Calculating asteroseismic diagrams for solar-like oscillations
Timothy R. White, Timothy R. Bedding, Dennis Stello, Jørgen Christensen-Dalsgaard, Daniel Huber, Hans Kjeldsen
aa r X i v : . [ a s t r o - ph . S R ] S e p Accepted by ApJ
Preprint typeset using L A TEX style emulateapj v. 11/10/09
CALCULATING ASTEROSEISMIC DIAGRAMS FOR SOLAR-LIKE OSCILLATIONS
Timothy R. White , Timothy R. Bedding , Dennis Stello , Jørgen Christensen-Dalsgaard , Daniel Huber ,and Hans Kjeldsen Accepted by ApJ
ABSTRACTWith the success of the
Kepler and
CoRoT missions, the number of stars with detected solar-likeoscillations has increased by several orders of magnitude, for the first time we are able to perform large-scale ensemble asteroseismology of these stars. In preparation for this golden age of asteroseismologywe have computed expected values of various asteroseismic observables from models of varying massand metallicity. The relationships between these asteroseismic observables, such as the separationsbetween mode frequencies, are able to significantly constrain estimates of the ages and masses of thesestars. We investigate the scaling relation between the large frequency separation, ∆ ν , and mean stellardensity. Furthermore we present model evolutionary tracks for several asteroseismic diagrams. Wehave extended the so-called C-D diagram beyond the main sequence to the subgiants and the red-giantbranch. We also consider another asteroseismic diagram, the ǫ diagram, which is more sensitive tovariations in stellar properties at the subgiant stages and can aid in determining the correct modeidentification. The recent discovery of gravity-mode period spacings in red giants forms the basisfor a third asteroseismic diagram. We compare the evolutionary model tracks in these asteroseismicdiagrams with results from pre- Kepler studies of solar-like oscillations, and early results from
Kepler . Subject headings: stars: fundamental parameters — stars: interiors — stars: oscillations INTRODUCTION
Asteroseismology promises to expand our knowledgeof the stars through the study of their oscillations. Thispromise has driven efforts to measure oscillations in solar-type stars with ground-based observations, but the re-quirement for precise measurements (at the level of m s − in radial velocity) that are well-sampled over a long pe-riod of time have limited the number of detections toonly a handful of stars (see Aerts et al. 2008; Bedding2011, for recent reviews).Space-based missions are ideal to ensure continuousdata sets and the CoRoT satellite has measured oscilla-tions in several stars (e.g. Michel et al. 2008). The
Ke-pler Mission is set to revolutionize the study of oscil-lations in main-sequence and subgiant stars by increas-ing the number of stars with high-quality observationsby more than two orders of magnitude (Gilliland et al.2010; Chaplin et al. 2011).With the large number of stars observed by
Kepler itbecomes possible to perform ensemble asteroseismologyof stars with solar-like oscillations. This includes con-structing asteroseismic diagrams, in which different mea-surements of the oscillation spectrum are plotted againsteach other, revealing features that are dependent uponthe stellar structure.For acoustic modes of high radial order, n , and lowangular degree, l , frequencies are well-approximated bythe asymptotic relation (Vandakurov 1967; Tassoul 1980; Sydney Institute for Astronomy (SIfA), School ofPhysics, University of Sydney, NSW 2006, Australia;[email protected] Australian Astronomical Observatory, PO Box 296, EppingNSW 1710, Australia Danish AsteroSeismology Centre (DASC), Department ofPhysics and Astronomy, Aarhus University, DK-8000 Aarhus C,Denmark
Gough 1986): ν n,l ≈ ∆ ν (cid:18) n + l ǫ (cid:19) − δν l . (1)Here, ∆ ν is the so-called large separation between modesof the same l and consecutive n , while δν l is the smallseparation between modes of different l , and ǫ is a di-mensionless offset. To a good approximation, ∆ ν isproportional to the square root of the mean density ofthe star (Ulrich 1986) and in Section 3 we investigatethe validity of this approximation. The small separa-tions, δν l , are sensitive to the structure of the core andhence to the age of the star, at least on the main se-quence. These somewhat orthogonal dependencies leadsto their use in the so-called C-D diagram, in which thelarge and small separations are plotted against each other(Christensen-Dalsgaard 1984). Calculating the C-D dia-gram is one of the main aims of this paper.Previous studies of the C-D diagram and its vari-ations have determined the expected evolution ofstars with varying mass and metallicity (Ulrich 1986;Gough 1987; Christensen-Dalsgaard 1988), and as-sessed the feasibility of applying the diagram to realdata (Monteiro et al. 2002; Ot´ı Floranes et al. 2005;Mazumdar 2005; Gai et al. 2009). However, none ofthese studies followed the evolution beyond the end ofthe main sequence. Recently, Montalb´an et al. (2010)computed the theoretical spectrum of solar-like oscilla-tions in red-giant stars, finding that the small separation δν depends almost linearly on ∆ ν , in agreement withthe red-giant results from Kepler (Bedding et al. 2010a;Huber et al. 2010). In Section 4 we bridge the gap, ex-tending the C-D diagram beyond main-sequence stars tothe subgiants and up towards the tip of the red-giantbranch.A complication with the C-D diagram for subgiants WHITE ET AL.and red-giant stars arises from mode bumping. As starsevolve, the convective envelope expands and the acousticoscillation modes ( p modes) decrease in frequency. Atthe same time, g -mode oscillations that exist in the coreof the star increase in frequency as the core becomesmore centrally condensed. Eventually, p - and g -modefrequencies overlap, resulting in oscillation modes thathave a mixed character, behaving like g modes in thecore and p modes in the envelope. The frequencies ofthese modes are shifted as they undergo avoided cross-ings (Osaki 1975; Aizenman et al. 1977), which leads tosignificant deviations from the asymptotic relation, equa-tion (1). This so-called mode bumping only affects non-radial modes, particularly l =1 but also l =2, and so itcomplicates the measurement of the small separations.Nevertheless, as we show, it is still possible to measureaverage separations that can be plotted in the C-D dia-gram.In this paper we also discuss an asteroseismic diagramthat uses the quantity ǫ (Section 5). Despite being in-vestigated by Christensen-Dalsgaard (1984), this dimen-sionless phase offset has since been largely overlooked forits diagnostic potential. Recently, Bedding & Kjeldsen(2010) suggested that it could be useful in distin-guishing odd and even modes when their identifi-cations are ambiguous due to short mode lifetimes(see, e.g., Appourchaux et al. 2008; Benomar et al. 2009;Bedding et al. 2010b). Using Kepler data, Huber et al.(2010) have found that ǫ and ∆ ν are related in red giants,implying that, like ∆ ν , ǫ is a function of fundamental pa-rameters. A similar analysis was done for CoRoT data byMosser et al. (2011b). We discuss the use of ǫ for modeidentification in Section 6.Finally, we discuss an asteroseismic diagram for red-giant stars. A recent breakthrough has been made withthe discovery of sequences of mixed modes in Kepler redgiants (Beck et al. 2011). Because these mixed modesexhibit g -mode behavior in the core of the star, theyare particularly sensitive to the core structure. Sub-sequently, Bedding et al. (2011) used the observed pe-riod spacings of these mixed modes, ∆ P obs , to distin-guish between red giants that are burning helium in theircore and those that are still only burning hydrogen in ashell. Mosser et al. (2011a) have found similar results in CoRoT red giants. In Section 7 we present the expectedevolution of ∆ P obs with ∆ ν in an asteroseismic diagramfor red giant stars. MEASURING ASTEROSEISMIC PARAMETERS FROMMODELS
A grid of 51000 stellar models was calculated fromthe ZAMS to almost the tip of the red-giant branchusing
ASTEC (Christensen-Dalsgaard 2008a) with theEFF equation of state (Eggleton et al. 1973). Weused the opacity tables of Rogers & Iglesias (1995) andKurucz (1991) for
T < K, with the solar mixtureof Grevesse & Noels (1993). Rotation, overshooting anddiffusion were not included. The grid was created withfixed values of the mixing-length parameter ( α = 1 . X i = 0 . . ⊙ witha resolution of 0 .
01 M ⊙ and metallicities in the range0 . ≤ Z ≤ .
028 with a resolution in log(
Z/X ) of0.2 dex. Figure 1 shows the H-R diagram for mod-
Fig. 1.—
H-R diagram of models with near-solar metallicity( Z = 0 . . ⊙ (magenta). Thesection of tracks that are gray are hotter than the approximate cooledge of the classical instability strip (Saio & Gautschy 1998). Thezero-age main sequence is indicated by the dotted red line. Otherred lines (solid, long dashed, dot-dashed and short dashed) relateto features in Figure 4. els of near-solar metallicity ( Z = 0 . . ⊙ . Models that are hotter than theapproximate cool edge of the classical instability strip(Saio & Gautschy 1998) are colored gray; they are notexpected to show solar-like oscillations because they donot have a significant convective envelope. Many of thesestars will show classical pulsations as δ Scuti, γ Doror roAp stars. However, we do note that Antoci et al.(2011) announced evidence for solar-like oscillations in a δ Scuti star.The model parameters chosen, and the physics in-cluded in the models, do have an impact on the modelfrequencies obtained. Several studies have already inves-tigated the impact of the choice of these parameters, in-cluding composition, mixing, overshooting and diffusion(e.g. Monteiro et al. 2002; Mazumdar 2005; Gai et al.2009), so we have not investigated the breadth of this pa-rameter space with our model grid. Our aim is to investi-gate the bulk behavior of asteroseismic observables acrossa wide range of evolutionary states, from the ZAMS tothe tip of the RGB.The adiabatic frequencies of every model were calcu-lated using
ADIPLS (Christensen-Dalsgaard 2008b), ad-justed to enable proper sampling of the extremely highorder eigenmodes that occur in red giants. Oscillationfrequencies determined from stellar models include moremodes than can be observed. It is therefore necessary todetermine asteroseismic parameters from them with careso that they are directly comparable to the parametersmeasured from data. Here we outline our approach.
Measuring ∆ ν and ǫ It is important that we treat the frequencies from mod-els and data the same, as much as possible, so that thecomparisons between the two may be validly made. Wetherefore consider the observed characteristics of oscilla-tions when deciding on a method for fitting to frequen-STEROSEISMIC DIAGRAMS 3
Fig. 2.—
An example of fitting the l = 0 ridge of a subgiantmodel. This model has M = 1 . ⊙ , Z = 0 . τ = 10 .
78 Gyr.The l = 0 modes are plotted with circles, l = 1 with trianglesand l = 2 with squares. The vertical red line shows the fit to theradial modes. The horizontal dashed red line indicates ν max , andthe dotted blue lines indicate the FWHM of the Gaussian envelopeused in the fit. Clearly visible is the distortion to the l = 1 ridgedue to avoided crossings that would complicate the measurement of δν . Filtering the l = 2 modes by their mode inertia successfullyremoves the modes most affected by avoided crossings (open graysquares). Fig. 3.—
An example of fitting the l = 2 ridge according toequation (3) for the same model as used in Figure 2. Pairwise sep-arations included in the fit are indicated by filled squares. Onceagain, filtering the l = 2 modes by their mode inertia successfullyremoves from the fit the modes most affected by avoided crossings.The near-horizontal red line shows the fit to the pairwise separa-tions. The dashed red line indicates ν max , and the dotted bluelines indicate the FWHM of the Gaussian envelope used in the fit. cies that can be consistently applied to both models andobservations.The amplitudes of solar-like oscillations are modu-lated by an envelope that is approximately Gaussian.The peak of this envelope is at ν max , the frequency ofmaximum power. Since it is near ν max that the os-cillations have the most power, and are therefore themost easily observed, we chose to measure the oscilla- tion properties about this point. To determine ν max formodels we used the scaling relation (Brown et al. 1991;Kjeldsen & Bedding 1995), ν max ν max , ⊙ = M/ M ⊙ ( T eff / T eff , ⊙ ) . L/ L ⊙ . (2)We must then choose which model frequencies to in-clude in our calculation of asteroseismic parameters. Wecould simply take the frequencies within a specified rangearound ν max , and fit to these frequencies. However, thiscauses difficulties when, as ν max varies, frequencies atthe top and bottom fall in and out of this range fromone model to the next. This will result in jumps in thederived quantities that are not physical. To overcomethis, we instead performed a weighted fit, using weightsthat decrease towards zero away from ν max . Inspired bythe approximately Gaussian envelope of the oscillationamplitudes, we weighted the frequencies by a Gaussianwindow centered on ν max . The width of this Gaussianneeds to be selected appropriately. The window shouldnot be so wide as to include model frequencies that areunlikely to be observed. On the other hand, a narrowerwindow is sensitive to departures from the asymptoticrelation as a result of acoustic glitches. We have foundthat a full-width-at-half-maximum of 0 . ν max is a goodcompromise.To measure ∆ ν and ǫ , we performed a weighted least-squares fit to the radial ( l = 0) frequencies as a functionof n . By equation (1), the gradient of this fit is ∆ ν and the intercept is ǫ ∆ ν . An example of a fit to the l = 0 frequencies of a stellar model is shown in Figure2 in ´echelle format, in which the frequencies are plottedagainst frequency modulo ∆ ν . In the ´echelle diagram,frequencies that are separated by precisely ∆ ν will alignvertically. The curvature in the l = 0 and l = 2 ridgesindicates variation in either ∆ ν or ǫ (or both) as a func-tion of frequency. Our choice of the fitting method couldpotentially impact on our measured values of ∆ ν and ǫ .We have tried different widths for the Gaussian envelopeand found that the measured value of ∆ ν does not varysignificantly. However, there is a substantial change inthe value of ǫ due to the influence of acoustic glitches(Gough 1990; Houdek & Gough 2007). As the Gaussianis made wider, more orders contribute to the fit, averag-ing over the curvature in the ´echelle diagram. NarrowerGaussians are more susceptible to curvature, which leadsto a significant change in ǫ , which in extreme cases canapproach a shift of 0.2 in ǫ . This is a greater concern forhigher-mass stars, for which models exhibit greater cur-vature. Our chosen Gaussian window (0 . ν max ) is wideenough to average over much of the curvature, while en-suring that no more frequencies are included in the fit tothe models than can reasonably be expected to be ob-served. More sophisticated approaches, involving a fit tothe curvature, are beyond the scope of this paper. Measuring δν There are several small separations that can be mea-sured. Only modes with l ≤ l = 3 modes are not easily observed. The l = 1 modes in subgiants can be significantly shifted infrequency due to avoided crossings, as exhibited by the WHITE ET AL. Fig. 4.—
The ratio ∆ ν/ ∆ ν ⊙ to ( ρ/ρ ⊙ ) / as a function of ∆ ν in models of near-solar metallicity ( Z = 0 . . ⊙ (magenta). Models that have effectivetemperatures hotter than the approximate cool edge of the classicalinstability strip are shown in gray. The zero age main sequence isindicated by the dotted red line. Sharp features in the tracks areindicated by the solid, long-dashed, dot-dashed and short-dashedred lines, which, for reference, are also labeled as such in Figure 1.The location of a solar model is marked by the Sun’s usual symbol. model in Figure 2. Although the l = 2 modes also un-dergo avoided crossings, those modes that are bumpedsignificantly in frequency have much greater inertia thannon-bumped modes and hence much smaller amplitudes.This contrasts with mixed l = 1 modes, for which evenstrongly bumped modes retain observable amplitudes. Itfollows that the l = 2 ridge in the ´echelle diagram is quitewell-defined, even in subgiant stars. For this reason, it isthe small separation δν , between modes of l = 0 and 2,that we have chosen to measure. To minimize the effectof bumped modes on our measurement of δν from themodels, we ignored those l = 2 modes (gray squares inFigure 2) whose inertia was significantly larger than thatof adjacent non-bumped modes, typically by at least anorder of magnitude.Another consideration is that δν decreases with fre-quency. This can be seen in the gradual decrease of theseparations between l = 0 and l = 2 modes, as shown inFigure 3. Guided by the Sun, for which δν decreasesapproximately linearly with frequency (Elsworth et al.1990), we fitted the changing small separation by δν = h δν i + k ( ν n, − ν max ) , (3)where k is a slope to be determined by the fit. Weweighted this fit by the same Gaussian envelope used inSection 2.1. As for ∆ ν , we did not find any significantchange in the determined value of δν when we variedthe envelope width. SCALING RELATION FOR ∆ ν Before discussing the C-D diagram, we shall first ex-plore the scaling relation for the large separation. Aspreviously mentioned, it is well-established that ∆ ν is ap-proximately proportional to the square root of the mean Fig. 5.—
The ratio ∆ ν/ ∆ ν ⊙ to ( ρ/ρ ⊙ ) / as a function of ef-fective temperature, T eff in models of near-solar metallicity ( Z =0 . . ⊙ (magenta). Thelocation of a solar model is marked by the Sun’s usual symbol. Thedashed black line shows the function given by equation 6. Fig. 6.—
The ratio ∆ ν/ ∆ ν ⊙ to ( ρ/ρ ⊙ ) / as a function of ef-fective temperature, T eff in metal-poor ( Z = 0 . Z = 0 . . ⊙ . The location of a solar model is marked by the Sun’s usualsymbol. density of the star. This leads to the scaling relation ρ ≈ (cid:18) ∆ ν ∆ ν ⊙ (cid:19) ρ ⊙ , (4)where ρ is the mean density of a star, ∆ ν is its large sepa-ration, and ρ ⊙ and ∆ ν ⊙ are the corresponding quantitiesfor the Sun. This rather simple relation allows an esti-mate of the mean density of any star for which ∆ ν canbe measured.Given its widespread use, it is important to validateequation (4). Observationally, this can only be done forthe handful of stars whose masses and radii have beenmeasured independently. However, we can at least askSTEROSEISMIC DIAGRAMS 5 TABLE 1Measurements of ∆ ν , δν and ǫ from published frequency lists. Star ν max ∆ ν δν ǫ Source( µ Hz) ( µ Hz) ( µ Hz) τ Cet 4500 169.47 ± ± ± α Cen B 4100 161.70 ± ± ± ± ± ± ± a ± a — Chaplin et al. (2010) α Cen A 2400 105.72 ± ± ± ± ± ± ± a ± a — Chaplin et al. (2010) µ Ara 2000 89.68 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± β Hyi 1000 57.34 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± η Boo 750 40.52 ± ± ± ± ± ± ± ± ± ± ± ± Kepler
Giants — 2 – 20 a a a Huber et al. (2010) a Value taken directly from source without fitting to a frequency list. how accurately equation (4) is followed by models. De-tailed model calculations have confirmed the scaling re-lation for main-sequence stars of low mass ( < . ⊙ ;Ulrich 1986). Stello et al. (2009) investigated the rela-tion up to 2 . ⊙ by calculating ∆ ν from the integralof the sound speed, [2 R d r/c ] − . However, they onlyshowed the relation between [2 R d r/c ] − and ∆ ν mea-sured from the frequencies for a restricted set of models.Basu et al. (2010) have also compared ∆ ν as calculatedfrom the scaling relation to the average ∆ ν from cal-culated model frequencies for a representative subset ofmodels of various masses and evolutionary states. Whilethey found the scaling relation broadly applies, they didnot investigate smaller scale deviations. With our calcu-lations of ∆ ν , we are able to investigate the validity ofthe scaling relation more directly over a broader range ofmasses and evolutionary states.We show in Figure 4 the ratio of ∆ ν/ ∆ ν ⊙ to ( ρ/ρ ⊙ ) / as a function of ∆ ν for models of mass 0.7–2.0 M ⊙ and initial metallicity Z = 0 . ν ⊙ froma solar model rather than the observed value in theSun. These values differ due to the offset between ob-served and computed oscillation frequencies. This off-set is known to arise from an improper modeling ofnear-surface layers (Christensen-Dalsgaard et al. 1988;Dziembowski et al. 1988; Christensen-Dalsgaard et al.1996; Christensen-Dalsgaard & Thompson 1997) andis presumably a problem for other stars as well(Kjeldsen et al. 2008). The offset increases with fre-quency, at least in the Sun, and so affects the large sepa-ration, with ∆ ν being ∼
1% greater in solar models thanobserved (Kjeldsen et al. 2008). We therefore adopted∆ ν ⊙ = 135 . µ Hz, derived from a fit to frequencies ofthe well-studied model S of Christensen-Dalsgaard et al.(1996). We note that the surface offset also has a signif-icant effect on δν (Roxburgh & Vorontsov 2003).From Figure 4 we can see that the scaling relation holdsquite well for lower-mass main-sequence stars, but thereis a significant deviation at other masses and evolution-ary states. These deviations can be as large as 3% forlow-mass red giants and are over 10% within the insta-bility strip.In Figure 5 we show the ratio against effective tem-perature, T eff . From this we see that the deviation fromthe scaling relation is predominantly a function of effec-tive temperature, which suggests that the scaling relationcould be improved by incorporating a function of T eff . Arough approximation to the models suggests a variationof the scaling relation of the form ρρ ⊙ = (cid:18) ∆ ν ∆ ν ⊙ (cid:19) ( f ( T eff )) − , (5) WHITE ET AL. Fig. 7.—
C-D diagram, with model tracks for near-solar metallicity ( Z = 0 . . ⊙ from 0 . ⊙ to2 . ⊙ as labeled. The section of the evolutionary tracks in which the models have a higher T eff than the approximate cool edge of theclassical instability strip (Saio & Gautschy 1998) are gray; they are not expected to show solar-like oscillations. Dashed black lines areisochrones, increasing by 2 Gyr from 0 Gyr (ZAMS) at the top to 12 Gyr at the bottom. Stars shown, as labeled, were observed by either CoRoT (orange triangles),
Kepler (red circles) or from the ground (purple diamonds). Gray circles are
Kepler red giants (Huber et al.2010). The Sun is marked by its usual symbol. where f ( T eff ) = − . (cid:18) T eff K (cid:19) + 4 . (cid:18) T eff K (cid:19) − .
35 (6)for stars with temperatures between 4700 and 6700 K,except main-sequence stars below ∼ . ⊙ , for whichit seems best to set f ( T eff ) = 1. The function in equa-tion (6) is the black dashed line shown in Figure 5. Mak-ing this adjustment improves the accuracy of the scalingrelation to approximately 1% for these models.Does metallicity also have an impact on the scalingrelation? In Figure 6 we show the ratio of ∆ ν/ ∆ ν ⊙ to( ρ/ρ ⊙ ) / as a function of temperature for metal-poor( Z = 0 . − . Z = 0 . . ν . A full investigation of this must await im-provements to the models, but meanwhile we recommendthat using equation (5) (or equation (4)) to estimate stel- lar densities from observed frequencies be based on the observed value of ∆ ν ⊙ (135 . µ Hz). THE C-D DIAGRAM
The C-D diagram is shown in Figure 7. The solidlines show the evolution of ∆ ν and δν for models witha metallicity close to solar and various masses. Starsevolve from the top-right of the diagram to the bottom-left. Isochrones are also shown as dashed lines. In ap-plying this diagram it should be recalled that both ∆ ν and δν are affected by the errors in the treatment ofthe near-surface layers. Modeling indicates that, e.g.,the ratio of δν to ∆ ν is less sensitive to surface layereffects (Roxburgh & Vorontsov 2003; Ot´ı Floranes et al.2005; Mazumdar 2005). Figure 8 shows a modified C-D diagram, which uses this frequency-separation ratio,although the surface dependency remains in ∆ ν . Theisochrones are close to horizontal in this figure, show-ing that this ratio is an effective indicator of age. Wenote that ∆ ν is typically measured from the l = 1 modeswhen calculating this ratio, but since the l = 1 modes de-part significantly from the asymptotic relation for moreevolved stars, we have determined ∆ ν using only l = 0STEROSEISMIC DIAGRAMS 7 Fig. 8.—
Modified C-D diagram using the ratio δν /∆ ν , with near-solar metallicity ( Z = 0 . Z = 0 . Z = 0 . modes. In the absence of avoided crossings, the differencebetween ∆ ν as measured from l = 0 and l = 1 modes issmall, so we expect this change in the definition of theratio δν /∆ ν to have little impact.Variations on the C-D diagram may be constructedby using different small separations in place of δν .Mazumdar (2005) and Montalb´an et al. (2010) have in-vestigated the C-D diagram using δν for main-sequenceand RGB stars, respectively. For subgiants, δν be-comes poorly defined due to avoided crossings causing amajor departure from the asymptotic relation (see Fig-ure 2 and Metcalfe et al. 2010; Campante et al. 2011;Mathur et al. 2011, for examples). We have thereforenot considered the δν –∆ ν C-D diagram here.The C-D diagram is clearly most useful for main-sequence stars, particularly for masses < . ⊙ , forwhich the evolutionary tracks are well separated. Asstars evolve off the main sequence, their tracks convergefor the subgiant and red-giant evolutionary stages. Thisconvergence of the tracks means that the C-D diagram isnot a good discriminant of age and mass for these stars.This behavior of the model tracks is consistent with earlyresults of red giants observed by Kepler (Bedding et al.2010a; Huber et al. 2010) and the modeling results ofMontalb´an et al. (2010), for which it was found that δν is an almost fixed fraction of ∆ ν . This also explains theobservation by Metcalfe et al. (2010), when modeling the Kepler subgiant KIC 11026764, that including δν in thefit to the models did not provide an additional constraintbeyond that provided by ∆ ν .To compare the models with observations we have mea-sured ∆ ν , δν and ǫ from the published frequency listsof 20 stars using the methods outlined in Section 2. Themethod described above was used for calculating δν ex-cept that, apart from the Sun, the data did not justifythe inclusion of k as an extra parameter in the fit tothe frequencies. We have therefore kept k fixed at thesolar value ( − . δν , or its uncertainty. The measured values are listedin Table 1, along with the values adopted for ν max whenfitting the frequencies of each star. Note that for two Kepler main-sequence stars and 470 red giants we usedpublished ∆ ν and δν values.For three of the stars, the correct mode identifica-tion is ambiguous, and so we report both of the pos-sible scenarios. In HD 49933, Scenario B is now acceptedas the correct identification (Benomar et al. 2009), andScenario 1 in HD 181420 seems most likely (Barban et al.2009; Mosser & Appourchaux 2009; Bedding & Kjeldsen WHITE ET AL. Fig. 9.—
C-D diagram, with model tracks for stars that are metal poor ( Z = 0 . Z = 0 . T eff than the approximate cool edge of the classicalinstability strip are not shown. Black symbols are for stars observed by CoRoT (triangles),
Kepler (circles) and from the ground (diamonds).Gray circles are
Kepler red giants (Huber et al. 2010).The Sun is marked by its usual symbol. ν and δν , the mass can be determined to a precision of a fewpercent, assuming that other model parameters, such asthe mixing-length parameter, are valid for the stars. Ageis more difficult to constrain, owing to the considerablylarger fractional uncertainty in δν . Extreme cases are α Cen B and τ Ceti, which have large uncertainties in δν due to having few reported l = 0 , Kepler stars alreadyhave quite small uncertainties in the small separation,and this will further improve as more data are collected.Until now, we have only considered the C-D dia-gram for models with near-solar metallicity. As shownby previous studies, the metallicity has a major im-pact on the tracks, as does the physics in the modelsused (Ulrich 1986; Gough 1987; Christensen-Dalsgaard1988; Monteiro et al. 2002; Ot´ı Floranes et al. 2005;Mazumdar 2005; Gai et al. 2009). The C-D diagram fora series of metal-poor models ( Z = 0 . Z = 0 . l = 2 modescan affect the measured small separation. Furthermore,the expected shift in the small separation due to metal-licity differences is similar in magnitude to the presentmeasurement uncertainties in the small separation, lim-iting the usefulness of the small separation as a proxy formetallicity in subgiants. THE ǫ DIAGRAM
Having extended the C-D diagram beyond the main se-quence, we have found that the evolutionary tracks con-verge for stars of different masses during the subgiantand red-giant phases. We now discuss the ǫ diagram,which breaks this degeneracy to some extent.Figure 10 shows the ǫ diagram with evolutionary tracksfor models of mass 0.7–2.0 M ⊙ and Z = 0 . Fig. 10.—
The ǫ diagram, with near-solar metallicity ( Z = 0 . . ⊙ from 0 . ⊙ to2 . ⊙ (green to magenta lines). The section of the evolutionary tracks hotter than the cool edge of the classical instability strip are gray.For clarity, isochrones are not shown. Symbols for stars are the same as for Figure 7. evolve from the top to the bottom in this diagram. Unlikein the C-D diagram, the evolutionary tracks in the ǫ di-agram remain well separated for subgiants. This raisesthe possibility of using this diagram to constrain massand age. However, some difficulties arise that make thischallenging, as we now discuss.There can be a large uncertainty in the measurementof ǫ , as is apparent for several stars shown in Figure10. This cannot be readily overcome by obtaining higherquality data because it is often due to the intrinsic curva-ture of the l = 0 ridge in the ´echelle diagram. To resolvethis, it may be necessary to fit to this curvature, althoughit may be difficult to do this consistently between mod- els and observations in which only a few radial orders areobserved.The value of ǫ from observations may also be ambigu-ous by ± n , of the modes is un-known (unlike for models). For the stars considered here,it seems that only Scenario A of Procyon has an ambigu-ous value of ǫ .An important feature of the ǫ diagram is the well-known offset between observed and computed oscilla-tion frequencies, as mentioned earlier, which manifestsitself as an offset in ǫ . This is the reason that the ob-served values of ǫ in Figure 10 are systematically offsetfrom the models. One way to address this issue is to0 WHITE ET AL. Fig. 11.—
The ǫ diagram, with model tracks for stars that are metal poor ( Z = 0 . Z = 0 . T eff than the approximate cool edge of the classical instability strip are not shown. Symbols for starsare the same as for Figure 9. correct the model frequencies empirically, as suggestedby Kjeldsen et al. (2008). A more satisfactory approachwould naturally be to improve the modeling of the near-surface layers.As in the C-D diagram, metallicity has an impact onthe evolutionary tracks in the ǫ diagram. Diagramsfor metal-poor stars ( Z = 0 . Z = 0 . TABLE 2Measured ǫ , modelled masses and spectroscopic metallicitiesof subgiants KIC Name ǫ M [Fe/H](M ⊙ ) (dex) β Hyi 1 . ± .
06 1 . ± . − . ± . . ± .
04 1 . ± . − . ± . . ± .
05 1 . ± .
07 +0 . ± . . ± .
03 1.13 or 1.23 +0 . ± . . ± .
05 1 . ± . − . ± . η Boo 1 . ± .
04 1.64—1.75 +0 . ± . . ± .
04 1 . ± .
11 — ders were observed the discrepancy would become clear.We have measured ∆ ν and ǫ consistently in models andobservations and conclude that the near-surface offset issignificant for red giants.Finally, we address whether ǫ really does discriminatebetween different stellar masses. Models indicate thatit does, but given the significant contribution to ǫ fromnear-surface layers, which are presently poorly modeled,it is not yet known how the near-surface offset varieswith mass, age or metallicity. It is therefore importantto verify that stars of higher mass really do have lowervalues of ǫ , as the models predict. To do this, we considerthe relative positions of stars with known masses in the ǫ diagram.Let us consider β Hyi, η Boo and the five
Kepler subgiants, since it is for subgiants that ǫ is poten-tially most useful. Masses for β Hyi and η Boo havebeen estimated from models by Brand˜ao et al. (2011)and Di Mauro et al. (2003), respectively. The metallic-ities adopted for this modeling were − . ± .
07 for β Hyi (Bruntt et al. 2010) and +0 . ± .
05 for η Boo(Taylor 1996). KIC 11026764 (Gemma), was modeledusing the individual mode frequencies as constraints byMetcalfe et al. (2010). The four other
Kepler subgiantswere modeled by Creevey et al. (2011) using ∆ ν and ν max as seismic constraints. The masses determined fromthe modeling and the spectroscopic metallicities are givenin Table 2, along with the the measured values of ǫ . Howdo these masses and metallicities compare with the ob-served values of ǫ ?In general, we do see a trend of decreasing ǫ with in-creasing mass, as expected from the evolutionary tracksin Figure 10. The only deviations from this trend areKIC 11395018 and η Boo, both of which are substantiallymore metal-rich than the other subgiants. Increased met-alicity results in a larger ǫ for a star of a given mass (Fig-ure 11), so it is no surprise that KIC 11395018 and η Boohave a slightly larger ǫ than KIC 11026764, despite beingmore massive. This qualitative comparison shows that ǫ does depend on fundamental stellar parameters, such asmass and metallicity and confirms that ǫ is useful as anadditional asteroseismic parameter. USING THE DIAGRAMS FOR MODE IDENTIFICATION
It has been difficult to establish the correct mode iden-tification in F stars. Short mode lifetimes in these starsresult in large linewidths, blurring the distinction be-tween the l =0,2 and l =1,3 ridges. It is common for bothpossible mode scenarios to be fitted. Usually, it is notedthat one of these identifications may be statistically morelikely, although the alternative cannot be ruled out. Fig. 12.— ǫ as a function of effective temperature for models with Z = 0 .
017 (gray lines) and observations of stars with secure modeidentifications. Symbols for main-sequence and subgiant stars arethe same as for Figure 7. Red giants which have been identifiedas hydrogen-shell burning RGB stars are indicated by blue circles,red clump stars by red diamonds, and secondary clump stars byorange squares (Bedding et al. 2011).
Fig. 13.— ǫ as a function of effective temperature for modelswith Z = 0 .
011 (blue) and Z = 0 .
028 (red).
Indeed, the initially favored ‘Scenario A’ of HD 49933(Appourchaux et al. 2008) was, with more data, foundto be less likely than ‘Scenario B’ (Benomar et al. 2009).Bedding et al. (2010b) favored the ‘Scenario B’ identifi-cation of the F5 subgiant Procyon. We note, however,that Huber et al. (2011) favored ‘Scenario A’ based upontheir analysis of the combined ground-based radial veloc-ity and MOST space-based photometric observations. Inmodeling Procyon, Do˘gan et al. (2010) found Scenario Ato be least problematic from a modeling perspective, butcould not rule out Scenario B.Bedding & Kjeldsen (2010) suggested that ǫ may beable to resolve this problem. They reasoned that, if ǫ varies slowly with stellar parameters then by scaling the2 WHITE ET AL. Fig. 14.—
Same as Figure 12 with the addition of stars withambiguous mode identifications. frequencies of one star whose identification is clear andcomparing to those of another should reveal the correctidentification of the second. However, the ǫ diagramspresented in this paper cast doubt upon the validity ofthe assumption that ǫ varies slowly with stellar param-eters. In some cases, ǫ is seen to vary quite rapidly asthe star evolves, particularly for higher-mass subgiantssuch as Procyon. Nevertheless, we have found that ǫ isstill a useful quantity for determining the most plausi-ble identification, especially if we plot it against effectivetemperature.Figure 12 shows the relation between ǫ and T eff formodels, and for stars in which the mode identificationis secure. In this figure the model tracks for differentmasses have much less spread than when plotted against∆ ν (Figure 10). Once again, the near-surface offset isapparent, with the models having a systematically lower ǫ than the observed stars. As shown in Figure 13, there islittle metallicity dependence in this relationship, exceptfor red giants. Therefore, given the effective temperatureof a star, ǫ can be very useful in deciding the correct modeidentification.In Figure 14 we again plot ǫ against T eff , but this timeadding three F stars for which mode identifications areambiguous: Procyon, HD 49933 and HD 181420. ForHD 49933, as previously mentioned, Scenario B is nowconsidered to be correct, and indeed this is the identi-fication that falls along the observed ǫ – T eff trend. Sce-nario 1 is the preferred identification in HD 181420, andagain this matches the trend. Unfortunately the situa-tion in Procyon is not completely clear. The trend withwhich we hope to identify the correct ǫ is still looselydefined due to the scarcity of F stars for which we havemeasured ǫ unambiguously. Scenario B appears to lietowards the top of any range we could expect for ǫ fora star of its effective temperature. On the other hand,Scenario A appears to be at the minimum. Expectinga correction for near-surface effects, we are inclined tobelieve Scenario B is more likely, but we cannot rule outthe alternative. We anticipate that a measurement of ǫ in many F stars observed by Kepler could help clearly
Fig. 15.—
Observed period spacing, ∆ P obs , as a function of thelarge separation ∆ ν in red giants. Model tracks show the evolu-tion of ∆ P obs for hydrogen shell-burning red giants of near-solarmetallicity ( Z = 0 . . ⊙ and 2 . ⊙ .The 2 . ⊙ track extends past the tip of the RGB to helium-burning phases. Black dots show ∆ P obs as measured from Kepler red giant branch stars by Bedding et al. (2011). Red and orangedots show red-clump and secondary-clump stars, as determined byBedding et al. (2011). Blue diamonds are model calculations onthe 2 . ⊙ track are equally spaced in time by 10 Myr, startingfrom 516.5 Myr. define the observational ǫ – T eff relation, clarifying the cor-rect mode identification in Procyon. GRAVITY-MODE PERIOD SPACINGS IN RED GIANTSTARS
Theoretical studies of red giants predict a verydense frequency spectrum of non-radial modes(Dziembowski et al. 2001; Christensen-Dalsgaard2004; Dupret et al. 2009; Montalb´an et al. 2010;Di Mauro et al. 2011; Christensen-Dalsgaard 2011).The vast majority are almost pure g modes that arelargely confined to the core of the star and thus havesurface amplitudes that are too low to be observable.Asymptotically, these modes are expected to exhibit anapproximately equal period spacing, which we denote∆ P g (Tassoul 1980).There also exist p modes in the convective envelope ofthe star that are equally spaced in frequency. As men-tioned in the Introduction, p and g modes of the sameangular degree may undergo coupling. The modes un-dergoing this interaction take on a mixed character, with p -mode characteristics in the envelope and g -mode char-acteristics in the core. The mixed modes with enough p -mode character will be observable due to their reducedmode inertias and we can expect to see a few l = 1 modesfor each p -mode order. Due to the weaker coupling be-tween l = 2 p and g modes, few additional mixed modesare likely to be observed. Early Kepler results revealedthat there were multiple l = 1 peaks due to mixed modesin each radial order (Bedding et al. 2010a), which haverecently been shown to have identifiable period spacings(Beck et al. 2011). Since these gravity-dominated mixedmodes propagate deeply into the core of the star, they arekey indicators of core structure. Bedding et al. (2011)found that the observed period spacing of the mixed l = 1modes can distinguish red giants burning helium in theircores from those still only burning hydrogen in a shell.STEROSEISMIC DIAGRAMS 13Similar results have also been found with CoRoT data(Mosser et al. 2011a).As mentioned, in the absence of any interaction with p modes, the g modes will be approximately equally spacedin period. However, due to mode bumping, the observedperiod spacing (∆ P obs ) is substantially smaller thanthe ‘true’, asymptotic g -mode period spacing (∆ P g ).Bedding et al. (2011) have shown that for red giants withthe best signal-to-noise ratio, in which several mixedmodes are observed in each radial order, the true pe-riod spacing may be recovered. However, for the vastmajority of stars in their sample, this was not possible.Instead they measured the average period spacing of theobserved l = 1 modes, ∆ P obs . We have done likewisewith models.To measure ∆ P obs from models it is first necessary todetermine which modes would be observable. The modewith the lowest inertia in each order has the greatestamplitude and we calculated the period spacing betweenthis mode and the two adjacent modes. We take the aver-age of these period spacings in the range ν max ± . ν max to calculate ∆ P obs . In general, there will be variation inthe number of l = 1 modes observable in each order anddetailed modeling of any star will have to take this intoaccount. Nevertheless, our method of including only thecentral three modes is sufficient to follow the expectedevolution and mass dependence of ∆ P obs .In Figure 15 the solid lines show ∆ P obs as a function of∆ ν for our red-giant models. Except for the highest massmodel ( M = 2 . ⊙ ), we have not evolved the modelspast the helium ignition. Stars on the RGB evolve fromright to left in this figure and we see a gradual decreasein ∆ P obs through most of this phase. For stars with M < . ⊙ we see only a weak dependence on mass.We confirm that ∆ P obs as determined from the mod-els for these lower-mass stars ( ∼
50 s) matches well withthe ∆ P obs seen in observations of red giants (40 −
60 s;Bedding et al. 2011), as shown in Figure 15. Higher-mass stars show larger ∆ P obs . This raises the possibil-ity that higher-mass RGB stars could be mistaken forhelium-core-burning stars, although we note that few ofthese higher-mass stars may be expected to be observeddue both to their lower abundance relative to lower-massstars and their rapid evolution.Following the evolution of low-mass stars past the tipof the red giant branch is difficult because they undergoa helium flash. Models computed from the helium mainsequence by Montalb´an et al. (in prep.) indicate pe-riod spacings around ∼
200 s, in agreement with observa-tions of core helium-burning stars (Bedding et al. 2011).We were able to compute the evolution of a higher-massstar (2 . ⊙ ), which undergoes a more gradual onset ofhelium-core burning. After the model reaches the tip ofthe RGB, the period spacing increases rapidly due to theonset of a convective core (Christensen-Dalsgaard 2011),followed by an increase in the large separation. The loopin the track during this phase occurs during a lull inthe energy generation from the core; helium ignition oc-curs in pulses for this higher-mass model. The modelthen settles onto the secondary clump, where it spendsa (relatively) long period of time, with period spacingincreasing slowly while the large separation remains rel-atively constant. The model period spacings agree wellwith the observations by Bedding et al. (2011). As the model begins to move up the asymptotic giant branch,the period spacing and large separation decrease oncemore. To help indicate the evolution through this dia-gram, the blue diamonds in Figure 15 are equally spacedin age by 10 Myr, starting from 516.5 Myr. Clearly, thesecondary-clump stage of evolution is the longest, whichis why we observe many more stars in this stage of theirevolution than in other stages (Girardi 1999). The agree-ment of this track with observations is excellent. SUMMARY OF CONCLUSIONS
We have investigated the evolution of several measure-ments of the oscillation spectra of stars that exhibit solar-like oscillations. We conclude:1. The standard scaling relation for ∆ ν with den-sity (equation (4)) does quite well for lower-massmain-sequence stars, being correct to within a fewpercent, but larger deviations are found for othermasses and evolutionary states. These deviationsare predominantly a function of effective tempera-ture and we suggest that the scaling relation maybe improved by including a function of T eff (equa-tion (5)).2. For main-sequence stars, their position in the C-Ddiagram is able to significantly constrain theirmass. Age is less well-constrained due to the un-certainty in δν generally being relatively largerthan that in ∆ ν . When stars evolve into subgiantsthe tracks converge, which limits the ability of therelationship between ∆ ν and δν to constrain themass and age.3. In the ǫ diagram, the degeneracy of the evolution-ary tracks during the subgiant stage is partiallybroken. This suggests that the position of stars inthis diagram could help constrain their masses andages, although difficulties in measuring ǫ and thenear-surface offset presently limit this diagram’susefulness. We find the near-surface offset is non-negligible for red giants, contrary to earlier claims.4. Measuring ǫ shows promise for mode identificationin stars where this is ambiguous due to short modelifetimes. In particular, ǫ is seen to depend mostlyon effective temperature. Provided T eff is known,the correct ǫ and therefore the correct mode iden-tification may be deduced.5. We have investigated the evolution of g-mode pe-riod spacings (∆ P obs ) in models of red giant stars.Models with masses below ∼ . ⊙ show a weakdependence of ∆ P obs on mass. For higher-massstars, the observed position of the secondary clumpin the ∆ P obs –∆ ν diagram is well matched by themodels.We acknowledge the support of the Australian Re-search Council. TRW is supported by an AustralianPostgraduate Award, a University of Sydney MeritAward, an Australian Astronomical Observatory PhDScholarship and a Denison Merit Award.4 WHITE ET AL.diagram is well matched by themodels.We acknowledge the support of the Australian Re-search Council. TRW is supported by an AustralianPostgraduate Award, a University of Sydney MeritAward, an Australian Astronomical Observatory PhDScholarship and a Denison Merit Award.4 WHITE ET AL.