Astron. Astrophys. 342, 153–166 (1999) ASTRONOMYAND

ASTROPHYSICS

Seismology ofδ Scuti stars in the Praesepe cluster

I. Ranges of unstable modes as predicted by linear analysis versus observations

E. Michel1, M.M. Hern andez2,?, G. Houdek3,4, M.J. Goupil1, Y. Lebreton1, F. Perez Hernandez2, A. Baglin1,J.A. Belmonte2, and F. Soufi1

1 Observatoire de Paris, DASGAL, CNRS URA 335, F-92195 Meudon, France2 Instituto de Astrofısica de Canarias, E-38200 La Laguna Tenerife, Spain3 Institut fur Astronomie, Universitat Wien, A-1180 Wien, Austria4 Institute of Astronomy, University of Cambridge, Cambridge, CB3 0HA, UK

Received 15 July 1998 / Accepted 23 October 1998

Abstract. We consider here a representative set ofδ Scuti starsobserved in the Praesepe cluster, thanks to recent efforts ofmulti-site networks. We show that the effect of rotation on thedetermination of fundamental parametersMbol andTeff cannotbe neglected when modelling fast rotators asδ Scuti stars. Wepropose a method to take this effect into account and gain someindication on the rotation rate of the stars.

We then use this unique homogeneous set of observationaldata to investigate the pulsational behaviour ofδ Scuti stars fordifferent masses, corresponding to different evolutionary stagesat the age of the Praesepe cluster. We propose here an alternativeapproach to seismology ofδ Scuti stars, trying to avoid specificbut questionable mode identifications. Via a large set of models,we compare the observed modes with the unstable modes asobtained from a linear stability analysis including a nonlocaltime-dependent treatment of convection.

We show that, in the framework of the linear analysisadopted here, a satisfying agreement can be reached between thepredicted ranges of unstable modes and the ranges derived fromobservations. However we found that for “low-mass”δ Scutistars (1.6-1.80 M�) the agreement is found only for a restrictedrange of values of the mixing length parameter. These stars thusbring a valuable observational constraint on the description ofthe convection in the outer layers of stars.

Key words: stars: oscillations – stars: rotation – stars: vari-ables:δ Sct – galaxy:open clusters and associations: individual:Praesepe

1. Introduction

Poor understanding and treatment of the transport mechanismsis currently identified as the main source of uncertainty in mod-elling intermediate mass stars (that is between1.5 and3 M�),in the Main Sequence phase and up to the red giant branch. Thisgoes mainly for the transport of heat and chemical species at

? Also associated scientist at institute 1

the edge of the convective core during Main Sequence, usuallytreated by a rough description of convective overshooting (forquantitative insight, see Maeder & Mermilliod 1981), but thetransport of angular momentum and potentially associated mix-ing must also be investigated as suggested by various surfaceanomalies, both in the giant phase and in the Main Sequencephase (see for instance Pinsonneault 1995).

Seismology ofδ Scuti stars is a very promising tool in thisrespect, sinceδ Scuti are representative of normal A-type starsin this range of mass, on the Main Sequence and in the rapidphase of broad shell hydrogen burning.

Great efforts have been made within the last decade on boththe observational and the theoretical aspects of the seismologyof δ Scuti stars. World-wide observational campaigns are nowregularly organized (see Michel et al. 1995, Breger & Audard1997, Mantegazza & Poretti 1994 and references therein) to ob-tain high quality long time series. New techniques have beenbrought in, like CCD rapid photometry (Kjeldsen & Frandsen1992, Gilliland et al. 1993), or line profile variations (Kennelly1994). On the theoretical part, the physics of the models hasbeen frequently updated. New physical processes have beenproposed: internal waves (Schatzman 1996), meridian circula-tion and rotational momentum transfer (Zahn 1992, Talon et al.1997). New numerical tools have been developed (Morel 1997,Morel & Schatzman 1996). An extensive review of the presentstate-of-the-art in stellar modelling can be found in Baglin &Lebreton (1995) and references therein.

The connection of these two parts presents an importantpotential scientific return. This is very well illustrated by the al-ready published tentative interpretations (Mangeney et al. 1991,Goupil et al. 1993, Perez Hernandez et al. 1995, Audard et al.1995, Guzik & Bradley 1995). In order to be fully exploitedhowever, this connection requires the identification of the de-tected modes and this remains the weak link of the chain. Itis generally possible to produce a consistent identification, butit is hardly unique. The relevance of the conclusions thereforeremains limited.

154 E. Michel et al.: Seismology ofδ Scuti stars in the Praesepe cluster. I

The problem comes from the complexity of the oscillationspectra ofδ Scuti stars. Despite all the progresses already madein this kind of observations, which reveal more and more de-tected frequencies (Breger et al. 1997), these spectra remain“lacunaires” (some modes are lacking), at least at the presentdetection level. Observations from space with COROT (Baglinet al. 1998) might change this view. These characteristics couldbe explained by non-linear coupling between modes (Buchler etal. 1995, Buchler et al. 1997), but, at the moment, no selectionrule has been noticed, which could help mode identification.

In addition to this, we know now that the simple characteris-tic pattern of symmetric multiplet split by rotation does not holdfor these stars. Their fast rotation (commonly100 km s−1 ≤V sin i ≤ 200 km s−1) makes this first order description of therotational splitting no longer sufficient. Second order effects in-duce strong asymmetry in the splitting of multiplets (Saio 1981,Dziembowski & Goode 1992) and shifts which cannot be ne-glected even for radial modes (Soufi et al. 1995).

Observational indications on mode identification are notconvincing as they are, mostly due to insufficient signal-to-noiseratios. They however constitute a good potential way to solvethis problem and recent efforts on this aspect of the observationsare very encouraging (Mantegazza & Poretti 1996, Kennelly etal. 1996, Viskum et al. 1998).

At the present time, it is thus very important to select objectsfor which we have complementary informations, as binaries orstars in clusters. The study ofδ Scuti stars in the same cluster al-lows to consider that all the objects have the same age, chemicalcomposition and distance, to a good extent. In addition to this,the parameters like metallicity or distance are better estimatedfor stars belonging to a cluster than for typical field stars.

The present work has been realized in this context. We intendto obtain reliable information on the pulsational behaviour ofδScuti stars, without assuming any identification of the modes.

We first introduce an effect of fast rotation hitherto neglectedin the seismological investigations ofδ Scuti stars. We show thatit has to be taken into account when estimating fundamentalparameters and we propose a method to take this effect intoaccount and obtain an estimation ofV andi, for δ Scuti starsin clusters. We then apply this method while studying sixδScuti stars of the Praesepe cluster. Varying different parameters,including rotation, we determine domains of possible values forradial eigenfrequencies associated with these stars.

Finally, we use this unique set of homogeneous data to con-front unstability predictions as derived from new linear growthrates calculations with observations.

This work set the stage for further seismological work (seeHernandez et al. 1998c, hereafter paper II).

In Sect. 2, we present the observational material as well asthe models, the calculation of oscillation frequencies and growthrates. In Sect. 3, we describe the effect of fast rotation on the de-termination of luminosity and effective temperature. In Sect. 4,we establish the link between our models and observations, viaa fit of isochrones. Sect. 5 contains the comparison of modelsand observations in terms of range of radial order of detectedmodes. Sect. 6 is devoted to the results of growth rates calcula-

Table 1. Observations for theδ Scuti considered here:V sin i, fre-quencies detected with a detection confidence level better than90%.Typical precision on frequencies:±0.2 µHz. The peak at68.3 µHz issubject to caution according toAlvarez et al. (1998).

star ν Amp. star ν Amp.(µHz) (mmag)

V sin i V sin i(km s−1)

BW 68.3∗ 1.4 BS 179.7 2.7170 138.7 1.2 135 197.2 6.1

139.1 1.6 396.2 1.4171.8 2.4223.0 1.0 BU 193.3 2.1260.6 1.0 205 195.1 3.4261.5 0.8 200.9 3.3307.2 1.6 215.5 3.8361.2 0.5 228.8 2.5

229.9 2.9

BN 263.6 1.6 EP 63.3 3.8130 266.5 1.8 146 67.0 1.8

279.5 1.2 75.4 2.3298.1 1.8300.2 1.7 BT 81.2 8.3327.2 2.9 154 113.2 10.6

tion. A comparative discussion of the results of Sect. 5 and 6 isgiven in Sect. 7. Sect. 8 is devoted to the conclusions.

2. Observations and modelling

2.1. The observational material

We consider here sixδ Scuti stars of the Praesepe cluster. Forfive of them, the observed oscillations frequencies have beenobtained from campaigns of the STEPHI network (Michel etal. 1995): BU Cnc and BN Cnc (HD 73576 and HD 73763, Bel-monte et al. 1994), BW Cnc, BS Cnc and BT Cnc (HD 73798,HD 73450 and HD 73575, Hernandez et al. 1998a, see alsoAlvarez et al. 1998, and Hernandez et al. 1998d). The sixth starEP Cnc has been observed and analyzed by Breger et al. (1994).

Frequencies and amplitudes are given in Table 1.The Geneva Photometric System catalog (Rufener 1988)

has been used to obtain cluster membership, values of JohnsonV magnitude and (B2-V1) color index in the Geneva system forall the stars considered here.

The V sin i values are from Centre de Donnees As-tronomiques de Strasbourg. They are included in Table 1, forthe sixδ Scuti stars considered here.

Cayrel de Strobel et al. (1992) determined a value of0 <∼[Fe/H] <∼ 0.2, which corresponds to a Population I cluster withmetallicity between solar (Z = 0.019) andZ = 0.03.

For confrontation between the observations and models, cal-ibration by Hauck (1984) and by Schmidt-Kaler (1982) wereused to deriveTeff andmbol respectively. Following Crawford& Barnes (1969), reddening was neglected.

E. Michel et al.: Seismology ofδ Scuti stars in the Praesepe cluster. I 155

As a result from the HIPPARCOS satellite, Mermilliod etal. (1997) recently determined the valued = 6.24 ± 0.12, forthe distance modulus associated to Praesepe.

2.2. The stellar models

Stellar models were calculated using the CESAM stellar evolu-tionary code (Morel 1997). We used input physics appropriateto the mass range covered byδ Scuti stars. The nuclear reactionrates are from Caughlan & Fowler (1988). The equation of stateis from Eggleton, Faulkner & Flannery (1973). The atmosphereis computed using the Eddington’sT(τ) law. We used the OPALradiative opacities (Iglesias et al. 1992) complemented at lowtemperatures (T ≤ 10000 K) by the Alexander & Fergusondata (1994). The mixture of heavy elements used in the opacitycalculations is the solar mixture from Grevesse (1991) whichwe also adopted in the stellar structure computations. A solarmetallicity Z = 0.019 is associated to this mixture. The iso-topic ratios are the same as in Maeder (1983). Convection isdescribed according to the classical mixing-length description(Bohm-Vitense 1958) in which the mixing-length is a free pa-rameter. A solar mixing-length parameter (α = l/Hp = 1.67,whereHp is the pressure scale-height) was used for all modelsfollowing investigations of visual binary systems with knownmasses and metallicity (Fernandes et al. 1996) resulting in simi-lar values ofα for a wide range of metallicities and ages. Modelstake into account an overshooting of the mixed convective coreover a distance of0.20Hp following the prescription of Schalleret al. (1992).

Detailed evolutionary sequences were calculated for massesbetween1.6M� and2.5M�, from the zero-age Main Sequenceto the subgiant branch. We considered two sets of chemicalcompositions in order to cover the Praesepe [Fe/H] observedrange: a solar composition with metallicityZ = 0.019 andhelium contentY = 0.286 in mass fraction and a metal enrichedcomposition withZ = 0.03 andY = 0.290. Representativeevolutionary states for each mass were extracted, and the Genevaisochrone program used to obtain isochrones with ages in therange500 − 900 Myr.

2.3. Theoretical oscillation frequencies

In order to calculate the oscillation eigenfrequencies of the pre-viously described models, we used the adiabatic oscillation codewritten by Joergen Christensen-Dalsgaard, following recom-mendations by Christensen-Dalsgaard & Berthomieu (1991).The effect of rotation up to second order included were addedfollowing Perez Hernandez et al. (1995) and based in the workby Saio (1981).

Although non-radial modes are obviously present in the ob-servations and currently computed for the models used here, wecarry out the comparison between predicted unstable modes (ac-cording to linear growth rates) and observed modes in terms ofrange of radial orders of radial modes. This is motivated by thefact that linear growth rates calculations (Dziembowski 1977,Stellingwerf 1979, Lee 1985, Milligan & Carson 1992, Dziem-

bowski 1993) suggest that a few consecutive radial and non-radial eigenmodes of low radial order are destabilized by the “κmechanism” (see Unno et al. 1989 and references therein). It isthus possible, in a first approximation at least and for a givenmodel, to define a frequency range of unstable modes, and to de-scribe it in terms of radial order of radial modes, as commentedin Sect. 5. This work will thus only refer to calculation of radialmodes.

For these modes, assuming that rotation is uniform and fol-lowing Saio (1981), the expression of a perturbed frequencyν′

n

is given by

ν′n = νn + (An + B)

(νrot)2

νn

whereνn is the unperturbed frequency,νrot is the rotation fre-quency,B is a constant with value 1.333 andAn is a coefficientthat depends on the eigenfunction and, hence, on the stellarmodel. However, for radial modesAn is mainly a function ofthe dimensionless frequencyσ = (2πν)/(R3/GM)

12 . We have

calculatedAn for the frequencies of our models by interpolatingfrom the coefficients reported by Saio (1981) for a polytropicmodel with index 3 to the dimensionless frequencies of ourmodels. For the fast rotation considered here, this correctionν′

n − νn is frequently of the order of the separation betweentwo consecutive radial modesνn − νn−1. Specific calculationsof this perturbation for realistic models have been performedrecently (Soufi et al. 1995), but the direct use of these resultsfor a large number of models, as required in our work, is notavailable yet. In any case, according to preliminary results, theconclusions inferred here in terms of range of radial modes arenot expected to be affected in a significant way by this morerefined treatment.

2.4. The linear growth rates

The outer layers ofδ Scuti stars exhibit convectively unstablelayers, where nonadiabatic effects, associated with the pulsa-tionally induced fluctuations of the convective heat flux, maybecome important. Balmforth (1992) introduced improved rou-tines for computing envelope models and their linear pulsations,using Gough’s (1976, 1977) nonlocal, time-dependent mixing-length prescription for convection. Using Balmforth’s code, wecomputed linear growth rates for models, considering the La-grangian perturbations of the convective heat fluxδFc, and theturbulent pressure fluctuationsδpt (pt = ρw2, ρ being themean density andw the average vertical velocity of the con-vective elements). The models were calculated by specifyingmass, luminosity and effective temperature as obtained fromthe full evolution sequences discussed in Sect.2.2, assumingthe same opacities and chemical composition. The equation ofstate included a detailed treatment of the ionization of C, N, andO, and a treatment of the first ionization of the next seven mostabundant elements (Christensen-Dalsgaard 1982). Pressure ion-ization was considered by the method of Eggleton, Faulkner& Flannery (1973). The integration of the equilibrium modelstarted at an optical depth oflog(τ) = −4 and was carried out

156 E. Michel et al.: Seismology ofδ Scuti stars in the Praesepe cluster. I

to a radius fraction of0.2. The atmosphere was treated as greyand was assumed to be plane-parallel. The temperature gradi-ent was corrected by using a spatially varying Eddington factor(Auer & Mihalas 1970) derived from model C of Vernazza etal. (1981). In both, the equilibrium and pulsation models, radi-ation was treated in the Eddington approximation. In Gough’snonlocal description of the convection, two more parameters,aandb, are introduced. They control the spatial coherence of theensemble of the turbulent parcels (a) and the degree to whichthe turbulent fluxes are coupled to the local stratification (b).We employed for the parametersa2 = b2 = 600 and for themixing-length parameterα a value of1.89, which provides, fora solar model, the same depth of the convection zone as obtainedfrom the evolution code, using the standard mixing-length de-scription (Bohm-Vitense 1958) withα = 1.67. Although themixing-length parameter does have the same intrinsic physi-cal meaning in both the standard and the nonlocal descriptionused here, its numerical value obviously depends on the detailsof the convection treatment. For the sake of clarity, we defineαNL to be the mixing-length parameter value associated withthe nonlocal model used in the stability computations. It shouldbe noted here that the momentum flux can only be implementedconsistently in a nonlocal formulation, whereas in a local con-vection model the equation for the turbulent pressure gradientexhibits singular points at the edges of the convective regions(Henyey etal. 1968, Gough 1976). While classical stability anal-yses usually fail to reproduce the observed red edge ofδ Scutiinstability strip (i.e. the stabilization of low-order modes whensurface temperature decreases), recent investigations (Houdek1997) showed that Balmforth’s description allows to reproducethe red-edge and revealed the Lagrangian perturbation of theturbulent pressureδpt as the key factor of this stabilization.

3. The effect of fast rotation on the positionin the HR diagram and correction for this effect

Fast rotation modifies the determination of the position of thestars in the HR diagram. The results presented here are basedon the work by Maeder and Peytremann (1970) (hereafter MP).MP estimate this effect using 3 interior models correspondingto 5, 2 and 1.4M� and atmospheric models with continuumand hydrogen-line (Balmer and Lyman) opacities. Consideringthe oblateness produced by the rotation as a Roche surface andusing the von Zeipel gravity-darkening law, they compute theintegrated flux for different filters. Further information about themodels and this procedure can be found in MP and referencestherein.

Collins and Smith (1985) extended this study to differentialrotation, but in the absence of any evidence for any specificrotation law, we have considered rigid rotation here.

3.1. Quantitative importance of this effect forδ Scuti stars

In order to illustrate the magnitude of this effect forδ Scuti starson the Main Sequence, we consider the results obtained by MPfor a 2M� ZAMS model.

Fig. 1. a(top): effect of the rotation on the position in the HR diagram.The asterisk corresponds to the star at rest.b (middle): iso-V sin iinterpolated froma. c (bottom): Mirror image of 1b. The asterisk cor-responds to the observed rotating star. The shaded region refer to theexample given in the text.

For convenience, we present the results in terms ofTeffrather than color index. The conversion of the original (B2-V1) color index in MP to effective temperature is based on thecalibration by Hauck (1984), which will be also used in theapplication given in Sect. 4.

We thus derive from MP a table of values ofδMv andδTefffor different rotation rates and for different inclinations. Thepattern associated with this table is depicted in Fig. 1a. Therotation rate is measured by the parameterω = νrot/νbr, where

E. Michel et al.: Seismology ofδ Scuti stars in the Praesepe cluster. I 157

Table 2a.Range of parameters associated with observed stars forZ = 0.03 (see text). BT and EP Cnc. are considered together, since theirassociated ranges of parameters are very similar, includingV sin i. For these two objects, three stages of evolution have been considered: corehydrogen burning phase (CHB), global contraction phase (GCP) and broad shell hydrogen burning phase (SHB)

star mass age ω radius νbr ω′min νrot.min νrot.max

(M�) (Myr) (%) (109 m) (µHz) (%) (µHz) (µHz)

BW 1.80 550–650 50–90 1.28–1.31 30–29 59–60 18–17 27–261.85 550–650 50–90 1.31–1.36 29–28 59–60 17–17 26–25

BS 1.80 550–650 ≤ 80 1.28–1.31 30–29 48–48 14–14 24–231.85 550–650 ≤ 80 1.31–1.36 29–28 48–49 14–13 23–22

BN 1.95 550–650 80–90 1.40–1.46 27–26 47–48 21–21 25–252.05 550–650 80–90 1.50–1.60 25–23 47–49 20–18 24–22

BU 2.10 550–650 ≤ 90 1.56–1.69 24–21 71–73 17–16 22–192.20 550–650 ≤ 90 1.71–1.90 22–19 72–75 15–14 20–17

BT-EP 2.50 550–613 ≤ 90 2.45–2.79 13–11 61–65 8–7 12–10CHB 2.55 550–575 ≤ 90 2.83–3.01 11–10 65–66 7–6 10–9

BT-EP 2.50 613–629 ≤ 90 2.79–2.68 11–12 65–64 7–7 10–11GCP 2.55 575–587 ≤ 90 3.01–2.76 10–11 66–64 6–7 9–10

BT-EP 2.50 629–636 ≤ 80 2.68–3.35 12–8 64–70 7–6 9–7SHB 2.55 587–590 ≤ 80 2.76–3.63 11–7 64–72 7–5 9–6

Table 2b.Same as Table 2a forZ = 0.019

star mass age ω radius νbr ω′min νrot.min νrot.max

(M�) (Myr) (%) (109 m) (µHz) (%) (µHz) (µHz)

BW 1.60 700–800 50–90 1.14–1.16 34–32 59–60 20–20 31–291.65 700–800 50–90 1.17–1.20 32–31 59–60 20–19 29–28

BS 1.60 700–800 ≤ 80 1.14–1.16 34–32 48–48 16–16 27–261.65 700–800 ≤ 80 1.17–1.20 32–31 48–48 16–15 26–25

BN 1.75 700–800 80–90 1.26–1.31 30–29 47–48 24–23 27–261.85 700–800 80–90 1.38–1.46 27–25 48–49 21–20 24–23

BU 1.90 700–800 ≤ 90 1.44–1.56 26–23 71–74 18–17 23–212.00 700–800 ≤ 90 1.62–1.81 22–19 73–77 16–14 20–17

BT-EP 2.20 700–762 ≤ 90 2.29–2.58 14–12 63–66 9–8 13–11CHB 2.25 700–716 ≤ 90 2.55–2.63 12–12 65–66 8–7 11–11

BT-EP 2.20 762–777 ≤ 90 2.58–2.29 12–14 66–63 8–9 11–13GCP 2.25 716–729 ≤ 90 2.63–2.38 12–13 66–63 7–8 11–12

BT-EP 2.20 777–781 ≤ 80 2.29–3.04 14–9 63–71 9–6 11–7SHB 2.25 729–733 ≤ 80 2.38–3.15 13–10 63–71 8–6 11–8

νrot is the rotational frequency of the star andνbr the break-uprotational frequency. For the inclinationi between the rotationalaxis and the observer’s line of sight, six values fori between0and90◦ are given in MP, for each rotation rate.

Instead ofω, we introduce the measurable quantityV sin i.We interpolate the previous table for selected values ofV sin i.Fig. 1b displays these iso-V sin i lines, by steps of20 km s−1

between50 and250 km s−1.Let us consider a star with a measuredV sin i between190

and210 km s−1. This range ofV sin i values is not uncommonamongδ Scuti stars. It corresponds necessarily (in the assump-

tion of rigid rotation) to a high rotation rate:νrot/νbr >∼ 50%.We will see in Sect. 5 that all theδ Scuti stars considered herehave rotation rates higher than50% (Table 2 a and b, Column 7).

It can be seen that, in that case, the rotational effect producesshifts of the order of0.1 to0.3 mag inMv and from200 to250 Kin Teff .

It should be emphasized that neglecting this effect couldruin definitely any tentative estimation of the global parame-ters through classical calibration, this effect being larger thanthe usual error box for field stars (currently±0.15 mag and±150 K).

158 E. Michel et al.: Seismology ofδ Scuti stars in the Praesepe cluster. I

3.2. The “twin-at-rest”

Since we want to compare our non-rotating models with theobserved stars, we need to “subtract” the effects that rotationhas on the resulting photometric indices. In other words, wewant to determine for each observed star a “twin-at-rest” whichcorresponds to a non-rotating star evolved up to the same age.Though mainly due to surface changes, the effect described inSect. 3.1 includes a small decrease in the intrinsic luminosityand polar radius due to rotation, estimated from interior modelsat the ZAMS. On the other hand, following Maeder (1971), weconsider that the effect of rigid rotation on the internal evolution(i.e. time scale for the core hydrogen burning) due to centrifugalinertial force is negligible compared with the surface photomet-ric effect discussed above. Other potential influences of rotationon the evolution (e.g. transport of chemical species) are not con-sidered here.

3.3. Correction for this effect

Having defined the “twin-at-rest” we shall describe now how weestimate itsMv andTeff in a practical situation of an observedrotating star with a givenV sin i measurement.

Let we mention first that in general, as remarked by Collinsand Smith (1985), the relation is not strictly unique and to givenδMv andδTeff can correspond different pairs (ω,i) (see Fig. 2of Collins & Smith 1985). However, at least in the case of rigidrotation and for typicalδ Scuti stars, this ambiguity is limitedto extreme rotation rates (ω > 90%), which are not consideredhere.

Since we usually cannot determine from the observationsneitherV nor i, but only the quantityV sin i, we cannot obtaina point in the HR diagram but rather a range of possible values,unless additional constraints are imposed. This is illustrated inFig. 1c, which is the mirror image of Fig. 1b. The asterisk hererepresents the observed rotating star. For instance, for a mea-suredV sin i between190 and210 km s−1, the shaded regionrepresents the possible positions for the “twin-at-rest”.

The pattern depicted in Fig. 1a is expected to be valid fora reasonably large set of masses and ages. In fact, for stars onor just off the Main Sequence, the effect discussed by MP isexpected to depend on the model at rest only slightly throughTeff and almost not on the surface gravityg (and thus on massand radius). We thus used it as representative for our stars.

Then, the interpolation inV sin i necessary to go fromFig. 1a to Fig. 1b requires ana priori estimation ofR(ω = 0),the radius of the “twin-at-rest”. This dependency is small. Fora20% change of radius, corresponding to the maximum uncer-tainty within a typical error box, the changes inV sin i in Fig. 1bremain lower than7%. Thus, for each observed rotating star, apattern like the one of Fig. 1b (and thus 1c) can be obtained withgood precision using a rough estimation of the radius.

In fact, we should keep in mind that this procedure is sen-sitive to uncertainties in the estimation of theTeff andMbol ofthe observed rotating star and will clearly benefit any substantialreduction of these uncertainties.

Fig. 2. HR diagram of Praesepe. ForZ = 0.03, the best fit is obtainedfor the isochrone600 Myr and a distance modulusd = 6.4 (shownhere).

When applied to clusters, this procedure benefits both thebetter determination of relativeMbol andTeff and the comple-mentary information brought by the possibility to determine anisochrone. This is commented in the following section, wherethis procedure is used to derive the global parameters of the“twin-at-rest” associated with each observed star and to initiatethe modelling.

4. Stellar parameters

In this section, we determine the domain of parameters repre-sentative for each star of the set, taking into account the effectof fast rotation discussed in Sect. 3.

4.1. Locating the stars in the HR diagram

We estimate the distance modulus and the age of the clusterby fitting, in the (mbol − Teff ) diagram, the isochrones asso-ciated with our models and observations presented herebefore.As shown in Fig. 2, the fit is made so that the isochrone coin-cides with the bottom envelope of the position of the clusterstars in the HR diagram. Doing this, we make sure we are freeof the potential binarity effect and of the fast rotation effect de-scribed in Sect. 3. Both effects are expected to induce systematicshifts toward higher luminosity and lower effective temperature,compared to a single non-rotating star. Two extreme metallic-ities were consideredZ = 0.03 andZ = 0.019. The best fitwas found for a distance modulusd = 6.40 and600 Myr, andd = 6.15 and 750 Myr respectively. These distances are inagreement with HIPPARCOS measurement (Mermilliod et al.1997) within 1.5σ.

According to Maeder (1971), such a fit might overestimatethe age of Praesepe by approximately10%, due to the fact thatstars of the turn-off are almost exclusively fast rotators or bi-naries. This will be taken into account further, for quantitativeapplications.

E. Michel et al.: Seismology ofδ Scuti stars in the Praesepe cluster. I 159

In Fig. 3, we present the correction for fast rotation obtainedfor the 6δ Scuti stars considered here. For this correction, weused the results of MP computed for a 2M� ZAMS model(Fig. 1a). From this diagram and for each of theseδ Scuti stars,we can determine the domain corresponding to its “twin-at-rest” in the HR diagram. Ideally, for each metallicity consideredhere, this domain should correspond to the region of possiblepositions for the “twin-at-rest” as defined in Sect. 3, restrictedto its intersection with the respective isochrone, e.g.600 Myrfor Z = 0.03 and750 Myr for Z = 0.019. We thus see thatthe efficiency of this method would ultimately be limited bythe precision with whichMbol, Teff and the isochrone can bedetermined. It is thus interesting to notice that, according toRufener (1988), the relative errors in the diagram presented hereare as low as±0.01 mag and±50 K, besides systematic errorsdue to the calibration inTeff and to the distance estimate (andbolometric correction).

Nevertheless, a more precise use of this information willfirst require complementary work in the line of the one by MPwith updated models adapted to each star individually.

In the present state of the art, we restrict ourselves to a firstlevel use of these results to derive for eachδ Scuti star consideredhere the conservative ranges of parameters presented in Table 2and commented in the next section.

We remark that the sixδ Scuti stars considered here are pre-senting a nice scan of various ranges of mass and evolutionarystages. BW Cnc and BS Cnc are representative of the “low mass”part of the set, while BN Cnc and BU Cnc are more massive butstill unambiguously on the Main Sequence. The evolution statusof BT Cnc and EP Cnc is more difficult to establish. Accordingto Maeder (1971) and references therein, EP Cnc and the twonearby stars in the HR diagram are objects more evolved thancore hydrogen-burning. Considering our isochrones and follow-ing Hernandez et al. (1998b), we estimate that EP Cnc is locatednear the “blue turn” (B-T) associated with the exhaustion of hy-drogen in the core and that BT Cnc is located around the “redturn” (R-T) characterizing stars at the end of Main Sequenceevolution and the beginning of the global contraction. In thisscenario, since the global contraction phase is very rapid, bothBT and EP Cnc have to be very close in mass (∆M ' 0.02M�according to our models). However, for completeness, the threestages of evolution are considered in the following discussion.

4.2. Range of parameters associatedwith models representative of the observed stars

Following Sect. 4.1, it is possible to obtain an estimate of theposition in the HR diagram of each “twin-at-rest”, for a givenmetallicity and a given fit of the isochrones.

Then, by comparison with our models, we assign to eachstar: a range of mass and a range of age and an indicative rangeof rotation rateω (respectively Column 2, 3, 4 of Table 2a andTable 2b). We arbitrarily limitω to values lower than90%, since,for extreme values, the calculation of the eigenfrequencies asdescribed in Sect. 2.3 hardly makes sense.

Fig. 3. Same as Fig. 2 with correction of the effect of fast rotation (seeSect. 4) given for eachδ Scuti star considered here. For each object,two lines are given, associated with extreme values of theV sin iestimation. The domain of possible “twin-at-rest” is delimited by thesetwo lines. Masses are indicated by steps of0.10M� with asterisks onthe isochrones.

From the models associated with each star, we deduce arange of radius (Column 5) for the corresponding “twin-at-rest”.

From the range of mass and the value of the radius, wethen deriveνbr, the break-up rotational frequency (Column 6,Table 2) as:

νbr =12π

[GM

R3eq(ω = 1)

]1/2

,

with Req(ω = 1) the equatorial radius at break-up,M the massandG the gravitational constant. The equatorial radius at thebreak up point is related to the polar radius at break-up byReq = 1.5 Rp (since we are assuming that the surface of the staris a Roche equipotential surface). The relation between the po-lar radius of the rotating star and the radius of the correspondingnon-rotating star depends only slightly on the details of the in-terior models used and following MP we adopt the expression:Rp(ω) = Rp(ω = 0)(1 − βω2), where for 2M�, β = 0.024.Then the break-up rotation is estimated by

νbr =12π

[GM

R3p(ω = 0)

]1/2

[2

3(1 − β)]3/2

' 0.562π

[GM

R3p(ω = 0)

]1/2

.

FromV sin i given in Table 1 and the estimate of the radius,we determineω′

min, a lowest limit for the quantityω, as:

νbr ω′min 2π Req(ω′

min) = Vmin = (1 − 10%) V sin i.

These value is given in Column 7. We notice that these lowestlimit ω′

min is more restrictive than theωmin obtained in Col-umn 4, except for BN Cnc. In the following, we define

ωinf = sup[ω′min, ωmin].

The associated valueνrot.min = ωinf νbr is given in Column 8of Table 2.

160 E. Michel et al.: Seismology ofδ Scuti stars in the Praesepe cluster. I

An estimate,νrot.max, of the highest limit forνrot is givenin Column 9. It corresponds to

νrot.max = ωmax νbr,

with ωmax from Column 4.For each metallicity, and for each star, we thus can associate

a restricted set of models, corresponding to the ranges of age,mass,νrot (andω).

5. Determination of the range of radial order nof the observed modes by comparison with models

In this section, we present the comparison between the observedand computed frequencies.

Considering the set of parameters associated with one starand given in Table 2, it is possible to obtain the range of possiblevalues for the different theoretical radial eigenfrequencies. Thisis shown in Fig. 4a and 4b.

We are facing a multidimensional problem with no simplecomplete graphic representation. Among the possible represen-tations, we selected the one presented hereafter because it isreasonably concise and well-suited for the analysis in terms ofrange of radial frequencies.

For each star, the comparison between observations andmodels is described in a triptych. Each panel presents, as ver-tical sticks, the observed frequencies (given in Table 1) for theconsidered star, with frequencies in abscissae and amplitudes inthe left coordinate axis.

Then, for each panel, each grey region represents the rangeof frequency encompassed by the 6 first radial modes (black forn = 1, fundamental mode, to lightest grey forn = 6), whenvarying parameters in the ranges described in Table 2.

In Fig. 4a, we consider stars in the phase of core hydrogenburning (CHB). The upper panel of each triptych describes thevariation of these frequency ranges, for a given uncertainty onmass, and when age and rotation vary, from the maximum ageand the maximum rotation (top, as mentioned in the upper title ofthe panel) to the minimum age and minimum rotation (bottom,as mentioned in the lower title of the panel). Both effects ofdecreasing age and rotation make the frequencies change in thesame direction. This is why we decided to vary both parameterssimultaneously.

The second (middle) panel of the triptych is dedicated tovariation inZ, betweenZ = 0.03 (top) andZ = 0.019 (bot-tom). As mentioned in upper and lower titles, the rotation rateis maintained to minimum, considering the values given in Ta-ble 2a and b.

The third (lower) panel is again dedicated to changes withage and rotation, but forZ = 0.019.

Fig. 4b shows the results for BT and EP in the phase ofoverall contraction (upper triptych) and in the phase of hydrogenbroad shell burning (bottom triptych).

Figs. 4a and 4b allow to derive the following informationabout the range of radial order over which observed modes aredetected.

For the two less massive stars BW and BS Cnc, the possibleranges of radial order found for the observed frequencies aresimilar and include systematically the fundamental for BW Cnc;It is also true in most of the cases for BS Cnc. The total rangefor both stars spans over 4 to 5 radial orders for BS Cnc and 3to 4 for BW Cnc.

The three lowest frequencies detected for BW always re-main below the value of the fundamental radial mode. With therotation rates considered here, the doublet around140 µHz canbe explained by p-modes with azimuthal orderm > 0 belongingto nonradial multiplets asymmetrically split by rotation.

On the other hand, the lowest peak at68 µHz (which exis-tence is subject to caution as mentioned in Sect. 2.1) can only beexplained by a g-mode, in the range of parameters consideredhere.

For the two “intermediate-mass” stars: BN and BU Cnc.,the results are also rather homogeneous. The possible rangesof radial order exclude systematically the fundamental (n = 1)and for BN, also the first overtone (n = 2). Observed peaks spanover a more restricted range of consecutive radial orders thanfor the two lower-mass stars since it can only be 2 consecutiven, betweenn = 2 andn = 6.

The case of the “high-mass” stars (BT and EP Cnc.), as al-ready commented in Sect. 4, is less straightforward, but someindications can be gained from our models, being aware of thedifficulties to model these phase of evolution with precise pa-rameters.

According to the results shown in Fig. 4a and 4b, the ob-served modes span over a range which includes 2 consecutiveradial modes in most of the cases.

For BT Cnc in core hydrogen burning or in the global con-traction phase, the observed range of radial modes can span 2consecutive radial modes withn between1 and4. If BT shouldbe in the phase of shell hydrogen burning, the potential rangewould go from [1,2] to [4,6].

In the case of EP Cnc, for phases of global contraction andcore hydrogen burning, the range is restricted to [1] or [1,2]. Itcan also be [1,3] in the case of shell hydrogen burning.

These results are summarized in Fig. 6. These ranges of ra-dial order are determined with an intrinsic error estimated as±1 radial order.

6. Radial order ranges for unstable modes accordingto growth rates

A linear stability analysis, as described in Sect. 2.4, has beenconducted for models representative of the sixδ Scuti starsconsidered in this paper, according to the ranges of parame-ters described in Table 2. The results are illustrated in Fig. 5afor two different values ofZ. They appear to be only slightlysensitive to metallicity and to age within the ranges consideredhere. We thus discuss them altogether, considering mainly thechanges with mass.

In the case of the “low-mass” stars BW and BS Cnc.(1.80 − 1.85 M� and 1.60 − 1.65 M� for Z = 0.03 andZ = 0.019 respectively), the range of unstable modes is large,

E. Michel et al.: Seismology ofδ Scuti stars in the Praesepe cluster. I 161

“low-mass stars”

“intermediate-mass stars”

“high-mass stars”

Fig. 4a. Comparison between observations and models in the phase of core hydrogen burning. Each star is associated to a given triptych whereobserved frequencies are compared with theoretical ones as described in the text. Each grey level corresponds to a specific radial order, fromn = 1 (fundamental mode, black) ton = 6 (lightest grey).

162 E. Michel et al.: Seismology ofδ Scuti stars in the Praesepe cluster. I

Global contraction phase

Shell hydrogen burning phase

Fig. 4b. Same as Fig. 4a but for global contraction phase (top) andshell hydrogen burning (bottom). Both triptychs are representative ofboth BT and EP Cnc, but, for sake of conciseness, the observed fre-quencies of BT Cnc. (respectively EP Cnc.) are only given in the upper(respectively lower) triptych.

spanning over5 to 8 consecutive radial orders. At the lowestmass (1.60 M�), it includes the fundamental. Then, when in-creasing mass, the first radial modes(n = 1, 2, ...) are rapidlystabilized, while the unstable range is globally shifted upward,toward higher radial ordersn.

For “intermediate” masses representative of BN andBU Cnc, we observe a narrow unstable range spanning over2 to 3 radial orders, which evolves slowly toward highern withincreasing mass.

In the case of the “high-mass” stars BT and EP Cnc.(2.50 − 2.55 M� and 2.20 − 2.25 M� for Z = 0.03 andZ = 0.019 respectively), the unstable band is still thin, withn values between4 and6 in CHB phase and almost similar inGCP and SHB phases.

The absence of unstable modes around the blue-turn (B-T) isinterpreted as models going out of the instability strip through

its blue edge, during the rapid excursion toward higherTeffassociated with global contraction phase (GCP).

These results as well as the ones of Sect. 5 are summarizedin Fig. 6 in order to make the comparison developed in the nextsection easier. As can be seen in Fig. 5 a and b, the transitionbetween stable and unstable modes is often not so sharp in thesense that it is difficult to classify as stable or unstable, modeswith growth rates values very near zero. Here again, we considerthat these ranges of radial orders are significant within±1 radialorder.

7. Comparative discussion of the results from growth-ratesand observations.

As mentioned before, the definition of the ranges given in Fig. 6does not pretend to be precise to better than plus or minus oneradial order. Nevertheless, in Fig. 6, we notice a significant dis-agreement between unstable ranges derived from observations(asterisks) and from stability analysis (rhombi), in the case of“low-mass” stars BW and BS Cnc. The width of unstable rangespredicted by growth rates is larger (∆ = 7, 8, 9) than the onederived from observations (∆ = 3, 4, 5). The ranges derivedfrom the observations systematically include the fundamental(n1 = 1) and never go further thann2 = 5, while ranges of un-stable modes suggested by growth rates systematically includemodes withn2 ≥ 8.

For the other stars of the set on the other hand, the agreementis satisfying, especially considering the width of the unstableranges∆. Ranges derived from observations and from stabilityanalysis are found on similar diagonal lines representative of agiven width∆. In fact, a common solution to both analyses canalways be found within the±1 error bars, except for EP Cnc.We nevertheless notice that the ranges obtained from stabilityanalysis are systematically shifted toward higher radial ordersn, compared with the ranges derived from observations.

As discussed before, we have considered very conservativeranges of values for parameters like rotation, mass and age inTable 2, and we believe the set of evolution models consideredhere to be representative of the observed stars within the frameof the input physics.

Considering the input physics of the evolution models, weestimate that the ranges admitted for mass, age and rotationmake negligible the potential errors classically admitted onopacities, nuclear reaction rates or equation of state.

The influence of the value chosen for the overshooting pa-rameterdov has been estimated. In the case of BU Cnc. Chang-ing the overshooting parameter from (dov = 0.2) to (dov = 0),induces a change of mass lower than2%, in the set of models rep-resentative of BU Cnc. For a given position in the HR diagram,this corresponds to a change less than1% in the eigenfrequen-cies of radial modes. Even considering the change in the slopeof the isochrone, this effect cannot be expected to have any sig-nificant influence on the result of this study for BU Cnc., havingin mind the error bars admitted for mass and age in Table 2. Thiseffect being even smaller for lower masses, we do not expectany significant change in the analysis when varying the over-

E. Michel et al.: Seismology ofδ Scuti stars in the Praesepe cluster. I 163

Fig. 5a. Contour plot of theoretical growth rates for the first10 radial modes as function of age, forZ = 0.030 (left) andZ = 0.019 (right) andfor masses representative of the stars considered here (see Table 2). Hatched regions correspond to unstable modes (see Fig. 5b for a detaileddescription of the different grey levels). For the most massive models, red-turn and blue-turn are indicated above the panels (R-T and B-Trespectively).

shooting parameter, in the case of the Main Sequence objects atleast. For the two most massive objects BT and EP Cnc., thingsmight be different. Changingdov might significantly change theset of models associated with these objects.

The outer convective region is very shallow for objectswithin the range of masses considered here. The parameterα is thus not expected to have any significant influence onthe global structure of the evolution models considered here,nor on the eigenfrequencies associated to these models, withinthe precision admitted here. On the other hand, as commentedin Sect. 2.4, the stability analysis can be strongly sensitive tothe treatment of these outer shells. The results of the stabilityanalysis presented here might thus be sensitive toαNL and toopacities. In fact, when reducingαNL from 1.89 to 1.5 in the

stability analysis, we obtain very similar results for high andintermediate-mass stars but strong changes appear for low-massobjects (see Fig. 5b). This illustrates that low-massδ Scuti starsmight bring some constraints on the theoretical description ofthe convection in the outer convective zone.

The results obtained withαNL = 1.5 are indicated withsquares in Fig. 6 (only for BS and BW Cnc, since no significantchange was found for other masses). We see that a lower valueof αNL leads to a better agreement with the results derived fromobservations. This suggests that, in the frame of the nonlocalmixing-length model used here, the description of the convec-tion near the surface might require a less efficient convectivetransport in the outer shells, characterized here by a value ofthe parameterαNL lower than the one taken as a reasonable

164 E. Michel et al.: Seismology ofδ Scuti stars in the Praesepe cluster. I

Fig. 5b. Contour plot of theoretical growth rates for the first10 radial modes as function of age and mass. The computations assumed theparametersα = 1.5 andZ = 0.030 (left panel) andZ = 0.019 (right panel). Hatched regions correspond to unstable modes, solid regionsindicate stable (damped) modes. The contour levels are similarly indicated.

initial value (αNL = 1.89, see Sect. 2.4). It is of course allur-ing to compare this relatively small value ofαNL (1.5) with thecurrently larger value required for obtaining the correct depthof the solar convection zone as measured accurately by helio-seismology (Christensen-Dalsgaard et al. 1991). However, dis-cussing these results in terms of the mixing-length parameterαalone does not describe the whole picture, mainly because ofthe large uncertainties in the phenomenological description ofthe mixing-length model. In this context, a crude comparisonof α values derived from observational constraints of differentnature probably makes poor sense.

The point which looks more important to us is thatδ Scutistars might bring valuable observational constraint on the de-scription of convection in the outer layers of stars and that thisnew observational constraints, together with already existingones will offer more possibilities to improve our description ofthese regions of the stars.

We thus obtain a satisfying agreement between the resultsof the stability analysis and the one derived from observationsif we assumeαNL = 1.5. This is especially true for the width∆ of the unstable ranges. We also notice that the ranges ob-tained from stability analysis are systematically shifted towardhigher radial ordern, compared with the ranges derived fromobservations. Our data seem to show an increase of this shiftwith mass. We do not see any way to explain this shift by a biasin our estimation of the unstable ranges derived from observa-tions. In the context of this work and taking into account fastrotation and asymmetric splitting, it would be possible to ex-plain a systematic underestimate ofn1, but hardly higher than 1order. In addition to this, we do not see any way to explain a sys-tematic underestimate ofn2, except if we accept the hypothesisthat the highest eigenfrequencies systematically correspond tosignificantly lower amplitude and are not detected.

In this work, we tried to give a rather complete description ofthe problem, taking into account several aspects of the influenceof rotation. However, the computation of oscillations in very

fast rotating stars is still a tricky problem under development.Further improvements in this field might contribute to make thiskind of analysis more precise.

8. Conclusions

We discuss here some potential advantages of performing seis-mological studies of stars in clusters and set the stage for furtherseismological interpretations of this kind of data.

We show that the effect of fast rotation on the position of astar in the HR diagram cannot be neglected in the modelling offast rotators like most ofδ Scuti stars and we propose a methodto correct this effect and gain some indication on the rotationrate.

We consider a representative set ofδ Scuti stars observedin the Praesepe cluster, thanks to recent efforts of multi-sitenetworks. We use this unique set of homogeneous observationaldata to investigate the pulsational behaviour ofδ Scuti stars fordifferent masses in the range of age of the Praesepe cluster(approximately600 − 750 Myr depending on the metallicity).

This work proposes an alternative approach toδ Scuti seis-mology, trying to avoid the use of precise but questionable modeidentifications in order to derive firm informations about the pul-sational behaviour of these objects. Comparing the observationswith a representative set of models, we show that it is possibleto constrain the range of radial order corresponding to the ob-served frequencies in each star (in terms of radial order of radialmodes).

We then compare these ranges with the ones expected fromlinear stability analysis as performed by one of us (G.H.) anddiscuss the agreement. A satisfying agreement can be foundfor most of the stars considered here. For low-mass objects[1.6 − 1.8 M�] however, we find a significant disagreement.This suggests that the precise treatment of the outer convectiveshells might be in cause. We show that, in the framework ofthe nonlocal mixing-length model used here, it can be solved

E. Michel et al.: Seismology ofδ Scuti stars in the Praesepe cluster. I 165

Fig. 6. Unstable ranges as derived from observations and from stabil-ity analysis. Each star is attributed one panel (two in the case of BTand EP Cnc, corresponding to the two plausible evolution stages). Inthese panels, each point of coordinates (n1,n2) corresponds to a rangespanning from n1 to n2. The ranges derived from observations are rep-resented with stars. The ranges obtained from stability analysis are rep-resented with rhombi. In the panels associated with “low-mass” stars,squares represent the results of the stability analysis withαNL = 1.5.Each diagonal line corresponds to a given width (∆) of the ranges,labeled by numbers around the top-right corner of the panels.

by setting a value of the mixing-length parameterαNL lowerthan our initial trial value, i.e. by reducing the efficiency of con-vective heat transport. This result illustrates that “low-mass”δScuti stars might bring valuable observational constraint on thedescription of the convection process in the outer layers of stars.The description of this physical process is currently consideredas the main source of uncertainties in the modelling of stars withlarge outer convective zone.

Once the mixing-length parameter is suitably chosen, weshow that the agreement is satisfactory for stars on the MainSequence (CHB), comforting the idea that theδ Scuti behaviourobeys, at least to a significant extent to linear analysis. We thusconclude that growth rates can bring trustworthy information formode identification of the observed modes, having in mind that,at the present time, identification of the modes is the missinglink to establish, in order to make possible the full potentialscientific return from stellar seismology ofδ Scuti stars.

The case of evolved stars BT and EP Cnc is also addressed.For EP Cnc, no agreement is found between growth ratesand observations, but a deeper study of the influence of theovershooting is necessary for these objects.

Acknowledgements.G.H. is grateful to Neil Balmforth for providinghis routines for computing the growth rates and acknowledges supportfrom the Particle Physics and Astronomy Research Council and inpart from the Austrian FWF (project S7303-AST). We thank AndreMaeder for helpful discussion. This work has been supported by theEuropean Union contract ANTENA, under which the continuouscollaboration between authors was possible. This research has madeuse of the SIMBAD database operated at CDS, Strasbourg (France).

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