Structural glitches near the cores of red giants revealed by oscillations in g-mode period spacings from stellar models
M.S. Cunha, D. Stello, P.P. Avelino, J. Christensen-Dalsgaard, R.H.D. Townsend
aa r X i v : . [ a s t r o - ph . S R ] M a r D RAFT VERSION A PRIL
21, 2018
Preprint typeset using L A TEX style emulateapj v. 2/19/04
STRUCTURAL GLITCHES NEAR THE CORES OF RED GIANTS REVEALED BY OSCILLATIONS IN G-MODEPERIOD SPACINGS FROM STELLAR MODELS
M. S. C
UNHA , D. S TELLO , P.P. A
VELINO , J. C
HRISTENSEN -D ALSGAARD , R. H. D. T OWNSEND Draft version April 21, 2018
ABSTRACTWith recent advances in asteroseismology it is now possible to peer into the cores of red giants, potentiallyproviding a way to study processes such as nuclear burning and mixing through their imprint as sharp structuralvariations – glitches – in the stellar cores. Here we show how such core glitches can affect the oscillationswe observe in red giants. We derive an analytical expression describing the expected frequency pattern in thepresence of a glitch. This formulation also accounts for the coupling between acoustic and gravity waves. Froman extensive set of canonical stellar models we find glitch-induced variation in the period spacing and inertia ofnon-radial modes during several phases of red-giant evolution. Significant changes are seen in the appearanceof mode amplitude and frequency patterns in asteroseismic diagrams such as the power spectrum and the ´echellediagram. Interestingly, along the red-giant branch glitch-induced variation occurs only at the luminosity bump,potentially providing a direct seismic indicator of stars in that particular evolution stage. Similarly, we findthe variation at only certain post-helium-ignition evolution stages, namely, in the early phases of helium-coreburning and at the beginning of helium-shell burning signifying the asymptotic-giant-branch bump. Based onour results, we note that assuming stars to be glitch-free, while they are not, can result in an incorrect estimateof the period spacing. We further note that including diffusion and mixing beyond classical Schwarzschild,could affect the characteristics of the glitches, potentially providing a way to study these physical processes.
Subject headings: stars: evolution — stars: oscillations — stars: interiors INTRODUCTION
The cores of red-giant stars hold the key to answering anumber of unresolved questions about fundamental physicsthat govern stellar evolution, such as mixing process, rota-tion, and the effect of magnetic fields. It has been knownfor over a decade that red giants show stochastically-drivenoscillations like the Sun (Frandsen et al. 2002), but onlywith recent data from space missions like
CoRoT and
Ke-pler , have asteroseismic investigations revealed details aboutthe cores of red giants. This advance has been possibledue to the fortunate circumstance that gravity waves (here-after, g modes) in the cores of red giants couple to acous-tic waves (hereafter, p modes) in the envelope, resulting inmixed modes whose information about the core propertiesis therefore observable at the surface (Dupret et al. 2009;Bedding et al. 2010). Recent findings include the distinc-tion between stars with inert cores from those that possesscore burning (e.g. Bedding et al. 2011), the measurement ofcore rotation rates much slower than predicted by current the-ory of angular momentum transport (e.g. Beck et al. 2012;Mosser et al. 2012c; Cantiello et al. 2014), and an ability todetermine the evolutionary stages of stars with unprecedentedprecision (Mosser et al. 2014). Despite these findings, the fullpotential of current asteroseismic data can only be realizedif all aspects are understood about how the internal structureof stars may influence the observed oscillations. To achieve Instituto de Astrof´ısica e Ciˆencias do Espac¸o, Universidade do Porto,CAUP, Rua das Estrelas, 4150-762 Porto, Portugal, [email protected] Sydney Institute for Astronomy (SIfA), School of Physics, University ofSydney, NSW 2006, Australia Stellar Astrophysics Centre, Department of Physics and Astronomy,Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark Departamento de F´ısica e Astronomia, Faculdade de Ciˆencias, Universi-dade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto, Portugal Department of Astronomy, University of Wisconsin-Madison, 2535 Ster-ling Hall, 475 N. Charter Street, Madison, WI 53706, USA this, it is necessary to explore how sharp structural variationsinside a red giant could impact its oscillation frequencies.Sharp structural variations can be found in stellar interiorsat the borders of convectively mixed regions, in regions ofionization of elements, or between layers that have acquireddifferent chemical composition as a result of nuclear burn-ing. The signatures they imprint on the oscillation frequen-cies have already been studied observationally and theoreti-cally in white dwarfs (e.g. Winget et al. 1991; Brassard et al.1992), main-sequence stars (Roxburgh & Vorontsov 2001;Miglio et al. 2008; Degroote et al. 2010) including thoselike the Sun (Monteiro et al. 2000; Monteiro & Thompson2005; Cunha & Metcalfe 2007; Cunha & Brand˜ao 2011;Houdek & Gough 2007; Mazumdar et al. 2014), and in sdBstars (Charpinet et al. 2000; Østensen et al. 2014), which areessentially the cores of previous red giants.In red giant stars, only the signature of the helium ioniza-tion zone has been studied (Miglio et al. 2010). This signa-ture arises from the stellar envelope and affects the acous-tic modes, but simulations indicate its application as a diag-nostic tool on single stars might be limited (Broomhall et al.2014). However, the effect of sharp variations occurring in thedeeper layers near the cores of red giants has neither been in-vestigated theoretically, nor been reported from observations.Given the mixed character of the waves in red giants, the studyof this phenomenon requires understanding the combined ef-fect of the sharp structural variation and of the coupling be-tween p and g modes.Here we present the first comprehensive study of the effecton the oscillation frequencies of red giants from sharp struc-tural variations located in their deeper layers. We illustratethe impact this may have on common asteroseismic diagramsand investigate where this effect might be relevant during thered giant evolution phase. While no assumption is made aboutthe degree of the modes in the analysis presented here, all ex-amples provided are for dipole modes, because these are the Cunha et al. F IG . 1.— One solar mass red-giant models considered for a detailed anal-ysis. (a) Position in the HR diagram: model 1a is on the red-giant branchjust below the luminosity bump and model 1b is between helium flashes.; (b) and (c) show, respectively, the helium profile and buoyancy frequency in theinner region of model 1a (solid curve) and model 1b (dashed curve). Thesudden decrease of the buoyancy frequency, at r/R ∼ . for model 1aand r/R ∼ . for model 1b, marks the lower boundary of the convectiveenvelope in the corresponding model. most promising from the observational point of view. STRUCTURE OF THE G-MODE CAVITY
Internal gravity waves have frequencies below the buoy-ancy (or Brunt-V¨ais¨al¨a) frequency and propagate only wherethere is no convection. While on the red-giant branch a staris powered by hydrogen burning in a shell surrounding an in-nert radiative helium core. The g-mode propagation cavityextends essentially from the stellar center to the bottom of theconvective envelope. Once stable core-helium burning starts,the central part of the core becomes convective, reducing thesize of the g-mode cavity. For massive stars the transitionbetween these two phases is smooth. However, according tocurrent standard 1D stellar models, in lower-mass stars with adegenerate helium core, this transition involves a successionof off-centered helium flashes (Bildsten et al. 2012) (see alsoSalaris et al. 2002, for a general overview of red-giant evolu- tion).The propagation speed of the gravity waves depends on thebuoyancy frequency. Consequently, variations in the buoy-ancy frequency inside the g-mode cavity may perturb the pe-riods of high-radial-order modes away from their asymptoticvalue. Sharp variations in the buoyancy frequency duringthe red-giant phase usually result from local changes in thechemical composition. Examples of these variations are il-lustrated in Figure 1 where we show two red-giant modelsat different evolution stages (panel a), prior to and duringthe helium-flash phase, respectively, and their correspondinghelium abundances (panel b) and buoyancy frequencies, N (panel c), for the core region, where N is defined by the rela-tion, N = g (cid:18) γ d ln p d r − d ln ρ d r (cid:19) . (1)Here, r is the distance from the stellar center in a spher-ical coordinate system ( r , θ , ϕ ) and g , γ , p and ρ are,respectively, the gravitational acceleration, the first adi-abatic exponent, the pressure, and the density in themodel. The models were computed with the evolu-tion codes ASTEC (Christensen-Dalsgaard 2008b) andMESA (Paxton et al. 2013), respectively. Two spikes are vis-ible in the buoyancy frequencies. The spikes located at rela-tive radii of ≈ . (model 1a) and ≈ . (model 1b) re-sult from the chemical-composition variation at the hydrogen-burning shell. The spike furthest out in model 1a, at a rela-tive radius of ≈ . , results from strong chemical gradientsleft behind by the retreating convective envelope which, dur-ing the first dredge-up, extended to the region where the gashad previously been processed by nuclear burning. As theconvective envelope retreats, the g-mode cavity expands toinclude the sharp variation in the chemical composition; thiseventually disappears, when reached by the hydrogen-burningshell which is moving out in mass as the helium core grows.In the case of low-mass stars, this takes place while the staris still on its way up the red-giant branch, when it reachesthe well-known luminosity bump. The bump shows itself as atemporary decrease in luminosity when the hydrogen-burningshell gets close to the sharp variation in the chemical compo-sition. As a result of the decrease in the average mean molec-ular weight in the region just above the shell, the luminosityof the hydrogen-burning shell decreases. This is followed bya return to increasing luminosity when the hydrogen-burningshell reaches the sharp variation. (Hekker and Christensen-Dalsgaard, in preparation). Finally, the innermost spike inmodel 1b, at a relative radius of ≈ . , results fromthe chemical composition variation caused by a helium flash.Spikes in the buoyancy frequency may have yet a different ori-gin from those discussed above. In particular, they can resultfrom sharp variations in chemical composition left by retreat-ing convective cores that were active either during the mainsequence or during the helium-core-burning phase. These willbe illustrated in section 5 where we look at sharp buoyancyvariations along the red-giant evolution more broadly. Since the model does not include diffusion, the dredge-up should leavebehind a discontinuity in composition. However, the numerical treatment ofthe mesh in the ASTEC calculation causes numerical diffusion which leadsto some smoothing of the composition profile and hence broadening and low-ering of the buoyancy-frequency spike, as is evident in Fig. 1. A similar butless pronounced effect appears to be present in the MESA models. For stars more massive than 2.2 M ⊙ helium burning is ignited before thehydrogen-burning shell reaches the discontinuity and no bump occurs on thered-giant branch. uoyancy glitches in the cores of red giants 3Whether or not the spikes in the buoyancy frequency aresufficiently sharp to produce a significant deviation of the fre-quencies of high-radial-order g modes from their asymptoticvalue depends on how the characteristic width of the spikescompares with the local wavelength. A comparison of thetwo scales is illustrated in Figure 2 for the two models pre-sented in Figure 1. The eigenfunction Ψ shown in this figureis related to the Lagrangian pressure perturbation (see sec-tion 3 for a precise definition). Clearly, the width of the innerspike is much larger than the local wavelength in both models.Hence, this spike is seen as a smooth variation by the waveand is well accommodated by asymptotic analysis. In con-trast, at the outermost spikes the buoyancy frequency variesat a scale comparable to or shorter than the local wavelength.We therefore may expect these features - hereafter glitches - to change the oscillation frequencies from their asymptoticvalue. In that case, the period spacing may also deviate fromthe fixed value predicted by the asymptotic theory (Tassoul1980). GLITCH EFFECT ON THE PERIOD SPACING: TOY MODEL
In this section we illustrate the effect of a buoyancy glitchon the oscillation frequencies and, consequently, on the periodspacing. To accomplish that we consider first an analytical toymodel in which the glitch is assumed to be infinitely narrowand well modeled by a Dirac delta function. In the analysiswe first introduce the analytical description of the problem,then consider the effect of the glitch on pure g modes and,finally, consider the same effect when the latter couple to theenvelope p modes.
Setting the problem
Our starting point for the analytic analysis is a second-order differential equation for the radial dependent part ofthe Lagrangian pressure perturbation, δp , derived from theequations that describe linear, adiabatic perturbations to aspherically symmetric star, under the Cowling approximation(hence neglecting the Eulerian perturbation to the gravita-tional potential). This equation can be written in the stan-dard wave-equation form (Gough 1993, 2007) by adopting Ψ = ( r /gρf ) / δp as the dependent variable, where f is a function of frequency and of the equilibrium structure(the f-mode discriminant defined by equation (35) of Gough(2007)). In terms of this variable, the wave equation takes theform, d Ψd r + K Ψ = 0 , (2)with the radial wavenumber K defined by, K = ω − ω c − L r (cid:18) − N ω (cid:19) . (3)Here, L = l ( l + 1) and l is the angular degree of the mode, c is the sound speed, and ω c and N are generalizations ofthe usual critical acoustic frequency and buoyancy frequency,respectively, which account for all terms resulting from thespherical geometry of the problem. The exact forms of thesequantities can be found in equations (5.4.8) and (5.4.9) ofGough (1993), which are reproduced in Appendix B of thispaper. The radii where K = 0 define the turning points ofthe modes. These separate the regions where waves can prop-agate (where K > ) from where they are evanescent (where K < ). For typical red giants, including the models discussed insection 2, there are two separate propagation regions de-fined by four turning points. We denote these points as r , r , r , and r . We note that for the models under consid-eration, r and r are essentially at the center of the star( r = 0 ) and at the stellar photosphere ( r = R ), respec-tively. The propagation regions and the turning points r , r , and r are illustrated in Figure 3 for a representativedipole mode in our model 1a, where we show the corre-sponding mode eigenfunction derived with the pulsation codeADIPLS (Christensen-Dalsgaard 2008a). The evanescent re-gion, which is located between the turning points r and r ,separates the two propagation cavities. To its left is the so-called g-mode cavity while to its right we have the p-modecavity.In practice, N (equation (1)) is a very good approximationof N everywhere except very close to the center of the starand in the evanescent region between the two cavities, wherethe latter diverges. These differences will be fully accountedfor in future work (Cunha et al., in preparation). Nevertheless,in the toy model presented here, we will approximate N by N from the outset. Despite this and other approximations thatwill follow, our toy model retains all important features seenin the full numerical solutions obtained with ADIPLS and,as will become clear in section 4, will be important for thecorrect interpretation of the results of the latter. Effect on pure g modes
The eigenvalue condition
To understand the impact of a glitch on the oscillation fre-quencies it is convenient to start by analyzing a simpler prob-lem in which we ignore any coupling between the g and pmodes. This coupling will be considered in section 3.3.In order to find the oscillation frequencies for the pure gmodes we need to impose adequate boundary conditions tothe solution of equation (2). Towards the center of the starthis condition is that Ψ decreases exponentially as r goes tozero. Moreover, because we are ignoring any coupling andbecause the g-mode cavity is located at such depth that thestellar atmosphere hardly influences the solutions, the condi-tion towards the envelope also needs to be that Ψ decreasesexponentially for r ≫ r . From the asymptotic analysis ofequation (2), which ignores the effect of the glitch, we knowthat the solution, Ψ in , satisfying the inner boundary conditionhas the form (Gough 1993), Ψ in ∼ ˜Ψ in K − / sin (cid:18)Z rr K d r + π (cid:19) , (4)in the region r ≪ r ≪ r , where, following the notation ofGough, we have used the symbol ∼ to indicate that the twosides of the equation are asymptotically equal. Here, ˜Ψ in is aconstant and the subscript on K indicates that we are not ac-counting for the glitch. This inner solution is illustrated in theinset in Figure 3 by the continuous yellow curve. Likewise,the asymptotic solution to equation (2), Ψ out , that satisfies theouter boundary condition can be written as (Gough 1993), Ψ out ∼ ˜Ψ out K − / sin (cid:18)Z r r K d r + π (cid:19) , (5)in the region r ≪ r ≪ r , where ˜Ψ out is also a constant.This outer solution is illustrated in Figure 3 (inset) by thedashed yellow curve. Cunha et al. −1•10 −1•10 Ψ ; N / π ( µ H z ) H−shell 1 st dredge−up signature (a) Model 1a −1•10 −1•10 Ψ ; N / π ( µ H z ) He−flashsignature H−shell(b) Model 1b F IG . 2.— Asymptotic eigenfunction (solid curve) and the buoyancy frequency (dashed curve) for: (a) model 1a and (b) model 1b. The eigenfunctions havearbitrary amplitude and are for characteristic eigenfrequencies of these models. The arrows mark the positions of the buoyancy frequency spikes discussed in thetext and seen also in Figure 1c. (r g ρ f ) / Ψ r r r g−modecavity evanes.region p−modecavity −3 −3 −3 −3 −3 −3 −0.6−0.4−0.20.00.20.40.6 close−up F IG . 3.— Normalized eigenfunction, as function of relative radius,for the dipole mode with frequency ν = 51 . µ Hz, computed withthe pulsation code ADIPLS for our model 1a. The chosen eigenfunction ( r gρf ) / Ψ = r δp has the dimensions of energy and is normalized tobe 1 at its maximum value. The vertical, blue dashed lines show the positionof r and r , the two turning points bounding the evanescent region. Theoutermost turning point, r , is also shown, while the innermost turning point, r , is outside the plotted range. The g-mode cavity is located between theunseen r and r and the p-mode cavity is located between r and r . Theclose-up shows a comparison between the numerical (in black) and analytical(in yellow) eigenfunctions in a particular region, well inside the g-mode cav-ity. The continuous yellow curve represents the inner solution derived fromequation (4), while the dashed yellow curve represents the outer solution de-rived from equation (5). Since equations (4) and (5) are both valid well inside theg-mode cavity, they must be the same. The requirement thatthey be the same provides the eigenvalue condition (the condi-tion that determines which oscillation (eigen)frequencies areallowed by the above boundary conditions). In this case, theeigenvalue condition translates to Z r r K d r = π (cid:18) n − (cid:19) , (6)where n is a positive integer. Hence, it is this condition thatensures the two yellow curves match (Figure 3 (inset)). Thephase shift that these solutions show in relation to the full ADIPLS solution (solid, black curve) is due to their not in-cluding the coupling to the p modes.Next, we include the effect from a glitch in the buoyancyfrequency. To keep the toy model simple we will initiallyassume that the glitch appears at a single position in radius, r = r ⋆ , well inside the g-mode cavity, such that the asymp-totic solutions (4) and (5) are still valid on either side of it (thisassumption will be relaxed in section 3.2.3). Accordingly, werepresent the glitch by a Dirac delta function, δ , such that thebuoyancy frequency becomes, N = N [1 + Aδ ( r − r ⋆ )] , (7)where A has dimensions of length and is a measure of thestrength of the glitch, and N is the glitch-free buoyancy fre-quency. By imposing continuity of the solutions given byequations (4) and (5) at r = r ⋆ we find, ˜Ψ in = sin (cid:16)R r r ⋆ K d r + π (cid:17) sin (cid:16)R r ⋆ r K d r + π (cid:17) ˜Ψ out . (8)Because under the approximation considered here the glitchis infinitely narrow, the first derivative of the solution is notcontinuous at r = r ⋆ . The condition to be imposed on thederivative can be found by integrating the wave equation (2)once in a finite region of width ǫ across the glitch and thentaking the limit when ǫ goes to zero. Accordingly, we have, Z r ⋆ + ǫr ⋆ − ǫ d Ψd r + Z r ⋆ + ǫr ⋆ − ǫ K Ψ = 0 , (9)where now K takes the glitch into account, differing from K only at r = r ⋆ , where N differs from N . Well inside theg-mode cavity K (equation (3), with N replaced by N ) maybe approximated by, K ≈ LNω r , (10)and, thus, we find, (cid:12)(cid:12)(cid:12)(cid:12) dΨ out d r − dΨ in d r (cid:12)(cid:12)(cid:12)(cid:12) r ⋆ = − AK ( r ⋆ ) Ψ ( r ⋆ ) , (11) Strictly speaking, the continuity condition is satisfied by δp . However,we have verified from the numerical solutions computed with ADIPLS thatthis condition is also very closely satisfied by Ψ . uoyancy glitches in the cores of red giants 5when ǫ → .By differentiating equations (4) and (5) and neglecting thesmall terms resulting from the derivatives of the amplitudes, K − / , we find, after substituting in equation (11), ˜Ψ out K / ( r ⋆ ) cos (cid:18)Z r r ⋆ K d r + π (cid:19) +˜Ψ in K / ( r ⋆ ) cos (cid:18)Z r ⋆ r K d r + π (cid:19) = A ˜Ψ out K / ( r ⋆ ) sin (cid:18)Z r r ⋆ K d r + π (cid:19) . (12) Using the continuity condition (8) and the fact that K ( r ⋆ ) ≈ LN ( r ⋆ ) /ωr ⋆ ≡ LN ⋆ /ω r ⋆ , equation (12) becomes sin (cid:18)Z r r K d r + π (cid:19) = A LN ⋆ r ⋆ ω sin (cid:18)Z r ⋆ r K d r + π (cid:19) sin (cid:18)Z r r ⋆ K d r + π (cid:19) . (13) Equation (13) provides us the eigenvalue condition in thepresence of a glitch. We note that this condition differs fromthose derived following similar principles by Brassard et al.(1992) and Miglio et al. (2008) for g modes in white dwarfsand main-sequence stars, respectively, in particular becausewe model the glitch by a Dirac delta rather than a step func-tion. To write the eigenvalue condition in a form that can be com-pared with the one derived without the glitch, we use the rela-tion Z r ⋆ r K d r + π Z r r K d r + π − Z r r ⋆ K d r − π . (14) Introducing equation (14) in equation (13) we find, after somealgebra, sin (cid:18)Z r r K d r + π (cid:19) = 0 . (15)In the above, the phase Φ is defined by the following systemof equations, B cos Φ = 1 − A LN ⋆ r ⋆ ω sin (cid:18)Z r r ⋆ K d r + π (cid:19) × cos (cid:16)R r r ⋆ K d r + π (cid:17) B sin Φ = A LN ⋆ r ⋆ ω sin (cid:18)Z r r ⋆ K d r + π (cid:19) , (16)where B is a function of frequency, also defined by the systemof equations (16). Thus, we arrive at the final form of theeigenvalue condition for our toy model when including theglitch in the buoyancy frequency, namely, Z r r K d r = π (cid:18) n − (cid:19) − Φ . (17)By comparing equations (6) and (17) we see that the frequen-cies of pure g modes are modified by the glitch through thefrequency dependent phase Φ only. We note that the mathematical derivation of the eigenvalue condition inthe present work differs substantially from that presented by Brassard et al.(1992) and Miglio et al. (2008), in that it is based on a single equation forthe variable Ψ , rather than on the equations for variables related to the radialdisplacement and the Eulerian pressure perturbation. Effect on the period spacing
Having considered the effect of the glitch on the g-modefrequencies, we now turn to the impact it has on the g-modeperiod spacing, defined as the difference between the periodsof two modes of the same degree and consecutive radial or-ders. A possible way to proceed would be to solve the eigen-value condition numerically (as done, e.g. , by Brassard et al.1992; Miglio et al. 2008) to find the oscillation frequenciesand, thus, compute the period spacings. Instead, we opt forderiving an analytical expression that directly describes theperiod spacings as a function of the oscillation frequency interms of the glitch parameters, which we find may be a use-ful path for the future comparison with the period spacingsderived from real data.Under the asymptotic approximation, the period spacing forhigh-radial order g modes, ∆ P as , is essentially constant andgiven by (Tassoul 1980), ∆ P as ≃ π ω g , (18)where, ω g ≡ Z r r LN r d r. (19)To see how the period spacing is modified from the asymp-totic value in the presence of the glitch, we first re-write theeigenvalue condition (17) as, ω g π P + Φ ≈ π (cid:18) n − (cid:19) , (20)where P = 2 π/ω is the oscillation period (and we recallthat Φ is itself a function of P ). In deriving the above, wehave used the fact that well within the g-mode cavity K ≈ LN /ωr to approximate R r r K d r by R r r LN /ωr d r . Be-cause K goes to zero towards the turning points, this ap-proximation leads to a slight overestimate of the value of theintegral. However, it allows us to derive a simple analyticalexpression for the period spacing.Next, we follow Christensen-Dalsgaard (2012) and definea function G ( P ) , by G ( P ) = ω g π P + Φ . (21)Using expression (20) and the definition of G , we can thenwrite π ≈ G ( P n +1 ) − G ( P n ) ≈ d G d P ∆ P, (22)where ∆ P = P n +1 − P n is the period spacing in the presenceof the glitch, or, equivalently, ∆ P ≈ π d G d P . (23)By differentiating G with respect to P and substituting inequation (23) we find that this period spacing is related to theasymptotic period spacing by ∆ P ≈ ∆ P as − ω ω g dΦd ω ≡ ∆ P as − F G . (24) Note that the analysis of the simplified model discussed in Section 4.2 ofthat paper contains two errors that fortuitously cancel. One is the neglect ofa singularity in the asymptotic expression (equation (1) of that paper) in theevanescent region. The second is a simple sign error in the analysis leadingto equation (22) of that paper. The combined effect of the errors is that theequation has the correct form, and the remaining analysis is still valid.
Cunha et al.The deviation of the period spacing from its asymptotic valueis reflected in the term F G . Its dependence on the glitch pa-rameters can be made explicit by solving the system of equa-tions (16) . Defining, ω ⋆ g ≡ Z r r ⋆ LN r d r (25)and making R r r ⋆ K d r ≈ ω ⋆ g we find, F G = ALN ⋆ r ⋆ ω g B (cid:26) ω ⋆ g ω cos (cid:18) ω ⋆ g ω (cid:19) + (cid:18) − ALN ⋆ ω ⋆ g r ⋆ ω (cid:19) sin (cid:18) ω ⋆ g ω + π (cid:19)(cid:27) , (26)and B = (cid:20) − ALN ⋆ r ⋆ ω cos (cid:18) ω ⋆ g ω (cid:19)(cid:21) + (cid:20) ALN ⋆ r ⋆ ω sin (cid:18) ω ⋆ g ω + π (cid:19)(cid:21) . (27)In the above, the dependance of the period spacing on thecharacteristics of the glitch is expressed by the parameters A (glitch strength) and r ⋆ (glitch position).The period spacing derived from expression (24) for ourmodel 1a is illustrated in Figure 4a (solid curve). It variesaround the asymptotic value (horizontal dotted line), form-ing relatively narrow dips that alternate with wider, less pro-nounced humps. The narrowing of the dips with decreasingfrequency is due to the /ω dependence of the argumentsof the sinusoidal functions present in expressions (26) and(27). Because all sinusoidal functions present in the definitionof F G and B (equations (26) and (27), respectively) can bewritten in terms of the argument ω ⋆ g /ω ( ≡ π − ω ⋆ g P ), weexpect the distance between dips to be constant in period andequal to π /ω ⋆ g . Thus, it provides a measure of the depth ofthe glitch in terms of the normalized buoyancy depth, ω r g ω g ≡ ω g Z r r LN r d r, (28)which is analogous to the normalized acoustic depth usedin studies of acoustic waves . Down to the middle of thecavity (located at ω r g /ω g = 0 . ), the deeper the glitch loca-tion, the smaller the spacing between dips. For yet deeperglitches ( ω r g /ω g > . ), the spacing between dips increasesagain, mirroring the separation found for a glitch positionedat − ω r g /ω g (e.g. Montgomery et al. 2003).For model 1a, if we take r ⋆ = 0 . R (the radius atwhich N − N is maximum) we find ω ⋆ g /ω g = 0 . .Hence the glitch is very close to the the edge of the cavitywhen measured in terms of ω r g /ω g (equation (28)).Figures 4b and 4c show the results of moving the glichdeeper inside the cavity, to ω ⋆ g /ω g = 0 . and ω ⋆ g /ω g = We emphasize that unlike the case of the dips caused by mode coupling,the glitch-induced dips are not associated with the presence of an extra mode.Thus, the decrease in the period spacing at the dips is fully compensated byits increase at the wider, less-pronounced humps. Here we adopted the notation of Montgomery et al. (2003), wherethe buoyancy depth is defined as the inverse of a period, resulting inthe sinusoidal part of the eigenfunction having approximately the form sin( π − ω r g P + π/ . However, we note that the term buoyancy depth issometimes used for L/ω r g (e.g. Miglio et al. 2008), instead. (d) close-up (c)(b)(a) F IG . 4.— Period spacing for pure g modes in model 1a. The horizon-tal dotted line shows the glitch-free period spacing, ∆ P as . (a) Results de-rived from expression (24) for the glitch parameters estimated for model 1a, r ⋆ = 0 . R (or ω ⋆ g /ω g = 0 . ) and A = 1 . × − R (see textfor details). (b) Results from expression (24), but for a slightly deeper glitchat ω ⋆ g /ω g = 0 . . (c) Results from expression (24), for an even deeperglitch, at ω ⋆ g /ω g = 0 . . The inset is a close-up of the region enclosed bythe dashed box. Diamonds show the individual modes. See text for details. (d) Results obtained from integrating the wave equation numerically, ignor-ing the coupling to the p modes (solid black curve). Overplotted is the resultderived from expression (24) for the glitch parameters estimated for model 1a(dotted-dashed red curve; same as solid black line in panel a) and the resultobtained from the same expression with the glitch parameters adjusted tothe numerical solution, namely r ⋆ = 0 . R (or ω ⋆ g /ω g = 0 . ) and A = 1 . × − R (red, dashed curve). The latter has been shifted infrequency by 1 µ Hz (see text for details). . , respectively. As expected, the spacing between thedips at fixed frequency gets smaller as the glitch is movedcloser to the center of the cavity (see also figures 8 and 15of Miglio et al. 2008, which show a similar effect for the g-mode period spacings in main-sequence classical pulsators).We note that in producing Figures 4b and 4c we have alsochanged A from the value used in Figure 4a. In Figure 4b, A was chosen such as to maintain the value of the effective glitch strength ˜ A ≡ ALN ⋆ /r ⋆ (see right hand side of condi-tion (13)) unchanged. Thus, the difference in the amplitudesof the patterns seen in Figures 4a and 4b results solely fromuoyancy glitches in the cores of red giants 7the difference in the location of the glitch. For Figure 4c, A was chosen such as to reduce the effective strength by one or-der of magnitude. In this limit of small effective strength theperiod spacing shows symmetric wiggles around the asymp-totic value, instead of the alternating dips and humps seen inthe other two cases. Interestingly, in Figure 4c we can iden-tify a modulation of the period spacing on a scale larger thanthe separation between wiggles. This modulation is more no-ticeable when the distance between adjacent modes becomescomparable with the distance between glitch-induced minima.It is simply a sampling effect, as can be confirmed through in-spection of the inset of Figure 4c. We note, however, that for aglitch positioned at ω ⋆ g /ω g = 0 . , the period spacing betweentwo minima is exactly twice the asymptotic period spacing,creating a perfect sawtooth diagram without the modulationseen in Figure 4c. The modulation introduced by the limitedsampling depends solely on ω ⋆ g /ω g , thus providing an alter-native way to measure the position of the glitch. This is im-portant, because due to the limited frequency resolution of theobservations, it might, in some stars, be easier to detect thislarger scale modulation than the series of glitch-induced vari-ations in the period spacingSince in reality the glitch is not infinitely narrow, estimatingthe parameters r ⋆ and A from a given model requires a littlethought. To estimate r ⋆ one may consider taking either thecenter of the glitch or the position of its maximum amplitude.However, to estimate A we need to consider how to trans-form the glitch in the stellar model into its infinitely narrowcounterpart while keeping the area under the glitch essentiallyunchanged. Recalling that the Dirac δ can be defined as thelimit, δ = lim ǫ → + ǫ √ π e − ( r − r ⋆ ) /ǫ , (29)and taking ǫ to be the characteristic half width of the glitch wefind, from equation (7), ∆ N N (cid:12)(cid:12)(cid:12)(cid:12) r ⋆ ≈ Aǫ √ π , (30)where ∆ N = N − N is the glitch induced deviation in thesquare of the buoyancy frequency. Taking r ⋆ to be the radiusat which ∆ N is maximum, we estimate that ǫ = 0 . × − R and A = 1 . × − R , for our model 1a. Numerical solution for pure g modes
In the next step we will move to a more realistic descrip-tion of the effect from a glitch on the period spacing. Figure 2shows that a Dirac δ function is not a realistic description ofthe glitch in our stellar model. In principle, the analyticalanalysis could include a more realistic function to describethe glitch. However, that would have increased the complexityof the analysis whose main purpose was to provide a simpleunderstanding of the seismic impact of the glitch. To obtainmore realistic results we therefore solve equation (2) numer-ically, for the case of pure g modes, by adopting N from thestellar structure model. By comparing the results with thosederived analytically, we can investigate the impact of the ap-proximations made in the analytical analysis and produce re-sults that are more directly comparable with the full numericalsolutions from ADIPLS that will be discussed in section 4.To find the numerical solutions for pure g modes we ap-proximate K in equation (2) by K = − L r (cid:18) − N ω (cid:19) . (31) The equation is then solved using a standard fourth orderRunge-Kutta method with adaptive step size control and theeigenfrequencies are found by imposing that the solutions sat-isfy the boundary conditions Ψ = 0 at r = 0 and r = R .The results are presented in Figure 4d (solid, black curve).Comparison with the analytical results derived in section 3.2.2for the glitch parameters estimated for model 1a (Figure 4a;also shown as dotted-dashed, red curve in Figure 4d) pro-vides a number of interesting conclusions. First, and mostimportantly, the general form of the period spacing variationis similar in the two cases, reemphasizing that the effect ofthe glitch is the formation of narrow dips that alternate withwider, much less pronounced humps. However, it is also clearthat both the depth of the dips and their separation in fre-quency are different in the analytical and numerical results.To understand these differences and their potential impact onglitch-parameter inferences based on the analytical model weadjust the glitch parameters such as to match the analytical tothe numerical results. The new analytical solution is shownby the red-dashed curve in Figure 4d. The solution had tobe shifted in frequency by 1 µ Hz because the approximation R r r ⋆ K d r ≈ ω ⋆ g made earlier introduces a phase shift betweenthe analytical and the numerical results. In practice, this maybe accounted for by adding a phase to the arguments of thesinusoidal functions in the analytical model, thus, increasingthe number of adjustable parameters by one.The rematched glitch location, r ⋆ , is almost unchanged(shifted by only ≈ of the glitch width), while the strengthof the glitch is about smaller. The latter reflects that theperiod spacing variations have a lower amplitude in the nu-merical results. This difference in the amplitudes and, morenotably, the fact that they vary in opposite ways with fre-quency, is a consequence of the non-negligible width of theglitch. Towards lower frequency the g-mode wavelength be-comes shorter. Seen by the wave, a spike in the buoyancyfrequency therefore appears smoother (less of a glitch). As aresult, the amplitude of the dips in the period spacing becomessmaller towards lower frequency. However, in the analyticalanalysis the spike is modelled as being infinitely narrow. Itis therefore always much narrower than the local wavelengthand, hence, no reduction of the amplitude is seen.A second striking difference seen in Figure 4d concernsthe small-scale variations that are present in the numerical re-sult, but absent in the analytical curve. Using our analyticalmodel (equation (24)) we found that these small-scale varia-tions would originate from a glitch at the hydrogen-burningshell. Given that the spike in the buoyancy frequency at thisposition is not seen as a glitch by the wave (as discussed insection 2) we inspected the derivatives of N and found thatthey show a high level of variation at much smaller scales thanthe local wavelength. By smoothing the derivatives and re-calculating the period spacing, the small-scale variations dis-appeared. We therefore conclude that their origin is purelynumerical and has no physical meaning. Coupling with the p modes
We now consider the same problem as in section 3.2, but in-clude the coupling between the g and p modes. That requiresreplacing solution (5), valid for pure g modes, by the solutionthat accounts for mode coupling.When we consider that waves can propagate also in the p-mode cavity, the asymptotic solution to equation (2) that isvalid well within the evanescent region is no longer an expo-nentially decaying function, but rather a linear combination Cunha et al.of an exponentially decaying and an exponentially growingfunction. The solution to equation (2) that matches the re-quired linear combination has the form (Gough 1993), Ψ out ∼ ˆΨ out K − / sin (cid:18)Z r r K d r + π ϕ (cid:19) , (32)in the region r ≪ r ≪ r , where ˆΨ out is a constant and ϕ is a frequency dependent phase, which is uniquely defined bythe coefficients of the linear combination mentioned above.Its form will be discussed below.Equations (4) and (32) provide us the eigenvalue conditionin the presence of mode coupling and no glitch, namely, Z r r K d r = π (cid:18) n − (cid:19) − ϕ. (33)The coupling phase ϕ can be obtained from the eigenvaluecondition derived by Shibahashi (1979) (see also Unno et al.(1989)), based on an asymptotic analysis of the equations forthe radial component of the displacement and for the Eulerianpressure perturbation, under the Cowling approximation. Be-cause the oscillation frequencies must be independent of thevariable used to express the pulsation problem, the eigenvaluecondition derived by Shibahashi (1979) must be equivalent toour eigenvalue condition (33). Comparing the two we find(see appendix A, for details), ϕ ≈ atan q tan (cid:18) ω − ω a ω p (cid:19) , (34)where q is a frequency dependent coupling factor that cantake values in the range ≤ q < / , where smaller val-ues imply a weaker coupling. Moreover, ω a are the oscilla-tion frequencies that would be obtained for p modes in theabsence of coupling (the acoustic resonant frequencies) and ω p = (cid:16)R r r c − d r (cid:17) − is approximately twice the asymptoticlarge separation. The corresponding period spacing, derivedas in section 3.2.2, is given by, ∆ P ≈ ∆ P as − ω ω g d ϕ d ω ≡ ∆ P as − F C . (35)The period spacing derived from expression (35) for ourmodel 1a is shown in Figure 5a. The dips associated with thecoupling to the p modes are equally spaced in frequency andlocated at the acoustic resonant (cyclic) frequencies ( ω a / π ).At these frequencies the denominator inside the arctan of (34)goes through zero and, as a consequence, ϕ varies rapidlywith frequency. The large derivative in frequency of ϕ therefore makes F C large, producing the dips in the pe-riod spacing. This is in agreement with the discussions byChristensen-Dalsgaard (2012) and Mosser et al. (2012b) andwith the period spacing derived from the analysis of real datafor red-giant stars (e.g. Beck et al. 2011).Next, we add the effect of the glitch. Following the samesteps as in section 3.2 we find that the eigenvalue condition inthe presence of mode coupling and a glitch is given by Z r r K d r = π (cid:18) n − (cid:19) − Φ − ϕ, (36) where Φ is now defined by the following system of equations, B cos Φ = 1 − A LN ⋆ r ⋆ ω sin (cid:18)Z r r ⋆ K d r + π ϕ (cid:19) × cos (cid:16)R r r ⋆ K d r + π + ϕ (cid:17) B sin Φ = A LN ⋆ r ⋆ ω sin (cid:18)Z r r ⋆ K d r + π ϕ (cid:19) . (37)We emphasize that both B and Φ now depend on ϕ . This isto be expected, since the effect of the glitch on the oscilla-tions depends critically on the phase of the eigenfunction atthe depth where the glitch is located, and that phase is influ-enced by the coupling. As a consequence, the relative devi-ation of the period spacings from the asymptotic value whenboth a glitch and mode coupling are present is different fromwhat would be found by simply adding the deviations gener-ated by the coupling and by the glitch separately. This factcan be readily seen in the period spacing derived from theeigenvalue condition (36), which has the form ∆ P ≈ ∆ P as − ω ω g (cid:20) dΦd ω + d ϕ d ω (cid:21) ≡ ∆ P as − F G , C . (38)As before, the deviation of the period spacing from its asymp-totic value is reflected in the second term, F G , C , present inthe denominator of expression (38). Its dependence on theglitch parameters can be made explicit by solving the systemof equations (37), from which we obtain F G , C = ω ω g d ϕ d ω (cid:26) ALN ⋆ r ⋆ ωB (cid:20) cos (cid:18) ω ⋆ g ω + 2 ϕ (cid:19) − ALN ⋆ r ⋆ ω sin (cid:18) ω ⋆ g ω + π ϕ (cid:19)(cid:21)(cid:27) + ALN ⋆ r ⋆ ω g B (cid:26) ω ⋆ g ω cos (cid:18) ω ⋆ g ω + 2 ϕ (cid:19) + (cid:18) − ALN ⋆ ω ⋆ g r ⋆ ω (cid:19) sin (cid:18) ω ⋆ g ω + π ϕ (cid:19)(cid:27) , (39)where, B is now given by B = (cid:20) − ALN ⋆ r ⋆ ω cos (cid:18) ω ⋆ g ω +2 ϕ (cid:19)(cid:21) + (cid:20) ALN ⋆ r ⋆ ω sin (cid:18) ω ⋆ g ω + π ϕ (cid:19)(cid:21) . (40)We see that F G , C has two terms, each marked by a set ofcurly brackets. If we assume there is no coupling, meaningthat ϕ is zero for all frequencies, the first term vanishes be-cause d ϕ/ d ω = 0 , and the second term becomes equal toequation (26), hence reducing F G , C to F G as expected. If in-stead we assume there is no glitch, which translates to A = 0 ,then the second term vanishes, and the first term becomesequal to F C defined in equation (35), reducing F G , C to F C ;again as one would expect. Finally, we consider that both cou-pling and glitch are present, but we look specifically at whathappens at the acoustic resonance where the coupling domi-nates the expression for F G , C . Here, the frequency derivativeof ϕ is very large, and hence the first term is generally muchuoyancy glitches in the cores of red giants 9
20 30 40 50 60020406020 30 40 50 60Frequency ( µ Hz)0204060 ∆ P ( s ) (a)
30 40 50 6065.066.568.0
20 30 40 50 60020406020 30 40 50 60Frequency ( µ Hz)0204060 ∆ P ( s ) (b)
30 40 50 6065.066.568.0 F IG . 5.— Period spacing derived from the analytical approach for model 1a. The inset shows a close-up around ∆ P as (dotted box). (a) case with modecoupling but no glitch, computed from expression (35). (b) case with mode coupling and glitch, computed from expression (38) (black) and case with glitch andno mode coupling, computed from expression (24) (red), adopting r ⋆ = 0 . R and A = 1 . × − R . larger than the second. However, we still have the two ex-tra cos and sin ‘glitch-induced’ terms within the first set ofcurly brackets compared to just F C ( = ω /ω g d ϕ/ d ω ). Thisshows that the mode coupling, and therefore also the dips lo-cated at the acoustic resonance frequencies, is influenced bythe glitch.The period spacing obtained from expression (38) is illus-trated in Figure 5b. Comparing with Figure 5a, we see that thecombined effect of the glitch and the coupling on the periodspacing is predominantly a change in the depth of the dips atthe acoustic resonant frequencies. Whenever a dip caused bythe glitch coincides with a dip caused by the coupling withthe p modes, the depth of the latter is reduced. But if a humpproduced by the glitch coincides with the dip caused by thecoupling, the depth of the dip increases. This behaviour isoposite to what would be found if the the combined effectwere simply the sum of the deviations to the asymtotic periodspacings caused by each effect separatly. The predicted be-haviour can be understood if we recall that the extent to whichg modes couple to a p mode depends critically on the proxim-ity of their frequencies (assuming everything else remains un-changed, which is the case here). A glitch-induced dip in theperiod spacings means the g modes are locally more denselypacked, as compared to the asymptotic case. Thus, if the dipcoincides with an acoustic resonant frequency the number of gmodes coupling to the p mode is greater, resulting in a widerand, consequently, shallower coupling dip. However, if anacoustic resonant frequency coincides with a glitch-inducedhump, the number of g modes coupling to the p mode is re-duced, resulting in a thinner, hence, deeper coupling dip. INTERPRETATION OF FULL NUMERICAL SOLUTIONS
In this section we consider the numerical solution of the fullpulsation equations, including the perturbation to the gravita-tional potential, for the models introduced in section 2 and in-terpret them in the light of the results found with the toy modelanalysis presented in section 3. The full numerical solutions were computed with the pulsation code ADIPLS. Care wastaken to have an adequate number of mesh points with appro-priate distribution to resolve the rapidly varying eigenfunc-tions in the g-mode cavity.The period spacing derived for our model 1a from the fullnumerical solutions is shown in Figure 6 (solid curve). Com-paring with Figure 5 we see that the dips associated with theacoustic resonant frequencies are closer in frequency in the re-sults from the full numerical solutions than in the toy model.This reflects the fact that the large separation computed fromthe eigenfrequencies is smaller (by about 5 % in the currentcase) than the corresponding asymptotic value, in accordancewith the results of previous studies (e.g. Stello et al. 2009;Belkacem et al. 2013; Mosser et al. 2013). Letting aside thatdifference, we see that in the full numerical solutions thedips associated with the acoustic resonant frequencies showa depth variation resembling what is seen using our toy model(Figure 5b). Comparison with the period spacing derived nu-merically considering only the g modes (section 3.2.3) (red,dashed curve in Figure 6) confirms that the glitch in thebuoyancy frequency is the cause of the larger-scale modula-tion seen in the full solution (see inset), and, thus, that thecombined effect of the glitch and mode coupling is to re-duce/increase the depth of the dips positioned at the acous-tic resonant frequencies when they coincide with a glitch-induced dip/hump.To illustrate how a glitch in the buoyancy frequencycould be revealed in observational data, Figure 7 shows(a) / √ Q nl representative of relative mode amplitude(Christensen-Dalsgaard 2004; Aerts et al. 2010), and both (b)a frequency- and (c) a period-´echelle diagram correspondingto model 1a. Here, Q nl is a measure of the inertia of a modeof radial order n and degree l , relative to that of radial modesdefined by Q nl = I nl I l =0 , (41)0 Cunha et al.
20 30 40 50 60020406020 30 40 50 60Frequency ( µ Hz)0204060 ∆ P ( s )
30 40 50 606566676869 F IG . 6.— Period spacings derived from ADIPLS for model 1a, includingthe effects of the glitch and of the coupling between the g-modes and the p-modes (solid curve), compared to the integration of the wave equation (2),ignoring the coupling with p modes (red, dashed curve). The inset shows aclose up around ∆ P as (dotted box). where I nl is the surface normalized mode inertia, I nl = R R s (cid:2) ξ + l ( l + 1) ξ (cid:3) ρr d rM ξ r ( R s ) , (42)and I l =0 is obtained by interpolating I n to the frequency ofthe mode under consideration. Moreover, ξ r and ξ h are the ra-dial and horizontal components of the displacement, respec-tively, M is the stellar mass, and R s is the surface radius.The frequency ´echelle shows the frequency spectrum (Fig-ure 7a) divided into segments of fixed length that are stackedone above the other. The length of segments equals the av-erage frequency separation between overtone radial modes, ∆ ν , found as the slope of a linear fit to the radial modesversus their order (Grec et al. 1983). In the period-´echellediagram we show only the dipole modes, and here the ab-scissa is the mode period modulo the asymptotic period spac-ing, ∆ P as (equation (18), see e.g., Bedding et al. (2011)). Forclarity, we show only modes of relative amplitude above 5%of the radial modes in the ´echelle diagrams. Noise set aside,the result seen in this model is a broadening of the clusters of‘observable’ dipole modes where the location of the dips co-incides with that of a cluster. We see this effect in Figure 7anear 28 µ Hz and 41 µ Hz (see also Figure 6). In the period-´echelle diagram, the same effect shows as a strong distortionof the usual ‘S’ shaped mode pattern seen between each radialmode order in a glitch-free case (see for example figure 1 ofBedding et al. (2011)).Next we consider the full numerical solutions for ourmodel 1b, located in the core-helium-flash evolution phase.The period spacings derived from the ADIPLS results for thismodel are shown in Figure 8 (solid curve). The existence ofclosely-spaced pronounced dips in the period spacing makesit harder to identify the dips associated with the acoustic reso-nant frequencies in this case. To help with that identification,we mark the frequencies of the radial acoustic modes (verti-cal lines) and recall that the dips produced by the coupling (a)(b) (c) F IG . 7.— Model 1a ( ν max = 45 µ Hz) - (a) Pseudo amplitude spectrumbased purely on mode inertia relative to the radial modes ( / √ Q nl ), (radialin black and dipole in red). (b) ´Echelle diagram including radial (circles) anddipole (triangles) modes. The abscissa is the frequency modulo ∆ ν . Symbolsize follows the peak heights from panel (a). The arrows indicate clustersof dipole modes affected by a glitch-induced dip in the period spacing. (c)Period ´echelle following the notation of panel (b). The abscissa is the periodmodulo ∆ P as . The solid curve connects all the dipole modes. between p and g dipole modes should be positioned roughlymid way between consecutive radial modes. Indeed, singleor double dips of greater depth than their neighbors are foundat the expected frequencies. Comparison of the period spac-ing derived from the full solutions (solid curve) with that de-rived considering only the g modes (red, dashed curve) showsthat the two are similar everywhere, except at the frequen-cies of these more pronounced dips. This confirms that themore pronounce dips are produced by the coupling betweenthe p and g modes and excludes that this coupling is the causefor the other dips. Using our analytical model for the caseof having a glitch but no coupling (equation (24)) we findthat the less pronounced dips are caused by the outer spike inthe buoyancy frequency (Figure 2b), that is, the glitch at thehydrogen-burning shell. Moreover, we also confirm, based onthe analytical model, that despite the glitch being located rela-tively far from the center of the cavity (at ω ⋆ g /ω g = 0 . ) thelarger-scale modulation seen in the period spacing (on a scaleof about 10 glitch-induced dips) is explained by the samplingeffect discussed in section 3.2.2.The separation between glitch-induced dips in model 1b issimilar to the width of the dips associated with the acous-tic resonant frequencies, which makes it difficult to interpretthe combined effect of the glitch and the mode coupling inuoyancy glitches in the cores of red giants 11
16 18 20 22 2460657075808590 16 18 20 22 24Frequency ( µ Hz)60657075808590 ∆ P ( s ) l=0 l=0 l=0 l=0 F IG . 8.— Period spacings derived from ADIPLS for model 1b, includingthe effects of the glitch and of the coupling between the g-modes and the p-modes (solid curve), compared to the integration of the wave equation (2),ignoring the coupling with p modes (red, dashed curve). The vertical, linesindicate the frequencies of the radial modes. this case. Nevertheless, the comparison between the dips at ≈ . µ Hz and . µ Hz , indicates that the combined ef-fect is the same as for model 1a. The latter dip is placed ata glitch-induced hump and, consequently, has its depth in-creased, while the former dip is placed at a glitch-induced dipand has its central depth reduced, forming a double-dip struc-ture. The observational impact of these double-dip structureswill be discussed in section 5. OBSERVING GLITCHES IN RED GIANTS
Next, we search an extensive set of stellar models of variousmasses to locate the stages of evolution where one could po-tentially observe the seismic signature from buoyancy glitchesin red giants. Our stellar models are derived using the ‘de-fault’ work inlist of MESA-v5271 (Paxton et al. 2011, 2013)with the only change that we turned off mass loss. Thesecanonical models do not include diffusion or extra mixing be-yond convection defined by the classical Schwarzschild crite-rion (Schwarzschild 1906). Our search comprises tracks rang-ing 1.0-3.0M ⊙ , all roughly with solar abundance, spanningthe entire evolution from the bottom of the red-giant branch tonear the end of the asymptotic-giant branch. To check that weobtain consistent results we also derived ASTEC tracks for1.0M ⊙ to near the tip of the red-giant branch and for 2.4M ⊙ to the end of helium-core burning. The frequency calcula-tions based on the full numerical solution shown in this sec-tion were made using GYRE (Townsend & Teitler 2013), butspot checks were made to verify that these results were con-sistent with what we obtained using ADIPLS.For each model, we calculate the dipole g-mode frequenciesby solving equation (2) numerically, with K defined by equa-tion (31) as described in section 3.2.3; hence, neglecting thecoupling to the acoustic cavity. The calculation is restricted towithin a ν max /2-wide range centered around the solar-scaled ν max ∝ g/ √ T eff , where T eff is the effective temperature and ν max is the frequency of maximum oscillations power. Thisrange is roughly equal to the full width at half maximum of the excess power observed for solar-like oscillations (Stello et al.2004; Kjeldsen et al. 2005; Mosser et al. 2012a). From theresulting frequencies we then derive the series of pairwise pe-riod spacings, ∆ P , and calculate an index of glitch-inducedvariation in ∆ P to determine if the g-modes are effected by aglitch. We tested two indices, both showing consistent results.One was simply the RMS of the period spacings and the otherwas the height of the strongest peak in the Fourier transformof the series of period spacings versus period. The latter isshown for a section of the 1M ⊙ track in Figure 9(a) indicat-ing a region bracketed by the vertical dotted lines where theindex is above twice the floor level. This phase is thereforeidentified as showing excess variation in ∆ P . Following theapproach described in relation to Figure 7b (section 4), wealso derive the large frequency separation for radial modes, ∆ ν . Before helium ignition
Along the red giant branch we find glitch-induced varia-tions only at one particular phase in evolution lasting roughly5-10 million years. Interestingly, this coincides with the lu-minosity bump. Figure 9(b-h) summarizes the results near thebump for low-mass models ( . ≤ M ≤ . ⊙ ). Modelsbeyond . ⊙ do not show the bump because the glitch fromthe first dredge-up is not reached by the hydrogen-burningshell until after the model is past the tip of the red-giantbranch. The thick red curves indicate the phases of excessvariation in ∆ P . This excess variation can be attributed to theglitch left by the dredge-up as illustrated by model 1a (Fig-ure 9b). The only exception to this picture is along the 2.0M ⊙ track, which shows an extra slightly earlier phase of excessvariation arising from a subtle but interesting combination ofeffects, also resulting in the extra luminosity bump we see forthis mass at 1.026 Gyr. During the main-sequence phase thegradually retreating convective core leaves a steep gradient inmolecular weight (hence a spike in the buoyancy frequency)where the convection reached its maximal extent at youngage. For models below 1.8M ⊙ , the gradient is smoothed awayby the hydrogen-burning shell, which is later established atalmost the same location. However, for models of roughly2.0M ⊙ , the hydrogen-burning shell starts at a smaller radiusrelative to this gradient, and the gradient therefore survives fora while. This allows the star to evolve to the point where thelocal wavelength becomes comparable to the scale of the asso-ciated spike in the buoyancy frequency, giving rise to the firstphase of excess variation in ∆ P that ends when the hydrogen-burning shell finally reaches the location of the spike. In moremassive stars that same spike is erased by the first dredge-upbefore the spike appears as a glitch for the gravity waves. After helium ignition
In Figure 10 we show the result for post-helium ignitiontracks with masses 1.0M ⊙ , 1.6M ⊙ , 2.2M ⊙ , and 2.8M ⊙ .Again, thick red curves indicate evolution phases showingexcess variation in ∆ P . The downward-pointing arrows in-dicate when the last off-center helium sub-flash and associ-ated convection zone reaches the center, signifying the start ofquiescent helium-core burning in the models with degeneratecores before helium ignition (Figure 10a,c). There is no suchequivalent for the higher-mass models, in which a more gen-tle at-center helium ignition starts immediately at the tip of thered giant branch. The upward-pointing arrows mark the end ofhelium-core burning at the so-called asymptotic-giant-branchbump, and the subsequent asymptotic-giant-branch phase. In2 Cunha et al. F IG . 9.— Panel (a): Maximum signal in the Fourier transform of the se-ries of period spacings along the 1M ⊙ track. Vertical dotted lines bracketthe region of glitch-induced variation in ∆ P . Panels (b-h): Close-up of theevolution near the red giant branch luminosity bump as a function of agefor models with fully or partially-degenerate cores. Thick red curves indi-cate phases of excess variation in ∆ P . A MESA- equivalent of model 1adiscussed in sections 2-4 is shown, in panels (a) and (b). the following we will discuss each phase in turn where we seeexcess variation in ∆ P . Low-mass stars
Along the 1.0M ⊙ track we see repeated intervals of excessvariation during the initial helium sub-flashing phase. Eachof these intervals are interspersed by short off-center heliumburning sub-flashes where the g-mode cavity is split in two F IG . 10.— Early (left) and late (right) stages of helium core burning. Thickred curves indicate phases of excess variation in ∆ P . The time in Myrs sincehelium ignition at the tip of the red giant branch is indicated along the topaxis of each panel. Down/Up-ward pointing arrows show the start/end ofquiescent helium-core burning (see text). The left- and right-side annotationof the ordinate applies to both panels. (Bildsten et al. 2012). If both g-mode cavities are taken intoaccount, the resulting effect on ∆ P during this cavity splitdiffers significantly from what is presented by Bildsten et al.(2012), who ignored the inner cavity in their analysis. Theresults including both cavities will be discussed in a forth-coming paper. The intervals with only one g-mode cavityare illustrated by model 1b, discussed in sections 2-4, andmodel 2 (Figure 10a). In Figure 11 we show a multi-facetedview of model 2 including its core structure and the glitchuoyancy glitches in the cores of red giants 13effect on the observed frequencies. Model 2 is similar tomodel 1b except that it has a lower luminosity, hence larger ν max and is therefore more likely to represent a case where ∆ P can be measured in observational data (Mosser et al.2014; Grosjean et al. 2014). As in model 1b, we see a series ofglitch-induced dips in ∆ P (Figure 11b). The associated dipsin mode inertia, or peaks in amplitude (Figure 11c), suggestthat some modes would be observable even if they are far fromthe acoustic resonant frequency. This decrease in the inertiaarises because some, almost pure, g-modes are trapped in theouter part of the g-mode cavity. As a result, we see a split ofthe l = 1 ridge in the ´echelle diagram (Figure 11d). That splitis most evident where one of the glitch-induced dips coincideswith an acoustic resonant frequency (a coupling-induced dip),splitting the coupling dip into two (Figure 11b).In the following quiescent helium-core burning phase wesee no significant variation in ∆ P for our canonical models.However, towards the end of core burning (Figure 10b), theretreating convective core leaves a sharp glitch, which resultsin very high-frequency variation in ∆ P at the asymptotic-giant-branch bump and the early helium-shell burning phase.Figure 12 shows the buoyancy frequency of model 3, whichis representative for models in this phase. The glitch is lo-cated closely to the center of the cavity (at ω ⋆ g /ω g ∼ . ).Hence, the induced period-spacing variations occur over ascale comparable to the separation between two consecutivemodes, which results in a low-frequency modulation in ∆ P on top of the high-frequency variation, as discussed in sec-tion 3.2.2 and illustrated by the analytical result in Figure 4c.The numerical result including adiabatic frequencies and in-ertias of such models is still under investigation and will bepresented in a forthcoming paper. The ∆ P variations indi-cated at the early helium-shell burning phase (Figure 10b, d,f, h) are similar for all masses that we investigated, and orig-inate from the same physical reasons as discussed above formodel 3. Moreover, all models that ignite helium in a degen-erate core, which include the models shown in Figure 10c,show quite similar behavior to the 1.0M ⊙ case, and will notbe discussed further. High-mass stars
Moving on to a case where helium ignites in a partiallydegenerate core, we see excess variation in the early stagesof helium core burning as illustrated along the 2.2M ⊙ track(Figure 10e). This variation originates from the glitch at thehydrogen-burning shell. We show a representative model inFigure 13.Finally, representative of stars igniting helium in a non-degenerate core, the 2.8M ⊙ track shows two phases of ex-cess variation during early stages of helium-core burning (Fig-ure 10g). The first phase is the ‘high-mass’ non-degenerate-core equivalent to what we saw in the low-mass degenerate-core models near the red-giant-branch bump, where the glitchfrom the first dredge-up ‘enters’ the g-mode cavity (Figures 9and 1c). As in the red-giant-branch bump cases, the ∆ P vari-ation vanishes when the hydrogen-burning shell reaches andsmooths out the glitch, but here this occurs after the modelhas become a quiescent helium-core burning clump star with ∆ ν ∼ . µ Hz. Figures 14, 15, and 16 show three exam-ples along this phase of evolution. Like in model 2, theglitch in these three models is expected to cause relative highamplitudes in the frequency spectrum for the almost pure gmodes located at dips in ∆ P (panels b and c). The ´echellediagram can therefore appear to show a dominant spacing be- tween strong modes that is significantly larger than the under-lying period spacing between adjacent modes (Figure 15d).A second phase of variation occurs due to a glitch that wasbuilt up near the edge of the convective core during heliumignition and its subsequent maximal extent. However, we donot show an example of this phase because, in our models, thevariation only shows up with a relatively low amplitude ( ∆ P ∼ sec) and dips in ∆ P that are rather broad and widelyseparated, making it very difficult to detect when the couplingto the acoustic modes is included. Discussion
Due to the glitch-induced variation in ∆ P around ∆ P as ,one can choose to use either ∆ P as (horizontal dotted linein panel b) or the maximum period spacing to generate theperiod ´echelle (see for example Figure 15b). We chose touse ∆ P as , which in some cases creates one overall ‘S’ shapeper radial mode order as in the glitch-free case (see high-frequency end of Figure 7; see also Fig.1c of Bedding et al.(2011)), but modulated with the glitch-induced variation ontop as in Figure 11e and 13e. Had we chosen to use themaximum value of ∆ P , we would obtain one ‘S’ shape forevery glitch-induced dip in ∆ P . In other cases using ∆ P as makes the period ´echelle look very complicated with no clearpattern, such as in Figures 14e, while the maximum value ap-pears to create a better aligned ´echelle, by straightening thezig-zag pattern. The latter might therefore be misinterpretedas the asymptotic period spacing of a glitch-free star. Thiscould potentially explain some of the more massive stars withobserved ∆ P as reported to fall significantly outside the mainensemble in the ∆ P as - ∆ ν diagram by Mosser et al. (2014), ifindeed real stars share the frequency behavior shown by ourmodels.In search for excess variation in ∆ P , as summarized in Fig-ures 9 and 10, we deliberately ignored the coupling with theenvelope (p-)modes to simplify and speedup the process. Inreal data, one would have to separate the variation in ∆ P caused by the coupling from the effect induced by the buoy-ancy glitches. We verified that our results presented here areconsistent with what we obtain if the p-mode coupling is in-cluded, which we did by first fitting and removing the cou-pling pattern from the mixed-mode frequencies derived fromthe full numerical solution (e.g. ADIPLS/GYRE), and sub-sequently deriving the RMS of the residual period-spacingvariation. Fitting and removing the coupling pattern was per-formed along 1.0M ⊙ and 2.4M ⊙ tracks using the toy modelfor the coupling presented by Stello (2012), but could aswell be done using equation (35) (see also equation (9) inMosser et al. 2012b). The analysis of glitch-induced varia-tions in ∆ P from real data will be presented in Stello et al.(in preparation).Although we verified that the results summarized in Fig-ures 9 and 10 are similar when based on ASTEC, ourASTEC models generally showed less high-frequency vari-ation in ∆ P due to the glitches being smoother, arising fromnumerical diffusion in ASTEC, as described in section 2. Itis also expected that including additional mixing processescould affect the buoyancy frequency significantly, and hencealter the signature in the frequencies, which would potentiallyprovide a way to test various prescriptions of mixing (Con-stantino et al., in preparation). CONCLUSIONS (a)(b)(c) (d) (e) F IG . 11.— Model 2 ( M = 1 . M ⊙ ; ν max = 31 µ Hz) - (a) Buoyancy frequency with key features indicated. (b) Period spacing of pure g modes (as insection 3.2.2) (red) and full numerical solution using GYRE (black). The vertical dotted lines indicate the approximate position of the dipole acoustic modes.They have been positioned relative to the nearest radial mode in agreement with Stello et al. (2014) (see also Huber et al. (2010); Montalb´an et al. (2010)).Horizontal dotted line marks ∆ P as . (c) Pseudo amplitude spectrum based purely on mode inertia, normalized to the radial modes. Dipole modes are shown inred and radial modes in black. (d) ´Echelle diagram. The abscissa is the frequency modulo ∆ ν . Symbol size follows the peak heights in panel (c). (e) Period´echelle diagram. Symbol sizes as in panel (d). The abscissa is the period modulo ∆ P as . Dotted lines indicate the approximate position of the dipole acousticmodes. Black curve connects all dipole modes.F IG . 12.— Model 3 ( M = 1 . M ⊙ ; ν max = 15 µ Hz) - Buoyancy fre-quency with key features indicated.
We have shown that structural glitches in the cores of redgiants can significantly affect the adiabatic properties of theirmixed modes – both mode inertias and frequencies. Themodulation in mode inertia can have strong consequences forwhich modes are observable. Moreover, the change in thefrequency pattern shows up as a variation in the underlyingperiod spacing of pure g-modes around the fixed asymptoticvalue of the glitch-free case. Hence, assuming the periodspacing follows the simple glitch-free asymptotic behavior(equation (35), see also equation (9) in Mosser et al. (2012b)),can hamper the estimate of the asymptotic (glitch-free) periodspacing, ∆ P as . This might explain some of the stars observedto show a period spacing that does not follow the main ensem-ble of stars both along the red-giant branch and the red clump(Mosser et al. 2014). We provide an approximate analytical solution to the waveequation in the presence of both a structural glitch and thecoupling between p and g modes. We find that the combinedeffect of a glitch and mode coupling is not merely the sumof the two. The combined effect is a modulation of the depthof the dips at the acoustic resonant frequencies and, in somecases, the split of these in two. The glitch-induced variationsin the period spacing are equally spaced in period, and reflectthe depth at which the glitch is located, while the amplitudeof the variation is a measure of the effective strength of theglitch.From an extensive set of evolution tracks of varying masswe find glitch-induced variation at the red-giant-branch lumi-nosity bump, at the early phases of helium-core burning, andat the asymptotic-giant-branch bump, which signifies the be-ginning of helium shell burning. We note that some of theseevolution stages last for a relatively short period of time, mak-ing the detection of glitches in such stars a strong indicator ofrelative age.uoyancy glitches in the cores of red giants 15 (a)(b)(c) (d) (e) F IG . 13.— Model 4 ( M = 2 . M ⊙ ; ν max = 84 µ Hz) - notation as in Figure 11. (a)(b)(c) (d) (e) F IG . 14.— Model 5 ( M = 2 . M ⊙ ; ν max = 34 µ Hz) - notation as in Figure 11.APPENDIX A
In this appendix we derive the explicit form of the cou-pling phase, ϕ , that appears in equation (33). This phase isuniquely determined by the coefficients entering the solutionof the wave equation in the evanescent region r ≪ r ≪ r .In principle these can be determined by matching the solu-tion in the evanescent region to that in the p-mode cavity and,subsequently, applying an appropriate boundary condition atthe photosphere. However, in red-giant models such as thoseunder study, we find that K defined by equation (3) goes to −∞ at some critical radius r c located in the evanescent re- gion between the two cavities. The analysis of the wave equa-tion across this singularity is rather cumbersome and will beconsidered in a separate paper (Cunha et al., in preparation).Here, we use, instead, the eigenvalue condition presented byShibahashi (1979), which accounts for mode coupling but notfor rapid variations in the structure (hence no glitch). Theireigenvalue condition is derived through the asymptotic anal-ysis of the pulsation equations, under the Cowling approxi-mation, for two pulsation variables, one related to the radialcomponent of the displacement and the other related to theEulerian pressure perturbation. The simultaneous use of thetwo equations allows the author to avoid having to match the6 Cunha et al. (a)(b)(c) (d) (e) F IG . 15.— Model 6 ( M = 2 . M ⊙ ; ν max = 65 µ Hz) - notation as in Figure 11. (a)(b)(c) (d) (e) F IG . 16.— Model 7 ( M = 2 . M ⊙ ; ν max = 100 µ Hz) - notation as in Figure 11. solutions across critical points similar to that referred above.The result is the eigenvalue condition (Shibahashi 1979, equa-tion (31)) cot (cid:18)Z r r κ d r (cid:19) tan (cid:18)Z r r κ d r (cid:19) = q (43)where q is often called the coupling factor and is given by q = 14 exp (cid:18) − Z r r | κ | d r (cid:19) , (44)where, as before, we used the subscript to indicate that thiscondition is valid in the absence of a glitch. In the above, κ is an approximation to the radial wavenumber appearing in theequations used by the author and is given by κ = ω − N c − L r (cid:18) − N ω (cid:19) . (45)Inside the g-mode cavity κ ≈ L/r p (1 − N /ω ) ≈ K .Using this fact, we can combine the conditions (33) and (43)to write, tan ( ϕ ) = q tan (cid:16)R r r κ d r (cid:17) . (46)uoyancy glitches in the cores of red giants 17Next, we note that the eigenvalue condition for pure p modesderived by Shibahashi (1979) (his equation (26)) is Z r r κ = mπ, (47)where m is an integer and the subscript “a” was added to in-dicate that this condition provides what would be the eigen-frequencies of acoustic waves in the absence of coupling.Writing κ ≡ κ + δκ and taking δκ ≈ δω/c (which isa good approximation throughout the p-mode cavity, exceptnear the turning points r and r ) we then have tan (cid:18)Z r r κ d r (cid:19) = tan (cid:18) mπ + Z r r δκ d r (cid:19) ≈ tan (cid:18) ω − ω a ω p (cid:19) , (48)where ω a are the eigenvalues that would be obtain for p modesin the absence of coupling and ω − = R r r c − d r . Finally,using (48) in equation (46) we find ϕ ≈ atan q tan (cid:18) ω − ω a ω p (cid:19) . (49) APPENDIX B
Below we reproduce the expressions for the generalizedbuoyancy frequency and critical frequency that appear inequations (5.4.8) and (5.4.9) derived by Gough (1993).In Gough (1993), the author expresses the equations de-scribing linear, adiabatic pulsations in terms of the Lagrangianpressure perturbation δp . After performing the Cowling ap-proximation, the resulting second order differential equationfor δp is reduced to the standard form by defining a new de-pendent variable Ψ = ( r /gρf ) / δp , where f is the f-modediscriminant given by f = ω rg + 2 + rH g − L gω r , (50)and H g is the scale height for the gravitational accelerationobtained following the general definition adopted by the au-thor that the scale height for a quantity q is H q = − d r d ln q . Thewave equation resulting from this variable transformation is d Ψd r + K Ψ = 0 , (51)with the radial wavenumber K defined by, K = ω − ω c − L r (cid:18) − N ω (cid:19) . (52) In the above, N is the generalized buoyancy frequencygiven by N = g (cid:18) H − gc − h (cid:19) , (53)where h is the scale height for g/r and is related to H g by h − = H − g + 2 r − and H is the scale height for gρf /r and is related to other relevant scale heights in the analysisby H − = H − ρ + H − f + h − + r − . Moreover, ω c is ageneralization of the critical frequency given by ω c = c H (cid:18) − H d r (cid:19) − gh (54)Using the definition for the density scale height and theequation for hydrostatic equilibrium, the buoyancy frequencydefined by expression (1) can be written as, N = g (cid:18) H ρ − gc (cid:19) . (55)Comparing this expression with expression (53) we see thatthe generalized buoyancy frequency has H in the place of H ρ and includes an additional term, − /h . As mentionedby Gough (1993) (and seen from the definitions of h and H ),these differences result from the geometry and self-gravity ofthe equilibrium state and, consequently, N reduces to N in the limit of a plane-parallel envelope under constant grav-itational acceleration. Comparison of the generalized criti-cal frequency with the one derived in that same limit (Gough2007), shows that the difference between the two is also solelythe outcome of geometry and self-gravity.Funding for this work was provided by the University ofSydney (IRCA grant) and by the ERC, under FP7/EC, throughthe project PIRSES-GA-2010-269194. MSC and PPA aresupported by the Fundac¸˜ao para a Ciˆencia e a Tecnologia(FCT) through the Investigador FCT contracts of referencesIF/00894/2012 and IF/00863/2012 and by POPH/FSE (EC)through FEDER funding through the program COMPETE.Funding for this work was also provided by the FCT grantUID/FIS/04434/2013. DS acknowledges support from theAustralian Research Council. Funding for the Stellar Astro-physics Centre is provided by The Danish National ResearchFoundation (Grant DNRF106). The research is supported bythe ASTERISK project (ASTERoseismic Investigations withSONG and Kepler) funded by the European Research Coun-cil (Grant agreement no.: 267864). RT acknowledges sup-port from NASA award NNX14AB55G and NSF award ACI-1339600. REFERENCESAerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. 2010, Asteroseismology(Springer)Beck, P. G., et al. 2011, Science, 332, 205—. 2012, Nature, 481, 55Bedding, T. R., et al. 2010, ApJ, 713, L176—. 2011, Nature, 471, 608Belkacem, K., Samadi, R., Mosser, B., Goupil, M.-J., & Ludwig, H.-G.2013, in Astronomical Society of the Pacific Conference Series, Vol. 479,Progress in Physics of the Sun and Stars: A New Era in Helio- andAsteroseismology, ed. H. Shibahashi & A. E. Lynas-Gray, 61Bildsten, L., Paxton, B., Moore, K., & Macias, P. J. 2012, ApJ, 744, L6 Brassard, P., Fontaine, G., Wesemael, F., & Hansen, C. J. 1992, ApJS, 80,369Broomhall, A.-M., et al. 2014, MNRAS, 440, 1828Cantiello, M., Mankovich, C., Bildsten, L., Christensen-Dalsgaard, J., &Paxton, B. 2014, ApJ, 788, 93Charpinet, S., Fontaine, G., Brassard, P., & Dorman, B. 2000, ApJS, 131, 223Christensen-Dalsgaard, J. 2004, Sol. Phys., 220, 137—. 2008a, Ap&SS, 316, 113—. 2008b, Ap&SS, 316, 138 Cunha et al.