Changing the ν_{\max} Scaling Relation: The Need For a Mean Molecular Weight Term
Lucas S. Viani, Sarbani Basu, William J. Chaplin, Guy R. Davies, Yvonne Elsworth
aa r X i v : . [ a s t r o - ph . S R ] M a y Draft version May 24, 2017
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CHANGING THE ν max SCALING RELATION: THE NEED FOR A MEAN MOLECULAR WEIGHT TERM
Lucas S. Viani , Sarbani Basu , William J. Chaplin , Guy R. Davies , and Yvonne Elsworth Department of Astronomy, Yale University, New Haven, CT, 06520, USA School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK Stellar Astrophysics Centre (SAC), Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C,Denmark
ABSTRACTThe scaling relations that relate the average asteroseismic parameters ∆ ν and ν max to the globalproperties of stars are used quite extensively to determine stellar properties. While the ∆ ν scalingrelation has been examined carefully and the deviations from the relation have been well documented,the ν max scaling relation has not been examined as extensively. In this paper we examine the ν max scaling relation using a set of stellar models constructed to have a wide range of mass, metallicity,and age. We find that as with ∆ ν , ν max does not follow the simple scaling relation. The most visibledeviation is because of a mean molecular weight term and a Γ term that are commonly ignored. Theremaining deviation is more difficult to address. We find that the influence of the scaling relationerrors on asteroseismically derived values of log g are well within uncertainties. The influence of theerrors on mass and radius estimates is small for main sequence and subgiants, but can be quite largefor red giants. Keywords: stars: fundamental parameters — stars: interiors — stars: oscillations INTRODUCTIONThe most easily determined asteroseismic parameters of a star are the large frequency separation, ∆ ν , and thefrequency at which oscillation power is maximum, ν max . These average asteroseismic parameters can be determinedeven in poor signal-to-noise data, and as a result are commonly used in asteroseismic analyses.What makes ∆ ν and ν max so useful is that they are related to the global properties of stars, their total mass, radius,and effective temperatures, through very simple relations, often known as scaling relations. The large frequencyseparation, ∆ ν , is the average frequency spacing between modes of adjacent radial order ( n ) of the same degree ( ℓ ).The theory of stellar oscillations shows that (see, e.g., Tassoul 1980; Ulrich 1986; Christensen-Dalsgaard 1993) ∆ ν scales approximately as the average density of a star, thus∆ ν ∝ √ ¯ ρ, (1)or in other words, we can approximate ∆ ν ∆ ν ⊙ ≃ s M/M ⊙ ( R/R ⊙ ) . (2)The situation for ν max is a bit more complicated. As mentioned earlier, ν max is the frequency at which oscillation poweris maximum, and thus should depend on how modes are excited and damped. Unlike the case of ∆ ν , we do not as ofyet have a complete theory explaining the quantity. There are some studies in this regard (e.g., see Belkacem et al.2011, 2013), but the issue has not been fully resolved. As explained in Belkacem (2012) and Belkacem et al. (2011)the maximum in the power spectrum can be attributed to the depression or plateau of the damping rates. This isalso discussed in Houdek et al. (1999), Chaplin et al. (2008), Belkacem et al. (2012), and Appourchaux et al. (2012).This depression of the damping rates can then be related to the thermal time-scale (Balmforth 1992; Belkacem et al.2011; Belkacem 2012) which in turn can be related to ν ac , however there is some additional dependence on the Machnumber (Belkacem et al. 2011; Belkacem 2012). [email protected] ν max carries diagnostic information on the excitation and damping of stellar modes, and hence must depend onthe physical conditions in the near-surface layers where the modes are excited. As assumed in Brown et al. (1991),the frequency most relevant to these regions is the acoustic cut-off frequency, ν ac . The sharp rise in ν ac close to thesurface of a star acts as an efficient boundary for the reflection of waves with ν < ν ac . Brown et al. (1991) argued that ν max should be proportional to ν ac because both frequencies are determined by conditions in the near surface layers.Kjeldsen & Bedding (1995) turned this into a relation linking ν max to near-surface properties by noting that under theassumption of an isothermal atmosphere the acoustic-cutoff frequency can be approximated as ν max ∝ ν ac = c πH , (3)where c is the speed of sound and H the density scale height (which under this approximation is also the pressurescale height). This can be further simplified assuming ideal gas as ν max ∝ ν ac ∝ gT − / , (4)where g is the acceleration due to gravity and T eff the effective temperature. This leads to the ν max scaling relation ν max ν max , ⊙ = (cid:18) MM ⊙ (cid:19) (cid:18) RR ⊙ (cid:19) − (cid:18) T eff T eff , ⊙ (cid:19) − / . (5)While the ν max scaling relation and the relation between ν max and ν ac have not been tested extensively, limited obser-vational studies as well as investigations using stellar models have been performed, suggesting that the approximationsare reasonable (e.g. Bedding & Kjeldsen 2003; Chaplin et al. 2008, 2011; Stello et al. 2008, 2009a; Miglio 2012; Bedding2014; Jim´enez et al. 2015; Coelho et al. 2015).Equations 2 and 5 have been used extensively, directly or indirectly, to determine the surface gravity, mass, radius,and luminosity of stars (e.g. Stello et al. 2008; Bruntt et al. 2010; Kallinger et al. 2010b; Mosser et al. 2010; Basu et al.2011; Chaplin et al. 2011; Silva Aguirre et al. 2011; Hekker et al. 2011; Chaplin et al. 2014; Pinsonneault et al. 2014,etc.). Estimates of stellar properties may be determined from Eqs. 2 and 5 by treating them as two equations withtwo unknowns (assuming T eff is known independently) which leads to RR ⊙ = (cid:18) ν m ax ν max , ⊙ (cid:19) (cid:18) ∆ ν ∆ ν ⊙ (cid:19) − (cid:18) T eff T eff , ⊙ (cid:19) / (6)and MM ⊙ = (cid:18) ν max ν max , ⊙ (cid:19) (cid:18) ∆ ν ∆ ν ⊙ (cid:19) − (cid:18) T eff T eff , ⊙ (cid:19) / . (7)Determining the mass and radius of a star in this manner is known as the “direct” method. Comparisons with radiusdeterminations made by other techniques shows that the asteroseismic radii determined from the scaling relationsusing the direct method hold to about 5% for subgiants, dwarfs, and giants (Bruntt et al. 2010; Huber et al. 2012;Silva Aguirre et al. 2012; Miglio et al. 2012; Miglio 2012; White et al. 2013). Examining red giants within eclipsingbinaries, Gaulme et al. (2016) found radii determined using the direct method to be about 5% too large. Thesedeviations motivate us to investigate the ν max scaling relation since the ∆ ν scaling relation deviations have alreadybeen examined.Masses determined using Eq. 7 have uncertainties on the order of 10–15% (Miglio 2012; Chaplin & Miglio 2013;Chaplin et al. 2014, and references therein), however it is more difficult to test the masses given that there are fewbinaries with asteroseismic data. Brogaard et al. (2012) used eclipsing binaries in NGC 6791 and found the mass ofred giant stars to be lower than the mass derived from studies which used the standard scaling relations (Basu et al.2011; Miglio et al. 2012; Wu et al. 2014). Gaulme et al. (2016) found that for red giant stars within eclipsing binariesthe direct method overestimated mass by around 15%. Epstein et al. (2014) examined 9 metal-poor ([M / H] < − α -rich red giant stars and found their masses calculated from the scaling relation to be higher than expected. Alsoexamining α -enriched red giants, Martig et al. (2015) determined masses using the scaling relations and stellar models.The lower mass limit for each red giant was then converted into a maximum age. Unexpectedly, Martig et al. (2015)found a group of stars in the sample that were both young and α -rich. Additionally, Sandquist et al. (2013), studyingNGC 6819, found that red giant masses in the cluster from asteroseismology are as much as 8% too large whileFrandsen et al. (2013) found indications that the detached eclipsing binary KIC 8410637 red giant star’s mass was lessthan asteroseismology indicated. These results further indicate that there are uncertainties with the direct methodand that the scaling relations in Eqs. 2 and 5 need to be carefully understood.Due to the wide use of the scaling relations in asteroseismology, the accuracy of the ν max and ∆ ν scaling relations arecrucial to obtain a better understanding of stellar properties. The scaling relations are a result of approximations, andare therefore not expected to be completely accurate. The deviation of the ∆ ν scaling relation has been studied quiteextensively (e.g. White et al. 2011; Miglio et al. 2013a; Mosser et al. 2013; Sharma et al. 2016; Guggenberger et al.2016; Rodrigues et al. 2017). It has been shown that the relation ∆ ν ∝ √ ¯ ρ holds only to a level of a few percent, andthat ∆ ν/ √ ¯ ρ instead of being equal to unity is a function of T eff and metallicity. At low log g , there also seems to be adependence of ∆ ν/ √ ¯ ρ on mass. It is easy to get rid of this error in stellar models, all that one needs to do is calculate∆ ν for the models using the oscillation frequencies, rather than Eq. 2. There are two approaches that can be used toaccount for ∆ ν errors in the direct method. One is to “correct” the observed ∆ ν using a correction determined frommodels (Sharma et al. 2016) and the other is to use a temperature and metallicity dependent reference ∆ ν instead ofthe solar value of ∆ ν in Eq. 2. Yıldız et al. (2016) claimed that the deviation of ∆ ν/ √ ¯ ρ from unity could be a resultof changes in the adiabatic index Γ , as this would affect the sound travel time. They found a linear relationshipbetween ∆ ν/ √ ¯ ρ and Γ which they used to tune the scaling relation.Unlike the ∆ ν scaling relation, the ν max scaling relation has not been tested as extensively. Additionally, the testshave been indirect. Coelho et al. (2015) tested the temperature dependence of the ν max scaling relation for dwarfs andsubgiants and determined the classical gT − / scaling held to ≃ T eff that was tested.Yıldız et al. (2016) examined the Γ dependence on ν max and found that the inclusion of a Γ term alone, from thederivation of ν ac , did not improve mass and radius estimates calculated using the scaling relations and in fact mademass and radius estimates worse than the traditional scaling relations (this is examined further in Sec. 3). Yıldız et al.(2016) found that additional tuning of the scaling relations (as a function of Γ ) was needed. Other tests of the ν max scaling relation depend on comparing the radius and mass results obtained by using Eqs. 2 and 5 with those obtainedfrom either detailed modeling of stars (Stello et al. 2009a; Silva Aguirre et al. 2015) or of independently determinedmasses and radii (e.g., Bedding & Kjeldsen 2003; Bruntt et al. 2010; Miglio 2012; Bedding 2014).In addition to the inaccuracies in the scaling relations, there is another problem with using Equations 6 and 7 instellar radius and mass determination. The basic equations (Eqs. 2 and 5) that link ∆ ν and ν max to the mass, radius,and temperature of a star assume that all values of T eff are possible for a star of a given mass and radius. However, theequations of stellar structure and evolution tell us otherwise — we know that for a given mass and radius, only a narrowrange of temperatures are allowed. Additionally, we know that the mass-radius-temperature relationship depends onthe metallicity of a star; the scaling relations do not account for that. Thus, an alternative to using Eqs. 6 and 7 isto perform a search for the observed ∆ ν , ν max , T eff , and metallicity in a fine grid of stellar models and to use theproperties of the models to determine the properties of the star. This is usually referred to as “Grid Based Modeling”(GBM) though it is more correctly a grid-based search and has been used extensively to determine stellar parameters(e.g., Chaplin et al. 2014; Pinsonneault et al. 2014; Rodrigues et al. 2014). There are many different schemes that havebeen used for GBM (e.g., Stello et al. 2009b; Basu et al. 2010; Quirion et al. 2010; Kallinger et al. 2010a; Gai et al.2011; Miglio et al. 2013b; Hekker et al. 2013; Creevey et al. 2013; Serenelli et al. 2013). While grid-based methodsgive more accurate results, they can give rise to model dependencies. Whether one uses the direct method to estimatemasses, radii, and log g , or used GBM, the results can only be as correct as the scaling relations.One of the most important applications of the asteroseismic scaling relations has been in estimating the surface gravityof stars. Spectroscopic surface-gravity measurements are notoriously difficult and inaccurate and affect metallicityestimates. It is becoming quite usual to use asteroseismic log g values as priors before determining the metallicity fromspectra (e.g., Bruntt et al. 2012; Brewer et al. 2015; Buchhave & Latham 2015).In this paper we examine the ν max scaling relation in a similar manner as to how the ∆ ν scaling relation has beentested. We use a set of stellar models to do so. We should note from the outset that we are not testing the basicassumption that ν max ∝ ν ac , which is beyond the scope of this paper, but whether ν ac (and hence ν max ) follows theproportionality in Eq. 4. We also examine the consequence of our results on asteroseismically derived stellar properties,in particular, values of log g , that are used so widely.The rest of the paper is organized as follows: we describe the models and ν max calculations in Section 2, the resultsare presented and discussed in Section 3. The consequences of the results are discussed in Section 4 and we give someconcluding remarks in Section 5. STELLAR MODELS AND ν max CALCULATIONS2.1.
The models
We use a grid of models to examine the ν max scaling relation. The models were constructed with the Yale RotatingEvolutionary Code (YREC) (Demarque et al. 2008) in its non-rotating configuration. Models were created for sevendifferent masses, M = 0 .
8, 1.0, 1.2, 1.4, 1.6, 1.8, and 2.0 M ⊙ beginning at the zero-age main sequence through to thered giant branch. Models were stopped at the point where ν max of the models calculated using Eq. 5 was 3 µ Hz, wherein Eq. 5 we adopted ν max , ⊙ = 3090 µ Hz. For each mass, models were constructed with eight metallicities, [Fe / H] = − − − − − τ relation in the atmosphere and one set with Model Cof Vernazza et al. (1981) (henceforth referred to as the VAL-C atmosphere). In the latter case the atmosphere wasassumed to be isothermal for τ ≤ . / H] = 0 is definedby (
Z/X ) ⊙ = 0 . Y we first constructed two standard solar models (SSM), one with Eddingtonand one with VAL-C atmosphere. The initial Y and Z needed to construct the SSM was translated to the Y – Z relation assuming that Z = 0 when Y has the primordial value of 0.248. Since the construction of the SSMs yieldssolar-calibrated mixing length parameters, those values were used to construct the models of the grid ( α MLT = 1 . N ( p, γ ) O reaction, where theFormicola et al. (2004) rate was used. 2.2. Calculating ν max We calculate ν max for our models assuming that ν max is proportional to the acoustic cut-off frequency, which isthe assumption that leads to the scaling relation in Eq. 5. The acoustic cut-off frequency is the frequency abovewhich waves are no longer trapped within the star. Waves of higher frequency form traveling waves and these highfrequency “pseudo modes” are visible in power spectra. While below ν ac the modes are a sum of Lorentzians nearlyequally spaced in frequency, above ν ac the pseudo mode shapes are more sinusoidal. These pseudo mode peaks arebelieved to be a result of interference between the waves that arrive at the observer having traveled different paths(e.g., Kumar & Lu 1991). These high frequency waves can either travel directly towards the observer, leaving the star,or travel into the star before being reflected and leaving the star. Since the two waves travel different paths on theirway to the observer, this results in constructive or destructive interference (depending on the path length differenceand wavelength) creating peaks in the power spectrum. These pseudo modes can be used to observationally determinethe acoustic cut-off frequencies of stars (Garc´ıa et al. 1998; Jim´enez et al. 2015, etc.).When it comes to the acoustic cut-off frequency of models, there are challenges. The acoustic cut-off divides modesinto those that are trapped inside a star, and the pseudo modes that are not. In the former case, the displacementeigenfunctions decay in the atmosphere, in the latter they do not. However, there is no clear boundary between thetwo, as is demonstrated in Fig. 1. Thus when it comes to the acoustic cut-off frequency of models, one relies on anapproximate theory (see e.g. Gough 1993) that shows that the acoustic cut-off is given by ν = c π H (cid:18) − dHdr (cid:19) , (8)where H is the density scale height. In the case of an isothermal atmosphere, this reduces to the expression in Eq. 3.As is clear from both Eq. 8 and Eq. 3, ν ac is a function of radius. The acoustic cut-off of a model is assumed to bethe maximum value of ν ac close to the stellar surface. The acoustic cut-off frequencies defined by Eq. 8 and Eq. 3 arereasonably similar (see Fig. 2). However, the frequency calculated using Eq. 8 has sharp changes close to the top ofthe convection zone where large variations of the superadiabatic gradient cause large changes in ν ac , making it difficultto determine what the cut-off frequency should be. It is difficult to determine ν ac from the eigenfunctions, since thechange from an exponential decay to an oscillatory nature is not sharp. However, they can guide us. Judging by thebehavior of the eigenfunctions shown in Fig. 1 and comparing the results with what we get as a maximum from Eq. 3for the same models (Fig. 2), using the isothermal approximation to calculate ν max should be adequate. In fact, this Figure 1 . The scaled eigenfunctions of modes with frequencies close to ν ac for (a) a solar model, (b) model of a subgiant ofmass 1.2 M ⊙ and [Fe / H] = − .
25, and (c) model of a red giant of mass 1.4 M ⊙ and [Fe / H] = 0 .
25. The legends indicate thefrequencies, in units of µ Hz, that correspond to the eigenfunctions. Note that the lower-frequency eigenfunctions in each caseshow a linear decay in log r , the higher frequency ones show a more oscillatory nature. Figure 2 . The acoustic cut-off frequency for the three models shown in Fig. 1 calculated as per Eq. 3 (red solid line) andEq. 8 (blue dot-dashed line). The cut-off frequencies for the three models using Eq. 3 are 4.94 mHz, 1.55 mHz and 0.07 mHzrespectively and quite consistent with the change in behavior of the eigenfunctions. is what is usually done.The ν max scaling relation is a proportionality and the Sun is used as the reference; in other words for any givenmodel, we can define the ratio R sc = ν max ν max , ⊙ = (cid:18) MM ⊙ (cid:19) (cid:18) RR ⊙ (cid:19) − (cid:18) T eff T eff , ⊙ (cid:19) − / . (9)For each of our models, we can also define R ac = ν ac ν ac , SSM , (10)where ν ac , SSM is the acoustic cut-off of a standard solar model constructed with the same input physics (particularatmospheres) as the models. The use of a solar model having the same physics to define the ratio allows us to minimize
Figure 3 . The ratio S (Eq. 11) for the sets of models with Eddington atmospheres (top row) and VAL-C atmospheres (bottomrow) plotted as a function of T eff and log g . The different colors and symbols refer to different metallicities. The dashed grayline at S =1 is provided for reference. Note the clear, systematic offset that is a function of metallicity. effects related to improper modeling of the surface layers. If the ν max scaling relation is perfect, the ratio S = R ac R sc (11)will be unity, if not, the scaling relation does not hold. We examine how S behaves in the next section. This S parameter is the inverse of the f ν parameter discussed in Yıldız et al. (2016). RESULTSFigure 3 shows the ratio S plotted as a function of T eff and log g separately for the Eddington and VAL-C models.Two features stand out immediately. First, that there is a metallicity dependence which results in a systematic offsetof S for models with non-solar metallicity. Secondly, that there is a deviation at all metallicities at low T eff and lowlog g , i.e., in evolved models.The origin of the metallicity dependence is easy to understand, and it is somewhat surprising that it has beenneglected for so long, even in grid-based modelling of average asteroseismic data. To understand the effect we need togo back to the origin of the scaling relation.Eq. 3 tells us that ν ac behaves as c/H . But for an isothermal atmosphere H = P/ ( ρg ). Since c ∝ p P/ρ then ν ac ∝ g r ρP . (12)The assumption of an ideal gas law tell us that Pρ = R Tµ , (13)where R is the gas constant, and µ the mean molecular weight. Substitution of Eq. 13 into Eq. 12 gives ν ac ∝ g r µT . (14) T eff (K) S (a) log g (b) Figure 4 . The ratio S calculated using Eq. 15 to calculate ν max for the models with Eddington atmospheres plotted as a functionof T eff (a) and log g (b). Symbols and colors correspond to metallicities as indicated in Fig. 3. The underlying light-gray pointsshow the ratio S calculated using the original scaling relation. Results for VAL-C atmospheres are similar and hence not shown.Note that the systematic offset has disappeared, but there is still a remaining departure from the scaling relation. It should be noted that Jim´enez et al. (2015) did include this term in their work.Does the √ µ term take care of the systematic difference seen for the non-solar metallicity models? To test this werecalculated R sc by modifying Eq. 5 to ν max ν max , ⊙ = (cid:18) MM ⊙ (cid:19) (cid:18) RR ⊙ (cid:19) − (cid:18) T eff T eff , ⊙ (cid:19) − / (cid:18) µµ ⊙ (cid:19) / (15)and calculated S using the resultant modified R sc . The results are shown in Fig. 4. It can be seen clearly that theaddition of the √ µ factor removes the difference between models with different metallicities. One explanation for theusual omission of the µ term is that the abundances of X , Y , and Z for an observed star, and therefore the value of µ ,can be difficult to determine. The application of the modified ν max scaling relation to observed stars will be discussedin Sec. 5. While the importance of the µ term in the ν max scaling relation might seem to contradict what was foundby Yıldız et al. (2016), it should be noted that Yıldız et al. (2016) use models with a much smaller range in µ thanthe models presented in this work.The main contribution to the difference in mean molecular weight between models with different metallicity is causedby differences in helium rather than metals. This means that in models with diffusion we should see a trend in theunmodified S as a function of evolution that is different from that for models without diffusion. To test this weconstructed Eddington models with diffusion for masses of M = 0 .
8, 1.0, 1.2, and 1.4 M ⊙ and compared them totheir corresponding non-diffusion models. The results, for diffusion models with initial metallicity of [Fe / H] = 0 . ν max scaling relation, we need to explicitly use the µ dependence in the expression. The µ term could also beincorporated in the direct method provided that the model’s µ value was known or could be calculated.Since c = Γ P/ρ we should also include a √ Γ in the scaling relation for ν max such that Eq. 15 becomes ν max ν max , ⊙ = (cid:18) MM ⊙ (cid:19) (cid:18) RR ⊙ (cid:19) − (cid:18) T eff T eff , ⊙ (cid:19) − / (cid:18) µµ ⊙ (cid:19) / (cid:18) Γ Γ , ⊙ (cid:19) / . (16)Fig. 6 examines the ratio S when R sc is calculated using Eq. 16. As can be seen, the inclusion of the Γ term alsoreduces the differences between the models of different metallicities and lessens the deviations from S = 1.Even when the main deviation from the scaling relation is removed, there is a residual difference at low T eff . Most ofthis deviation can be explained by the fact that the maximum value of ν ac does not occur at r = R but at a differentradius. As seen in Fig. 7, this deviation is significantly lessened if Eq. 16 is modified so that ν max is instead scaled as ν max ν max , ⊙ = (cid:18) M max M max , ⊙ (cid:19) (cid:18) R max R max , ⊙ (cid:19) − (cid:18) T max T max , ⊙ (cid:19) − / (cid:18) µ max µ max , ⊙ (cid:19) / (cid:18) Γ , max Γ , max , ⊙ (cid:19) / , (17) T eff (K) S (a) T eff (K) (b) No DiffusionWith Diffusion
Figure 5 . A comparison of the ratio S for models with and without diffusion of helium and heavy elements. Panel (a) showsthe results for the original scaling relation while Panel (b) shows the results with the µ -term included. For the sake of clarityonly diffusion models of initial metallicity [Fe / H] = 0 . T eff (K) S Figure 6 . The ratio S when the effects of both µ and Γ are included. The underlying light-gray points show the ratio S calculated using the original scaling relation. Symbols and colors correspond to metallicities as indicated in Fig. 3. where R max , M max , T max , µ max , and Γ , max are the radius, mass, temperature, mean molecular weight, and Γ at theradius where ν ac is the maximum (note that for all models M max = M since the atmosphere is usually assumed tobe massless), and R max , ⊙ , M max , ⊙ , T max , ⊙ , µ max , ⊙ , and Γ , max , ⊙ are the same quantities for a solar model with thesame physics. As can be seen in Fig. 7, when taking into account that the maximum value of ν ac is not at r=R, theremaining deviation in the scaling relation is dramatically reduced. To compare the differences between the values of T max and T eff and R max and R , refer to Fig. 8. CONSEQUENCES OF THE ERROR ON THE ν max SCALING RELATIONBecause the ν max scaling relation is used extensively along with the ∆ ν scaling relation to estimate stellar properties,any deviations from the scaling relation will add to systematic errors in the estimates. In this section, using the errorsin the ν max scaling relation implied from our previous analysis, we examine the consequences on stellar log g , radius,and mass estimates.As mentioned earlier, asteroseismic estimates of log g are often used as priors in spectroscopic analyses used to T eff (K) S Figure 7 . The ratio S calculated using Eq. 17 to calculate ν max for the models with Eddington atmospheres. The underlyinglight-gray points show the ratio S from Eq. 16. Note that all systematic errors have been reduced. Symbols and colors correspondto metallicities as indicated in Fig. 3. T eff (K) −0.160−0.158−0.156−0.154−0.152−0.150 ( T m a x − T e ff ) / T e ff (a) T eff (K) ( R m a x − R ) / R (b) Figure 8 . The fractional differences between (a) T max and T eff and (b) R max and R for each model with an Eddington atmosphere.Symbols and colors correspond to metallicities as indicated in Fig. 3. estimate atmospheric properties and parameters. Thus errors in asteroseismic estimates of log g because of ν max errorsis a troubling matter. To test what systematic errors could result, we calculate log g for the models from ν max usingthe usually accepted relation for ν max , but with ν max of the models calculated with the acoustic cut-off frequency, i.e., gg ⊙ = ν ac ν ac , SSM (cid:18) T eff T eff , ⊙ (cid:19) / , (18)and compare that to the actual log g of the models. The results are shown in Fig. 9(a). As can be seen, there is indeeda systematic error, but for the metallicity range of stars for which asteroseismic log g values have been measured, thesystematic error is well within the uncertainty range of data uncertainties (of the order ± .
01 dex). The systematiceffects are somewhat larger in the low temperature range that corresponds to red giants. This error can be made much0 T eff (K) −0.06−0.04−0.020.000.02 l og ( g / g ⊙ ) s c a li n g − l og ( g / g ⊙ ) m o d e l (a) T eff (K) −0.06−0.04−0.020.000.02 l og ( g / g ⊙ ) s c a li n g − l og ( g / g ⊙ ) m o d e l (b) −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 [Fe/H] −0.06−0.04−0.020.000.02 l og ( g ) E q . − l og ( g ) E q . (c) GS98 Model∆Y/∆Z=2, (Z/X) ⊙ from GS98∆Y/∆Z=1, (Z/X) ⊙ from GS98AGSS09 Model∆Y/∆Z=2, (Z/X) ⊙ from AGSS09∆Y/∆Z=1, (Z/X) ⊙ from AGSS09 Figure 9 . (a) The error made in log g estimates when the original scaling relation is used and (b) when the µ term is included(colored points) and when the µ and Γ terms are included (background gray points). Symbols and colors correspond tometallicities as indicated in Fig. 3. (c) The difference between log g estimates if the µ term is included or not, plotted as afunction of [Fe/H] for models with different values of ∆ Y / ∆ Z . In each panel the gray dashed lines at ± g , with a dotted gray line at 0.0 for reference. smaller if a µ term or a µ and Γ term are included, i.e., if g is calculated as gg ⊙ = ν ac ν ac , SSM (cid:18) T eff T eff , ⊙ (cid:19) / (cid:18) µµ ⊙ (cid:19) − / (19)or if also including the Γ term, gg ⊙ = ν ac ν ac , SSM (cid:18) T eff T eff , ⊙ (cid:19) / (cid:18) µµ ⊙ (cid:19) − / (cid:18) Γ Γ , ⊙ (cid:19) − / . (20)The effects of calculating log g using Eq. 19 or 20 can be seen in Fig. 9(b). The addition of the µ term or the µ and Γ terms lessens the deviations between models of different metallicities and brings the value of log g from scaling moreinto agreement with the actual log g values of the models.In Fig. 9(c) we examine the difference in log g estimates if the µ term is included or not as a function of [Fe/H]. Weignore the Γ term here, as determining the value of Γ for an arbitrary star with a given [Fe/H] is not as clear asdetermining µ for that star. So, in Fig. 9(c) we are examining the difference between log g calculated with Eqs. 18 and19. Since the ratio µ/µ ⊙ depends on the Y – Z relationship, we include different values of solar metallicity and differentvalues of ∆ Y / ∆ Z as a function of [Fe/H]. In Fig. 9(c) both the Grevesse & Sauval (1998) value of ( Z/X ) ⊙ = 0 . Z/X ) ⊙ = 0 .
018 (AGSS09) are used. The added uncertainty becauseof the uncertainty in ∆
Y / ∆ Z is small (less than ± .
01 dex) at low metallicity, but increases with an increase inmetallicity. We are yet to gather asteroseismic data for stars with [Fe/H] larger than about 0.5, thus the errors forobserved stars are expected to be quite low and smaller than typical log g uncertainties. While this is a reassuringconfirmation that the original scaling relation has produced trustworthy log g estimates, Fig. 9(b) shows that log g estimates are improved if Eqs. 19 or 20 are used.What of the errors in radius and mass estimates that arise due to deviations in the ν max scaling relation? Using Eq. 6and 7 each model’s radius and mass was determined, where ν max /ν max , ⊙ was calculated using ν ac /ν ac , ⊙ . Here ∆ ν wascalculated using the scaling relation in Eq. 2, as opposed to calculating ∆ ν for each model using mode frequencies.This was done in order to avoid introducing errors in the radius and mass estimates from the ∆ ν scaling errors. Thesame exercise but with ∆ ν values calculated from mode frequencies will be performed later in the paper. However,for now we just want to examine the effects of the errors due to ν max scaling deviations. The results are shown inFigs. 10(a) and 11(a). As can be seen, there are systematic errors in both mass and radius estimates. Errors in bothestimates are reduced substantially when Eqs. 6 and 7 are modified to include the effect of the mean molecular weightand Γ , i.e., RR ⊙ = (cid:18) ν max ν max , ⊙ (cid:19) (cid:18) ∆ ν ∆ ν ⊙ (cid:19) − (cid:18) T eff T eff , ⊙ (cid:19) / (cid:18) µµ ⊙ (cid:19) − / (cid:18) Γ Γ , ⊙ (cid:19) − / , (21)1 T eff (K) −0.04−0.020.000.020.04 ( A c t u a l R a d i u s − I n f e rr e d R a d i u s ) / A c t u a l R a d i u s (a) T eff (K) −0.04−0.020.000.020.04 (b) Figure 10 . (a) The error made in radius estimates when the original ν max scaling relation is used. (b) The same when the ν max scaling relation is modified to include the µ and Γ terms. Symbols and colors correspond to metallicities as indicated inFig. 3. ∆ ν in both cases was calculated using the scaling relation to avoid introducing errors in the radius estimates from the∆ ν scaling errors. T eff (K) −0.15−0.10−0.050.000.050.100.15 ( A c t u a l M a ss − I n f e rr e d M a ss ) / A c t u a l M a ss (a) T eff (K) −0.15−0.10−0.050.000.050.100.15 (b) Figure 11 . Same as Fig. 10 but shows errors in mass estimates. and MM ⊙ = (cid:18) ν max ν max , ⊙ (cid:19) (cid:18) ∆ ν ∆ ν ⊙ (cid:19) − (cid:18) T eff T eff , ⊙ (cid:19) / (cid:18) µµ ⊙ (cid:19) − / (cid:18) Γ Γ , ⊙ (cid:19) − / . (22)The errors in results obtained with these expressions are shown in Figs. 10(b) and 11(b).The total error in radius and mass estimates obtained using the scaling laws are of course a combination of errors inthe ∆ ν scaling relation as well as the ν max scaling relation. To determine what that is, instead of determining radiusand mass using input ∆ ν values calculated using the scaling relation, we calculated the input ∆ ν using the frequenciesof ℓ = 0 modes assuming Gaussian weights around ν max with FWHM of 0 . ν . as from Mosser et al. (2012). Oncethe value of ∆ ν from mode frequencies was calculated for each model, Eq. 21 and 22 were again used to determinethe error in radius and mass estimates. The results are show in Fig. 12 which also shows the errors in radius andmass estimates when the µ and Γ terms are not included. Including the µ and Γ terms helps reduce the deviationssomewhat, but there is still substantial error in mass ( ± − ± −0.20−0.15−0.10−0.050.000.050.100.15 (a) −0.20−0.15−0.10−0.050.000.050.100.15 (b) T eff (K) −0.20−0.15−0.10−0.050.000.050.100.15 (c) T eff (K) −0.20−0.15−0.10−0.050.000.050.100.15 (d) ( A c t u a l R a d i u s − I n f e rr e d R a d i u s ) / A c t u a l R a d i u s ( A c t u a l M a ss − I n f e rr e d M a ss ) / A c t u a l M a ss Figure 12 . The combined effect of the deviation of both ∆ ν and ν max on radius (a,c), and mass (b,d) estimates. The upperpanels (a,b) show the fractional differences using the original scaling relation and the lower panels (c,d) show the deviationsonce the µ and Γ terms are taken into account. Symbols and colors correspond to metallicities as indicated in Fig. 3.5. DISCUSSION AND CONCLUSIONSWe used a large set of models to test how well the ν max scaling holds, and find that just as in the case of ∆ ν , thereare significant departures from the scaling law. The largest source of the deviation is the neglect of the mean molecularweight and Γ terms when approximating the acoustic cut-off frequency. The deviations in the scaling relations causesystematic errors in estimates of log g , mass, and radius. The errors in log g are however, well within errors caused bydata uncertainties and are therefore not a big cause for concern, except at extreme metallicities.The results from our work would suggest we should start using the µ and Γ terms explicitly in the scaling relation,as in Eq. 16. Additionally, when using the scaling relations to determine radius and mass the µ and Γ terms shouldbe included, as in Eqs. 21 and 22. For stellar models, ideally the best method is to use the actual value of µ and Γ calculated in each model. For models where the values can be determined the modified scaling relation can be easilyimplemented. For models where Γ is not readily accessible, we suggest still including the µ term in the ν max thescaling relation, which as seen in Fig. 4 is an improvement over the traditional ν max scaling relation.For observational data, incorporating these terms is not as straight forward. Even ignoring the Γ term and justdetermining µ for an observed star is complicated. One possible way to implement the µ term into the scaling relationfor observed stars would be to create stellar models and estimate the value of µ in this manner. For observed starsimplementing the modified ν max scaling relation into the direct method (Eqs. 21 and 22) is not recommended due tothe difficulty of determining the Γ and µ terms from observational data. However, for observed stars a grid basedmethod gives more precise estimates of radius and mass and should be used over the direct method. So, the difficultyin applying this result to observed stars is less critical.Furthermore, we should treat the ν max scaling the way we have begun to treat ∆ ν scaling, i.e., either calculatecorrections to the relation or determine a reference ν max that depends on T eff to replace ν max , ⊙ as the constant ofproportionality. For the non-diffusion Eddington atmosphere models a correction formula as a function of T eff isprovided in Appendix B. For grid-based modeling, we would suggest that ν max for the grid of models be calculatedfrom the ratio ν ac /ν ac , SSM to avoid most of the systematic errors.The authors would like to thank Joseph R. Schmitt for the use of some of the software he had written and AndreaMiglio for helpful comments and suggestions. This work has been supported by NSF grant AST-1514676 and NASAgrant NNX16AI09G to SB. WJC, GRD, and YE acknowledge the support of the UK Science and Technology FacilitiesCouncil (STFC). Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation(Grant DNRF106).
Software:
YREC (Demarque et al. 2008)3 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 [Fe/H] −0.12−0.10−0.08−0.06−0.04−0.020.000.02 ( ν m a x − ν m a x , c o rr ec t e d ) / ν m a x , c o rr ec t e d GS98 Model∆Y/∆Z=2, (Z/X) ⊙ from GS98∆Y/∆Z=1, (Z/X) ⊙ from GS98AGSS09 Model∆Y/∆Z=2, (Z/X) ⊙ from AGSS09∆Y/∆Z=1, (Z/X) ⊙ from AGSS09 Figure A1 . The fractional difference between the traditional ν max value and the µ corrected ν max as a function of [Fe / H] fordifferent values of ∆
Y / ∆ Z and ( Z/X ) ⊙ . APPENDIX A. FRACTIONAL DIFFERENCE BETWEEN THE ORIGINAL AND MODIFIED SCALING RELATION AS AFUNCTION OF [Fe/H]The effects of the µ correction can be seen if we plot the fractional difference between the traditional value of ν max ,as in Eq. 5, and the value of ν max , corrected which includes the µ term as in Eq. 15. So, examining ν max − ν max , corrected ν max , corrected . Bycomparing Eq. 5 and Eq. 15 it can be seen that ν max − ν max , corrected ν max , corrected = ( µ/µ ⊙ ) − / − . (A1)The fractional difference between the traditional ν max and ν max , corrected is shown in Fig. A1 as a function of [Fe / H] fordifferent values of ∆
Y / ∆ Z and ( Z/X ) ⊙ . B. ν max CORRECTION AS A FUNCTION OF TEMPERATUREHere we provide a correction formula for the non-diffusion Eddington models from Fig. 3, solely as a function of T eff .For the Eddington atmosphere models, Figure B2 plots the difference between ν max determined from the acoustic-cutoff frequency and ν max determined using Eq. 16 (which includes the µ and Γ terms) as a function of T eff . A fifthorder polynomial was fit to the data, giving the relationship, (cid:18) ν max ν max , ⊙ (cid:19) ac − (cid:18) ν max ν max , ⊙ (cid:19) Eq . = (4 . × − ) T − (1 . × − ) T + (1 . × − ) T − (4 . × − ) T + (0 . T eff − (1 . . (B2)While this correction formula may be useful, the best and most accurate method to apply the µ and Γ corrections isto calculate µ and Γ for individual models and apply Eq. 16 to calculate ν max .REFERENCES Adelberger, E. G., Austin, S. M., Bahcall, J. 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