Probing core overshooting using asteroseismology
PProbing core overshooting using asteroseismology
S´ebastien Deheuvels IRAP, Universit´e de Toulouse, CNRS, CNES, UPS, (Toulouse), France
Abstract:
Modeling properly the interface between convective cores and radiative interiors is one themost challenging and important open questions in modern stellar physics. The rapid development ofasteroseismology, with the advent of space missions partly dedicated to this discipline, has providednew constraints to progress on this issue. We here give an overview of the information that can beobtained from pressure modes, gravity modes and mixed modes. We also review some of the mostrecent constraints obtained from space-based asteroseismology on the nature and the amount of mixingbeyond convective cores.
Keywords:
Stellar evolution – convection – asteroseismology
At the occasion of the workshop ”How much do we trust stellar models?” organized in Li`ege tocelebrate the 75 th birthday of Arlette Noels, I was asked to review the recent results obtainedwith asteroseismology to better understand the interface between convective cores and radiativeinteriors. This topic is both one of the most pressing open questions for stellar physics anda subject that is dear to Arlette’s heart. The impact on stellar physics is clear. The mixedregion associated to the convective core plays the role of a reservoir for nuclear reactions andknowing its extent is crucial to accurately model stellar evolution, in particular to estimatestellar ages. Over the last decade, the advent of spatial asteroseismology has yielded preciousconstraints on the size of the mixed core for stars of various masses and stages of evolution. Theinterpretation of these seismic data has greatly benefitted from the work of Arlette Noels andher collaborators in Li`ege on the physical processes responsible for the extension of convectivecores (overshooting, semiconvection) and their asteroseismic signature (see, e.g., Noels et al.2010).Among the processes that can extend convective cores, overshooting is the most often cited.Formally, the limit of the convective core is set by the Schwarzschild criterion and it correspondsto the layer above which upward-moving convective blobs start to be braked. However, thiscriterion does not take into account the inertia of the ascending blobs, which can in fact overshoot over a certain distance inside the stable region. This is expected to extend the size ofthe mixed core. Despite the large number of studies dedicated to this phenomenon, the detailsof how it operates remain very uncertain. Three physical quantities need to be determined inorder to properly model core overshooting: ”How Much do we Trust Stellar Models?”, held in Li`ege (Belgium), 10-12 September 2018 a r X i v : . [ a s t r o - ph . S R ] J a n . The distance d ov over which chemical elements are mixed beyond the formal limit of theconvective core. Theoretical studies wildly disagree on the value of d ov , with predic-tions ranging from 0 to several units of local pressure scale height H P (e.g., Saslaw &Schwarzschild 1965, Shaviv & Salpeter 1971, Roxburgh 1978, Zahn 1991).2. The nature of the extra-mixing beyond the convective core. Overshooting can be mod-eled either as an instantaneous mixing (all chemical elements being homogeneous in theovershooting region), or as a diffusive process where the turbulent velocities are generallyassumed to decay exponentially in the overshoot region (Herwig 2000).3. The temperature stratification in the extra-mixing region. According to the Schwarzschildcriterion, the temperature gradient should correspond to the radiative gradient ( ∇ = ∇ rad ) in the overshoot region. However, the convective blobs that penetrate inside thestable regions could heat these layers and bring the temperature gradient closer to theadiabatic gradient ∇ ad . The latter case is usually referred to as penetrative convection and by opposition, the case of an inefficient penetration that does not alter the temper-ature gradient ( ∇ = ∇ rad in the extra-mixed region) is referred to as non-penetrativeconvection .The situation is even more complicated because other poorly-understood processes can alsoextend the size of convective cores, such as rotation-induced mixing (e.g. Maeder 2009) orsemiconvection (e.g. Langer et al. 1985).The combined effects of all these phenomena are generally modeled in stellar evolution codesby a simple extension of the mixed core over a distance considered as a free parameter. Thisdistance is often referred to as the overshooting distance and denoted as d ov , even though oneshould keep in mind that the extension of the core may in fact be caused by several distinctprocesses, not only core overshooting. We also use this terminology in this review. The detailsof how the core extension is implemented vary from one evolution code to another. The codesassuming an instantaneous mixing in the overshoot region usually take d ov as a fraction α ov ofthe pressure scale height H P at the core boundary. Core overshooting can also be implementedas a diffusive process and in this case the diffusion coefficient is generally taken as D ov ( r ) = D conv exp " r − r s ) f ov H P (1)where r s is the radius of the Schwarzschild boundary, D conv is the MLT diffusion coefficientsome distance below r s , and f ov is an adjustable parameter controlling the distance of over-shooting. The temperature gradient is chosen to be either ∇ ad (penetrative convection) or ∇ rad (non-penetrative convection). Another important aspect is the treatment of the extensionof “small” convective cores, for stars with masses around 1.2 M (cid:12) . The pressure scale heightdiverges in the center, so for small cores, the classical implementation described above gener-ates unrealistically large core extensions that can reach the size of the core itself. Here again,evolution codes have different ways of remedying this problem. For instance, Cesam2k definesthe overshooting distance as d ov = α ov min( H P , r s ). By default, MESA adopts the definition d ov = α ov min( H P , r s /α MLT ) where α MLT is the mixing length parameter. Considering these def-initions, small convective cores can have extensions over distances that vary by a factor α MLT Note that the initial terminology proposed by Zahn (1991) was to reserve the term overshooting for the caseof an inefficient penetration. However, since the term overshooting is widely used to refer to the general process,regardless of the temperature stratification, we prefer to use the term non-penetrative convection instead in thisreview. α ov in the two codes. Other studies chose to impose a linear dependenceof α ov on stellar mass in this mass range (e.g., Pietrinferni et al. 2004, Bressan et al. 2012).One should be aware of these differences, which prevent direct comparisons of overshootingefficiencies between codes that adopt different prescriptions.The diversity of these implementations is due to the current lack of observations that couldhelp constrain the physical properties of the extra-mixing beyond convective cores. So far,constraints on core overshooting were obtained mainly from the modeling of eclipsing binaries(e.g. Stancliffe et al. 2015, Claret & Torres 2018) and from the color-magnitude diagrams ofclusters (e.g., Maeder & Mermilliod 1981, VandenBerg et al. 2006). These observational dataare essentially sensitive to the distance of the extra-mixing d ov . Asteroseismology can directlyprobe the size of the mixed core at current age because oscillation modes are sensitive to thesharp gradient of chemical composition at this location. As will be shown in the followingsections, seismic constraints can also be obtained on the chemical profile within the region ofextra-mixing and on the temperature stratification, which opens the interesting prospect oftesting more complex models of core overshooting.In this review, we present a selection of the most recent seismic constraints on the physicalproperties of the boundaries of convective cores. Our aim is not to be exhaustive, but to givean overview of the latest developments made possible thanks to space-based asteroseismology.For this purpose, we focus on four types of stars. We start on the main sequence with resultsobtained for solar-like pulsators using pressure modes (Sect. 2) and for slowly pulsating B(SPB) stars using gravity modes (Sect 3). We then show that mixed modes can also placestrong constraints on core overshooting in subgiants (Sect. 4) and core-helium burning giants(Sect. 5). Figure 1: Variations in the ratio r as a function of frequency for 1.15- M (cid:12) main sequencemodels with α ov = 0 (green), 0 . .
15 (cyan), 0 . c s . Assuming an ideal gas law, c = Γ R T /µ ,where Γ is the adiabatic exponent, T is the temperature, and µ is the mean molecular weight.At the boundary of the mixed core, a strong µ -gradient develops, which creates a near discon-tinuity in the sound speed velocity. This generates an acoustic glitch for pressure modes (thespatial scale of the variations in c s is smaller than the mode wavelength), which produces a clearsignature in the frequencies of these modes. It is well known that acoustic glitches generate aperiodic modulation of the mode frequencies (Gough 1990). The amplitude of the modulationdepends on the intensity of the glitch (sharpness of the µ -gradient) and the period depends onthe location of the glitch (the deeper the boundary of the mixed core, the longer the period).In principle, pressure modes thus convey information about the size of the mixed core and thenature of the mixing in the overshoot region. Note that acoustic glitches are also produced bythe bottom of the convective envelope (e.g., Christensen-Dalsgaard et al. 2011) and the zoneof ionization of helium (e.g., Mazumdar et al. 2014, Verma et al. 2014).Although the periodic modulation due to the acoustic glitch is present in the mode frequen-cies themselves, it is more convenient to use combinations of mode frequencies instead. Moststudies use small differences d or second differences dd built with radial and dipolar modes d = 12 ( − ν ,n − + 2 ν ,n − ν ,n ) (2) dd = 18 ( ν ,n − − ν ,n − + 6 ν ,n − ν ,n + ν ,n +1 ) . (3)It has indeed been shown that these quantities are particularly sensitive to the structure ofthe core (e.g., Provost et al. 2005). Besides, the ratios r defined as dd / ∆ ν , where ∆ ν corresponds to the large separation of dipolar modes (∆ ν ,n = ν ,n − ν ,n − ), have been shownto be largely insensitive to the structure of the outer layers, which makes them almost immuneto the well-known near-surface effects (Roxburgh & Vorontsov 2003).As an illustration, Fig. 1 shows the variations in the ratio r with frequency for 1.15 M (cid:12) main sequence models. Different extensions of the convective core were considered, rangingfrom α ov = 0 to α ov = 0 .
2, and models were evolved until the same age. For α ov > . α ov increases), the period of the oscillation decreases. Fig. 1 also shows the approximate rangeof frequencies where p-modes are expected have detectable amplitudes. It appears that thisinterval is much shorter than the period of the modulation, which unfortunately prevents usfrom getting model-independent measurements of the size of the mixed core. However, thebehavior of r in the range of observed modes changes significantly as α ov is varied, showingthat the extent of the convective core can be determined using model-dependent analyses. Inparticular, the coefficients of a linear regression of r ( ν ) have been shown to efficiently constrainthe amount of extra mixing (Deheuvels et al. 2010b, Silva Aguirre et al. 2011). Several analysesof this type have been recently obtained. HD49933:
HD49933 is an F5-type main sequence star and was the first solar-like pulsator tobe observed with the
CoRoT satellite. It benefitted from 180 days of nearly continuous obser-vations and the properties of its oscillation modes were determined by Benomar et al. (2009).The identification of the degree of the detected modes initially caused problems, an ambiguityarising between the l = 1 rotationally split modes and the overlapping l = 0 and l = 2 modes.4his problem is now known to occur for all F-type pulsators owing to their large mode width,and several methods have been proposed to remedy this issue (e.g., Bedding & Kjeldsen 2010).The mode identification for HD49933 is now robust, and Goupil et al. (2011) performed a mod-eling of the star. They found that HD49933 has a stellar mass of in the range 1.05-1.18 M (cid:12) andan age in the range 2.9-3.9 Gyr. They showed that to reproduce the behavior of the observedsmall differences d , an extension of the convective core over a distance d ov ≈ . H P needsto be invoked. They also calculated models of the star including microscopic diffusion androtationally-induced mixing using the code CESTAM (Marques et al. 2013). They found thatthese models fail to reproduce the slope of d ( ν ) and that some amount of core overshoot needsto be included to produce a good agreement with the seismic data. KIC12009504 (Dushera):
The
Kepler satellite has provided us with nearly four years of contin-uous observations during the nominal mission. An early analysis of the
Kepler main sequencetarget KIC12009504 (dubbed Dushera) already permitted to find evidence that the star has aconvective core and to place constraints on its extent (Silva Aguirre et al. 2013). The authorsmodeled the star using nine months of
Kepler data analyzed by Appourchaux et al. (2012).They found that the star has a stellar mass of 1 . ± . M (cid:12) , a radius of 1 . ± . R (cid:12) andan age of 3 . ± .
37 Gyr. They also showed that the observed ratios r could be reproducedonly by models with a convective core that extends beyond the Schwarzschild boundary (seeFig. 2). Optimal fits were obtained when the limit of the mixed core is located at an acousticradius equal to ∼ .
4% of the total acoustic radius .Figure 2: Ratios r of KIC12009504 (opencircles). The colored symbols correspondto models computed with various evolutionscodes and input physics (see Silva Aguirreet al. 2013). Figure 3: Amount of core overshooting re-quired for the eight stars studied by De-heuvels et al. (2016) plotted as a function ofthe stellar mass. Blue squares (resp. graycircles) indicate models computed without(resp. with) microscopic diffusion. Verticalarrows indicate upper values of α ov for fiveother stars. Dependence of the amount of overshoot with stellar mass:
As illustrated by the examples pre-sented here, several asteroseismic studies were led on individual stars, which all reported the The acoustic radius is defined as τ ≡ R r d r/c s . It corresponds to the wave travel time from the center to aradius r . Kepler solar-like pulsators in a consistent way, using the coefficients of a 2 nd -order polynomialfit to the ratios r to probe the mixed core. Within this sample, 10 stars were found to bealready on the post-main-sequence. Among the other targets, the authors detected a convectivecore in eight stars and they were able to estimate the size of their mixed core, finding a goodagreement with the two evolution codes Cesam2k and
MESA (using identical prescriptionsfor core overshooting). It was necessary to include significant extensions of the mixed core inall the considered targets. The optimal values of α ov obtained for these eight stars are shown asa function of stellar mass in Fig. 3. As can be seen in this figure, there seems to be a tendencyof core overshooting to increase with stellar mass in the considered mass range, although moredata points will be required to confirm this trend. Interestingly an increase of the efficiencyof core overshooting with mass was also found using constraints from double-lined eclipsingbinaries by Claret & Torres (2018), although this result is currently debated (Constantino &Baraffe 2018). One should also beware that the stars studied by Deheuvels et al. (2016) arein the range of mass where the radius of the convective core is smaller than the pressure scaleheight at the core edge during most of the main sequence evolution. The efficiency of theextra-mixing beyond the convective core parameterized by α ov thus depends on the treatmentthat they adopted for “small” convective cores ( d ov redefined as α ov r s when H P > r s in thisstudy). Gravity modes are expected to be excellent probes of the region of extra-mixing beyond theconvective core, through their dependence on the Brunt-V¨ais¨al¨a frequency N (see Sect. 3.1).The CoRoT and
Kepler missions have produced exquisite photometric data for g-mode classi-cal pulsators, in particular slowly pulsating B (SPB) stars and γ Doradus stars, thus providinginformation about core properties for stars of intermediate masses.
High-order gravity modes (in the asymptotic regime) are expected to be equally spaced inperiod. The asymptotic period spacing of g modes of degree l is approximately given by∆Π l ≈ π L Z r o r i (cid:18) Nr d r (cid:19) − , (4)where L = l ( l + 1) and the radii r i and r o are the inner and outer turning points of the g-modecavity. The Brunt-V¨ais¨al¨a frequency directly depends on the temperature stratification andthe µ -gradient in the g-mode cavity through the relation N = gδH P (cid:18) ∇ ad − ∇ + ϕδ ∇ µ (cid:19) (5)where ∇ µ ≡ (d ln µ/ d ln P ), δ = ( ∂ ln ρ/∂ ln T ) P,µ , and ϕ = ( ∂ ln ρ/∂ ln µ ) P,T .As stars evolve, the hydrogen content in the convective core decreases and a region ofincreasingly large µ -gradient develops above the boundary of the core (see Fig. 4). This6igure 4: Behavior of the hydrogen abundance profile (left), of the Brunt-V¨ais¨al¨a frequency(center) and of the l = 1 g-mode period spacing (right) in a 1.6- M (cid:12) model computed with(thick lines) or without (thin lines) overshooting (from Miglio et al. 2008).generates a buoyancy glitch at the outer edge of the µ -gradient region, where ∇ µ varies ona length scale that is shorter than the mode wavelength. This glitch produces a periodicmodulation of ∆Π l , whose period depends on the location r µ of the glitch within the cavity(the deeper the glitch, the longer the period, as can be seen in the right panel of Fig. 4). Theamplitude of the modulation depends on the intensity of the glitch, i.e., on the smoothness ofthe chemical profile outside the convective core. For stars massive enough for the CNO cycleto dominate during their main-sequence evolution, the convective core recedes, which increasesthe size of the µ -gradient region. As a result, the outer edge of the µ -gradient region movesoutwards and the period of the modulation decreases, as can be seen in the right panel of Fig.4. The characteristics of this periodic modulation give direct constraints on the properties ofthe extra-mixing beyond the convective core. As shown by Miglio et al. (2008), adding coreovershooting to stellar models changes the size of the µ -gradient region and thus modifies theperiod of the modulation (see Fig. 4). The nature of the mixing in the overshoot region can alsobe tested. When core overshooting is treated as a diffusive process, ∇ µ varies more smoothlythan when an instantaneous mixing is assumed and the glitch produced in the Brunt-V¨ais¨al¨afrequency is less steep (see left panel of Fig. 5). This makes a difference for g-modes withhigher periods. These modes have shorter wavelengths, which eventually become smaller thanthe length scale of the sharp feature in ∇ µ as mode period increases. Thus, higher-periodg modes do not “feel” this feature as a glitch and we expect the amplitude of the periodicmodulation in ∆Π l to decrease as mode period increases. The situation is different for anactual discontinuity in the Brunt-V¨ais¨al¨a frequency, for which all the modes have wavelengthslonger than the length scale of the glitch.To test this quantitatively, Pedersen et al. (2018) calculated a reference model of 3.25 M (cid:12) with diffusive overshooting and tried to see if its seismic content could be reproducedby models computed with an instantaneous overshooting. For this purpose, they generated agrid of models with instantaneous mixing in the overshoot region, with varying masses, initialhydrogen abundances, central hydrogen contents, and overshooting efficiencies. They showedthat no model of the grid was able to reproduce the period spacings of the reference modelcomputed with diffusive overshoot (see Fig. 6). This shows that for this type of star, oneshould be able to distinguish between an instantaneous and a diffusive overshoot. This is nolonger true for more evolved models nearing the end of the main sequence (Pedersen et al.7igure 5: Profile of the Brunt-V¨ais¨al¨a frequency (red dashedcurves) where overshooting istreated as a diffusive process (left)or as an instantaneous mixing(right) (from Moravveji et al. 2016). Figure 6: Comparison between the period spacings ofa reference model with diffusive overshoot (red curve)and the 15 best matching models of a grid computedwith instantaneous mixing. The colors indicate thelevel of agreement with the reference model (fromPedersen et al. 2018).2018).In principle, information could also be obtained about the temperature stratification inthe overshooting region. Indeed, with penetrative convection ( ∇ = ∇ ad ), the Brunt-V¨ais¨al¨afrequency vanishes in the overshooting region, whereas with non-penetrative overshooting ( ∇ = ∇ rad ), it remains strictly positive. The inner turning point r i of the g-mode cavity is thereforelocated deeper in the latter case. This should have an impact on the buoyancy radius of thesharp µ -gradient, defined as Π µ = (cid:16)R r µ r i Nr d r (cid:17) − , and thus on the period of the oscillatorybehavior of ∆Π l , which corresponds to the ratio between the buoyancy radius of the glitch andthe total buoyancy radius of the cavity. This remains to be theoretically addressed.To use the information conveyed by γ Doradus and SPB stars about the core properties,one difficulty arises: these stars are usually fast rotators and the effects of rotation need tobe taken into account to properly identify and interpret the periodic modulation caused bythe µ -gradient region. This issue has been extensively studied and goes beyond the scope ofthe present review. However, we can mention that the validity of the so-called traditionalapproximation of rotation (TAR, Eckart 1960) has been shown (Ballot et al. 2012). Thishas made it possible to successfully identify the modes and analyze the oscillation spectra offast-rotating γ Doradus and SPB stars (Bouabid et al. 2013).
HD50230:
This star is a hybrid pulsator, oscillating both as an SPB star (gravity modes) anda β Cephei star (pressure modes) orbserved with
CoRoT . It is also a slow rotator, which sim-plifies the interpretation of its oscillation spectrum. In the g-mode region of the spectrum, agroup of eight modes with nearly constant period spacing was found by Degroote et al. (2010).The period spacings of these modes show a periodic modulation that the authors attributed tothe edge of the mixed more. The authors found that the period of this modulation can only be This approximation consists in assuming a spherical shape for the star and neglecting the horizontal compo-nent of the rotation vector. This way, the problem remains separable in the radial and latitudinal coordinates,as it is for slow rotators. . H P .Interestingly, the amplitude of the modulation seems to decrease with increasing period, whichthe authors interpreted as an evidence for a smooth gradient of chemical composition at theboundary of the mixed core. KIC10526294:
KIC10526294 is an SPB star observed with
Kepler . A series of 19 dipolar grav-ity modes with consecutive radial orders were detected by P´apics et al. (2014) for this star,making it a particularly interesting target to search for periodic modulation induced by theconvective core. Rotational splittings could be measured for the star, which indicated that itis a very-slow rotator (average rotation period of ∼
188 days). The period spacings ∆ P ofthe detected modes exhibit a clear deviation from the asymptotic period spacing. Moravvejiet al. (2015) performed a detailed modeling of this target. They showed that the variations of∆ P with mode period are better reproduced with core overshooting implemented as a diffusiveprocess than with an instantaneous mixing in the overshoot region. They found optimal valuesof the overshoot parameters of f ov between 0.017 and 0.018 (see Eq. 1). They also claim thatincluding an extra-mixing in the radiative interior outside the overshooting region can signifi-cantly improve the agreement between the models and the observations. It should however beremarked that the optimal models are still far from giving a good statistical agreement with the Kepler observations (see Fig. 7, left panel). This suggests that the models might be missingsome important ingredient.
KIC7760680:
This star is a moderately-rotating SPB star observed with the
Kepler satellite.It exhibits a series of 36 consecutive gravity modes, in which a clear periodic modulation can bedetected (see Fig. 7, right panel). It is also apparent that the period spacings of KIC7760680show an almost linear decrease with mode period. This is the clear signature of moderaterotation for prograde modes (Bouabid et al. 2013). Moravveji et al. (2016) modeled the star,considering different assumptions for the mixing within the overshooting region. They consid-ered a solid-body rotation for the star and for each model, they optimized the rotation rate toreproduce the slope of the period spacings as a function of the mode period. As was the casefor HD50230 and KIC10526294, they found that a diffusive overshoot reproduces the periodicmodulation in the period spacings better than an instantaneous overshoot. With both imple-mentations, the optimal models include a sizable overshooting region ( f ov = 0 . ± .
001 in9he case of a diffusive overshoot and α ov ∼ .
32 for an instantaneous overshoot). Here again,the optimal solutions are quite far from the observations, yielding reduced χ of the order of2000. The bottom right panel of Fig. 7 shows that there is clear structure in the residuals(periodic modulation for mode periods larger than ∼ P differs between the models and the observations, especially for largemode periods. This is likely indicating that improvements could be made in the modeling ofthe chemical composition profile in the overshooting region. γ Doradus stars:
Recently, long series of consecutive g modes were also revealed in the spectraof γ Doradus stars (Van Reeth et al. 2016, Christophe et al. 2018). These stars are generallymoderate to fast rotators. However, once the signature of rotation has been correctly identified,an oscillatory behavior of the period spacings has been reported for some γ Doradus stars(Christophe et al. 2018). These stars could therefore also provide precious information on theproperties of the extended mixed cores in the near future.
When stars evolve past the end of the main sequence, their inner layers contract as hydrogenstarts burning in a shell. This causes the frequencies of gravity modes to increase owing tothe increasing Brunt-V¨ais¨al¨a frequency in the core. In the meantime, the envelope expends asstars become subgiants. The mean density of the star decreases and therefore the frequenciesof pressure modes also decrease. As a result, the frequencies of the lowest radial order g modesbecome of the same order of magnitude as the frequencies of the p modes that are stochasticallyexcited in the outer part of the convective envelope. At this point, non-radial modes develop a mixed nature , behaving as g modes in the core and as p modes in the envelope. This phenomenonarises because of the coupling exerted between the two cavities by the evanescent zone thatseparates them. Mixed modes have a large potential because they convey information aboutthe core properties while having detectable amplitudes at the surface.
The helium core of subgiants is radiative because hardly produces any luminosity. So even ifthe star had a convective core during the main sequence, convective mixing has ceased whenthe star becomes a subgiant. Nevertheless, the main sequence convective core leaves an imprintin the chemical composition profile of young subgiants. Since mixed modes are sensitive to theBrunt-V¨ais¨al¨a profile, and thus to the profile of µ , they can bring indirect information aboutthe extent of the core and the nature of the mixing at its edge.The oscillation spectra of young subgiants contain only a few g-dominated modes, i.e., modesthat are trapped mainly in the g-mode cavity. However, in subgiants, the coupling between thep- and g-mode cavities is strong for dipolar modes, and the frequencies of p-dominated modesare significantly affected by this coupling (Deheuvels & Michel 2010). Mixed modes conveyinformation about the core properties through two channels: • The frequencies of g-dominated modes.
As is apparent from Eq. 4, they depend essentiallyon the integral R r r N/r d r , where r and r are the inner and outer turing points of theg-mode cavity. Fig. 8 shows the Brunt-V¨ais¨al¨a profile of a 1 . M (cid:12) model in the subgiantphase. In the outer part of the g-mode cavity (below r ), the Brunt-V¨ais¨al¨a frequency is10igure 8: Propagation diagram of a 1 . M (cid:12) subgiant. The Brunt-V¨ais¨al¨a frequency(black curve) is split into its thermal part(blue dashed line) and its chemical part (redsolid line). The l = 1 Lamb frequency isshown by the black dashed line. The hor-izontal line indicates the frequency of an l = 1 mixed mode with dotted lines showingevanescent regions. Figure from Deheuvels& Michel (2011). Figure 9: Variations in the large separationof l = 1 modes as a function of mode fre-quency for HD49385. The black dots cor-respond to CoRoT data, the red solid lineshows ∆ ν for the best-fit model, obtainedwith α ov = 0 .
19, and the blue dashed linecorresponds to the best model with α ov =0 .
1. Figure from Deheuvels & Michel (2011).dominated by the contribution of the µ -gradient ( N µ = gϕ ∇ µ /H P , red solid line), whoseshape depends on the extent of the main sequence convective core. • The intensity of the coupling between the p- and g-mode cavities.
The coupling essentiallydepends on the Brunt-V¨ais¨al¨a profile in the evanescent zone ( r (cid:54) r (cid:54) r in Fig. 8). Itthus conveys information about the µ -gradient above r , as can be seen in Fig. 8. Theintensity of the coupling can be estimated observationally by observing its effect on the p-dominated modes. For low coupling intensities, their frequencies will hardly deviate fromthe asymptotic frequencies of p modes, whereas if the coupling is strong, large deviationsare expected. The star HD49385 was observed with the
CoRoT satellite during 137 days and its oscillationspectrum was analyzed by Deheuvels et al. (2010a). Fig. 9 shows the variations in the large sep-aration ∆ ν of dipolar modes as a function of mode frequency. At low frequency, ∆ ν stronglydeviates from the roughly constant value that is expected from asymptotic developments. Itwas later established that this was caused by the presence of a g-dominated mixed mode in thelower-frequency part of the spectrum, which coupled to the detected p modes and altered theirmode frequencies (Deheuvels & Michel 2010).Deheuvels & Michel (2011) proposed a new optimization technique adapted to the modelingof stars with mixed modes, which they applied to HD49385. They found that the star has amass of 1 . ± . M (cid:12) and an age of 5 . ± . d ov . They found twodifferent families of solutions: one with a small amount of overshooting ( α ov < .
05) and the11ther with a moderate amount of overshooting ( α ov = 0 . ± . l = 1 modes are shown in Fig. 9. Deheuvels & Michel (2011) showed that this bimodality ofthe solutions is due to the strong dependence of the mode coupling to the stellar mass (thehigher the mass, the lower the coupling). Only models with masses around 1 . M (cid:12) are ableto produce the correct coupling and thus reproduce the observed frequencies of l = 1 modes.The optimal mass was found to vary non-linearly with the amount of overshooting. Only low( α ov < .
05) or moderate ( α ov = 0 . ± .
01) values of overshooting correspond to a stellarmass of about 1 . M (cid:12) . Models with α ov ∼ . α ov > . Kepler and couldalso provide constraints on the size of main sequence convective cores. The study of thesetargets is under way.
Giant stars with masses M (cid:38) . M (cid:12) eventually start burning helium in their core. Thishappens either quietly in a non-degenerate core (for stars with masses M (cid:38) M (cid:12) ) or in a flashfor stars with masses M (cid:46) M (cid:12) , whose core is degenerate when it reaches the temperature atwhich He starts burning. In both cases, the star then develops a convective core. Measuringthe extent of the mixed core at this evolutionary stage can bring complementary informationabout the interface between convective and radiative regions. We start by briefly introducingthe challenges posed by the modeling of convective cores in core-helium burning (CHeB) stars(Sect. 5.1) and we then present the constraints derived from asteroseismology (5.2). The modeling of mixing in the core of low- and intermediate-mass stars during the CHeB phaseis notoriously challenging. Depending on the criterion that is adopted for convective stability,evolutionary codes predict very different values for the size of the He-burning convective core,and thus also for the duration of the CHeB phase (see Fig. 10). The situation is more compli-cated than during the main sequence because C and O, which accumulate as He is burnt in thecore, are more opaque than He. As a result, the radiative gradient increases in the convectivecore, and a discontinuity of the radiative gradient tends to develop at the boundary of theconvective core. We here briefly describe some of the choices made to treat this in evolutionarycodes and refer the interested reader to the review by Salaris & Cassisi 2017 for more details.In what is usually referred to as the bare Schwarzschild (BS) model, the Schwarzschildcriterion is applied on the radiative side of the convective boundary (panel (a) of Fig. 11).Since the radiative gradient is hardly modified over time on the radiative side, the core sizeremains roughly constant during the whole CHeB phase (see black curve in Fig. 10, labeled asthe “no ivershooting” case). Meanwhile, the radiative gradient increases in the core, and thequantity ∇ rad − ∇ ad thus increases on the convective side of the core boundary. As establishedby Schwarzschild (1958) and reminded by Castellani et al. (1971b) and Gabriel et al. (2014),this situation is in fact unphysical because the convective velocities are expected to vanish atthe edge of the convective core. As a result, the total flux should be equal to the radiative12igure 10: Size of the mixed core during the CHeB phase with different modelings for theboundary of the convective core. Figure from Constantino et al. (2015).flux at this layer, and one should have ∇ rad = ∇ ≈ ∇ ad there. The BS model is therefore anincorrect implementation of the Schwarzschild criterion.Another way of understanding the inadequacy of the BS model is to realize that it isunstable to any mixing beyond the core boundary. Indeed, let us assume a mild extra-mixing,such that the first layer above the convective core is mixed with the convective core. In thislayer, the abundance in carbon and oxygen increases, the opacity increases and hence theradiative gradient increases above the adiabatic gradient. The layer then becomes definitivelyconvective. At the next time step, the layer above the enlarged convective core will in turnbecome convective. This process stops only when ∇ rad = ∇ ad on the convective side of the coreboundary. Panel (b) of Fig. 11 thus shows the correct implementation of the Schwarzschildcriterion. In practice, this is implemented in evolution codes by including a small amount ofcore overshooting (the extension of the convective core that it produces is sometimes referredto as induced overshooting ) or by checking at each time step whether the layers above theconvective core would become convective if they were mixed with the core, and by adding theselayers to the core if it is the case.However, a complication occurs when the mass fraction of helium in the core drops below ∼ .
7. Then, a minimum appears in the profile of the radiative gradient in the core, as can beseen in panel (a) of Fig. 12. For low and intermediate amounts of overshooting, the outwardmixing brings fresh helium into the core and thus induces a decrease of ∇ rad in the wholeconvective core (panel (b) of Fig. 12). The minimum of ∇ rad eventually drops below ∇ ad andthe convective core is split in two convective regions separated by an intermediate radiativezone. The outer convective region rapidly vanishes because of the decrease in ∇ rad . Theconvective core is thus comprised only of the inner convective region and it has shrunk. Ashelium is burnt in the core, the abundance of carbon and oxygen increases again, ∇ rad increasesand eventually has again a minimum within the core. We are then brought back to panel (a) ofFig. 12 and the situation reiterates. As a result, the boundary of the convective core goes backand forth (see the case labeled as standard overshooting in Fig. 10), leaving behind step-likefeatures in the helium abundance profile. In this case, the behavior of the convective core is infact independent of the amount of core overshooting that is included (this is no longer true forlarge amounts of overshooting as explained below).The treatment of the intermediate radiative region that appears in the vicinity of the min-imum of ∇ rad has been the subject of several studies. It is generally thought that it undergoesa partial mixing that enforces convective neutrality ( ∇ rad = ∇ ad ) in this zone (Castellani et al.1971a, Castellani et al. 1985), as can be seen in panel (c) of Fig. 12. The partially mixed regionshares similar features with a semi-convective layer, and this mechanism has been referred to13igure 11: Schematic behavior of the tem-perature gradient near the boundary of theconvective core with ∇ rad = ∇ ad imposedon the radiative side (a) and on the convec-tive side (b) of the boundary (from Castellaniet al. 1971b). Figure 12: Schematic behavior of ∇ rad af-ter it has reached a minimum in the con-vective core. Panel a (reps. b) shows theevolution with standard overshooting and anincreasing (resp. decreasing) radiative gradi-ent. Panel c: evolution with semiconvection.Figure from Castellani et al. (1971a).14s induced semi-convection . Modeling this intermediate region as a semi-convective layer pro-duces core sizes that are very similar to those obtained with overshooting (see how the cyanand orange curves nearly overlap in Fig. 10), but without the back-and-forth motion of thecore boundary, and therefore with a smoother chemical composition profile.It was also found that when applying large amounts of core overshooting at the boundaryof the convective core, the extra-mixed region becomes large enough to prevent the formationof a semi-convective region (Bressan et al. 1986, Bossini et al. 2015). In this case, the size ofthe mixed core depends on the amount of core overshooting that is imposed. Thanks to the space missions
CoRoT and
Kepler , mixed modes have now been detected intens of thousands of red giants. The frequencies of these modes can be identified using theirasymptotic expression, which was first developed by Shibahashi (1979). By fitting this analyticexpression to the observed mode frequencies, one can obtain estimates of various global seismiccharacteristics of the star, including the asymptotic period spacing ∆Π of its dipolar gravitymodes (see Eq. 4). The fitting procedure is challenging because of the large number of modesand it is made much more complicated by the splitting of mixed modes due to rotation. Mosseret al. (2015) have proposed a convenient method, based on the calculation of corrected modeperiods (called stretched periods ), which made it possible to perform an automatic fitting ofred giants. Using this method, Vrard et al. (2016) were able to measure the asymptotic periodspacing ∆Π of 6100 Kepler red giants.This database constitutes an unprecedented opportunity to probe the core of red giants.Bedding et al. (2011) showed that the period spacing ∆Π can be used to reliably distinguishCHeB giants from H-shell burning giants, which are ascending the red giant branch (RGB).The reason for this is evident from Eq. 4. In contrast with RGB stars, CHeB giants have aconvective core. Their g-mode cavity is therefore smaller and they have larger values of ∆Π .For CHeB giants, Montalb´an et al. (2013) showed that there is a nearly linear relation betweenthe size of the convective core and the asymptotic period spacing ∆Π . Indeed, if the convectivecore expends, the g-mode cavity becomes smaller and ∆Π increases. The Kepler data thushave a great potential to measure the size of the mixed core in CHeB stars.The asymptotic period spacings can also convey information about the temperature strat-ification. Indeed, in the case of penetrative convection, we have ∇ = ∇ ad and thus N = 0 inthe extra-mixed region. As a result, gravity waves do not propagate in the overshoot region.On the contrary, with non-penetrative overshooting, N = N T > than models computed with penetrative convection. For models withsemi-convection above the convective core, N = N µ > is also expected to be smaller than with penetrative overshooting.As described in Sect. 3.1, sharp variations in the Brunt-V¨ais¨al¨a frequency (buoyancyglitches) induce periodic modulations in the period spacings of g modes. Such features couldbe measured from the frequencies of mixed modes and give strong constraints on the chemicalcomposition profile near the core boundary. We come back to this in more detail in Sect. 5.2.3. Constantino et al. (2015) and Bossini et al. (2015) both led studies to compare the observed15igure 13: Evolution in the ∆ ν -∆Π plane of 1 M (cid:12) (left figure) and 2.5 M (cid:12) (right figure) CHeBmodels computed with different mixing schemes (“bare Schwarzschild” in black, low overshootin cyan, semi-convection in orange, maximal overshoot in magenta). The grey dots correspondto the observed period spacings of Kepler giants restricted to stars that have undergone a Heflash ( M (cid:54) M (cid:12) ) in the left figure, and stars that triggered He-burning quietly ( M (cid:62) M (cid:12) ).Panels (b) of both figures show probability density curves with the same color code (observationsin thick grey dashes). Figures from Constantino et al. (2015).distribution of period spacings of CHeB giants to the distributions that would be predicted withdifferent mixing schemes beyond the convective core. They found generally consistent results.The “bare Schwarzschild” models have the smallest convective cores because the (incorrect)implementation of the Schwarzschild criterion on the radiative side prevents the core fromgrowing. The highest period spacings ∆Π predicted by these models are around 250 s (seeblack symbols in Fig. 13), well below the maximum observed period spacings, which are around340 s. Bossini et al. (2015) reach the same conclusion. This confirms that the convective coresof the bare Schwarzschild models are much too small.Models that include low amounts of core overshooting or semi-convection also have periodspacings that appear to be too small compared to the observations (cyan and orange symbols inFig. 13). This means that their convective cores are too small. We already mentioned in Sect.5.1 that models computed with semi-convection and models computed with low overshootinghave very similar core sizes (Fig. 10). Yet Fig. 13 shows that the latter models have largerperiod spacings. According to Constantino et al. (2015), this is justified by the fact that large µ -gradients develop in models computed with overshooting, owing to the back-and-forth motionof the core boundary. This is enough to create efficient mode trapping inside the partially mixedregion. As a result, the observed period spacing corresponds to the asymptotic expression ofEq. 4 calculated excluding the region of µ -gradient. It is thus larger than for models computedwith low overshooting than for models computed with semi-convection, for which the chemicalcomposition profile is smooth and such mode trapping does not occur.The Kepler data clearly point in favor of an extended mixed core, larger than the oneproduced with semi-convection or standard amounts of overshooting. To reproduce the seis-mic data, Bossini et al. (2015) calculated models with high amounts of overshooting. Theyfound that models with non-penetrative convection over a distance of 1 H P or with penetrativeconvection over a distance of 0.5 H P could roughly reproduce the distribution of the observedperiod spacings. They gave their preference to the latter models because they also match theluminosity of the asymptotic-giant-branch (AGB) bump, which can be measured from Kepler data. Constantino et al. (2015) calculated models with a modified implementation of core over-shooting. They prevented at all time the splitting of the convective core that occurs because of16igure 14: CHeB giants of NGC 6791 and NGC 6819 shown in the ∆ ν -∆Π plane. Predictionsfrom models with different mixing schemes are overplotted: intermediate (MOV) and high(HOV) non-penetrative convection, intermediate (MPC) and high (HPC) penetrative convec-tion. Figure from Bossini et al. (2017).the minimum in ∇ rad . This model, which they refer to as maximal overshooting has no phys-ical justification but aims at building convective core with maximal sizes. The authors foundthat these models produce period spacings that are consistent with the bulk of the low-massobservations (see magenta line in Fig. 13).Additional information was recently obtained from the measurement of period spacings inthe CHeB giants of the two old open clusters NGC 6791 and NGC 6819 (Bossini et al. 2017). Fig.14 shows the location in the ∆ ν -∆Π plane of the CHeB-members of these two clusters. Theauthors calculated models with the same physical properties as the CHeB giants of both clustersand using different mixing schemes at the core boundary. They found that models computedwith a moderate amount of overshooting can reproduce the range of observed period spacings.Interestingly, the models computed with penetrative convection (adiabatic stratification in theextra-mixed region) predict too large period spacings for the stars at the beginning of the CHeBphase in NGC 6819, which led the authors to favor the non-penetrative convection scenario.Naturally more evidence is required to be more conclusive. Further constraints could also be obtained in the near future by detecting the signature ofbuoyancy glitches in the period spacings of g modes in CHeB giants. The mixing schemespresented in Sect. 5.1 predict very different abundance profiles in the region above the fullymixed core. For instance, models computed with standard overshooting show step-like featuresin the helium abundance above the core, while models computed with semi-convection havesmooth helium profiles. Sharp variations of µ are expected to be felt as buoyancy glitches by gmodes, which should produce an oscillatory component in the period spacing, as was describedin Sect. 3.1. The occurrence of buoyancy glitches in the cores of red giants and their seismicsignature in the period spacing of g modes has been extensively addressed by Cunha et al.(2015) using stellar models. Detecting these modulations in ∆Π is more complicated for CHeBgiants than for main sequence g-mode pulsators because of the mixed character of the modes.17igure 15: Stretched period ´echelle diagram (see explanations in Mosser et al. 2015) of a Kepler
CHeB giant. The blue squares indicate high peaks in the oscillation spectrum of the star.In the absence of a buoyancy glitch, they are expected to line up on a straight ridge. Here,a modulation is observed (orange dashed line), which is compatible with a buoyancy glitch.Figure from Mosser et al. (2015).Nonetheless, the method of Mosser et al. (2015) can be used to recover the period spacings ofpure gravity modes and thus reveal potential periodic modulations produced by glitches (seeFig. 15). Glitches produced by sharp µ -gradients above the mixed core are located deep withinthe g-mode cavity and are thus expected to produce long-period modulations. A systematicsearch for such features in the oscillation spectra of CHeB giants observed with Kepler shouldbring strong constraints on the way chemical elements are mixed above the convective core.
The advent of space asteroseismology has yielded numerous novel constraints on the propertiesof convective cores for stars with various masses and evolutionary stages. We started thisreview by mentioning that three physical quantities needed to be known to progress in ourmodeling of the boundary of convective cores. We conclude by summarizing the recent findingsof asteroseismology for each of them:1.
Distance over which mixed cores are extended:
We here presented only a small selection ofall the seismic studies that provided constraints on the extent of the mixed core. The greatmajority of them concluded that an extension of the mixed core beyond the Schwarzschildlimit needed to be invoked. These studies also showed that large star-to-star variationsexist for the distance of the extra-mixing. Nevertheless, tendencies can be found in theavailable data. Main sequence intermediate-mass stars seem to require extensions of theorder of 0.2-0 . H P . For lower-mass stars (1 . (cid:46) M/M (cid:12) (cid:46) . . H P , r s ), where r s is the formal boundary of the convectivecore). In this mass range, a potential increase of the distance of extra-mixing with stellar18ass has been reported but needs to be confirmed. In this review, we have focused onlow- and intermediate-mass stars, which have so far benefitted more from space-basedasteroseismology, but seismic constraints have also been obtained on the core propertiesof massive stars. The seismic analyses of β Cephei pulsators (8 to 20 M (cid:12) ), essentiallywith ground-based observations, have shown quite large variations in the extent of theextra-mixed region from one star to another, typically ranging from 0 to 0.3 H P (e.g.,Dupret et al. 2004, Ausseloos et al. 2004, Aerts et al. 2011, Briquet et al. 2012). Finally, ithas been found that the convective core of core-helium-burning stars needs to be extendedover even larger distances, likely in the range of 0.5-1 H P .2. Nature of the mixing in the core extension:
Seismology is currently the only tool totest how efficient the mixing of chemicals is beyond the edge of the convective core.Gravity modes, through their sensitivity to the gradient of µ , are particularly well suitedfor this purpose. The seismic study of three SPB stars has consistently shown that adiffusive overshooting modeled with an exponentially decaying diffusion coefficient yieldsbetter agreement with seismic observations than an instantaneous mixing in the overshootregion. Other constraints on the nature of the mixing could be brought in the near futureby using mixed modes in subgiants.3. Temperature stratification in the region of extra-mixing:
Measuring this quantity is par-ticularly difficult. However, having penetrative ( ∇ = ∇ ad ) or non-penetrative ( ∇ = ∇ rad )convection changes the propagation of gravity modes in the overshoot region. This mod-ifies the period spacing of g modes. Hints in favor of non-penetrative convection wereobtained from the core-helium burning giants of an old open cluster. Further constraintscould be obtained from SPB and γ Doradus stars. We here note that constraints havebeen obtained on the temperature stratification at the bottom of the envelope convectionof the Sun. Christensen-Dalsgaard et al. (2011) found evidence for a smooth transitionfrom ∇ = ∇ ad to ∇ = ∇ rad in the overshoot region.We note that in this review, we have focused exclusively on results obtained with theforward modeling approach. Seismic inversions also have a large potential to bring informationon the properties of convective cores. Recent studies have shown promising results for solar-likepulsators (Bellinger et al. 2017, Buldgen et al. 2018) and new, model-independent constraintscould come from such analyses in the near future.The number of targets for which the edge of the mixed core could be seismically probedis increasing rapidly. We are starting to build large enough samples so that trends can besearched in the properties of the extra-mixed region as a function of global stellar parameters.On the short term, this can help us calibrate more refined models of convective core extensionsin evolutionary codes. This could provide us with more reliable stellar ages, which is crucial fordisciplines that require high-precision stellar modeling, such as the characterization of exoplan-ets, with the upcoming PLATO mission, or galactic archaeology. Even more challenging willbe the task of disentangling the contributions from the different physical processes to the ex-tensions of convective cores. So far a pragmatic approach has generally been adopted, wherebythe effects of all these processes are modeled together in a parametric way. To establish thecontribution of rotational mixing, it would be very interesting to search for correlations betweenthe amount of mixing beyond convective cores and the rotational properties of stars. Stars forwhich seismology can provide measurements of the size of the mixed core and the internal ro-tation profile would be particularly useful. Magnetic fields are also expected to play a role byinhibiting rotational mixing through the damping of differential rotation in radiative interiors.19or instance, this might be happening in the β Cephei pulsator V2052 Ophiuci, which hostsa fossil magnetic field with B pol ∼
400 G. Through a seismic modeling of the star, Briquetet al. (2012) found that it indeed has an unexpectedly low amount of extra-mixing beyond theconvective core. More studies of this type are needed to progress in our understanding of theprocesses that can extend the size of mixed cores. In this context, the
TESS and
PLATO missions are particularly welcome. They will provide us with seismic data with a nearly all-skycoverage, which will greatly increase the number of targets for which seismic constraints on thecore properties can be derived. In particular, with
PLATO data, we will be able to performmuch more meaningful statistical studies of the extent of the mixed core in solar-like pulsators.
Acknowledgements
I am thankful to Marc-Antoine Dupret for inviting me to the Li`ege Workshop organized inhonor of Arlette Noels. I also take the opportunity to express my deep gratitude to Arlette,for all the very enlightening discussions that I have had with her. She is a great source ofinspiration for me. I also acknowledge support from the project BEAMING ANR-18-CE31-0001 of the French National Research Agency (ANR) and from the Centre National d’EtudesSpatiales (CNES).
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