Issue |
A&A
Volume 508, Number 2, December III 2009
|
|
---|---|---|
Page(s) | 849 - 854 | |
Section | Stellar structure and evolution | |
DOI | https://doi.org/10.1051/0004-6361/200912417 | |
Published online | 21 October 2009 |
A&A 508, 849-854 (2009)
Asteroseismic study of helium and heavy element diffusion in solar-type stars![[*]](/icons/foot_motif.png)
N. Gai1 - S. L. Bi1 - Y. K. Tang2 - L. H. Li3
1 - Department of Astronomy, Beijing Normal University, Beijing 100875, PR China
2 -
Department of Physics, Dezhou University, Dezhou 253023, PR China
3 -
Department of Astronomy, Yale University, New Haven, CT06520, USA
Received 3 May 2009 / Accepted 18 September 2009
Abstract
Aims. Element diffusion is a basic physical element
transport mechanism which induces the redistribution of chemical
elements. Using the asteroseismic method, we study the effects of
helium and heavy element diffusion on the internal structure and
stellar evolution of solar-type stars. We also provide asteroseismic
parameters for a grid of models which would be useful for direct
comparison with the Kepler mission observations.
Methods. We construct a grid of solar-type stellar models with various masses (from 0.8
to 1.2
)
and metallicities (Zi =
0.03, 0.025, 0.02, 0.015, 0.01, 0.005) with and without helium and
heavy element diffusion. We compute ``second differences'' and ``small
separations'' of the solar-metallicity models (Zi = 0.02)
to analyze the effects of diffusion on the convection zone, helium
abundance and the evolutionary sequence of the star. In order to study
the asteroseismic property of models with and without diffusion, we
compute the p-mode oscillation frequencies of low-degree modes for a grid of models and construct the (
,
)
asteroseismic diagram.
Results. We find that the element diffusion could speed up the
evolution of the star, especially in the main sequence. The results
show that it could enlarge the convective core and change the base of
the convection envelope. In addition, the helium and heavy element
diffusion make the models evolve to lower large and small separations
in the asteroseismic diagram. This effect is more efficient at lower
metallicity.
Key words: stars: evolution - stars: interiors - stars: oscillations
1 Introduction
Element diffusion is a basic physical transport mechanism that results
in a microscopic redistribution of stellar matter. Helium and metal
elements settle to the core and deplete their abundances in the outer
layers, while the hydrogen abundance is enriched at the surface. The
main diffusion mechanism includes pressure diffusion, thermal diffusion
and concentrational diffusion.
Radiative acceleration is also another very important mechanism
predicted to be important for solar-metallicity stars with masses
greater than approximately 1.2
(Chaboyer et al. 2001) or for main sequence stars with
(Michaud et al. 1976). In the present work, we do not consider the effect of radiative acceleration.
Helioseismic inferences have demonstrated the importance of including
element diffusion in solar models (Christensen-Dalsgaard et al. 1993).
Following the success of helioseismology, asteroseismology can help us
to know the stellar internal structure and understand stellar
evolution. Due to oscillation frequencies depending on density,
temperature, and other properties of the stellar interior, it can give
information about regions where the sound speed change rapidly. In the
stellar outer regions, the partial reflection of acoustic waves occurs
at the bottom of the outer convective zone, because there is a helium
element gradient which is induced by diffusion (Vauclair & Théado 2004; Théado et al. 2005; Castro &
Vauclair 2006; Mazumdar 2005). In the second helium ionization zone, the depression of
can also cause an oscillatory signal in the frequencies, which can be
used to determine the helium abundance in the envelope (Basu &
Antia 1995; Basu et al. 2004). These are generally studied with the help of the ``second differences'', defined as
=
+
- 2
(Gough 1990).
In the stellar internal regions, small separations have been
successfully applied to extract information about the stellar interior.
They are determined predominantly by the sound-speed
gradient in the core. Using the asymptotic theory of p-modes, the small separation is given by dll+2 = -
(see, e.g., Basu et al. 2007). It could be used to derive the size of the convective core (Roxburgh & Vorontsov 2001; Mazumdar et al. 2006), and give evidence for the presence or absence of overshooting (Bi et al.
2008). In addition, it is sensitive
to the helium core, and gives hints as to its evolutionary state. For
stars at the end of the main sequence, or at the beginning of the
sub-giant branch, the small separations dll+2 computed between l = 0 and l = 2 modes
may become negative. The reason is that the helium accumulates in the
convective core, which leads to a slightly steeper variation of the
sound speed than in the case of convection alone (Vauclair
2007; Soriano et al. 2007; Soriano & Vauclair 2008).
Asteroseismology is a powerful method for testing both the
stellar internal structure and evolution through the asteroseismic
observational data directly. Many ground telescopes and the space
missions, like the CoRoT mission (Baglin et al. 2006), are providing extensive data of very high quality on oscillations in a broad range of stars. NASA's Kepler mission was launched in 2009. The Kepler
mission will soon yield precise high-cadence time-series photometry of
hundreds of pulsating stars every few months for at least
3.5 years (Christensen-Dalsgaard et al. 2007; Metcalfe 2009). Except for the Sun, seismic observations of distant stars is likely to permit the detection of
only low degree modes of oscillations (l = 0-3). The most traditional asteroseismic diagram, which was proposed by Christensen-Dalsgaard (1984, 1988, 1993),
is a very useful technique combining useful features of the frequency
spectrum of stars to extract information about physical processes, for
example helium and heavy element diffusion, in the stellar interior,
that affect the frequencies. This well-known asteroseismic diagram
exploits the fact that the average large separation of radial modes
reflects the gross properties of a
star, like its mass and radius, and the average small separation
is more sensitive to the innermost layers of the star, and therefore to its evolutionary state (Mazumdar 2005; Tang et al. 2008).
In our work, firstly we compute evolutionary tracks with various masses
and metallicities, with and without helium and heavy element diffusion.
Using the ``second differences'' and ``small separations'', we choose
the solar-metallicity models to study the effects of helium and heavy
element diffusion on the internal structure and stellar evolution.
Secondly, in order to study the asteroseismic property of models with
and without diffusion and test the effect of diffusion by using real
seismic data, we construct the (
,
)
asteroseismic diagram. In the online material, we provide the asteroseismic parameters
,
,
for all the models with various masses and metallicities to compare them directly with frequency separations from the Kepler mission observations.
![]() |
Figure 1:
Evolutionary tracks computed for solar metallicity Z = 0.02 with masses of 0.8 |
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In Sect. 2, we give details of the evolutionary models and the computational method. The effects of diffusion on the stellar internal structure, evolution and its asteroseismic property also are analyzed in this section. We summarize our conclusions in Sect. 3.
2 Computation and analysis
2.1 Computation
Using the Yale stellar evolution code (YREC; Guenther et al. 1992; Demarque et al. 2008), we compute evolutionary sequences for models with masses 0.8 ,
0.9
,
1.0
,
1.1
,
1.2
,
the initial metallicities Zi = 0.03, 0.025, 0.02, 0.015, 0.01, 0.005, fixed helium abundance Yi = 0.28 and mixing-length parameter
=
1.75. The initial zero-age main sequence (ZAMS) models are derived from
pre-main-sequence evolution models. The input physics are described as
follows:
Diffusion: diffusion of both the helium and heavy element abundances (by weight Y and Z respectively) is considered in the stellar model computations, using the diffusion coefficients of
Thoul et al. (1994).
The diffusion of all metal elements are assumed to diffuse at the same
rate as fully ionized iron. The differences in the diffusion velocities
among the metal elements are small for the low mass stars (M
1.3
solar-metallicity stars). The 1.1
model especially behaves much like a solar model and all elements are affected by
gravitational settling at a similar pace (Turcotte et al. 1998a).
Radiative acceleration: we note that the Thoul et al. (1994)
diffusion coefficients include the effects of pressure diffusion
(i.e., gravitational settling), thermal diffusion
and concentrational diffusion, but exclude the effect of radiative
acceleration. Turcotte and collaborators have researched the effects of
radiative acceleration in detail in the Sun, solar metallicity F-stars,
AmFm stars, metal-poor stars and so on (Richer et al. 1998; Turcotte et al. 1998a,b; Richer et al. 2000; Richard et al. 2002a,b).
Radiative acceleration is another important chemical transport
mechanism. It is predicted to be important for main sequence stars
with
> 6000 K (Michaud et al. 1976). The above mentioned studies have shown further that radiative acceleration is predicted
to be important for solar metallicity stars with masses greater than approximately 1.2
(Chaboyer et al. 2001; Turcotte et al. 1998a). The metallicity dependence of the effects of atomic diffusion was studied by Richard et al. (2002b). They calculated evolutionary models for population II stars of 0.5-1.2
and [Fe/H] = -4.31
-0.71 to determine the effect of metallicity on the size of abundance
anomalies to be expected from gravitational settling, thermal diffusion
and radiative acceleration. From their discussion, especially Fig. 3 of Richard et al. (2002b),
we found that radiative acceleration could only counteract
about 10% of the gravitational diffusion even for their
metal-richest models Zi = 0.0067 (
-1.22). Since radiative acceleration is reduced for higher metallicity,
it is unlikely that radiative acceleration
is very important in the stars that we will model in this paper and we
concentrate on the effect of pressure diffusion, thermal diffusion and
concentrational diffusion.
Opacity: we use the OPAL GN93 opacities (Iglesias & Rogers 1996) along with low temperature opacities (Ferguson et al. 2005). The low temperature opacities are used for < 4.4 and the OPAL opacities are used for
> 4.5. In the transition region for 4.4
4.5, the low temperature opacities tables and the OPAL opacities are used.
Both the OPAL opacities and low temperature opacities will automatically interpolate to the required Z. The opacity routines, with YREC, perform a four-point Lagrangian interpolation scheme over a 4-dimensional grid of Z, X, T and .
The temperature range of OPAL GN93 is 3.75 <
< 8.7 and of low temperature opacity is 2.7 <
< 4.5, the radius range is -8.0 <
< 1.0. According to the formula
=
R(T/10-6)3, the opacity tables span
nearly all of the temperature and density space at all metallicities
needed for the model grid and the extremes of evolution state that we
considered.
Equation-of-state: the OPAL equation-of-state tables EOS2003 (Rogers & Nayfonov 2002) are used in the computation. In the outer regions ( <
5.8), the equation-of-state (i.e., EOS) routines determine
particle densities by solving the Saha
equation for the single ionization state of H and metals, and the
single and double ionization states of He. In the inner regions (
> 6.3),
the gas is assumed to be fully ionized and the Saha equation is not
used in the EOS calculation. In the transition region between these two
regions (5.8
6.3), the EOS interpolates between the partially and fully ionized
solutions with a ramp function and is averaged to obtain a smooth
transition.
Boundary condition: the two inner boundary conditions set the values of the radius and luminosity variables to zero at the innermost mass shell which is not at the very center, but in a shell chosen close to the center.
The surface boundary condition use a triangulation method as described in Kippenhahn et al. (1967). In order to specify the surface boundary condition, three inward envelope
integrations are constructed. For all the stellar models the Eddington relation are used. The inward integration starts from an optical depth near
= 10-10 to
= 2/3 at which the temperature reaches
(Demarque et al. 2008).
Model zoning and time step: as we computed the evolution of stars of various masses, the main sequence model has about 1500 mass shells in the interior, approximately 600 shells in the envelope and 600 shells in the atmosphere. We have YREC automatically determine the optimum time step during the evolution.
In order to investigate the influence of diffusion on the internal structure and evolution of the models, we also computed two series of models with and without helium and heavy element diffusion for a given stellar mass and metallicity. The subsequent pulsation analysis of low degree p-modes (l= 0-3) for models in each given mass is implemented using the Guenther & Demarque pulsation code under the adiabatic approximation (Guenther et al. 1992).
2.2 Solar-metallicity stars
Figure 1 displays the computed evolutionary tracks with Zi =
0.02. For each evolutionary track, we plot several groups of models. In
each group, the dot sign represents the model without diffusion, and
the asterisk represents the model with helium and heavy element
diffusion. Furthermore, at the same group, the models of 1.2
and some other groups of subgiant models have similar positions in the
H-R diagram, i.e. the similar luminosities and effective
temperatures, but they have different ages, which can be seen in
Fig. 1. In the case of 0.8
,
0.9
,
1.0
and 1.1
,
each group of main sequence models has the same age. But the increase
of the opacity in the envelope due to the settling of helium and metal
elements provides evolutionary tracks cooler than those without
diffusion. From Fig. 1
we found that element diffusion could speed up the evolution of the
star, especially in the main sequence. However, when stars evolve past
the turn off point, the surface convection zone rapidly deepens, and
the diffusion is suppressed.
In order to study the effect of helium and heavy element
diffusion on the internal structure (not only outer layers but
also the internal core), we selected the relatively high mass star
1.2 .
Model A and model B for 1.2
are also shown in Fig. 1. They are at the similar positions in the H-R diagram, i.e.
= 0.44,
= 0.42 and
= 3.79,
= 3.79, but their ages are different. The age of model A is 3.80 Gyr (without diffusion) and model B is 3.28 Gyr (with diffusion). Then, we use the Guenther & Demarque pulsation code (Guenther
1992) to compute the low degree p-mode oscillation frequencies (l= 0-3) for the above two models, and show their oscillation characteristics in Figs. 2 and 3, respectively.
![]() |
Figure 2: Upper panel: the second differences of the oscillation frequencies. The errorbars are obtained using the error propagation formula and assuming errors of 1 part in 104 in frequencies. Lower panel: the Fourier transform of the second differences as a function of the acoustic depth for model A and model B. Solid lines describe model A and dash dot lines describe model B. |
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![]() |
Figure 3: Upper panel: buoyancy frequency, middle panel: sound speed gradient profiles, lower panel: small separations for model A and model B. The black line describes model A and the red line describes model B. |
Open with DEXTER |
![]() |
Figure 4:
(
|
Open with DEXTER |
In the framework of Tassoul's asymptotic theory (Tassoul 1980), Gough (1990) mentioned the ``second differences'' which give interesting information on the stellar outer
layers. Stellar acoustic p-modes with low l values
propagate deep inside the stars. But the rapid change in the
derivatives of the sound speed, which may be due to the boundary of the
convective zone, to the helium ionization region or to element
gradients, produces a periodic variation in the phase shift
of acoustic waves as a function of frequency with a period 2
,
where
is the time needed for the acoustic waves to travel between the surface and the considered region (acoustic depth), i.e.:
![]() |
(1) |
These effects on the frequencies can be clearly seen in the second differences and their sinusoidal modulation appears as a peak in the Fourier spectrum (Fig. 2), for a timescale t = 2

We estimate the errors in
through the error propagation formula and assuming errors of 1 part in 104 in frequencies. Such an error margin in the frequencies is consistent
with the expectations from the asteroseismic space missions MOST (Walker et al. 2003) and CoRoT (Baglin 2003) and ground-based seismic observations
(e.g., Bedding & Kjeldsen 2003; Bouchy & Carrier 2002) (Mazumdar 2005). We also use the exact frequencies of model A, B and frequencies for which
=
is added to the exact frequencies to fit the Fourier transform of
.
From the lower panel of Fig. 2, i.e. the Fourier transform of the second differences, we can see the difference between model A and B. The peaks around t
2000 s are caused by HeII ionisation zones which lead to depression in the thermodynamical coefficient
.
For model B
with diffusion, the feature due to the helium ionisation zone is
smaller; helium depletion occurs in the HeII zone by diffusion.
The boundary of the convection envelope also leads to characteristic
features at t
6400 s (A) and t
5800 s (B). The peak due to the bottom of the convective zone of model B is much higher than A.
That is because the helium and heavy elements are accumulated and
formed a gradient at the base of the convective zone by diffusion.
Diffusion can change the convective depth and the helium and heavy
element gradient also affects the sound velocity.
Figure 3 presents the buoyancy (or Brunt-Väisälä) frequency (upper panel), the sound speed gradient (W(r) =
)
profiles (middle panel) and the small separations (lower panel) for model A and B. The differences between the models with and without diffusion are clearly visible.
Firstly, we compute the square of the buoyancy frequency N2. This parameter is very sensitive to the convection zone. In this region N2 < 0, hence the frequency N is imaginary (Guenther 2002). For plotting purposes, we set the buoyancy frequency N to zero in the convection zone. From the upper panel of Fig. 3, we can see that there are bumps at the boundary of the convective core for both model A and B. The surface convection zone is also revealed in the propagation diagram. The frequency N again drops quickly to zero (actually becomes imaginary) at the base of the convection envelope.
Secondly, in the middle panel of Fig. 3,
there is a discontinuity in the sound speed gradient profile at the
boundary of the convective core of the star which is about
= 0.042 (model A) and 0.053 (models B).
For a solar-metallicity model with diffusion, the size of the
convective core is greater than the case without diffusion by
about 26
.
The sound speed gradient of model A is steeper than that of B.
Thirdly, as shown in the lower panel of Fig. 3, the values of the small separations of model A are smaller than those of B. As the age of the star increases, the values of the small separations decrease. Model B is younger than A.
In addition, the small separations of model A for the degrees l=0 - l=2 become negative at a frequency of about 3.6 mHz. Soriano & Vauclair (2008) showed that the small separations may become negative at the end of the main sequence for the d02 separations. This effect is directly related to the presence of a helium core; it was also induced by convection in the central stellar regions. At the end of the main sequence, the chemical gradient induces a much larger sound velocity gradient than convection alone, and leads to negative small separations. So we can see that model B is at the main sequence, while model A is at the end of the main sequence.
2.3 Asteroseismic diagram
In order to study the asteroseismic property of models with and without diffusion, we compute the p-mode oscillation frequencies of low-degree modes for a grid of models under the adiabatic
approximation. These models include the initial metallicity Zi = 0.03, 0.025, 0.02, 0.015, 0.01, 0.005 and the masses range from 0.8
to 1.2
in steps of 0.1
.
Then we construct the (
,
)
asteroseismic diagram in Fig. 4.
For a given metallicity and mass, we have shown
two sets of models with and without helium and heavy element diffusion.
The solid lines and circles represent the no-diffusion models; the
dot-dashed lines and asterisks represent the diffusion
models.
The diagram shows that helium and heavy element diffusion make the
models evolve to lower large and small separations. For a given
metallicity, as the evolutionary ages become shorter for larger
masses and because of the very long diffusion time scale, the effect of
diffusion reduces with larger mass. For the models with the same mass,
since the outer convective envelope grows thinner for lower metallicity
models, the effect of diffusion becomes more efficient for models with Zi = 0.01 and 0.005. But models with mass greater than 1.0
with Zi = 0.005 and much poorer-metallicity models (for example, Zi = 0.001) have a much thinner surface convective envelope (
< -7).
These models are hard to analyze with asteroseismology since they do
not obey the asymptotic frequency relation (Tassoul
1980; Karoff et al. 2009).
In Fig. 4, for a given
metallicity and mass, we also labeled the same number of circles and
asterisks on the no-diffusion and diffusion tracks, which indicates the
age of models from 0.05 Gyr and in steps of 0.05 Gyr in the
main sequence stage. The asteroseismic parameters, i.e. the
average large and small separations
,
,
of these models are calculated averaging over n = 10, 11, 12, ..., 30 and listed in Tables 1-6.
For each group of models with the same ages, the large and small
separations of diffusion model are smaller than ones of the
no-diffusion model. From the formulas (Ulrich 1986)
![]() |
(2) |
![]() |
(3) |
we find that the diffusion could enlarge the radius. The small separation is very sensitive to the variation in the sound speed in the center part of the star, and may provide information about the composition of the star in its nuclear region. The smaller d02 indicats that the diffusion could speed the evolution of stars.
Thus, the helium and heavy element diffusion has very important effects
on the internal structure and evolution of a star. Many ground
telescopes and asteroseismic space missions, like MOST, CoRoT, and especially Kepler,
will soon provide accurate seismic data on solar-type stars. On the one
hand, we expect to have enough accurate seismic data to test the effect
of diffusion. On the
other hand, the (
,
)
asteroseismic diagram
and asteroseismic parameters in our Tables 1-6 can compare results to frequency separations from the Kepler
mission observations directly. They also can help in estimating
physical
parameters of targeted stars efficiently. In addition, they may be
useful in deducing the common age and composition of a cluster
of stars.
3 Conclusion
Element diffusion is a basic physical element transport mechanism
which is driven by pressure diffusion, thermal diffusion and
concentrational diffusion. In our paper, we study the effects of
helium and heavy element diffusion on solar-type stars with metallicity
Zi = 0.03, 0.025, 0.02, 0.015, 0.01, 0.005 and masses from 0.8
to 1.2
.
We computed ``second differences'' and ``small separations'' of the solar-metallicity models to analyze the effects of diffusion on the convection zone, helium abundance and the evolutionary sequence of the star.
We also constructed a (
,
)
asteroseismic
diagram to study the asteroseismic property of models with and without diffusion.
We find that:
- (1)
- The helium and heavy element diffusion could speed up the
evolution of the star, especially in the main sequence; diffusion could
enlarge the convective core by about 26
for solar-metallicity stars and change the base of the convection envelope.
- (2)
- Helium and heavy element diffusion made the models evolve to lower large and small separations in the asteroseismic diagram. This effect is more efficient at lower metallicity. We expect that the effect of diffusion will be tested by accurate seismic data from ground telescopes and space missions.



We thank the anonymous referee for constructive comments that lead to significant improvements of this paper. We wish to thanks Sarbani Basu for her comments and help with the software. This work was supported by The Ministry of Science and Technology of the People's Republic of China through grant 2007CB815406, and by NSFC grants 10773003, 10778601 and 10933002. Y. K. Tang acknowledge the support of Shandong Nature Science Foundation and Dezhou University Foundation(08RC11).
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- Walker, G., Matthews, J., Kuschnig, R., et al. 2003, PASP, 115, 1023 [NASA ADS] [CrossRef]
Online Material
Table 1:
The average large and small separations
,
,
for models with Zi = 0.03.
Table 2:
The average large and small separations
,
,
for models with Zi = 0.025.
Table 3:
The average large and small separations
,
,
for models with Zi = 0.02.
Table 4:
The average large and small separations
,
,
for models with Zi = 0.015.
Table 5:
The average large and small separations
,
,
for models with Zi = 0.01.
Table 6:
The average large and small separations
,
,
for models with Zi = 0.005.
Footnotes
- ... stars
- Tables 1-6 are only available in electronic form at http://www.aanda.org
All Tables
Table 1:
The average large and small separations
,
,
for models with Zi = 0.03.
Table 2:
The average large and small separations
,
,
for models with Zi = 0.025.
Table 3:
The average large and small separations
,
,
for models with Zi = 0.02.
Table 4:
The average large and small separations
,
,
for models with Zi = 0.015.
Table 5:
The average large and small separations
,
,
for models with Zi = 0.01.
Table 6:
The average large and small separations
,
,
for models with Zi = 0.005.
All Figures
![]() |
Figure 1:
Evolutionary tracks computed for solar metallicity Z = 0.02 with masses of 0.8 |
Open with DEXTER | |
In the text |
![]() |
Figure 2: Upper panel: the second differences of the oscillation frequencies. The errorbars are obtained using the error propagation formula and assuming errors of 1 part in 104 in frequencies. Lower panel: the Fourier transform of the second differences as a function of the acoustic depth for model A and model B. Solid lines describe model A and dash dot lines describe model B. |
Open with DEXTER | |
In the text |
![]() |
Figure 3: Upper panel: buoyancy frequency, middle panel: sound speed gradient profiles, lower panel: small separations for model A and model B. The black line describes model A and the red line describes model B. |
Open with DEXTER | |
In the text |
![]() |
Figure 4:
(
|
Open with DEXTER | |
In the text |
Copyright ESO 2009
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