The Astrophysical Journal, 749:109 (13pp), 2012 April 20 doi:10.1088/0004-637X/749/2/109C© 2012. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

TOWARD A NEW KIND OF ASTEROSEISMIC GRID FITTING

M. Gruberbauer1, D. B. Guenther1, and T. Kallinger2,31 Institute for Computational Astrophysics, Department of Astronomy and Physics, Saint Mary’s University, B3H 3C3 Halifax, Canada

2 Instituut voor Sterrenkunde, K.U. Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium3 Institute for Astronomy, University of Vienna, Turkenschanzstrasse 17, 1180 Vienna, Austria

Received 2011 November 16; accepted 2012 February 10; published 2012 March 28

ABSTRACT

Recent developments in instrumentation (e.g., in particular the Kepler and CoRoT satellites) provide a newopportunity to improve the models of stellar pulsations. Surface layers, rotation, and magnetic fields imprint erraticfrequency shifts, trends, and other non-random behavior in the frequency spectra. As our observational uncertaintiesbecome smaller, these are increasingly important and difficult to deal with using standard fitting techniques. Toimprove the models, new ways to compare their predictions with observations need to be conceived. In this paper, wepresent a completely probabilistic (Bayesian) approach to asteroseismic model fitting. It allows for varying degreesof prior mode identification, corrections for the discrete nature of the grid, and most importantly implements atreatment of systematic errors, such as the “surface effects.” It removes the need to apply semi-empirical correctionsto the observations prior to fitting them to the models and results in a consistent set of probabilities with which themodel physics can be probed and compared. As an example, we show a detailed asteroseismic analysis of the Sun.We find a most probable solar age, including a 35± 5 million year pre-main-sequence phase, of 4.591 billion years,and initial element mass fractions of X0 = 0.72, Y0 = 0.264, Z0 = 0.016, consistent with recent asteroseismic andnon-asteroseismic studies.

Key words: methods: statistical – stars: oscillations – Sun: helioseismology

Online-only material: color figure

1. INTRODUCTION

The success of recent space missions CoRoT and Kepler,designed for the discovery of exoplanets and the analysis ofstellar pulsation, have produced a large number of high-qualitylight curves (Chaplin et al. 2010). With these data sets, obtainedover long time bases of several months, we are able to detectvariability with semi-amplitudes down to a few parts per million.These observations have now firmly established the existenceof solar-type pulsation in a large number of solar-like and redgiant stars. Moreover, observations of an unprecedented numberof δ Scuti stars and other types of pulsators have also revealedrich mode spectra.

These data are now causing a paradigm shift for manytopics in stellar astrophysics. In particular, the determination offundamental stellar parameters, and any inferences regarding thephysics of stellar interiors, have for a long time been restrictedto testing theoretical models using classic observables such asphotometric indices or spectroscopic data. Even though thesemethods have become more advanced, for instance by applyingcomplex Bayesian methods to determine stellar ages (Pont& Eyer 2004; Jørgensen & Lindegren 2005) and to evaluatecompeting models (Takeda et al. 2007; Bazot et al. 2008), thevalue of additional information provided by pulsation modesis tremendous, as they directly probe the whole star. Already,the asteroseismic community is successful in extracting generalcharacteristics of the mode spectra for many different types ofstars (e.g., Mathur et al. 2010; Kallinger et al. 2010c) and also indevising promising tools for a comparative interpretation of theobservations (e.g., Bedding & Kjeldsen 2010). Average modeparameters, such as the large and small frequency separations,and the frequency of maximum power, have been shownto successfully constrain stellar parameters although certaincorrelations remain as a source for uncertainty (see, e.g.,Kallinger et al. 2010b; Huber et al. 2011; Gai et al. 2011). These

have been incorporated into the current advanced probabilisticpipelines to investigate stellar model grids (Quirion et al. 2010)and already been applied to recent observations (Metcalfe et al.2010). The next step to improving our knowledge about stellarinteriors is to analyze individual pulsation modes in an equallyrigorous way to see where our models agree or disagree.

In the past, χ2-minimization techniques (Guenther & Brown2004), or equivalent Bayesian analyses (e.g., Kallinger et al.2010a), have been introduced to find the pulsation model thatmost closely reproduces the observed frequencies within alarge and dense grid of models. The Bayesian analysis, in thiscontext, only provides an additional framework for constrainingsolutions to models that match our prior knowledge about thestars’ fundamental parameters. Due to the rich informationprovided by the pulsation frequencies, these approaches shouldbe successful in many cases, which is why they are beingapplied also to the most recent Kepler data sets. For instance,Metcalfe et al. (2010) test various approaches from differentmodelers with different methods that actually use the individualfrequencies. However, there are currently (at least) three majorproblems when applying these techniques.

Stellar rotation, at all but the slowest rotation speeds, hasbeen shown to produce rotational splittings which are incom-patible with the traditional linear approximations. It even per-turbs the values of the axisymmetric (m = 0) frequencies (e.g.,see Deupree & Beslin 2010 and references therein). In order tocorrectly take this into account, the rotation speed as a func-tion of stellar depth needs to be known, and extensive compu-tations would be necessary to do these effects justice. Giventhe large variety of possible rotation profile characteristics, thiswould greatly expand the dimensionality and size of the pul-sation model grid. This implies currently insurmountable com-putational expenses for the types and sizes of grids that arenecessary for a comprehensive asteroseismic analysis of manystars.

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The Astrophysical Journal, 749:109 (13pp), 2012 April 20 Gruberbauer, Guenther, & Kallinger

0 20 40 60 80 100frequency mod 137.24 μHz

1000

2000

3000

4000

5000

freq

uenc

y [μ

Hz]

l=0l=1l=2l=3

Figure 1. Echelle diagram of solar p modes taken from Broomhall et al. (2009)(filled circles) and an appropriate solar model constructed using YREC. Thehigher-order model frequencies are increasingly deviating from the observationsdue to deficiencies in modeling the upper stellar layers. The systematic errorsof the models are much bigger than the random observational uncertainties.

(A color version of this figure is available in the online journal.)

For stars with a convective envelope, model frequencies athigh radial orders differ from observations due to problemsin modeling the outer layers (see Figure 1). These so-calledsurface effects can be compensated by looking at ratios offrequency differences (Roxburgh 2005) or by “correcting” theobserved frequencies through calibration of the surface effectsseen for the Sun as proposed by Kjeldsen et al. (2008). It islikely that the surface correction as calibrated for the Sun is notuniversally applicable, and evidence for this has been mounting(e.g., Bedding et al. 2010). Moreover, neglecting (or correctingfor) the surface effects in the observed frequencies is onlyreasonable when studying properties of the star for which theouter layers are unimportant. However, if we want the theoreticalmodels to more closely reflect reality, we need to include moreand better physics to bring the computed frequencies closer tothe (uncorrected) observed ones.

Furthermore, the fact that static asteroseismic grids canonly have a finite resolution in parameter space is often ne-glected. If the error bars of the observed frequencies are smallcompared to the differences between calculated frequenciesin adjacent grid points, the likelihood of having a model inthe grid that corresponds to the best model one’s code coulddeliver decreases rapidly. The problem of finding the “true”model and the actual uncertainties with respect to the grid be-comes apparent. Even grids with adaptive resolution have thesame problem in principle, as the decision for further refiningthe resolution of a particular region in parameter space mustalways depend on a number of discrete grid points. This prob-lem is much more severe if our aim is to calculate probabili-ties (or some summary statistics) to compare different modelgrids.

In this paper, we present a new approach to asteroseismicmodel grid fitting. Our goal is to find a new way of puttingour model physics to the test that can handle all of theaforementioned difficulties. Even restricted to models that areunable to produce all the details of the observations, we wantto know which models are most “correct” (i.e., consistent withappropriate fundamental parameters and physics), and how wellthe solution is constrained. We show how to quantitatively assessour model grids as a function of the observational uncertainties,the uncertainties of the calculated frequencies, and our generalprior knowledge about the star and possible shortcomings of ourmodels.

2. BAYESIAN TREATMENT OF SYSTEMATIC ERRORS

2.1. Basics of Bayesian Inference

Bayes’ theorem, applied to the problem of inference, statesthat the probability of a particular hypothesis after obtaining newdata (i.e., the posterior) is proportional to the probability of thehypothesis prior to obtaining the new data (i.e., the prior) timesthe likelihood of obtaining the new data, under the assumptionthat the hypothesis is true (i.e., the likelihood function). Thisapproach to inference is derived from the product and sumrules of probability theory that have shown to be necessaryand sufficient for consistent, quantitative logical reasoning4 (seeJaynes & Bretthorst 2003).

In this paper, we stay as close as possible to the generalnotation used in Jaynes & Bretthorst (2003) or Gregory (2005).We start with Bayes’ theorem applied to the problem ofcomparing observations with the predictions of a model M.If the predictions of a model M are governed by a set of nparameters θ = {θ1, . . . , θn}, and we define the observationsto be represented by the symbol D (for data), it is commonlyformulated by expressing the posterior probability

P (θ |M,D, I ) = P (θ |M, I )P (D|θ ,M, I )

P (D|M, I ). (1)

The symbol I is equivalent to the prior information about theproblem that is investigated. The first term in the numera-tor of Equation (1) is the prior probability of a particular setof parameter values θ , given the model M and our prior in-formation I about the problem. It is independent of any newdata which are supposed to be analyzed. The second termin the numerator is called the likelihood. It gives the like-lihood of obtaining the observed data under the assumptionthat the predictions of model M are correct, given the par-ticular choice of its parameter values θ . The denominator inEquation (1) is called the global likelihood, or evidence, andis the sum (or integral) of the numerator over the whole pa-rameter space of model M. It therefore acts as a normalizationconstant. Most importantly, if the prior probabilities are ade-quately normalized, it also represents the likelihood of obtainingthe data given the whole model M, independent of the particularchoice of θ . Thus, it can be used as a likelihood for comparisonsamong different alternative models.

More details on the application of Bayes’ theorem, in partic-ular with respect to data analysis in astronomy, can be found inGregory (2005).

4 Strictly speaking, Bayes’ theorem is only one result that derives from theserules. Consistent use of Bayes’ theorem, in particular the assignment of thevarious terms in Equation (1), also requires knowledge of its origin andconsistent application of the product and sum rule. However, for the sake ofbrevity we will simply call our approach in this manuscript “Bayesian” ratherthan “based on probability theory as extended logic.”

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The Astrophysical Journal, 749:109 (13pp), 2012 April 20 Gruberbauer, Guenther, & Kallinger

2.2. Systematic Errors in the Bayesian Framework

One of the strengths of the Bayesian framework is that aparameter θn, known to be necessary to describe a model M,can be marginalized by applying the sum rule. In the case ofcontinuous parameters, the sum turns into an integral, and byintegrating the full posterior over the parameter range of θn, oneobtains the marginal posterior

P (θ1, ..., θn−1|M,D, I ) =∫

P (θ1, ..., θn−1, θn|M,D, I ) dθn.

(2)The marginal posterior retains the overall effects of includingparameter θn in the model, but is independent of any particularchoice of its value. In other words, θn is “removed” from thedetailed analysis. This is similar to what is done for calculatingthe evidence in the denominator in Equation (1). The onlydifference is that the evidence is the marginal likelihood overall parameters of the model, weighted by the prior.

The reason this is useful is that if the data and the modelare known to show systematic differences, like shifts or trends,such “systematic errors” can simply be encoded introducingadditional parameters to the model M, so that M is able tomodel these effects as well. By subsequently marginalizing overthese “fudge parameters,” one is then able to perform a standardBayesian analysis without any need for knowing the exact valueof the systematic error(s). However, even though the exactvalue is unknown, the presence of the error is being consideredin the evaluation of the posterior probabilities. Furthermore,an increasing number of “fudge parameters” comes at a costbecause it potentially decreases the evidence for the model dueto the increase in prior volume. As mentioned in Section 2.1, theevidence is used as the value for the likelihood of obtainingthe data in Bayesian model comparison. It is therefore possibleto compare models with and without “fudge parameters.”Improved models that do not need them, but are able to explainthe observations just as well, will be favored.

3. TOWARD A BAYESIAN SOLUTION TOASTEROSEISMIC MODEL FITTING

3.1. Review and Problems of the Standard Approach

The general problem of asteroseismic model fitting is to matchobserved frequencies fi,o to those calculated from models fi,m.If the nobs observed frequencies have individual uncertaintiesσi,o, and the model frequencies have random uncertainties σi,mthen a χ2-statistic can be calculated according to

χ2 = 1

nobs

nobs∑i=1

(fi,o − fi,m)2

σ 2i,o + σ 2

i,m

. (3)

Searching a large grid of N stellar models Mj with fundamentalparameters close to those estimated for the observed star willproduce a minimum in χ2 (=best-fit model). In addition,uncertainties can be estimated from the change in χ2 as thedistance in parameter space to the best fit increases. Calculatedwith adequate stellar evolution and pulsation codes, it should bepossible to infer details about the stellar interior and to obtainprecise fundamental parameters.

In order to consistently encode prior information about thefundamental parameters and other model properties, and tomake use of all the additional advantages that come with theBayesian approach (all of which will become clear in the next

section), it is much easier to perform the model fitting usingprobabilities. Assuming that the random uncertainties σi,o andσi,m are compatible with Normal distributions, one can define

σ 2i = σ 2

i,o + σ 2i,m. (4)

This leads to the likelihood for observing the data (= the specificvalues of fi,o), given a single observed and calculated frequency

P (fi,o|fi,o�→i,m,Mj , I ) = 1√2πσi

exp

[−

(fi,o − fi,m

)2

2σ 2i

].

(5)Here, fi,o�→i,m stands for the proposition “The observed modefi,o corresponds to the calculated mode fi,m.”5 Naturally, wewant our models Mj to reproduce all observed frequencies. As-suming that each observed frequency is a statistically indepen-dent data point, this leads to a product for the likelihood ofobtaining all observed frequency values given that the model iscorrect

P (D|Mj, I ) =nobs∏i=1

P (fi,o|fi,o�→i,m,Mj , I ). (6)

Here, D stands for the complete set of observed frequencies andtheir uncertainties. This can then be incorporated in the usualframework for Bayesian inference.

Alas, both of the mentioned, straightforward approachesabove suffer from the following problems.

1. The most appropriate model is not necessarily the one thatminimizes Equation (3) or maximizes Equation (6). Thereare many possible scenarios where this would be the case(e.g., due to surface effects, stellar activity, magnetic fieldeffects, rotational effects). Straightforward application ofthe formalism above will then lead to wrong or nonsensicalresults in both best fit and derived uncertainties. Even worse,this would propagate into our assessment of the modelphysics that were used to produce the models.

2. In case of such systematic differences, we need to take intoaccount that for each observed pulsation frequency, multi-ple model frequencies are possible candidates (not neces-sarily only the closest one). This is particularly problematicin cases were no prior mode identification is available.

3. As the observational uncertainties decrease, the contrast inχ2 (and even more so the contrast in probabilities) betweendifferent models increases. If the model that minimizes/maximizes Equation (3)/Equation (6) is not the correctmodel due to missing physics, this increase in fittingcontrast is misleading and unwarranted.

4. In static grids the finite grid resolution increases the riskof missing the most adequate model that the code couldproduce. If there are systematic differences between eventhe most adequate mode and the observations, the “contrastenhancement effect” will be magnified. For the samereason, adaptive grids run into the same problem and willmiss the correct parameter space region to finer resolve inthe first place.

5 Although the explicit notation seems clumsy at first glance, it is actuallyone of the major assets of the Bayesian approach. It visualizes exactly whichpropositions we are evaluating, and under which conditions the probabilitiesare calculated. Slightly different propositions or conditions can yield vastlydifferent results. If the notation is explicit, there are no hidden variables orassumptions.

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As a consequence of all these shortcomings, it is clear thata method is needed that considers the possibility of systematicdifferences. It is also mandatory to consider the finite resolutionof our model grids. Solutions to these problems are presentedin the following sections.

3.2. The Argument for Probabilities

There are obvious benefits to quantifying the best fit andthe uncertainties in terms of probabilities. With probabilitiesfor each specific model, we automatically obtain probabilitydistributions for each parameter of the model itself. We canfurthermore consistently compare different grids and see whichset of input physics is more probable, given all our currentinformation and the data.

However, there are much stronger arguments for a probabilis-tic approach. Marginalization allows us to consistently treat nui-sance parameters, while the sum and product rules allow us toclearly formulate the question we are asking. This question is“Given the observed frequencies, our knowledge about the starand model physics, which model(s) best represent the star interms of its fundamental parameters and general physical prop-erties as probed by the pulsation modes?” In reality, this generalquestion has to be further refined as we encounter more compli-cated situations such as: “We have model frequencies that couldpotentially show negative or positive systematic offsets, or nosuch offset at all, when compared to our observations. Theycould be influenced by rotation or actually be rotationally splitfrequencies themselves. They could be bumped l = 1 modes orl = 0 modes. Given all of these possibilities, which model is themost adequate one, and how well is the solution constrained?”From the viewpoint of probability theory, the only way to treatsuch a set of possibilities and get meaningful answers is to usethe sum rule and product rule, as we will show in the nextsection.

3.3. Ambiguous Mode Identification

As a first improvement to the general approach of asteroseis-mic model fitting, we can involve the sum rule to consistentlyconsider uncertainties (or even ignorance) in our mode identi-fication. In essence, if there is no unique proposition fi,o�→i,m,Equation (6) changes to

P (D|Mj, I ) =nobs∏i=1

{nmatch∑k=1

P (fi,o, fi,o�→k,m|Mj, I )

}(7)

with

P (fi,o, fi,o�→k,m|Mj, I )

= P (fi,o�→k,m|Mj, I )P (fi,o|fi,o�→k,m,Mj , I ). (8)

Here the sum over the index k means that all possible and mu-tually exclusive assignments nmatch of one observed mode toa number of calculated frequencies fk,m have to be taken intoaccount as an “or” proposition.6 Note that due to the prod-uct rule of probability, the terms in each sum now include theconditional prior probabilities P (fi,o�→k,m|Mj, I ). These haveto be normalized so that

∑nmatchk=1 P (fi,o�→k,m|Mj, I ) = 1. The

most conservative assignment is to assign equal probabilitiesP (fi,o�→k,m|Mj, I ) = 1/nmatch to each possible scenario. How-ever, if more information is available (e.g., a mode could be

6 Hereafter, a possibly ambiguous frequency assignment will always bedenoted as fi,o�→k,m.

identified to be either l = 0 or l = 2 with specific probabilitiesfor both cases as found by some peak-bagging program), thiscan easily be encoded at this stage.

The end result is a product of weighted sums of probabilities,where the weights are given by the respective prior probabil-ities.7 This product is the correctly normalized likelihood forobtaining the data, given the proposition that any one of the pro-posed scenarios is correct. Note that if there is an unambiguousassignment fi,o�→i,m for every observed frequency, each priorprobability P (fi,o�→i,m|Mj, I ) = 1 and Equation (7) simplifiesto Equation (6). Now that we have included our uncertaintiesconcerning the assignment of model frequencies and observedfrequencies, we will deal with uncertainties in the validity of themodel frequencies themselves in the next section.

3.4. Treatment of Systematic Errors

As a next step, we now show how to treat the problem of im-perfect models. As mentioned before, applying standard tech-niques that rely on minimizing the quadratic differences betweenthe observations and the models will give incorrect results ifsystematic differences exist. The alternative of correcting forsuch imperfections prior to modeling is also undesirable if thecorrection is not known to be universally applicable.

To treat any systematic deviation from the model frequenciesdue to unmodeled physical effects, we simply expand the modelsMj by considering an additional systematic error parameter foreach tested frequency. The aim is to construct new values fi,Δto compare with the observations according to

fi,Δ = fi,m + γ Δi . (9)

Here, Δi is the absolute value of the systematic error. γ = 1or γ = −1 and determines whether the model frequencyis expected to be systematically higher or lower than theobserved frequency. To keep our notation from occupying toomuch space, we will implicitly assume the value of γ to beconstant throughout the following derivations and attribute thisto our prior information I. Δi is an unknown parameter butas long as its lower and upper boundaries can be roughlyestimated, it can be treated fully consistently in the probabilisticframework.

In the following, we will again work out an example ofonly one observed and calculated frequency. Therefore, forthe derivation the assignment fi,o�→i,m is unique. We will thenprovide the extension to multiple frequencies and ambiguousmode identifications.

Using the new parameters, the equivalent to Equation (5) is

P(fi,o, Δi |fi,o�→i,m,MΔ

j , I)

= P(Δi |fi,o�→i,m,MΔ

j , I)P

(fi,o|Δi , fi,o�→i,m,MΔ

j , I)

= P(Δi |fi,o�→i,m,MΔ

j , I)

× 1√2πσi

exp

[− (fi,o − fi,m − γ Δi)2

2σ 2i

]. (10)

Here the symbol MΔj simply denotes the model Mj aug-

mented by the new parameter Δi . Self-evidently, the prod-uct rule again requires that we introduce a prior probability

7 A common misconception is that these “priors” are only there to allow us toincorporate prior information. In reality, they are formally required by theproduct rule and ensure that the result of Equation (7) is always properlynormalized.

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The Astrophysical Journal, 749:109 (13pp), 2012 April 20 Gruberbauer, Guenther, & Kallinger

P (Δi |fi,o�→i,m,MΔj , I ). This can either encode prior information

about the expected behavior of the error or be simply assignedby considerations of symmetry. Again it is required that theintegral over the prior

∫P (Δi |fi,o�→i,m,MΔ

j , I ) dΔi = 1.It would now be possible to try to find the Δi that maximizes

P (fi,o, Δi |fi,o�→i,m,MΔj , I ) in Equation (10). However, this is

completely irrelevant for our needs. In case of multiple observedfrequencies it would also quickly lead to a highly dimensionalparameter space that we are not interested in navigating. Instead,we are interested in finding the probabilities of the models MΔ

j .To do this it is necessary to integrate out Δi , which we have justintroduced. We obtain the marginal likelihood

P(fi,o|fi,o�→i,m,MΔ

j , I)

=∫ Δi,max

Δi,min

P(fi,o, Δi |fi,o�→ i,m,MΔ

j , I)dΔi . (11)

This integral naturally depends on the shape of the prior prob-ability distribution for Δi and can easily be evaluated nu-merically.8 It represents the likelihood of obtaining the valueof the observed frequency fi,o given that Mj predicts a fre-quency fi,m but that there is a possibility of a systematic dif-ference Δi , between Δi,min and Δi,max. Furthermore, it is fun-damentally constrained and properly weighted by the priorwe assigned. This result is now easily extended to multi-ple modes and ambiguous mode identifications. Equation (7)becomes

P(D|MΔ

j , I) =

nobs∏i=1

{nmatch∑k=1

P(fi,o, fi,o�→k,m|MΔ

j , I)}

(12)

and

P(fi,o, fi,o�→k,m|MΔ

j , I)

= P(fi,o�→k,m|MΔ

j , I)P

(fi,o|fi,o�→k,m,MΔ

j , I). (13)

In summary, we have to calculate a product of weighted sumsof integrals in the form of Equation (11), where the summationis performed over every possible assignment fi,o�→k,m.

Note that even when we choose to consider systematicdeviations, we usually do not expect them to be significant for allfrequencies. For good models some frequencies should alreadymatch well “right out of the box.” In particular, this is true for allfrequencies in the idealized case where we have (finally) founda way to correctly model all the effects that previously causedsystematic deviations.

One might think that this is taken care of by setting Δi,min = 0.However, unless the prior P (Δi |fi,o�→k,m,MΔ

j , I ) is a δ functionat Δi = 0, it is much more likely that Δi > 0. This meansthat a priori a model will be preferred which shows at leasta small deviation from the observations, depending on theobservational uncertainties and the steepness of the prior.The limiting case, however, the δ function, corresponds to awhole different model which is simply the standard modelwithout systematic deviations, Mj. Thanks to the sum rule,there is an elegant solution for taking this alternative intoaccount.

8 For several simple shapes, such as the beta prior introduced in the nextsection, analytical solutions also exist.

For the mutually exclusive logical propositions9 MΔj and Mj

we can calculate

P(fi,o, fi,o�→k,m|MΔ

j + Mj, I)

= P(MΔ

j , fi,o, fi,o�→k,m|I)+ P (Mj, fi,o, fi,o�→k,m|I )

P(MΔ

j |I)+ P (Mj |I )

= P(MΔ

j |I)P

(MΔ

j |I)+ P

(Mj |I

)P(fi,o, fi,o�→k,m|MΔ

j , I)

+P (Mj |I )

P (MΔj |I ) + P (Mj |I )

P (fi,o, fi,o�→k,m|Mj, I ). (14)

Note that here MΔj + Mj means “MΔ

j or Mj is true.” This is thelikelihood of observing the frequency value fi,o, given that asystematic deviation either does or does not exist. The principleof indifference as the most conservative approach for the priorprobabilities obviously demands P (MΔ

j |I ) = P (Mj |I ) = 0.5,but if more information is available, it can be encoded here.This result is also easily generalized to the case of multiplefrequencies and ambiguous mode identification.

3.5. The Choice of the Prior for Δi

A very important detail to consider when extending the mod-els with systematic error parameters is their prior probabilitiesP (Δi |fi,o�→k,m,MΔ

j , I ). There is a basic choice between two pos-sibilities. The first is to use uninformative (or ignorance) priors,or alternatively, maximum entropy priors. Uninformative pri-ors can be derived from arguments of invariance to specifictransformations, while maximum entropy priors should satisfythe maximum entropy criterion for a given set of constraints.The other possibility is to use priors derived from heuristic orphysical arguments.

The specific form of the prior probabilities of Δi are part ofthe model that is evaluated, as indicated by the notation.10 Theyare not necessarily “prior” as in a sense of “before obtainingobservations,” but conditional probabilities required for thecorrect normalization, as demanded by the product rule ofprobabilities. They encode specific ways in which we expectΔi to behave, given our grid of frequencies and our information(which of course can be influenced by previous observations).For instance, if we expect our best model to minimize thesystematic deviations, the prior should assign larger probabilitydensities to smaller Δi , so that models with smaller deviationswill be more probable. On the other hand, if we expect our bestmodel to show more erratic deviations, a flat uninformative prioris a better choice. After a complete evaluation of the probabilitiesand likelihoods, the Bayesian evidence will indicate whether thestate of information encoded by the priors is supported by thedata or not.

As a first important example of an uninformative prior,consider a uniform prior

P(Δi |fi,o�→k,m,MΔ

j , I) = 1

Δi,max − Δi,min= constant. (15)

This means that all values of Δi are equally likely. With such aprior, every model that predicts frequencies at any value between

9 Note that from a logical standpoint even if Δ = 0, MΔj is still a different

model than Mj because the prior is not a δ function. Therefore, they are alwaysmutually exclusive.10 The prior is described by P (Δi |fi,o�→k,m,MΔ

j , I ) rather than, e.g., P (Δi |I ).

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The Astrophysical Journal, 749:109 (13pp), 2012 April 20 Gruberbauer, Guenther, & Kallinger

fi,o + Δi,min and fi,o + Δi,max has the same maximum likelihood(i.e., the same maximum value for Equation (10)).

On the other hand, a Jeffreys prior,

P(Δi |fi,o�→k,m,MΔ

j , I) = 1

Δi ln(Δi,max/Δi,min), (16)

assigns equal probability per decade and, in terms of theprobability density, favors smaller values of Δi . This prior isobviously not defined for Δi = 0, i.e., it requires Δi,min > 0.This is problematic for, e.g., surface effects that approach zeroat low orders. However, when Δi,min = 0, one can use a modifiedversion of this prior given by

P(Δi |fi,o�→k,m,MΔ

j , I) = 1

(Δi + c) ln[(Δi,max + c)/c], (17)

where c is a small constant. For values smaller than c, this prioracts more or less like a uniform prior, while for higher values itbehaves like the usual Jeffreys prior. This prior is nowadaysoften used in “peak-bagging” algorithms (e.g., Gruberbaueret al. 2009; Benomar et al. 2009; Handberg & Campante 2011).However, there is no objective criterion for how to set c, andvarious tests we conducted with our grid fitting code have shownthat the choice of c can have a large impact on the evidencevalues.

Consequently, we argue that any priors used for a systematicerror parameter Δi = [0, Δi,max] should be functions that areclearly defined by the parameter limits, without additionalparameters that have large effects on the evidence. The uniformprior11 is such a prior, as are priors derived from the betadistribution (given in units of our problem)

P(Δi |fi,o�→k,m,MΔ

j , I) ∝

(Δi

Δi,max

)α−1 (1 − Δi

Δi,max

)β−1

.

(18)With α = 1 and β = 2 this simply leads to a linearly

decreasing probability density

P(Δi |fi,o�→k,m,MΔ

j , I) =

( √2

Δi,max

)2

(Δi,max − Δi). (19)

It is the only prior that allows for a linearly decreasingprobability density, is properly normalized, and reaches zeroat Δi = Δi,max. It also leads to an analytical solution for theintegral in Equation (11). Thus, it satisfies all our requirementsfor a prior with which to minimize systematic errors.

We have compared the results obtained from Equation (19)(hereafter the beta prior) with several other plausible choices,such as an exponential distribution with expectation valueΔi,max/2 and a modified Jeffreys prior with c = σi,m, and wefind them to yield comparable results and evidence values. Dueto the clarity of its definition and lack of additional parameters,we therefore argue that the beta prior is an appropriate choicefor a non-flat prior. We will show how to use it in Section 4 in aworked example.

Last, priors based on heuristic or physical arguments ob-viously vary strongly with the specific problem to which thefitting method is applied. As an example, when modeling sur-face effects on p-mode frequencies, the prior could be Gaus-sian, following the heuristic frequency correction proposed by

11 In fact, the uniform prior is consistent with a beta distribution withα = β = 1.

Kjeldsen et al. (2008). It would be a function of frequency,expecting greater deviations toward higher-order modes. Thewidth of the Gaussian, however, would be again a rather arbi-trary choice, leading to potentially different evidence values.Such priors clearly need to have a strong basis either in theoryor prior observations.

3.6. Bumped Modes and Finite Grid Resolution

Equation (12) represents the final likelihood for obtainingthe observed frequencies given our (extended) model MΔ

j . Thismodel still represents only a single point in a discrete grid.12

However, the probability is small that a single model in the gridcorresponds to the “true best model” our code can produce. Theproblem becomes worse as the grid resolution is lowered, or asmode frequencies are changing quickly or unpredictably fromone model to the next (e.g., avoided crossings, magnetic shifts).The probabilities (or χ2-values) we obtain will not be a fairassessment of the model physics, even at higher grid resolutions.Even worse, the overall evidence for the grid will be finely tunedto the positions of all models in the grid. This makes it difficultto compare different grids with different physics. We will nowshow how to improve on this.

In a sequence of models along a single evolutionary track,except for the first and last models, each model Mj has twoneighboring models Mj−1 and Mj+1. In most cases theseadjacent models will contain the same modes, and their changingvalues can be traced from Mj−1 to Mj and Mj+1. Now we declarethe difference between observed and calculated frequency as anew free parameter:

δfi = fi,o − fi,m. (20)

This value is fixed if only a single grid point is considered.However, we can split the evolutionary tracks into segments inbetween grid points, and define

δfi,j− = fi,o − fi,Mj−1 + fi,Mj

2(21)

and equivalently

δfi,j+ = fi,o − fi,Mj+ fi,Mj+1

2. (22)

Adding δfi as a new parameter to the equations derived in theearlier sections, we change our focus to evaluate probabilitiesof model track segments T Δ

j centered around the models MΔj .

To do this, we again use marginalization to integrate out bothΔi and δfi to obtain the marginal likelihood. We obtain

P(fi,o|fi,o�→i,m, T Δ

j , I)

=∫ Δi,max

Δi,min

∫ δfi,j+

δfi,j−P

(fi,o, Δi , δfi |fi,o�→i,m, T Δ

j , I)dΔi dδfi .

(23)

If the priors for Δi do not vary greatly from one model to thenext, Δi and δfi can be considered to be independent parameters.It is therefore possible to use the product rule to separate the

12 Note that this is also the case for approaches using an adaptive grid, sinceeach iteration of an adaptive scheme is based on a discrete grid.

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The Astrophysical Journal, 749:109 (13pp), 2012 April 20 Gruberbauer, Guenther, & Kallinger

98 100 102 104frequency mod 137.10 μHz

1800

1900

2000

2100

2200

2300

freq

uenc

y [μ

Hz]

δi, j+

δi, j-

40 60 80 100 120

1000

2000

3000

4000

Figure 2. Example for the definition of δi,j− and δi,j+ (see the text). Fourradial orders of l = 1 modes from three adjacent models in a high-resolutiongrid of solar models are shown. The triangles represent the central model. Thefrequencies for the adjacent models along the evolutionary track sequence aredepicted as squares and white circles. Black circles (and error bars) indicateobserved frequencies published in Broomhall et al. (2009). δi,j− and δi,j+ fora single mode are represented using arrows. The insert shows an unzoomedversion of the l = 1 and l = 3 ridge.

conditional probabilities

P(fi,o, Δi , δfi |fi,o�→i,m, T Δ

j , I)

= P(Δi |fi,o�→i,m, T Δ

j , I)

× P(δfi |fi,o�→i,m, T Δ

j , I)

× P(fi,o|Δi , δfi, fi,o�→i,m, T Δ

j , I). (24)

Furthermore, since we evaluate the complete evolutionarytrack segment we can assume a uniform prior probabilityP (δfi |fi,o�→i,m, T Δ

j , I ) = 1/(δfi,j+ − δfi,j−). With these defini-tions, the integral over δfi can easily be calculated analytically.The equivalent to Equation (10) becomes

P(fi,o, Δi |fi,o�→i,m, T Δ

j , I)

= P(Δi |fi,o�→i,m, T Δ

j , I)

× 1

2(δfi,j+ − δfi,j−)

×[

erf

(δfi,j+ − γ Δi√

2σi

)− erf

(δfi,j− − γ Δi√

2σi

)], (25)

where erf is the error function. The remaining integral overΔi again has to be carried out numerically. Figure 2 shows anexample for the definitions introduced above, given three modelsin a solar evolutionary track.

We have now used the free parameter δfi to “trace” each modethrough segments of the evolutionary track, and compare it tothe observed frequencies, retaining the possibility of systematicdifferences. Note that our only assumption here is that the modefrequencies change smoothly between the frequencies givenby the constraining models. In principle, this approach can becarried out in multiple dimensions (e.g., not only along theevolutionary track in stellar age but also between tracks in mass).As before, an extension to multiple frequencies and ambiguousmode identifications is straightforward.

We stress that this approach only locates the region ofhighest probability given the current grid, and given unspecifiedbehavior of frequencies in between grid points. It is thus bestused for frequencies whose behavior is difficult to capture, e.g.,due to mode bumping or for a first general assessment of a verycoarse grid. Given a dense enough grid, regular frequenciesthat are expected to change approximately linearly from onegrid point to the next need to be treated using interpolation,since the integration over the model gaps for individual modes,independently of all other modes, would allow for highlyunphysical models.

Therefore, in order to obtain a final best model and uncer-tainties for the model parameters, the regions of substantialprobability should be further refined after the track probabilitieshave been calculated. Eventually, the grid is resolved enough sothat well-defined distributions arise. In dense enough grids, thiscan easily be accomplished by interpolation of the frequenciesin between grid points without violating the condition of hydro-static equilibrium. This can also be done during run-time witharbitrary precision using probabilities, by interpolating betweengrid points and using the sum rule to calculate a probabilityrepresentative of the original grid resolution using the interpo-lated models. Naturally, modes that change erratically, shouldbe excluded from such an interpolation routine and treated asshown above instead.

3.7. Model Probabilities

So far we have only shown how to calculate the likelihoodfor standard pulsation models, models that contain systematicdifferences, and also evolutionary track segments. In orderto obtain the probabilities for individual models (or tracksegments), we want to use Bayes’ theorem, assign model priors,and calculate the total evidence for each model grid. Thesimplest method to assign model priors in the absence of anyother prior information is to use the principle of equipartitionand assign a uniform prior

P (Mj |I ) = 1.0/NM, (26)

where NM is the number of models (or, equivalently, NT wouldbe the number of track segments) that are analyzed.

Although each model or track only predicts a number offrequencies, it implicitly represents values or ranges for fun-damental parameters like Teff or L, which can be compared to(and constrained by) different and non-seismic observations.For instance, assuming our prior photometric and spectroscopicobservations of a pulsating star indicate Teff = Tspec ±σspec thenthe prior probability density for the model temperature is

P (Teff,j |I ) = k exp

[− (Tspec − Teff,j )2

2σ 2spec

]. (27)

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The Astrophysical Journal, 749:109 (13pp), 2012 April 20 Gruberbauer, Guenther, & Kallinger

This example assumes that the uncertainty in Teff follows aGaussian distribution. k is a normalization constant dependingon the absolute lower and upper plausible limits of Teff .

All the different implicit parameters for which prior observa-tions or other fundamental constraints are available, and henceprior knowledge exists, then can be used for prior probabili-ties which combine into an overall prior for model Mj. As anexample

P (Mj |I ) = P (Teff,j |I )P (Lj |I )P ([Fe/H]j |I ) . . . , (28)

if we assume separable priors for simplicity. If probabilities oftrack segments are calculated, such a prior could be approx-imated by a product of separable integrals, which are easilyevaluated analytically. Again, in order to obtain a proper priorand therefore proper values for the evidence, the integral of theprior probability over all possible models/tracks in a grid shouldbe 1.

By calculating, e.g.,

P(MΔ

j |D, I) = P

(MΔ

j |I)P

(D|MΔ

j , I)

∑NMk=1 P

(MΔ

k |I)P

(D|MΔ

k , I) (29)

or, e.g., in the case of rapidly changing modes with or withoutsystematic errors,

P(T Δ

j + Tj |D, I) = P

(T Δ

j + Tj |I)P

(D|T Δ

j + Tj , I)

∑NTk=1 P

(T Δ

k + Tj |I)P

(D|T Δ

k + Tj , I) ,

(30)

we obtain the probability of MΔj or, respectively, T Δ

j + Tj givenour prior knowledge (or lack thereof), our grid, and the setof observed frequencies. Note that the denominators of theseequations represent the evidence or likelihood for the grid as awhole. We can therefore use these as likelihoods when we wantto compare different grids with different input physics.

4. APPLICATION TO SURFACE EFFECTS

As mentioned in the introduction, shortcomings in modelingthe outer stellar layers produce systematic deviations in com-parison to the observations. These deviations seem to be suchthat model frequencies tend to be higher than the observed fre-quencies, and therefore γ = −1 (see Equation (9)). Kjeldsenet al. (2008) have proposed to calibrate a power-law descriptionof the deviations by measuring the surface effects in the Sunand then fitting this relation to frequencies of other stars. Theircorrection expressed in terms of our definitions has the generalform of

γ Δi ≈ a

(fi,m

fref

)b

, (31)

where fref is some reference frequency, typically the frequencyof maximum power νmax, and a and b are parameters to befitted. From their fits, Kjeldsen et al. determined b ≈ 4.90in the Sun, which has subsequently been used for other starsby a number of authors. A comprehensive implementation ofthis formalism into a χ2-fitting algorithm was presented ina recent study by Brandao et al. (2011). However, even inthis more advanced approach, there is still a choice of a andb required. Moreover, complications for modes of differentspherical degree and also bumped modes arise because theydo not necessarily conform to this relation. The authors propose

to alleviate these problems by introducing additional model-dependent parameters that approximately correct for some ofthese deviations. While this approach is a great improvementover applying a fixed surface-effect correction (or no correctionat all), our approach is much more powerful. It allows for amuch greater flexibility and leads to clearly defined probabilisticresults.

4.1. Priors for Surface Effects

As we want to treat systematic errors of more or lessunknown magnitude, the most general approach is to usethe flat uniform prior (Equation (15)). Imposing only minoradditional constraints, as we argued in Section 3.5, the betaprior (Equation (19)) can also be used to give more weightto models which minimize these unknown errors. We can useboth priors and compare the Bayesian evidence to tell us whichinterpretation of the surface effect is better supported by the data,given our model and everything we know. Moreover, irrespectiveof which prior is chosen, we also always allow for the possibilityof no surface effects at all, as discussed in Section 3.4.

This now gives us enough flexibility to consider a possiblyfrequency-dependent behavior of the surface effects. However,instead of “predicting” the behavior of Δi as is done by modelingthe surface deviations through a power law, we will prescribethe behavior of its upper limit Δi,max. In contrast, the lowerlimit should always remain 0, since our ultimate goal is to findmodels that correctly describe the surface layers (and thereforeapproach Δi → 0).

The choice of the largest allowed Δmax is not unique, but itshould be sensible and used consistently throughout the analysis.A reasonable strategy is to use sup(Δmax) = Δν, the largefrequency separation of each specific model, as a sensible upperlimit. If the systematic differences between observations andmodels are larger than the average distance between modes ofadjacent radial order, we no longer recognize this as a validfrequency assignment.13 With this upper limit defined, we nowwant to model different types of surface effects. If we have nopreference for any frequency-dependent trend (i.e., all we knowis that observed frequencies are lower than model frequencies)we require that all frequencies have equal Δmax.

On the other hand we can also use a more specific model,such as Equation (31), but retain the same flexibility. The surfaceeffect as shown in Equation (31) depends on two parameters.The power-law exponent b determines how quickly the surfaceeffect increases as we move to higher frequencies, whereas ais simply a scaling factor. We are not interested in the scalingparameter, since the scaling (i.e., the magnitude of the offset) isgoverned by our condition that for each model sup(Δmax) = Δν.It is taken care of by the fact that we are marginalizing overΔi anyway. Since the largest surface effects are expected atthe highest frequency fmax,m in the model, it follows that for aspecific b

Δmax,i = Δν

(fi,m

fmax,m

)b

. (32)

Figure 3 shows how these definitions affect the prior probabilitydensity P (Δi |fi,o�→i,m,MΔ

j , I ) as we increase the value of Δi

for both the constant and the power-law approach. With allthe Δmax,i set, we can then use the priors as discussed abovefor all our calculations. Note that we can also easily evaluatenew composite propositions at this stage and compute the

13 This condition may be relaxed at the highest radial orders.

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The Astrophysical Journal, 749:109 (13pp), 2012 April 20 Gruberbauer, Guenther, & Kallinger

0 20 40 60 80 100 120

frequency mod 137.18 μHz

1000

2000

3000

4000

5000fr

eque

ncy

[μH

z]

0.001

0.010

0.100

1.000

prob

abili

ty d

ensi

ty

0 20 40 60 80 100 120

frequency mod 137.18 μHz

1000

2000

3000

4000

5000

freq

uenc

y [μ

Hz]

0.001

0.010

0.100

1.000

prob

abili

ty d

ensi

ty

Figure 3. Behavior of the beta prior for systematic offsets in an echelle diagram. The squares represent model frequencies, while the shaded “trails” indicate the priorprobability density for varying Δi . The left panel uses equal Δmax, whereas the right panel uses a power-law Δmax with exponent b = 4.9 (see Section 4.1). Note thatthe uniform prior is not shown, since it simply assigns a constant probability density.

Table 1Parameter Ranges for the Solar Grid

Parameter Range Step Size

Mass 0.95–1.05 0.005X0 0.68–0.74 0.01Z0 0.016–0.022 0.001αml 1.8–2.4 0.1

Notes. Masses are given in units of solar masses; αml is the mixing-lengthparameter.

probability for a hypothesis that allows for, e.g., a rangeb = {4.4, 4.65, 4.9, . . .}. This is done in the same way as wasexplained earlier (see Equation (14)).

4.2. Detailed Analysis of the Sun

As an example for how to implement the surface-effecttreatment, we will consider the solar l = 0, 1, 2, and 3p-modes obtained by using BiSON data (Broomhall et al. 2009).For our models, we used a large and dense solar grid obtainedwith YREC (Demarque et al. 2008). The model grid spansmasses from 0.95 M� to 1.05 M� in steps of 0.005 M�, initialhydrogen mass fractions from 0.68 to 0.74 in steps of 0.01, initialmetal mass fractions from 0.016 to 0.022 in steps of 0.001, andmixing-length parameters from 1.8 to 2.4 in steps of 0.1. Theseparameters are also summarized in Table 1.

Our model tracks begin as completely convectiveLane–Emden spheres (Lane 1869; Chandrasekhar 1957) andare evolved from the Hayashi track (Hayashi 1961) through thezero-age main sequence (ZAMS) to 6 Gyr with each track con-sisting of approximately 2500 models. Only models between 4.0and 6.0 Gyr are included in the model grid. Constitutive physicsinclude the OPAL98 (Iglesias & Rogers 1996) and Alexander& Ferguson (1994) opacity tables using the GS98 mixture(Grevesse & Sauval 1998), and the Lawrence Livermore 2005equation of state tables (Rogers 1986; Rogers et al. 1996). Con-vective energy transport was modeled using the Bohm-Vitense

mixing-length theory (Bohm-Vitense 1958). The atmospheremodel follows the (T–τ ) relation by Krishna Swamy (1966).Nuclear reaction cross-sections are from (Bahcall et al. 2001).The effects of helium and heavy element diffusion (Bahcallet al. 1995) were included. Note that our atmosphere mod-els and diffusion effects have been shown to require a largervalue of mixing-length parameter (αml ≈ 2.0–2.2) than standardEddington atmospheres (αml ≈ 1.7–1.8) (Guenther et al. 1993).

The pulsation spectra were computed using the stellar pul-sation code of Guenther (1994), which solves the linearized,non-radial, non-adiabatic pulsation equations using the Henyeyrelaxation method. The non-adiabatic solutions include radia-tive energy gains and losses but do not include the effects ofconvection. We estimate the random 1σ uncertainties of ourmodel frequencies to be of the order of 0.1 μHz.

We analyzed our grid using adiabatic and non-adiabatic fre-quencies, and employed three different surface-effect models:

M1: frequency-independent surface effects withΔi,max = Δν,

M2: frequency-dependent, “canonical” surface effects withΔi,max following Equation (32) with b = 4.90, and

M3: same as M2, but with b as a free parameter marginal-ized from b = 3.0 to b = 6.0.

For each frequency evaluated throughout our model grid, ir-respective of the surface-effect model, we also considered thepossibility of no surface effect, i.e., we consistently calculatedP (MΔ

j + Mj |D, I ). To take into account the discrete nature ofthe grid, we interpolated along the evolutionary tracks duringrun-time by a factor of 20, thereby increasing the effective “fre-quency resolution” of the grid to below the random uncertaintiesof the model frequencies. All models were evaluated with

(a) a uniform prior for all track segments,

(b) a prior using normal distributions for the constraintsM = 1.0000 ± 0.0002 M�, log Teff = 3.7617 ± 0.01,and log (L/L�) = 0.00 ± 0.01, where YREC uses the

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The Astrophysical Journal, 749:109 (13pp), 2012 April 20 Gruberbauer, Guenther, & Kallinger

Table 2Evidence for the Solar Grid Using the BiSON Data Set

Surface HRD log10(evidence) log10(evidence)Model Prior (adiabatic) (non-adiabatic)

M1 a −233.4 −229.9M2 a −189.8 −186.7M3 a −189.8 −187.2

M1 b −235.5 −231.6M2 b −190.8 −187.1M3 b −191.4 −187.9

M1 c −236.7 −235.1M2 c −193.5 −190.7M3 c −192.6 −189.4

Notes. See the text for the definition of models and prior a, b, and c. Results formodels M2a, M2b, and M2c, which are analyzed in more detail, are indicatedin bold face. Note that small numbers are expected.

following adopted values for M� = 1.9891±0.0004·1033g(Cohen & Taylor 1986) and L� = 3.8515 ± 0.009 ·1033erg s−1 (the average of the ERB-Nimbus and SMM/ARCRIM measurements; Hickey & Alton 1983), and

(c) same as (b) but with an additional Gaussian constrainton the age of 4.603 ± 0.0075 Gyr, derived from theestimated age of the solar system found by Bouvier &Wadhwa (2010) and an average pre-main-sequence phaseof our models of 35 ± 5 Myr.

For the Δi we consistently used beta priors, as discussed inthe previous section. Our calculations yield the most probablemodels and uncertainties for all these approaches, and they alsogive the Bayesian evidence for each approach. The results aresummarized in Table 2. We also computed the probabilitiesusing uniform priors, but found similar results with lowerevidence (several orders of magnitude) values than for thecorresponding beta prior analysis.

The non-adiabatic frequencies consistently produce largerevidence values than for the respective adiabatic case. This is nosurprise, as the non-adiabatic frequencies are in general betterat reproducing the higher frequencies. Overall, model M2ashows the largest evidence, followed by M2b and M3a. Notethat M1a, M1b, and M1c, which use frequency-independentpriors for the surface effects, and therefore are extremelyflexible, fail compared to the other models. Also, M3a andM3b cannot beat their M2 counterparts. These are examplesof how marginalization and the consistent normalization ofprobabilities work together to penalize more flexible modelsif they cannot generate considerably better results. Model M3chas a greater evidence than M2c, but the most probable stellarmodels are the same in both cases, suggesting that these modelsfit well, but do not necessarily adhere in detail to the standardsurface correction.

At first glance it might be unsettling that M2a has a slightlygreater evidence than M2b (and significantly greater evidencethan M2c). This indicates that there are models in our gridwhich reproduce the pulsation spectrum very well but do notmatch the solar fundamental parameters. A correctly calibratedgrid would produce higher evidences with a prior restrictedto the true solution. However, regardless of whether or notwe include the fundamental parameter constraints, we are stillfinding models that match the oscillations constraints reasonablywell. Furthermore, recall that the evidence is only the likelihood

Table 3Most Probable Models for the Complete BiSON Data Set Model Fitting

Model Mass Age X0 Z0 Zs αml Probability

M2a 1.015 4.885 ± 0.006 0.73 0.017 0.0153 2.2 0.541.005 4.713 ± 0.006 0.72 0.017 0.0153 2.2 0.21

M2b 1.000 5.017 ± 0.006 0.72 0.018 0.0161 2.1 0.681.000 4.983 ± 0.006 0.71 0.019 0.0160 2.2 0.17

M2c 1.000 4.591 ± 0.005 0.72 0.016 0.0144 2.2 0.951.000 4.562 ± 0.005 0.71 0.017 0.0153 2.3 0.05

Notes. Age is given in billion years and is computed from the pre-main-sequencebirth line. The age from the ZAMS is 35 ± 5 million years less. X0 and Z0 areinitial hydrogen and metal mass fractions, Zs is the metal mass fraction inthe envelope. Probabilities are given with respect to the specific surface-effectmodel and prior combination.

of obtaining the data, given that the approach is correct.14 Weknow that the a prior is misrepresenting our state of information.The solar prior approach b more correctly encodes what weknow about the Sun, and the age prior c puts even tighterconstraints on the pulsation models. Ignoring this information(using prior a and setting equal conditional probabilities) is aninteresting and necessary exercise to study the consistency ofthe results, and how the different models, approaches, and priorswork. For an actual detailed study of the solar model physics,however, it is not appropriate. We can nonetheless compare theresults, restricting ourselves to the non-adiabatic frequenciesand the on average best model for each prior, M2. The resultingparameters are displayed in Table 3.

The results obtained without using our prior knowledge ofthe Sun for model M2a are spread over several models inthe parameter space that can fit the observations quite well.However, for the most probable models, the mass and age areinconsistent with our prior knowledge. These models seem toproduce smaller surface effects and are therefore preferred. Formodel M2b the situation is similar. Although the mass is nowfixed to the true solar value, we do not obtain models that areconsistent with the solar age.

For M3c a single combination of physical parameters dom-inates the results and manages to fit well all the constraintswe impose (mass, luminosity, Teff , pulsation frequencies, andage). Loosening the conditions on Teff and the luminosity doesnot significantly change the result. We have also tested slightvariations of up to 20 million years in the age prior and donot find the result to be affected. In all cases, we recover atightly constrained most probable model with Z0 = 0.016 andZs = 0.0144, and an age of 4.591 ± 0.005 Gyr. We thereforefind a result similar to Houdek & Gough (2011). Given that ourmodels take 35 ± 5 Myr to reach the main sequence, our resultis also consistent with meteoritic age determinations of the so-lar system to within several million years (see, e.g., Bouvier &Wadhwa 2010). However, we also recover X0 = 0.72, whichleads to an initial helium mass fraction of Y0 = 0.264(1). Thisis different compared to the value of Y0 = 0.250(1) that wasfound by Houdek & Gough, but more consistent with Asplundet al. (2009).

14 In order to obtain correctly normalized probabilities for the differentapproaches themselves, we have to introduce conditional probabilities like,e.g., P (M2a|I ) or P (M2b|I ) and use Bayes’ theorem. Only comparing theevidence amounts to setting these conditional probabilities to be equal for alltested hypotheses (e.g., P (M2a|I ) = P (M2b|I )).

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The Astrophysical Journal, 749:109 (13pp), 2012 April 20 Gruberbauer, Guenther, & Kallinger

0 20 40 60 80 100(frequency + 5 μHz) mod 137.25 μHz

1000

1500

2000

2500

3000

3500

4000

freq

uenc

y [ μ

Hz]

l=1l=3l=0

l=2

Figure 4. Non-adiabatic (shaded symbols) and adiabatic (open symbols)frequencies of the most probable solar model from evaluating the BiSONfrequencies (black circles + error bars) using approach M2c. Note thatfrequencies have been shifted upward by 5 μHz before calculating the x-axisvalues in order to prevent the l = 2 modes from wrapping around.

Fitting the observations to the adiabatic frequencies, includ-ing the age prior, we also recover the exact same model. We alsotested how sensitive the grid is to the prior constraints in orderto estimate the actual impact of the pulsation frequencies on theprobabilities. If we only evaluate the combined priors, ignoringthe frequencies but including the prior on the age, we obtainX0 = 0.71 ± 0.01, Z0 = 0.019 ± 0.002, Zs = 0.017 ± 0.002,age = 4.603 ± 0.008 Gyr, and αml = 2.1 ± 0.2. This leadsus to conclude that the frequencies have a decisive impact andactually select the low-metallicity models, no matter whetheradiabatic or non-adiabatic model frequencies are used.

However, it has to be stressed again that the evidence drops byalmost two orders of magnitude when we introduce the age prior.This can be understood by the fact that the solution is so wellconstrained and at the edge of our current parameter space in Z0,and that many other models can also produce similar pulsationspectra. It could also suggest that we might not have covered thetrue best model parameters yet in our current grid. Therefore,our next goal will be to extend the grid to lower metallicities, andalso include different abundance mixtures, but this is beyond thescope of this paper.

Figure 4 compares the BiSON observations with our mostprobable model at the correct solar age. Even with non-adiabatic frequencies, significant surface effects can still befound. The measured surface effects themselves are shown inFigure 5, together with least-squares fits following the relationproposed by Kjeldsen et al. (2008). The magnitude of thesurface deviations depends on whether the non-adiabatic orthe adiabatic frequencies are used for the fit. Nonetheless,our method manages to identify the same exact model to bethe most probable, even using the same surface-effect model,thanks to the power of marginalization. However, the non-

-10

-5

0 l=0

-10

-5

0

freq

uenc

y di

ffere

nce

γΔi [μ

Hz] l=1

-10

-5

0 l=2

1000 1500 2000 2500 3000 3500 4000frequency [μHz]

-10

-5

0 l=3

Figure 5. Measured surface effects for non-adiabatic (filled circles) andadiabatic (open circles) frequencies of the most probable solar model fromevaluating the BiSON frequencies using approach M2c. The uncertainties ofthe differences are smaller than the symbols. Least-squares power-law fits (seeEquation (31)) to the surface effects for the adiabatic (solid line) and non-adiabatic (dashed line) frequencies are also shown.

adiabatic models are vastly preferred in terms of the Bayesianevidence. This is an example for how the presented approachcan be used to iterate toward improved stellar model physics,while still recovering meaningful stellar parameters from currentasteroseismic investigations.

We also determined surface-correction power-law exponentsfor every spherical degree via least-squares fits. For both the non-adiabatic and adiabatic frequencies the best-fitting exponents aremarkedly different from b = 4.9 which was both advocatedby Kjeldsen et al. (2008) and also used as the basis forour probabilistic surface model M2. This is also the reasonwhy the M3c models have a greater evidence than their M2ccounterparts. The fitted values range from b = 4.23 for non-adiabatic (l = 0) frequencies to b = 5.13 for adiabatic(l = 3) frequencies. Moreover, the power-law fits do notmatch the deviations very well at intermediate radial ordersnear 2400 μHz. From our point of view, fixing the exponent tob = 4.9 for a least-squares fit, as for instance done by Brandaoet al. (2011), is therefore a potential problem since it does noteven match the Sun very well, in particular when improved(e.g., non-adiabatic) physics are implemented. The probabilisticprocedure has no problem with these deviations, even though itformally assumes an exponent of b = 4.9, since the magnitudeof the surface effects is marginalized for every frequency.

4.3. Asteroseismic Analysis of a Sun-like Star

To investigate the applicability of our method to current aster-oseismic investigations, we also performed an “asteroseismic”analysis of a Sun-like star, simulated by artificially “degrading”

11

The Astrophysical Journal, 749:109 (13pp), 2012 April 20 Gruberbauer, Guenther, & Kallinger

the set of observed BiSON frequencies to a precision and ac-curacy expected from current space-based missions for averageSun-like solar-type oscillators. We first multiplied the uncertain-ties of the BiSON observations by a factor of 20 and then addedcorresponding random errors to the frequency values. Further-more, we did not assume to have detailed prior information onthe fundamental parameters. Instead, we fitted the “degraded”data set with a completely flat prior to the same grid as before,again using our surface effect model M2.

Although a different most probable model is identified,the overall results are comparable to our findings for M2a.They show a slightly larger spread of the model probabilitiesacross the grid. Summarizing the uncertainties for the mainparameters by calculating the first and second central momentsof the probability distribution in our grid we approximatelyobtain M = 1.015 ± 0.007 M�, age = 4.76 ± 0.10 Gyr,X0 = 0.72±0.01, Z0 = 0.017±0.001, Zs = 0.0148±0.0005,and αml = 2.3 ± 0.1.

However, these results become worse if we systemati-cally remove lower-order modes which are crucial to “an-chor” the surface-effect relation. To illustrate, we further de-graded our data set by only keeping 13 l = 0 modes from1950 to 3580 μHz, 12 l = 1 modes from 2020 to 3505 μHz, 10l = 2 modes between 2080and 3300 μHz, and 8 l = 3 modesfrom 2270 to 3220 μHz. Similar data sets from Kepler andCoRoT with comparable uncertainties and numbers of modeshave recently been analyzed in the literature. The results for themodel parameters become M = 1.046 ± 0.007 M�, age =4.80 ± 0.43 Gyr, X0 = 0.72 ± 0.01, Z0 = 0.021 ± 0.01,Zs = 0.019 ± 0.001, and αml = 2.3 ± 0.1. Although the valuesare still within ∼ 5% we are almost at the border of our param-eter space, and higher-mass models systematically outperformlower-mass models.

We know from investigating the BiSON data using our gridthat we require αml = 2.2 to fit all solar observables. Therefore,in an analysis of a Sun-like star, we can constrain the fit to allmodels with this value or use a prior based on the marginalposterior probability for αml as determined from the fit to theSun. In this case we obtain M = 1.04 ± 0.01 M�, age =4.41 ± 0.29 Gyr, X0 = 0.72 ± 0.01, Z0 = 0.020 ± 0.002,Zs = 0.018 ± 0.001. This is an improvement, but still notcomparable to the results obtained when using the full data set.

Thanks to the probabilistic method, however, we can alsoeasily add new observables as further constraints, such as thefrequency of maximum power, which can also be inferred froma power spectrum analysis and which approximately scales forSun-like stars as

νmax ≈ M/M�(Teff/Teff,�)3.5

L/L�νmax,�, (33)

with νmax,� = 3120 ± 5 μHz (Kallinger et al. 2010b).Assuming an observed value of νmax,obs and calculating νmax,modfor each model according to Equation (33) we can then multiplythe probability for each model with

P(νmax,obs|MΔ

j , I) = 1√

2πσν

exp

[−

(νmax,obs − νmax,mod

)2

2σ 2ν

],

(34)

where σν = √σ 2(νmax,obs) + σ 2(νmax,mod).

With νmax,obs = 3120 ± 20 μHz for our simulated Sun-like star, we then obtain M = 1.02 ± 0.01 M�, age =

4.39 ± 0.28 Gyr, X0 = 0.72 ± 0.01, Z0 = 0.019 ± 0.002,and Zs = 0.017 ± 0.002. Finally, if we were able to determineνmax,obs to about solar precision, the results would be M =1.008 ± 0.006 M�, age = 4.39 ± 0.30 Gyr, X0 = 0.71 ± 0.01,Z0 = 0.019 ± 0.002, and Zs = 0.017 ± 0.002. Therefore, ifour observations provide precise additional information suchas νmax, it can easily be implemented with our method. It thenseems possible to obtain reasonably accurate results for Sun-like stars, even in the absence of low-order modes and withouta fixed surface-effect correction.

5. CONCLUSIONS

In this paper, we have derived a new, completely probabilisticframework for asteroseismic grid fitting. We explicitly usedmarginalization and the formulation of combined propositionsto allow for the quantitative evaluation of the model grid physics.While computationally more intensive than the standard χ2

evaluation, this approach has several benefits in that it

1. allows for the treatment and analysis of systematic errorssuch as the surface effects, therefore removing the need toapply corrections prior to fitting,

2. easily implements uncertainties in the mode identification,3. takes into account the fact that grids are discrete representa-

tions of a continuous parameter space, which is especiallyimportant for rapidly varying bumped modes,

4. provides a consistent framework to use prior knowledgeabout stellar fundamental parameters or to evaluate addi-tional observables such as νmax, and

5. produces correctly normalized probabilities and likeli-hoods, respectively evidences, which can be used to assessthe model grid physics and the calibration of the grids.

While the above was explicitly derived using the example of astatic grid, the probabilistic approach would also be suited for anadaptive grid approach. The Bayesian evidence could be used asa formidable criterion to decide whether an adaptive grid needsto be further refined or not.

We also showed how to apply our method to study the Sun.The analysis based on our current grid and our prior informationmatches well the findings of Houdek & Gough (2011), and ingeneral fits the up-to-date picture of the Sun. The age of ourbest model (measured from the pre-main-sequence birth line) isconsistent with the meteoritic solar age. The solar model arriveson the ZAMS approximately 35 ± 5 Myr after appearing on thebirth line. We found the same best model whether non-adiabaticor adiabatic frequencies were used. This shows that our methodcan adequately deal with different shapes of surface effects,even when using the same (flexible) surface-effect model. Onerequirement, however, is that enough lower-order modes existto “anchor” the fit.

To our knowledge, this work is also the first completelygrid-based asteroseismic analysis of the Sun, using all theinformation provided by the frequencies and prior knowledgeabout the solar fundamental parameters, that results in the needfor initial hydrogen, helium, and metal mass fractions moreconsistent with Asplund et al. (2009) than the traditional higher-metallicity models. At least for our current grid, these values arerequired to produce a model that “looks” like the Sun, pulsateslike the Sun, and has the correct solar age. We stress that a formalχ2 fit to the Sun’s oscillation frequencies (Guenther & Brown2004) or even targeted nonlinear inversion of the oscillationfrequencies (Marchenkov et al. 2000) will not necessarily yieldthe same model as our approach. With χ2 fits it is difficult

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The Astrophysical Journal, 749:109 (13pp), 2012 April 20 Gruberbauer, Guenther, & Kallinger

to provide an unbiased correction for surface effects that atthe same time does not overly weight the deeper penetratingmodes. Some of the deeper penetrating modes are sensitive tothe base of the convection zone where the effects of convectiveovershoot and turbulence, introduced by rotation shears, arenot included in the standard models. Inversion methods, wherea standard base model is perturbed to fit the oscillations,are also distinct because even though the perturbed modelobtained from inversions reveal regions of the standard modelthat are inadequate, e.g., the base of the convection zone, theinversion model is not an actual standard model in the sense thatit is constrained and generated by the model physics.

We know that our best-fit model is inaccurate at the surfaceand we suspect that it is inaccurate at the base of the convectionzone (the latter suspicion based on the inadequate model physicsfor this region). Regardless, the model is probabilistically thebest model from the current model grid that matches all theknown constraints. We speculate that preferring fits that matchthe oscillation frequencies at the expense of the other physicalconstraints may be the reason that helioseismologists have beenunable to reconcile the observed solar p-mode frequencies withfrequencies derived from standard models based on the Asplundmixture and metal abundance (Serenelli et al. 2009; Guzik &Mussack 2010). We will pursue these matters in a future studywhere we include model grids based on the Asplund mixture.

While the purpose of our analysis of the Sun is to test thedetails of our model physics, our method can also be usedin general asteroseismic investigations. When applying ourtechnique to stars other than the Sun, e.g., recent asteroseismictargets from the Kepler mission, tight prior constraints as in thesolar case are generally not available. However, the probabilityformalism can simply assign uninformative (e.g., uniform)priors for the unknown parameters and still retain all theremaining benefits like treatment of missing mode identificationand of finite grid resolution.

For current asteroseismology, however, the most importantfeature is the flexible treatment of the surface effects thatdiffers from the usual approach of employing the empiricalcorrection by Kjeldsen et al. (2008) to the frequencies. Insteadof measuring the empirical correction for the Sun with the helpof a reference model, we use a flexible probabilistic modelthat allows us to measure surface effects in any star given ourcurrent asteroseismic grids. We do not rely on the validity ofthe solar surface-effect correction and can test new surface-effect models that deviate from the solar power-law approach.Correctly treating the impact of the surface effects on the modelprobabilities, this also yields correctly propagated uncertainties,and therefore a less biased (but model-dependent) assessmentof the stellar fundamental parameters.

The results presented in the previous section indicate thatthe accuracy of such current asteroseismic analyses is stillan open question and heavily dependent on the number ofunaffected, lower-order modes. If there are not enough lower-order modes the surface effect will lead to systematic errorsin the fundamental parameter determination. However, even insuch a case, by looking at how the evidence changes as betterphysics are included in the models, our method can be used toiterate toward improved models, hopefully solving the surface-effect problem eventually.

We are very grateful to Werner W. Weiss for his valuableinput and fruitful discussions. We also thank the referee for

improving the quality of the manuscript. M.G. and D.G. ac-knowledge funding from the Natural Sciences & EngineeringResearch Council (NSERC) Canada. T.K. is supported by theFWO-Flanders under project O6260-G.0728.11.

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