Mixed modes in red-giant stars observed with CoRoT
B. Mosser, C. Barban, J. Montalban, P.G. Beck, A. Miglio, K. Belkacem, M.J. Goupil, S. Hekker, J. De Ridder, M.A Dupret, Y. Elsworth, A. Noels, F. Baudin, E. Michel, R. Samadi, M. Auvergne, A. Baglin, C. Catala
aa r X i v : . [ a s t r o - ph . S R ] J un Astronomy&Astrophysicsmanuscript no. 16825 c (cid:13)
ESO 2018July 2, 2018
Mixed modes in red-giant stars observed with CoRoT ⋆ B. Mosser , C. Barban , J. Montalb´an , P.G. Beck , A. Miglio , , K. Belkacem , , M.J. Goupil , S. Hekker , , J. DeRidder , M.A Dupret , Y. Elsworth , A. Noels , F. Baudin , E. Michel , R. Samadi , M. Auvergne , A. Baglin , and C.Catala LESIA, CNRS, Universit´e Pierre et Marie Curie, Universit´e Denis Diderot, Observatoire de Paris, 92195 Meudon cedex, France;e-mail: [email protected] Institut d’Astrophysique et de G´eophysique, Universit´e de Li`ege, All´ee du 6 Aoˆut, 17 B-4000 Li`ege, Belgium Instituut voor Sterrenkunde, K. U. Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom Institut d’Astrophysique Spatiale, UMR 8617, Universit´e Paris XI, Bˆatiment 121, 91405 Orsay Cedex, France Astronomical Institute ‘Anton Pannekoek’, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The NetherlandsPreprint online version: July 2, 2018
ABSTRACT
Context.
The CoRoT mission has provided thousands of red-giant light curves. The analysis of their solar-like oscillations allows usto characterize their stellar properties.
Aims.
Up to now, the global seismic parameters of the pressure modes remain unable to distinguish red-clump giants from membersof the red-giant branch. As recently done with Kepler red giants, we intend to analyze and use the so-called mixed modes to determinethe evolutionary status of the red giants observed with CoRoT. We also aim at deriving di ff erent seismic characteristics depending onevolution. Methods.
The complete identification of the pressure eigenmodes provided by the red-giant universal oscillation pattern allows us toaim at the mixed modes surrounding the ℓ = Results.
We have identified the mixed-mode signature separation thanks to their pattern compatible with the asymptotic law ofgravity modes. We have shown that, independent of any modelling, the g-mode spacings help to distinguish the evolutionary statusof a red-giant star. We then report di ff erent seismic and fundamental properties of the stars, depending on their evolutionary status.In particular, we show that high-mass stars of the secondary clump present very specific seismic properties. We emphasize that starsbelonging to the clump were a ff ected by significant mass loss. We also note significant population and / or evolution di ff erences in thedi ff erent fields observed by CoRoT. Key words.
Stars: oscillations - Stars: interiors - Stars: evolution - Stars: mass loss - Methods: data analysis
1. Introduction
The CoRoT and
Kepler missions have revealed solar-like oscil-lation in thousands of red-giant stars. This gives us the opportu-nity to test this important phase of stellar evolution, and providesnew information in stellar and galactic physics (Miglio et al.2009; Bedding et al. 2011). Thanks to the dramatic increaseof information recently made available (De Ridder et al. 2009;Hekker et al. 2009; Mosser et al. 2010; Kallinger et al. 2010;Huber et al. 2010), we have now a precise view of pressuremodes (p modes) corresponding to oscillations propagating es-sentially in the large convective envelopes. Gravity modes (gmodes) may exist in all stars with radiative regions. They resultfrom the trapping of gravity waves, with buoyancy as a restor-ing force. In red giants, gravity waves propagating in the corehave high enough frequencies to be coupled to pressure wavespropagating in the envelope (Dupret et al. 2009). The trappingof such waves with mixed pressure and gravity character gives
Send o ff print requests to : B. Mosser ⋆ The CoRoT space mission, launched on 2006 December 27, wasdeveloped and is operated by the CNES, with participation of theScience Programs of ESA, ESAs RSSD, Austria, Belgium, Brazil,Germany and Spain. the so-called mixed modes. The coupling insures non-negligibleoscillation amplitudes in the stellar photosphere, hence the pos-sible detection of these mixed modes.Observationally, mixed modes were first identified in white-dwarf oscillation spectra (Winget et al. 1991). They have alsobeen observed in sub-giant stars, first with ground-based ob-servations (Carrier et al. 2005; Bedding et al. 2007, 2010b),then recently in
Kepler and CoRoT fields (Chaplin et al. 2010;Deheuvels et al. 2010). In red giants, they were first suspected byBedding et al. (2010a). Their presence significantly complicatesthe fit of the p modes observed in CoRoT giants (Hekker et al.2010; Barban et al. 2010) and they were identified as outliers tothe universal red-giant oscillation spectrum (Mosser et al. 2011).Recent modelling of a giant star observed by the
Kepler mis-sion had to take their presence into account (Di Mauro et al.2011). Finally, they have been firmly identified in
Kepler data(Beck et al. 2011), making the di ff erence clear between giantsburning hydrogen in shell or helium in the core (Bedding et al.2011).The theoretical analysis of these mixed modes in red giantshas been performed prior to observations (Dziembowski et al.2001; Christensen-Dalsgaard 2004; Dupret et al. 2009). Usinga non-radial non-adiabatic pulsation code including a non-local B. Mosser et al.: Mixed modes in red-giant stars time-dependent treatment of convection and a stochastic exci-tation model, Dupret et al. (2009) have computed the eigenfre-quencies and the mode heights in several red-giant models. Theyhave shown that mixed modes have much larger mode iner-tias than p modes, hence present longer lifetimes and smallerlinewidths. They were able to identify di ff erent regimes, de-pending on the location of the models on the red-giant branch.Recently, Montalb´an et al. (2010) proposed to use the oscilla-tion spectrum of dipole modes to discriminate between red-giant branch (RGB) and central-He burning (clump) evolution-ary phase of red giants. This illustrates the fact that observing p-g-mixed modes and identifying their properties give us a uniqueopportunity to analyze the cores of red giants, since the g compo-nent is highly sensitive to the core condition (Dupret et al. 2009).In this paper, we present and validate an alternative methodto Bedding et al. (2011) to detect and identify mixed modes inred giants. We use it to analyze red-giant stars in two di ff er-ent fields observed by CoRoT and show that they present di ff er-ent properties: mixed modes do not only allow us to distinguishdi ff erent evolutionary status, they can also show di ff erent pop-ulation characteristics. We also assess clear observational dif-ferences between the fundamental and seismic properties of thered-giant stars, depending on their evolutionary status. We showfor instance that the observation of mixed modes opens a newway to study the mass loss at the tip of the RGB. We have alsoclear indication that mixed modes in red giants are sensitive tochemical composition gradients in the deep interior, as they arein SPB and γ Doradus stars (Miglio et al. 2008). Their study willdefinitely boost both stellar and galactic physics.
2. Analysis
According to Dupret et al. (2009), each ℓ = T n g ,ℓ = π ( n g + α ℓ ) √ ℓ ( ℓ + "Z core N BV r d r − = ( n g + α ℓ ) ∆ T g √ ℓ ( ℓ +
1) (1)with n g the gravity radial order, α ℓ an unknown constant and N BV the Brunt-V¨ais¨al¨a frequency. The period spacing ∆ T g is theequivalent for g modes of the large separation for p modes. We have first restricted our attention to ℓ = ℓ = ℓ = ℓ = T n g , = h n g + α i ∆ T , with ∆ T = ∆ T g / √ . (2)The period spacing ∆ T , linked to the integral of the Brunt-V¨ais¨al¨a frequency by Eq. 1, can be deduced from the frequency spacing in the Fourier spectrum. To measure this spacing, weneed to focus on the mixed modes. We are able to do this us-ing the method introduced in Mosser et al. (2011) which allowsfor a complete and automated mode identification. We then usethe envelope autocorrelation function (Mosser & Appourchaux2009) to derive the period spacing. The method is described inthe Appendix. As for the method presented by Bedding et al.(2011), it derives a period ∆ T obs less than ∆ T , due to the bump-ing of mixed modes. We have estimated that, for the 5-monthlong CoRoT time series, we measure ∆ T obs ≃ ∆ T / .
15 (seeAppendix).
The analysis has been performed on CoRoT red giants previ-ously analyzed in Hekker et al. (2009) and Mosser et al. (2010).These stars were observed continuously during 5 months in2 di ff erent fields respectively centered on the Galactic coordi-nates (37 ◦ , − ◦ ′ ) and (212 ◦ , − ◦ ′ ). Targets with a toolow signal-to-noise ratio were excluded (see the Appendix). Wetherefore only considered stars with accurate global seismic pa-rameters and labelled as the N set in Mosser et al. (2010).Thanks to the method exposed in the Appendix, we have an-alyzed all red-giant spectra in an automated way. We have thenchecked all results individually. This allowed us to discard a fewfalse positive results, and then to verify that the asymptotic ex-pansion of the g modes (Eq. 2) gives an accurate description ofthe mixed modes since it is able to reproduce their spacings. Theirregularities of the spacings are discussed in Section 3.1 andin the Appendix. A few examples are given in Fig. 1: we haveoverplotted on red-giant oscillation spectra the expected loca-tion of the p-mode pattern derived from Mosser et al. (2011) andwe have indicated the frequency separation of the mixed modesderived from the asymptotic description. Because of the modebumping due to the coupling, the g-mode asymptotic expressionis not able to derive the exact location of the mixed-mode eigen-frequencies, but we see that the spacings reproduce the obser-vations. This agreement is certainly related to the fact that, ac-cording to Eq. 2, very high values of the gravity radial orders aremeasured, typically above 60 and up to 400 (Table 1). We werethen able to provide a diagram representing the period separation ∆ T obs of the mixed modes as a function of the large frequencyseparation ∆ ν of the pressure modes (Fig. 2).The 5-month long CoRoT time series provide a frequencyresolution of about 0.08 µ Hz, accurate enough for detecting themixed-modes in the red clump but limiting their possible detec-tion at lower frequency. Therefore, we have taken care of pos-sible artefacts. We have excluded the region of the h ∆ ν i - ∆ T obs diagram limited by the frequency resolution (Fig. 2). We havealso taken care of the possible confusion with the small separa-tion δν , since in given frequency ranges its signature can mimica g-mode spacing (dotted line in Fig. 2). Due to the low visibilityof ℓ = . Mosser et al.: Mixed modes in red-giant stars 3 Fig. 1.
Shaded regions of these bar code spectra reproduce the universal red-giant oscillation pattern and identify the location ofthe di ff erent harmonic degrees for three CoRoT targets (ID given on the right side); the more dense the background, the higher theprobability to have the short-lived p mode realized there. Black dashes in the ℓ = Table 1.
Typical parameters of mixed modes h ∆ ν i ν max δν env ∆ T obs ∆ T proxy n g ; ∆ n g n g ; ∆ n g n g ; ∆ n g T at ν max − δν env / ν max at ν max + δν env / µ Hz µ Hz µ Hz s s dayRed-clump stars4.0 32 13 250 287 137; 5 108; 3 89; 2 787.0 73 28 150 173 97; 3 78; 2 66; 1 25Red-giant branch stars2.5 19 8 800 920 73; 3 57; 2 46; 1 704.0 32 13 80 92 429;17 339;10 281; 7 2456.0 60 23 60 69 300; 9 241; 6 201; 4 939.0 102 38 60 69 174; 5 142; 3 119; 2 32The frequency ν max corresponds to the maximum oscillation signal; δν env indicates the full width at half-maximum of the observed excess oscillationpower (Mosser et al. 2010). The proxies of the period spacing ∆ T are estimated as 1 . ∆ T obs . The values of n g are derived from Eq. 2, with theparameter α ℓ = = ∆ n g corresponds to the maximum number of observable mixed modes in a 0 . ∆ ν -wide interval around each pure p mode. T indicates the observational length required to measure the g-mode spacing according to the Shannon criterion. Finally, we have identified an ℓ = . ∆ ν around the expected pure p-mode eigenfrequencies ν n , .This represents typically, at the peak of the red clump distribu-tion ( ∆ ν ≃ µ Hz and ν max ≃ µ Hz), from two to ten mixed modes (Table 1). Due to the frequency dependence derived fromEq. 2, the number of observable modes varies very rapidly, asobserved by Beck et al. (2011) and made explicit in Table 1.According to Dupret et al. (2009), ℓ = ℓ = ∆ T ≃ ∆ T / √ ℓ = B. Mosser et al.: Mixed modes in red-giant stars
Fig. 2.
Period separation of mixed modes ∆ T obs as a function ofthe mean frequency separation of the pressure modes h ∆ ν i . Blue(purple) crosses represent the targets observed in the center (an-ticenter) direction. The dashed line indicates the frontier belowwhich the frequency resolution is not fine enough for deriving ∆ T obs and the dotted line represents the location of the spuri-ous signature that would correspond to the small separation δν (Mosser et al. 2011). The two separate domains correspond tothe location of ∆ T expected by Montalb´an et al. (2011) and ob-served by Bedding et al. (2011). The vertical thick line indicatesthe mean 1- σ error bars (the error bar on the large separation ∆ ν is in fact very small).of the radial gravity orders observed, makes it possible to achievethis high-resolution analysis. We observe in most cases groups ofonly two ℓ = . ∆ ν . A further consequence of the observa-tion of ℓ = δν by Huber et al. (2010) andMosser et al. (2011); they are correct, but only indicative of thebarycenter of the ℓ =
3. Discussion
The measurement of mixed-mode spacings varying as 1 /ν (Eq. A.2) validates the use of the asymptotic law of g modes,even if a detailed view of the mixed modes indicates irregulari-ties in their spacings (Fig. 3). These shifts may be interpreted asa modulation due to the structure of the core with a sharp densitycontrast compared to the envelope, as observed in white dwarfs.The determination of ∆ T will allow modelers to define the sizeof the radiative core region. The direct measurement of the in-dividual shifts ∆ T ( n g ) will give access to the core stratification(Miglio et al. 2008), as well as the observation (or not) of ℓ = ff erent cavity.The method developed in this paper presents many in-teresting characteristics compared to the method used byBedding et al. (2011). First, it is directly applicable in theFourier spectrum and does not require the power spectrum tobe expressed in period. The method is fully automated, sinceit is coupled to the identification of the spectrum based on theuniversal pattern, and it includes a systematic search of periodicspacings that are not related to the p-mode pattern. Based onan H test, it intrinsically includes a statistical test of reliability. Fig. 3. ´Echelle diagrams around ν max for di ff erent targets sortedby increasing large separations. The red squares indicate the ob-served peaks, selected with a height-to-background ratio above6. Dark blue crosses indicate the expected location of the ℓ = ℓ =
2. Similarly to Fig. 1, crosses and di-amonds indicate the mean spacing of the mixed modes only, nottheir exact location.This test defines a threshold level that makes the method e ffi cienteven at low signal-to-noise ratio. Its most interesting propertycertainly consists in its ability to derive the measurement of thevariation of ∆ T with frequency, as the EACF method used for pmodes (Mosser & Appourchaux 2009; Mosser 2010). In fact, themethod was developed in parallel to the one mainly presented inBedding et al. (2011), and gave similar results that confirmed thedetection of mixed modes in Kepler giants.
Red-clump stars have been characterized in previous work (e.g.Fig. 5 of Mosser et al. 2010). They contribute to a distributionin ∆ ν with a pronounced accumulation near 4 µ Hz (equivalentto the accumulation near ν max ≃ µ Hz). However, populationanalysis made by Miglio et al. (2009) has shown that this pop-ulation of stars with ∆ ν ≃ µ Hz e ff ectively dominated by red- . Mosser et al.: Mixed modes in red-giant stars 5 clump stars also contains a non-negligible fraction ( ≃
30 %) ofRGB stars. None of the global parameters of p mode (neither ∆ ν nor ν max ) is able to discriminate between RGB and red-clumpstars. However, as shown by Bedding et al. (2011) with Kepler data, the examination of the relation between the p-mode and g-mode spacings shows two regimes (Fig. 2). The signature of thered-clump stars piling up around h ∆ ν i ≃ µ Hz is visible with ∆ T obs ≃
250 s. Another group of stars has ∆ T obs values lowerthan 100 s. Regardless of any modelling, this gives a clear signa-ture of the di ff erence between red-clump stars that burn heliumin their core and stars of the red-giant branch that burn hydrogenin a shell (Montalb´an et al. 2010). The agreement with theoreti-cal values is promising (Montalb´an et al. 2011).The contribution of the red-clump stars in the ∆ ν - ∆ T obs dia-gram presented in Fig. 2 is unambiguous. Stars with large sep-aration larger than the clump value (about 4 µ Hz) are locatedon the ascending red-giant branch or members of the secondaryclump (Girardi 1999; Miglio et al. 2009). At lower frequencythan the clump, using mixed modes to disentangle the evolu-tionary status is more di ffi cult. According to the identificationof the clump provided by the distributions of the large sep-aration (Mosser et al. 2010), we assume that the giants with ∆ ν ≤ . µ Hz and ∆ T obs ≥
200 s belong to the RGB. The fol-lowing analysis shows further consistent indications.We finally note that the detection of RGB stars having a largeseparation similar to the peak of the clump stars ( ≃ µ Hz) isonly marginally possible, due to insu ffi cient frequency resolu-tion. According to Table 1, more than 200 days are necessary toresolve the mixed modes, while CoRoT runs are limited to about150 days. We have remarked that, as predicted by Dupret et al. (2009), thelifetimes of the mixed modes trapped in the core are much largerthan the lifetimes of radial modes (Fig. 1). In most cases, themixed modes are not resolved. When the mixed modes are iden-tified, they corresponds in most cases to a comb-like pattern ofthin peaks, without a larger and broader peak that could cor-respond to the pure ℓ = ℓ = The possibility to distinguish the evolutionary status allows us torefine the ensemble asteroseismic analysis made on CoRoT redgiants, especially the mass and radius distributions (Fig. 12 and13 of Mosser et al. 2010). Benefitting from the same calibrationof the asteroseismic mass and radius determination as done inthis work, we have investigated the mass-radius relation, havingin mind that without information of the evolutionary status, thereis no clear information (Hekker et al. 2011b).We note here that the mass distribution is almost uniform inthe RGB, contrary to clump stars (Fig. 4). The number of high-mass stars in the RGB, above 1 . M ⊙ as defined in Bedding et al.(2011), is about half of that in the clump, consistent with theirexpected more rapid evolution. Similarly, stars with masses be-low 1 M ⊙ are significantly rarer (by a factor of six to one) inthe RGB by comparison with the clump. If we assume that thescaling relations, valid along the whole evolution from the mainsequence to the giant class, remain valid after the tip of the RGB, Fig. 4.
Asteroseismic mass as a function of the asteroseismic ra-dius, with indication of the evolutionary status derived from themixed-mode spacing: blue squares for RGB stars, red diamondsfor clump stars; stars without clear measurement of ∆ T obs aremarked with a small cross. The rectangle indicates the mean 1- σ error bars. The dotted lines at respectively 1 and 1.8 M ⊙ corre-spond to the limits defined in the text. Fig. 5.
Mean mode height as a function of ν max . Same color codeas in Fig. 4. The rectangle indicates the mean 1- σ error bars.this comparative study proves that low-mass stars present in theclump but absent in the branch have lost a substantial fractionof their mass due to stellar winds when ascending up to the tipof the RGB. Therefore, after the helium flash, these stars showlower mass. The quantitative study of this mass loss will requirea careful unbiased analysis, out of the scope of this paper. Onthe contrary, high-mass stars, even if they lost mass too, can beobserved as clump stars since they spend a much longer time inthe core-helium burning phase than while ascending the RGB.Most of these stars belong to the secondary red clump (Girardi1999; Bedding et al. 2011). We note that the secondary-clumpstars present a larger spread in the mass-radius distribution, cer-tainly due to the fact that those stars that have not undergone thehelium flash present di ff erent interior structures. On the otherside, the low-mass stars of the clump present a clear mass-radiusrelation: the helium flash shall have made their core largely ho-mologous. B. Mosser et al.: Mixed modes in red-giant stars
Fig. 6.
Proxy of the number of observable modes, given by theratio δν env / ∆ ν , as a function of ν max . Same color code as in Fig. 4.The rectangle indicates the mean 1- σ error bars. We can also address the influence of the evolutionary status onthe energetic parameters of the red-giant oscillation spectrum.When plotting the mean height of the excess power Gaussianenvelope as a function of the frequency ν max , we remark thatclump stars and RGB stars present very similar height for ν max less than 40 µ Hz. However, above 40 µ Hz, the oscillations inclump stars present systematically lower heights than RGB stars(Fig. 5). The contrast is more than a factor of 2. On the other side,the number of modes, estimated from the ratio δν env / ∆ ν where δν env is the full width at half-maximum of the total oscillationexcess power envelope, is similar below 40 µ Hz, but more than30 % larger for clump stars above this limit (Fig. 6). Despite thesomewhat arbitrary limit in ν max we see here evidence of the sec-ondary clump (Girardi 1999), which consists of He-core burningstars massive enough to have ignited He in non-degenerate con-ditions. These stars have, at ν max > µ Hz, similar radii as RGBstars, and larger masses. Hence, they have a smaller L / M ra-tio, so that the excitation of oscillation is expected to be weaker.With a larger ν max , they also present a solar-like oscillation spec-trum with modes excited in a broader frequency range.We suggest that, in case the measurement of ∆ T obs is madedi ffi cult by a low signal-to-noise ratio, the oscillation amplitudeand the size of the Gaussian envelope with noticeable amplitudemay be used to help the determination of the evolutionary status.Finally, we note that the secondary-clump population is cer-tainly underestimated, owing to the fact that the low amplitudesmake their detection quite complex in CoRoT data. In fact, suchstars are observed with a low signal-to-noise ratio since theirFourier spectra are dominated by a white-noise contribution. The number of detections of solar-like oscillation towards theanticenter direction is lower than towards the center, due to alower red-giant density in the outer galactic regions and to dim-mer magnitudes (Mosser et al. 2010). Hence, the number of reli-able measurements of ∆ T obs is lower too, but with the same pro-portion of 40 %. The sample is large enough to make sure thatthe di ff erence between the distributions is significant (Fig. 2). The anticenter field shows a significant deficit of red-clumpstars below ∆ ν = . µ Hz, e.g. with small mass. Another simi-lar deficit is observed at low frequency in the red-giant branch;it should indicate populations with di ff erent ages. We also notethat secondary-clump stars of the anticenter direction presentslightly higher g-mode spacings; it should indicate populationswith di ff erent mass distributions. This illustrates the interest tocompare the di ff erent fields in order to analyze di ff erent popula-tions (Miglio et al., in preparation).
4. Conclusion
The clear identification of the p-mode oscillation pattern has al-lowed us to identify in CoRoT observations the pattern of mixedmodes behaving as gravity modes in the core and pressure modesin the envelope. We have verified that this pattern is very close tothe asymptotic expression of g modes. Benefitting from the iden-tification of ℓ = ℓ = Kepler willallow us to investigate this relation in more detail. Benefittingfrom the fact that CoRoT provides observations in two fields, to-wards the Galactic center and in the opposite direction, we havenow a performing indicator for making the population studymore precise. This will be done in future work.These data confirm the power of red-giant asteroseismology:we have access to the direct measure of the radiative central re-gions. Even if observations only deliver a proxy of the large pe-riod separation, comparison with modelling will undoubtedly bevery fruitful. In a next step, the dedicated analysis of the bestsignal-to-noise ratio targets will allow us to sound in detail thered-giant core. Modulation of the large period spacing, observedin many targets, will give a precise view of the core layers.
Appendix A: Method
Identifying ℓ = ∆ ν and thento derive the complete identification of the p-mode oscillationpattern. Complete identification means that all eigenfrequencies,their radial order and their degree, are unambiguously identified,as for instance the expected frequencies of the pure pressure ℓ = ν n ,ℓ = = [ n + / + ε ( ∆ ν ) − d ] ∆ ν, (A.1)with ε ( ∆ ν ) representing the surface term and d accounting forthe small separation of ℓ = ν n , in the . Mosser et al.: Mixed modes in red-giant stars 7 Fig. A.1.
Zoom on the bar code spectrum of the target CoRoT 100752538, with the large separation provided by the adjustmentderived from the red-giant oscillation universal pattern. Di ff erent narrow filters centered on di ff erent pure ℓ = ff erent line styles. They allow us to measure a local g-mode frequency spacing in each filter. Fig. A.2.
Mean period spacings measured around the expectedpure ℓ = ν max of maximum oscillation ampli-tude (Mosser & Appourchaux 2009). We deliberately chose theEACF method since it has proved to be e ffi cient at very lowsignal-to-noise ratio (Mosser et al. 2009; Gaulme et al. 2010;Hekker et al. 2011a), thanks to a statistical test of reliabilitybased on the null hypothesis.In order to only select mixed modes, the full width at half-maximum of the filter is fixed to ∆ ν/ δν g , = ν n g , ∆ T ≃ ν n , ∆ T . (A.2)For an individual measure centered on a given p mode, the fre-quency spacings selected by the narrow filter can be consideredas uniform, since the ratio ∆ n g / n g that represents the relativevariation of the gravity radial order within the filter is less thanabout 1 /
25 (Table 1). Then, when comparing the di ff erent mea-sures around di ff erent pressure radial orders, obtaining mean fre-quency spacings varying as ν n ,ℓ = validates the hypothesis of aTassoul-like g-mode pattern (Fig. A.2).We have checked that the method can operate with a filternarrow enough to isolate the mixed modes. This is clearly a limit, since the performance of the EACF varies directly withthe width of the filter (Mosser & Appourchaux 2009). We havealso checked that such a filter is able to derive the signature of ℓ = ℓ = ℓ = ∆ ν/ ff erent frequencyranges (Fig. A.1 and A.2), we have chosen a thresholdvalue corresponding to the rejection of the H0 hypothesis atthe 10 % level. For the characteristics of mixed modes, dif-ferent to the characteristics of the p modes considered inMosser & Appourchaux (2009), this corresponds to a normal-ized EACF of about 4.5 at ν max . In practice, stellar time serieswith a low signal-to-noise ratio are excluded by this thresholdvalue. In case of reliable detection, error bars can be derived fol-lowing Eq. A.8 of Mosser & Appourchaux (2009).The measurement of the g-mode frequency spacing δν g , isnot only validated by a correlation signal larger than the thresh-old level: we only selected results with δν g , measured in at leasttwo frequency ranges and verifying Eq. A.2 within 20 %. Thisflexibility allows us to account for possible discrepancy to theexact asymptotic relation (Eq. 2), as produced by the avoidedcrossings resulting from coupling of the p mode in the stellarenvelope to the g modes in the core. It also accounts for the pos-sible modulation of the period due to a composition gradient inthe core (Miglio et al. 2008).Deriving an estimate ∆ T obs of ∆ T from the spacings δν g , isthen direct. Due to avoided crossings, ∆ T obs is close to but lessthan ∆ T (Althaus et al. 2010; Beck et al. 2011; Bedding et al.2011). This is called mode bumping and results from the factthat the mixed modes around ν n , present necessarily smallerspacings than pure g modes since the mixing of the g modeswith one p mode gives one supernumerary mixed mode per ∆ ν frequency interval, as shown in Beck et al. (2011). The ra-tio ∆ T / ∆ T obs ≃ .
15 is derived from the examination, whenpossible, of the g-mode spacing far from the expected p mode,assuming as in Bedding et al. (2011) that this spacing is unper-turbed by the mode bumping and corresponds to the asymptoticg-mode spacing. The ratio di ff ers from the value obtained withthe Kepler data, since the frequency resolutions and the analysismethods are di ff erent. First, the frequency resolution is 2.5 timesless fine for CoRoT data; as a consequence, the influence of the B. Mosser et al.: Mixed modes in red-giant stars mode bumping is more smooth. Second, and more importantly,the envelope autocorrelation method is able to derive a meanvalue of the spacing in a larger frequency range than the methodexposed in Bedding et al. (2011) thanks to the ∆ ν/ Acknowledgements.
This work was supported by the Centre National d’ ´EtudesSpatiales (CNES). It is based on observations with CoRoT. The researchhas made use of the Exo-Dat database, operated at LAM-OAMP, Marseille,France, on behalf of the CoRoT / Exoplanet program. KB acknowledges fi-nancial support from CNES. SH acknowledges financial support from theNetherlands Organisation for Scientific Research (NWO). PB received fund-ing from the European Community’s 7th Framework Programme, ERC grantn227224 (PROSPERITY).