Bayesian Seismology of the Sun
aa r X i v : . [ a s t r o - ph . S R ] M a r Mon. Not. R. Astron. Soc. , 1– ?? (2013) Printed 18 July 2018 (MN L A TEX style file v2.2)
Bayesian Seismology of the Sun
M. Gruberbauer ⋆ , D.B. Guenther † Institute for Computational Astrophysics, Department of Astronomy and Physics, Saint Mary’s University, Halifax, NS B3H 3C3, Canada
Accepted Received
ABSTRACT
We perform a Bayesian grid-based analysis of the solar l =0,1,2 and 3 p modes ob-tained via BiSON in order to deliver the first Bayesian asteroseismic analysis of thesolar composition problem. We do not find decisive evidence to prefer either of thecontending chemical compositions, although the revised solar abundances (AGSS09)are more probable in general. We do find indications for systematic problems in stan-dard stellar evolution models, unrelated to the consequences of inadequate modellingof the outer layers on the higher-order modes. The seismic observables are best fit bysolar models that are several hundred million years older than the meteoritic age ofthe Sun. Similarly, meteoritic age calibrated models do not adequately reproduce theobserved seismic observables. Our results suggest that these problems will affect anyasteroseismic inference that relies on a calibration to the Sun. Key words:
Sun:oscillations – Sun:abundances – Sun:fundamental parameters
The study of solar-type pulsation with its reliance on scalingrelations (e.g., Huber et al. 2011) and calibrations of funda-mental free parameters in stellar models (i.e., mixing lengthparameter and helium abundance) is ultimately anchoredby what we know about the Sun and by how well seismol-ogy performs at identifying the Sun’s key properties. Re-cent asteroseismic investigations of sun-like pulsators (e.g.,Metcalfe et al. 2010; Mathur et al. 2012) are able to giveprecise model-dependent constraints but it is difficult to as-sess their accuracy. Inferences from certain asteroseismic ob-servables are not necessarily model dependent as can be ver-ified using spectroscopy or interferometry (e.g., Huber et al.2012). However, full asteroseismic analyses that determinestellar ages and compositions, or decide among different im-plementations of how to model important physical processes(e.g., different approaches to convection) rely on a thoroughcalibration of the properties and parameters of the model.Several incompatibilities remain between solar mod-elling and the results inferred from helioseismology that canpotentially affect our calibrations (for a recent comprehen-sive review see Christensen-Dalsgaard 2009). For example,many investigators find that models that use the previousgeneration (Grevesse & Sauval 1998) abundances fit helio-seismic observables better than the current revised solarabundances (Asplund et al. 2005, 2009). Consequently, thehelium abundance and the resulting value for the ratio of the ⋆ E-mail: [email protected] † E-mail: [email protected] metal mass fraction to hydrogen mass fraction at the sur-face, ( Z s /X s ) ⊙ , is uncertain. We do know that inadequatemodelling of the outer layers leads to the so-called “surfaceeffects” (see, e.g., Kjeldsen et al. 2008; Gruberbauer et al.2012) that worsens the model fit to higher order frequen-cies. Uncertainties in opacities, equations of state, nuclearreaction rates, and other global parameters also influencethe properties of the solar model and, as a consequence, itsseismic calibration. Recently, for instance, an increase in theopacities (Serenelli et al. 2009) and different accretion sce-narios (Serenelli et al. 2011) have been identified as possibleremedies for the disagreement between the results of he-lioseismic inversion and models based on the previous andcurrent generation of chemical compositions.Previous studies testing different model configurations,for example, different chemical mixtures, have often reliedon the direct comparison of non-seismic observables and gen-eral properties inferred from helioseismology to stellar mod-els calibrated to the non-seismic observables: age, radius,mass, luminosity, and in some cases also surface abundances.More recent approaches (Basu et al. 2007; Chaplin et al.2007; Serenelli et al. 2009) compared low-degree p modes,or rather various spacings derived from them, to modelswith solar characteristics. The result again suggests thatthey cannot be reconciled with the revised solar abundances.Houdek & Gough (2011) also developed an approach thatuses quantities derived from the observed modes to infersolar model properties via iterative calibration procedures.What is missing, though, is a test of the solar modelwith a tool that takes into account all the informationgiven by the low-degree solar p modes and other con- c (cid:13) Gruberbauer and Guenther straints and which then results in a quantitative compar-ison of how much certain model properties are actually pre-ferred on a global, probabilistic level. In our previous pa-per (Gruberbauer et al. 2012, hereafter Paper I) we intro-duced a new Bayesian method that uses prior informationand properly treats known systematic effects (i.e., “surfaceeffects”). We performed a state-of-the-art, albeit, abbrevi-ated grid-based asteroseismic analysis of the solar model. Inthis paper we build on and extend our solar modelling bytesting various chemical compositions and nuclear reactionrates. Our goal is to answer the following questions:(i) Which models fit the solar modes and other observ-ables the best?(ii) Is there a clear preference for any of the chemicalcompositions and reaction rates?(iii) How do the surface effects affect the fit?(iv) How do our results affect the calibrations for astero-seismology?We approach our analysis, leaning more toward utilisingthe techniques applicable to asteroseismology than helioseis-mology. Specifically we will only utilise the lower l -valued pmodes and we will allow all parameters except for the massto remain unconstrained. We are, therefore, setting out tomodel the global properties of the Sun as a star, hence, toperform asteroseismology of the Sun. As in Paper I, we fit our models to the activity-corrected so-lar l = 0, 1, 2, and 3 p modes obtained by using BiSON data(Broomhall et al. 2009). For our prior probabilities on othersolar observables, we take an investigative approach by usingboth broad and narrow priors for the most important solarquantities: T eff , L , and age. This will help us to study thesystematic dependencies of our results on the imposed con-straints. For the general properties of the Sun, we use both abroad prior with log T eff = 3 . ± .
01, and log (
L/L ⊙ ) =0 . ± .
01, or alternatively a more realistic but still conser-vative prior with log T eff = 3 . ± . L/L ⊙ ) =0 . ± . L ⊙ = 3 . ± . · erg s − (the aver-age of the ERB-Nimbus and SMM/ARCRIM measurements;Hickey & Alton (1983)). For the solar mass, we use M ⊙ =1 . ± . · g (Cohen & Taylor 1986). As a referencefor the solar age we take the result from Bouvier & Wadhwa(2010) who determined a meteoritic age of the solar systemof τ ≈ . R BCZ = 0 . ± . R ⊙ (Christensen-Dalsgaard et al.1991; Basu & Antia 1997). All priors are assumed to be nor-mal distributions. Just as in Paper I, our aim was to employ YREC(Demarque et al. 2008) and produce a set of dense grids cov-ering a wide range in initial hydrogen mass fractions X , ini-tial metal mass fractions Z , and mixing length parameters α ml . For this study we kept all model masses constrained to1 M ⊙ , but we additionally varied the chemical compositionand the nuclear reaction rates.Our model tracks begin as completely convective Lane-Emden spheres (Lane 1869; Chandrasekhar 1957) and areevolved from the Hayashi track (Hayashi 1961) through thezero-age-main-sequence (ZAMS) to 6 Gyrs with each trackconsisting of approximately 2500 models. Only models be-tween 4.0 and 6.0 Gyrs are included in the model grid.Constitutive physics include the OPAL98 (Iglesias & Rogers1996) and Alexander & Ferguson (1994) opacity tables,as well as the Lawrence Livermore 2005 equation ofstate tables (Rogers 1986; Rogers et al. 1996). Convectiveenergy transport was modelled using the B¨ohm-Vitensemixing-length theory (B¨ohm-Vitense 1958). The atmospheremodel follows the ( T - τ ) relation by Krishna Swamy (1966).For each grid, we varied the chemical composition andtested two different nuclear reaction rates. We consideredthe GS98 mixture (Grevesse & Sauval 1998), the AGS05mixture (Asplund et al. 2005), and the AGSS09 mixture(Asplund et al. 2009). Nuclear reaction cross-sections weretaken from Bahcall et al. (2001) and the nuclear reactionrates from Table 21 in Bahcall & Ulrich (1988). In ad-dition, we also calculated grids using the NACRE rates(Angulo et al. 1999). The effects of helium and heavy el-ement diffusion (Bahcall et al. 1995) were included. Notethat our atmosphere models and diffusion effects have beenshown to require a larger value of mixing length parame-ter ( α ml ≈ . − .
2) than standard Eddington atmospheres( α ml ≈ . − .
8) (Guenther et al. 1993). The model gridspans: X from 0.68 to 0.74 in steps of 0.01, Z from 0.014to 0.026 in steps of 0.001, and α ml from 1.3 to 2.5 in stepsof 0.1.The pulsation spectra were computed using the stel-lar pulsation code of Guenther (1994), which solves the lin-earized, non-radial, non-adiabatic pulsation equations usingthe Henyey relaxation method. The non-adiabatic solutionsinclude radiative energy gains and losses but do not includethe effects of convection. We estimate the random 1 σ un-certainties of our model frequencies to be of the order of0 . µ Hz. These uncertainties are properly propagated intoall further calculations.
Our Bayesian fitting method is explained in detail in Paper I.To briefly summarize, we compare theoretical and observedfrequencies by calculating the likelihood that the two valuesagree were it not for the presence of random and systematicerrors. These likelihoods are then combined using the sumrule and product rules of probability theory, and weightedby priors to arrive at correctly normalised probabilities. Therandom errors are assumed to be independent and Gaussian.Although frequency uncertainties are likely to be somewhatcorrelated depending on the data set quality and extractiontechnique, independence is a fundamental necessity to allowthe independent treatment of surface effects. In the solarcase the observational uncertainties are rather small, and sorandom errors in the model frequencies due to the modelshell resolution ( ∼ . µ Hz), the influence of priors, and thesurface effect treatment will outweigh the influence of corre- c (cid:13) , 1– ?? ayesian Seismology of the Sun lations . The systematic errors in the case of solar-like starsare assumed to be similar to “surface effects”. At higherorders, observed frequencies are systematically lower thanmodel frequencies, and the absolute frequency differencesincrease with frequency. This is modelled by introducing asystematic difference parameter, ∆, between observed andcalculated frequency so that f obs , i = f calc , i + γ ∆ i . (1)In the case of surface effects, γ = −
1. These ∆ i are thenallowed to become larger at higher frequencies. The upperlimit at each frequency is determined by the large frequencyseparation and a power law similar to the standard correc-tion introduced by Kjeldsen et al. (2008). The ∆ parameteris incorporated in a completely Bayesian fashion, using a β prior to prefer smaller values over larger ones (see Paper Ifor more details). In addition, we always allow for the pos-sibility that a mode is not significantly affected by any kindof systematic error. Altogether, this allows us to fully prop-agate uncertainties originating from the surface effects intoall our results, and at the same time gives us more flexibilitythan the standard surface-effect correction.We obtain probabilities for every evolutionary track inour grids, and within the tracks also for every model. We alsoobtain the correctly propagated distributions for systematicerrors so that the model-dependent surface effect can bemeasured. In order to fully resolve the changes in stellarparameters and details in the stellar-model mode spectra,we oversample the evolutionary tracks via linear interpola-tion until the (normalised) probabilities no longer changesignificantly. Eventually, we obtain so-called evidence val-ues, equivalent to the prior-weighted average likelihood, forevery grid as a whole. These evidence values are also iden-tical to the likelihood of the data (i.e., the solar frequencyvalues) given the particular grids as conditional hypotheses.Just as the likelihoods for individual stellar models or, onestep further, for evolutionary tracks can be used to comparetheir probabilities and evaluate the stellar parameters, theevidence values as likelihoods for whole grids can thereforebe employed to perform a quantitative comparison betweendifferent input physics used in the grids . This exemplifiesthe hierarchical structure of Bayesian analysis, which is dis-cussed in more detail in Paper I and also in the more generalliterature (e.g., Gregory 2005). The advantage of the Bayesian analysis method from PaperI is that many different approaches to fitting the same dataset can be compared using the evidence values. Our goal isto see if there is a strong preference for either the GS98,AGS05, or AGSS09 mixture. We also want to test whether Moreover, if frequency errors are derived from their marginaldistribution as in Bayesian peak-bagging (e.g., Gruberbauer et al.2009; Handberg & Campante 2011), they can be treated as inde-pendent. Other hypothesis modifications (e.g, different shapes of system-atic errors) can in principle also be compared. or not the NACRE nuclear reaction rates are an improve-ment. The corresponding grids will be designated as GS98N,AGS05N, and AGSS09N . We use priors for the HRD po-sition and age, as well as R BCZ , to see which of these gridsare more consistent with well-known solar properties otherthan the frequencies. Also, by turning off the priors we cantell which solar-mass models best reproduce the frequenciesirrespective of their fundamental parameters.We will start our analysis without any priors and suc-cessively increase the prior information we use, to answerthe questions outlined in Section 1. For example, if we wereto find that the best solar models are much too old and lumi-nous, or if the evidence values decrease when the priors areturned on, we will then have evidence that the model physicscannot reproduce an accurately calibrated solar model.It should be noted that all results presented in the fol-lowing sections are highly dependent on the models used(i.e., what was described in Section 2.2). We therefore cannotclaim that our results represent the real Sun, as indeed weperform our analysis to investigate the similarities and sys-tematic differences between models and observations. How-ever, as explained in Paper I, our approach is capable tocompare different grids produced from different codes andthus draw probabilistic inferences about systematic differ-ences between these codes as well.
For the first test we did not use any of the luminosity, tem-perature, age, or R BCZ constraints as formal priors. In thiscase the only effective prior is provided by the selection ofthe model grid parameters and the restriction to solar-massmodels, hence, every model in the grids was given equal priorweight.Fig. 1 compares the grids in terms of the logarithm ofthe evidence. Note that differences between these values areequivalent to the logarithm of the posterior probability ra-tios for the grids as a whole under the condition that theyhave equal prior probabilities.Following the guidelines provided by Jeffreys (1961),differences of up to 0.5 (or likelihood ratios up to 3) areconsidered “barely worth mentioning”. Differences between0.5 and 1.0 indicate “substantial” strength of evidence. Onlywhen the differences rise above 1.0 (i.e., likelihood ratios >
10) should the strength of evidence be considered strong.Accordingly, the GS98-mixture models are not significantlybetter than AGSS09-mixture models. However, there is sub-stantial evidence that the AGS05, AGS05N, and AGSS09Nmodels do not reproduce the solar frequencies adequately,i.e., the GS98, GS98N, and AGSS09 are significantly betterthan the AGS05, AGS05N, and AGSS09N models. This in-dicates that there are problems with the AGS05 mixture andit also suggests that the NACRE rates have a detrimentaleffect on the model frequencies. Inspection of the frequenciesfor AGSS09N and AGSS05N reveals that, compared to the Statements that are valid for both reaction rates will refer toboth grids at once using the notation GS98(N), AGS05(N), orAGSS09(N)c (cid:13) , 1– ?? Gruberbauer and Guenther -190.4-190.2-190.0-189.8-189.6-189.4-189.2-189.0-188.8-188.6 l og ( e v i den c e ) GS98 GS98N AGS05 AGS05N AGSS09 AGSS09Nno priors
Figure 1.
Model grid performance without HRD, age or R BCZ priors. The thick double-sided and thin arrows indicate strengthof evidence that is “barely worth mentioning” and “substantial”respectively. Differences larger than the thin arrow can be con-sidered “strong” evidence (see text). corresponding models in the AGSS09 and AGSS05 grids, thelower order modes do not fit as well and the surface effectalso increases . For instance, when just considering the bestevolutionary track for AGSS09, adopting the NACRE ratesfor the same track leads to decrease in probability by a fac-tor of ∼ ∼ R BCZ from 0.7164to 0.7182.In Fig. 2 we show the mean values and uncertainties ofsome model properties, corresponding to the grids in Fig. 1.Note that these uncertainties are caused by spreading theprobabilities over a few different evolutionary tracks withmodels that fit the frequencies best. If we were to restrictthe parameter space by using priors as described in the nextsections, then the probabilities will be mostly concentratedon only one or two evolutionary tracks and, consequently,the formal uncertainties will be reduced.Table 1 contains more details for the most probablemodel parameters of the best and second-best evolution-ary tracks in all grids. Considering the metallicities and thelocations of the base of the convection zone, the results aresimilar to the general picture that has emerged in the litera-ture. The GS98 and GS98N models requires higher metallic-ities and a deeper base of the convection zone. Concerningthe latter, the uncertainties are such that both AGSS09 andGS98(N) are in general agreement with R BCZ . None the less,the GS98(N) models fit this value a little bit better. Usingthe R BCZ prior in the next sections will put a formal con-straint on this as well.It is disturbing, however, to see that all of the bestmodels greatly overestimate the age of the Sun by severalhundred million years. Furthermore, most of the models donot match the solar T eff and luminosity very well. Thereforeour next step is to “switch on” either the broad or the morerealistic priors constraining the Sun’s position in the HRdiagram. As will be shown in Section 4.3, the former is usually moreimportant than the latter. age [Gyrs] -190.5-190.0-189.5-189.0-188.5 l og ( e v i den c e ) GS98AGS05AGSS09 R BCZ -190.5-190.0-189.5-189.0-188.50.019 0.02 0.021 0.022 0.023 Z S /X S -190.5-190.0-189.5-189.0-188.5 l og ( e v i den c e ) Z S -190.5-190.0-189.5-189.0-188.5 Figure 2.
Grid evidence versus mean values and uncertainties ofsome model properties when fitting the observed frequencies with-out any priors. Open symbols denote the corresponding NACREgrids. -190.5-190.0-189.5-189.0-188.5-188.0 l og ( e v i den c e ) -191.0-190.5-190.0-189.5-189.0-188.5 l og ( e v i den c e ) GS98 GS98N AGS05 AGS05N AGSS09 AGSS09NGS98 GS98N AGS05 AGS05N AGSS09 AGSS09N
Figure 3.
Model grid performance with the broad (top panel)and the realistic (bottom panel) HRD prior. T eff and L priors As in Paper I we now use normal distributions as priorsfor log T eff and log ( L/L ⊙ ) (hereafter: HRD prior). Moreweight is put on models that match the solar position in theHRD. Note that this does not mean that the best modelswill match the solar values. In this paper we employ slightlydifferent HRD priors, using either a broad prior or a morerealistic narrow prior based on current observational uncer- c (cid:13) , 1– ?? ayesian Seismology of the Sun Table 1.
Most probable parameters without priors. The quoted probabilities refer to the probability of the evolutionary track withineach grid. X , Z : initial hydrogen and metal mass fractions; Z s : metal mass fraction in the envelope; R BCZ : fractional radius of thebase of the convection zone; α ml : mixing length parameter.grid T eff [ K ] L/L ⊙ R/R ⊙ Age X Z Z s Z s /X s R BCZ α ml ProbabilityGS98 5718 0.958 1.0001 5.022 0.72 0.018 0.0161 0.0214 0.7116 2.1 0.895802 1.016 0.9998 4.656 0.71 0.018 0.0162 0.0218 0.7139 2.2 0.05GS98N 5660 0.920 1.0000 5.046 0.72 0.019 0.0170 0.0226 0.7114 2.0 0.525816 1.025 0.9997 4.637 0.71 0.018 0.0161 0.0217 0.7160 2.2 0.34AGS05 5711 0.953 0.9997 4.967 0.72 0.016 0.0143 0.0190 0.7173 2.1 0.515754 0.983 1.0000 4.975 0.71 0.017 0.0152 0.0204 0.7139 2.2 0.38AGS05N 5694 0.942 1.0000 5.041 0.71 0.018 0.0161 0.0216 0.7139 2.1 0.505647 0.911 0.9997 5.029 0.72 0.017 0.0152 0.0202 0.7165 2.0 0.26AGSS09 5718 0.958 0.9998 4.932 0.72 0.016 0.0143 0.0190 0.7164 2.1 0.705761 0.988 1.0000 4.941 0.71 0.017 0.0152 0.0205 0.7132 2.2 0.26AGSS09N 5701 0.947 1.0001 5.006 0.71 0.018 0.0161 0.0216 0.7128 2.1 0.675654 0.916 0.9998 4.993 0.72 0.017 0.0152 0.0202 0.7155 2.0 0.11 tainties. As we show below, the differences between the morerealistic prior and the broad prior enable us to distinguishthe chemical compositions. The resulting grid evidences areshown in the two panels of Fig. 3.For the broad HRD prior, an increase in evidence for allgrids can be seen. This indicates that the models that aresomewhat consistent with the solar values do include the ma-jority of the best fit models. Since the evidence is a weightedaverage of the likelihood, however, most of the increase inevidence is caused by putting less weight on the many mod-els that are clearly outside the solar values and do not matchthe solar frequencies at all. The relative likelihood ratios re-main comparable to the “no prior” case, but now AGSS09is actually slightly more probable than the GS98(N) mod-els. As before, the evidences of the three best grids are notdifferent enough to clearly prefer one grid over the other.Table 2 again gives information on the best fitting evolu-tionary tracks within each grid for the broad HRD prior.About half of the best or second-best models from the “noprior” analysis remain among the most probable models butonly GS98 shows the same models and ranking as before. Itis interesting that the best-fitting model from the AGSS09grid, which also is the overall best fit using the broad HRDprior, now matches the observed base of the convection zoneclosest from all models considered. Except for GS98N, theNACRE grids again perform worse than their counterparts.Note, however, that with the broad HRD prior the mostprobable basic model parameters are the same whether ornot NACRE rates are used.For the realistic HRD prior, on the other hand, theGS98(N) grids receive an evidence penalty. Here, the pref-erence for AGSS09(N) is more pronounced, and the previ-ous decrease in evidence for AGSS09N is now compensatedby its much closer match to the solar HRD position. As isshown in Table 3, the most probable models for AGS05(N)and AGSS09(N) remain the same. However, the best AGS05model underestimates luminosity and effective temperatureand therefore its evidence decreases compared to the broadHRD prior.All the conclusions drawn from the “no prior” approach still apply, i.e., the model fits give us no clear indication for,e.g., preferring GS98(N) over AGSS09, but they do showsignificant evidence against AGS05 and for the detrimentaleffect of the NACRE rates.Lastly, we turn on the R BCZ prior in tandem with theHRD priors, which puts stronger constraints on a properfit to the interior. The results are shown in Fig. 4 and thecorresponding model parameters for the realistic HRD priorare summarized in Table 4. Interestingly, for both HRD pri-ors, AGSS09 manages to increase the probability contrastto the other models. The evidence rises once more, whichsignifies that the models that fit the pulsation frequenciesalso are among those that fit best to R BCZ . This is alsoconfirmed by Table 4 which shows that the most probablemodels for AGSS09(N) and AGS05(N) have not changed.
For these mixtures the models that are best at reproducingthe pulsation and broad HRD constraints also fit the realisticHRD constraints and the base of the convection zone.
This isalso responsible for producing the enormous concentrationof probability on the best evolutionary tracks. The bottompanel in Fig. 4 also indicates that, with the realistic HRDprior and the R BCZ constraint, there is formally strong ev-idence for the AGSS09 mixture to provide the overall mostrealistic solar model.Nonetheless, all ages are still too high compared to thewell-established meteoritic age estimate. We cannot con-sider these models to be properly calibrated to the Sun,even though the frequencies clearly prefer these solutions.We therefore now turn to age priors to avoid the solutionsthat are clearly too old (or too young).
In Paper I, we used a similar approach to rule out oldermodels and employed a Gaussian prior centred on the me-teoritic solar age but allowed for a few tens of millions ofyears of PMS evolution. In this paper, however, we chose totake a more careful approach.Different authors often use different definitions for theage of their solar model (e.g., age from the birthline or age c (cid:13) , 1– ?? Gruberbauer and Guenther
Table 2.
Same as Table 1 but with the broad HRD priors.grid T eff [ K ] L/L ⊙ R/R ⊙ Age X Z Z s Z s /X s R BCZ α ml ProbabilityGS98 5718 0.958 1.0001 5.022 0.72 0.018 0.0161 0.0214 0.7116 2.1 0.775802 1.016 0.9998 4.656 0.71 0.018 0.0162 0.0218 0.7139 2.2 0.20GS98N 5816 1.025 0.9997 4.637 0.71 0.018 0.0161 0.0217 0.7160 2.2 0.855732 0.967 1.0000 5.002 0.72 0.018 0.0161 0.0214 0.7127 2.1 0.10AGS05 5754 0.983 1.0000 4.975 0.71 0.017 0.0152 0.0204 0.7139 2.2 0.845711 0.953 0.9997 4.967 0.72 0.016 0.0143 0.0190 0.7173 2.1 0.15AGS05N 5768 0.992 0.9999 4.957 0.71 0.017 0.0152 0.0204 0.7161 2.2 0.465725 0.962 0.9996 4.951 0.72 0.016 0.0143 0.0189 0.7189 2.1 0.36AGSS09 5761 0.988 1.0000 4.941 0.71 0.017 0.0152 0.0205 0.7132 2.2 0.665718 0.958 0.9998 4.932 0.72 0.016 0.0143 0.0190 0.7164 2.1 0.33AGSS09N 5775 0.997 1.0000 4.923 0.71 0.017 0.0152 0.0204 0.7149 2.2 0.695701 0.947 1.0001 5.006 0.71 0.018 0.0161 0.0216 0.7128 2.1 0.23
Table 3.
Same as Table 1 but with the realistic HRD priors.grid T eff [ K ] L/L ⊙ R/R ⊙ Age X Z Z s Z s /X s R BCZ α ml ProbabilityGS98 5767 0.992 1.0002 4.980 0.71 0.019 0.0170 0.0229 0.7096 2.2 0.835802 1.016 0.9998 4.656 0.71 0.018 0.0162 0.0218 0.7139 2.2 0.15GS98N 5780 1.001 1.0002 4.959 0.71 0.019 0.0170 0.0228 0.7109 2.2 0.9975769 0.992 0.9995 4.660 0.72 0.017 0.0152 0.0203 0.7184 2.1 2.6e-3AGS05 5754 0.983 1.0000 4.975 0.71 0.017 0.0152 0.0204 0.7139 2.2 0.905789 1.006 0.9995 4.848 0.72 0.015 0.0134 0.0178 0.7205 2.2 0.07AGS05N 5768 0.992 0.9999 4.957 0.71 0.017 0.0152 0.0204 0.7161 2.2 0.999965779 1.000 0.9997 4.680 0.70 0.018 0.0161 0.0220 0.7177 2.2 3.7e-5AGSS09 5761 0.988 1.0000 4.941 0.71 0.017 0.0152 0.0205 0.7132 2.2 0.99965796 1.011 0.9996 4.814 0.72 0.015 0.0134 0.0178 0.7191 2.2 3.0e-4AGSS09N 5775 0.997 1.0000 4.923 0.71 0.017 0.0152 0.0204 0.7149 2.2 0.9999985787 1.005 0.9998 4.646 0.70 0.018 0.0161 0.0220 0.7158 2.2 1.0e-6
Table 4.
Same as Table 1 but with the R BCZ and realistic HRD priors.grid T eff [ K ] L/L ⊙ R/R ⊙ Age X Z Z s Z s /X s R BCZ α ml ProbabilityGS98 5802 1.016 0.9998 4.656 0.71 0.018 0.0162 0.0218 0.7139 2.2 0.835789 1.007 1.0000 4.941 0.72 0.017 0.0152 0.0202 0.7130 2.2 0.14GS98N 5780 1.001 1.0002 4.959 0.71 0.019 0.0170 0.0228 0.7109 2.2 0.999985746 0.977 0.9998 4.694 0.71 0.019 0.0171 0.0230 0.7142 2.1 6.2e-6AGS05 5754 0.983 1.0000 4.975 0.71 0.017 0.0152 0.0204 0.7139 2.2 0.999965798 1.014 1.0001 4.947 0.70 0.018 0.0161 0.0219 0.7119 2.3 1.8e-5AGS05N 5768 0.992 0.9999 4.957 0.71 0.017 0.0152 0.0204 0.7161 2.2 0.99999975779 1.000 0.9997 4.680 0.70 0.018 0.0161 0.0220 0.7177 2.2 1.2e-7AGSS09 5761 0.988 1.0000 4.941 0.71 0.017 0.0152 0.0205 0.7132 2.2 0.99999975773 0.996 0.9998 4.664 0.70 0.018 0.0162 0.0221 0.7139 2.2 2.6e-7AGSS09N 5775 0.997 1.0000 4.923 0.71 0.017 0.0152 0.0204 0.7149 2.2 0.99999985787 1.005 0.9998 4.646 0.70 0.018 0.0161 0.0220 0.7158 2.2 1.6e-7c (cid:13) , 1– ?? ayesian Seismology of the Sun -190.0-189.5-189.0-188.5-188.0-187.5-187.0-186.5 l og ( e v i den c e ) -192.0-191.0-190.0-189.0-188.0-187.0 l og ( e v i den c e ) GS98 GS98N AGS05 AGS05N AGSS09 AGSS09NR
BCZ priorGS98 GS98N AGS05 AGS05N AGSS09 AGSS09N
Figure 4.
Model grid performance with the R BCZ prior, as wellas the the broad (top panel) and the realistic (bottom panel)HRD prior. from the ZAMS). Therefore, Fig. 5 presents the age-relateddetails in our solar model evolution. The meteoritic ageis measured from the time when the initial abundance ofthe isotopes used to date the meteorites are no longer keptin equilibrium. This probably occurs at some point on theHayashi track. We take the zero age of our models to coin-cide with the birthline as defined in Palla & Stahler (1999).This introduces an uncertainty of ∼ R BCZ priors.
The broad age prior is a uniform prior that rules out veryold or young models. We designed it to allow for an agerange of 4.4 – 4.7 Gyrs. This removes most of our previousbest fits, but retains the good GS98(N) models which have ≈ .
65 Gyrs. Fig. 6 shows the results in terms of evidence.Clearly, the AGS05 and AGSS09 mixture have suffered a se-vere penalty for their older models are now outside the range l og L / L s un eff Birthline -0.20-0.15-0.10-0.050.000.05 l og L / L s un eff ZAMS @ 48 MyrSun @ 4.62 Gyr Convective CoreAppears @ 28 MyrFirst Gravitational EnergyDrop Below 1% @ 36 Myr Convective CoreDisappears @ 84 Myr
Figure 5.
Evolution of a one solar mass model. Evolution isstarted above the birthline. The model crosses the birthline af-ter < . allowed by the prior. The GS98 and GS98N models, on theother hand, show an increase in evidence compared to Fig. 3and therefore the evidence contrast has increased markedly.The realistic HRD prior does affect and slightly decreasethis contrast, but since the AGS05(N) and AGSS09(N) gridshave lost their previous best models to the age prior, the ef-fect is not as pronounced as in Fig. 3. In terms of the strengthof evidence, this result would amount to decisive evidencefor the GS98(N) grids. Since the best models are the samefor the broad and the realistic HRD prior, we only list theresults for the latter in Table 5.For GS98, the probability is now concentrated in thebest model from Table 4. Note that both the best GS98and second best GS98N models have the same fundamen-tal parameters, differing in their nuclear reaction rates. TheAGS05(N) and AGSS09(N) grids all find the same basic c (cid:13) , 1– ?? Gruberbauer and Guenther -198.0-197.0-196.0-195.0-194.0-193.0-192.0-191.0-190.0-189.0-188.0-187.0-186.0 l og ( e v i den c e ) -197.0-196.0-195.0-194.0-193.0-192.0-191.0-190.0-189.0-188.0 l og ( e v i den c e ) GS98 GS98N AGS05 AGS05N AGSS09 AGSS09Nbroad age + R
BCZ priorsGS98 GS98N AGS05 AGS05N AGSS09 AGSS09N
Figure 6.
Model grid performance with the R BCZ and broadage prior, as well as the broad (top panel) and realistic (bottompanel) HRD prior. model (except for the different mixture and reaction rates)with intermediate Z = 0 .
018 proving to be the most prob-able.Without the R BCZ prior (not shown) the best GS98 andGS98N models are the same, and the overall evidence distri-bution is very similar as in Fig. 6. However, the AGS05(N)and AGSS09(N) grids would prefer models with low metal-licity ( Z = 0 . R BCZ well out-side the range supported by the inversion results . For allgrids, the ages of the best models are still too high by up to150 Myrs. In order to see how fully age-constrained solar models in theGS98(N) grids compare to the AGS04 and AGSS09 models,we restricted the age even further by employing a narrowuniform age prior that only allows ages of 4.52 – 4.62 Gyrs.As shown in Fig. 7, the narrow age prior has a big effect onthe analysis. Since it is interesting to see whether models atthe correct age can fit the base of the convection zone, wefirst perform the analysis without the R BCZ prior.Compared to Fig. 6 and for the broad HRD prior, thenarrow age constraint strongly decreases the evidence forthe GS98(N) models, while increasing the evidence for theother models. GS98 still comes out to be the most probableby an order of magnitude. The remaining grids show more These models will nonetheless turn out to be the most proba-ble when we make the age constraint even stronger in the nextsection. -192.0-191.5-191.0-190.5-190.0 l og ( e v i den c e ) -202.0-200.0-198.0-196.0-194.0-192.0-190.0 l og ( e v i den c e ) GS98 GS98N AGS05 AGS05N AGSS09 AGSS09Nnarrow age priorGS98 GS98N AGS05 AGS05N AGSS09 AGSS09N
Figure 7.
Model grid performance with the narrow age prior, aswell as the broad (top panel) and realistic (bottom panel) HRDprior. or less comparable evidences, but AGS05N and AGSS09Nare worse than their non-NACRE counterparts. All solu-tions for AGS05 and AGSS09 favour the same basic modelparameters. Table 6 lists the corresponding most probablemodels. Ultimately, the narrow age prior has led to mod-els which are very close to the meteoritic solar age withoutconstraining them too strongly (as would be the case for anon-uniform, e.g., Gaussian, age prior) so that we do notrule out completely the possibility of systematic errors inthe stellar model age. All of the best models, irrespective ofmixture or reaction rates, have X = 0 . Z < . R BCZ > . Z < . R BCZ prior. The evidenceresults are depicted in Fig. 8. For the broad HRD prior, theevidence present a similar picture as before, but the contrastbetween GS98(N) and the AGS05(N) and AGSS09(N) mod-els has intensified. Furthermore, the most probable mod-els for GS98N, AGS05(N) and AGSS09(N) now have higher c (cid:13) , 1– ?? ayesian Seismology of the Sun Table 5.
Same as Table 1 but with R BCZ , realistic HRD and broad age priors.grid T eff [ K ] L/L ⊙ R/R ⊙ Age X Z Z s Z s /X s R BCZ α ml ProbabilityGS98 5802 1.016 0.9998 4.656 0.71 0.018 0.0162 0.0218 0.7139 2.2 0.99985755 0.983 0.9996 4.678 0.72 0.017 0.0153 0.0203 0.7167 2.1 1.0e-4GS98N 5746 0.977 0.9998 4.694 0.71 0.019 0.0171 0.0230 0.7142 2.1 0.435816 1.025 0.9997 4.637 0.71 0.018 0.0161 0.0217 0.7160 2.2 0.30AGS05 5766 0.990 0.9997 4.697 0.70 0.018 0.0162 0.0220 0.7149 2.2 0.99999975795 1.010 0.9994 4.613 0.71 0.016 0.0143 0.0193 0.7204 2.2 2.9e-7AGS05N 5779 1.000 0.9997 4.680 0.70 0.018 0.0161 0.0220 0.7177 2.2 1.005733 0.968 0.9995 4.687 0.71 0.017 0.0152 0.0205 0.7194 2.1 1.8e-13AGSS09 5773 0.996 0.9998 4.664 0.70 0.018 0.0162 0.0221 0.7139 2.2 0.9999995802 1.015 0.9995 4.580 0.71 0.016 0.0144 0.0193 0.7192 2.2 9.9e-7AGSS09N 5787 1.005 0.9998 4.646 0.70 0.018 0.0161 0.0220 0.7158 2.2 0.9999985767 0.991 0.9999 4.671 0.69 0.020 0.0180 0.0248 0.7121 2.2 1.0e-6 metal mass fraction as before. The overall most probablemodel of the GS98 grid, which far outweighs the others interms of evidence, still is the same as in Table 6.For the realistic HRD prior, which most closely reflectsall of our prior knowledge of the Sun, the verdict is clearas well. However, for this prior, it is the AGS05 and theAGSS09 models which are preferred.
The evidence contrastbetween these two grids and the others is the highest contrastmeasured in all analyses performed in this paper . The influ-ence of the realistic HRD prior is again substantial and evenproduces a null result for the GS98N grid because of ournumerical thresholds. The parameters for the most proba-ble models are given in Table 7. All best models now have X < . Z = 0 . α ml = 2 .
2, and R BCZ > . X = 0 .
71 aswell.
Our detailed analysis using various priors has shown: • Without priors, the frequencies fit best to models withsignificantly underestimated luminosities and ages of about5 Gyrs. There is no clear preference for any specific compo-sition. • Models that are constrained by the solar L , T eff and R BCZ prefer the revised composition but are still too old.Except for the age, they can reproduce all known parame-ters, as well as the frequencies, quite well. • Models that are tightly constrained by our informationon the solar age suffer a strong degradation in their qualityof fit. Depending on whether L and T eff are included as tightconstraints, there is a either a clear preference for the oldor the revised composition. In any case, for solar-age mod-els T eff is overestimated while the stellar radius is slightlyunderestimated, producing a significantly overestimated lu-minosity. The model values for R BCZ are too high and welloutside the observational uncertainties. -204.0-202.0-200.0-198.0-196.0-194.0-192.0-190.0 l og ( e v i den c e ) -210.0-208.0-206.0-204.0-202.0-200.0-198.0 l og ( e v i den c e ) GS98 GS98N AGS05 AGS05N AGSS09 AGSS09Nnarrow age + R
BCZ priorsGS98 GS98N AGS05 AGS05N AGSS09 AGSS09N
Figure 8.
Model grid performance with the R BCZ and narrowage priors, as well as the broad (top panel) and realistic (bottompanel) HRD prior.
In the following section, we will consider the questions for-mulated at the beginning of the paper.
Our first question was “Which models fit the solar modesand other observables the best?”. This can be answered bylooking at the evidence values for all the grids we tested inthe previous section.Considering only the p-mode frequencies, i.e., no priors, c (cid:13) , 1– ?? Gruberbauer and Guenther
Table 6.
Same as Table 1 but with broad HRD and narrow age priors.grid T eff [ K ] L/L ⊙ R/R ⊙ Age X Z Z s Z s /X s R BCZ α ml ProbabilityGS98 5872 1.065 0.9997 4.566 0.71 0.017 0.0153 0.0205 0.7162 2.3 0.985829 1.034 0.9995 4.591 0.72 0.016 0.0144 0.0191 0.7192 2.2 0.02GS98N 5885 1.075 0.9996 4.547 0.71 0.017 0.0152 0.0205 0.7183 2.3 0.9995843 1.043 0.9994 4.573 0.72 0.016 0.0143 0.0190 0.7212 2.2 4.3e-4AGS05 5795 1.010 0.9994 4.613 0.71 0.016 0.0143 0.0193 0.7204 2.2 0.99995837 1.040 0.9997 4.605 0.70 0.017 0.0153 0.0208 0.7180 2.3 1.6e-5AGS05N 5809 1.019 0.9994 4.596 0.71 0.016 0.0143 0.0192 0.7220 2.2 0.99995851 1.050 0.9996 4.588 0.70 0.017 0.0152 0.0207 0.7194 2.3 2.5e-5AGSS09 5802 1.015 0.9995 4.580 0.71 0.016 0.0144 0.0193 0.7192 2.2 0.99995845 1.045 0.9997 4.572 0.70 0.017 0.0153 0.0208 0.7162 2.3 2.5e-5AGSS09N 5816 1.024 0.9994 4.564 0.71 0.016 0.0143 0.0193 0.7209 2.2 0.99995858 1.055 0.9997 4.555 0.70 0.017 0.0152 0.0207 0.7180 2.3 4.9e-5
Table 7.
Same as Table 1 but with R BCZ , realistic HRD and narrow age priors.grid T eff [ K ] L/L ⊙ R/R ⊙ Age X Z Z s Z s /X s R BCZ α ml ProbabilityGS98 5829 1.034 0.9995 4.591 0.72 0.016 0.0144 0.0191 0.7192 2.2 1.05788 1.004 0.9991 4.580 0.73 0.015 0.0135 0.0177 0.7224 2.1 1.0e-18AGS05 5795 1.010 0.9994 4.613 0.71 0.016 0.0143 0.0193 0.7204 2.2 1.05741 0.974 0.9998 4.538 0.68 0.022 0.0199 0.0278 0.7101 2.2 5.1e-70AGS05N 5809 1.019 0.9994 4.596 0.71 0.016 0.0143 0.0192 0.7220 2.2 1.0AGSS09 5802 1.015 0.9995 4.580 0.71 0.016 0.0144 0.0193 0.7192 2.2 1.05762 0.986 0.9991 4.547 0.72 0.015 0.0135 0.0179 0.7227 2.1 6.8e-26AGSS09N 5816 1.024 0.9994 4.564 0.71 0.016 0.0143 0.0193 0.7209 2.2 0.9999975834 1.039 0.9999 4.607 0.69 0.019 0.0171 0.0235 0.7134 2.3 2.9e-6
Fig. 1 shows that GS98, GS98N, and AGSS09 all performcomparably well, as they are all able to reproduce the ob-served solar frequencies. Taking into account some of ourprior knowledge by using the broad HRD prior, we find asimilar result in Fig. 3. This also increases the evidence ofall models. For a tighter, more realistic HRD prior the ev-idence for AGSS09(N) is significantly higher than for theother models.However, we have to reject these results, as the modelsare clearly too old. By removing the older models via uni-form age priors, we first see a large drop in the evidence val-ues for the AGS05(N) and AGSS09(N) grids. Indeed, whenwe employed the narrow age prior, which is still compar-atively broad (100 Myrs) to allow for systematic errors inthe model evolution, the GS98(N) grid evidences suffer thesame effect. We are forced to conclude that the model fre-quencies are getting worse as we approach the (presumably)correct solar age. A similar conclusion was reached in PaperI, but here we have shown that this is not affected by thecontested different chemical compositions or the two differ-ent nuclear reaction rates.The different compositions onlyproduce clearly different results when using additional con-straints, as discussed in the next section.In Fig. 9 we have plotted the relative difference betweenthe solar sound speed profile as measured from inversion (Basu et al. 2009) and as determined from some of our bestfit models. The Model S from Christensen-Dalsgaard et al.(1996) is plotted as well. This reflects our summary fromabove, concluding that models constrained to the solar ob-servables are worse at reproducing the observed frequenciesand therefore the solar sound speed profile. The figure alsoshows, contrary to what is commonly reported in the litera-ture, that when using all our prior information the best GS98model performs worse than the the models with the revisedcomposition. It would be interesting to include the soundspeed profile information in the fitting procedure as well, butthe systematic differences between observations and calcula-tions are substantial and the analysis would be non-trivial.It is therefore beyond the scope of this paper and should betargeted for future work.To summarize, the argument for or against the gridspresented in this paper cannot be made by simply claimingthat one grid produces better frequencies in one particularsetup of priors and observables. As we have shown, the gridsare able to deliver similar fits in various conditions, and allgrids actually have problems to fit both seismic and solar pa-rameters. Therefore, we cannot identify a clear “best fittingmodel”. c (cid:13) , 1– ?? ayesian Seismology of the Sun ( c i n v - c m od ) / c i n v JCD model SNo Priors GS98 N X72Z19A20 GS98 X72Z18A21 AGSS09 X72Z16A21HRD+Base Conv.Env. GS98 N X71Z19A22 AGSS09 X71Z17A22 AGSS09 N X71Z17A22HRD+Base Conv.Env.+Age AGSS09 X71Z16A22 AGSS05 X71Z16A22 GS98 X72Z16A22
Figure 9.
Relative difference between the solar sound speed as measured from inversion and determined from our various models. Thelegend indicates which models and priors were used. Only the realistic HRD and narrow age prior results are plotted. N denotes theNACRE reaction rates. (A colour version of this figure is available in the electronic version of the paper)
Contrary to most studies in the literature, our results leadus to reject any clear preference for any of the contestedchemical compositions over the others. Looking at the fre-quencies alone, no composition is clearly preferred, but thereis very strong evidence against AGS05. Ignoring the modelages but using our other priors leads to a significant pref-erence for AGSS09(N). Employing the R BCZ , narrow age,and broad HRD prior leads to decisive evidence for GS98.On the other hand, using all priors leads to a clear pref-erence for AGS05 and AGSS09. Hence, which compositionbetter represents the Sun depends on the consideration oftight constraints on R BCZ , T eff , L , and age. This suggeststhat our models are not calibrated well enough to the Sun,so that prior information is playing an important role com-pared to the observed frequencies. The latter do have aneffect, however, in selecting the models that are compati-ble with our prior information, and thus the results cannotsimply be dismissed.We have to conclude that without solving the generalproblem of how to produce solar-age models that look likethe Sun and produce adequate frequencies, any discussionof the contending compositions has to remain unresolved.Therefore, we also have to refute the claim that the AGSS09composition is incompatible with helioseismic results.We want to exemplify this, and contrast it to argumentsused in the past, by looking at some of the solar parametersobtained from the fits. As shown in Table 7, the best GS98model when subject to all our prior knowledge constraints,has X = 0 .
72 and Z = 0 . Y = 0 . Y s = 0 . Y s = 0 . ± . X = 0 .
71 and Z = 0 . Y = 0 . Y s = 0 . σ uncertainties.Judging from the goodness of fit to the frequencies, as well asfrom the agreement to these helioseismically determined val-ues, we would have to conclude that AGSS09 outperformsGS98. Naturally, this is only true if we ignore the overes-timated luminosities, and R BCZ values. Also, as shown inFig. 9 both models produce some of the strongest deviationsfrom the solar sound speed profile.In a similar case, when ignoring the age and using justthe R BCZ and realistic HRD prior we find Y s = 0 . R BCZ = 0 . Y s = 0 . R BCZ = 0 . c (cid:13) , 1– ?? Gruberbauer and Guenther
As mentioned in the introduction, previous studies mostlylooked at frequency differences and spacings, due to the sur-face effect problem. We have employed the Bayesian formal-ism to take the surface effects into account while still usingthe full information provided by each frequency. We nowwant to analyse how they affect the analysis and to whatextent they influence our conclusions. This is possible be-cause, as discussed in Paper I, our method provides the mostprobable systematic deviations between observed and theo-retical frequencies, as well as their uncertainties, for everyobserved mode.One explanation for the higher evidences at older agescould be that the older models show smaller surface effects.Such a situation would pose a difficult problem, because wecannot assume that our models are already good enough inthe outer layers. The determination of the basic solar pa-rameters would again depend on the surface layers whichis not what we want. Fig. 10 shows that, fortunately, theopposite is the case. It compares the surface effects as de-termined from our GS98 models with the “no prior” ap-proach to respective values from the narrow age prior ap-proach. The mean most probable deviation for the latteramounts to h γ ∆ i i = − . h γ ∆ i i = − . l =0 modes in particular, the“no prior” approach is more consistent with the γ ∆ = 0baseline.In a similar comparison, it is also interesting to probethe differences between the surface effects for the differ-ent compositions. Fig. 12 and 13 show a comparison of theAGSS09 and GS98 results obtained with the R BCZ , realisticHRD, and narrow age prior. The AGSS09 model exhibits alarger surface effect at the lowest orders and therefore getspenalized in the probability terms for these modes. However,it also fits better on average at the lowest l = 0, l = 1, and l = 3 modes and, at the highest orders, has slightly smallersurface effects.Comparing the systematic differences plotted in Fig. 11and Fig. 13 reveals that the most important component ofthe frequency fit is indeed the overall goodness of fit at thelower order modes. While the “no prior” frequencies do havesignificantly larger surface effects above ∼ µ Hz, the sys-tematic differences are significantly smaller between 2000and 3000 µ Hz.In conclusion, our surface effect treatment performsfavourably by allowing low-order modes to dominate the fit-ting process, while still being flexible enough to allow us toproperly measure the most probably surface effect at higherorders for every frequency. Note that the surface effects are always measured with respectto specific models and using, e.g., adiabatic rather than non-adiabatic frequencies will result in different surface effects. -10-50-10-50 f r equen cy d i ff e r en c e γ ∆ i [ µ H z ] -10-501000 1500 2000 2500 3000 3500 4000 frequency [ µ Hz] -10-50 l=0l=1l=2l=3
Figure 10.
Most probable systematic deviations and uncertain-ties as measured using the GS98 grid. The black circles representthe “no prior” approach, while open circles derive from the broadHRD + narrow age prior. Note that the uncertainties are domi-nated by the theoretical frequency uncertainties (0 . µ Hz) exceptat the highest orders. The “no prior” approach results in largersurface effects. The dashed guide line shows a frequency differenceof zero. -1-0.500.51-1-0.500.51 f r equen cy d i ff e r en c e γ ∆ i [ µ H z ] -1-0.500.511200 1400 1600 1800 2000 2200 2400 2600 frequency [ µ Hz] -1-0.500.51 l=0l=1l=2l=3
Figure 11.
Same as Fig. 10 but zoomed in on the lower-ordermodes. In addition, long dashed and solid lines represent linearfits to the open and black circles. Note that the increase in slopetowards higher spherical degree is an artifact due to missing lower-order modes. c (cid:13) , 1– ?? ayesian Seismology of the Sun -10-50-10-50 f r equen cy d i ff e r en c e γ ∆ i [ µ H z ] -10-501000 1500 2000 2500 3000 3500 4000 frequency [ µ Hz] -10-50 l=0l=1l=2l=3
Figure 12.
Same as Fig. 10 but for GS98 (open circles) andAGSS09 (black circles). Both results are based on the realisticHRD + narrow age + R BCz prior analysis. -1-0.500.51-1-0.500.51 f r equen cy d i ff e r en c e γ ∆ i [ µ H z ] -1-0.500.511200 1400 1600 1800 2000 2200 2400 2600 frequency [ µ Hz] -1-0.500.51 l=0l=1l=2l=3
Figure 13.
Same as Fig. 12 but zoomed in on the lower-ordermodes. In addition, long dashed and solid lines represent linearfits to the open and black circles.
General properties of Sun-like stars can now be inferredvia scaling laws using high-quality frequencies from spacemissions and other asteroseismic observables (see, e.g.,Huber et al. 2012). While the uncertainties of the currentsolar frequency sets are still smaller than those of the best Kepler targets, the asteroseismology community expects tobe able to go beyond scaling laws and probe details of thestellar physics (e.g., determining ages and chemical compo-sitions). Our results suggest that in order to obtain accurateresults, more work is needed to first understand the proper-ties of the Sun. As we know from meteoritic data, we obtainsolar ages that are wrong by hundreds of millions of yearsunless we restrict the model space. Furthermore, when weperform a full grid-based analysis, we cannot yet properlydistinguish between the competing chemical compositionswhich have an effect on all the involved quantities.The impact of our analysis also extends beyond thepurely asteroseismic applications. For instance, for ourbest models presented in this paper we obtain values for( Z s /X s ) ⊙ ranging from 0.0190 to 0.0230. If we constrainourselves to our models at the approximate solar age, werequire ( Z s /X s ) ⊙ < . / H] and Z s . In addition, uncertainties and systematic er-rors in the metallicity and helium abundance will naturallypropagate into the results of other fields (e.g., the study ofGalactic abundances) that rely on the solar calibration. In this paper we have reported on our extensive grid-based“asteroseismic” investigation of the Sun using the Bayesianformalism developed in Paper I. We extended our previ-ous study by using different grids with competing chemicalcompositions (GS98, AGS05 and AGSS09) and nuclear re-action rates. We found that we cannot accurately reproducethe solar properties by fitting the frequencies alone with-out using prior information. On the other hand, when usingprior information, we observe a strong degradation in thegoodness-of-fit for the frequencies. This leads us to concludethat we cannot yet give preferential weight to either of thecompeting chemical compositions (or nuclear reaction ratesfor that matter) since the evidence values contradict ourprior information. In other words, the grids are not prop-erly calibrated and some parts of the fundamental modelphysics are inappropriate. Our work does not suggest thatthe revised compositions are any more incompatible with he-lioseismology in some systematic way than the traditionalGS98 abundances. We have also established that it is notthe outer layers which cause the problem, as our Bayesiantreatment of surface effects all but removes their impact.The meteoritic age of the solar system of about 4.568Gyrs is very well established (even if we allow for a system-atic error of perhaps a few Myrs) and its relation to thesolar model age, although not precisely known, cannot beexpected to introduce a larger uncertainty than the dynam-ical time scales associated with evolution down the Hayashitrack. Yet, if we do not constrain the solar age, we obtainvalues around 4.9 to 5 Gyrs, which is an error of about 10percent. Systematic errors in the models are well below 100Myrs, and therefore below the discrepancy between the as-teroseismic solar age and the meteoritic age . So although The age discrepancy predates the present work. Thestandard, often used, reference solar model Model S byChristensen-Dalsgaard et al. (1996) uses an age of 4.6 Gyrs mea-c (cid:13) , 1– ?? Gruberbauer and Guenther ultimately this may not be the best way to untangle our re-sults and characterise, in a simple way, what is wrong withthe models, we have come to see the problem as one relatedto, or at least indicated by, the age. Unfortunately, nearlyevery model assumption, e.g., opacities (especially of themetals), primordial abundances (especially of helium, neon,carbon, oxygen, and nitrogen), convective transport theoryand the modelling of the surface convective envelope, diffu-sion of helium and heavier elements, winds, mass loss, mag-netic fields, rotational shear at the base of the convectionzone, can affect the model age.Our conviction is that the problems reported in this pa-per are not caused by inadequate frequencies or the generalinability to use asteroseismology in the way we have pre-sented. Rather, we think that all the tools and the data arenow at an adequate level to show us the limitations of ourmodels. Indeed, we would like to emphasise that evidence-based Bayesian studies are an excellent way to accurately as-sess future developments in solar modelling. They provide afully consistent framework to test observables, treat system-atic errors (e.g., surface effects) and use prior information, inorder to iterate towards more accurate model physics. Suchis necessary to both better understand our Sun and to reapthe full benefits of asteroseismology.
ACKNOWLEDGMENTS
We would like to thank the referee for improving the qualityand truly expanding the scope of this paper. The authors ac-knowledge funding from the Natural Sciences and Engineer-ing Research Council of Canada. Computational facilitieswere provided by ACEnet, which is funded by the CanadaFoundation for Innovation (CFI), Atlantic Canada Opportu-nities Agency (ACOA), and the provinces of Newfoundlandand Labrador, Nova Scotia, and New Brunswick.
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