The Current Status of Asteroseismology
aa r X i v : . [ a s t r o - ph ] M a r Solar PhysicsDOI: 10.1007/ ••••• - ••• - ••• - •••• - • The Current Status of Asteroseismology
C. Aerts · J. Christensen-Dalsgaard · M. Cunha · D.W. Kurtz c (cid:13) Springer ••••
Abstract
Stellar evolution, a fundamental bedrock of modern astrophysics, is drivenby the physical processes in stellar interiors. While we understand these processesin general terms, we lack some important ingredients. Seemingly small uncertaintiesin the input physics of the models, e.g. , the opacities or the amount of mixing andof interior rotation, have large consequences for the evolution of stars. The goal ofasteroseismology is to improve the description of the interior physics of stars bymeans of their oscillations, just as global helioseismology led to a huge step forwardin our knowledge about the internal structure of the Sun. In this paper we present thecurrent status of asteroseismology by considering case studies of stars with a varietyof masses and evolutionary stages. In particular, we outline how the confrontationbetween the observed oscillation frequencies and those predicted by the models allowsus to pinpoint limitations of the input physics of current models and improve themto a level that cannot be reached with any other current method.
Keywords:
Oscillations, Stellar; Interior, Convective Zone; Interior, Core; Instru-mentation and Data Management
1. Introduction
Despite extensive research in recent decades, we lack detailed knowledge of some im-portant physical processes relevant for the description of stellar interiors. The reasonis that, in general, the existing observations do not yet allow a detailed confrontationwith the description of the physical properties of either the stellar material in thedeepest internal layers, or of the dynamics of the outer stellar envelope. At firstsight, seemingly small uncertainties in the input physics of the models have large Instituut voor Sterrenkunde, Katholieke UniversiteitLeuven, Celestijnenlaan 200 D, 3001 Leuven, Belgium;Afdeling Sterrenkunde, Radboud University Nijmegen, POBox 9010, 6500 GL Nijmegen, The Netherlands. email: [email protected] Institut for Fysik og Astronomi, Aarhus Universitet,Aarhus, Denmark. email: [email protected] Centro de Astrofísica da Universidade do Porto, Rua dasEstrelas, 4150-762, Porto, Portugal. email: [email protected] Centre for Astrophysics, University of Central Lancashire,Preston PR1 2HE, UK. email: [email protected] . Aerts et al. consequences for the whole duration and end of the stellar life cycle. The lack of agood understanding of interior transport processes, caused by different phenomenasuch as rotation, gravitational settling, radiative levitation, magnetic diffusion, etc. ,is particularly acute when it comes to precise predictions of stellar evolution, andthe galactic chemical enrichment accompanying it.Given that global helioseismology led to a huge step forward in the accuracy ofthe internal structure model and of the transport processes inside the Sun, astero-seismology aims to obtain similar improvements for different types of stars by meansof their oscillations. Stellar oscillations indeed offer a unique opportunity to probethe internal properties and processes, because these affect the observable frequencies.The confrontation between the frequencies measured with high accuracy and thosepredicted by models gives insight into the limitations of the input physics of modelsand improves them. In fact, stellar oscillation frequencies are the best diagnosticknown that can reach the required precision in the derivation of interior stellarproperties.At present, the unknown aspects of the physics and dynamics are dealt withby using parameterized descriptions, where the parameters are tuned from observa-tional constraints. These concern, e.g. , the treatments of convection, the equationof state, diffusion and settling of elements. When a lack of observational constraintsoccurs, solar values are often assigned, e.g. , to the mixing length parameter in thedescription of convection, or phenomena are ignored, e.g. , convective overshootingand the diffusion of heavy elements. It is hard to imagine, however, that one singleset of parameters is appropriate for very different types of stars. Similarly, rotationis either not included or is included only with a simplified treatment of the evolutionof the rotation law in stellar models ( e.g. , Maeder and Meynet, 2000). Fortunately,rotation also modifies the frequencies of the star’s modes of oscillation ( e.g. , Gough,1981; Saio, 1981). An adequate seismic modelling of rotation inside stars thereforeis within reach with high-precision measurements and mode identifications of stellaroscillations.The origin and physical nature of stellar oscillations, as well as their mathematicalproperties, were thoroughly discussed by Cunha et al. (2007), to which we referthe reader for the theoretical considerations of asteroseismology. Extensive recentoverviews of the occurrence of stellar oscillations across the entire HR diagram,and asteroseismic applications thereof, are already available in Kurtz (2004, 2006),Cunha et al. (2007), and Aerts, Christensen-Dalsgaard, and Kurtz (in preparation forSpringer-Verlag). Rather than repeating such an observational overview here in a farmore concise format, we have opted to outline the current status of asteroseismologyby focusing on a few carefully chosen examples that show the merits this researchfield has brought to the improvement of stellar modelling, i.e. , we confine ourselvesto cases where quantitative measures of internal structure parameters have beenachieved. We start by considering examples of stars that oscillate similarly to theSun, and then move on to pulsators excited by a heat mechanism for a discussion ofconvective overshooting and rotation inside stars. The rapidly oscillating Ap stars,being pulsators with very strong magnetic fields, are of particular interest to solarastronomers, so are discussed in a separate paper in this volume (Kurtz, 2008). aerts_rev3.tex; 25/10/2018; 2:40; p.2 he Current Status of Asteroseismology
Figure 1.
HR diagram showing the stars in which solar-like oscillations have been detected(with HR 2530 = HD 49933). The discoveries for 171 Pup, HR 5803 (HD 139211) and τ PsA areunpublished (Carrier et al. , in preparation). Figure courtesy of Fabien Carrier.
2. From the Sun to Stars: the Properties of Solar-Like Pulsators
As the oscillations of the Sun are caused by turbulent convective motions near itssurface, we expect such oscillations to be excited in all stars with significant outerconvection zones. Solar-like oscillations are indeed predicted for the lowest-massmain-sequence stars up to objects near the cool edge of the classical instability stripwith masses near 1.6 M ⊙ ( e.g. , Christensen-Dalsgaard, 1982; Christensen-Dalsgaardand Frandsen, 1983; Houdek et al. , 1999) as well as in red giants (Dziembowski et al. , 2001). Such stochastically excited oscillations have very tiny amplitudes, whichmakes them hard to detect, particularly for the low-mass stars. Velocity amplitudeswere predicted to scale roughly as L/M , where L and M are the luminosity and massof the star, before the first firm discoveries of such oscillations in stars other than theSun (Kjeldsen and Bedding, 1995). This scaling law was later modified to ( L/M ) . from excitation predictions based on 3D computations of the outer convection zones aerts_rev3.tex; 25/10/2018; 2:40; p.3 . Aerts et al. of the stars (Samadi et al. , 2005), resulting in lower amplitudes compared with thosefound for 1D models.The modes observed in the Sun at low spherical harmonic degree l , and hencesolar-like oscillations observable in distant stars, are high-order acoustic modes. Theysatisfy an approximate asymptotic relation ( e.g. , Vandakurov, 1967; Tassoul, 1980;Gough, 1993), according to which, to leading order, ν nℓ ∼ ∆ ν (cid:18) n + ℓ ǫ (cid:19) , (1)where ν nl is the cyclic frequency of a mode of radial order n and degree l and ǫ is afunction of frequency determined mainly by the conditions near the stellar surface.Also, ∆ ν = Z R d rc ! − (2)is a measure of the inverse sound travel time over a stellar diameter, r being thedistance to the stellar centre, R the surface radius of the star and c is the adiabaticsound speed. From simple considerations it follows that the large frequency separation satisfies ∆ ν ∝ ( M/R ) / and hence is a measure of the mean density of the star.Departures from Equation (1) can be characterized by the small frequency separation : δν nℓ = ν nℓ − ν n − ℓ +2 . This quantity is mainly sensitive to the sound speed inthe core of the star and hence provides a measure of the evolutionary state. Thesensitivity of ∆ ν and δν on stellar properties allows a calibration of stellar modelsin terms of (∆ ν, δν ) to estimate the mass and evolutionary stage of the star ( e.g. ,Christensen-Dalsgaard, 1984, 1988; Ulrich, 1986).The search for solar-like oscillations in stars in the solar neighbourhood has beenongoing since the early 1980s. The first indication of stellar power with a frequencydependence similar to that of the Sun was obtained by Brown et al. (1991) in α CMi(Procyon, F5IV). The first detection of individual frequencies of solar-like oscillationswas achieved from high-precision time-resolved spectroscopic measurements only in1995 for the G5IV star η Boo (Kjeldsen et al. , 1995); Brown et al. (1997), however,could not establish a confirmation of this detection from independent measurements,but it was subsequently confirmed by Carrier, Bouchy, and Eggenberger (2003) andKjeldsen et al. (2003). It took another four years before solar-like oscillations weredefinitely established in Procyon (Martić et al. , 1999). While there was a recentcontroversy about this detection (Matthews et al. , 2004; Bedding et al. , 2005) whichwe do not discuss in detail here, the results of Martić et al. (1999) have been con-firmed (Mosser et al. , 2008) and an in-depth asteroseismic investigation based on alarge multisite campaign is presently being conducted. Subsequent to Martić et al.’sresults, solar-like oscillations were found in two more stars: the G2IV star β Hyi(Bedding et al. , 2001) and the solar twin α Cen A (Bouchy and Carrier, 2001). Theseimportant discoveries led to several more subsequent detections, a summary of whichwas provided by Bedding and Kjeldsen (2007). The positions of confirmed solar-likepulsators in the HR diagram are displayed in Figure 1. The detected frequencies andfrequency separations for all stars behave as expected from theoretical predictionsand from scaling relations based on extrapolations from helioseismology.The oscillation frequencies and frequency separations detected in solar-like pul-sators provide additional constraints with which to test models of stellar structure aerts_rev3.tex; 25/10/2018; 2:40; p.4 he Current Status of Asteroseismology and evolution in conditions slightly different from those provided by the Sun. Suchstudies generally involve a fit of theoretical models, characterized by a number ofmodel parameters, to the set of seismic and non-seismic data available for a givenpulsator. Theoretical modelling of solar-like pulsators using this direct fitting ap-proach has been carried out for several stars, including η Boo (Carrier, Eggenberger,and Bouchy, 2005; Guenther et al. , 2005), Procyon (Eggenberger et al. , 2004a;Eggenberger, Carrier, and Bouchy, 2005; Provost et al. , 2006), and α Cen A andB ( e.g. , Eggenberger et al. , 2004b; Miglio and Montalbán, 2005; Yildiz, 2007). Sofar, the main results of these fits are estimates of the stellar masses, ages, and initialmetallicities, even though in some cases the results are still controversial, and callfor better sets of data.Among the solar-like pulsators, the binary star α Cen A and B provides a par-ticularly interesting test-bed for studies of stellar structure and evolution, due tothe numerous and precise seismic and non-seismic data that are available for bothcomponents of the binary. Studies of α Cen A and B, including seismic and non-seismic data for both components, indicate that the age of the system is likely tobe between 5.6 and 7.0 Gyr, the value derived being dependent, in particular, onthe seismic observables that are included in the fits. Moreover, the same studiespoint to a significant difference in the values of the mixing-length parameter ( α MLT ,see Section 3 for a definition), for the two stars, although the sign is uncertain andpossibly dependent on the detailed treatment of the effects of the near-surface layersin the analysis of the observed frequencies. Eggenberger et al. (2004b) and Miglio andMontalbán (2005) found that the value for α Cen B is larger than that for α Cen A,whereas Teixeira et al. (in preparation) found that α MLT was slightly smaller for α Cen B than for α Cen A. The latter study also found that the best-fitting model for α Cen A was on the border of having a convective core (see Christensen-Dalsgaard,2005): even a slight increase in the mass of the model led to a significant convectivecore and hence a model that was quite far from matching the observed properties.Despite the successful case studies just outlined, the detailed seismic studies ofstars with stochastically-excited modes are currently still in their infancy comparedwith global helioseismology. However, given the recent detections and the contin-uing efforts to improve them, we expect very substantial progress in the seismicinterpretation of such targets in the coming years. In particular, the CoRoT ( e.g. ,Michel et al. , 2006) and
Kepler ( e.g. , Christensen-Dalsgaard et al. , 2007) missionswill give data of very high quality on solar-like oscillations. As seen in the example of α Cen A above, it is noteworthy that the class of main-sequence solar-like oscillatorsencompasses transition objects regarding the development of a convective core onthe main sequence ( M ⊙ < M < . M ⊙ ). Asteroseismology will surely refine thedetails of the yet poorly understood physics that occurs near the core of the objectsin this transition region. Also, data of the expected quality will provide informationabout the depth and helium content of the convective envelope ( e.g. Houdek andGough 2007a), as well as more reliable determinations of stellar ages (Houdek andGough 2007b).Additional information from solar-like oscillations is available in the cases ofrelatively evolved stars, beyond the stage of central hydrogen burning. Here the fre-quency range of stochastically-excited modes may encompass mixed modes behavingas standing internal gravity waves, or g modes, in the deep chemically inhomogeneousregions, thus providing much higher sensitivity to the properties of this region. In aerts_rev3.tex; 25/10/2018; 2:40; p.5 . Aerts et al. fact, there is some evidence that such modes have been found in the subgiant η Boo(Christensen-Dalsgaard, Bedding, and Kjeldsen, 1995).Evidence for rotational splitting (see Section 4 for a definition) has been foundin α Cen A (Fletcher et al. , 2006; Bazot et al. , 2007). However, it has not yet beenpossible to map the interior rotation of a solar-like pulsator, since the present fre-quency multiplet detections are insufficient. We note that Lochard, Samadi, andGoupil (2004) found, with simulated data, that the presence of mixed modes in astar such as η Boo may allow some information to be derived about the variation ofthe internal rotation with position.For a few pulsators excited by the heat mechanism, data are already availablethat provide such information, albeit only very roughly. We discuss this further inSection 4, but first we highlight in the next section some case studies through whichthe properties of core convection have been tuned by asteroseismology.
3. Seismic Derivation of Convective Overshooting inside Stars
The standard description of convection used in stellar modelling is the Mixing LengthTheory (MLT) of Böhm-Vitense (1958). In this theory, the convective motions aretreated as being time-independent. In the absence of a rigorous theory of convec-tive motions based on first principles, the convective cells are assumed to have amean-free-path length of α MLT H p , where H p is the local pressure scale height. Themixing-length parameter depends on the physics considered in the model and onthe specific formulation of the MLT used. Its value for Model S for the Sun ofChristensen-Dalsgaard et al. (1996) is α MLT ≃ . , using the Böhm-Vitense (1958)MLT formulation.In the context of stellar evolution, it is of crucial importance to quantify theamount of matter in the fully mixed central region of the star. This amount isusually derived from the Schwarzschild criterion, which states that convection occursin regions where the adiabatic temperature gradient is smaller than the radiativegradient. However, from a physical point of view, it is highly unlikely that convectiveelements stop abruptly at the boundary set by the Schwarzschild criterion. Rather,their inertia causes them to overshoot into the adjacent stable area where radiativeenergy transport takes place. The amount of such overshooting is, however, largelyunknown. For this reason, it is customary to express it as α ov H p where α ov isexpected to be a small fraction of α MLT .The inability to derive a value for α MLT and α ov from a rigorous theoreticaldescription is highly unsatisfactory, particularly for stars with a convective core,because the total mass of the well-mixed central region of the star determines itsstellar lifetime. This is the reason why great effort has been, and is being, made toquantify α ov , keeping in mind that we have already a fairly good estimate of α MLT from the Sun. We describe here the power of asteroseismology to determine α ov .In the solar case helioseismic analyses have provided constraints on the over-shoot from the solar convective envelope, assuming that this results in a nearlyadiabatic extension of the convection zone followed by an abrupt transition to theradiative temperature gradient (Zahn, 1991). Assuming also, as usual, a sphericallysymmetric model, such a behaviour introduces a characteristic pattern in the fre-quencies in the form of an oscillatory variation of the frequencies as functions ofthe mode order. From the observed amplitude of this signal, an overshoot region aerts_rev3.tex; 25/10/2018; 2:40; p.6 he Current Status of Asteroseismology of the nature considered must have an extent less than around . H p (Basu, An-tia, and Narasimha, 1994; Monteiro, Christensen-Dalsgaard, and Thompson, 1994;Christensen-Dalsgaard, Monteiro, and Thompson, 1995). It was found by Monteiro,Christensen-Dalsgaard, and Thompson (2000) that a similar analysis can be carriedout on the basis of just low-degree modes, such as will be observed in distant stars.Owing to their sensitivity to the core structure, the low-degree solar-like oscilla-tions should in principle be sensitive to overshoot from convective cores. Models of η Boo without and with overshoot were considered by Di Mauro et al. (2003, 2004).Although the present observed frequencies are not sufficiently accurate to providedirect information about the properties of the core, it was found that for α ov ≥ . models could be found in the central hydrogen-burning stage which matched theobserved location in the HR diagram. In such models, mixed modes are not ex-pected; thus the definite identification of mixed modes would constrain the extentof overshoot in the star. Straka, Demarque, and Guenther (2005) considered modelsof Procyon with various types of core overshoot to determine the extent to whichovershoot could be asteroseismically constrained.For the p -mode diagnostics considered, little sensitivity to overshoot was found,while the, perhaps unlikely, detection of g modes in Procyon, such as have beenclaimed in the Sun, would provide much stronger constraints on the overshoot dis-tance. As in the case of η Boo, the definite identification of the star as being onthe subgiant branch, e.g. , from the properties of the oscillation frequencies, wouldprovide strict constraints on the extent of overshoot during the central hydrogen-burning phase. Mazumdar et al. (2006a) made a detailed analysis of the sensitivityof suitable frequency combinations to the properties of stellar cores and found thatthe mass of the convective core, possibly including overshoot, could be determinedwith substantial precision, given frequencies with errors that should soon be reached.Cunha and Metcalfe (2007) developed diagnostics of small convective cores that mayin principle also provide information about the properties of overshoot; the detailedsensitivity still needs investigation, however.Quantitative measures of the core convective overshooting parameter have beenachieved by fitting the frequencies of some of the β Cep stars. This group of young,Population I, near-main-sequence pulsating B stars has been known for more than acentury. They have masses in the range 8 – 18 M ⊙ , and they oscillate in low-order p and g modes with periods in the range 2 – 8 hours. These oscillations are excited bya heat mechanism acting through opacity features associated with elements of theiron group ( e.g. , Dziembowski and Pamiatnykh, 1993; Pamyatnykh, 1999; Miglio,Montalbán, and Dupret 2007). A recent overview of the observational properties ofthe class was provided by Stankov and Handler (2005). Most of the β Cep stars showmultiperiodic light and line-profile variations and most rotate at only a small fractionof the critical velocity.Significant progress in the detailed seismic modelling of the β Cep stars hasoccurred over the last few years and has led to quantitative estimates of the core over-shooting parameter α ov for several class members with slow rotation (see, e.g. , Aerts,2006, for a summary). We illustrate this here for the star θ Oph, whose frequencyspectrum was determined from a multisite photometric campaign and is representedin Figure 2 (Handler, Shobbrook, and Mokgwetsi, 2005). An additional long-term,high-resolution spectroscopic campaign revealed that this star is a member of aspectroscopic binary with an orbital period of 56.71 days and an eccentricity of 0.17(Briquet et al. , 2005), and allowed the identification of the spherical wavenumbers aerts_rev3.tex; 25/10/2018; 2:40; p.7 . Aerts et al.
Figure 2.
The schematic frequency spectrum of the β Cep star θ Oph for the Strömgren u filter as derived from a multisite photometric campaign. The measured photometric amplituderatios led to an identification of the frequencies ν , ν , ν , and ν as, respectively, ℓ = 2 , , , .(Figure reproduced from Handler, Shobbrook, and Mokgwetsi, 2005). Figure 3.
The rotational kernels defined in Equation (4) as a function of radial distance insidethe star ( x = r/R ), for the identified ℓ = 1 , p mode (solid line) and ℓ = 2 , g mode (dashedline) of the β Cep star θ Oph. The vertical dotted line marks the position of the boundary ofthe convective core, including the overshoot region. Note that the kernels also approximatelyrepresent the relative sensitivity of the mode frequencies to other aspects of the stellar interior.(Figure reproduced from Briquet et al. , 2007). aerts_rev3.tex; 25/10/2018; 2:40; p.8 he Current Status of Asteroseismology
Table 1.
The identification of the pulsation modes of the β Cep star θ Oph derived from multicolour photometric andhigh-resolution spectroscopic data. Positive m -values repre-sent prograde modes. The amplitudes of the modes are givenfor the Strömgren u filter and for the radial velocities. Tablereproduced from Briquet et al. (2007).ID Frequency (d − ) ( ℓ, m ) u ampl. RV ampl.(mmag) (km s − ) ν (2 , − ν (2 , +1) ν (2 , +2) ν (0 , ν (1 , − ν (1 , ν (1 , +1) ( ℓ, m ) of the seven detected frequencies from the line-profile variations induced bythe oscillations (see Table 1 reproduced from Briquet et al. , 2007).Because the frequency spectra of β Cep stars are so sparse for low-order p and g modes compared with those of solar-like pulsators (see Figure 2), one does not havemany degrees of freedom to fit the securely-identified modes. This led to the identi-fication of the radial order of the modes of θ Oph as g for the frequency quintupletcontaining ν , ν , ν , the radial fundamental for ν and p for the triplet ν , ν , ν .Fitting the three independent m = 0 frequencies results in a relation between themetallicity and the core-overshooting parameter, because the stellar models for main-sequence B stars typically depend on the five parameters ( X, α ov , Z, M, age) ifwe ignore effects of diffusion. Note that α MLT is usually fixed to the solar value; forB stars, with their extremely thin and inefficient outer convection zones, changing α MLT within reasonable limits does not change the characteristics of the models.In this way, one finds α ov = 0 . ± . from a detailed high-precision abundancedetermination for θ Oph (Briquet et al. , 2007).The reason why we can derive the core overshooting and the rotation (see Sec-tion 4), and provide a quantitative measure of these parameters for this star, is thedifferent probing ability of the non-radial modes. This can be illustrated by plottingprobing kernels of the modes. Different types of such kernels are used, dependingon the kind of behaviour under investigation. This is illustrated in Figure 3, wherewe show the rotational splitting kernels K ( x ) (which will be defined in Equation (4)below) of θ Oph for the two non-radial modes. It can be seen that the g mode’skernel behaves differently near the boundary of the core region, and thus probesthat region in a different way than the p mode, allowing the derivation of detailsof the rotational properties as explained below. A similar figure holds for the energydistribution, which allows probing the extent of the core region. A comparable resultwas obtained for V836 Cen whose frequency spectrum is almost a copy of that of θ Oph (see Figure 5) and also for ν Eri (Pamyatnykh, Handler, and Dziembowski,2004).The combination of low-order p and g modes thus turns out to be a very pow-erful tool to derive the internal structure parameters of massive stars. Additional aerts_rev3.tex; 25/10/2018; 2:40; p.9 . Aerts et al. measures of the core overshooting have been obtained for the β Cep stars β CMa(Mazumdar et al. , 2006b) and δ Ceti (Aerts et al. , 2006). For all these β Cep stars, α ov ranges from 0.1 to . , although these values depend somewhat on the adoptedmetal mixture (Thoul et al. , 2004). It is remarkable that the frequencies of just twowell-identified oscillation modes that have sufficiently different kernels allow one toderive the overshooting parameter with a precision of typically . expressed in H p .Adding just a few more well-identified modes should drastically reduce this error forspecific input physics of the models.The seismically derived estimates of core overshooting in β Cep stars are compat-ible with the quantitative results for eight detached double-lined eclipsing binariesobtained by Ribas, Jordi, and Giménez (2000), who found α ov to range from 0.1to . for primary masses ranging from 1.5 to 9 M ⊙ . Another way of determiningthe amount of overshooting from data is by fitting stellar evolutionary tracks tothe dereddened colour-magnitude diagrams of clusters, e.g. , α ov = 0 . ± . forthe intermediate age open cluster NGC3680 (Kozhurina-Platais et al. , 1997) and α ov ≈ . for the old open cluster M67 (VandenBerg and Stetson, 2004). In thesetwo methods, essentially the same five unknown structure parameters occur as forthe seismic modelling, since stellar evolution models are used to fit the positionof the binary components and of the cluster main-sequence turn-off point in theHR diagram, respectively. The uncertainty on the overshoot distance derived fromthe light curve analysis of an accurately modelled eclipsing binary or from fittingof a cluster turn-off point is typically between 0.05 and . H p provided that themetallicities are known. It is interesting, although perhaps fortuitous, that all thesequantitative measures of the amount of overshooting are in agreement with thetheoretical predictions by Deupree (2000) from 2D hydrodynamic simulations ofzero-age main-sequence stars with a convective core.
4. Seismic Derivation of the Internal Rotation Profile of Stars
The rotation of a star implies a splitting of the oscillation frequencies compared withthe case without rotation. Hence, rotation becomes apparent in frequency spectraas multiplets of ℓ + 1 components for each mode of degree ℓ . Ignoring rotationaleffects higher than order one in the rotational frequency as well as the influence of amagnetic field, the frequency splitting becomes ν m = ν + m Z R K ( r ) Ω( r )2 π d rR , (3)where ν m is the cyclic frequency of a mode of azimuthal-order m , and Ω is theangular velocity which we here assume to depend only on the distance ( r ) to thecentre. The rotational kernels are defined as K ( r ) = (cid:0) ξ r − ξ r ξ h + [ ℓ ( ℓ + 1) − ξ h (cid:1) r ρ Z R d rR h ξ r + ℓ ( ℓ + 1) ξ h i r ρ , (4)with ξ r and ξ h the radial and tangential components of the displacement vector ξ = ( ξ r e r + ξ h ∇ h ) Y mℓ . (5) aerts_rev3.tex; 25/10/2018; 2:40; p.10 he Current Status of Asteroseismology Figure 4.
Frequency splittings | ∆ ν | = | ν m − ν | of the triplets detected for GD 358 as afunction of radial overtone n . The full line connects the values for the m = +1 componentscorresponding to prograde modes and the dotted line those for m = − representing retrogrademodes. (Figure reproduced from Winget et al. , 1994). If such multiplets are observed, their structures are a great help in mode iden-tification, as non-radial modes with a given value of ℓ have ℓ + 1 multiplet peakscorresponding to the different values of m , although not all peaks may be visibleowing to the geometry of the modes or excitation of the components, while radialmodes show no multiplet structure. The recognition of the multiplet structure is fareasier for very slow rotators, where “slow” here means that the rotational frequencyis far lower than the frequency spacing for m = 0 components of modes of adjacentradial order n .Below, we describe two types of pulsators for which a quantitative measure ofdifferential interior rotation has been established.4.1. White DwarfsThe first detection of differential ( i.e. , non-rigid) rotation inside a star besides theSun was achieved for the DBV white dwarf GD 358 from a multisite campaign bythe Whole Earth Telescope organisation (Winget et al. , 1994). The multiperiodicvariations of DBV white dwarfs are due to low-degree, high-order g modes, excitedby the heat mechanism active in the second partial ionization zone of helium. Theiroscillation periods range from 4 to 12 minutes and their photometric amplitudesare relatively large, from a few mmag to 0.2 mag ( e.g. , Bradley, 1995). Among themore than 180 significant frequency peaks detected in the white-light photometriclightcurve of GD 358 covering 154 hours of data, 27 are the components of well-identified triplets. These frequency splittings are shown as a function of radial order n in Figure 4. It can be seen that larger splittings occur for higher radial order, aerts_rev3.tex; 25/10/2018; 2:40; p.11 . Aerts et al. while one would expect these splittings to be constant for rigid rotation inside thewhite dwarf. Since the modes of higher radial-order probe predominantly the outerlayers and those of lower radial-order the inner parts, the rotation of GD 358 mustbe radially differential. The mean splitting for the modes of n = 16 and 17 leads toa rotation period of 0.89 days through Equations (3) and (4), while for n = 8 , therotation period is 1.6 days. Winget et al. (1994) therefore concluded that the innerparts of GD 358 rotate 0.6 times more slowly than its outer layers, where “inner”and “outer” refer to those regions probed by the detected triplets. However, Kawaler,Sekii, and Gough (1999) found that the data were not yet of sufficient quality to allowa more detailed inversion for the variation of the internal rotation with depth.The detailed seismic modelling of GD 358 followed that achieved previously for theprototypical DOV white dwarf PG1159-035 (GW Vir) described in the seminal workby Winget et al. (1991), which was again based on data collected by the Whole EarthTelescope consortium (Nather et al. , 1990). This led to 125 significant frequencies forGW Vir, of which 101 were identified as components of rotationally split triplets andquintuplets. Unlike the case for GD 358, all multiplets for a given ℓ showed the samefrequency splitting within the measurement errors, allowing Winget et al. (1991)to deduce a constant rotation period of approximately . ± . day throughoutthe white dwarf. Classical spectroscopy can in no way reveal the rotation periods ofsingle compact stellar remnants with such high precision, not even in the case wherethe inclination angle can be estimated from independent information.The deviation of GD 358’s splittings for m = +1 with respect to those for m = − in Figure 4 was interpreted by Winget et al. (1994) in terms of a weak magnetic fieldof ± G, which causes splittings ∼ | m | in addition to the rotational splittinggiven in Equation (3) ( e.g. , Dziembowski and Goode, 1984; Jones et al. , 1989). Theeffect of a magnetic field could not be established for the frequency multiplets ofPG 1159-035, which led to an upper limit of 6000 G for that object’s magnetic field(Winget et al. , 1991). It is noteworthy that the magnetic-field strength that can beprobed by classical spectroscopy of white dwarfs through the Zeeman effect requiresfields roughly a factor of 1000 stronger than what can be found from asteroseismology.The case studies of GD 358 and PG 1159-035 by the Whole Earth Telescopeconsortium implied not only a first test case for the technique of asteroseismology,but at the same time a real breakthrough in the derivation of white dwarf structuremodels. It not only led to estimates of internal rotation and magnetic field strength,but also allowed a high-precision mass estimate ( . ± . M ⊙ for PG1159-035and . ± . M ⊙ for GD 358). It also proved that the outer layers of white dwarfsare compositionally stratified. This was derived from deviations of the frequencyspacings due to mode trapping compared with spacings for unstratified models. Massestimates with such high precision cannot be achieved from other means, except forrelativistic effects in binary pulsars. These two seismic studies of white dwarfs pavedthe road for many others of their kind, but none of the more recent ones have ledto more accurate internal rotation rates than those for PG1159-035 and GD 358. Werefer to Kepler (2007) and Fontaine and Brassard (in preparation) for recent reviewpapers on white-dwarf seismology.4.2. Main-Sequence StarsThere are presently only three main-sequence stars, besides the Sun, for which anobservational constraint on the internal-rotation profile has been derived. In all three aerts_rev3.tex; 25/10/2018; 2:40; p.12 he Current Status of Asteroseismology Figure 5.
The schematic frequency spectrum of the β Cep star V836 Cen derived from sin-gle-site Geneva U data spanning 21 years. The dotted lines are frequencies that are not yetfirmly established; these were not used in the seismic modelling. (Figure reproduced from Aerts et al. , 2004). cases, it was achieved through asteroseismology of β Cep stars. Several of these aresuitable targets to attempt mapping of their interior rotation because their rotationalfrequencies are well below the frequency spacing between multiplets. These starsare particularly interesting targets for this purpose, because the largest uncertaintyin stellar evolution models for massive stars is precisely concerned with rotationalmixing effects.The first seismic proof of differential rotation in a massive star was obtained forthe B3V star V836 Cen (HD 129929; Aerts et al. , 2003). This result was derivedfrom the well identified (parts) of one rotationally split frequency triplet and onequintuplet, as shown in Figure 5. This star’s detected frequency spectrum is obviouslyvery similar to the one of θ Oph (compare Figures 2 and 5), except that V836 Cenis a slower rotator than θ Oph. Given that only two multiplets were available forV836 Cen, Dupret et al. (2004) assumed a linear rotation law and concluded thatthe rotational frequency near the stellar core is 3.6 times higher than at the surface.It was possible to derive this because the kernels of the g and p modes probedifferently the rotational behaviour near the stellar core, just as for θ Oph (seeFigure 3). A very similar result, the rotation of the deep interior exceeding thesurface rotation by a factor between three and five, was obtained by Pamyatnykh,Handler, and Dziembowski (2004) from the g and p ℓ = 1 modes of the B2III β Cep star ν Eri (HD 29248). These results are compatible with the assumption oflocal angular-momentum conservation. Both V836 Cen and ν Eri are – for uppermain-sequence stars – very slow rotators, with surface rotation velocities of 2 km s − (V836 Cen) and 6 km s − ( ν Eri). This made the seismic derivation of the interiorrotation possible, because the splitting of the multiplets does not interfere with the aerts_rev3.tex; 25/10/2018; 2:40; p.13 . Aerts et al. frequency separation between different multiplets. The rotation profile itself couldnot be tuned further, given that only parts of very few multiplets were available.Classical spectroscopy, even at extremely high resolution, could never have led tothe proof of differential interior rotation, as it can only measure the surface rotation.Moreover, the intrinsic line broadening of such stars is typically of order ≈ km s − ,which is larger than the surface rotation velocity of these two stars, preventing aderivation of the projected equatorial rotation velocity to better than 1 km s − .For the star θ Oph, which is a twin of V836 Cen as far as the detected frequencyspectrum is concerned, rigid interior rotation could not be excluded from comparisonof the frequency spacing in its triplet and its quintuplet (Briquet et al. , 2007). Itsfrequency precision is two orders of magnitude lower that for V836 Cen and one orderof magnitude lower than for ν Eri. In any case, strong differential rotation is excludedfor that star as well.
5. Expected Future Improvements p modes were discovered a decade ago(Kilkenny et al. , 1997), those with g modes were discovered more recently (Green et al. , 2003). The existence of sdBVs was predicted independently and simultane-ously with their observational discovery (Charpinet et al. , 1996). An opacity bumpassociated primarily with iron-group elements turns out to be turns out to be anefficient driving mechanism. The atomic diffusion processes that are at work in sdBstars – radiative levitation and gravitational settling – cause iron (and also zinc)to become overabundant in the driving zone, thus exciting low-order p and g modes(Charpinet et al. , 1997; Jeffery and Saio, 2006). The details of the diffusion processesare, however, still uncertain. These may also be relevant for the SPBs and β Cep stars(Bourge et al. , 2006), for which diffusion processes have been ignored so far in theseismology. Such processes are dominant in the atmospheres of the roAp stars, as isdiscussed by Kurtz (2008).From an evolutionary point of view, the sdB stars are poorly understood. Theireffective temperatures are in the range 23 000 – 32 000 K, and their log g in the range5 – 6. They all have masses below 0.5 M ⊙ which implies that they have lost almosttheir entire hydrogen envelope at the tip of the red-giant branch. Their thin hy-drogen layer does not contain enough mass to burn hydrogen, making them evolveimmediately from the giant branch towards the extreme horizontal branch. While itis clear they will end their lives as low-mass white dwarfs, it is yet unclear how theyexpelled their envelopes. All scenarios that have been proposed involve close binaryinteraction (Han et al. , 2003; Hu et al. , 2007).The currently known sdB pulsators have multiple periods in the range 80 – 600seconds and amplitudes up to 0.3 mag. Their amplitude variability and faintness haveprevented unambiguous mode identifications so far, limiting the power of seismicinference to tune the diffusive and rotational processes. Rapidly rotating cores havebeen claimed for some of the sdBVs in order to explain their dense frequency spectra aerts_rev3.tex; 25/10/2018; 2:40; p.14 he Current Status of Asteroseismology in terms of low-degree modes (Kawaler and Hostler, 2005). Firm observational proofof that is not yet available, but, given the impressive efforts undertaken to understandthe internal and atmospheric structure of these stars as well as their evolutionarystatus, we expect rapid progress in the near future. For a recent overview of thestatus of sdB seismology, we refer to Charpinet et al. (2007).Recently, a seven-year study of the sdBV star V391 Peg (Silvotti et al. , 2007) usedthe extreme frequency stability of two independent pulsation frequencies to show thepresence of a ≈ Jupiter planet that had moved from about 1 AU out to 1.7 AUduring the red giant phase of the sdB star precursor, allowing the planet to survive,much as the Earth may survive the Sun’s red giant phase in about 7 Gyr. This novelapplication of asteroseismology highlights the close relation and mutual interests ofhelioseismology, asteroseismology, planet-finding and solar system studies.5.2. Heat-Driven Pulsators along the Main SequenceOvershooting parameters and internal rotation profiles have not yet been determinedfor the other heat-driven pulsators known along the main sequence (besides the β Cepstars), such as the slowly pulsating B stars (SPBs) and the A- and F-type δ Sct and γ Dor stars. The main obstacles to overcome are the limited number of detectedoscillation frequencies of the g modes for the SPB stars and γ Dor stars, and reliablemode identification for those as well as for the δ Sct stars. While the pioneeringspace missions WIRE (Buzasi, 2002; Bruntt and Southworth, 2008) and MOST(Matthews et al. , 2004; Walker, 2008) led to an impressive and unprecedented numberof oscillation modes for several such stars, the time base of the data was limited toa few weeks and unique mode identifications are not available for these mission’starget stars. It is to be expected that the uninterrupted photometry obtained by theCoRoT (five months time base, launched 26 December 2006) and
Kepler (3.5 yearstime base, to be launched in 2009) space missions, along with their ground-basedspectroscopy programmes, will result in the necessary frequency precision and modeidentification. This should imply big steps forward for the seismic modelling of thesetype of stars.Thus, even with several space missions in operation, ground-based efforts to in-crease the number of heat-driven pulsators with (preferably simultaneous) long-termmulticolour photometric and high-resolution spectroscopic data for mode identifica-tion should definitely be intensified. It was this type of extensive data that yieldedsudden and immense progress in the β Cep star seismology discussed in this paperand that also advanced significantly the interpretation of the oscillation spectrum ofthe prototypical δ Sct star FG Vir (Zima et al. , 2006). Only systematic and dedicatedobserving programmes can bring us to the stage of mapping and calibrating theinternal-mixing processes inside stars across the HR diagram.5.3. Solar-Like PulsatorsA new dimension in the progress for stochastically-excited pulsators is expected fromthe combination of asteroseismic and interferometric data, as explained by Cunha et al. (2007). The CoRoT and
Kepler missions will also provide a large improvementin the data for solar-like oscillators. In particular,
Kepler will yield data over severalyears for more than a hundred stars, together with three-month surveys of manymore stars. The very extended observations may reveal possible frequency variations aerts_rev3.tex; 25/10/2018; 2:40; p.15 . Aerts et al. associated with stellar magnetic cycles, as has been observed in the Sun, and henceimprove our understanding of such cycles. Also, the identification and interpretationof mixed modes, which have both a p -mode and a g -mode character due to a highlycondensed stellar core, would be a great help to tune stellar evolution models towardsthe end of, and after, the central hydrogen-burning phase.Data of even higher quality on solar-like oscillators can be obtained with dedicatedm s − -precision radial-velocity campaigns, since the intrinsic stellar noise backgroundis much lower, relative to the oscillations, in velocity than in photometry (e.g. Harvey1988). This is the goal of the SIAMOIS (Mosser et al. , 2007) and SONG (Grundahl et al. , 2007) projects. SIAMOIS will operate from the South Pole, while SONG aimsat establishing a global network of moderate-sized telescopes. Both are dedicated toobtain high-precision radial-velocity observations. This will increase substantially thenumber of (rotationally split) detected modes, particularly at relatively low frequencywhere the mode lifetime is longer and the potential frequency accuracy is higher.With the high-quality data expected from CoRoT, Kepler , SIAMOIS and SONG,we may hope to carry out inverse analyses (e.g., Basu et al. , 2002, Roxburgh andVorontsov 2002) to infer the detailed properties of stellar cores.These projects, and others further into the future, covering stars across the HRdiagram will provide an extensive observational basis for investigating stellar interi-ors. Together with the parallel development of stellar modelling techniques we mayfinally approach the point, in the words of Eddington (1926), of being ‘competent tounderstand so simple a thing as a star’.
Acknowledgements
CA is supported by the Research Council of the Catholic Univer-sity of Leuven under grant GOA/2003/04. MC is supported by the EC’s FP6, FCT, andFEDER (POCI2010) and through the project POCI /CTE-AST /57610 /2004. The authorsacknowledge support from the FP6 Coordination Action HELAS.
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