WHY BOTHERING TO MEASURE STELLAR ROTATION WITH COROT ?

M.J. Goupil1, A. Moya1, J.C. Suarez1, J. Lochard1, C. Barban 1, J. Dias do Nascimento3, M.A.Dupret1, R. Samadi1, A. Baglin1, J.P. Zahn4, A.M. Hubert5, S. Brun6, L. Boisnard7, P. Morel8,R. Garrido9, S. Mathis4, E. Michel1, J. Renan de Medeiros3, A. Palacios10, F. Lignières2, and

M. Rieutord2

1LESIA, UMR CNRS 8109, Observatoire de Paris, France2Observatoire de Midi-Pyrénées, Toulouse, France

3 DFTE- Natal Departamento de Fsica Terica e Experimental, Brazil4LUTH, Observatoire de Paris, France

5GEPI, UMR CNRS 8111, Observatoire de Paris, France6CEA, Saclay, France

7CNES, Toulouse, France8Observatoire de la Côte d’Azur, Nice , France9Instituto de Astrophysica de Andalousia, Spain

10GRAAL, Universite de Montpellier, France

ABSTRACT

One important goal of the CoRoT experiment isto obtain information about the internal rotationof stars, in particular the ratio of central to sur-face rotation rates. This will provide constraints onthe modelling of transport mechanisms of angularmomentum acting in radiative (rotationally inducedturbulent) and convective zones (plumes, extensionbeyond convectively instable regions). Relations be-tween the surface rotation period and age, magneticactivity, mass loss and other stellar characteristicscan also be studied with a statistically significant setof data as will be provided by Corot. We present var-ious theoretical efforts performed over the past yearsin order to develope the theoretical tools which willenable us to study rotation with Corot.

1. INTRODUCTION

All stars rotate more or less rapidely and many areactually rotating quite fast. Many issues concerningrotating stars are not fully understood yet and un-der investigation : what is the structure of a rotatingstar? what is its rotation profile? what are the trans-port mechanisms of angular momentum and of chem-ical elements ? As some mixing is driven by rotation,to what extent does it depend upon the rotationalhistory of the star? Which physical processes con-trol the behavior of rotation along the HR diagram?What is the effect of rotation on mass loss? Whatare the consequences of rotationally induced mixingon the yields and then on the formation of succes-sive star generations? How does magnetic braking

affect rotation ? Does rotation control dynamo pro-cess and stellar activity? More generally what arethe consequences of the interactions between rota-tion, magnetic field and atomic diffusion processes onthe structure and the evolution of stars? Hence, rota-tion can seldomly be studied alone as it is connectedwith many other processes such as mixing, activityand compete with many other processes: turbulence,gravitational settling and/or radiative diffusion. Asfor the Sun, seismology is expected to provide someconstrains on these various physical processes at playin stars and obtain clues about how to model them.Seismology is expected to provide information on therotation itself and on the structure of the star. How-ever, rotation is able to modify the oscillation fre-quencies and changes differ from one type of star toanother, depending on the magnitude of the rota-tion rate and the type of excited oscillation modes.Determining effects of rotation on stellar oscillationscan then be seen as a field of research on its own.

Many observational and theoretical efforts have beencarried out over the past decade to provide answersto some of the above questions. The aim of thepresent paper is to give an overview of various the-oretical developments which have been carried outin recent years in preparation of the scientific re-turn from CoRot experiment. Stellar rotation andactivity are two quantities which are accessible toCorot accurate measurement capabilities for manystars of various classes (ie wide mass and age ranges).The interested Corot community has therefore gath-ered in a team, AcRoCoRot, in order to coordinatetheir efforts about these two general issues. Twomeetings have taken place during the Corot Brazilworkshops in 2004 and 2005 as Bresilian are deeplyinvolved in AcRoCoRot projects. Presentations

given during these meetings can be found on lineat the site http://ace.dfte.ufrn.br/corot/index.htmlSeveral presentations during both meetings con-cerned rotation as will be mentionned later on inthis review.

The paper is divided into 4 parts: Part I presentssome additional programs related to stellar rotationwhich have been proposed. These projects basicallyrequire the determination of the surface rotation pe-riod for as large a number of stars as possible. As forthe core program, proposals related to rotation willneed the knowledge of internal rotation gradients forinstance and will require the accurate measurementsof as large a number of frequencies as possible for afew stars.

Part II briefly explains the tools which have been de-velopped in order to compute oscillation frequenciesof rotating stars. Indeed, differences between com-puted and observed frequencies in fine must comefrom differences between the stellar model and thereal structure of the star and not from inaccuracyin determining the theoretical oscillation frequencies.Different levels of approximation are available anddiscussed depending on the magnitude of the rota-tion, whether it is fast or not. One important issuethen is to determine how fast is fast in the presentframework.

In Part III, we consider imtermediate mass and mas-sive stars, more precisely A-B type stars such as δScuti and β Cephei stars which are suitable for test-ing rotational mixing of type I . Effects of these pro-cesses upon oscillation frequencies must be quantifiedby means of comparison with frequencies of nonro-tating models. As these stars do not lose angular mo-mentum, they rotate fast and effects of rotation onfrequencies are important. Perturbation techniquesas well as nonpertubative approaches have been de-velopped in order to be able to study the oscillationsof these stars and will be presented. The domain ofvalidity of perturbation techniques will also be dis-cussed.

In part IV, we discuss the case of solar-like stars. Asthe Sun, these stars oscillate with high frequency p-modes which are stochastically excited by turbulentconvection in outer layers. Their oscillation spectraare characterised by a large number of small ampli-tude frequencies, a number of equal spacings. Thesestars are usually known as slow rotators. Howeverwe will see that effects of rotation on some frequencyspacings at the needed accuracy level are not neg-ligible and must be taken into account for furthercorrect seismic interpretations of the data.

Finally some conclusions are drawn in Sect.9

Determinations of rotational splittings and inclina-tion angles are left to another paper in this volume.Some parts of the present paper are not exhaustive

Figure 1. top: Distribution of v sin i in the HRdiagram obtained with Coravel measurements (DeMedeiros, Mayor, 1999). Tracks are from do Nasci-mento et al. (1999) (taken from Do Nascimento andDe Meideros, 2004, AcRotCoRoT) . bottom pan-nels: v sin i histograms for F-G-K stars (taken fromGoupil et al. (2003), based on data from Cutispotoet al. (2002)).

as they are treated in other contributions in this vol-ume. Because of an expanding amount of work aboutstellar rotation, the present paper cannot be exhaus-tive and is therefore biased by the interests of theauthors which do apologize for that.

Part I: What is at stake ?

1D standard models reproduce the gross features ofevolution. However it is well known that hydrody-namical processes such as turbulent convection arenot well modelled. In addition, observations re-quire to go beyond standard models, particularlythey show a need to include additional mixing. Twodevelopement directions beyond standard evolutionmodelling are currently being followed:

- modelling transport processes and their conse-quences with prescriptions derived from physical

Figure 2. v sin i histograms fro A-B stars (taken fromGoupil et al (2003), based on data from Royer et al.2002a,b)

first principles, 3D numerical simulations, labora-tory experiments or observations. The results of suchmodelling is implemented 1 D evolutionary models.Along this line, effects of rotation are taken into ac-count in several evolutionary codes (Geneva code,Maeder et Meynet, 2004; STAREVOLV, Palacios etal. 2003; Cesam, Moya et al 2006).

- 2D numerical approach. A few 2D evolutionarymodels are currently being developed with inclusionof microphysics to become more realistic (Roxburgh2004, ESTER code: Rieutord et al. 2005, Rieutord2006)

Outcome of these modellings must be tested with ob-servations. One information at our disposal can bethe surface rotation rate. This enables to study rela-tions between surface rotation (determined throughmicrovariability) and other stellar parameters (Sect.2below). More information about stellar rotation canin principle come from seismology. However rotationcan significantly affect stellar pulsations. The cen-trifugal force distorts the shape of the star and mod-ifies its equilibrium structure. Coriolis force directlyaffects the oscillatory motions. In order to take fullyadvantage of pulsations, one must first establish towhat extent they are affected by rotation. A firstapproach assumes that rotation is slow enough thatit can be treated as a perturbation both for the equi-librium structure and for the oscillations (Sect.3.3.1below). The 2nd approach is valid for any rotationand consists in solving a 2D numerical system whichinclude effects of rotation (Sect.3.3.2). The approachhas been developed to apply to pressure modes (Es-pinosa et al, 2004; Lignières et al 2006; Reese et al2006) as well as to gravito-inertiel modes (Dintrans,Rieutord, 2000).

2. ONE SINGLE INFORMATION :PROT,SURFACE FOR MANY STARS

Fig.1 (presented by Do Nascimento, 2005 AcRot-CoRoT) displays the distribution des vsini in the HRdiagram obtained from Coravel instrument (De Mei-deros, 1999; Do Nascimento and de Meideros 2005).Some Corot additional programs propose to deter-mine the surface rotation period for a large sampleof stars (Do Nascimento, de Meideros et al.; Favata etal., AcroCorot 2005). This will enable to study thebehavior of the surface rotation of solar-type starsfrom the main sequence to the giant branch. Statis-tical studies will be used to obtain relations betweenthe surface rotation rate and other stellar quantitiessuch as luminosity, mass, age, and chemical elements.

Determination of the surface rotation period will beobtained with CoRoT through periodic variabilityindicative of the presence of spots at the surface ofthe stars. Using a model with multispots at differ-ent latitudes, it could even be possible to study thelatitudinal surface differential rotation (Petit et al.,2004, 2006; Donati et al. 2003; Marsden etal. 2006;Reiners et al 2003; Streissmeier et al. 2003) fromthe (micro)variability of the star as measured withCoRoT as explained by Cutispoto (2005, AcRot-CoRoT)

2.1. Age-activity -rotation relation for latetype stars

The basic idea is that chromospheric activity is trig-gered and sustained by rotation in convective enve-lope of solar like stars. Because of rotation breakingby a magnetised stellar wind, rotational velocitiesdecreases with age as well as activity for single stars.

Low mass stars at the very beginning of the mainsequence show a high dispersion in rotational veloc-ity and therefore in their chromospheric-activity levelas a result of their different PMS history. Howevermagnetic breaking forces very rapidely uniform ro-tational velocities. Issues then are for instance ’doesa ta law hold for late type stars and more massivestars as well? which value must be chosen for a ?’

Another important issue is to study the relationbetween rotation, activity and convection in stel-lar outer layers (Aigrain et al., Baudin et al., 2005,AcRoCoroT). This relation can be studied in a di-agram activity level versus Rossby number wherethe Rossby number is the ratio of the rotation pe-riod (obtained with Corot) and the turn over timeof the turbulent eddies (Fig.3). The activity levelcan be assessed with photometric variability whichfrom ground is detected at the level of 10−2 − 10−3(Fig.3). A level down to 10−3 − 10−4 is expected tobe reached with Corot. This will allow to extend the

Figure 3. Observed activity-rotation relation for lowmass stars. Both activity and Rossby number (rota-tion period) are measured through photometric vari-ability.

study of the magnetic activity to stars earlier thanG8.

Figure 4. A schematic representation of darkeningeffect due to fast rotation. Polar regions are hot-ter than the equatorial ones. The line profiles seenfrom the poles and from the equator differ. This de-pendence upon the inclination angle must be takeninto account when deriving stellar parameters frommodel atmospheres. (from Hubert el al., 2004, AcRo-CoRoT)

2.2. Mass loss-rotation relation for massivestars

For massive stars (9 − 20M�) (O-B stars), the evo-lution of surface rotation is affected by mass lossand/or internal transport mechanisms (Meynet &Maeder 2000; Meynet& Maeder, 2005). Of particularinterest here are Be stars as presented by Hubert etal. (2004) at the 1st Corot Brazil workshop. Be starsare main sequence or slightly evolved B stars sur-rounded by an equatorially concentrated envelope,fed by discrete mass loss events. Be stars usually

rotate rapidly (Ω/Ωc ≥ 0.80) however their rotationrates do not reach the break-up velocity. The is-sue is therefore to identify the mechanism at the ori-gin of the nonregular mass losses in these stars. Asthese stars are non radial pulsators, one question iswhether pulsations do trigger direct mass loss events.Be stars also show stellar and circumstellar activ-ity. Projected rotational velocities have been deter-mined for hot and massive (O type) stars (Pennyet al 2004; Gies, Huang 2004). One often encoun-ters stars with v/vcrit ∼ 0.9 (Townsend et al., 2004)which means that vesc ∼ cs the escape velocity is ofthe same order of magnitude than the sound speed,this indicates that a nonradial pulsation driven windcan indeed be efficient for these stars (Owocki et al.,2004). Magnetic field (several hundred of gauss) havebeen detected at the surface of these stars (Neiner etal 2003). Activity of magnetic origin has also beenproposed as a mechanism that could give rise to theadditional amount of angular momentum needed toeject material. Metallicity can also play a role in thesense that at low metallicity, mass loss is smaller,hence less loss of angular momentum and the starsrotate faster and could reach more easily the breakup velocity than stars with same age and mass butwith solar metallicity (Maeder, Meynet, 2001). Ob-servations are carried out with the purpose to con-firm these statements (Martayan et al. 2006).

In order to investigate the interrelation between fastrotation, magnetic field, pulsations, metallicity andmass loss in Be stars, accurate determination of stel-lar parameters (location in the HR diagram, v sin i,metallicity) is required. This is currently done for anumber of Be stars in the set of Corot possible targetstars (Fremat et al., 2006).

These stars are fast rotators and gravitational dark-ening effect are no longer negligible (Fig.4). Thisis confirmed by recent interferometric observationswhich starts to yield information about the oblateshape of rotating stars and gravitational darkening(Domiciano de Souza, 2005). As the poles are hotterthan the equator, some lines will tend to form moreeasily at the poles rather than at the equator; hencethe line profiles are not the same when seeing fromregions near the pole or equatorial regions. They areinclination angle dependent and one has to take thiseffect into account in the models when deriving thestellar parameters of rapid rotating stars (Zorec et al2005, Fremat et al 2005, Martayan 2005 phD). Whenstudying the Be star ω Orionis, Neiners et al (2003)for instance used the code FASTROT (Fremat et al2005) which has been developped for providing gridsof theoretical spectra taking into account the effectsof a rapid rotation and inclination angle on the lineprofiles.

Meynet and Maeder (2005) stress that stellar mod-els with rotation predict that for stars with massless than < 12M�, mass loss is small and internalmechanisms transports are dominant in controlling

- + 0 30 60 90latitude [deg]

0.12

0.14

0.16

Fe

r/R

rc

ro

Figure 5. Numerical simulation of the inner 30% inradius of A type 2 M� star. The rotating convectivecore extends over approximately 15% of the radius .left: a 3-D rendering of the radial velocity is shownnear the equatorial seen from above at a given time.Blue regions: descending flows; red ones: ascendingflows. Overturning motions have time scales of or-der 1 month ie about the rotation period. right: en-thalpy flux: a latitude dependence is clearly visible.The mixed central region when including effects ofboth penetration and overshoot keeps a nearly spher-ical shape in average. Extension of the mixed centralregion is in average ∼ 0.2Hp. (taken from Browninget al 2004)

the time evolution of the surface rotation velocity.They suggest to study of the behavior Prot in func-tion of the distance from the Zams which can provideconstrains on the tranasport processes.

3. ONE STAR: MANY OSCILLATION PE-RIODS

As mentionned earlier, open questions are whetherthe rotation profile is shellular, what is the strengthof the radial differential rotation and the extention ofthe central mixed regions when they are convectiveand rotating Hence with Corot, goals are – to iden-tify regions of uniform rotation and regions of differ-ential rotation (depth, latitude dependences) insidethe star, – to determine Ωcore/Ωsurf that-is whetherthe core is rotating faster than the surface and inthis case by what amount. This type of informationcan in primciple best come from seismology, providedthere are enough oscillation frequencies detected andaccurately measured. These strong requirements metfor a few stars in the Corot observing program.

3.1. Extension of rotating convective core

Intermediate mass stars and massive stars have aconvective core which real extension is not reallyknown. Indeed the turbulent eddies do not stop atthe radius given by the Schwarzschild criterium butpenetrate in the adjacent convectively stable regionswhere they are decelerated.

It is important to include this extension in stel-lar modelling as the nuclear fuel, hence the evolu-tion and the stellar parameters are modified. Thiscan have important consequences in studies of age -abundances relations in chemical evolution of galax-ies; in extragalactic studies based on massive starswhich have convective core and rotate rapidely.This has also consequences on the relation mass-luminosity for Cepheids for instance which serve asdistance indicators.

The mixing length is not able to predict such an ex-tension; it has been empirically determined by com-parison with observations. It is expected that Corotwill provide, at least for a few specific stars, informa-tion on the rotation and rapid variation of the soundspeed near the convective core of intermediate massand more massive stars. This will give us some quan-titative information on the extension of the centralmixed regions.

Intermediate and massive stars rotate rapidely. Fromground, spectroscopic determinations of the pro-jected rotational velocity v sin i for a large set of A-B stars have been obtained (Royer et al 2000, 2002).Histograms of v sin i show that the bulk of A, B starshave velocity around 100 km/s with a decreasing tailwhich extends up to v sin i 300 km/s i.e. near breakup velocity (Fig.2). It is expected that the interac-tion between convection and rotation modifies thepenetration of the eddies in the overlaying radiativelayers. This has been studied with 3D numerical sim-ulations of a rotating spherical shell which is a sim-plified representation of a rotating convective core fora A type star and a rotation rate of 1/10 t 4 timesthe solar rotation rate Ω� = 2.6µHz (Browning etal 2004). Fig.5 shows that indeed rotation stronglyinfluences the convective motions which in turn re-distribute angular momentum. The rotation is dif-ferential both in latitude and in radius in a rotatingconvective core. Sizes of ascending and descendingflows are similar as the density contrast is small (only2-3) unlike the case of the solar outer convective re-gion. Convective motions overshoot into the radia-tive region above, which makes it oblate (prolate iealigned with the rotation axis). The extension ofovershooting appears to vary with latitude creatingasymetries in the extent of the penetration. In aver-age however, the entire mixed central region seemsto remain spherically symmetric. For this simulationrSch = 0.14R and rov = 0.16R, R is the stellar ra-dius i.e. with an extension into the stable layers ofabout 0.02R = 0.2Hp.

This is in agreement with empirical determination forthese types of stars by fitting the main sequence turnoff and isochrones. It is important to note that whenrotation is increased, the mixed central region getslarger. As this extension depends on the rotation ofthe star, it can vary from one type of star to another,and even from one star to another.

Browning et al explain that overshoot extension de-pends on the stiffness of the entropy gradient. In caseof a weaker stiffness, the penetration extends furtherinto the radiative part. Increasing Ω without alter-ing the viscosity results in a more laminar convec-tion due to the stabilizing effect of the rotation andtherefore to a greater overshooting: rotation seemsto enhance overshooting. The authors caution thatdue to various simplifications of the physical descrip-tion, extention of overshoot is likely to be somewhatoverestimated .

In a recent paper, Brun et al. 2005 have modelled theconvective core of a magnetized A star. They foundthat the differential rotation when Lorentz forces areaccounted for in the redistribution of the angular mo-mentum is greatly reduced and becomes much moretime dependent. However their results indicate thatthe inclusion of magnetic effect does not seem tomodify the amount of overshooting.

3.2. Transport of angular momentum in ra-diative regions

Another piece of information we are expected isthe location and the magnitude of rotation gradientwithin the star, and particularly in radiative regionsof solar like stars as this would allow testing varioustransport mechanisms of angular momentum and as-sociated chemical mixing.

Stellar rotation profile results from angular momen-tum transport that-is the angular momentum redis-tribution from one region of the star to another whichcan have several origins:

• caused by evolution: expansions and contractionsof stellar regions

• caused by hydrodynamical processes. In radiativeregions, (differential) rotation causes large scale mo-tions, i.e. meridional circulation, and hydrodynam-ical instabilities which generate turbulence which inturn contribute to the redistribution of angular mo-mentum. In convective region, interaction betweenlarge scale convective motions and rotation is quiteefficient in transporting angular momentum and gen-erating differential rotation.

• caused by surface losses by stellar winds (dynamo,thermal radiative) or surface gain from surrounding(accretion)

• magnetic field or internal waves (Mathis, Zahn,2005; Talon et al 2002; Talon 2006)

In radiative regions, turbulence is expected to behighly anisotropic due to vertical stratification whichacts as a stabilizing agent. As a result, it canbe reasonably assuned that the rotation is shellular(Ω = Ω(r)) which enables a 1D description. The ra-dial transport of specific angular momentum in theshellular assumption is then governed by the equa-tion (Zahn, 1992):

ρd

dt

(r2Ωr

)mr

= 15r2∂∂r

(ρr4U(r) Ωr

)+

1r2

∂∂r

(ρr4 Dv

∂Ωr∂r

)(1)

where Ω(r) is the rotation rate of the assumed shel-lular rotation; U(r) is the velocity of the merid-ional circulation. The description is lagrangiennewith dmr = 4πρr2dr and mr the mass enclosed ina sphere of radius r, hence expansion and contrac-tion effects are automatically taken into account.The first term on the right hand side is advectionby meridional circulation and the second term isdiffusion by weak vertical turbulence (Zahn, 1992;Maeder, Zahn 1998). Turbulence here is assumeddriven by shear due to differential rotation (verti-cal shear Ω(r)). Onset for turbulence occurs only ifthe flow satisfies a criterium (modified Richardson)which is still debated as it must include various sta-bilizing and destabilizing processes (µ gradient- ra-diative diffusion...). Horizontal shear Ω(θ) acts tosuppress the instability. Due to a vertical stablestratification, turbulence is highly anisotropic andhorizontal turbulence is much more efficient in ho-mogeneize the meddium hence turbulent transportcoefficient must satisfy Dh >> Dv.

These processes cause chemical transport (directelyor indirectly) which in turn affects the structure andthe evolution of the star. Transport of chemical ele-ments is given by the equation :

ρ(dci

dt

)mr

+ 1r2

∂ρr2 ciUdiffi

∂r =

1r2

∂∂r

(ρr2 (Dv + Deff ) ∂ci∂r

)(2)

where ci is the concentration and Udiffi the micro-

scopic diffusion velocity of the chemical element i.The second term on the left hand side represents ef-fect of atomic diffusion. The two terms on the righthand side model effect of rotationally induced trans-port.

While the assumption of strong anisotropic turbu-lence generating a shellular rotation profile Ω(r)likely holds true, the difficulties lie in determiningthe expressions of the turbulent diffusion coefficients

Dv, Dh, Deff in Eq.1, Eq.2 and establishing a real-istic criterium for the onset of turbulence in radia-tive regions (Zahn, 1992; Maeder, 2003; Mathis, et al2004). These are still uncertain, advances are basedon results of numerical 3D calculations (Browning etal. 2004) or laboratory experiments and could wellafford information and observational constraints asCorot precisely could bring up.

Two types of rotationally induced processes of mix-ing have been identified: mixing of type I where bothangular momentum and chemical elements are trans-ported by rotationally induced mechanisms and mix-ing of type II where again chemical elements aretransported by rotationally induced processes butfor transport of angular momentum, other processesare more efficiently operating: internal gravity wavesand/or magnetic field). Effect of rotation in trans-porting angular momentum here arises through Cori-olis force which causes different behavior (in partic-ular damping properties) between prograde et retro-grade waves and therefore a net non zero momentumflux (Mathis 2004, phD).

The circulation is driven by gain or loss of angularmomentum in surface. Hence if in absence of angu-lar momentum loss, there is no need to transport an-gular momentum and the circulation remains weak:this happens to be the case for normal A type stars.On the other hand, important losses of angular mo-mentum from the surface through a wind forces theneed for a strong circulation in order to transportangular momentum to the surface. As far efficiencyof transport of angular momentum in radiative re-gions is concerned which is responsible for uniformor nonuniform rotation profiles, we must thereforeconsider several cases of single stars (binarity is an-other issue not discussed here) according to their in-ternal structure which itself depends on mass andage (Fig.6):

• Main sequence stars

- Low mass stars (M ≤ 1.4M�) have an extendedouter convective region which lose angular momen-tum through a dynamo driven wind (F-G-K stars)during their main sequence life time. The inner partis entirely radiative or for the most massive ones witha small convective core. By analogy with the Sun,we must expect a nearly uniform rotation profile inthe inner radiative layer and latitudinal dependentrotation in the overlying convective region. As forthe Sun, these stars have been speed down whenevolving from their very youth and are much slowerrotators than the intermediate mass stars. - Inter-mediate mass (1.4 − 1.5 ≤ M� ≤ 2.5 − 3M�) arenot expected to lose angular momentum nor mass asthey have no extended external convective region foran efficient dynamo process and are too cool to havea radiatively driven wind. Because of evolution ofthe structure with expansion and contraction of massshells, one expects that they rotate internally with a

Figure 6. Structure (convective (circled ar-rows)/radiative (straight arrows) regions) of a stardepending on its mass or age, whether it is a main se-quence low mass star (right), an intermediate mass,main sequence star (middle) or a giant star (left)

differential rotation profile ie radius dependence. Wehowever ignore the stiffness of the rotation gradient.

- massive stars (M� > 3M� which lose mass througha radiatively driven wind (O-B stars). This mass loss(onset, magnitude, geometry) is strongly affected byrotation (Meynet & Maeder 2004; Maeder, 2006)

For massive and intermediate mass main sequencestars, rotationally induced mixing of type I has beenfound quite successful in reproducing observation(Talon & Zahn 1997; Maeder & Meynet 2000).

However for late type stars, this processus predictsa too fast rotating core for the Sun which seems tobe in disagreement with heliosesimology (Matias &Zahn, 1998). Fig.7 shows results of calculations byCharbonnel, Talon (2005) : the rotation profile inthe solar radiative region obtained from a 1D stellarassuming mixing of type I; the rotation if far fromuniformity. When the rotation profile in the solarradiative region is obtained from a 1D stellar by as-suming mixing of type II that-is including all trans-port processes of type I and including internal gravitywave transport but no for magnetic field, the redis-tribution of angular momentum is efficient enoughto enforce uniform rotation in the radiative regionof the Sun as observed by helioseismology. In ad-dition, the observed behavior of the surface lithiumabundance in function of the effective temperature ofthe star (Li gap) is now reproduced by these models(Talon, Charbonnel 1998, 2003; Talon, 2006).

• Evolved stars: rotation on main sequence has astrong influence on the evolution of the structure andsurface abundances in later stages when the star be-comes a red giant

• PMS stars ;

Their fast rotation provides the initial condition forevolution of rotation on the main sequence.

Figure 7. top: Rotation profile in the solar radiativeregion obtained from a 1D stellar model assumingmixing of type I. bottom: rotation profile in the so-lar radiative region obtained from a 1D stellar modelassuming mixing of type II (Zahn presentation 2005,AcRoCoRoT, taken from Charbonnel, Talon (2005)).Colors represent different ages.

Part II Rotation and oscillations

It is expected that the space mission CoRoT (Baglinet al 1998) will measure oscillation frequencies withan accuracy of 0.5 µHz (20 days observation) and0.08 µHz (150 days observations) for coherent modesie self excited modes such as those of A-B type stars(δ Scuti, γ Dor, β Ceph)(Fig.8). For solar like stars,this will depend on the signal to noise ratio henceon the apparent magnitude of the star (see Goupilet al., this volume; Appourchaux et al HH exercisesthis volume).

These accurate measurements of the oscillation fre-quencies and the associated rotational splittings(Eq.5 below) will enable to infer information aboutdepth and perhaps also latitude dependences of theinternal rotation of a star. Forward techniques con-sist in comparing measured splittings with computedsplittings assuming a rotation profile. Inversion tech-niques in principle do not assume a priori a rotationprofile and determine either locally by suitable com-binations of rotational splittings or globally by leastsquared fit minimisation its behavior with depth andlatitude at least in regions where the modes associ-ated with the detected frequencies propagate.

Figure 8. HR diagram showing several types ofoscillating stars: two large classes are emphasizedas special Corot target classes: intermediate massA-B stars and solar like stars. A third class isnow included in the favorite Corot target choices:red giants stars (diagram taken from Do Nasci-mento’stalk,2005 AcRotCoRoT, itself taken fromBoisnard 2004, AcRotCoRoT)

3.3. Effects of rotation on oscillation fre-quency

Let ν0,n,�,m be the frequency (in Hz) for a given os-cillation mode when no rotation effect is included.The geometry of an oscillation mode is described byone single spherical harmonics, Y m� , and the mode is2 + 1 degenerate and the frequency is m indepen-dent ν0,m ≡ ν0 (Fig.9). When rotation is included,the centrifugal force distorts the structure of the starand therefore indirectly influences the wave propaga-tion ie frequencies; on the other hand, Coriolis forcehas a direct effect on the wave propagation and onthe oscillation frequencies. As a result, the sym-metry breaking lifts the frequency degeneracy andthe frequencies become m dependent. For symmetryreasons, it is usually assumed an axisymetric rota-tion Ω(r, θ). The equilibrium structure then remainsaxisymetric; the modes can be labelled with a welldefined azimuthal indice m.

The eigenmodes are computed using a basis whichmust be appropriately chosen in case of rapid ro-tation (Lignières et al 2006, Reese et al 2006) andwhich reduces to the usual vectors in spherical co-ordinates when rotation is slow enough to authorizeperturbation expansion as in Sect.3.3.1 below.

For later discussions, it is convenient to distinguishbetween fast, moderate and slow rotation classified

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Ω

ν�� ν0 ν�

Figure 9. Schematic representation of effects of rota-tion on oscillation frequencies. From top to bottom, adegenerate mode n, in absence of rotation; a multi-plet when rotation lifts the m degeneracy: first ordereffect due to Coriolis force gives rise to an equallyspaced multiplet; the equla spacing is destroyed bysecond order effects mainly due to centrifugal distor-sion; finally corrections due to near-degeneracy mustbe applied to modes with degrees , + 2 and sameazimuthal indice m. Correcting for near degeneracyresults most often in repelling the close (ie near de-generate) frequencies

according to two parameters:

� = Ω2/(GM/R3) µ = Ω/ω (3)

where � represents the centrifugal over gravitationalstrengths and µ compares the rotation time scale(Coriolis force) compared to the oscillation period.Then

• slow rotation here means �, µ �max, µmax 2D equilibrium modelsand non perturbative approach for computing theoscillation frequencies are required.

The upper limits �max, µmax depend on the staras it depends on the frequency range of its ex-cited modes. More rigorously speaking, perturbationmethods cease to be valid whenever the wavelengthof the mode reaches the order of ∼ � R where R is thestellar radius that-is for p-modes with high enoughfrequencies or g-modes with low enough frequencies

3.3.1. Effect of rotation as a perturbation onoscillation frequencies

We focuse here on spheroidal modes only and we con-sider only the depth dependence of rotation henceΩ(r) =< Ω(r, θ) >horizontal. When rotation is slowenough, � and/or µ are small and perturbation meth-ods can be used. However, with increasing rotationrates, one must includes higher and higner order cor-rections to the frequency for a given accuracy (suchas Corot frequency measurement accuracy)

• Slow rotators ; O(Ω)

Slow rotation, as in the solar case for instance, canbe treated as a perturbation and only the first ordercorrection (from Coriolis force) is retained. Oscilla-tion frequencies then are given by

νn,�,m = ν0,n,� + mΩs2π

(Cn,� − 1) (4)

in an inertial frame, with Ωs is the surface rotationrate (in rad/s) and and Cn,� is the Ledoux constant(DG92).

One defines the linear and the generalized rotationalsplittings respectively as:

δ0,m =νn,�,m − ν0,n�,m

mδm =

νn,�,m − νn,�,−m2m

(5)When Ω is uniform, then δm/(Cn,� − 1) = Ω is aconstant with Cn,� = 1I

∫K(r)dr. Eq.5 is an integral

equation for a shellular rotation rate as

νn,�,m = ν0,n,� + mΩs2π

∫ R0

K(r)ΩΩs

dr (6)

where the kernel K(r) depends on the eigenmode, ris the radius. This relation can be inverted to yieldΩ(r).

• For moderate rotation, O(Ω2, Ω3)

For moderate rotation, one has to take into accounthigher order (second and third order) corrections.Several degrees of approximation for realistic mod-els have been studied over the years (Dziembowski,Goode (1992, DG92), Soufi et al 1998(SGD98),Suarez et al 2006, Karami et al 2005)

Figure 10. Oscillation frequencies corrected for ro-tation effects for two modes which are a = 0 and = 2 modes when no rotation is included. The modelis a 1.8 M�. Rotation included as a perturbation. Auniform rotation rate is assumed with v= 93 km/s.From bottom to top, successive higher order correc-tions are added to the zeroth order frequency (bottom)to provide the final correct frequency up to 3rd order(top) (from Goupil et al. 1999)

Oscillation frequencies must then be corrected ac-cording to

νn,�,m = ν0,n,� + mΩs2π (Cn,� − 1) +

Ω2s2πω0 Dn,�,m +

Ω3s2πω20

m Tn,�,m (7)

where ω0 = 2πν0,n,� and Cn,� is the Ledoux con-stant Expressions for the Dn,�,m, Tn,�,m coefficients(SGD98) show that they verify Dn,�,m = Dn,�,−mand Tn,�,m = Tn,�,−m which indeed is a general prop-erty resulting from the symmetry between a star andits mirror image with respect to any meridional planeas demonstrated by Reese et al. (2006).

The ’zeroth order’ frequency ν0 in Eq.7 actuallyincludes the effect of the centrifugal force averageover the surface (spherically symmetric distorsion)through an effective gravity geff ; It does also in-cludes some contribution of Coriolis force (if derivedas in SGD98) and strictly speaking is m dependent.This dependence is omitted here for sake of clarity.The second term is the first order (Coriolis) cor-rection and the last two terms represent effects ofnonspherical distorsion of the star on the oscillationand third order correcton due to both centrifugal andCoriolis forces.

At 2nd order (O(Ω2)) the centrifugal force in distort-ing the shape of the star is responsible for the factthat a pure radial mode, for instance, of a nonrotat-ing star can no longer exist when the star rotates. Ifrotation is low enough, the mode can keep its mainnature as radial but with a contamination from othereven modes. The contamination by the = 2 mode

mainly depends on the ratio of the inertia of bothmodes. Conversely the mode = 2 is also contam-inated by the radial mode. This happens when fre-quencies of both modes are close to each other. Moregenerally, for large rotation, the modes can no longerbe described with a single spherical harmonics. Aneffect which we refer to as near degeneracy. Thisis taken into account in writing that the mode is infact a linear combination of two modes a and b whichsatisfy the conditions: b = a +2, ma = mb, ωa ∼ ωb

Let two modes labelled a and b with δ = νa − νb ∼ 0then mode a is contaminated by mode b ie νobsa ;mode b is contaminated by mode a ie νobsb then theirfrequencies are given by :

νobsa = ν̄ −12

√δ2ab + H

2ab (8)

νobsb = ν̄ +12

√δ2ab + H

2ab (9)

with ν̄ = (1/2)(νa + νb) is the mean frequency andδ = νa−νb is the small separation ; Hab is a couplingcoefficient (see DG92, SGD98, Goupil et al. 2000,2002; Suarez et al 2006, Karami et al 2005)

Such near degenerate frequency corrections can reachthe level of 0.5%−2%. This is illustrated in Part III.

3.3.2. Effects of fast rotation on oscillationfrequencies

For rapidly rotating stars, a perturbative approachis no longer applicable. One must turn to a 2D nu-merical approach to compute both the equilibriummodel and the eigenmodes. Lignières et al. (2006)and Reese et al. (2006) have built a 2D numeri-cal technique to compute nonperturbative oscillationfrequencies. The solution is searched for as a sum-mation over the spherical harmonics (up to = 80 inEq.4) for the angular dependence and decomposedover Chebyshev polynonials for the radial depen-dence of the eigenmodes. The equilibrium model inthese studies is a self-gravitating uniformly rotatingpolytrope of indice 3 with rotation taken in a rangefrom 0 to 0.59 ΩK where ΩK is the break up velocityΩK =

√GM/R3e with Re the equatorial radius. In-

vestigation concerned first the effect of the centrifu-gal force alone (Lignières et al 2006) then both thecentrifugal and Coriolis forces were included (Reeseet al 2006). For details concerning the adopted nu-merical techniques, see Lignieres et al. (2006), Reeseet al. (2006) and references therein. These authorshave then studied the properties of p-modes in thefrequency range of δ Scuti stars. A brief summary ofsome of their results is given in Sect.6 below.

When the star is non rotating, modes are easily clas-sified as one assigns a single spherical harmonics withone given . When the star is rapidly rotating andthe mode structure is given by a sum of many spher-ical harmonics, it is more difficult to classify the

modes, especially when they are involved in avoidedcrossings. In order to adress this difficulty, Lignièreset al. (2006) followed several modes from zero rota-tion to 0.59 Ωk. They found that even at fast rota-tion rates, it is still possible to classify the modes andassign them a value, based on the correspondancewith modes in a non-rotating star.

Part III A-B type stars

Corot in its SISMO field will observe A-B stars suchas β Cep (B type, ie massive main sequence stars), δScuti and γ Dor stars (A, intermediate mass main se-quence stars) (Fig.8). Results from seismology fromground are still very sparse for these stars but seemto confirm the fact that the central regions of A-Bstars rotate at least 2 or 3 times faster than theirsurface layers.

δ Scuti stars (PMS, MS post MS) have projected ro-tational velocities in the range v sin i = 70−250km/s(Fig.2). This identifies them as moderate to rapid ro-tators in a � − µ diagram (Fig.24). For these stars,rotation strongly affects the location in HR diagram,the mode visibility and identification, mode excita-tion and selection (for a review, see Goupil et al.2004 and references therein).

β Cephei stars are more massive stars. Their rota-tions are slow or moderate and they have no convec-tive envelope, which makes their modelling easier.Two stars illustrate well the success found in theirseismic modelling: HD 129929 (Aerts et al. 2004,Dupret et al 2004) and ν Eri (Pamyatnykh et al.2004, Ausseloos et al. 2004). These two stars havewell identified pulsation modes (at least 8 for ν Eriand 6 for HD 129929). Their seismic analyses givestrong constrains on the location of the convectivecore boundary and overshooting parameter dov � 0.1in one case and dov � 0.3 for the other one. Ineach of these stars, 2 rotational multiplet structuresare clearly identified. Their interpretation allows thedetermination of their internal rotation compared tothe surface. In both cases, the core rotates about3-3.5 faster than the surface. Many more modes(higher degrees, more multiplets) are expected to beobserved in the β Cep stars observed with COROT.

γ Dor stars are g-modes pulsators. The excitation oftheir modes is well explained by non-adiabatic mod-els (Dupret et al. 2005). Their long periods (0.3 to3 days) are of the same order as the rotation ones.Hence, rotation is expected to affect strongly the fre-quency pattern of these stars (Dintrans & Rieutord,2000). As for δ Scuti and β Cephei stars, withoutmode identification based on other observables thanfrequencies, no unique solution is found. Taking ac-curately the effect of rotation into account is the nextstep for feasible seismic studies of γ Dor stars.

4. EFFECT OF ROTATION ON THESTRUCTURE AND EVOLUTION OFA-B TYPE STARS

Talon, Zahn (1997) studied the effects of transportof internal stellar angular momentum and inducedtransport of chemical elements on the evolution-ary track and structure of a 9M� model accordingto Eq.1, Eq.2 above with Dv, Dh, Deff coefficientsgiven by Zahn (1992). Following this study, sev-eral works such as Meynet & Maeder (2000) investi-gate consequences of rotationally induced transportin massive stars on their structure, evolution and in-teraction with their environments (mass loss, yields).For a review see Meynet, Maeder (2004).

Using the same evolutionary code with the sameassumptions as in Talon, Zahn (1997), calculationsfor less massive models showed how rotationally in-duced transport affects the evolutionary track ofa ∼ 1.75M� model (Goupil, Talon 2002). Fourtypes of models were built assuming 1) no rotation,no overshoot - 2) no rotation, overshoot includedwith a classical value for the overshoot extensiondov = 0.2Hp with Hp the pressure scale height. -3) No overshoot but rotationally induced transportassuming a rotational velocity of v = 70, 100 km/s.The corresponding evolutionary tracks are shown inFig.11. As no mass and angular momentun losses areincluded, no net torque is applied at the surface ofthe star, hence total angular momentun is conserved.In that case, only a weak circulation is taking place.It is however able to transport inward enough hy-drogen to increase the main sequence track as doesthe overshoot although with a quite different resultin the evolution in the vicinity of the TAMS.

Models were then chosen to represent the δ Scutistar FG Vir with masses adapted to keep the modelsat the same location in the HR diagram. The struc-tures of these models, as represented in Fig.11 by theBrunt-Vaissala frequency N . Although the evolutionof a rotating model in the HR diagram is close to thatof a nonrotating model with overshoot, the Brunt-Vaissala frequency inner maximum remains close tothe one of the model without overshoot nor rotation.This can have important consequences on the fre-quencies of g and mixed modes which are sensitiveto the properties of those inner regions for such astar. Fig.11 also shows the rotation profiles of therotating models. The ratios Ωc/Ωsurf differ by 14%between the two models which is enough to mod-ify significantly the oscillation frequencies of modeswhich have significant amplitudes in the vicinity ofthe rotation gradient.

As mentionned earlier, uncertainties remain in theexpressions for the transport coefficients (mainly Dh)as well as for the criterium for the onset of turbu-lence. Several works adressed these issues and gaverise to different forms for these coefficients and cri-

Figure 11. Top: Evolutionary tracks in a HR dia-gram for evolution of models (representative of FGVir): standard models i.e without rotation with nocore overshoot (blue curve) and including core over-shoot (0.2Hp) (black curve); models including rota-tionally induced mixing, no overshoot (surface rota-tional velocity v = 100km/s (red curve) and v =70km/s (green curve) at the stage labelled with a tri-angle). bottom left: Inner maximum of the Brunt-Vaissala frequency for the selected models shown withtriangles on the top pannel with the same line styles.bottom right: rotation rate normamized to a sur-face value corresponding to v = 100km/s (red curve)and to v = 70km/s (yellow curve). (from Goupil,Talon 2002)

terium (Mathis et al. 2004 and references therein).Mathis et al. (2004) studied the effect of enhancingthe horizontal turbulent transport on the evolution of1.5 Msol models compared to previous calculations.This results in increasing the vertical transport andmixing of chemical elements.

In view of future applications to Corot target stars,rotationally induced transport according to Eq.1 andEq.2 has also been recently implemented in theevolutionary code Cesam assuming transport coef-ficients given in Mathis & Zahn (2004). Moya etal (2006) computed evolutionary tracks for 1.5M�models assuming no overshoot but different othertypes of transport: 1) atomic diffusion alone 2) uni-form rotation alone (assuming conservation of totalangular momentum with evolution) 3) rotationallyinduced mixing following the approach of Mathis &Zahn (2004) 4) same as 3) with atomic diffusion in-cluded

Fig.12 shows that the tracks are not significantlymodified at the beginning of the main sequence. Theauthors then selected models which fall very close toeach other in an observational box (with classicaluncertainties on the effective temperature and lumi-nosity). Important differences betwen these mod-els can be seen on the hydrogen profile for instance(Fig.12). Atomic diffusion significantly modifies theH abundance in surface (Morel, Thevenin 2002) com-pared to the H abundance in absence of atomic dif-fusion. In turn, rotationally induced transport whichis important in the surface layers smoothens the ef-fects of atomic diffusion and decreases back the hy-drogen abundance. We expect from these struc-tural differences large differences in the oscillationfrequencies at ’zeroth order’ level (ν0,n,� in Eq. 7) forthose modes which are sensitive to surface propertiesnamely high frequency p-modes as seen in Fig.13.Relative differences between frequencies ν0,n� of areference model (ie without rotation nor atomic dif-fusion) and of models which include atomic and/orrotation are almost independent of the frequency ex-cept at low frequencies (mixed modes). Differenceswith frequencies of a model including only atomicdiffusion amount to ∼ 0.015 whereas differences aremuch larger when the model includes uniform rota-tion (with conservation of total angular momentum)with changes of about 5%. When effects of rota-tion are included, magnitude of the changes are inbetween at the level of 4%.

Finally comparisons between a (mid)main sequencemodel evolved from a homogeneous initial model anda (mid)main sequence model which is evolved withPMS calculation of models including rotation (Moyaet al. 2006) show little differences in the evolution-ary tracks and in the rotation profile. Again herewe expect nevertheless significant differences in theoscillation frequencies.

These preliminary results must be taken with caution

0.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

3.834 3.836 3.838 3.84 3.842 3.844 3.846 3.848 3.85

log

L/L

o

log Teff

HR diagram

mpzov=0

only difmpz dif0

globmodels Xc=0.5

Figure 12. Evolutionary tracks of !.5 M� main se-quence models in a HR diagram. Colors correspondto different assumptions about internal transport pro-cesses: all models are without overshoot. The trackof standard models (ie with no rotation, no atomicdiffusion) is represented in green; the track of mod-els with diffusion only is pink; the black curve cor-responds to models with atomic diffusion and uni-form rotation with time independent total angularmomentum; the red curve represents tracks for mod-els with rotationally induced transport according toMathis, Zahn, 2004 with atomic diffusion and andwithout atomic diffusion (light blue curve). (takenfrom Moya et al 2006)

as comparisons are presently being carried out withresults from another evolutionary code (STARE-VOLV) where rotationally induced transport hasbeen already implemented and tested (Palacios etal. 2003). The comparison seems to indicate thatour present implementation yields a too strong hor-izontal turbulent diffusion.

5. MODERATE ROTATORS: PERTUR-BATIVE APPROACH

Studies in this section concern modes with frequen-cies in the range where perturbation techniques arevalid that-is low frequency p-modes and high fre-quency g-modes

For δ Scuti stars with � = 0.014−0.14 and µ = 0.01−0.2 for instance, it is necessary to include O(Ω2, Ω3)order corrections due to rotation when computingoscillation frequencies.

5.1. Moderate uniform rotation:

Using SGD98 approach, Goupil et al (1999) investi-gated the effects of a moderate rotation on the os-cillation frequencies of a typical δ Scuti star modelwith 1.8M�, logTeff = 3.876, a photospheric radiusof 2.125R� v = 10 to 100 km/s. From such a study,several conclusions were drawn:

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X

x=r/R

mpz

mpz dif 0

only dif

glob

ov=0

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

200 400 600 800 1000 1200 1400 1600 1800 2000(σ

ov0-σ

X)/σ

ov0

Frequency in µHz

Frequency comparison with respect no rotation nor diffusion

mpz dif0only dif

globalmpz

Figure 13. top: Hydrogen profiles of Xc = 0.5models of Fig.12. Bottom: Comparison betweenfrequencies for = 2, m = 0 modes computed fora reference model (assuming no rotation, no over-shoot, no atomic diffusion) and frequencies of modelscomputed with different assumptions about internaltransport processes (from Moya et al 2006)

The ratio µ2/�2 ∼ 1 − 15 rapidly decreases with in-creasing frequencies (σ >> 1) and distorsion due tothe centrifugal force dominates over the Coriolis ef-fect for high frequency p-modes while it is the oppo-site for low frequency g-modes.

The aspherical distorsion shifts all frequencies by aquantity (Ω2s/ω0)Dn,�,m which depend on m, for ax-isymmetric modes .

For high frequency p-modes, one can neglect effectsof horizontal displacement in front of the verticalones. Using this approximation in expressions forDn,�,m (which can be found in DG92, SGD8, Karamiet al. 2005, Suarez et al. 2006), one finds

Ω2s2πω0

Dn,�,m ∼ Q�,m Vn,� (10)

where the geometrical factors are given by:

Q�,m =Λ − 3m2

4Λ − 3Q�,�+2,m =

32

β� β�+2 (11)

with

Λ = ( + 1) β� =( 2 − m2

42 − 1

)1/2(12)

The Vn,� coefficient can be approximated as

Vn,� ∼ �2 ω0 < S2 > (13)

where

< S2 >=1I

∫ R0

y2n (2rdu2dr

+ u2) ρ r2 dr

with yn,� the relative fluid displacement eigenfunc-tion in Eq.?? (defined as y1 in Unno et al.1989).The quantity u2 is related to the perturbation ofthe gravitational potential by rotation, φ22 u2 =φ22/(r2Ω2s) + (1/3)(Ω/Ωs)

2. φ22 is defined byφ(r, θ) = φ(r) + φ22(r)P2(cos θ) is solution of a per-turbed Poisson equation, P2(cos θ) is the 2nd orderLegendre polynomial (DG92, Soufi et al 98) Thestructural quantity < S2 > is nearly model inde-pendent (D/3 of Fig.8 in Goupil et al. 1999).

For two degenerate modes labelled a = (n, ) andb = (n′, + 2), one can assume Va ∼ Vb ∼ Vab asyn,� ∼ yn′,�+2 in the outer parts which contributemost to Vab. Hence one approximately has:

Hab,m ∼ Q�,�+2,mVa (14)

For high radial order modes, the coupling coefficientHab,m, like Dnlm, increases linearly with the fre-quency. Degenerate coupling is found to dominatefor v > 50km/s.

A measure of the asymetry of a multiplet is given by:

∆ν = νn,�,m=0 −νn,�,m+νn,�,−m

2

= −ν0�2 6m2

4Λ−3 < S2 > (15)

The asymetry in the multiplet also increases linearlywith the frequency ν0 and the squared rotation rateΩ2.

Cubic order effects are at the relative level of 10−2−10−3 that-is that they are non negligible at the levelof Corot accuracy for moderate rotating δ Scuti starsand more important for non axisymmetric modesthan centroid modes.

Using Eq.7 in Eq.5, the generalized splittings for non-degenerate modes take the form:

δm = mΩs(Cn,� − 1) + mΩ3sν20

Tn,�,m (16)

In absence of cubic effects, the quantities dm, (m =1, ) such that

dm =δm

(Cn,� − 1)(17)

are the same and yield the true rotation rate. Whencubic effects are nonnegligible, the correct rotationvalue cannot be recovered directly from Eq.17. How-ever it is possible to combine the different values ofdm so as to eliminate the high order perturbationsand to recover the rotation rate (Sect. 5.3 below).

Figure 14. Rotation frequency through the star nor-malized to its surface value. The continuous line rep-resent a shellular rotation profile and the dashed lineto a uniform rotation profile. Models are built withthe same surface rotation frequency Ωs = 9.98 µHz.Figure taken from Suarez et al. 2006.

For a given mode, avoided crossing, which is a signa-ture of the inner gradient of Brunt-Vaissala frequency(due to µ gradients in the vicinity of a convectivecore), arises at cooler temperatures for rotating mod-els than for no rotating models. of same mass andage (Goupil, Talon 2002).

Difficulties in identifying the modes of δ Scuti starscan be somewhat overcome in using histogram fre-quencies and statistical analyses, provided that alarge number of modes are detected as expected withCorot data. However for moderate and fast rotatingstars, histograms of frequencies are strongly affectedby rotation as shown with the case of FG Vir andfor high rotation rates the peaks representative ofthe rotation at low frequency is smeared out when vreaches 70-100 km/s (Goupil et al. 1999)

Finally frequencies of a given multiplet obey a nearresonance of the type ∆ν ∼ 0 (Eq.15). Such a res-onance can be responsible for a nonlinear couplingwhich can cause the amplitudes of a given multipletto be asymetric and enforces This effect seems to bedistroyed for fast rotation and therefore would ex-ist in a limited range of rotation rate; amplitudesthen become time dependent over a time scale whichcan be much shorter that the Kelvin Helmoltz time.This is one of the important issues we expect Corotto provide clues about.

5.2. Shellular rotation

The effect of radial differential (shellular) rota-tion on adiabatic oscillation has been studied forintermediate-mass rotating stars (δ Scuti stars,γDor) in the framework of the preparation of theCorot mission (see CW4) (Suarez et al., 2006) fora typical δ Scuti star. Equilibrium models are builtwith the evolutionary code CESAM (Morel 1997).Rotation is included by considering an effective grav-ity modified by the effects of the centrifugal force.

Figure 15. Weighted radial displacement eigenfunc-tions for a mixed mode (top) and a p mode (bottom)in function of the radial distance r (normalized tothe radius of the star R). Solid and dashed lines rep-resent the f = ρ r4 y201 functions computed for a dif-ferentially rotating model with ρ the density. Dash-dotted lines represent the f function computed fora uniformly rotating model. Dotted lines representthe rotation profile given by the radial function η0(r)(the scale has been adapted for sake of clarity) (takenfrom Suarez et al. 2006)

Figure 16. The plot displays mode-to-mode frequencydifferences between differentially and uniformly ro-tating 1.8M� models. Frequencies include near de-generacy corrections. Symmetric solid line branchesrepresent from top to bottom, differences for m = −1and m = +1 mode frequencies respectively. Form = 0 modes, differences are represented by a dot-ted line. In both panels, the shaded region representsan indicative frontier between the region of g and gpmodes (left side) and p modes (right side). Figuretaken from Suarez et al. (2006)

That is, only the spherically symmetric terms areconsidered. This is known by pseudo-rotation as iscalled in SGD98. Although the non-spherical com-ponent of the deformation of the star is not con-sidered, its effects are included through a perturba-tion in the oscillation equations (Suarez, 2002 PhD).Rotationally-induced mixing and transport of angu-lar momentum as described in previous sections arenot included in the models used here. Two illustra-tive cases when prescribing the rotation profile Ω(r):1) either instantaneous transport of angular momen-tum in the whole star (global conservation) whichthus yields a uniform rotation, or, for sake of sim-plicity and illustrative purpose, 2) local conservationof the angular momentum (shellular rotation). InFig. 14 both cases are depicted for two 1.8 M� mod-els with the same rotation velocity ( 100 km s−1) atthe stellar surface and very close in the HR diagram.

Adiabatic eigenfrequencies were computed followingthe perturbation formalism of DG92 and SGD98. upto second order in the rotation rate Ω allowing a ra-dial dependence of Ω = Ω(r). The effect of shel-lular rotation on the eigenfunctions is illustrated inFig. 15, for a high radial order p mode (Fig. 15b) andfor comparison in Fig. 15a for a mixed mode. As canbe seen, for gp modes, the eigenfunctions present aninner maximum near the core r/R ∼ [0, 0.2]. As thekinetic energy of these modes are large in core-closeregions, large seismic differences between the uniformand the shellular rotation cases are reasonably ex-pected. For high radial order p modes, implicit ef-fects of shellular rotation on eigenfunctions are repre-sented by dashed lines and can be compared to those

Figure 17. Recovering the rotation profile in the caseof a moderately fast rotating 1.8M� star. A uniformrotation has been assumed corresponding to a value15.3 µHz (a rotational velocity of 144.5 km/s). Theblue curves represent d1 and d2 (Eq.18) for = 2multiplets and the red curve is d1 + d2/2. The largeexcursion of d1 and d2 are due to the presence ofmixed modes in this frequency range

for a uniform rotation. Maximum amplitudes of suchmodes are located near the stellar surface, where theeigenmodes do not differ much between the uniformand the shellular cases.

5.2.1. Effect of shellular rotation on thefrequencies

Figure 16 displays = 1 mode to mode frequencydifferences between frequencies computed assuminga uniform rotation and a shellular rotation (with aratio Ωcore/Ωsurf ∼ 2). Both models have the samesurface rotation velocity v = 100km/s. Frequenciesare computed according to (Eq.9). Differences can belarger than 1 µHz. Large effects due to near- degener-acy can be seen for gp modes, and for high-frequencyp modes, which can reach up to 3 µHz. For the for-mer modes, such effects are expected because therotation profile rapidly varies in the inner layers atthe edge of the convective core where gp modes havelarge amplitudes (see Fig. 15. For high-frequency pmodes such effects can be explained by the structurespherical deformation caused by the centrifugal force,which mainly affects the zeroth-order oscillation fre-quencies. Indeed although high frequency p-modeshave small amplitudes in the vicinity of the core,the rotation variation which is significant only in thevicinity of the core is felt by these modes stronglyenough to modify significantly their frequencies. Itis found that, although only a few triple-mode (orhigher) interaction is present in moderately rotat-ing δ Scuti stars the effect of taking such interac-tion into account instead of double interaction canbe quite different depending on the configuration ofthe modes (closeness of the frequencies). We find

Figure 18. The reduced 2nd order coefficient D1/ω0(Eq.19) decreases toward a constant with increasingradial order n for modes with different degree (solidline: Coriolis force is included; dashed line: Coriolisforce is not included). At high frequency, as Cori-olis effects are small and effects of the centrifugalforce dominate, differences between both calculationsare quite small. This is less true at lower frequencywhere Coriolis plays a nonnegligible role (taken fromReese et al. (2006)

that the effect can be occasionally amount up to afew µHz.

Hence, with such an accuracy, effects of shellular ro-tation are likely to detectable with Corot data, pro-vided numerical eigenfrequencies reach this level ofprecision.

5.3. Recovering the rotation profile frommoderately rotating star oscillation fre-quencies

As outlined by Dziembowski & Goupil (1998) andGoupil et al. (2004), it is possible to remove con-taminations of high-order (second and higher) effectsof rotation from the rotational splitting in order torecover the true rotation profile. Using Eq.16, split-tings with different m are combined in order to elim-inate cubic order pollution and to recover the rota-tion profile. Goupil, Talon (2002) illustrate the casewith a 1.8M� model with a uniform rotation corre-sponding to 15.3 µHz. The plot shows the behaviorof frequency difference dm (Eq.17 above) for m = 1and m = 2 = 2 modes in function of the associatedm = 0 mode. The red curve δ1 + δ2/2 is found to benearly uniform and give the value of the input uni-form rotation rate. This will be efficient for modeswhich exist in nearly uniform rotating regions.

Figure 19. Relative differences between nonpertur-bative frequencies and perturbative frequencies com-puted according to Eq.19 for modes with radial or-der n = 6, 10 (dotted lines). Relative differencesbetween nonperturbative frequencies and frequenciescomputed using a truncated formulation involving 2poloidal and 2 toroidal modes (solid lines). These rel-ative differences are plotted in function of increasingrotation rate Ω/ΩK with ΩK the break up rotationrate. from Reese et al 2006

In case of non uniform rotation or when the rotationis fast enough that all modes are degenerate, it ismore difficult to remove high order perturbative ef-fects of rotation and it is likely that one will have touse some a priori simplified rotation profiles or someiterative processes based on

δm =∫ R

0

drK1(r) Ω(r) +∫ R

0

K2(r)Ω3(r) (18)

6. RAPID ROTATION : NON PERTUR-BATIVE APPROACH

As already mentioned, Lignières et al. (2006) andReese et al. (2006) have studied the effects of fast ro-tation for a self-gravitating uniformly rotating poly-trope of index 3 with a rotation rate taken in a rangefrom 0 to 0.59 ΩK (� = 0 to 0.35)

6.1. Effets of the centrifugal force

The centrifugal force induces a decrease of the fre-quency because a rotating star has a larger volumeand a smaller sound speed. As found by Lignières etal (2006), this decrease is in average proportionalto the frequency, the rate of decrease being closeto that of the mean density. Nevertheless, besides

Figure 20. Evolution of the structure of a given mul-tiplet (here = 1, n = 4) when rotation is increased.For ω/ΩK > 0.2, the differences δω is larger thanthe value of the splitting itself and the m �= 0 are notaffecetd the same way. (from Reese et al. (2006))

this average behavior, the frequency decrease is and n dependent and this differential effect modifiesthe structure of the frequency spectrum. It followsthat the structure of a non-rotating frequency spec-trum is destroyed. In particular, the small differenceνn,� − νn−1,�+2 is no longer small above a certain ro-tation rate. Actually this quantity befomes nearlyconstant and characterizes together with the largedifference νn,� − νn−1,� the new structure of the fre-quency spectrum at large rotation rates.

Rotation also affects the properties of mode visibil-ity since the angular dependence is no longer givenby a unique spherical harmonic (SGD98, Daszynska-Daszkiewicz et al., 2002). Lignières et al (2006) showthat the amplitude of the low frequency p-modes pro-gressively concentrates near the equator as rotationincreases. This effect strongly modifies the variationof the disk-averaging factor with the degree of themode. In a non rotating star, the cancellation be-tween positive and negative surface perturbations re-sults in a strong decrease of the disk-averaging factorwith the degree number. This property does not holdin rapidly rotating stars where the equatorially con-centrated modes have similar surface distribution.

6.2. Comparison between full and perturbativecalculations

An important issue is to establish the domain ofvalidity of perturbative calculations. Accordingly,Reese et al (2006) compared results from full 2D cal-culations and from perturbation calculations. Thelast ones were carried out assuming that frequencies

take the form

ωn,�,m = ω0,n,� + mΩ(Cn,� − 1)+(D1 + m2D2)Ω2 + m(T1 + m2T2)Ω3 (19)

with ω = 2πν. The coefficients D1, D2, ... were de-termined by a least squared fit from results of full 2D calculations assuming low rotation rates. Includ-ing both the centrifugal and Coriolis force, Reese etal (2006) computed frequencies of = 0 to 3 andradial order n = 1 to 10. The studied polytrope cor-responds to a 1.9M� and a polar radius of 2.3 R�typical of a δ Scuti star. Fig.19 shows the relativedifferences δω/ω = ω(full) − ω(pert,3)/ω in functionof the rotation rate normalised to the break up rota-tion rate ΩK . ω(full) are frequencies obtained witha full 2D numerical calculation and ωpert,3 frequen-cies computed up to third order according to Eq.19above. The differences are seen to increase with therotation rate and the radial order. They are mainlydue to the centrifugal force as, as mentionned earlier,Coriolis force has a negligible effect at high frequen-cies. The differences become quite significant whenΩ/ΩK < 0.1 (� < 0.01)

Here significant means larger than the measurementaccuracy of rotational splittings expected with Corotdata. The comparison shows that the full calculationis required for vsini > 50km/s for Corot 150 daysaccuracy.

Fig.20 shows the evolution of the structure of a givenmultiplet (here = 1, n = 4) when rotation is in-creased. For ω/ΩK > 0.2, the differences δω arelarger than the values of the splittings themselves.The m and −m components are not affected the sameway.

Fig.21 displays the frequencies computed whenadding successively higher order corrections accord-ing to Eq.19. At zero rotation rate, the mode is mdegenerate. At small rotation rates (first order cor-rection in green) nice evenly-spaced multiplets can beseen. At higher rotation rates, multiplets stars to beno longer equally spaced and multiplets start to over-lap (second order corrections in blue). At even higherrotation rates, the initial structures within the mul-tiplets and between multiplets are lost (third ordercorrections in red; full nonperturbative calculationsin black)

In Fig.19, black lines represent relative differencesbetwen frequencies computed with a full nonpertur-bative calculation (with Lmax ∼ 80 in Eq.??) andfrequencies obtained when truncating the summa-tion to Lmax = 2 in Eq.?? that is includind twopoloidal and 2 toroidal contributions. The relativedifferences become significant for slightly larger ro-tation rates than when a simple perturbation expres-sion, Eq.19, is assumed. For instance, a relative fre-quency difference of is reached for � ∼ 0.18 when a

Figure 21. Plot of pulsation frequencies as a func-tion of the rotation rate Ω/ΩK. Modes have radialorders n = 1 to 10 and degrees = 0 to 3, m = −lto l. The top horizontal axis represents the flatten-ing 1 − Rp/Re where Rp is the polar radius and Rethe equatorial radius. Green curves indicate the do-main of validity of 1st order perturbative calculationsthat-is allowing a 0.6 µHz differences (which corre-sponds to a Corot 20 day observing program) betweennonperurbative and perturbative calculations . Blue(resp. red) curves indicate the domain of validityfor 2nd order (resp.3rd order) perturbation calcula-tions. Black curves corresponds to the nonpertur-bative frequencies. (from Reese, 2005 at 2nd CorotBrazil workshop)

perturbation (Eq.19) is assumed and for � ∼ 0.25when a truncated summation is used. To fix ideashowever, for typical frequencies of excited modes ofδ Scuti stars in the range 500 − 1000µHz a relativedifference of 2.510−3 corresponds to a frequency dif-ference of 1.25− 2.5µHz significantly larger than theaccuracy expected on splittings measurements withCorot data. Note that the truncated calculationscorrespond to pertubation calculations when system-atically including near-degeneracy (Eq.9).

6.3. Conclusion

It is quite reassuring that a number of qualitativefrequency characteristics found when using pertur-bation calculations subsist when rotation is real fast.Modes with a high radial order n are less affected bythe Coriolis force than by the distorsion due to thecentrifugal force is the main cause for deviations be-tween frequencies in non-rotating stars, frequenciesbased on the perturbative approach and thise basedon the non-perturbative approach.

Quantitatively, the non-perturbative approach tellsus the limit of validity of perturbative methods. Inthe present case , the comparison with Corot mea-surement precision shows that perturbative resultsare no longer valid for Ω/ΩK > 0.2 for radial ordern ∼ 6 and higher and for Ω/ΩK > 0.3 for radialorder n ∼ 1, 2. For a rotation rate of 0.59ΩK (cor-responding to a rotational velocity of 240 km/s for1.8M�, 2R�), the frequency spectrum stemmed froma nonperturbative calculation is very different thanthe frequency spectrum obtained from a perturbativecalculation.

Finally, we list some Corot selected stars which arerotating fast and which quantitative seismic studieswill require nonperturbative calculations.

Rapidly rotating primary targets for CoRoTStar identification Stellar type vsini (in km s−1)HD 181555 δ Scuti 170HD 49434 γ Doradus 90HD 171834 F3V 72

Some of CoRoT’s rapidly rotating secondary targetsStar identification Stellar type vsini (in km s−1)HD 170782 δ Scuti 198HD 170699 δ Scuti > 200HD 177206 δ Scuti > 200

The vsini measurements are based on Poretti et al.,2005 and Corotsky.

Part IV Solar like oscillators : are theyreally slow rotators ?

Stars of spectral type F,G,K are usually consideredas slow rotators. Custipoto et al. (2002, 2003) havedetermined values of vsini of a large sample of F,G,Kstars in the solar neighborhoud. The stars were se-lected specifically for their high v sin i values. Rein-ers & Schmid (2003) have also obtained values ofv sin i for a sample of F-G-K type PMS and MS stars.These stars have often v sin i exceeding 15 km/s. Asseen in Fig.1, for F,G,K stars, the bulk of stars isabout 10 km/s- 15 km/s but the tail extends up to30 km/s, the tail is more pronounced for K stars.These measurements show that solar like stars areslow rotators in comparison with more massive (A-B) stars but are often rotating more rapidely thanthe Sun.

Solar like oscillation high frequency p-modes are ex-cited by stellar outer convective regions. For a grow-ing number of solar like stars, high frequency p-modes have indeed been detected and in some caseswith an enough large number of frequencies that thefirst seismic models have been built for α Cen A, B,

Procyon, η Boo (see references section). No split-ting have been detected for solar like oscillating starsother than the Sun. We expect that some Corottarget stars rotate fast enough that their rotationalsplittings will be detected, This means a 20−30 km/sfor a 1.5 M� (Goupil et al, this volume; Lochard etal 2005)

7. INFLUENCE OF ROTATION ONTHE SMALL SEPARATIONS ANDECHELLE DIAGRAMS

Even though solar-like oscillators are considered asslow rotators, the m = 0 mode frequencies are af-fected by rotation for these stars (Goupil et al 2003).Lochard et al. (2006) show that the reppelling ef-fect of near degeneracy (Fig.9) can significantly affectthe frequency separation of modes and + 2 withsuccessive radial order as they have close frequen-cies which is referred to as small separation. Thishas been stressed by SGD98, Dziembowski (1997),Dziembowski, Goupil (1998). The small separationis often used to characterise the evolutionary stage(age) of the star. The small separations between = 0, 2 and that between = 1, 3 become so smallthat rotation, even at a modest rate of 15 km/s, leadsto significant degeneracy effect.

The echelle diagram is useful to identify the valueof modes associated with the detected frequencies.The echelle diagram is also affected by rotation tothe same extent that the small separations are. Thisis illustrated in Fig.22 for 3 different values of the sur-face rotational velocity. Deviations from nonrotatingmodel echelle diagram can be seen for v > 25km/sparticularly at high frequency. When only m = 0modes are used, the effect is not large enough tolead to an error in mode identification. However de-viations from echelle diagram for non rotating starmust be taken into account when fits of the frequencycurves in echelle diagram are determined in order tobuild a seimic models or when attributing the de-parture between observed and theoretical curves toa physical deficiency of the models such as turbulencein the outer layers (Straka et al. 2006).

At v = 50km/s the effects are huge. However,the corresponding �, µ tells us that the perturbationtechnique is probably no longer valid for computingthe oscillation frequencies in that case and one mustturn to techniques for rapid rotators (as mentionnedSect.6)

8. RECOVERING THE SMALL SEPARA-TION FREE FROM ROTATION CON-TAMINATION

It possible to remove most of the contamination ofhigher order effects of the rotation for high frequencyp-modes by combining suitably the splittings pro-vided they are available. For solar like stars, rotationis indeed small enough that cubic order frequencycorrections in Eq.7 are negligible. For high frequencyp-modes, as is the case of solar like oscillations, thehorizontal displacement is negligible in front of thevertical one. Using the approximate expressions forDa,m, Db,m and Hab,m given in Eq. 10-14, it can beshown (Goupil et al. 2003) that the ’true’ small sepa-ration δn,� ≡ ν

(Ω=0)a −ν

(Ω=0)b for m = 0 modes - that

is the small separation free from rotation pollution -is given by

δn,� ∼((νobsa − νobsb )

2 − 4 Q2�,�+2,0V2a

)1/2−(Q�,0 − Q�+2,0) Va (20)

where the geometrical factors are given in Eq.11.Hence the small separation increases linearly withthe frequency and with Ω2.

The structure quantity Va ∼ Vb itself can be obtainedfrom the observed frequencies as

Va = 3(sa,2 − sa,3)

withsa,m =

12

(νobsa,m + ν

obsa,−m

)

for mode a = (n + 1, = 3) and mode b = (n, l = 1)for instance.

Note that we remove here the contamination of thefrequency due to direct influence of rotation on thewave propagation hence the oscillation. The smallseparation still includes the indirect effect of rotationon the structure which is precisely what we wantto probe. Also Va is a measurable seismic quantitywhich can be inverted to provide information aboutthe distorted structure.

Goupil et al (2003) condider the case of a 1.4M�with solar chemical composition, log Teff = 3.810with rotation velocity v = 10 to 40 km/s (includedfor the spherical distorsion mean centrifugal force viaan effective gravity). Modes are considered as sys-tematically degenerate. Frequencies including rota-tion effects according to Eq.9 for modes = 0 to 3with radial order n = 6 − 25 have been computed.For this model, rotation increases the small sepa-ration by 1.2 µHz (Fig.23). From this value, onewould then deduce that the star is younger than inreality by 1 Gyr when plotted in the mean smallseparation- mean large seperation diagram as pro-posed by Christensen-Dalsgaard (1993).

Fig.23 shows the small separations for modes = 1, 3computed without including rotation effects (= ’true’small separation) and including rotation effects forvelocities 10,20,30 35 km/s respectively. Departureof the curves from the zero velocity one increaseswith the rotation rate Ω. From the ’polluted ’ smallseparation ν(obs) − ν(obs), one computes the ’true’small separation according to Eq.20. Fig.23 showsthat the process is very efficient in recovering thesmall separation free of rotation effects.

9. CONCLUSIONS

If one compares with results of heliseismology, seis-mology of other stars has not yet been so successfulexcept for white dwarfs and a few cases of main se-quence stars. There are several reasons for this:

• Stellar parameters such as mass, age, chemicalcomposition are not known with enough preci-sion and must be considered as free parameters.These free parameters must be determined withenough precision on the basis of the seismic con-straints. Only in that case, it can be possible toprobe the internal physics of the star.

• For solar-like stars (α Cen, procyon, η Boo)there are still uncertainties on the frequency setand/or mode identification although they can beidentified to some extent with techniques suchas echelle diagrams. However the excited modesare low degree, high freauency that-is quite re-dondant.

• For stars other than solar like stars, excitedmodes are difficult to identify among the pos-sible linearly unstable modes for a given star.Often, their mixed p-g nature and the effect offast rotation complicate the frequency pattern.

Hence seismic studies for stars other than the Sunmust be carried out with different tools than for theSun. These problems will also exist for seismologyof Corot stars. Problems and prospects related torapid rotation was the topic of the present paper.It is important to realize that the classification ofslow, moderate or fast rotation, when oscillations areconcerned, depends on the rotation rate of the starand on the frequency range of its excited oscillations,that-is on the type of stars.

For instance intermediate mass stars such as δ Scutistars, have their excited frequencies and rotationrates which yield � in the range 0.1-0.5. Fast rota-tion corresponds approximatively to �max ∼ 0.2−0.3The region they fall in the � − µ diagram (Fig.24)shows that for the lowest rotating δ Scuti stars rota-tion can still be treated as a perturbation provided

higher order contributions (O(Ω2), O(Ω3) and near-degeneracy (or truncated formulation) rather thansimple perturbation approximation are included.For the fatstest rotators, on the other hand v >150km/s, a non perturbative approach is necessaryfor quantitative comparison with observations.

For SPB and γ Dor stars, although their ritationrate is small, excited modes are low frequency g-modes. The consequence is that whereas the cen-grifugal force can be neglected, µ is large (Fig.24)and the effect of the Corioilis force cannot be treatedas a perturbation (Dintrans, Rieutord, 2000)

For many solar like stars, the rotation is expectedto be slightly larger than for the Sun but they arestill rotating much slower than A-F stars. Excitedmodes of solar like oscillators are high frequency p-modes which are confined to the surface where therotation effects are larger. Strictly speaking, val-ues of � and µ are small for these stars. One musthowever keep in mind that distorsion effects due tothe centrifugal force linearly increase with frequencyand become significant for high frequency modeswhen the required accuracy on the computed fre-quencies must match that of (CoRoT) observationsFor these stars, higher order frequency correctionsdue to rotation must be taken into account when� > �max ∼ 0.035 − 0.04 (for a 1.4M�). µ is toosmall and corrections due to Coriolis are negligible.As shown in Fig.epsmu, a few stars are located atthe limit � ∼ 0.035 − 0.04. As the � − µ values havebeen obtained for a lower limit of the rotation rate(ie v sin i), it is likely that for some of them includinghigh order frequency corrections due to rotation willbe necessary.

The hope is that seismology will help to detect local-ized rotation gradients and a measure of their mag-nitudes. This will be even more efficient if infor-mation about the surface rotation rate is availableindependently. A large amount of work remains tobe done before seismic studies can be as fruitful asexpected: deeper studies about effect of rotation onmode visibility, mode identification, mode excitationand mode selection; studies of fast rotation for real-istic stellar models, quantitative studies of PMS evo-lution of the rotation profile and transport of angularmomentum to name a few.

Acknowledgements: Most of the work described inthe present paper has been financially supported bythe french PNPS (Programme de Physique stellaire).We also thank the french-spanish cooperation for fi-nancing visit exchanges.

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