quasi constant large separation (in normalized units) with frequency.
Differences in the inner layers of pms and postZams models can also be
detected by investigating the behavior of the
small separations
as a function of
the frequency
.
In the asymptotic regime i.e. for high radial order modes (solar like
oscillations) the small separation is defined as
.
In the low radial order range, avoided crossings
perturb the regular sequence of alternations of
and
modes.
We therefore find more convenient to define a small separation as
the difference between a
mode (radial order n')
and its closest
neighbor (radial order n which in some cases
at low frequency differs from n'+1).
Such small separations are plotted in Fig. 9. When the associated models are
very close to the Zams as for instance models M13-M16,
no difference exists between the small separations of the
models. For these models, as no
gradient has developed yet,
there is no avoided crossing in the excited frequency range.
On the other
hand,
for associated models located further away from the Zams,
for instance M4-M23, the structures of the
inner layers are significantly different;
this leads to a different behavior of
the small separations of the two models
at low frequency.
An avoided crossing is seen to occur for modes
with
Hz (
)
for model M23 which is not present for its associated pms model M4
(Fig. 9). At lower frequencies (below
Hz corresponding to
), a second feature is seen
for both models M4 and M23: this
sharp variation of the small separation is
due to modes
with frequencies close to the value of the secondary inner
maximum of the Brünt-Vaissälä
frequency (the only inner maximum for pms models)
near the edge of the convective core.
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Figure 8:
Large separations
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Pre-main sequence models of 1.7-2 crossing the classical instability strip on their way to the Zams, are
found to be vibrationally unstable, destabilized by the same mechanisms as
the
Scuti stars, as main sequence or postZams stars.
For a given mass and a given location in the HR diagram,
the frequencies of radial modes with same radial order
remain nearly
identical. As the outer layers in both stages are almost the
same, oscillations in
pms stars should have the same level of amplitudes as
in
Scuti stars; this means that
some modes are expected to be detectable from ground.
Low radial order modes with dual (gravity and pressure) nature penetrate in the upper layers of the star and there is a good chance to observe them. We have shown that they can act as discriminators, as they are very sensitive to the structure of the inner layers, which differ between pms and postZams stars. This is particularly interesting for this region of the HR diagram where classical indicators of youth as accretion disks or remaining nebulosities have disappeared.
Frequency shifts and differences in the frequency distributions of pms and main sequence oscillators can be detected if a sufficient number of modes are observed at low frequencies in the vinicity of the fundamental radial mode.
Important progress has been made in detecting
larger sets of frequencies from the ground using
networks of telescopes covering all longitudes. Nevertheless,
ideal conditions will only be
achieved in space.
The COROT experiment (Baglin et al.
1998), to be launched in 2004, will
continuously observe several stars over 20 to 150
days; this will provide an increase of a thousand for
the signal to noise ratio and a frequency resolution from 0.5
down
to 0.1
.
The large number of modes
which is expected to be detected will help to provide, at least, a
partial mode identification. Frequency histograms
for instance can provide the mean density
(Breger et al. 1999; Baglin et al. 2000;
Barban et al. 2001) and the structure of the frequency spectrum
will tell whether the star is a pms or a postZams star.
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Figure 9:
Small separations
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Pms stars are expected to be fast rotators
and it will be necessary to include
effects of rotation on the frequencies in the same way
as proposed by Soufi et al. (1998) for fast rotating
Scuti stars. This should enable to put constraints
on the internal rotation of these stars (Goupil et al. 1996).
It will then be possible to trace the history - and to better understand the
process - of transfer of angular momentum inside stars with
in their early stages of evolution.
Acknowledgements
We are grateful to P. Morel for making available the CESAM code. M. D. Suran acknowledges Paris-Meudon Observatory for providing him a grant. We thank an anonymous referee for his comments which helped to improve the writing of this manuscript.