Open issues in probing interiors of solar-like oscillating main sequence stars: 2. Diversity in the HR diagram

Abstract

We review some major open issues in the current modelling of low and
intermediate mass, main sequence stars based on seismological studies. The
solar case was discussed in a companion paper, here several issues specific to
other stars than the Sun are illustrated with a few stars observed with CoRoT
and expectations from Kepler data.

Full text

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arXiv:1102.0252v1 [astro-ph.SR] 1 Feb 2011
Open issues in probing interiors of solar-like
oscillating main sequence stars:
2. Diversity in the HR diagram
MJ Goupil1, Y. Lebreton1, J.P. Marques1, S. Deheuvels2, O.
Benomar3, J. Provost4
1Observatoire de Paris, UMR 8109, Paris, France
2Astronomy Department, Yale University, New Haven CT, U.S.A.
2Sydney Institute for Astronomy (SIfA), School of Physics, University of Sydney, NSW 2006,
Australia
3Universit´ e de Nice-Sophia Antipolis, CNRS UMR 6202, Observatoire de la Cte d’Azur, Nice,
France
E-mail: mariejo.goupil]@obspm.fr
Abstract.
mass, main sequence stars based on seismological studies. The solar case was discussed in a
companion paper, here several issues specific to other stars than the Sun are illustrated with a
few stars observed with CoRoT and expectations from Kepler data.
We review some major open issues in the current modelling of low and intermediate
1. Introduction
After more than two decades of helioseismology, almost four years of asteroseismology with
CoRoT [1] and almost two years of intensive asteroseismology with KEPLER [2], we review
some current open issues about the internal structure of solar-like oscillating stars. We focus on
low and intermediate mass, main sequence stars. We started with the Sun and stars that have
a similar structure than the Sun in a companion paper. Here open issues not encountered with
the Sun will be discussed and illustrated with a few individual stars observed by CoRoT and
from ground. We end with a brief discussion about expectations from KEPLER data. For sake
of shortness, we decided to include only unpublished figures and to cite published figures in the
text. Several reviews exist on the topic, for instance [3], [4].
2.
We focus on low and intermediate mass, main sequence stars that-is stars with masses up to 1.5
M⊙corresponding to F, G, K spectral types. These stars have an external convective region
and can oscillate like the Sun with high frequency p modes. For these stars, one encounters
the same problems as for the Sun, namely surface effects when comparing absolute values
of the frequencies.But although we refer to these stars as solar-like stars in the present
framework for shortness, these stars can differ from the Sun by several aspects: mass, age,
surface metallicity and helium abundances, initial conditions for the chemical abundances, the
rotation and magnetic properties. For masses larger than about 1.2 M⊙, they have a convective
From the Sun to solar-like oscillating MS stars:

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Figure 1. Evolutionary tracks of models built with several assumptions as indicated in the plot. All
nuclear reaction rates are from NACRE except for the reaction14N(p,γ)15of the CNO cycle which is
from LUNA when specified. The other nuclear rates are all from NACRE. The helium and metallicity
are solar.
core unlike the Sun. These differences therefore lead to additional open questions about their
modelling. The major problem concerns dynamical processes occuring inside stars that have
a significant impact on the mean structure of the stars and their ages. In particular, the 3D
multiscaled turbulent convective transport and related instabilities is taken into account by
means of a 1D crude formulation that involves free parameters which cannot be derived from
first principles. As a consequence they are calibrated with observations (the Sun, binaries) but
these values have no predictive quality.
The input parameters such as mass, age, initial chemical composition Y0,(Z/X)0, rotation
profile are usually not well known. A first order of magnitude for mass and age is obtained via
the location of the star in the HR diagram based on photometric and spectroscopic information.
Uncertainties on this location and the chemical abundances give rise to a large number of
possible models which makes difficult to probe in detail the internal structure of the star. This
number is significantly reduced when seismic diagnostics such as the large separation and the
small spacings are used. However free parameters used to describe the convective transport and
related instabilities increase the family of acceptable models. The resulting solution therefore
remains input physics dependent.
2.1. Observational constraints and seismic diagnostics
Efforts are therefore currently put at obtaining seismic constraints that are discriminating and as
model independent as possible. A first information that is looked for is the large separation, its
mean value as well as its variation with frequency [5], [6]. Other combinations of low degree mode
frequencies are built to locate the base of the upper convective zone and obtain properties about
He ionization regions [7]; [8]; [9]; [10] and surface helium content [11]. Theoretical developements
are carried also out to devise diagnostics and appropriate methods to determine the age ([12]
and references therein); to probe the core [13], [14],[15]; [16]; [9], [17] with specific attention to
tiny convective core properties [18] of low mass stars using only low degree modes; to investigate

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mixing beyond the convective core [19] or semiconvection [20]. For instance, the spacing d01
is sensitive to convective core properties [21], [22]; ; [23]; [24]. [24] shows the variations of the
seismic quantity d01with frequency over a large range of frequency for a model representing the
star HD 203608. The slope of the observed d01over the observed frequency range is directly
related to the slow period of oscillation seen on the theoretical d01that corresponds to a time
scale related to the acoustic radius of core convective radius.
2.2. CoRoT solar-like stars
Most CoRoT solar-like stars differ from the Sun either because they are more massive and faster
rotators such as the F2 star HD181420 [25]; [26] and the F5 star HD170987 [27] or because they
are evolved and have an isothermal core such as the G0 star HD49385 [28],[29] or they have a
mass similar to that of the Sun but differ in their metallicity such as HD49933 [30], HD 52265,
or the young magnetic star HD 46375 [31]. These stars can also differ significantly from the Sun
by their level of magnetic activity and their magnetic cycles [32].
2.2.1. Which star for which diagnostic?
suited to probe different physical processes.
Initial abundances and chemical mixture: Fig.1 shows some of these stars in a HR diagram
together with evolutionary tracks of models built assuming no microsocopic diffusion; nuclear
reaction rates are from NACRE completed with recent LUNA determinations. Two mixtures
have been used : AGS09 and GN93. The effect of changing the mixture has a clear impact on
a star such that the F star HD181906 [33] which has a convective core in the first case and no
convective core in the second case. As the existence of a convective core manifests itself with a
large slope of d01, this can be in principle determined with this seismic diagnostic.
Nuclear reaction rate:CNO cycle and14N(p,γ)15burning: The change from NACRE [34] to
LUNA [35] reaction rate led to reduce the CNO cycle efficiency. As a result, the tracks for the
most massive stars (i.e. with central temperature high enough for CNO to dominate) are slightly
shifted upward. This of course coincides with stars having a convective core. For a 1.2M⊙and
Z = 0.01 model, the convective core is smaller at given mass and appears at higher mass [34].
Microscopic diffusion, surface helium abundance and initial metallicity: Due to gravitational
settling and atomic diffusion in the radiative region below the convective envelope, the surface
helium decreases with time. This decrease is larger for higher mass stars because the convective
envelope is thiner. The decrease is also larger for lower metallic stars which are more compact
and - due to smaller opacities and therefore smoother radiative temperature gradient- have also
a thiner convective envelope. The disparition of helium in such thin convective zones is therefore
very rapid. If microscopic diffusion acts alone, the envelope of HD49933 for instance would be
fully depleted of helium. Mechanisms opposite to diffusion must therefore be at work such as
turbulent diffusion and/or radiative acceleration, rotationally induced mixing and must therefore
be included in the modelling.
Due to their differences, some of these stars are better
2.2.2. Mode degree identification
is that the observed frequencies can be attributed to modes with given degree l and azimuthal
number m values in a spherical harmonics description. However in some cases, the l = 1 split
multiplets can be mistaken with overlapping l = 0 and l = 2 modes in the fitting process. Hence
some ambiguity can exist in the determination of l = 0 and l = 1 ridges in an echelle diagram.
This is particularly true for the hottest (F type) stars because of their large mode linewidhs
and their relatively fast rotation. Exemples are HD49933 [36], HD181420 [25] and HD181906
[33] for which 2 scenarii are proposed for the identification of the l = 0 and l = 1 modes. In
one case, fitting the data imposes a quite large core overshoot whereas in the other case a usual
intermediate core overshoot amount is sufficient. This ambiguity that exists when the data sets
Seismic inferences assume that the modes are identified that-

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Figure 2. (Left top:) Large separation ∆νn,lin function of the frequency νn,land (left bottom:) the
small spacings d01= νn,0−(νn,1+νn−1,1)/2 and d10= −νn,1+(νn+1,0+νn,0)/2 in function of the frequency
νn,0for HD49933. Data (open circles and crosses) are from [38]. Models (solid curves) are built assuming
AGS05, no diffusion; αcgm = 0.6, αov = 0.27 (magenta); AGS05, diffusion, αcgm = 0.6, αov = 0.21,
Y = 0.27, Z/X = 0.0079 (yellow) AGS09, diffusion (green); AGS05, diffusion,αov = 0.2, rotationally
induced transport (blue); AGS05, diffusion, αov = 0., rotationally induced transport, Y = 0.27,
αcgm = 0.6 (chocolate).(Right:)Echelle diagram for HD49933. Blue dots represent the observations
and red ones the model.
are too short can be lifted when using sophisticated data analysis treaments [37], [38], [39], [5].
To lift the ambiguity in mode identification when it exists, [40] have proposed to use scaling
relations (ν and ∆ν scale as < ∆ν >) to build scaled echelle diagrams with reference to a star
similar to the studied one. The authors tested this procedure with two sets of twin stars Sun and
18 Sco - τ Ceti and α Cen B and used it to determine the most probable scenario for two CoRoT
stars HD181420 and HD181906 using HD49933 as the reference star. This scaling procedure has
then been recently used on the ground based observed star, HD203608 [41]. Indeed for this F8V
star, two possible scenarii have been found with again important consequences on conclusions
that can be drawn from the mode identification [28]; [24]. Echelle diagrams built assuming the
scaling according to [40] coincide with the observed echelle diagram in the case of one of the
two possible scenarii and definitely rejects the alternative scenario [41]. This scenario favours a
mild overshoot and the survical of convective core despite the small mass and old age of the star
due its low metallicity [29]. Mode identification based on scaling relations therefore appears as
a potential interesting method that must nevertheless be studied further on theoretical ground
before it can be used as a proper decision method.
2.2.3.
seismic data analyses agree to provide the same scenario for the l = 0,1 mode identification
for this star [30].Seismic modelling of HD49933 illustrates the difficulty one encounters
because of the degeneracy in input parameter space composed of mass, age, (Z/X)0,Y0,
αcgm,αov, initial rotation and transport coefficients. The free parameters αcgm,αov represent
the convective mixing length for the adopted CGM formulation [42] and the convective core
HD49933: a low metallicity star
With 180 days of observation with CoRoT, all

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overshoot parameter respectively. All models discussed below are calibrated so that the mean
large separation < ∆ν > and the mean small spacings d01,d10agree with the observations as
well as the observed location in a HR diagram. This lifts only partially the degeneracy. The
phase of oscillation of the large separation is found to be quite sensitive to values of αcgmand
Y0and therefore such a constraint must be added in the minimisation process to reduce the
number of acceptable models although this has not been done here. Individual frequencies have
not been fitted (see below). The surface metallicity of the star is taken in the observed range
[Fe/H] = −0.4 ± 0.1, significantly lower than the Sun [43].
Fig.2 compares the mean large separation and the small spacings d01,d1,0for models built
assuming either AGS05 or AGS09 mixtures and various assumptions about microscopic diffusion,
rotationally induced transport, convective core overshoot as listed in the caption. All models
are computed with CESAM2k [44]. Models including rotationally induced transport of angular
momentum as implemented by J. Marques in the code CESTAM (a modified version of cesam2k)
have been computed assuming no loss of angular momentum. The initial angular rotation on
the PMS has been set in order to fit the observed rotation period of the star, P = 3.4 days at
the age of HD49933. Details on the modelling of this star will be published elsewhere.
For all these different assumptions, one can find a mass and age that fit the mean large
separation oscillation by ajusting αcgm and Y0 although this has not been done yet for our
rotating models. The mass is found in the range 1.05-1.18 M⊙and the age in the range 2900-3900
Myr depending on the assumptions in the physical description and the chemical abundances. As
a result of these various calculations, we find that a) when the AGS05 mixture is assumed, it is
difficult to find a model satisfying all the observational constraints when (Z/X)0is on the smaller
part of the authorized interval. This is less the case with the less extreme AGS09 mixture. It
is important to stress that the star being metallic deficient compared with the Sun, the above
mixtures taken from the Sun might not well be suited; b) because this star is low metallic, its
thin convective envelope is rapidly devoid of helium when microscopic diffusion is included if one
starts with solar initial helium abundance when one assumes AGS05 mixture. One then needs
to start with a large initial helium abundance Y0or one must include some turbulence in the
radiative zone. Starting then with a large Y0= 0.35, one still obtains a small Ysurf= 0.10 value
for HD49933. One obtains a less extreme surface helium abundance Ys= 0.18 when using the
less extreme AGS09 mixture; c) without including microscopic diffusion nor rotationally induced
transport, no model fits d01for an overshoot smaller that 0.25−0.3 Hp whatever the mixture; d)
for AGS05 mixture, when diffusion is included with or without rotationally induced transport,
some intermediate amount of overshoot (≈ 0.2 Hp) still remains necessary. As a conclusion,
microscopic diffusion and rotationally induced transport as modelled here are not enough to
render count of the slope of d01variation with AGS05. One needs to include also some amount
of convective core overshoot as a proxy for true overshoot and/or additional mixing process.
Absolute frequencies and echelle diagram The echelle diagram displayed in Fig.2 shows a
systematic shift between the observed and theoretical l = 1 ridges roughly independent of the
frequency when the mean separation is taken to be the same in building the observed and
theoretical echelle diagrams. This discrepancy comes from the fact that absolute values of the
frequencies have not been included in the optimisation procedure to find an optimal model.
Increasing slightly the mean separation for building the echelle diagram for the model enables
to perfectly match the low frequency part of the observed ridge as seen in Fig.2. The deviation
of the theoretical ridge with the observed ones remains at high frequencies and might reflect
surface effects that are not properly taken into account in the numerical frequencies.
2.2.4. HD181420: a fast rotator
by [25] and [39] who found two possible scenarii for the mode identification. Results based on
a Bayesian approach favors scenario 1 Benomar (2010, priv.com). This is also the conclusion of
Analyses of the light curve of this star has been performed

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Figure 3. Echelle diagrams for a model of HD181420: rotational velocities v = 10 km/s (left) and 25
km/s (right) are assumed when computing the frequencies. From left to right in each pannel, ridges for
l = 2,0,1 respectively appear.
[40] using scaling properties. Focusing then on scenario 1, one reproduces the large separation
and the small spacing d01 with a 1.36M⊙ stellar model assuming a core overshoot of 0.2Hp
as well as with a 1.37M⊙stellar model assuming no overshoot, everything else being the same
in particular microscopic diffusion and rotation are not included [45]. A secondary oscillation
component is seen in the observed large separation that is not reproduced by any models [45],
Michel (2010 priv.com). The ’period’ of this oscillation corresponds to the base of the convective
zone of the above models, but the reality of this secondary component is not confirmed [26],
Mosser (2010, priv. comm). This star is a ’rapid’ rotator compared to the Sun. Indeed with
a radius R = 1.66R⊙and the observed mean rotational splitting νsplit= (3. ± 1)µHz [25], one
obtains a rotational velocity v = 21.97.3 km/s that corresponds to a ratio of the centrifugal to
the gravitational accelerations of ǫ = Ω2/(GM/R3) = 320ǫ⊙! Perturbation methods to compute
the effect of rotation on the frequencies nevertheless remain valid for this rotation rate [46]. The
non-spherically centrifugal distortion causes asymetries of split multiplets that are seen in echelle
diagrams already for v = 10 km/s at high frequency (Fig.3, frequencies including rotating effects
have been performed with the WarM oscillation code). Assuming a uniform rotation velocity of
25 km/s, the m = 2 components of the l = 2 modes coincide with the l = 0 mode for the lowest
frequencies whereas at high frequencies both m = 1 and m = 2 components are mixed with the
l = 0 frequency. Asymetries increase with frequency and are therefore larger at high frequencies
where they contribute to surface effects! The difference between the rotation frequency measured
at low frequency in a power spectrum [47] and the mean splitting [25] is found ot be compatible
with a latitudinal dependence of the surface rotation of the star (Ouazzani, 2010 priv. com).
2.2.5.
[29]. [29] clearly put in evidence the existence of an l = 1 avoided crossing and showed that
it produces a characteristic distortion of the l = 1 ridge in the neighbour of this mode in an
echelle diagram. A clear explanantion of the deformation of the ridge by the presence of an
avoided crossing for a l = 1 mode has been provided by [48]. The distorsion is caused by the
HD49385: an evolved star
The light curve of this star has been analysed by [48] and

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Figure 4. Frequency νmax in function of the mean large separation < ∆ν >. Models (solid curves)
have been computed with cesam2k code. Scaling laws are computed according to [49] and [50]. Data for
a few ground based observed stars and some CoRoT targets plotted with their errors bars represented
by crosses.
fact that the modes propagate as p mode in a surface cavity and as g mode in a central cavity.
For each mode, both cavities are separated by an evanescent region which acts as a coupling
between the two regions. The magnitude of the ridge deformation is related to the properties of
the evanescent region. The deformation of the ridge is then fitted to constrain the stellar model.
A model satisfying all the constraints simultaneously then is difficult to obtain. Only a change
in the chemical mixture from AGS05 to AGS09 is able to affect the evanescent region so as to
provide a correct ridge deformation [41].
3.
The seismic part of the space mission Kepler produces seismic data for a huge number of stars
for which stellar parameters are in general not well known. Ensemble investigations to derive
stellar masses and radii then rely on scaling seismic properties [51]. Fig.4 shows νmax(frequency
at maximum oscillation power in a power spectrum) in function of the mean large separation
< ∆ν > for series of stellar models (open circles) assuming different chemical compositions
along evolutionary tracks from ZAMS (top right corner) to TAMS (down left corner). Data for
a few ground based observed stars are overplotted as well as CoRoT targets. Due to NASA data
policy, Kepler data are not shown. large The error bars indicate that one cannot distinguish
between metallicities Z = 0.017 and Z = 0.04 (with Y = 0.26) for instance, nor between
Y = 0.23 and Y = 0.30 (Z = 0.017). This leads to small uncertainties on masses and radii
derived from these scalings. This is a crucial issue for the ESA project PLATO [52] that aims at
detecting and studying earth-type exoplanets and therefore requires the determination of stellar
masses of exoplanet host stars with an accuracy of about 10-15%. This means that it will be
mandatory to determine the metallicity and helium abundances Z/X and Y by other means;
for instance Z/X by spectroscopy and Y by the properties of the frequency dependence of the
large separation. Another important issue is to establish the physical origin of the νmaxscaling
which sofar has only been conjectured [53] but has been validated with observations [50]. A
better understanding of this origin could provide the explicit dependence of the scaling relations
Kepler data and ensemble seismic investigations

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on metallicity and chemical composition, rotation etc ...
4.
With the wealth of data from CoRoT and Kepler, we are confronted with seismologial studies
of a rich variety of low, and intermediate mass, main sequence stars that differ from the
Sun in their stellar parameters and internal structure. Although this will enrich considerably
our understanding of stellar physics and evolution, this also generates several difficulties not
encountered with the Sun. Some were expected such as inaccurate determination of stellar
parameters, degeneracy in parameter space and resulting non unicity of stellar models but some
other problems were not really expected such as ambiguities in the mode identification due
to broad linewidths and fast rotation (compared to the Sun) for F stars. So as far as probing
internal structure by means of seismology of solar-like stars is concerned, we are therefore still at
the beginning of the learning phase. Significant advances in stellar physics and solid conclusions
about the open issues discussed in the present paper will require homogeneous detailed seismic
studies for a larger number of individual stars that has been done so far. New roads also
develope such as ensemble studies and scaling procedures and additional observational seismic
constraints start to exist such as accurate mode amplitudes and linewitdhs and their variations
with frequency that were not available from ground. Eventually studying a star and its planets
as a global system is certainely the issue that must adressed in the future. In that framework,
the perspective offered by the ESA project PLATO which has precisely this aim is a strong
motivation for the seismic community to pursue its efforts and in turn PLATO will grandly
benefit from all the forthcoming advances in the field.
Conclusion:
Acknowledgments
We gratefully thank our colleagues K. Belkacem, J. Ballot, B. Mosser, C. Barban, T. Corbard,
D. Reese, O. Creevey, A. Baglin, E. Michel, T. Appourchaux, C. Catala, A. Mazumdar for
providing useful information and fruitful discussions when preparing this review. We also thank
P. Morel and J. Christensen-Dalsgaard for providing public evolutionary and oscillation codes
respectively that were used in the present review. We ackowledge financial support from CNES
and the ANR SIROCO
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