arXiv:1001.0032v1  [astro-ph.SR]  30 Dec 2009Draft version November 15, 2018
Preprint typeset using L ATEX style emulateapj v. 08/22/09
ASTEROSEISMIC INVESTIGATION OF KNOWN PLANET HOSTS IN THE KEPLER FIELD
J. Christensen-Dalsgaard1,2, H. Kjeldsen1,2, T. M. Brown3, R. L. Gilliland4, T. Arentoft1,2, S. Frandsen1,2,
P.-O. Quirion1,2,5, W. J. Borucki6, D. Koch6, and J. M. Jenkins7
Draft version November 15, 2018
ABSTRACT
In addition to its great potential for characterizing extra-solar p lanetary systems the Kepler mis-
sionis providing unique data on stellar oscillations. A key aspect of Keplerasteroseismology is the
application to solar-like oscillations of main-sequence stars. As an ex ample we here consider an ini-
tial analysis of data for three stars in the Keplerfield for which planetary transits were known from
ground-based observations. For one of these, HAT-P-7, we obt ain a detailed frequency spectrum and
hence strong constraints on the stellar properties. The remaining two stars show definite evidence for
solar-like oscillations, yielding a preliminary estimate of their mean dens ities.
Subject headings: stars: fundamental parameters — stars: oscillations — planetary systems
1.INTRODUCTION
The main goal of the Kepler mission is to character-
ize extra-solar planetary systems, particularly Earth-like
planets in the habitable zone (e.g., Borucki et al. 2009).
The mission detects the presence of planets through the
minute reduction of the light from a star as a planet
crosses the line of sight. Several observations of such
reductions at fixed time intervals for a given star, and
extensive follow-up observations, are used to verify that
the effect results from planet transits and to characterize
the planet. To ensure a reasonable chance of detection
Keplerobserves more than 100,000 stars simultaneously,
in a fixed field in the Cygnus-Lyra region. Most stars
are observed at a cadence of 29.4 min, but a subset of
up to 512 stars can be observed at a short cadence (SC)
of 58.85s. Keplerwas launched on 6 March 2009 and
data from the commissioning period and the first month
of regular observations are now available.
The very high photometric accuracy required to detect
planet transits (Borucki et al. 2010; Koch et al. 2010)
also makes the Keplerobservations of great interest for
asteroseismic studies of stellar interiors. In particular,
the SC data allow investigations of solar-like oscillations
in main-sequence stars. Apart from the great astrophys-
ical interest of such investigations they also provide pow-
erful tools to characterize stars that host planetary sys-
tems (Kjeldsen et al. 2009).
In stars with effective temperature Teff<∼7000K we
expect to see oscillations similar to those observed in the
Sun (e.g., Christensen-Dalsgaard 2002), excited stochas-
ticallybythe near-surfaceconvection. Theseareacoustic
modes of high radial order; in main-sequence stars such
1Department of Physics and Astronomy, Aarhus University,
DK-8000 Aarhus C, Denmark: e-mail jcd@phys.au.dk
2Danish AsteroSeismology Centre
3Las Cumbres Observatory Global Telescope, Goleta, CA 93117
4Space Telescope Science Institute, 3700 San Martin Drive, B al-
timore, MD 21218
5Canadian Space Agency, 6767 Route de l’A´ eroport, Saint-
Hubert, QC, J3Y 8Y9 Canada (present address)
6NASA Ames Research Center, MS 244-30, Moffett Field, CA
94035, USA
7SETI Institute/NASA Ames Research Center, MS244-30, Mof-
fett Field, CA 94035, USAmodes approximately satisfy the asymptotic relation
νnl≃∆ν0(n+l/2+ǫ)−l(l+1)D0 (1)
(Vandakurov 1967; Tassoul 1980). Here νnlis the cyclic
frequency, nis the radial order of the mode and lis
the degree, l= 0 corresponding to radial (i.e., spher-
ically symmetric) oscillations. Also, ∆ ν0is essentially
the inverse sound travel time across the stellar diameter;
this is closely related to the mean stellar density ∝angbracketleftρ∗∝angbracketright:
∆ν0∝ ∝angbracketleftρ∗∝angbracketright1/2.D0depends sensitively on conditions
near the center of the star; for stars during the central
hydrogenburningphasethisprovidesameasureofstellar
age. Finally, ǫis determined by conditions near the stel-
lar surface. This regular form of the frequency spectrum
simplifies the analysis of the observations, and the close
relation between the stellar properties and the param-
eters characterizing the frequencies make them efficient
diagnostics of the properties of the star. This has been
demonstrated in the last few years through observations
of solar-like oscillations from the ground and from space
(for reviews, see Bedding & Kjeldsen 2008; Aerts et al.
2009; Gilliland et al. 2010a).
Even observations allowing a determination of ∆ ν0
provide useful constraints on ∝angbracketleftρ∗∝angbracketright. With a reliable de-
termination of individual frequencies ∝angbracketleftρ∗∝angbracketrightis tightly con-
strained and an estimate of the stellar age can be ob-
tained. This can greatly aid the interpretation of obser-
vations of planetary transits (e.g., Gilliland et al. 2010b;
Nutzman et al. 2010). We note that photometric obser-
vations such as those carried out by Keplerare predom-
inantly sensitive to modes of degree l= 0−2. As indi-
catedbyEq.(1)thesearesufficienttoobtaininformation
about the core properties of the star.
Ground-based transit observations have identified
three planetary systems in the Keplerfield: TrES-2
(O’Donovan et al. 2006; Sozzetti et al. 2007), HAT-P-7
(P´ al et al. 2008), and HAT-P-11 (Dittmann et al. 2009;
Bakos et al. 2010). These systems have been observed
byKeplerin SC mode. Their properties (cf. Table 1)
indicate that they should display solar-like oscillations
at observable amplitudes, and hence they are obvious
targets for Keplerasteroseismology. Here we report the
results of a preliminary asteroseismic characterization of2 Christensen-Dalsgaard et al.
TABLE 1
Properties of transiting systems.
Name KIC No Teff(K) [Fe/H] L/L⊙log(g) (cgs) vsiniSource
(kms−1)
HAT-P-7 10666592 6350 ±80 0.26±0.08 4 .9±1.1 4.07±0.06 3.8±0.5 (a)
6525±61 0.31±0.07 4 .09±0.08 (b)
HAT-P-11 10748390 4780 ±50 0.31±0.05 0.26±0.02 4.59±0.03 1.5±1.5 (c)
TrES-2 11446443 5850 ±50−0.15±0.10 1.17±0.10 4.4±0.1 2 ±1 (d)
5795±73 0.06±0.08 4 .30±0.13 (b)
Note. — Sources: (a): P´ al et al. (2008); (b): Ammler-von Eif et al . (2009); (c): Bakos et al. (2010); (d): Sozzetti et al. (2007 ). In some
cases asymmetric error bars have been symmetrized.
the central stars in the systems, based on the early Ke-
plerdata.
2.OBSERVATIONS AND DATA ANALYSIS
We have analyzed data from Kepler for three
planet-hosting stars using a pipeline developed for
fast and robust analysis of all Keplerp-mode data
(Christensen-Dalsgaard et al. 2008; Huber et al. 2009).
Each time series contains 63324 data points. SC data
characteristics and minor post-pipeline processing are
discussed in Gilliland et al. (2010c). In addition a limb-
darkened transit light curve model fit has been removed
and 5-σclipping applied to remove outlying data points
from each of the time series. The frequency analysis con-
tains four main steps:
1. We calculate an oversampled (factor of four) ver-
sion ofthe power spectrum by using a least-squares
fitting. We smoothed the spectrum to 3 µHz reso-
lution to remove the fine structure caused by the
finite mode lifetime.
2. We correlated the smoothed power spectrum with
an equally spaced comb of delta functions, sepa-
ratedby∆ ν0/2,andconfinedtoaGaussian-shaped
band with a full width at half maximum of 5∆ ν0.
We adopted the maximum of this convolution over
lags between 0 and 0.5 ∆ ν0as the filter output for
each ∆ν0.
3. After identifying the peak correlation for the best
matched model filter and extracting the large sep-
aration corresponding to this peak we calculate the
folded spectrum (see Fig. 1b), i.e., the sum of the
power as a function of frequency modulo the opti-
mumlargeseparation(theonecorrespondingtothe
peak correlation). The summed power is used to
locate the p-mode structure and identify the ridges
corresponding to the different mode degrees (based
on the asymptotic relation).
4. From the asymptoticrelationandthe identification
of mode degrees we finally identify the position of
the individual p-mode frequencies in the smoothed
version of the power spectrum; when more than
one mode is seen near the expected frequency we
use the power-weighted average of the two peaks.
Those extracted frequencies and the mode identifi-
cations are used in the modeling.
For observations with low signal-to-noise ratio it may
not be possible to identify the individual frequencies. In       0        1        2
Fig. 1.— (a)PowerspectrumofHAT-P-7forfrequencies between
300 and 3000 µHz. The spectrum is smoothed with a gaussian filter
with a FWHM of 3 µHz. The noise level at high frequencies corre-
sponds to 1.1 ppm in amplitude. The white curve is a smoothed
power spectrum with a gaussian filter (150 µHz FWHM). A fit to
the background (dashed white curve) is also shown. The exces s
power and the individual p-modes are evident. (b) Folded pow er
spectrum, between 750 and 1500 µHz, for HAT-P-7 for a large sep-
aration of 59 .22µHz. Indicated are the positions corresponding to
radial modes ( l= 0) and non-radial modes with l= 1 and 2. The
measured positions are used to identify the individual osci llation
modes in panel (a). (c) ´Echelle diagram (see text) for frequencies
of degree l= 0, 1, and 2 in HAT-P-7; a frequency separation of
59.36µHz and a starting frequency of 10 .8µHz were used. The
filled symbols, coded for degree as indicated, show the obser ved
frequencies, while the open symbols are for Model 3 in Table 2 ,
minimizing χ2
ν.3
such cases the analysis is carried through step 2, to de-
termine the maximum response and hence an estimate of
the large separation.
Results on the three individual cases are presented in
§4.
3.MODEL FITTING
Stellar evolution models and adiabatic oscillation
frequencies were computed using the Aarhus codes
(Christensen-Dalsgaard 2008a,b), with the OPAL
equation of state (Rogers et al. 1996) and opacity
(Iglesias & Rogers 1996) and the NACRE nuclear reac-
tion parameters (Angulo et al. 1999). In some cases (see
below) diffusion and settling of helium were included,
using the simplified formulation of Michaud & Proffitt
(1993). Convection was treated with the B¨ ohm-Vitense
(1958) mixing-length formulation, with a mixing length
αML= 2.00 in units of the pressure scale height roughly
corresponding to a solar calibration. In some models
with convective cores, overshoot was included over a dis-
tance of αovpressure scale heights. Evolution started
from chemically homogeneous zero-age models. The ini-
tial abundances by mass X0andZ0of hydrogen and
heavy elements were characterized by the assumed value
of [Fe/H], using as reference a present solar surface com-
position with Zs/Xs= 0.0245 (Grevesse & Noels 1993)
and assuming, from galactic chemical evolution, that
X0= 0.7679−3Z0.
From the observed ∆ ν0, effective temperature and
composition an initial estimate of the stellar parame-
ters was obtained using the grid-based SEEK pipeline
(Quirion et al., in preparation). Smaller grids were then
computed in the vicinity of these initial parameters, to
obtaintighterconstraintsonstellarproperties. ForHAT-
P-7 the analysis of the observations yielded frequencies
of individually identified modes; here the analysis was
based on
χ2
ν=1
N−1/summationdisplay
nl/parenleftBigg
ν(obs)
nl−ν(mod)
nl
σν/parenrightBigg2
,(2)
whereν(obs)
nlandν(mod)
nlare the observed and model fre-
quencies, σνis the standard error in the observed fre-
quencies (assumed to be constant) and Nis the num-
ber of observed frequencies. In addition, we considered
χ2=χ2
ν+χ2
T, whereχ2
Tis the corresponding normalized
square difference between the observed and model effec-
tive temperature. When χ2
νwas available we minimized
it along each evolution track and considered the result-
ing minimum values, and the corresponding value of χ2,
as a function of the parameters characterizing the mod-
els (see Gilliland et al. 2010b, for details). When only
the large separation ∆ ν0could be determined from the
observations, we identified the model along each track
which matched ∆ ν0and considered the resulting χ2
Tas
a function of the model parameters.
4.RESULTS
4.1.HAT-P-7
The observed power spectrum for HAT-P-7 is shown
in Fig. 1a. The presence of solar-like p-mode peaks, with
a maximum power around 1.1mHz, is evident. At high
frequency the noise level in the amplitude spectrum is1.1 parts per million (ppm), with some increase at lower
frequency, likely due to the effects of stellar granulation.
Carrying out the correlation analysis described in §2
we determined the large separation as ∆ ν0= 59.22µHz.
Figure 1b shows the resulting folded spectrum. This
clearly shows two closely spaced peaks, identified as cor-
responding to modes of degree l= 0 and 2, and single
peak separated from these two by approximately ∆ ν0/2,
corresponding to l= 1. On this basis we finally deter-
mined the individual frequencies, identifying the modes
from the asymptotic relation; the final set includes 33
p-mode frequencies, determined with a standard error
σν= 1.4µHz. These frequencies, corresponding to ra-
dial orders between 11 and 24, are illustrated in Fig. 1c
in an ´ echelle diagram (see below).
A grid of models was computed for masses between
1.41 and 1 .61M⊙, [Fe/H] between 0.17 and 0.38, and
αov= 0,0.1 and 0.2, extending well beyond the end of
central hydrogen burning. The modeling did not include
diffusion and settling. At the mass of this star the outer
convection zone is quite thin, and as a result the set-
tling timescale is much shorter than the age of the star.
Including settling, without compensating effects such as
partial mixing in the radiative region or mass loss, leads
to a rapid change in the surface composition which is
inconsistent with the observed [Fe/H]; for simplicity we
therefore neglected these effects for HAT-P-7.8
The computed frequencies were corrected according to
the procedure of Kjeldsen et al. (2008) for errors in the
modeling of the near-surface layers, by adding a(ν/ν0)b
wherea= 0.1158µHz,ν0= 1000µHz andb= 4.9. As
discussed in §3, for each evolution track, characterized
by a set of model parameters, we minimized the depar-
tureχ2
νof the model frequencies from the observations,
defining the best model for this set.
We first consider χ2
νas a function of the effective
temperature of the models (Fig. 2a). It is evident
that there is a clear minimum in χ2
ν; this is consistent
with the determination of Teffby P´ al et al. (2008) but
not with the somewhat higher temperature obtained by
Ammler-von Eif et al. (2009) (see also Table 1). Thus
in the following we use the observed quantities from
P´ al et al. (2008).
Since the frequencies to leading order are determined
by the mean stellar density ∝angbracketleftρ∗∝angbracketright, Fig. 2b,c show χ2
νand
χ2as functions of ∝angbracketleftρ∗∝angbracketright. It is evident that the best-fitting
modelsoccupyanarrowrangeof ∝angbracketleftρ∗∝angbracketright, withawell-defined
minimum. Fittingaparabolato χ2inpanel(c)weobtain
the estimate ∝angbracketleftρ∗∝angbracketright= 0.2712±0.0032gcm−1. In Fig. 2d
χ2is shown against model age. Here the variation with
model parameters is substantially stronger, resulting in
a greater spread in the inferred age; in particular, it is
evident, not surprisingly, that the results depend on the
extent of convective overshoot. From the figure we esti-
matethattheageofHAT-P-7isbetween1.4and2.3Gyr.
Examples of evolution tracks are shown in Fig. 3; pa-
rameters for these models are provided in Table 2. They
were chosen to give the smallest χ2
νfor each of the three
values of αovconsidered. Also shown are the locations
8Artificially suppressing settling in the outer layers, whil e in-
cluding diffusion and settling in the core, leads to results t hat are
very similar to those presented here.4 Christensen-Dalsgaard et al.
TABLE 2
Stellar evolution models fitting the observed frequencies for HAT-P-7.
No M ∗/M⊙Age Z0X0αovR∗/R⊙/angbracketleftρ∗/angbracketrightTeffL∗/L⊙χ2
νχ2
(Gyr) (gcm−3) (K)
1 1.53 1.758 0.0270 0.6870 0.0 1.994 0.2718 6379 5.91 1.08 1.2 1
2 1.52 1.875 0.0290 0.6809 0.1 1.992 0.2708 6355 5.81 1.04 1.0 4
3 1.50 2.009 0.0270 0.6870 0.2 1.981 0.2718 6389 5.87 1.00 1.2 4
Note. — Models minimizing χ2
ν(cf. Eq. 2) along the evolution tracks, illustrated in Fig. 3 . The models have been selected as providing
the smallest χ2
νfor each of the three values of the overshoot parameter αov. The smallest value of χ2
νis obtained for Model 3.
Fig. 2.— Results of fitting the observed frequencies to a grid of
stellar models (see text for details). Plusses, stars and di amonds
correspond to modelswith αov= 0 (no overshoot), 0.1, and 0.2. (a)
Minimum mean square deviation χ2
νof the frequencies (cf. Eq. 2)
along each evolution track, against the effective temperatu reTeff
of the corresponding models. The vertical dashed and dotted lines
indicate the effective temperatures found by P´ al et al. (200 8) and
Ammler-von Eif et al. (2009). (b) Minimum mean square devia-
tionχ2
νagainst the mean density /angbracketleftρ∗/angbracketrightof the corresponding models.
(c) As (b), but showing the combined χ2. (d)χ2against the age
for the models that minimize χ2
ν; the different ridges correspond to
the different masses in the grid, the more massive models resu lting
in a lower estimate of the age.Fig. 3.— Theoretical HR diagram with selected evolutionary
tracks, corresponding to the models defined in Table 2. The ’+ ’ in-
dicate the models along the full set of evolutionary sequenc es mini-
mizing the difference between the computed and observed freq uen-
cies. The box is centered on the LandTeffas given by P´ al et al.
(2008), with a size matching the errors on these quantities.
of the models minimizing χ2
νalong each of the computed
tracks; these evidently fall close to a line in the HR di-
agram, corresponding to the small range in ∝angbracketleftρ∗∝angbracketright. The
range of luminosities, from P´ al et al. (2008), is based on
modeling and hence has not been used in our fit; even
so, it is gratifying that the present models are essentially
consistentwiththesevalues. Also,asindicatedbyFig.2a
andTable 2, the best-fitting models areclose to the value
ofTeffobtained by P´ al et al. (2008).
The match of the best-fitting model (Model 3 of Ta-
ble 2) to the observed frequencies is illustrated in a so-
called´ echelle diagram (Grec et al. 1983) in Fig. 1c. In
accordance with Eq. (1) the frequency spectrum is di-
vided into slices of length ∆ ν, starting at a frequency of
10.8µHz; the figure shows the location of the observed
(filled symbols) and computed (open symbols) frequen-
cies within each slice, against the starting frequency of
the slice; the model results extend to the acoustical cut-
off frequency, 1930 µHz, of the model. There is clearly a
very good overall agreement between model and obser-
vations, including the detailed variation with frequency
which reflects the frequency dependence of the large sep-
aration, as a possible diagnostics of the outer layers of
the star (e.g., Houdek & Gough 2007).
We have finally made a fit of the inferred ∝angbracketleftρ∗∝angbracketright, as
well asTeffand [Fe/H] from P´ al et al. (2008), to com-
puted evolutionary tracks from the Yonsei-Yale compi-
lation (Yi et al. 2001). This was based on a Markov
Chain Monte Carlo analysis to obtain the statistical
properties of the inferred quantities (see Brown 2010,
for details). This resulted in M= 1.520±0.036M⊙,
R= 1.991±0.018R⊙and an age of 2 .14±0.26Gyr.
We note that the age estimate reflects the specific as-5
sumptions in the Yonsei-Yale evolution calculations; as
indicated by Fig. 2d the true uncertainty in the age de-
termination is likely somewhat larger.
4.2.HAT-P-11
For HAT-P-11 the oscillation amplitudes were much
smaller than in HAT-P-7, as expected from the general
scaling of amplitudes with stellar mass and luminosity
(e.g., Kjeldsen & Bedding 1995). Thus with the present
short run of data it has only been possible to determine
thelargeseparation∆ ν0= 180.1µHzfromthemaximum
in the correlation analysis. We have matched this to a
grid of models, including diffusion and settling of helium,
with masses between 0.7 and 0 .9M⊙and [Fe/H] between
0.21 and 0.41. These models provide a good fit to the
observed TeffandL/L⊙; note that in the presentcase the
luminosity is based on a reasonably well-determined par-
allax. We havedetermined an estimateof ∝angbracketleftρ∗∝angbracketrightbyaverag-
ing the results of those models which match the observed
∆ν0and lie within 2 standard deviations ( ±100K) from
the value of Teffprovided by Bakos et al. (2010); the re-
sult is∝angbracketleftρ∗∝angbracketright= 2.5127±0.0009gcm−3. Although the for-
mal error is extremely small, owing to a tight relation
between the large separation and the mean density for
stars in this region in the HR diagram, the true error is
undoubtedly substantially larger. In particular, we ne-
glected the error in the determination of ∆ ν0and these
data have not allowed a correction for the systematic
errors in the modeling of the near-surface layers of the
star.
4.3.TrES-2
Here also we were unable to determine individual fre-
quencies from the present set of data. The expected am-
plitudes are smaller than for HAT-P-7, and the noise
level higher due to the fainter magnitude of TrES-2.
The correlation analysis yielded two possible values of
∆ν0: 97.7µHz and 130 .7µHz. For this star ∝angbracketleftρ∗∝angbracketrighthas
been determined from the analysis of the transit light
curve. Sozzetti et al. (2007) obtained ∝angbracketleftρ∗∝angbracketright= 1.375±
0.065gcm−3, while Southworth (2009) found ∝angbracketleftρ∗∝angbracketright=
1.42±0.13gcm−3. From the scaling with ∝angbracketleftρ∗∝angbracketright1/2thesmaller of the two possible values of ∆ ν0is clearly incon-
sistent with these values of ∝angbracketleftρ∗∝angbracketright, while ∆ ν0= 130.7µHz
yields models that are consistent with the observed Teff
and log(g) of Sozzetti et al. (2007) as well as with these
values of the mean density. Here we considered a grid
of models with helium diffusion and settling, masses be-
tween 0.85 and 1 .1M⊙and [Fe/H] between −0.25 and
−0.05. Determining again the mean value of ∝angbracketleftρ∗∝angbracketrightfor
those models that matched ∆ ν0and had Teffwithin two
standard deviations of the value of Sozzetti et al. (2007)
we obtained ∝angbracketleftρ∗∝angbracketright= 1.3233±0.0027gcm−3. As in the
case of HAT-P-11 the true error is likely substantially
higher.
5.DISCUSSION AND CONCLUSION
The present preliminary analysis provides a striking
demonstration of the potential of Keplerasteroseismol-
ogyanditssupportingroleintheanalysisofplanethosts.
Thesestarswill undoubtedly be observedthroughout the
mission and hence the quality of the data will increase
substantially. For HAT-P-7 the detected frequencies are
already close to what will be required for a detailed anal-
ysis of the stellar interior, beyond the determination of
the basic parameters of the star. Thus here we can look
forward to a test of the assumptions of the stellar mod-
eling; the resulting improvements will further constrain
the overall properties of the star, in particular its age.
Also, given the observed vsiniwe expect a rotational
splitting comparable to that observed in the Sun, and
hence likely detectable with a few months of observa-
tions. For the other two stars there is strong evidence
for the presence of solar-like oscillations; thus continued
observations will very likely result in the determination
of individual frequencies and hence further constraints
on the properties of the stars.
Funding for this Discovery mission is provided by
NASA’s Science Mission Directorate. We are very grate-
ful to the entire Keplerteam, whose efforts have led to
this exceptional mission. The present work was sup-
ported by the Danish Natural Science Research Council.
Facilities: The Kepler Mission
REFERENCES
Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. 2009,
Asteroseismology, Springer, Heidelberg, in the press
Ammler-von Eif, M., Santos, N. C., Sousa, S. G., Fernandes, J .,
Guillot, T., Israelian, G., Mayor, M., & Melo, C. 2009, A&A,
507, 523
Angulo, C., et al. 1999, Nucl. Phys. A, 656, 3
Bakos, G. ´A., et al. 2010, ApJ, in the press (arXiv:0901.0282v2)
Bedding, T. R., & Kjeldsen, H. 2008, in Proc. 14thCambridge
Workshop on Cool Stars, Stellar Systems, and the Sun, ed.
G. T. van Belle, ASP Conf. Ser., San Francisco, 384, 21
B¨ ohm-Vitense, E. 1958, ZAp, 46, 108
Borucki, W., et al. 2009, in Proc. IAU Symp. 253, Transiting
Planets, eds F. Pont, D. Sasselov & M. Holman, IAU and
Cambridge University Press, 289
Borucki, W., et al. 2010, Science, submitted
Brown, T. M. 2010, ApJ, in the press (arXiv:0912:1639)
Christensen-Dalsgaard, J. 2002, Rev. Mod. Phys., 74, 1073
Christensen-Dalsgaard, J. 2008a, Ap&SS, 316, 13
Christensen-Dalsgaard, J. 2008b, Ap&SS, 316, 113Christensen-Dalsgaard, J., Arentoft, T., Brown, T. M., Gil liland,
R. L., Kjeldsen, H., Borucki, W. J., & Koch, D. 2008, in Proc.
HELAS II International Conference: Helioseismology,
Asteroseismology and the MHD Connections, eds L. Gizon &
M. Roth, J. Phys.: Conf. Ser., 118, 012039
Dittmann, J. A., Close, L. M., Green, E. M., Scuderi, L. J., &
Males, J. R. 2009, ApJ, 699, L48
Gilliland, R. L., et al. 2010a, PASP, in the press
Gilliland, R. L., McCullough, P. R., Nelan, E. P., Brown, T. M .,
Charbonneau, D., Nutzman, P., Christensen-Dalsgaard, J., &
Kjeldsen, H. 2010b, ApJ, submitted
Gilliland, R. L., et al. 2010c, ApJL, in the press
Grec, G., Fossat, E., & Pomerantz, M. 1983, Sol. Phys., 82, 55
Grevesse, N., & Noels, A. 1993, in Origin and evolution of the
Elements, eds N. Prantzos, E. Vangioni-Flam & M. Cass´ e
(Cambridge: Cambridge Univ. Press), 15
Houdek, G., & Gough, D. O. 2007, MNRAS, 375, 861
Huber, D., Stello, D., Bedding, T. R., Chaplin, W. J., Arento ft,
T., Quirion, P.-O., & Kjeldsen, H. 2009, Comm. in
Asteroseismology, 160, 74
Iglesias, C. A., & Rogers, F. J. 1996, ApJ, 464, 943
Kjeldsen, H., & Bedding, T. R. 1995, A&A, 293, 876 Christensen-Dalsgaard et al.
Kjeldsen, H., Bedding, T. R., & Christensen-Dalsgaard, J. 2 008,
ApJ, 683, L175
Kjeldsen, H., Bedding, T. R., & Christensen-Dalsgaard, J. 2 009,
in Proc. IAU Symp. 253, Transiting Planets, eds F. Pont, D.
Sasselov & M. Holman, IAU and Cambridge University Press,
309
Koch, D., et al. 2010, ApJL, submitted
Michaud, G., & Proffitt, C. R. 1993, in Proc. IAU Colloq. 137:
Inside the stars, eds A. Baglin, & W. W. Weiss, ASP Conf.
Ser., San Francisco, 40, 246
Nutzman, P., Gilliland, R. L., McCullough, P. R., Charbonne au,
D., Christensen-Dalsgaard, J.,Kjeldsen, H., Nelan, E. P.,
Brown, T. M., & Holman, M. J. 2010, ApJ, submitted
O’Donovan, F. T., et al. 2006, ApJ, 651, L61P´ al, A., et al. 2008, ApJ, 680, 1450
Rogers, F. J., Swenson, F. J., & Iglesias, C. A. 1996, ApJ, 456 ,
902
Southworth, J. 2009, MNRAS, 394, 272
Sozzetti, A., Torris, G., Charbonneau, D., Latham, D. W.,
Holman, M. J., Winn, J. N., Laird, J. B., & O’Donovan, F. T.
2007, ApJ, 664, 1190
Tassoul, M. 1980, ApJS, 43, 469
Yi, S., Demarque, P., Kim, Y.-C., Lee, Y.-W., Ree, C. H.,
Lejeune, T., & Barnes, S. 2001, ApJS, 136, 417
Vandakurov, Yu. V. 1967, AZh, 44, 786 (English translation:
Soviet Ast., 11, 630)