# Automatic Determination of Stellar Parameters via Asteroseismology of
Stochastically Oscillating Stars: Comparison with Direct Measurements

Pierre-Olivier Quirion1,2,3, and Jørgen Christensen-Dalsgaard2,3, Torben
Arentoft2,3 1 Canadian Space Agency, 6767 Route de l’Aéroport, Saint-Hubert,
QC, J3Y 8Y9 Canada
2 Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C,
Denmark
3 Danish AsteroSeismology Center  pierre-olivier.quirion@asc-
csa.gc.ca,jcd<EMAIL_ADDRESS>
###### Abstract

Space-based projects are providing a wealth of high-quality asteroseismic
data, including frequencies for a large number of stars showing solar-like
oscillations. These data open the prospect for precise determinations of key
stellar parameters, of particular value to the study of extra-solar planetary
systems. Given the quantity of the available and expected data it is important
to develop efficient and reliable techniques for analyzing them, including the
determination of stellar parameters from the observed frequencies. Here we
present the SEEK package developed for the analysis of asteroseismic data from
the Kepler mission. A central goal of the package is to obtain a fast and
automatic determination of the stellar radius and other parameters, in a form
that is statistically well-defined. The algorithms are tested by comparing the
results of the analysis with independent measurements of stellar radius and
mass, for a sample of well-observed stars. We conclude that the SEEK package
fixes stellar parameters with accuracy and precision.

methods: statistical – stars: fundamental parameters – stars: oscillations –
stars: solar-type

††slugcomment: The Astrophysical Journal; Accepted

##  1 ASTROPHYSICAL CONTEXT

Solar-like oscillation occurs in stars with convective envelopes and are seen
as acoustic modes (p modes) with low angular degree ${l}\lesssim 3$ and
intermediate to high radial order $10\lesssim n\lesssim 30$. After early work
by Ando & Osaki (1975) and Goldreich & Keeley (1977a, b), evidence has been
found that the excitation of solar-like oscillation is caused by turbulent
convection in the uppermost part of the envelopes of stars (see for example
Balmforth, 1992; Goode et al., 1992; Goldreich et al., 1994). An important
aspect of the driving caused by turbulent convection is that the excitation of
the mode occurs at random times, and hence the process is known as stochastic
excitation. One of the effects of this stochastic behavior is that the
eigenmode phases are changing with time. In the Sun, the modes with the
largest amplitudes, around 3000 $\mu$Hz, have a coherence time of less than 12
days (Gelly et al., 2002; Chaplin et al., 1997). We can contrast this value to
classical pulsators where the coherence time is of the order of the stellar
evolution time scale. Short but recent introductions to the topic of
stochastic modes excitation, and references to modern approaches to that
problem, can be found in Houdek (2006) and Samadi et al. (2008).

The first star other than the Sun showing evidence for individual frequencies
of solar-like oscillation, $\eta$ Boötis, was observed by Kjeldsen et al.
(1995), using a technique measuring the equivalent widths of the stellar
Balmer lines over time. Most subsequent observations of solar-like oscillation
have been made with Doppler-velocity measurements using a laboratory reference
(typically an iodine cell) to measure the shift in a stellar spectrum. This
technique finally permitted the first unambiguous detection of solar-like
oscillation in $\alpha$ Centauri A (Bouchy & Carrier, 2001, 2002). Since then,
p-mode excitation has been detected with ground-based observations in more
than a dozen main-sequence and subgiant stars. Lists of these solar-like
oscillators can be found in Bedding & Kjeldsen (2008) and Bruntt et al.
(2010).

Doppler-velocity measurement is the technique of choice to observe solar-like
oscillation, providing the least interference from stellar background
contributions to the ‘noise’ in the observations. The ratio between the signal
and the stellar background noise, that is non-periodic signals coming, for
instance, from granulation, is far higher in Doppler-velocity than in
photometric intensity measurements. Solar-like oscillations can also be
observed in photometric intensity measurement, albeit with a higher stellar
background noise, but with the advantage of allowing simultaneous observation
of a large number of stars. However, owing to noise induced by the Earth’s
atmosphere such observations are essentially only possible from space (Harvey,
1988). Space-based photometric data for solar-like oscillations have started
to be obtained with the help of the CoRoT (Baglin et al., 2006) and Kepler
(Borucki et al., 2010) missions. Results coming from these satellites, in
particular the Kepler satellite (Gilliland et al., 2010), are indeed a
motivation for the work undertaken in the present study. The development of an
objective technique to interpret results from asteroseismology, more than
being intrinsically desirable, is now made necessary by the era of high-speed
space photometry, where data bases containing large numbers of light curves
ready for asteroseismic study are made available. By objective
asteroseismology we mean a technique that is not directed at every step by
subjective choice of a scientist and that results in an estimate of global
stellar parameters along with realistic, if not completely objective,
uncertainties on these parameters. The need for efficient and reliable,
automatic analysis of asteroseismic data will be further emphasized with the
ESA PLATO mission (Catala, 2009) which, if finally selected, will be launched
around 2018.

The Sun is evidently the canonical example of solar-like oscillation. As shown
in detail in the review on helioseismology by Christensen-Dalsgaard (2002) the
study of the Sun’s oscillations has proven that physical information can be
extracted from the understanding of its vibration modes. The extraction of
information from other solar-like stars has started to occur and the sustained
modeling efforts by several groups has yielded many asteroseismic studies
(e.g. Thoul et al., 2003; Eggenberger et al., 2004; Miglio & Montalbán, 2005;
Carrier et al., 2005; Eggenberger et al., 2005, 2008; Mosser et al., 2008).
This has positively confirmed the promise of asteroseismology: to dramatically
improve knowledge about stellar parameters. This prospect, especially the
promise to determine stellar radii within a 3 % error, is very attractive for
the exoplanet groups involved in the Kepler mission. For example, Baines et
al. (2007) showed how the interferometric radius measurement of HD 189733 has
dramatically improved and clarified the value for the radius a nd mean density
of its planetary companion. We expect radius measurement from asteroseismology
to have similar effect. An early indication of the power of Kepler
asteroseismology to characterize exoplanet hosts was provided by Christensen-
Dalsgaard et al. (2010).

Photometric measurement of planetary transits, given that a precise modeling
of the limb darkening is achieved (Prsa et al. 2010, in preparation), permits
to identify four parameters of the star-planet systems: its orbital period
$\Pi_{\rm orb}$, its inclination $i$ toward the observer and the fractional
radii of the star and planet, which are defined as

$\displaystyle r_{\rm A}=\frac{R_{\rm A}}{a}{\rm,}$ $\displaystyle{\rm and}$
$\displaystyle r_{\rm b}=\frac{R_{\rm b}}{a}\;,$ (1)

respectively. We clearly see that the knowledge of the stellar host radius
$R_{\rm A}$ and its associated error directly leads to a determination of the
desired planet radius $R_{\rm b}$, as well as the orbital semi-major axis $a$.
Further details on the influence of the precision of stellar parameters on the
inference of physical properties of planetary system were provided by
Southworth (2009). One of the goals of the present study is to test the
reliability of asteroseismology in determining the radius of stars by
comparing the results with available interferometric measurements.

This work presents an objective procedure to extract information from solar-
like oscillations. The procedure, named SEEK, determines fundamental stellar
parameters with the use of quantities inspired by the p-mode asymptotic
asteroseismic relations, namely the large and small separations, $\Delta\nu$
and $\delta\nu$, respectively (Tassoul, 1980). It can also make use of
standard astrophysical input, the parallax $\pi$, the $V$ band magnitude and
the reddening $E_{B-V}$, as well as input derived from atmosphere modeling,
the effective temperature $T_{\rm eff}$, the gravity $\log g$ and finally the
iron to hydrogen ratio [Fe/H]. Therefore, SEEK is not a purely asteroseismic
procedure that only uses the frequency spectra of a pulsating star to
constrain its fundamental parameters, but rather a hybrid procedure that uses
both traditional and seismic input.

The procedure is part of the Kepler Asteroseismic Science Consortium (KASC)
pipeline that provides in priority stellar parameters to be used by exoplanet
seekers of the Kepler team. Its aims are also to analyze every star in which
$\Delta\nu$ and/or $\delta\nu$ has been fixed and to provide an extended and
homogeneous overview of the disk population observed by Kepler. A description
of SEEK and of its main characteristics is made in the following sections. We
tested SEEK on a set of well-studied stars. Of course the Sun is the golden
standard, but we also tested the procedure against stars with diameters
determined through interferometric measurements and/or with masses derived
from Kepler’s third law, thus testing our procedure against the closest we
have to absolute quantities. The stars studied comprise $\beta$ Hydri (HD
2151), $\tau$ Ceti (HD 10700), Procyon A (HD 61421), $\eta$ Boötis (HD
121370), $\alpha$ Centauri A and B (HD 128620 and HD 128621), and $70$
Ophiuchi A (HD 165341).

##  2 THE SEEK PROCEDURE

###  2.1 Stellar models

The SEEK procedure makes use of a large grid of stellar models computed with
the Aarhus Stellar Evolution Code (ASTEC). It compares an observed star with
every model of the grid and makes a probabilistic assessment, with the help of
Bayesian statistics, about the structure of that star.

Details of ASTEC and its input physics were described by Christensen-Dalsgaard
(2008a). Out of the possible options offered by ASTEC, we used, for all our
computations: the OPAL equation of state (Swenson et al., 1996) along with the
OPAL plus Ferguson & Alexander opacity tables (Iglesias & Rogers, 1996;
Alexander & Ferguson, 1994), the element to element ratios in the metallic
mixture of Grevesse & Sauval (1998), and convection treated with the mixing-
length formulation of Böhm-Vitense (1958); the mixing length to pressure scale
height ratio $\alpha$, characterizing the convective efficacy, was treated as
a variable parameter in the SEEK fits. Neither diffusion nor overshooting was
included.

A brief introduction to a beta version of SEEK was presented along with a
series of tests using synthetic observations in $\S$ 5 of Stello et al.
(2009). The beta version was built with an ad hoc probabilistic approach that
is quite different from the one used in the present version. There is a major
difference in the way errors on stellar parameters is obtained. This
difference lies in the fact that the new grids have been extended and are
dense enough to permit the use of Bayesian statistic. While SEEK yields
similar results with both approaches, the Bayesian method used here is more
rigorous, more robust and more comprehensive than the technique developed for
the beta version of SEEK.

The core of SEEK is a grid of models used to fit observations. We have
calculated 7,300 evolution tracks or 5,842,619 individual models. Each track
calculation was stopped after the track had reached the giant branch or an age
of $\tau=15\times 10^{9}$ yr. The tracks are separated into 100 subsets using
a different combination of metallicity $Z$, initial hydrogen content $X_{0}$
and mixing-length parameter $\alpha$. These combinations are separated into
two regularly spaced and interlaced subgrids. The first subgrid comprises
tracks with $Z=[0.005,0.01,0.015,0.02,0.025,0.03]$,
$X_{0}=[0.68,0.70,0.72,0.74]$, and $\alpha=[0.8,1.8,2.8]$ while the second
subset has $Z=[0.0075,0.0125,0.0175,0.0225,0.0275]$, $X_{0}=[0.69,0.71,0.73]$,
$\alpha=[1.3,2.3]$. Every subset is composed of 73 tracks spanning from 0.6 to
3.0 $M_{\rm\odot}$. The spacing in mass between the tracks is 0.02
$M_{\rm\odot}$ from 0.6 to 1.8 $M_{\rm\odot}$ and 0.1 from 1.0 to 3.0
$M_{\rm\odot}$. A slice of the grid is presented in Figure reffig:slice for
$Z=0.015$, $X_{0}=0.72$ and $\alpha=1.8$ in a $\log g-\log T_{\rm eff}$
diagram.

The structure of the SEEK procedure is quite flexible. It will be easy in
future to expand the grid with regularly or irregularly spaced tracks. In
addition, we have started to test the possibility of interpolating models
along the age-mass plane of the grid and between grid values of $Z$ and
$X_{0}$. The results are quite promising and it seems that the grid is dense
enough with regards to these parameters to realize precise interpolation.
However, interpolation tests in the $\alpha$ direction were not convincing and
it seems that the grid would have to be refined significantly with respect to
that parameter to permit precise interpolation.

Some care has been required in defining the parameters of the grid, including
the specific range and spacing we have chosen. Several factors influenced the
definition of the composition. Recent analyses have led to substantial
revisions of the solar surface composition (Asplund, 2005; Grevesse et al.,
2007; Asplund et al., 2009); since the solar composition is used as reference
to relate the observed [Fe/H] to $Z/X$ this potentially affects the assumed
stellar composition. Here we have chosen to use the older value of Grevesse &
Sauval (1998). This can be justified based on the fact that solar models using
the old values are closer to the helioseismic inferences than those using the
more recent abundances (see, for example Basu et al., 2007; Basu & Antia,
2008). We also emphasize that for most stars, atmospheric parameters in the
literature are still only available with older metallic mixtures. Homogeneity
of our analysis requires the use of these mixtures. The metallicity range used
in the grid, $-0.61\lesssim{\rm[Fe/H]}\lesssim 0.20$, covers most ($\gtrsim
90\%$) of the stellar population of the disk if the Hipparcos color-magnitude
diagram is taken as a reference (Reid, 1999; Chabrier, 2001). This should
permit the analysis of a large majority of the Kepler stars.

The careful reader will also remark that relatively high values of
$Y_{\odot}=0.2713$ and $Z_{\odot}=0.0196$ have been used in the standard
definition of [Fe/H]. This value is used to calibrate solar models of ASTEC to
the right luminosity (Christensen-Dalsgaard, 1998). This value is used in all
conversions done here.

The mass step of 0.02 $M_{\rm\odot}$ from 0.6 to 1.8 $M_{\rm\odot}$ has first
been chosen to be small enough to permit interpolation in the $\log g-\log
T_{\rm eff}$ diagram. However, we realized that this small step in mass has
the virtue of making the grid sufficiently dense that we do not have to rely
on interpolation to make our probabilistic estimate of the stellar parameters.
The lower limit to the range results from the fact that molecules start to
play an important role in these stars and that our evolution code does not
include them. This should not be a handicap to SEEK since the low absolute
magnitude of the stars not included in the grid, with $M<0.5$ $M_{\rm\odot}$,
makes them improbable Kepler targets. The upper limit of 1.8 $M_{\rm\odot}$ is
approximately the highest mass of the grid at which a model on the main
sequence has at least 1% of its radius as a convective envelope. We wanted to
keep a high density of models up to that limit where it is most likely to
observe stars with stochasticall y excited oscillations. The models at higher
masses are put as a safeguard permitting us to pin down a star with a very
thin convective zone showing solar-like oscillations. We also remark that
typically models with higher $\alpha$ and/or higher $X_{0}$ and/or higher $Z$
will tend to keep a convective envelope at slightly higher masses. Similarly,
these models develop a convective core, leading to the typical hook path in
the H-R diagram at higher masses.

The grid mid-value of $\alpha=1.8$ was chosen in accordance with the solar
calibration. The span of $\alpha$ is a trade-off between trying to get a wide
range of values, from 0.8 to 2.8, and to limit the quantity of computed models
in the grid. We had a bias towards testing a wide range of convective
efficiency since it is not clear how the mixing-length theory extends from the
Sun to other main-sequence stars. However, this choice is not ideal since, as
mentioned, the steps are too large to permit interpolation between the grid
points.

The role of SEEK is more to measure the size of the “valley of good solutions”
produced by Kepler observations than it is to probe the deepest abyss of that
valley. Specifically, the quality of a solution is measured by how well the
computed observables in the grid, $\\{q_{i}^{\rm g}\\}$, match the observed
values $\\{q_{i}^{\rm obs}\\}$; this is quantified in terms of

$\chi^{2}={\displaystyle\sum_{i=0}^{n}}\left(\frac{{q}_{i}^{\rm
obs}-{q}_{i}^{\rm g}}{\sigma_{i}}\right)^{2}\;,$ (2)

where $\sigma_{i}$ is the estimated error for the observation $q_{i}^{\rm
obs}$ and $n$ the number of observables. SEEK does not claim to find the best
model for an observed star; its aim is to draw the contour of good solutions
which is located around $\chi^{2}_{r}=\chi^{2}/P\sim 1$, $P$ being the number
of degrees of freedom of the problem. SEEK also outputs the stellar parameters
with reliable error bars. We do not want to underestimate the errors by
restricting the size of our grid. The approach is especially well suited to
our problem since our solutions coming from the observation of $\Delta\nu$ are
always heavily degenerate. This means that we do not expect a single and small
region of the parameter space to yield a result that is much better than any
other in terms of $\chi^{2}$. Instead we have, as illustrated in Figure 2 for
the star $\alpha$ Centauri A, a large region of the parameter space with
$\chi^{2}_{r}\sim 1$. In Figure 2, we present for drawing convenience
$\log\chi^{2}_{r}$ as a function of the mass $M$ and the normalized age
defined as

$\tau_{\rm N}=\frac{\tau}{\min(\tau_{\rm RGB},14{\rm Gyr})}\;,$ (3)

where $\tau$ is the age of a model on a specific track and $\tau_{\rm RGB}$ is
the age of the model following the same track when it reaches the red-giant
branch (RGB). In Figure 2, this parameterization is useful for visual purposes
as it permits a smooth transition of the $\chi^{2}$ values from the lower-mass
models not reaching the RGB to the models reaching it. The exact position of
$\tau_{\rm RGB}$ on the RGB is not critical since the evolution time scale on
the RGB is much faster than on the main sequence. We stress that this
parameterization was used for cosmetic purposes and is not involved in the
SEEK computations. The models showing higher $\chi_{r}^{2}$ to the left of the
valley are caused by the appearance of convective cores in heavier models;
this makes the automatic computation of the small separation somewhat
difficult and creates these artifacts.

Two works presenting pipelines to do asteroseismology of solar-like stars have
recently been published, Metcalfe et al. (2009) and Basu et al. (2010). In the
former, the authors combine a pipeline using a genetic algorithm to find the
best model in the parameter space and a local analysis, based on linearization
around this solution, to refine the fit and estimate the error in that
solution. Basu et al. (2010) find the best model with a grid of stellar
models, not unlike SEEK but using a smaller grid, while the errors are
determined with a frequentist approach, using a Monte Carlo simulation of
synthetic observations. We certainly recognize the desirability of a genetic
algorithm, or a similar technique aiming at the absolute minimum in the
parameter space, while also acknowledging the large computation power needed
for such techniques. However, we note that a local error estimator used by
Metcalfe et al. (2009) assumes that the solution is linear within the error
bars and that no nearby local minima can be deep enough to contribute to the
error analysis; this can lead to underestimation of the errors. In our view,
only an error estimator, Bayesian or frequentist, looking at various models
around the best solutions may provide fully reliable results. This is
particularly true in a problem, as the present, where the valley of solution
is extended. This preference for Bayesian statistics has motivated our use of
it in the development of SEEK.

###  2.2 The Observables

The grid permits the mapping of the model physical input parameters
$\boldsymbol{p}\equiv\\{M,\tau,Z,X_{0},\alpha\\}$ into the grid of observable
quantities $\boldsymbol{q}^{\rm g}\equiv\\{\Delta\nu,\delta\nu,T_{\rm
eff},\log g,{\rm[Fe/H]},M_{V},...\\}$, defining the transformation

$\boldsymbol{q}^{\rm g}=\mathcal{K}(\boldsymbol{p}).$ (4)

We use these quantities and compare them to the actual observed quantities
$\boldsymbol{q}^{\rm obs}$.

Some of the observables used for the subsequent fit are easily extracted from
the grid like $T_{\rm eff}$, $\log g$, [Fe/H]. However, $M_{V}$, $\Delta\nu$,
and $\delta\nu$ need some extra attention. We compute absolute magnitude
$M_{V}$, in the Johnson $V$ band, of our model by applying the bolometric
correction $B_{\rm c}(V)$ of the VandenBerg & Clem (2003) tables to our
luminosity $L$,

$M_{V}=M_{\rm Bol,\odot}-2.5\log\frac{L}{L_{\rm\odot}}-B_{\rm c}(V),$ (5)

where $M_{\rm Bol,\odot}$ is the VandenBerg & Clem (2003) prescription of 4.75
for the Sun’s bolometric magnitude. We can then compare this value to its
observed counterpart defined as

$M^{\rm obs}_{V}=5\log\pi-(V-A_{v})+5.$ (6)

The conversion is done using the following relation between the absorption
$A_{V}$ and the interstellar reddening $E_{B-V}$: $A_{V}=3.1E_{B-V}$.

We computed p-mode frequencies for all models on the main sequence and
subgiant branch using the ADIPLS code (Christensen-Dalsgaard, 2008b); the
models on the red-giant branch are not included in this asteroseismic study.
The large separation $\Delta\nu$ is computed with modes of $l=0$ only,

$\Delta\nu=\langle\nu_{n,0}-\nu_{n-1,0}\rangle\;,$ (7)

while $\delta\nu$ is the combination of modes with $l=0$, $2$ only,

$\delta\nu=\langle\nu_{n,0}-\nu_{n-1,2}\rangle\;;$ (8)

here $\nu_{n,l}$ is the frequency of the pulsation mode with angular degree
$l$ and radial order $n$. We know that these asymptotic values change with the
radial order $n$ in models. This is especially true in the case of
$\delta\nu$. To compare observed separations with model values we have
computed the average separations with up to 8 modes having different $n$ and
centered on a predetermined value of $\nu_{\rm central}$. The quantity
$\nu_{\rm central}$ is related to the maximum power $\nu_{\rm max}$ seen in
the Fourier transform of light curves of solar-like stars (see Stello et al.,
2009, for comments on $\nu_{\rm max}$). Thus, in our grid the large and small
separations are functions of the observed $\nu_{\rm max}$.

The computation of the average large and small separations is done by finding
the maximum of the cross-correlation function of the frequencies selected
around a suitable value $\nu_{\rm central}$ near $\nu_{\rm max}$. Figure 3
shows how the small separation is computed for a specific model of the grid.
In panel (a), the function $\psi_{0}$ corresponding to the $l=0$
eigenfrequencies is shown in blue and the function $\psi_{2}$ for $l=2$ in
red. These functions are convolved with the triangular distribution $T$ of
panel (b). The base of the triangular distribution is one fifth of the
smallest distance between any combination of the $l=0$ eigenfrequencies. The
convolution

$\displaystyle D_{i}(x)\equiv\int_{-\infty}^{\infty}\psi_{i}(s)T(x-s)ds$
$\displaystyle i=0,2$ (9)

is presented in panel (c) while the resulting cross-correlation,

$\displaystyle(D_{0}\ast
D_{2})(x)\equiv\int_{-\infty}^{\infty}D_{0}(s)D_{2}(x+s)ds$ , (10)

is in panel (d). The maximum of the correlation is at $\delta\nu$. It is easy
to see that the extra $l=2$ mode around $\nu\sim 750$ $\mu$Hz is only
producing a small bump in the correlation function and has no effect on the
value of $\delta\nu$. For $\Delta\nu$, the location of the maximum of the
auto-correlation $\left(D_{0}\ast D_{0}\right)$, apart from the maximum at
zero frequency shift, is taken. This technique is more robust than using the
mean value of Equation (7) and especially (8) over different values of $n$.
The cross-correlation is not influenced by missing or extra eigenmodes,
usually nonradial, produced by ADIPLS or by ambiguous determination of the
radial order $n$ of a mode. The method is especially robust when it comes to
computing $\delta\nu$ for models having a convective core, or for models with
avoided crossing. Robustness is very desirable since it is not possible to
control the ADIPLS output for the 6 million models included in th e grid.

In the present version of SEEK, we computed all adiabatic modes with $l=0$,
$2$ in a fixed range of the dimensionless frequency, $10\leq\sigma^{2}\leq
2800$, where $\sigma$ is related to the angular mode frequency $\omega$ by

$\sigma^{2}=\frac{R^{3}}{GM}\ \omega^{2}\;,$ (11)

where $G$ is the gravitational constant. This range makes it possible to cover
observed solar-like pulsation in known stars on the main sequence and subgiant
branch. These pulsations cluster around the value of $\nu_{\rm max}$ which
tends to follow the scaling relation of Kjeldsen & Bedding (1995)

$\nu_{\rm max}\propto\frac{M}{R^{2}T^{1/2}}.$ (12)

To accelerate the SEEK procedure and for models showing eigenmodes in these
region, we precalculated 25 different sets of $\Delta\nu$ and $\delta\nu$ for
values of $\nu_{\rm central}$ equally spaced on a logarithmic scale, ranging
from the Nyquist frequency of the short cadence of Kepler, 8330 $\mu{\rm Hz}$
(corresponding to a period of 2 min) to 27.8 $\mu{\rm Hz}$ (corresponding to a
period of 10 h). (For any given model the actual range in $\nu_{\rm central}$
was restricted to values corresponding to acoustic modes for that model.) If,
for example, we want to compare the grid value of $\delta\nu$ with the solar
value of $\delta\nu_{\odot}$ at $\nu_{\rm max,\odot}\sim 3333$ $\mu{\rm Hz}$,
our grid automatically chooses the set $\delta\nu(\nu_{\rm central})$ which is
the closest to $\nu_{\rm max,\odot}$ at $\nu_{\rm central}=3220$ $\mu{\rm
Hz}$.

##  3 BAYESIAN APPROACH

The Bayesian statistics method of SEEK and the notation used here were
inspired mainly by the work of Pont & Eyer (2004) and Jørgensen & Lindegren
(2005). These investigations compared the Bayesian approach to other means
used for the determination of stellar age. We refer the reader to these papers
and to the textbook by Gregory (2005) for technical questions involved in the
use of Bayesian statistics.

Let us first define the maximum likelihood function

$\mathcal{L}=\left({\displaystyle\prod_{i=0}^{n}}\frac{1}{\sqrt{2\pi}\sigma_{i}}\right)\exp(-\chi^{2}/2),$
(13)

with $\chi^{2}$ defined in Equation (2). A maximum-likelihood estimate of
stellar parameters can be obtained by finding the maximum of $\mathcal{L}$,
which, in case of Gaussian errors, is equivalent to minimizing $\chi^{2}$. It
can be argued that a maximum likelihood estimator is often enough to estimate
stellar parameters. However, the highly non-linear mapping function
$\mathcal{K}$ makes it necessary to provide a set of priors, especially if a
realistic estimate of the errors on the stellar parameters is to be made. The
prior can be seen as a specific weight attached to every point of the grid
determining the probability of that point to be observed. The most obvious
example of the utility of priors is well studied in Jørgensen & Lindegren
(2005). It arises when one has to choose between different stellar models at
different evolutionary stages but showing the same observables. In their
example, one model is slowly evolving on the main sequence while the other is
rushing up the subgiant branch. An experienced astronomer would naturally
choose the slowly evolving star as the most probable model even if the
resulting $\mathcal{L}$ value is the same for both models. For the astronomer
using Bayesian tools, a larger prior weight $f_{0}(\boldsymbol{p})$ given to
the slowly evolving model automatically makes it a better candidate because
the resulting probability, or posterior $f$ defined at each point $j$ as

$f^{j}\propto f_{0}^{j}\mathcal{L}^{j}\;,$ (14)

is bigger for that model. The grid overall posterior can be described as a
density function,

$f(\boldsymbol{p})\propto
f_{0}(\boldsymbol{p})\mathcal{L}(\mathcal{K}(\boldsymbol{p}))\;.$ (15)

¿From the grid presented in the previous section, the computation of
$\mathcal{L}$ at every point is straightforward. On the other hand, the
definition of the priors $f$ requires some insight in the problem.

The most natural way to write the prior density is as a function of
$\boldsymbol{p}$, the model parameters,

$f_{0}=\Psi(\tau)\Phi(Z|\tau)\zeta(X_{0}|Z,\tau)\xi(M|X_{0},Z,\tau)\beta(\alpha|X_{0},Z,\tau,M)$
(16)

where $\Psi(\tau)$ is the star formation rate (SFR) through time, and
$\Phi(Z|\tau)$ and $\zeta(X_{0}|Z,\tau)$ describe the metallicity and initial
hydrogen mixture as a function of age and as a function of age and
metallicity, respectively. The initial mass function (IMF)
$\xi(M|X_{0},Z,\tau)$ is function of the element mixture as well as a function
of time, and the prior related to the mixing-length parameter
$\beta(\alpha|X_{0},Z,\tau,m)$ can also depend on the other stellar
parameters.

The distribution and the correlations can be built with the help of
assumptions made from observations and from stellar and galactic models. In
our case, since we are mainly interested in the Kepler satellite field of view
which is well documented through the Kepler Input Catalog (KIC), we could use
Equation (4) to map a color-color distribution, or any other relation found in
the KIC, into the prior of Equation (16). However, we shall keep here a more
conservative approach, or as Jørgensen & Lindegren (2005) put it, a non-
committal approach and assume that $\tau$, $Z$, $X_{0}$, $M$ and $\alpha$ are
independent. Thus we write

$f_{0}=\Psi(\tau)\Phi(Z)\zeta(X_{0})\xi(M)\beta(\alpha)\;,$ (17)

and also take $\Psi(\tau)$, $\Phi(Z)$, $\zeta(X_{0})$, and $\beta(\alpha)$ to
be flat. This means that the prior density is constant through these
dimensions. We only make an assumption on the IMF and choose the so-called
IMF1 model from Chabrier (2001) where

$\displaystyle\xi(M)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}0.019\ M^{-1.55}&\text{if }M\leq
1.0M_{\rm\odot}\\\ 0.019\ M^{-2.70}&\text{if
}M>1.0M_{\rm\odot}.\end{array}\right.$ (20)

In a grid, flat priors can be easily illustrated by $N$ equally spaced points,
every point having an equal probability $1/N$. Of course, if the grid is not
regularly spaced the weight assignment can get a bit more complicated. For a
non-flat prior, we could distribute the $N$ grid points so that they follow
the density prescribed for example by Equation (20) and give every point a
weight of $1/N$ and meet the non-flat IMF prior prescription. However, and
this is especially true in the $\tau$ direction, we have an irregularly spaced
grid. Also, we do not wish to interpolate our grid for the sake of precision
and for the same reason it is not necessarily reasonable to make the grid
sparser in the region where the prior is relatively small. A star observed in
that sparser region, even if less likely, would not be analyzed in the same
detail by SEEK.

We have solved this problem by dividing the parameter space into a number of
small 5-D bins of (penta-)volume $V^{i}$ containing each a small number of
point $n^{i}\geq 1$. Then, a prior weight $f^{i}_{0}$, where

$f^{i}_{0}=\int_{V^{i}}\Psi(\tau)\Phi(Z)\zeta(X_{0})\xi(M)\beta(\alpha)dV^{i}\;,$
(21)

is distributed through the $n^{i}$ point contained in the bin such that the
weight $f_{0k}^{i}$ given to model $k$ is inversely proportional to the
evolution speed of that model,

$\displaystyle
f_{0k}^{i}=\frac{\Delta\tau_{k}}{\sum_{k^{\prime}}\Delta\tau_{k^{\prime}}}f_{0}^{i}$
$\displaystyle{\rm where}$ $\displaystyle k=1,2,3,...,n^{i}\;.$ (22)

Here $\Delta\tau_{k}$ is the time taken by model $k$ to evolve to the next
point in its evolution track. This forces the slowly evolving models of the
bin to have larger priors. These two equations are central pieces of SEEK.
They ensure that the probability given to a model reflects the size of the
parameter space where no other models are computed around it. These empty
regions of the parameter space will be seen as gaps in some of the probability
distributions of the next section.

##  4 RESULTS

In practice, once $f$ is fixed, the computation of the posterior probability
$\mathcal{G}$ on an arbitrary set of parameters is easily done. In one
dimension, the grid is sliced along the desired parameter and all models
entering a bin from $s$ to $s+\Delta s$ are added

$\mathcal{G}(s_{i})=C{\displaystyle\sum_{k=0}^{K}}f^{k}(s_{i})\;,$ (23)

where the $f^{k}(s_{i})$ are the $K$ points of the grid lying from $s_{i}$ to
$s_{i+1}=s_{i}+\Delta s$, and $C$ is a constant ensuring that the total
probability of the problem is 1 (see Equation 24). The results of this
computation is a probability distribution $\mathcal{G}(s)$ that is best seen
as a histogram. The results for four parameters $R$, $M$, $\tau$, and $T_{\rm
eff}$, computed for 70 Ophiuchi A, are presented in Figure 4. For convenience,
there and in the following figures, we have normalized the histograms so that
${\rm max}(\mathcal{G}(s))=1$.

Another important quantity is the running integral of $\mathcal{G}$ seen as
the solid red line in Figure 4. The running integral over the entire parameter
space is the total probability of the grid. This should be 1 since we assume
that the observed star lies within the grid. We use that relation to fix the
constant $C$ in Equation (23)

${\displaystyle\sum_{i}}\mathcal{G}(s_{i})=1\;.$ (24)

Similarly, we can integrate over a section of the distribution to get a $1$,
$2$, or $3\,\sigma$ probability range for $s$. The error interval
$[s^{(1)},s^{(2)}]$ output by SEEK is by default at the $1\,\sigma$ level and
is determined based on the probability integral of $\mathcal{G}$.
Specifically,

$\int_{s_{0}}^{s^{(1)}}\mathcal{G}(s)ds=0.1585\;,\qquad\int_{s_{0}}^{s^{(2)}}\mathcal{G}(s)ds=0.8415\;,$
(25)

where $s_{0}$ is the initial value of the parameter, and the integrals are
represented by suitable sums, as in Equation (24). (Note that
$0.8415-0.1585=0.6830$ corresponds to $1\,\sigma$ for a normal distribution.)
The solution SEEK outputs is the mid-point $(s^{(1)}+s^{(2)})/2$ of this
interval. In this way the error bars are symmetric; this would not be the case
if the median value $s^{(m)}$, with

$\int_{s_{0}}^{s^{(m)}}\mathcal{G}(s)ds=0.5\;,$ (26)

had been chosen. This $1\,\sigma$ section of the running integral is depicted
in blue (between the dots) on the running integral of Figure 4.

These histograms along with the net numbers output by SEEK are useful to
interpret the results. In the case of 70 Oph A, the solutions $R=0.86\pm 0.01$
[$R_{\rm\odot}$] and $T_{\rm eff}=5306\pm 36$ [K] are well constrained and
quasi Gaussian. The histograms visually confirm the validity of the numerical
solutions. The result for the mass $M=0.90\pm 0.04$ $M_{\rm\odot}$ reveals
some of the grid’s limitation as there are gaps in the otherwise well defined
structure of $\mathcal{G}(M)$. These gaps are artifacts of the 0.02
$M_{\rm\odot}$ mesh used in the grid below 1.8 $M_{\rm\odot}$. However, since
the prior density is properly weighted through Equations (23) and (24), the
overall value of the mass, with its error bars, is correct. For the age,
$\tau=8.65\pm 3.64$ Gyr, a large error is revealed by the output and the
histogram shows the probability distribution to be populated all along $\tau$
from 2 Gyr all the way to 13 Gyr. In that case a more reasonable answer is a
lower limit of $\tau\gtrsim 2$ Gyr. The histogram for $\tau$ also shows gaps
that are due to the finite resolution of the grid.

Table Automatic Determination of Stellar Parameters via Asteroseismology of
Stochastically Oscillating Stars: Comparison with Direct Measurements shows
our selection of solar-like oscillators and observables found in the
literature. These observables are processed by SEEK and produce the outputs in
Table Automatic Determination of Stellar Parameters via Asteroseismology of
Stochastically Oscillating Stars: Comparison with Direct Measurements and
Table 3. In Table Automatic Determination of Stellar Parameters via
Asteroseismology of Stochastically Oscillating Stars: Comparison with Direct
Measurements, we compared SEEK results with direct measurements of radius via
interferometry and of mass via Kepler’s third law. For SEEK, the fits have
been made for two different posterior sets, the full set

$\boldsymbol{q}^{\rm g,f}=\\{\Delta\nu,\delta\nu,T_{\rm eff},\log
g,{\rm[Fe/H]},V,\pi,E_{B-V}\\}\;,$ (27)

and one using only a selected subset of the same observables,

$\boldsymbol{q}^{\rm g,s}=\\{\Delta\nu,T_{\rm eff},{\rm[Fe/H]}\\}\;.$ (28)

The subset may be more representative of what we can expect from a typical
Kepler observation when no ground-based follow-up has been done for a star,
and where the asteroseismic data have not yielded a reliable determination of
$\delta\nu$. This can occur in a stochastically oscillating star when the
width of observed modes is larger than the small separation itself. In that
case, we have $\Delta\nu$ from the satellite observations, while $T_{\rm eff}$
and [Fe/H] can be obtained from the KIC.

The results of Table Automatic Determination of Stellar Parameters via
Asteroseismology of Stochastically Oscillating Stars: Comparison with Direct
Measurements show that the 3% promise on the radius precision mentioned in the
introduction is generally reached with both the full set and the subset of
observables. More striking and certainly more important than the precision
itself is the very good accuracy reached by SEEK. It is on this basis that our
method, or any other technique, should be judged, provided that independent
measurements are available. SEEK has pinned down all stellar radii at the 3%
level when the extended set of observable was used. The same level of accuracy
is generally reached if the subset of observables (Eq. 28) is used.

We note that in well-posed problems, where parameters are actually strongly
constrained, the value of $\mathcal{L}$ is much larger than $f_{0}$ close to
the best solution. It is only when we get away from the best parameter values
that the prior gets more important. The details of the priors used here only
have an influence on the wing of the $\mathcal{G}$ distribution. Thus, the
choice of priors has an effect on the errors, especially if the errors are
large, but little effect on the solution itself.

##  5 DISCUSSION

We clearly need to understand the exceptions to the general success of SEEK;
in particular, we note that $\beta$ Hyi was a bit further from the direct
observations with an offset of 6%. An extra tool offered by SEEK can be used
to study the $\beta$ Hyi radius offset when compared to direct measurements.
Figure 5 shows a two-dimensional projection $\mathcal{G}(R,Z/X)$ of the
posterior computed for $\beta$ Hyi with only $\Delta\nu$ and $T_{\rm eff}$ as
input observables. The metallicity is not included as an observable in the
computation as a trick to have a clearer view of the correlation between that
observable and $R$. Several possibilities are offered to explain the
discrepancy between SEEK’s radius $R/R_{\odot}=1.92\pm 0.05$ and the direct
measurement of $1.814\pm 0.017$. The first possibility is that the observed
$Z/X$ is too high. Reducing [Fe/H] by $\sim 0.2$ dex, to $Z/X\sim 0.0163$,
would reconcile the asteroseismic and interferometric radii. This can be
concluded from a visual examination of Figure 5. In fact, lowering the
metallicity by this amount would put it back to the level measured in $\beta$
Hyi by Dravins et al. (1998). We have made an a posteriori test using the
value of metallicity and temperature published by Dravins et al. (1998),
$T_{\rm eff}=5800\pm 100$ K, and [Fe/H]$=0.2\pm 0.1$ instead of the more
recent values of Table Automatic Determination of Stellar Parameters via
Asteroseismology of Stochastically Oscillating Stars: Comparison with Direct
Measurements (da Silva et al., 2006). These inputs processed by SEEK place the
radius $R/R_{\odot}=1.85\pm 0.03$ in much better agreement (2%) with the
observations. We note that Dravins et al. (1998) used the older Noels &
Grevesse (1993) metal mixtures in the computation. This underlines that
accuracy can only be reached if the input metallicity is selected with some
care.

The offset in radius for $\beta$ Hyi can be explained otherwise. The extension
of the grid to lower values of $X_{0}$ would to some extent reconcile SEEK and
the interferometric measurement. It can be seen in Figure 5 that the left-hand
side of the correlation function $\mathcal{G}$ is in fact the edge of the grid
where $X_{0}=0.68$. The extension of the grid would likely expand the existing
ridge of solutions to the left in Figure 5, at lower radii. A simple eye-ball
estimate in the figure can be made to indicate that the combination, $Z\sim
0.0165$, $X_{0}\sim 0.64$ would yield the right metallicity and the right
radius. However, it seems that a hydrogen content that low is not reasonable
and perhaps this possibility should not be considered seriously. The effect of
the hydrogen content, or should we say helium content, on the radius is in
fact the main factor compromising the precision of $R$ in SEEK or any other
technique using the large sepa ration to determine the radii. The error on the
radius is influenced by the size of the grid in the $X_{0}$ direction. The
imperative we had not to underestimate the errors on stellar parameters
influenced our choice of having a large range of values for that parameter. It
also means that raising the precision on the metallicity ratio $Z/X$ to fix
the radius with better precision has some intrinsic limitations, unless the
helium abundance can be fixed independently. We also note that the gaps
between the islands of Figure 5, like the one for the 1D histograms, are
caused by the finite resolution of the grid. The gaps in this figure also
reflect that if the grid is regularly spaced in $X_{0}$ and $Z$ it is not in
$Z/X$ since the gaps are not regularly spaced along that direction.

One other result obtained by SEEK is worth examining in more detail. For
Procyon A, if the subset of observables is used to do the fit, the values of
$M$ and $R$ are less precise, which is in accordance with expectation, but
more accurate than the case where all observables are used. This could reveal
some inconsistency in the observables as well as some of the limitations of
the SEEK procedure. Indeed forcing SEEK to look for a model of Procyon A with
the right astrometry, $V=0.363$ and $\pi=134.07$ mas, reduces the accuracy of
the results, compared with using only $T_{\rm eff}$, [Fe/H] and $\Delta\nu$.
This discrepancy could come from inhomogeneity between the tool used in the
spectroscopic study giving the observables and in our models. The metallicity
and temperature taken as input for Procyon A are from 3D modeling (Allende
Prieto et al., 2002), while we used the conversion of the VandenBerg & Clem
(2003) tables using 1D models to get our value of $M_{V}$. Using older
atmospheric parameters, or a f ully consistent conversion, could perhaps
improve the fit.

We also remark that the behavior of our results for the mass of $\alpha$ Cen B
presents similarity to those of Procyon A. The results are slightly more
accurate if $V$, $\pi$ and $\delta\nu$ are dropped from the fit. It seems that
this reveals some of the limitation of our grid when extremely precise
measurements are available for a star. The small box defined within $\sim
1-\sigma$ of all $\alpha$ Cen B observables only includes a few models. This
number is too small and can only produce a weak probabilistic assessment over
the value of the mass. Computing more models to cover with more precision the
parameters space of $\alpha$ Cen B would most likely resolve the small
deviation over the value of the mass. This underscores the fact that modeling
dedicated to individual stars becomes useful once classical observables of a
star are very precise.

In any case, Table Automatic Determination of Stellar Parameters via
Asteroseismology of Stochastically Oscillating Stars: Comparison with Direct
Measurements shows that SEEK is certainly capable of pinning down the radius
of solar-like oscillators with a minimum effort and that it is also able to
fix the mass of these stars with an accuracy not attainable when only non
asteroseismic inputs are available. This improvement in accuracy is especially
potent in case the distance to the star is not known.

In Table 3, all primary stellar parameters $\boldsymbol{p}$ of the modeled
stars are shown as well as some selected secondary values. We address here the
question of the age of stars fixed by SEEK, and especially the age of the Sun,
since it is the only star where an independent determination of the age is
available. A fairly good accuracy, 3.96$\pm$0.41 Gyr (13%), compared to the
meteoritic value of 4.57$\pm$0.02 Gyr (Wasserburg, in Bahcall & Pinsonneault,
1995), is reached in the case where $\Delta\nu$ and $\delta\nu$ are known. A
look at the histogram in the top panel of Figure 6 confirms visually that the
solution is well constrained. This is quite promising in the case where the
small separation is known. It seems that we can expect a precision of 5 to 20%
for the age when $\Delta\nu$, $T_{\rm eff}$, [Fe/H], and especially
$\delta\nu$ are known with reasonable precision.

If only the three observables in the subset are used, the result varies from
star to star. For the Sun, the lower panel of Figure 6 shows for the age two
distinct and very probable solutions. This specific example shows how useful
the histograms are since the output of Table 3 does not reveal the presence of
two solutions. One thing not shown here, but that we were able to see from the
computation of the probability distribution $\mathcal{G}(\tau,\alpha)$, is
that the two islands of solution for the age are at two different value of
$\alpha$, the young solution corresponding to the standard value of
$\alpha=1.8$ and the older one to $\alpha=2.8$. As seen in the lower panel of
Figure 6 higher values of $\alpha$ are acceptable if only the subset of
observables is used.

Figure 7 illustrates another type of effect on the age when the subset of
observables is used. The precision on the age of $\beta$ Hyi is degraded by
20%, but the overall answer, or accuracy, is the same. As mentioned previously
in the description of Figure 4, 70 Oph A shows a third type of scenario where
not much can be concluded on the age of the star as the posterior is showing
solutions at almost any age. Since the relation between the age and the
observables is highly non-linear, it is very difficult to estimate beforehand
what will be the precision obtained on the age depending on the type of star
and/or on the precision of the observables, especially when $\delta\nu$ is not
known.

We also note that any age determination based on stellar modeling is
susceptible to changes in basic aspects of the modeling. In the solar case
helioseismic fits to low-degree modes result in a solar age very close to the
meteoritic value, for models including diffusion and settling of helium and
heavy elements, and when the ‘old’ solar composition is used, whereas use of
the Asplund et al. (2009) composition induces a significant shift (see
Christensen-Dalsgaard, 2009, for a review). Also, the age determination based
on fits to the Kepler data for HAT-P-7, which has a convective core, was
understandably rather sensitive to the extent to which convective-core
overshoot was included (Christensen-Dalsgaard et al., 2010). Such potential
systematic effects must clearly be taken into account in the interpretation of
results of fits such as those carried out with SEEK. Of course the long-term
goal is to reduce these effects through an improved understanding of stellar
structure and evolution based on more detailed asteroseismic investigations.

##  6 CONCLUSIONS

We have shown the details involved in the SEEK procedure and we have tested
the validity of the approach with all independent measurement known to us.
This is the first step before Kepler observations of solar-like stars, now
becoming available in large quantity (Gilliland et al., 2010), are analyzed.
We hope that SEEK will be able to process these observation with ease and lead
to the publication of a catalog that includes a homogeneous sample of stars
revealing their radius, mass, age, and other stellar parameters.

We expect that the results of SEEK can be a useful starting point in more
detailed asteroseismic analyses in the, likely frequent, cases where extensive
sets of oscillation frequencies are available. Such detailed investigations
can further refine the basic stellar parameters, including the age which, as
shown above, is sometimes not significantly constrained by SEEK. Also,
detailed analyses of the frequencies are likely to uncover evidence for the
need for improvements in stellar modeling.

The SEEK model grid extends to the base of the red-giant branch and hence SEEK
can be used to determine properties of stars on both the main sequence and the
subgiant branch. In fact, the results obtained for the subgiants $\eta$ Boo
and $\beta$ Hyi indicate the potential of SEEK for the investigation of
subgiants, common amongst the Kepler asteroseismic targets. On the other hand
we acknowledge the substantially different diagnostic potential between
centrally hydrogen burning stars and subgiants. In particular, $\delta\nu$
provides a direct measure of stellar age in the former case, while the
diagnostic potential of $\delta\nu$ for subgiants is more subtle. In the
latter case, detailed analysis of individual frequencies of mixed modes may
provide much more stringent constraints (e.g., Metcalfe et al., 2010). These
issues of optimizing the asteroseismic diagnostics certainly require further
investigations.

In parallel, further tests and development of SEEK are required. A first step
will be to test the level of systematic errors that are introduced by the
neglect of relevant physical effects in the stellar modeling. An obvious
example is diffusion and settling which have been demonstrated to have a
substantial effect on solar modeling, as tested with helioseismology. Also,
effects of convective core overshoot could influence the results for stars
slightly more massive than the Sun. New grids of models including such effects
can then be computed for inclusion in SEEK.

A related issue, with less obvious solutions, concerns the systematic effects
of the frequency errors introduced by the failure to model properly the
outermost layers of the star, which are known to dominate the difference
between observed and modeled frequencies in the solar case (e.g., Christensen-
Dalsgaard et al., 1996) and which also have potentially significant effects on
the large and small separations. It is important at least to obtain an
estimate of the extent to which these effects may influence the results
obtained with SEEK. One may hope that improvements in the modeling of the
relevant effects, combined with detailed analyses of Kepler data for a broad
range of stars, will eventually allow us to reduce or eliminate such potential
systematic errors.

POQ would like to thank H. Bruntt, J. Rowe, G. Doğan, T. Campante and J.
Dupuis for their interest in this work. This project was supported by the
European Helio- and Asteroseismology Network (HELAS), a major international
collaboration funded by the European Commission’s Sixth Framework Programme,
by the Danish Natural Science Research Council and by the Natural Sciences and
Engineering Research Council of Canada through a Canadian Space Agency
Visiting Fellow grant.

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Figure 1:  Slice of the SEEK grid projected in the $\log g-\log T_{\rm eff}$
diagram for $Z=0.015$ $X_{0}=0.72$ and $\alpha=1.8$. Three isochrones are also
drawn. The computation was stopped after the star had reached 15 billion years
or an arbitrary point on the RGB branch. The low-mass models are separated by
steps of 0.02 $M_{\rm\odot}$, the more massive ones by 0.10 $M_{\rm\odot}$.
Figure 2:  Slice of the solution valley obtained by SEEK for $\alpha$ Cen A at
$Z=0.025$, $X_{0}=0.70$, and $\alpha=1.8$. Note the logarithmic color scale
showing the value of $\chi^{2}_{r}$ as a function of $M$ and $\tau_{\rm N}$
(see Eq. 3). The solution is centered on the constant radius line of
$R=1.25\,R_{\rm\odot}$, shown as a solid red line. The solid black lines are
also radius isocurves. Models older than 13 Gyr, on the upper right region of
the figure, and the models on the ZAMS, with $\tau_{\rm N}=0$, were excluded
from the fit and given values of $\log\chi^{2}_{r}=6.0$. The slight deviation
in the constant-radius isocurves with $R\geq 1.5\,R_{\rm\odot}$ and $\tau_{\rm
N}\gtrsim 0.7$ marks the position of the hook in the evolution tracks.  Figure
3:  Description of the $\delta\nu$ computation. The functions
$\psi_{0},\psi_{2}$ corresponding to the computed frequencies for $l=0,2$,
shown in blue and red respectively in panel (a), are convolved with the
distribution $T$ in panel (b) to create the distribution $D_{0},2$ in panel
(c). The resulting cross-correlation function ($D_{0}\ast D_{2}$) fixing the
$\delta\nu$ at its maximum is shown in panel (d). The extra bump of the cross-
correlation is caused by the extra $l=2$ mode at $\nu\sim 760$ $\mu$Hz and has
no influence on the computed separation.  Figure 4: SEEK’s posterior for 70
Oph A. All the observables of Table Automatic Determination of Stellar
Parameters via Asteroseismology of Stochastically Oscillating Stars:
Comparison with Direct Measurements are used in the fit. The probability
distribution $\mathcal{G}(p)$ for $R$, $M$, $\tau$ and $T_{\rm eff}$,
(normalized to a maximum of 1, black histogram) is plotted along with the
probability integral (solid red line). The delimitation of the $1\,\sigma$
probability of the distribution, determined by the interval
$[s^{(1)},s^{(2)}]$ (cf. Eq. 25), is plotted between the blue dots.  Figure 5:
Correlation between $Z/X$ and $R$ for $\beta$ Hyi. The two thick solid curves
through peaks in the correlation function $\mathcal{G}(R,Z/X)$ follow the
constant metallicity ridges for $Z=0.025$ and 0.015. Along these curves the
hydrogen mass ratio varies from the upper left peak to the lower right one as
$X_{0}=0.68$, 0.70, 0.72, and 0.74. A clear general trend is that $R$
increases with $Z$ and that it also increases with increasing $X_{0}$. The
solid horizontal line is the $\beta$ Hyi metallicity ratio of $Z/X=0.0258\pm
0.003$ (da Silva et al., 2006), with its error bars as dashed lines, while the
solid and dashed vertical lines show the direct interferometric measurement of
$R=1.814\pm 0.107$. Note that the merging of the constant metallicity ridges
at lower values is an artifact of the drawing procedures.

$\begin{array}[]{c}\includegraphics[angle={0},width=341.43306pt]{./fig6a.eps}\\\
\\\ \\\ \includegraphics[angle={0},width=341.43306pt]{./fig6b.eps}\end{array}$

Figure 6: SEEK’s posterior for the Sun. The top 9 panels represent the
probability distribution (normalized to 1) as well as the probability integral
for the model using all the observables of Table Automatic Determination of
Stellar Parameters via Asteroseismology of Stochastically Oscillating Stars:
Comparison with Direct Measurements. The lower panels are similar but use a
subset of observables ($\Delta\nu$, $T_{\rm eff}$, and [Fe/H]).

$\begin{array}[]{c}\includegraphics[angle={0},width=341.43306pt]{./fig7a.eps}\\\
\\\ \\\ \includegraphics[angle={0},width=341.43306pt]{./fig7b.eps}\end{array}$

Figure 7: Same as Figure 6 for $\beta$ Hyi.  Table 1: Observables

star |  |  $\Delta\nu$ $[\mu\text{Hz}]$ |  $\delta\nu$ $[\mu\text{Hz}]$ |  $\nu_{\rm max}$ $[\mu\text{Hz}]$ |  $T_{\rm eff}$ [K] | [Fe/H] |  $\log g$ [cgs] | $V$ |  $\pi$ [mas] | $E_{B-V}$ | Reference
---|---|---|---|---|---|---|---|---|---|---|---
$\beta$ Hyi |  | $57.2\pm 0.5$ | $5.3\pm 0.5$ | $1000$ | $5964\pm 100$ | $-0.03\pm 0.05$ | $-$ | $2.8\pm 0.01$ | $134.07\pm 0.11$ | $0.01$ | 1 ; a
$\tau$ Cet |  | $169.6\pm 0.5$ | $12.7\pm 1.2$ | $4500$ | $5264\pm 100$ | $-0.5\pm 0.03$ | $4.36\pm 0.15$ | $3.39\pm 0.01$ | $-$ | $0$ | 2 ; b
Procyon A |  | $55.5\pm 0.5$ | $-$ | $1000$ | $6514\pm 27$ | $-0.05\pm 0.05$ | $-$ | $0.363\pm 0.003$ | $285.93\pm 0.88$ | $0$ | 3 ; c
$\eta$ Boo |  | $39.9\pm 0.9$ | $4.0\pm 1.0$ | $750$ | $6007\pm 255$ | $0.09\pm 0.01$ | $-$ | $2.68\pm 0.03$ | $88.17\pm 0.75$ | $0$ | 4 ; d
$\alpha$ Cen A |  | $105.5\pm 0.5$ | $5.6\pm 0.7$ | $2410$ | $5847\pm 27$ | $0.24\pm 0.03$ | $4.34\pm 0.12$ | $-0.01\pm 0.01$ | $747.1\pm 1.2$ | $0$ | 5 ; e
$\alpha$ Cen B |  | $161.5\pm 0.5$ | $10.1\pm 0.6$ | $4100$ | $5316\pm 28$ | $0.25\pm 0.04$ | $4.44\pm 0.15$ | $1.35\pm 0.01$ | $747.1\pm 1.2$ | $0$ | 6 ; e
70 Oph A |  | $161.7\pm 0.8$ | $-$ | $4500$ | $5300\pm 50$ | $0.04\pm 0.05$ | $-$ | $4.19\pm 0.014$ | $194.2\pm 1.2$ | $0$ | 7 ; f
Sun |  | $134.8\pm 0.5$ | $9.1\pm 0.2$ | $3034$ | $5778\pm 20$ | $0\pm 0.01$ | $4.44\pm 0.01$ | $-$ | $-$ | $-$ | 8 ; g

References. — Asteroseismology (1) Bedding et al. 2007; (2) Teixeira et al.
2009; (3) Eggenberger et al. 2005; (4)Carrier et al. 2005; (5)Bouchy & Carrier
2002; (6) Kjeldsen et al. 2005; (7) Carrier & Eggenberger 2006; (8) Thiery et
al. 2000

References. — Other observables (a) da Silva et al. 2006; (b) Soubiran et al.
1998; (c) Allende Prieto et al. 2002; (d) Morossi et al. 2002; (e) Porto de
Mello et al. 2008; (f) Eggenberger et al. 2008 ; (g) Grevesse & Sauval 1998

Table 2: Comparison With Direct Measurements

|  |  $R$ $[R_{\rm\odot}]$ |  |  $M$ $[M_{\rm\odot}]$ |
---|---|---|---|---|---
|  | SEEK |  |  |  | SEEK |  |  |
Star |  | Selected | All | Direct/Interferometry | Reference |  | Selected | All | Direct/Kepler’s Law | Reference |
$\beta$ Hyi |  | $1.92\pm 0.05$ | $1.87\pm 0.01$ | $1.814\pm 0.017$ | 1 |  | $1.27\pm 0.11$ | $1.16\pm 0.02$ |  |  |
$\tau$ Cet |  | $0.79\pm 0.01$ | $0.81\pm 0.01$ | $0.790\pm 0.005$ | 2 |  | $0.77\pm 0.03$ | $0.83\pm 0.01$ |  |  |
Procyon A |  | $2.04\pm 0.05$ | $2.12\pm 0.02$ | $2.048\pm 0.025$ | 3 |  | $1.45\pm 0.10$ | $1.59\pm 0.05$ | $1.497\pm 0.037$ | 4 |
$\eta$ Boo |  | $2.65\pm 0.08$ | $2.69\pm 0.06$ | $2.672\pm 0.028$ | 5 |  | $1.62\pm 0.13$ | $1.71\pm 0.08$ |  |  |
$\alpha$ Cen A |  | $1.23\pm 0.04$ | $1.25\pm 0.01$ | $1.224\pm 0.003$ | 6 |  | $1.09\pm 0.09$ | $1.11\pm 0.01$ | $1.105\pm 0.007$ | 6 |
$\alpha$ Cen B |  | $0.87\pm 0.01$ | $0.87\pm 0.01$ | $0.863\pm 0.005$ | 6 |  | $0.92\pm 0.04$ | $0.89\pm 0.01$ | $0.935\pm 0.006$ | 6 |
70 Oph A |  | $0.86\pm 0.02$ | $0.86\pm 0.01$ |  |  |  | $0.89\pm 0.06$ | $0.89\pm 0.05$ | $0.890\pm 0.020$ | 7 |
Sun |  | $1.02\pm 0.03$ | $1.01\pm 0.01$ | $1$ |  |  | $1.03\pm 0.08$ | $1.01\pm 0.02$ | $1$ |  |

References. — (1)North et al. 2007; (2)di Folco et al. 2007; (3)Kervella et
al. 2004; (4)Girard et al. 2000 ;(5)van Belle et al. 2007; (6)Miglio &
Montalbán 2005; (7)Eggenberger et al. 2008;

Table 3: SEEK Output  |  |  | Stellar Parameters
---|---|---|---
|  | Primary |  | Secondary |
Star | Observables |  |  $M$ $[M_{\rm\odot}]$ |  $\tau$ [Gyr] | $Z$ | $X_{0}$ | $\alpha$ |  |  $R$ $[R_{\rm\odot}]$ | $\log(L/L_{\rm\odot})$ |  $\log g$ [cgs] |  $T_{\rm eff}$ [K] | $Y$
$\beta$ Hyi | All |  | $1.16\pm 0.02$ | $5.41\pm 0.89$ | $0.018\pm 0.001$ | $0.708\pm 0.022$ | $1.74\pm 0.60$ |  | $1.87\pm 0.01$ | $0.58\pm 0.02$ | $3.964\pm 0.004$ | $5845\pm 38$ | $0.274\pm 0.023$
| Selected |  | $1.27\pm 0.11$ | $3.94\pm 1.45$ | $0.019\pm 0.002$ | $0.710\pm 0.022$ | $1.69\pm 0.62$ |  | $1.92\pm 0.05$ | $0.62\pm 0.05$ | $3.97\pm 0.02$ | $5959\pm 111$ | $0.271\pm 0.024$
$\tau$ Cet | All |  | $0.83\pm 0.01$ | $0.75\pm 0.03$ | $0.007\pm 0.001$ | $0.730\pm 0.003$ | $1.30\pm 0.17$ |  | $0.81\pm 0.01$ | $-0.32\pm 0.02$ | $4.547\pm 0.006$ | $5450\pm 34$ | $0.263\pm 0.004$
| Selected |  | $0.77\pm 0.03$ | $6.31\pm 2.18$ | $0.007\pm 0.001$ | $0.729\pm 0.004$ | $1.30\pm 0.17$ |  | $0.79\pm 0.01$ | $-0.36\pm 0.03$ | $4.53\pm 0.01$ | $5286\pm 71$ | $0.264\pm 0.005$
Procyon A | All |  | $1.59\pm 0.05$ | $1.83\pm 0.24$ | $0.021\pm 0.003$ | $0.731\pm 0.012$ | $2.42\pm 0.46$ |  | $2.12\pm 0.02$ | $0.88\pm 0.02$ | $3.991\pm 0.010$ | $6548\pm 37$ | $0.248\pm 0.015$
| Selected |  | $1.45\pm 0.10$ | $2.25\pm 0.64$ | $0.018\pm 0.002$ | $0.710\pm 0.023$ | $1.93\pm 0.62$ |  | $2.04\pm 0.05$ | $0.82\pm 0.02$ | $3.98\pm 0.02$ | $6513\pm 64$ | $0.272\pm 0.025$
$\eta$ Boo | All |  | $1.71\pm 0.08$ | $1.90\pm 0.38$ | $0.025\pm 0.001$ | $0.729\pm 0.013$ | $1.32\pm 0.60$ |  | $2.69\pm 0.06$ | $0.97\pm 0.02$ | $3.81\pm 0.02$ | $6166\pm 80$ | $0.246\pm 0.014$
| Selected |  | $1.62\pm 0.13$ | $2.24\pm 0.62$ | $0.025\pm 0.001$ | $0.728\pm 0.013$ | $1.33\pm 0.64$ |  | $2.65\pm 0.08$ | $0.88\pm 0.08$ | $3.80\pm 0.02$ | $5974\pm 280$ | $0.247\pm 0.014$
$\alpha$ Cen A | All |  | $1.11\pm 0.01$ | $4.95\pm 0.04$ | $0.030\pm 0.001$ | $0.680\pm 0.003$ | $1.80\pm 0.17$ |  | $1.25\pm 0.01$ | $0.18\pm 0.02$ | $4.299\pm 0.001$ | $5850\pm 34$ | $0.290\pm 0.004$
| Selected |  | $1.09\pm 0.09$ | $8.04\pm 4.05$ | $0.030\pm 0.001$ | $0.698\pm 0.020$ | $2.25\pm 0.59$ |  | $1.23\pm 0.04$ | $0.20\pm 0.03$ | $4.30\pm 0.01$ | $5850\pm 37$ | $0.272\pm 0.021$
$\alpha$ Cen B | All |  | $0.89\pm 0.01$ | $6.95\pm 0.03$ | $0.030\pm 0.001$ | $0.680\pm 0.003$ | $1.80\pm 0.17$ |  | $0.87\pm 0.01$ | $-0.32\pm 0.02$ | $4.521\pm 0.001$ | $5250\pm 34$ | $0.290\pm 0.004$
| Selected |  | $0.92\pm 0.04$ | $6.99\pm 4.45$ | $0.029\pm 0.001$ | $0.709\pm 0.030$ | $2.23\pm 0.52$ |  | $0.87\pm 0.01$ | $-0.26\pm 0.03$ | $4.53\pm 0.01$ | $5309\pm 66$ | $0.262\pm 0.020$
70 Oph A | All |  | $0.89\pm 0.05$ | $8.65\pm 3.70$ | $0.023\pm 0.003$ | $0.716\pm 0.018$ | $2.06\pm 0.34$ |  | $0.86\pm 0.01$ | $-0.27\pm 0.02$ | $4.525\pm 0.006$ | $5311\pm 65$ | $0.261\pm 0.021$
| Selected |  | $0.89\pm 0.06$ | $7.54\pm 4.54$ | $0.022\pm 0.003$ | $0.710\pm 0.026$ | $1.99\pm 0.33$ |  | $0.86\pm 0.02$ | $-0.28\pm 0.03$ | $4.52\pm 0.01$ | $5304\pm 71$ | $0.268\pm 0.029$
Sun | All |  | $1.01\pm 0.02$ | $3.96\pm 0.41$ | $0.020\pm 0.001$ | $0.707\pm 0.011$ | $1.80\pm 0.17$ |  | $1.01\pm 0.01$ | $0.02\pm 0.02$ | $4.432\pm 0.005$ | $5753\pm 36$ | $0.273\pm 0.012$
| Selected |  | $1.03\pm 0.08$ | $6.80\pm 5.43$ | $0.020\pm 0.001$ | $0.729\pm 0.013$ | $2.18\pm 0.53$ |  | $1.02\pm 0.03$ | $0.03\pm 0.03$ | $4.44\pm 0.01$ | $5757\pm 39$ | $0.251\pm 0.014$