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THEORETICAL ASPECTS OF ASTEROSEISMOLOGY: SMALL STEPS TOWA RDS A GOLDEN FUTURE

Maria Pia Di Mauro

INAF- Osservatorio Astrofisico di Catania, Via S. Sofia 78, 95123 Catania, Italy

ABSTRACT

The current status of asteroseismic studies is here re-viewed and the adequate techniques of analysis availabletoday for the study of the oscillation frequencies are pre-sented. Comments on prospects for future investigationsthrough the possibility of getting ever more precise aster-oseismic observations from ground and space are given.

Key words: asteroseismology; stellar oscillations; solar-type stars.

1. INTRODUCTION

The present conference took place at a period whichmarks a significant milestone in the development of stel-lar physics: asteroseismology, which aims to infer thestructural properties of stars which display multi-modepulsations, has entered in a new golden age. In fact,ground-based observations have reached such level of ac-curacy to allow investigation of several type of oscillat-ing stars; the satellite MOST (Walker et al. 2003), thefirst space mission totally dedicated to the observationof stellar oscillations, has been successfully launched inJune 2003 and has just started to reveal primary resultson the analysis of pulsations seen outside earth’s atmo-sphere. In order to make full use of these observationalsuccesses, several theoretical techniques have been devel-oped or adopted from helioseismology to probe the inter-nal characteristics of stars other than the Sun.

Here I will review on the theoretical aspects of astero-seismology, considering methods and tools available to-day to manipulate observed frequencies of oscillation inorder to investigate the evolutionary and structural prop-erties of the stars. Example of application to real dataset will be given for each asteroseismic diagnostic toolconsidered. Some recent results will be shown and com-ments will be given on perspectives for the future. Forbasic concepts on the theory of stellar oscillations I referto classical books (e.g. Cox 1980; Unno et al. 1989); fortheoretical methods and developments in asteroseismol-ogy see reviews by, e.g., Christensen-Dalsgaard (2003,

2004), Christensen-Dalsgaard and Dziembowski (2000),Dziembowski (2003b), Gough (2003), Paterno, Di Mauroand Ventura (2003), Roxburgh (2002, 2004).

2. FROM HELIO- TO ASTEROSEISMOLOGY

During the last decades, numerous observational and the-oretical efforts in the study of the acoustic modes of solaroscillations, has brought to a detailed knowledge of theinterior of the Sun.

The experience acquired in helioseismology on handlingpulsations frequency data provides a good starting pointfor asteroseismic investigations. However, inferencesof the interior of stars other than the Sun appear to bemuch more complicated and less outstanding in terms ofachievable results. The large stellar distances, the point-source character of the stars, the low amplitude of theoscillations and the effect of the earth’s atmosphere onthe signal, restrict the asteroseismic studies to the use ofsmall sets of data often characterized by modes with onlylow harmonic degrees (l � 4). However, the main differ-ence between helio- and asteroseismology is that globalparameters of the Sun, are much better known than theyare for any other star. Luminosity, effective temperature,surface composition andv sin i are obtained, within a cer-tain error, from spectrum analysis of distant stars; age andcomposition are estimated approximately only for stars inclusters; masses and radii are measured only for spectro-scopic binaries. In addition, all these measurements canbe affected by unknown effects such as loss, accretion,or diffusion of mass. Thus, the structure of the model ofthe stars cannot be so well constrained such as that of theSun.

All these problems make asteroseismic inferences quitedifficult, so that the successes reached by helioseismicstudies are beyond the possibility of asteroseismology.Nevertheless, several attempts have been made, duringthe last years, with the aim to identify oscillations in dis-tant stars and also to model the stellar pulsational phe-nomena. In fact, oscillations have several advantagesover all the other observables: pulsational instability hasbeen detected in stars in all the evolutionary stages and of

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different spectral type; frequencies of oscillations can bemeasured with high accuracy and depend in very simplyway on the equilibrium structure of the model (only ontwo independent variables, e.g. the local adiabatic speedof sound and density); different modes are confined andprobe different layers of the interior of the stars. Thus,accurate observations of the acoustic frequencies, can beused not only to study the physics of the stellar interior,but also to constrain theories of stellar evolution.

3. PULSATING STARS

The historical pulsating stars, like the classical Cepheids,W-Virginis and RR-Lyrae, characterized by a large lumi-nosity variations, show only one or two pulsation modes,limiting the knowledge of the structural properties to themean density or in better cases to the mass and radius.Asteroseismology is understood as the study of pulsa-tions on stars in which many oscillation modes are ex-cited at once. This definition is broad enough to includemany types of pulsating stars. Multi-mode oscillationshave been detected in stars which are spread over a sig-nificant part of the HR-diagram, reflecting different evo-lutionary stages from main-sequence to the white dwarfcooling sequence. Luminosity and effective temperatureof the major classes of pulsating stars are shown in theHR-diagram of Fig. 1.

We can divide the stars on which it is possible to applyseismic techniques into two groups. A group of starsshow oscillations with intensity amplitude in the rangeof millimagnitudes and includes white dwarf,Æ Scutistars, the rapidly rotating Ap stars, the�-Cephei stars,the slowly pulsating B (SPB) stars and the Doradusstars. Stars of these classes show modes excited by�-mechanism due to a perturbation of the opacity, causedby structural conditions typical of each considered class.The second group includes the stars with tiny oscillationsamplitude (around10�magnitudes or less) and includesthe so-called solar-like stars, in which, like the Sun, os-cillations are excited by turbulent convection in the outerconvective envelope.

For the purposes of this review, however, I will concen-trate manly on solar-type stars and I will give just a sum-mary on some theoretical results obtained by asteroseis-mology on some other pulsating targets at the end ofthis review. The principal reason for doing so is due tothe excitement of the community for the recent observa-tional successes obtained on some solar-type stars (e.g.,�-Can Min and�-Cen) which have given the possibil-ity, for the first time, of handling large sets of accurateoscillation frequencies to investigate the internal struc-ture of the stars. Moreover, it should be pointed out, thatmethods and techniques developed and adapted for solar-type stars, are also valid for the other pulsating stars. Adetailed and updated description of the characteristics ofthe main asteroseismic targets can be found in recent re-views (e.g. Dziembowski 2003b; Kurtz 2004; Paterno,Di Mauro and Ventura 2003).

Figure 1. HR diagram showing several classes of pulsat-ing stars. Courtesy of J. Christensen-Dalsgaard.

3.1. Properties of pulsations

Many of the oscillation modes which can be detected instars are limited to only the lowest degree ones, owing tothe point-like character of these sources. If the conditionthat l � n is satisfied, the excited p-modes can be de-scribed in terms of the asymptotic theory (Tassoul 1980),which predicts that oscillation frequencies�n;l of acous-tic modes, characterized by radial ordern, at harmonicdegreel should satisfy the following approximation:�n;l = �� �n+ l2 + �+ 14�+ �n;l ; (1)

where� is a function of the frequency determined by theproperties of the surface layers,�n;l is a small correctionwhich depends on the conditions in the stellar core.��is the inverse of the sound travel time across the stellardiameter: �� = 2 Z R0 dr !�1; (2)

where is the local speed of sound at radiusr andR isthe photospheric stellar radius. The property expressedby Eq. (1) may provide almost immediate asteroseismicinferences on stellar parameters and constraints on theo-retical models for a variety of solar-like stars in a widerange of evolutionary stages. For a givenl, the acousticspectra show a series of equally spaced peaks between pmodes of same degree and adjacentn, whose frequencyseparation represents the so called large separation whichis approximately equivalent to��:�� ' �n+1;l � �n;l � ��l : (3)

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The spectra are characterized by another series of peaks,whose narrow separation is�l, known as small separa-tion: �l � �n;l � �n�1;l+2 = (4l+ 6)D0 (4)

where D0 = � ��4�2�n;l " (R)R � Z R0 d dr drr # : (5)

Instead of the small separations one could consider theso-called ’second differences’ of frequencies:Æ2�l � �n+1;l � 2�n;l + �n�1;l : (6)��, and hence the general spectrum of acoustic modes,scales approximately as the square root of the mean den-sity, that is, for fixed mass, asR�3=2. On the other hand,the small frequency separation and the second difference,are both sensitive to the chemical composition gradientin central regions of the star and hence to its evolutionarystate. Thus, the determination of both large and small fre-quency separation,��l andÆ�l, provides measures of themass and the age of the star (e.g. Christensen-Dalsgaard1988). See review by Gough (2003) on the potential offrequencies separations as asteroseismic diagnostic.

Analogous properties have been derived for the g-modes.Tassoul’s theory shows that in the asymptotic regime theg-modes are nearly uniformly spaced in period. g-modeswill be discussed in more details in Sec. 9, since the onlytypes of stars in which these are commonly observed arerather special cases.

3.2. Sub-surface effects and a new seismic indicator

Seismic analysis of observed acoustic frequencies is anextremely powerful tool for the investigation of the inter-nal structure of the stars, but the use of large and smallseparations can be misleading if not accurately consid-ered. Theoretical pulsation frequencies, essential for as-teroseismic investigation, are calculated on theoreticalmodels of stars which are inevitably affected by errors.In particular, the structure of the near-surface regions ofstars is quite uncertain: there are still substantial ambi-guity in modelling the convective flux, considering thesources and the mechanisms of excitation and damping ofthe oscillations, defining an appropriate equation of stateto describe the thermodynamic properties of the stellarstructure, as well as in the treatment of non-adiabatic ef-fects. Limiting the investigation to the use of small sep-arations do not solve the problem. In fact, the small sep-aration which is determined principally by condition inthe core, also retains some sensitivity to the mean densityand to the detailed properties of the stellar envelope.

To solve these problems the use of a new seismic indica-tor was introduced by Roxburgh and Vorontsov (2003a),which compared pulsating properties of several modelswith exactly the same interior structure, but different

Figure 2. HR diagram showing location in luminos-ity and effective temperature of the solar-type pulsatingstars. Asterisks indicate targets on which solar-type os-cillations have been already detected. The Sun is shownas a full circle. The solid lines are evolutionary trackscalculated for increasing values of mass. Courtesy of G.Houdek.

outer envelopes, obtained by keeping constant the firstadiabatic exponent all along the radius or by introducinglinear variation of the polytropic index in the structureof the envelope. They showed that the ratios of small tolarge separations, rl = �l=��l ; (7)

are independent of the structure of the outer layers andhence can be used as diagnostic of the interior of stars.

4. SOLAR-TYPE STARS

Solar-type stars are F, G and possibly K main-sequenceand sub-giants stars in which oscillations are excitedstochastically by vigorous near-surface convection in abroad spectrum of low amplitude p-modes, as in the Sun.There is also good evidence, both from photometry withthe WIRE satellite (Buzasi et al. 2000) and ground-basedvelocity studies, for solar-like oscillations in G, K andsemi-regular M giants. The oscillation periods are ex-pected to be in the range from the typical 5-min (main-sequence stars), as in the Sun, up to about a few days(sub-giant and giant stars). Fig. 2 shows location in HR-diagram of the solar-type pulsating stars and of some se-lected targets.

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4.1. Amplitudes and life-time of solar-like oscilla-tions

One of the greatest deficiencies in modelling oscilla-tions in stars with surface convection zones is the lackof a proper theory to describe convection and its interac-tion with pulsations. Several attempts have been madein recent years to address this problem. Christensen-Dalsgaard and Frandsen (1983) made the first effort togive a rough prediction of amplitudes of solar-like oscil-lations. They found velocity and luminosity amplitudesincreasing with age and with mass along the main se-quence. Simple scaling laws to estimate the amplitudes invelocity have been proposed by, e.g., Kjeldsen and Bed-ding (1995, 2001).

Impressive progress has been made on hydrodynamicalsimulations of convection, and its interaction with pulsa-tions, leading Houdek et al. (1999) to give estimates ofthe pulsation properties, including the damping rates, ofmain-sequence stars by using models based on mixing-length treatment of convection. According to theoreti-cal estimations (e.g. Houdek et al. 1999, Kjeldsen andBedding 1995) the expected luminosity variations are ofabout a few mmag in giant stars while in main-sequenceand sub-giant stars are so small (from10�3 up to a few10�2mmag) to be well below the detection limit forground-based photometric observations. The radial ve-locity amplitudes are expected to be at most50�60m s�1in K giants,1 � 2m s�1 in F and G sub-giants and evensmaller in main-sequence solar-type stars. These ampli-tudes are at the edge of the photon noise limits of mostof the available spectrographs. The theoretical predic-tion have been tested and compared with observationalresults, revealing a fairly good agreement for the severalsolar-like stars already detected. However, calculation ofdamping rates are still a problem. Theoretical predictionhave been able to reproduce observed lifetimes of oscil-lations only in the case of the Sun. Observations of starswhich are somewhat more evolved than the Sun, like Pro-cyon (Barban et al. 1999),�Cen A (Bedding et al. 2004),� Hyi (Bedding et al. 2002) and� Hya (Stello et al. 2004)have shown mode lifetimes shorter than the values pre-dicted by the theoretical models (Houdek et al. 1999).This inconsistency with observations seems to indicatethat there is still some contributions to damping so farignored in the theory.

5. STELLAR MODELLING

Reliable calculation of stellar models is the basic pre-requisite for all asteroseismic investigations of stellarin-ternal properties. The structure models are produced byemploying up-to-date physical information and the mostupdated input parameters (composition, mass, mixing-length) in order to reproduce all the spectroscopic andphotometric observations of the considered stars. Evolu-tionary sequences are calculated by including additionaleffects such as overshoot from the convective core (e.g.

Di Mauro et al. 2003) during the main-sequence phase,diffusion and settling of helium and heavy elements (e.g.Vauclaire 2003).

In the case of solar-type stars, detailed classical modelshave been produced on several well observed targets:�Cen (e.g. Guenther and Demarque 2000; Morel et al.2000, Thevenin et al. 2002; Thoul et al. 2003; Eggen-berger et al. 2004b);� Boo (Christensen-Dalsgaard, Bed-ding and Kjeldsen 1995; Guenther and Demarque 1996;Di Mauro et al. 2003, 2004; Kervella et al. 2003;Guenther 2004), Procyon A (e.g. Barban et al. 1999;Chaboyer, Demarque and Guenther 1999; Di Mauro,Christensen-Dalsgaard and Weiss 2000, Kervella et al.2004, Provost, Martic and Berthomieu these proceed-ings),� Hydri (Di Mauro, Christensen-Dalsgaard and Pa-terno 2003; Fernandes and Monteiro 2003; Dravins, Lin-degren and Vandenberg 1998);� Hydrae (Teixeira et al.2003; Christensen-Dalsgaard 2004).

Unfortunately, modelling of distant stars involves phys-ical effects which substantially complicates the calcula-tion. One of the most crucial problem in computing stel-lar structure is related to the modelling of the convec-tive transport of energy in the interior and the associatedphysical effects such as the evolution of the compositionin presence of convective regions (e.g. Dupret, these pro-ceedings ). See the reviews by Demarque and Robinson(2003) and D’Antona, Montalban and Mazzitelli (2004)about sophisticated models of convection.

Another trouble is represented by the fact that effects ofrotation in stellar modelling are often neglected. The ro-tation may affect oscillation frequencies forming rathercomplex power spectra. In order to identify and repro-duce the observed modes it is necessary to develop an ap-propriate treatment of rotation in the evolutionary modelsand in the calculation of oscillation frequencies. To date,rotational effects have been taken into account mainly us-ing perturbation theory approaches. The Coriolis and thecentrifugal forces, which in the case of moderate and fast-rotators cannot be neglected, have been taken into properaccount as first-, second- and even third order perturba-tions. Many authors (Saio 1981; Gough and Thompson1990; Dziembowski and Goode 1992; Soufi, Goupil andDziembowski 1998; Karami, Christensen-Dalsgaard andPijpers 2003) have approached the problem under var-ious approximations, also including radial and latitudi-nal dependence of rotation and magnetic fields, whichremove the degeneracy of the modes. However pertur-bation theory might be inadequate for the most rapidlyrotating stars.

Calculation of the evolution and pulsations of two- orthree-dimensional models, represents a solution for allthese problems and turns out to be strongly recom-mended, but at present, very difficult to be attained (e.g.Demarque and Robinson 2003).

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Figure 3. The dashed blue lines show evolution tracks plotted in HR-diagrams for� Cen A and B, respectively on theleft and right panel. The tracks have been calculated with the Liege evolution code by A. Miglio and J. Montalban byassuming OPAL EOS. Input values of masses, age and mixing length parameters (�MLT) are indicated. The rectanglesdefine the one-sigma error box for the observed luminositiesand effective temperatures and the green lines show theone-sigma error box for the observed radius. The symbol starindicates the location of the selected models. Courtesy ofA. Miglio.

Figure 4. Evolution tracks for� Cen A and B plot-ted in a HR-diagram calculated with the evolution codeof Christensen-Dalsgaard, withMA = 1:1095M� andMB = 0:9301M�, the OPAL EOS and an age of7:382Gyr. The rectangles define the one-sigma error boxesaround the observed luminosities and effective temper-atures (asterisks). The triangles indicate the locationof the selected models. Courtesy of J. Christensen-Dalsgaard.

5.1. Model fitting method and application to the�Cen system

One of the most recent strategy introduced in order tofind the models which better reproduce the observationsis represented by an optimization procedure known as’model fitting’ method. Given a set of input parameters,such as mass, mixing-length parameter and compositionof a star, the method consists in searching, among all thepossible models, the one which best fits all the known ob-servables. This can be obtained by performing a�2 min-

Table 1. Mass, luminosity, effective temperature, radiusand logarithmic value of the iron abundance relative tothe solar one for the two solar-type components of the�Cen system, adopted from Eggenberger et al. (2004b).� Cen A � Cen BM=M� 1:105� 0:007 0:934� 0:006L=L� 1:522� 0:030 0:503� 0:020Te�(K) 5810� 50 5260� 50R=R� 1:224� 0:003 0:863� 0:005

[Fe/H] 0:22� 0:05 0:24� 0:05imization, looking for the parameters which minimizes:�2 = NXi �Omodi �Oobsi�i �

(8)

whereOobsi are all the observables - luminosity, effectivetemperature, radius, oscillation frequencies, large andsmall separations, ratio of small to large separation - andOmodi are the theoretical values calculated on the models.The observed errors of the parameters are�i. Extensiveexplanation of the method can be found in Guenther andBrown (2004) and Roxburgh (2004). Although in princi-ple the method might represent an useful strategy, it stillpresents some problems: the solution depends on the ini-tial inputs and therefore it does not explore the globalspace of parameters. To solve this problem , Metcalfeet al. (these proceedings) have suggested to use geneticalgorithms, well developed and applied for modelling thewhite dwarfs, also for the case of the solar-type stars .

’Model fitting’ is particularly valuable when applied toa set of stars in a cluster or in a binary system: the in-crease of parameters to be fitted, by assuming same age

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and composition for all the components of the system, en-hance the quality of the minimization technique. An im-mediate application of this optimized procedure has beenperformed, for example, in the case of the� Cen sys-tem by several authors (e.g. Eggenberger et al. 2004b;Guenther and Brown 2004). Here, I present the resultsnot yet published obtained by A. Miglio and J. Montal-ban which implemented the Code Liegeois d’EvolutionStellaire, and by J. Christensen-Dalsgaard, whose codewas implemented by T. Teixeira. Many spectroscopicmeasurements of both components of the� Cen systemcan be found in literature; for a summary see Morel etal. (2000) and Eggenberger et al. (2004b). Taking intoaccount the numerous observational constraints listed inTable 1, including new seismological observations for� Cen A (Bedding et al. 2004) and� Cen B (Carrierand Bourban 2003) a common solution was obtained bythe two groups, and respectively shown in Figs. 3 and4. The results indicate that a best fit with all the avail-able observations can be obtained by assuming a massof aboutMA = 1:11M� for � Cen A and of aboutMB = 0:93M� for � Cen B, identifying both compo-nents as being in the main-sequence phase. The compo-sition was assumed such that the initial hydrogen abun-dance wasX = 0:7 and the metallicityZ = 0:03.

6. THE POTENTIAL OF THE MIXED MODES

The properties of the solar-like oscillations are expectedto change as the stellar structure evolves as a consequenceof the hydrogen exhaustion in the core. According toEq. (1), oscillation frequencies of a given harmonic de-gree should decrease as the star evolves and the radius in-creases and should be almost uniformly spaced by�� ateach stage of evolution. However, while the radial modesseem to follow closely Eq. (1), the frequencies of somenon-radial modes appear to increase suddenly at certainstages of evolution (Christensen-Dalsgaard, Bedding andKjeldsen 1995; Guenther and Demarque 1996). This isdue of the occurrence to the so called ’avoided crossing’.The core contraction, as the star evolves and the radiusexpands, causes an increase of the local gravitational ac-celeration and of the gradients in hydrogen abundance,and hence of the buoyancy frequency in the deep interiorof the star. As a consequence g modes with high fre-quencies are allowed to propagate and can interact with ap mode of similar frequency and same harmonic degree,giving rise to a mode with mixed character, which be-haves as a g mode in the interior and as a p mode in theupper layers. The interaction can be explained as the cou-pling of two oscillators of similar frequencies. The effectof the coupling becomes much weaker for modes withhigher harmonic degree, since in these cases the gravitywaves are better trapped in the stellar interior and bet-ter separated from the region of propagation of acousticwaves.

6.1. Mixed modes in� Bootis: main-sequence orsubgiant star?

As an example of the use of mixed modes as asteroseis-mic diagnostic in the stars, I consider the case of� Boo-tis. In Di Mauro et al. (2004) authors built a grid of stellarevolutionary sequences tuned to match the position of�Boo in the theoretical HR diagram, considering the in-clusion of convective overshooting, and testing the useof different equations of state and of different treatmentsof convective transport. They concluded that present ob-servations are consistent with two possible evolutionaryscenarios for� Boo: (i) a subgiant star in post-main-sequence phase, whose oscillation spectrum contains fre-quencies of nonradial modes with mixed character dueto avoided crossings; (ii) a less evolved star in the main-sequence phase which show p modes with no mixed char-acter and frequencies which follow the asymptotic theory.Fig. 5 shows the echelle diagrams of the observed fre-quencies in comparison with the theoretical frequenciescomputed for Model 1 and 2 of Table 2, respectively inpost-main-sequence and main-sequence phases. At thisstage it can be concluded that the accuracy of the exist-ing frequency observations does not allow a clear distinc-tion between the two proposed scenarios, although it isevident, but hardly decisive, that the post-main-sequencephase is a rapid phase of the evolution, which might beless likely to be observed. For the future there is hope thatmore accurate observations will help to detect presence orabsence of mixed modes and then to identify uniquely thepresent evolutionary state and clarify the properties of theinterior of this star.

6.2. Mixed modes in the red giant� Hya

The analysis of the properties of solar-type oscillationshas so far confined to stars which lie in the main se-quence or subgiant phase. However there have been re-ports of detections of solar-like oscillations also in giantstars, and in particular in few red giants (e.g. Buzasi etal. 2000; Merline 1999, Retter et al. 2003; Frandsen etal. 2002; Stello et al. 2004; Retter et al. 2004; Barban etal. these proceedings). Although there are difficulties inthe identification of the observed modes and severe prob-lems of theoretical interpretation, Guenther et al. (2000)and Dziembowski et al. (2001) for� Uma and Teixeiraet al. (2003) and Christensen-Dalsgaard (2004) for� Hyasucceed to draw important conclusions on the structuraland pulsational properties of red giant stars. They foundthat observations of these stars are consistent with mod-els in the hydrogen shell-burning phase, or more proba-bly, in the longer lasting successive phase of core heliumburning. These stars are characterized by a deep con-vective envelope and a small convective core. Since thedensity in the core is quite large, the buoyancy frequencycan reach very large values in the central part, with somemaxima rising from the steep changes in the molecularweight. In these conditions, g modes of high frequen-cies can propagate and might eventually interact with pmodes giving rise to modes of mixed character. Fig. 6

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Table 2. Relevant parameters, the massM , theAge, the luminosityL, the effective temperatureTe� , the surface radiusR,the overshooting parameter�ov and the locationr b of the base of the convective envelope in units ofR, for two modelsof � Boo, computed with the OPAL EOS andZ = 0:04. Model 1 and 2 represent evolutionary structure respectively inpost-main and main-sequence phases.

Model M=M� Age (Gy) L=L� Te� (K) R=R� �ov r b=R1 1:71 2:44 9:07 6072 2:73 0:1 0:852 1:82 1:96 8:93 6009 2:76 0:2 0:84

Figure 5. Echelle diagrams for Model 1 (left panel) and Model2 (right panel) of Table 2, which are respectively inthe post-main-sequence and main-sequence phases. The opensymbols represent the computed frequencies. The filledsymbols with error bars show observed frequencies and errors (Kjeldsen et al., 2003). Circles are used for modes withl = 0, triangles forl = 1, squares forl = 2, and diamonds forl = 3. Theoretical and observed frequencies are plottedwith �� = 40:47�Hz. The size of the open symbols indicates the relative surfaceamplitude of oscillation of the modes.Crosses are employed for modes with small predicted amplitude (e.g. g modes). For details see Fig. 4 of Di Mauro et al.(2004).

shows the eigenfunctions for a pure radial mode and fora mode with l=2. The eigenfunction of a mixed modeis characterized by a large p-mode like displacement am-plitude in the upper layers and also g mode behaviour inthe core. Since the radiative damping scales as the thirdderivative of the displacement, it appears clear that non-radial modes are more easily damped than radial ones. Asa consequence, the spectrum of the red-giant stars is quitecomplex, with a sequence of peaks uniformly spaced, dueto radial modes, which have high amplitude and hencehave more probability to be detected, and other series ofpeaks with low amplitude, due to the mixed modes.

7. ASTEROSEISMIC INVERSION TO INFERTHE INTERNAL STRUCTURE OF STARS

The asteroseismic inversion is a powerful tool which al-lows to estimate the physical properties of the stars, bysolving integral equations expressed in terms of the ex-perimental data. Inversion techniques are well known andapplied with success to several branches of the physics,from geophysics to the radiation theory. Applications tothe helioseismic data have been studied extensively andinversion methods and techniques have been reviewed

and mutually compared by several authors, leading to ex-traordinary results about the global properties of the Sun(see review by, e.g., Di Mauro 2003).

The experience acquired in helioseismology on invert-ing mode frequency data provides a good starting pointfor asteroseismic inversion. The several techniques de-veloped for asteroseismic inversions support ’linear’ or’non-linear’ approaches. The linear inversion is basedon the linearization of the equations of stellar oscilla-tions about a known reference model, under the assump-tion that perturbations are small. This results in integralequations which can be used to determine the correctionswhich have to be imposed on the reference model in orderto obtain the observed oscillation frequencies. See the re-view of Basu (2003) about the applications and problemsof linear inversions. The non-linear inversion based onthe analytical resolution of the integral equations have re-cently been discussed and compared with the linear inver-sions by Roxburgh and Vorontsov (2003b). The choice ofone or the other approach is not an easy task. The linearinversion, although is a convenient approach to study theinnermost part of the stars, represents a questionable ap-proximation. The global parameters of the stars, such asmass, radius, luminosity and chemical composition arenot well known, so that the structure of the model can-

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Figure 6. Radial variation of eigenfunctions in a model of� Hya. The dashed line represents a mode with l=0 andfrequency� = 72:1�Hz, while the solid line representsa mode with l=2 and frequency� = 71:3�Hz. Courtesyof J. Christensen-Dalsgaard.

not be well constrained and hence might be very differentfrom the observed star. In these conditions linearizationcannot be valid. On the other hand the statistical prop-erties, such as the propagation of errors, are much moredefined in the linearized approach than in the non-linearinversions.

Earlier attempts in generalizing the standard helioseismicdifferential methods to find the difference of structure be-tween the observed star and a model have been appliedon artificial data with encouraging results by Gough andKosovichev (1993) and Roxburgh et al. (1998). Morerecently, Berthomieu et al. (2001) carried out a care-ful analysis of the results to be expected in inversion forstars, showing that the kernels and hence the solutionscan be well concentrated only in the inner core. Pa-terno, Di Mauro and Ventura (2003) demonstrated thatif a more realistic set of low degree modes is used, suchas the accurate observations for the Sun, the inversion al-lows the inference of the inner core of the stars below0:3R�. In addition, Basu, Christensen-Dalsgaard andThompson (2002) demonstrated that the success of theinversion results depends strongly on the choice of vari-ables to be inverted, and in the case of solar-type stars apreferable choice seems to be represented by the pairu; Y(Basu 2003), whereu is the squared isothermal soundspeed andY is the helium abundance in the convectivezone. Further aspects can be also found in Thompson

Table 3. Mass, luminosity, effective temperature, radiusand logarithmic value of the iron abundance relative tothe solar one for Procyon A and relative references.

Parameters ReferencesM=M� 1:497� 0:037 Girard et al. (2000)M=M� 1:42� 0:06 Allende Prieto et al. (2002)L=L� 7:0� 0:2 Pijpers (2003)Te�(K) 6500� 80 Pijpers (2003)R=R� 2:048� 0:025 Kervella et al. (2004)[Fe/H] �0:015� 0:029 Taylor (2003)

and Christensen-Dalsgaard (2002).

7.1. Inversion for Procyon A

The Procyon binary system (� Cmi) consists of a F5 sub-giant primary and a white-dwarf secondary. Procyon A,the primary, has already attracted the attention of stel-lar seismologists for its proximity and brightness. Re-cently, Martic et al. (2004), by observing this star withthe CORALIE spectrograph, have been able to identify55 p-mode frequencies in the range250�1400�Hz, withharmonic degreesl = 0 � 2, characterized by an aver-age large frequency separation�� = 53:6 � 0:5�Hzand an average small frequency separation� = 5:1 �0:5�Hz. These results appear to be in good agreementwith those obtained by Eggenberger et al. (2004a) who,from Doppler-velocity observations, have been able toidentify 23 p modes in the same range of observed fre-quencies and with an average large separation of�� =55:5�Hz. The observables listed in Table 3 have been

Figure 7. Evolution tracks of Procyon A plotted in a HRdiagram for several values of masses and metallicity cho-sen within the observed intervals. The rectangle definesthe one-sigma error box for the observed luminosity andeffective temperature.

9

Table 4. Basic parameters,M , Age, Z, L, Te� , the surface radiusR, small and large separations for two models ofProcyon A, computed with the OPAL EOS (1996). Model 3 and 4 represent evolutionary structure respectively in main-sequence and in late main-sequence phases.

Model M=M� Age (Gy) Z L=L� Te� (K) R=R� �0 (�Hz) ��0 (�Hz)3 1:47 1:78 0:016 6:88 6501 2:07 4:2 53:64 1:42 2:51 0:020 6:72 6481 2:05 4:2 53:6

Figure 8. Echelle diagrams for Model 3 (left panel) and Model4 (right panel) which are respectively in the main-sequenceand the late-main-sequence phases. Open symbols representthe computed frequencies. The filled symbols show observedfrequencies by Martic et al. (2004); blue filled symbols correspond to old identified frequencies given between bracketsin Martic et al. (2004). Theoretical and observed frequencies are plotted with�� = 53:6�Hz and a reference frequency�0 = 635�Hz. Circles are used for modes withl = 0, triangles forl = 1, squares forl = 2, and diamonds forl = 3.The size of the open symbols indicates the relative surface amplitude of oscillation of the modes. Crosses are employedfor modes with small predicted amplitude (e.g. g modes).

assumed to constrain a grid of stellar models of Pro-cyon A based on a recent physics, including the OPALopacities (Iglesias and Rogers 1996), the Bahcall, Pinson-neault and Wasserburg (1995) nuclear cross sections andthe OPAL equation of state (Rogers, Swenson and Igle-sias 1996). Stellar models shown here have been calcu-lated by using evolution code developed by Christensen-Dalsgaard et al. (1982), taking into account diffusion ofhelium and heavy elements from the convection zone intothe adjacent radiative layers, but suppressing diffusion inthe outer layers of the model to be sure that the heliumand heavy elements were not totally drained out of theenvelope. Models shown here have been obtained forseveral heavy-element abundances, and assuming mix-ing length parameter calibrated on a solar model that usesthe same opacities, and an initial hydrogen mass fractionX = 0:7As in the case of� Boo, it appears difficult to constrainthe evolution state of this star by considering only modelresults. The location of the star in the HR diagram identi-fies Procyon A as being in the late-main-sequence phaseof core hydrogen burning if the mass is taken in the rangeM = (1:42� 1:46)M�. However, it is also possible thatProcyon A is still in the core hydrogen-burning main-

sequence phase, if models are computed by assuminga mass higher by about5%, as shown by the evolutiontracks plotted in Fig. 7.The problem of identifying the evolutionary state of thisstar can be approached by studying the pulsational char-acteristics of the computed models. The relevant param-eters are given in Table 4 for two models of ProcyonA selected for the pulsation analysis, which have sur-face radius, large and small theoretical separations con-sistent with the observed values. The pulsational fea-tures are clearly illustrated by looking at the echelle di-agrams for the two different models (Fig. 8). It is ev-ident that both the two considered models show a verygood agreement with the observed frequencies. The onlydifference which can be noticed between the two mod-els is that for the model in the late-main-sequence phase(Model 4), the nonradial modes withl = 1 show consid-erably more scatter than the radial ones. This aspect isassociated with the avoided crossings which introduce aless regular structure in the frequency spectrum. On thecontrary, the modes of the models in the main-sequencephase (Model 3) show no occurrence of avoided cross-ing. Fig. 8 shows that the observed frequencies forl = 1do not seem to indicate the presence of some irregulari-ties similar to those seen for the computed frequencies of

10

Figure 9. The relative isothermal squared sound-speed differences between Procyon A and the two stellar models ofTable 4 as obtained by inversion of Martic et al. (2004) data. Panel on the left shows results for Model3, panel on theright shows the comparison between the inversion results obtained for Model 3 (red dots) and those obtained for Model 4(green dots).

the more evolved model. If this were to be confirmed bymore accurate sets of observed data, Procyon A would bea star definitely identified as in the main-sequence phase.

On considering the large amount of data available, in or-der to obtain additional information on the internal char-acteristics of Procyon A, it has been thought to applyinversion techniques to the observed frequencies of thisstar (Martic et al. 2004). The results shown here havebeen obtained by applying a linear inversion procedure.If it is assumed that the equation of state of the staris known, the relative differencesÆ!i=!i between thefrequencies of Procyon A and the model are related tothe differences (Æu=u; ÆY ) in squared isothermal soundspeedu = p=� and helium abundanceY in the convec-tive zone, between the structure of the star and the refer-ence model, through the following integral equation (e.g.,Dziembowski, Pamyatnykh and Sienkiewicz 1990):Æ!i!i = Z R0 Kiu;Y Æuu dr + Z R0 KiY;u ÆY dr + "i ; (9)

whereKiu;Y andKiY;u are the kernels calculated for eachmodei = n; l of the set considered. Equation (9) hasbeen solved by applying the so called SOLA method (Pi-jpers and Thompson 1992, 1994) well used in helioseis-mology. Fig. 9 shows the relative differences inu be-tween Procyon A and the two selected models (Models3 and 4 of Table 4) as functions of the fractional radius.The results indicate that the difference between Model 3and Procyon A are extremely small, below the5%, whileModel 4 shows very large deviations indicating that liner-izations cannot be applied to this model, whose propertiesare extremely different from those of the observed star.

From both the inversion results and analysis of the echellediagrams, it can be concluded that present observations ofoscillation frequencies of Procyon A seem to indicate thatthis star is in main-sequence phase. It is worth to mentionthat Provost et al. (these proceedings) arrived exactly to

the opposite conclusion by considerations on the recentmeasurements of radius by Kervella et al. (2004). It ishoped that in the future more accurate observations willallow a detailed test of stellar modelling and evolutiontheories of Procyon A, in order to identify its present evo-lutionary phase.

8. SHARP FEATURES

Another important property of the oscillation spectra isthat sharp variations localized at certain acoustic depthin the structure of pulsating stars produce a distinctivequasi-periodic signal in the frequencies of oscillation.The characteristics of such signal are related to the lo-cation and thermodynamic properties of the layer wherethe sharp variation occurs. Sources of sharp variations arethe borders of convection zones and regions of rapid vari-ation in the first adiabatic exponent�1, such as the onethat occurs in the region of second ionization of helium.Fig. 10 shows the variation of�1 in the upper layers for aZAMS model of a star of1:2M� and its relative periodicsignal as seen in the large separation.

A general expression (valid for low degree modes) forthe signal generated is of the form (e.g. Monteiro,Christensen-Dalsgaard and Thompson 1994, 1998, 2000;Roxburgh and Vorontsov 1994):Æ!n;l ' A(!) os(2!n;l�d + �0) ; (10)

whereA(!) is an amplitude as function of frequency!,which depends on the properties of the sharp variation;�0 is a constant phase;�d is the acoustic depth of thefeature. Several attempts have been tried in order to iso-late the generated oscillatory components directly fromthe frequencies of oscillations or from linear combinationof them (large separations, second differences, etc). Thecommon approach consists in removing a smooth com-ponent from the frequencies and to fit the residual signal

11

to a theoretical expression, like that of Eq. (10), which isrelated to the properties of the sharp feature.

This method can be applied, for example, to determinethe properties of the base of the convective envelope(Monteiro, Christensen-Dalsgaard and Thompson 2000;Ballot, Turck-Chieze and Garcıa 2004) and in particu-lar to put limits on the extension of the convective over-shoot (Monteiro, Christensen-Dalsgaard and Thompson2002), or to investigate the border of the convective core(Mazumdar and Antia 2001; Nghiem et al. 2004). Butin particular, this peculiar property of the oscillation fre-quencies can be used to infer the helium abundance in thestellar envelope, by studying the variation of�1 in the re-gion of second ionization of helium. Several authors havetried to develop or refine their own fitting function to in-fer the helium abundance (Lopes et al. 1997; Monteiroand Thompson 1998; Perez Hernandez and Christensen-Dalsgaard 1998; Miglio et al. 2003; Basu et al. 2004 andthese proceedings; Houdek and Gough in these proceed-ings). Unfortunately this technique cannot be applied tostars with mass larger than1:4M�, owing to the con-tamination in the oscillatory signal coming from the firsthelium ionization zone. In fact, one of the main prob-lem in the application of such an approach rises from thefact that signals coming from different sharp features inthe interior of the star might overlap generating a com-plex behaviour, as it was noticed by Mazumdar and Antia(2001). A similar effect was also considered by Mont-gomery, Metcalfe and Winget (2003) in the case of gmodes observed in white dwarfs. Very recently Miglioand Antonello (2004) have shown that in the solar-typestars the problem of distinguishing between a signal gen-erated near the core or near the surface in the star can beeasily solved by considering modes of different degree ordifferent seismic indicators. Unfortunately this methodcannot successfully be applied in the case of the whitedwarfs. At the moment, L. Mantegazza et al. (privatecommunication) is working on a new technique basedon the Principal Component Analysis (Golyandina et al.2001; Ghil et al. 2002) in order to isolate all the differ-ent oscillatory components directly from the oscillationfrequencies.

It was already pointed out in Miglio et al. (2003) that amore general analysis of the effect of the bump in�1 canbe obtained by performing an inversion of the observedfrequencies, by using the integral equation by Gough andThompson (1991). At the moment the method has beentested only in order to find differences between a ref-erence equilibrium model and a fictitious model whichdiffers from the reference one only because�1 has beensmoothed in the region of second helium ionization. Inthis case, the inversion of the artificial data set of modeswith 0 � l � 3 is able to reconstruct the variations in�1for models with masses from low up to2M�, if the in-verted set includes also few modes with low radial order(n = 4 � 8), whose kernels seem to have considerableweight in the upper regions.

9. THEORETICAL RESULTS ON OTHER PUL-SATING STARS

9.1. White dwarfs

The results obtained by applying seismic techniques towhite dwarfs represent probably the biggest success ofasteroseismology (e.g. Vauclair 1997; Kawaler 1998;O’Brien 2003). In fact, they represent the stars, other thanthe Sun, in which the largest number of oscillation fre-quencies has been detected. The white-dwarf stars (WDs)represent the most common end point of stellar evolu-tion. Their physical structure is very simple: a degen-erate carbon-oxygen core and an extremely thin, mostlynon-degenerate, surface layer of pure H or He. Pulsationsin WDs represent a short-lived phase during their evolu-tion: the majority of them do not show pulsations exceptin very narrow regions of instability located along theircooling track in HR-diagram: the planetary nebulae hotnuclei (PNN), the hot DO (H deficient pre-white-dwarfs),the warm DB (WDs with He rich photospheres) and thecool DA (WDs with H rich photospheres) stages. WDvariables are mainly multi-periodic pulsators in whichhigh-order g-modes of low angular degree are excited,but the mechanism triggering the pulsations is still matterof discussion (e.g., Cox 2002).

Asteroseismology has been applied to identify the struc-tural differences between different types of white dwarfs,and there is hope that it will permit in the future also theidentification of the various origins. In fact, while theo-retical models can be easily calculated without major ef-forts, it is still not possible to say which stellar structuremight represent the precursor of each of the several typesof white dwarfs.

Asymptotic theory, which gives a clear prediction of pe-riod spacing for these stars, has helped to determine thetotal mass, the rotation rate, the magnetic field strengthand even the mass of the outer layers of some selectedtargets. In particular, since the rate of evolution of whitedwarfs depends primarily on the thickness of their sur-face layers, it might be possible to measure the galac-tic age through the luminosity function (e.g. Wood andOswalt, 1998). However, the most interesting character-istics studied by asteroseismology are the structure andcomposition of the C/O cores of white dwarfs and theevidence for crystallization of the cores in the most mas-sive stars (DAV). As shown by Montgomery and Winget(1999), crystallization leaves a typical signature on theperiod spectrum with an amplitude which increases asthe fraction of crystallized matter increases. Recently,two recent debates have been opened on the possibility tomeasure the fraction of crystallized matter (see Metcalfe,Montgomery and Kanaan 2004; Brassard and Fontainethese proceedings) and the12C(�; )16O reaction ratesin white dwarfs (see Metcalfe, Salaris and Winget 2002;Fontaine and Brassard, 2002) by studying their pulsa-tional properties .

Some WD variables have, or are expected to have, spec-

12

Figure 10. First adiabatic exponent�1 for a ZAMS stellar model of1:2M� (left panel), where the dip corresponds tothe second helium ionization region. The relative oscillatory behaviour it is shown on the right panel where the largefrequency separations is plotted as function of the frequency.

tra rich enough in modes to provide useful constraints ontheir rotational rates. Attempts to derive the internal rota-tional profiles by using the observed splitting and adopt-ing forward calculations and inversion techniques havebeen presented in several works (e.g. Kawaler, Sekii andGough 1999, Vauclair et al. 2002).

Finally, another asteroseismic achievement is representedby the prediction that, during the pre-white-dwarf stage,the dominant energetic process is the neutrino cooling.Since evolutionary changes affect the pulsation periods,the detection of secular period changes in the hottest WDpulsators allow to measure their cooling rates, providingexperimental constraints on the theories of neutrino emis-sion in dense plasmas and thermal neutrino productionrates (O’Brien and Kawaler 2000).

9.2. Æ Scuti starsÆ-Scuti stars are typically population I, A and F main-sequence or slightly post main-sequence objects of about2�2:5M�. They are located in the lower part of the clas-sical Cepheid instability strip where the�-mechanismis expected to drive low-order radial and non-radial p-modes, modes with a mixed p- and g-mode character,and possibly g-modes. Most of theÆ-Scuti stars aremoderate or rapid rotators with surface velocities up to100� 200 km s�1.The identification of the oscillation modes is a very com-plex task for these stars, since the asymptotic theorydoes not apply to the excited modes (low-order p-modes).However, the main problem depends on the scarcity ofobserved modes, among the hundreds of excited pulsa-tions predicted by theory. Rapid rotation and possible dif-ferential rotation produce a rather complex power spec-trum, making more difficult the identification of modeswithout additional information.Thus, at the present, asteroseismology is able to put onlysome constraints on the internal structure ofÆ Scuti stars.

Many interesting and comprehensive reviews on astero-seismology ofÆ-Scuti stars can be found in the recent lit-erature (e.g., Breger and Montgomery 2000; Goupil andTalon 2002; Paparo, these proceedings).

9.3. Rapidly oscillating Ap (roAp) stars

The rapidly oscillating Ap (roAp) variables, a subgroupof the chemically peculiar A-type stars, are H core-burning stars of mass' 2 M�, characterized by strongdipole magnetic fields of the order of a few kG, which arethe most intensive fields in main-sequence stars. Theirposition in the H-R diagram overlaps theÆ-Scuti star in-stability strip (see Fig. 1), but the two groups of variablesdiffer significantly in the behaviour of the excited modes.The roAp stars pulsate in high-order (n � 20), low-degree p-modes. Most of the roAp stars pulsate in almostpure dipole (l = 1) modes. The reason why stars withsimilar luminosity and effective temperature may showso different pulsation characteristics is not completely un-derstood, even though it is clear that the strong magneticfields could play an important role in selecting the excitedmodes.

The parallax measurements have provided a primary op-portunity to check independently the predictions of roApstars seismology. The results pointed out the tendencyfor seismic predicted values to be systematically slightlysmaller than the Hypparcos ones (Matthews et al. 1999).

The observed pulsation properties of roAp stars wereoriginally explained in terms of the so-called oblique pul-sator model (Kurtz 1982), in which pulsation and mag-netic axes are mutually aligned, but tilted with respectto the rotation axis. The oblique pulsator model has beenimproved in order to better reproduce the observed powerspectra of roAp stars (e.g. Dziembowski and Goode1996) and more recently revisited by Bigot and Dziem-bowski (2002) which found out that the rotational effectsseem to be so strong to result in a pulsation axis inclined

13

with respect to both the rotation and the magnetic axes.However, the new theory still needs to be tested.

The excitation mechanism for the roAp stars is stillan unresolved problem, although extensively debatedover the years. The�-mechanism in the He II ion-ization region which drivesÆ-Scuti star oscillations hasbeen demonstrated to be inadequate for exciting high-frequency modes in roAp stars, even though the twogroups of variables share the same location in the H-Rdiagram. A great deal of alternative models were thenproposed, each of them studying various effects whichcould affect the�-mechanism giving rise to mode insta-bility, such as a reduced convective efficiency due to theaction of the magnetic field (Balmforth et al 2001), set-tling of He (Dolez and Gough 1982), presence of a stel-lar wind (Dolez, Gough and Vauclair 1988) or a chromo-sphere (Gautschy, Saio, Harzenmoser 1998).

10. CONCLUSION

Asteroseismology constitutes a unique approach to thedirect investigation of stellar structure and evolutionwhich can significantly improve our knowledge of astro-physics, giving information on the structure, dynamicsand evolutionary stage of the stars. Asteroseismology isstill far from the great results of helioseismology, but itis clear that its success will depend on the amount andquality of data collected by the ground-based observa-tories, the significant improvement in the accuracy ex-pected from the future space missions, and the progressesin the development of the application to the stars of tech-niques for the data inversion and study of the effects ofsharp features on the oscillation frequencies.

ACKNOWLEDGMENTS

I am grateful to the SOC and in particular to SarbaniBasu, for the kind invitation and to ESA for the fi-nancial support which allowed me to participate to theSOHO14/GONG2004conference with the present contri-bution. I am very grateful to J. Christensen-Dalsgaard, M.Cunha, J. Daszynska-Daszkiewicz, A. Miglio, J. Montal-ban, M. J. P. F. G. Monteiro and L. Paterno for fruitfuldiscussions, comments and help to realize and improvethis review. Finally, I wish to thank D. Cardini for thekind hospitality at the institute IASF of Rome where Ihad the possibility to accomplish the preparation of thepresent paper.

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