The CoRoT target HD 49933: 2- Comparison of theoretical mode amplitudes with observations
R. Samadi, H.-G. Ludwig, K. Belkacem, M.J. Goupil, O. Benomar, B. Mosser, M.-A. Dupret, F. Baudin, T. Appourchaux, E. Michel
aa r X i v : . [ a s t r o - ph . S R ] N ov Astronomy & Astrophysics manuscript no. 11868 c (cid:13)
ESO 2018November 9, 2018
The CoRoT ⋆ target HD 49933: R. Samadi , H.-G. Ludwig , K. Belkacem , , M.J. Goupil , O. Benomar , B. Mosser , M.-A. Dupret , , F.Baudin , T. Appourchaux , and E. Michel Observatoire de Paris, LESIA, CNRS UMR 8109, Universit´e Pierre et Marie Curie, Universit´e Denis Diderot, 5 pl.J. Janssen, F-92195 Meudon, France Observatoire de Paris, GEPI, CNRS UMR 8111, 5 pl. J. Janssen, F-92195 Meudon, France Institut d’Astrophysique et de G´eophysique de l’Universit´e de Li`ege, All´e du 6 Aoˆut 17 - B 4000 Li`ege, Belgium Institut d’Astrophysique Spatiale, CNRS UMR 8617, Universit´e Paris XI, 91405 Orsay, France.November 9, 2018
ABSTRACT
Context.
The seismic data obtained by CoRoT for the star HD 49933 enable us for the first time to measure directly the amplitudes and linewidths of solar-like oscillations for a star other than the Sun. From those measurements it ispossible, as was done for the Sun, to constrain models of the excitation of acoustic modes by turbulent convection.
Aims.
We compare a stochastic excitation model described in Paper I with the asteroseismology data for HD 49933, astar that is rather metal poor and significantly hotter than the Sun.
Methods.
Using the seismic determinations of the mode linewidths detected by CoRoT for HD 49933 and the theoreticalmode excitation rates computed in Paper I for the specific case of HD 49933, we derive the expected surface velocityamplitudes of the acoustic modes detected in HD 49933. Using a calibrated quasi-adiabatic approximation relating themode amplitudes in intensity to those in velocity, we derive the expected values of the mode amplitude in intensity.
Results.
Except at rather high frequency, our amplitude calculations are within 1- σ error bars of the mode surfacevelocity spectrum derived with the HARPS spectrograph. The same is found with respect to the mode amplitudesin intensity derived for HD 49933 from the CoRoT data. On the other hand, at high frequency ( ν & . Conclusions.
These results validate in the case of a star significantly hotter than the Sun and α Cen A the mainassumptions in the model of stochastic excitation. However, the discrepancies seen at high frequency highlight somedeficiencies of the modelling, whose origin remains to be understood. We also show that it is important to take thesurface metal abundance of the solar-like pulsators into account.
Key words. convection - turbulence - atmosphere - Stars: oscillations - Stars: individual: HD 49933 - Sun: oscillations
1. Introduction
The amplitudes of solar-like oscillations result from a bal-ance between excitation and damping. The mode linewidthsare directly related to the mode damping rates. Once we canmeasure the mode linewidths, we can derive the theoreticalvalue of the mode amplitudes from theoretical calculationsof the mode excitation rates, which in turn can be comparedto the available seismic constraints. This comparison allowsus to test the model of stochastic mode excitation investi-gated in a companion paper (Samadi et al. 2009, hereafterPaper I).As shown in Paper I, a moderate deficit of the sur-face metal abundance results in a significant decrease ofthe mode driving by turbulent convection. Indeed, by tak- ⋆ The CoRoT space mission, launched on December 27 2006,has been developped and is operated by CNES, with the contri-bution of Austria, Belgium, Brasil, ESA, Germany and Spain.
Correspondence to : [email protected] ing into account the measured iron-to-hydrogen abundance([Fe/H]) of HD 49993 ([Fe/H]= − . P ex-pected for this star. The resulting value of P is found to beabout two times smaller than for a model with the samegravity and effective temperature, but with a solar metalabundance (i.e. [Fe/H]= 0).The star HD 49933 was first observed in Doppler ve-locity by Mosser et al. (2005) with the HARPS spectro-graph. More recently, this star has been observed twiceby CoRoT. A first time this was done continuously duringabout 61 days (initial run, IR) and a second time contin-uously during about 137 days (first long run in the cen-ter direction, LRc01). The combined seismic analysis ofthese data (Benomar et al. 2009) has provided the modelinewidths as well as the amplitudes of the modes in inten-sity. Then, using mode linewidths obtained for HD 49933with the CoRoT data and the theoretical mode excitationrates (obtained in Paper I), we derive the expected valuesof the mode surface velocity amplitudes. We next compare R. Samadi et al.: The CoRoT target HD 49933: 2 - Comparison with observations these values with the mode velocity spectrum derived fol-lowing Kjeldsen et al. (2005) with seismic data from theHARPS spectrograph (Mosser et al. 2005).Mode amplitudes in terms of luminosity fluctuationshave also been derived from the CoRoT data for 17 radialorders. These data provide us with not only a constrainton the maximum of the mode amplitude but also with thefrequency dependence. The relative luminosity amplitudes δL/L are linearly related to the velocity amplitudes. Thisratio is determined by the solution of the non-adiabatic pul-sation equations and is independent of the stochastic excita-tion model (see Houdek et al. 1999). Such a non-adiabaticcalculation requires us to take into account, not only theradiative damping, but also the coupling between the pulsa-tion and the turbulent convection. However, there are cur-rently very significant uncertainties concerning the model-ing of this coupling (for a recent review see Houdek 2008).We relate further for the sake of simplicity the mode lumi-nosity amplitudes to computed mode velocity amplitudesby assuming adiabatic oscillations as Kjeldsen & Bedding(1995). Such a relation is calibrated in order to reproducethe helioseismic data.The comparison between theoretical values of the modeamplitudes (both in terms of surface velocity and intensity)constitutes a test of the stochastic excitation model witha star significantly different from the Sun and α Cen A.In addition it is also possible to test the validity of thecalibrated quasi-adiabatic relation, since both mode ampli-tudes, in terms of surface velocity and intensity, are avail-able for this star,This paper is organized as follows: We describe in Sect. 2the way mode amplitudes in terms of surface velocity v s are derived from the theoretical values of P and from themeasured mode linewidths (Γ). Then, we compare the the-oretical values of the mode surface velocity with the seismicconstraint obtained from HARPS observations. We describein Sect. 3 the way mode amplitudes in terms of intensityfluctuations δL/L are derived from theoretical values of v s and compare δL/L with the seismic constraints obtainedfrom the CoRoT observations. Finally, Sects. 4 and 5 arededicated to a discussion and conclusion respectively.
2. Surface velocity mode amplitude
The intrinsic rms mode surface velocity v s is related tothe mode exitation rate P ( ν ) and the mode linewidth Γ( ν )according to (see, e.g., Baudin et al. 2005): v s ( r h , ν ) = s P π M h Γ (1)where P is the mode excitation rate derived as described inPaper I, Γ is the mode full width at half maximum (in ν ), ν = ω osc / π the mode frequency and M h is the mode massdefined as: M h = Iξ ( r h ) (2)where I is the mode inertia (see Eq. (2) of Paper I), ξ r theradial mode eigendisplacement, r h ≡ R + h the layer in theatmosphere where the mode is measured in radial velocity, R the radius at the photosphere (i.e. at T = T eff ) and h theheight above the photosphere. In Sect. 2.2 we will compare estimated values of v s withthe seismic constraint obtained by Mosser et al. (2005) withthe HARPS spectrograph. We therefore need to estimate v s at the layer h where the HARPS spectrograph is themost sensitive to the mode displacement. As discussed bySamadi et al. (2008a), the seismic measurements obtainedwith HARPS spectrograph are likely to arise from the opti-cal depth τ
500 nm ≃ . h associated with theoptical depth τ
500 nm (Christensen-Dalsgaard & Gough1982). For the model with [Fe/H]= 0 (resp. [Fe/H]= − h ≃
390 km (resp. h ≃
350 km).For the mode linewidth Γ we use the seismic measure-ment obtained from the seismic analysis of the CoRoT dataperformed by Benomar et al. (2009). This seismic analy-sis combined the two CoRoT runs available for HD 49933.Two different approaches were considered in this analysis:one based on the maximum likelihood estimator and thesecond one using the Bayesian approach coupled with aMarkov Chains Monte Carlo algorithm. The Bayesian ap-proach remains in general more reliable even in low signal-to-noise conditions. Nevertheless, in terms of mode ampli-tudes, mode heights and mode linewidths, both methodsagree within 1- σ . We will consider here the seismic param-eters and associated error bars obtained on the basis of theBayesian approach. The seismic analysis in velocity has been performed byMosser et al. (2005) using data from the HARPS spectro-graph. The quality of these data is too poor to performa direct comparison between the observed spectrum andthe calculated amplitude spectrum ( v s , Eq. (1)). Indeed,the observed spectrum is highly affected by the day aliases.Furthermore, the quality of the data does not allow to iso-late individual modes, in particular modes of a different an-gular degree ( ℓ ). A consequence is that energies of modeswhich are close in frequency are mixed.In order to measure the oscillation amplitude in away that is independent of these effects, we have followedthe method introduced by Kjeldsen et al. (2005, see alsoKjeldsen et al. (2008)). This method consists in derivingthe oscillation amplitudes from the oscillation power den-sity spectrum smoothed over typically four times the largeseparation (i.e. four radial orders). Next, we multiply thissmoothed spectrum by a coefficient in order to convert the apparent amplitudes into intrinsic amplitudes. This coef-ficient takes into account the spatial response function ofthe angular degrees ℓ =0, 1, 2 and 3 (see Kjeldsen et al.2008). We have checked that the sensitivity of the visi-bility factor with the limb-darkening law is significantlysmaller in comparison with the error associated with theMosser et al. (2005) seismic measurements. The amplitudespectrum v HARPS derived following Kjeldsen et al. (2005) isshown in Fig. 1. The 1- σ error bar associated with each val-ues of v HARPS is constant and equal to ∆ V HARPS = 7 cm/s.The maximum of v HARPS reaches V max = 50 . ± ±
10 cm/s, which once converted into intrinsic ampli-tude represents a maximum of 42 ±
10 cm/s. The differ-ence between the two values is within the 1- σ error bars. . Samadi et al.: The CoRoT target HD 49933: 2 - Comparison with observations 3 The different value found by Mosser et al. (2005) can beexplained by the way the maximum of the mode ampli-tude was derived. Indeed, Mosser et al. (2005) have con-structed synthetic time series based on a theoretical low de-gree p-modes eigenfrequency pattern and theoretical modelines widths (Houdek et al. 1999). The maximum ampli-tudes were assumed to follow a Gaussian distribution infrequency. Using a Monte-Carlo approach, the maximumamplitude was then determined in order to obtain com-parable energy per frequency bin in the synthetic and ob-served spectra. On the other hand, except for the mode re-sponse function, the method by Kjeldsen et al. (2005) doesnot impose a priori constraints concerning the modes. Thismethod can then be considered to be more reliable thanthe method by Mosser et al. (2005).We compare in Fig. 1 v HARPS with the calculated modesurface velocity v s (Eq. (1)). However, in order to have aconsistent comparison, we have smoothed v s quadraticallyover four radial orders. We note ∆ v s the 1- σ error bars as-sociated with v s . They are derived from ∆Γ, the 1- σ errorbars associated with Γ. As pointed out in Paper I, the un-certainty related to our knowledge of the metal abundance Z for HD 49933 results in an uncertainty about the deter-mination of P . However, in terms of amplitude, this uncer-tainty is of the order of 5 % ; this is negligible compared tothe uncertainty that arises from ∆Γ (ranging between 25 %to 50 % in terms of amplitude).The difference between computed values and observa-tions is shown in the bottom panel of Fig. 1. This differencemust be compared with σ v , the 1- σ interval resulting fromthe errors associated with v s and this in turn associatedwith v HARPS , that is σ v ≡ p ∆ v s + ∆ v . As seen inFig. 1, except at high frequency ( ν & . v s lie well in the 1- σ v domain. However, there is aclear disagreement at high frequencies where the computedmode surface velocities overestimate the observations. Thisdisagreement is attributed to the assumptions in the theo-retical model of stochastic excitation (see Sect. 4.5).Assuming the 3D model with the solar abundance re-sults in significantly larger v s . In that case the differencesbetween computed v s and the seismic constraint are in gen-eral larger than 2- σ v . This shows that ignoring the metalabundance of HD 49933 would result in a larger discrepancybetween v s and v HARPS .
3. Amplitudes of mode in intensity
Fluctuations of the luminosity L due to variations of thestellar radius can be neglected since we are looking at high n order modes; accordingly the bolometric mode intensityfluctuations δL are mainly due to variations of the effectivetemperature, that is: δLL = 4 δT eff T eff (3)As in Kjeldsen & Bedding (1995), we now assume that δT eff is proportional to the variation of the temperature inducedby the modes at the photosphere (i.e. at T = T eff ). Thisassumption is discussed in Sect. 4.3. Assuming further lowdegree ℓ and adiabatic oscillations, one can derive a relation Fig. 1. Top:
Intrinsic mode surface velocity as a function ofthe mode frequency ( ν ). The filled circles connected by thethick solid line correspond to the mode surface velocity ( v s )derived for HD 49933 according to Eq. (1), where the modeexcitation rates P are derived as explained in Paper I andthe mode linewidths and their associated error bars are de-rived by Benomar et al. (2009) from the CoRoT data. Thethick dashed line corresponds to the mode velocity asso-ciated with the model with [Fe/H]= 0. The thick and redsolid line corresponds to the amplitude spectrum derivedfrom the seismic observations obtained with the HARPSspectrograph (see text). The dotted line corresponds to the1- σ domain associated with this measurement. Bottom:
Differences between v s and v HARPS . The 1- σ error bars cor-respond to σ v ≡ p ∆ v s + ∆ v (see text).between δT eff / T eff and the radial mode velocity v that is: δT eff T eff = (Γ − (cid:12)(cid:12)(cid:12)(cid:12) ω osc ξ r d ξ r d r (cid:12)(cid:12)(cid:12)(cid:12) v (4)where Γ = ∇ ad Γ + 1, ∇ ad is the adiabatic temperaturegradient, Γ = (cid:16) ∂ ln P g ∂ ln ρ (cid:17) s , ξ r the radial mode eigendisplace-ment, and v the mode velocity at the photosphere . Finally,according to Eqs. (3) and (4), one has: (cid:18) δLL (cid:19) = 4 β (Γ − (cid:12)(cid:12)(cid:12)(cid:12) ω osc ξ r d ξ r d r (cid:12)(cid:12)(cid:12)(cid:12) v (5) R. Samadi et al.: The CoRoT target HD 49933: 2 - Comparison with observations where v is computed using Eq. (1) with h =0 (the photo-sphere), that is: v = s P π M Γ (6)where M is the mode mass evaluated at the photosphere( h =0).In Eq. (5), β is a free parameter introduced so thatEq. (5) gives, in the case of the solar p modes, the cor-rect maximum in δL/L . Indeed, Eq. (5) applied to thecase of the solar p modes, overestimates by ∼
10 times themode amplitudes in intensity. This important discrepancyis mainly a consequence of the adiabatic approximation.From the SOHO/GOLF seismic data (Baudin et al.2005), we derive the maximum of the solar mode (intrinsic)surface velocity, that is 32.6 ± ξ r , weinfer the maximum of mode velocity at the photosphere,that is 18.5 ± ± β = 0 . ±
10 %. We have checked that this calibrationdepends very little on the choice of the chemical mixture(see also Sect. 4.3). We then adopt this value for the caseof HD 49933.
The seismic analysis by Benomar et al. (2009) provides theapparent amplitude A ℓ of the ℓ =0, 1 and 2 modes andthe associated error bars. However, the CoRoT measure-ments A ℓ correspond to relative intensity fluctuations inthe CoRoT passband. Furthermore, the observed (appar-ent) mode amplitudes depend on the degree ℓ . Therefore,to transform them into bolometric and intrinsic intensityfluctuations normalised to the radial modes, we divide themby the CoRoT response function, R ℓ , derived here for ℓ =0,1 and 2, following Michel et al. (2009). The adopted valuesfor R ℓ are: R = 0 . R = 1 .
10, and R = 0 .
66. We finallyderive the bolometric intensity fluctuations normalised tothe radial modes according to:( δL/L ) CoRoT = vuut (cid:18) A R (cid:19) + (cid:18) A R (cid:19) + (cid:18) A R (cid:19) ! . (7)We shall stress that the differences between theamplitudes derived by Benomar et al. (2009) and byAppourchaux et al. (2008) are smaller than the 1- σ errorbars. Furthermore, these amplitudes are in agreement withthose found by Michel et al. (2008), using a different tech-nique. We compute the mode amplitudes in terms of bolomet-ric intensity fluctuations, δL/L , according to Eqs. (5) and(6) (see Sect. 3.1). As for v s , the uncertainty associatedwith the measured mode linewidths, Γ, put uncertain-ties on the theoretical values of δL/L . Furthermore, theuncertainty associated with the calibrated factor β (seeSect. 3.1) also puts an additional uncertainty on δL/L .From here on, ∆( δL/L ) will refer to the 1- σ uncertainties associated with δL/L . Accordingly, we have ∆( δL/L ) =( δL/L ) q(cid:0) ∆Γ / Γ (cid:1) + (∆ β/β ) , where ∆Γ (reps. ∆ β ) isthe 1- σ uncertainty associated with Γ (resp. β ).Figure 2 compares, as a function of the mode frequency, δL/L to the CoRoT measurements: ( δL/L ) CoRoT . The dif-ference between our calculations and the observations isshown in the bottom panel. As for the velocity, this differ-ence must be compared with σ L , the 1- σ interval resultingfrom the association of the 1- σ error bars ∆( δL/L ) andthe 1- σ error, ∆( δL/L ) CoRoT , associated with the CoRoTmeasurements. Accordingly, we have σ L ≡ √ a + b where a ≡ ∆( δL/L ) and b ≡ ∆( δL/L ) CoRoT .As seen in Fig. 2, below ν . . δL/L are within approximately 1- σ L in agreement with( δL/L ) CoRoT . However, above ν ∼ . δL/L and ( δL/L ) CoRoT exceed 2- σ L .Assuming a solar abundance ([Fe/H]= 0) results in aclear overestimation of ∆( δL/L ) CoRoT . Furthermore, calcu-lations which assume the Grevesse & Noels (1993) chemicalmixture result in mode amplitudes larger by ∼
15 %.Both in terms of intensity and velocity, differencesbetween the calculated mode amplitudes and those de-rived from the observations (CoRoT and HARPS) are ap-proximately within the 1- σ domain below ν ∼ . δL/L ) peaks at ν max ≃ . v s at ν max ≃ . δL/L ) CoRoT peaks at ν max ≃ . v HARPS peaksat ν max ≃ . ν max between theobservations (CoRoT and HARPS) and the model can bepartially a consequence of the clear tendency at high fre-quency toward over-estimated amplitudes compared to theobservations.
4. Discussion
Uncertainties in the knowledge of T eff and log g place un-certainties on the theoretical values of P and hence on themode amplitudes ( v s and δL/L ). However, estimating theseuncertainties would require the consideration of 3D mod-els with a T eff and a log g that depart more than 1- σ fromthe values adopted in our modeling, i.e. T eff = 6750 K andlog g = 4 .
25. This is beyond the scope of our efforts sincesuch 3D models are not yet available.
As discussed in details in Samadi et al. (2008a), the com-puted mode surface velocities v s significantly depend on thechoice of the height h in the atmosphere where the modemasses are evaluated. According to Samadi et al. (2008a),seismic measurements performed with the HARPS spectro-graph reflect conditions slightly below the formation depthof the K line. Accordingly, we have evaluated by defaultthe mode masses at the optical depth where the K line isexpected to be formed (i.e. τ
500 nm ≃ . h . Indeed, evaluating the mode mass . Samadi et al.: The CoRoT target HD 49933: 2 - Comparison with observations 5 Fig. 2. Top:
Mode bolometric amplitude in intensity as afunction of the mode frequency ( ν ). The filled circles con-nected by the thick solid line correspond to the mode am-plitudes in intensity, δL/L , derived for HD 49933 accord-ing to Eq. (5) and Eq. (1) where the mode surface veloc-ity v is evaluated at the photosphere. The thick dashedsolid line corresponds to the mode amplitude in intensityassociated with the model with [Fe/H]= 0. The red tri-angles and associated error bars correspond to the modeamplitudes in intensity, ( δL/L ) CoRoT , obtained by fromthe CoRoT data (Benomar et al. 2009). These measure-ments have been translated into bolometric amplitudes fol-lowing Michel et al. (2009).
Bottom:
Same as top for thedifference between δL/L and ( δL/L ) CoRoT . The 1- σ errorbars correspond here to √ a + b where a ≡ ∆( δL/L ) and b ≡ ∆( δL/L ) CoRoT (see text).at the photosphere results in values of v s which are about15 % lower and hence would reduce the discrepancy withthe HARPS observations. On the other hand, evaluatingthe mode mass one pressure scale height ( ∼
300 km at thephotosphere) above h = 350 km results in an increase of v s of about 10 %. A more rigorous approach to derive the dif-ferent heights in the atmosphere where the measurementsare sensitive would require a dedicated modeling (see a dis-cussion in Samadi et al. 2008a). Sensitivity to the location:
The derivation of Eq. (4) (or equivalently Eq. (5)) is basedon the assumption that δT eff ∝ δT | T = T eff (see Sect. 3.1).This is quite a arbitrary simplification. In order to checkhow sensitive our results are to this assumption, we havecomputed Eq. (5) and Eq. (6) for two different positionsin the atmosphere. The first position, h = h , is chosenone pressure scale height ( ≃
300 km) above the photo-sphere, which corresponds to an optical depth of τ ∼ . h = h , is chosen one pressure scaleheight beneath the photosphere, that is around τ ∼ ∼ . ∼ . ∼
20 % when h = h and are inturn almost unchanged when h = h . Since the fluctuationsof L induced by the oscillations are mostly due to temper-ature changes that occur around an optical depth of theorder of the unit, we can conclude that our calculations arealmost insensitive to the choice of the layer in the visibleatmosphere where δT is evaluated. Non-adiabatic effects:
The modes are measured at the surface of the star wherenon-adiabatic interactions between the modes and convec-tion as well as radiative losses of the modes are important.Assuming Eq. (4) is then a crude approximation. In fact,it is clearly non-valid in the case of the Sun since it re-sults in a severe over-estimation of the solar mode ampli-tudes in intensity (see Sect. 3.1). Avoiding this approxima-tion requires non-adiabatic eigenfunctions computed witha time-dependent convection model. However, such models(e.g. Grigahc`ene et al. 2005; Balmforth 1992) are subjectto large uncertainties, and there is currently no consensusabout the non-adiabatic mechanisms that play a signifi-cant role (see e.g. the recent review by Houdek 2008). Forinstance, parameters are usually introduced in the theoriesso that they cannot be used in a predictive way.In the present study, we adopt by default the adiabaticapproximation and introduce in Eq. (5) the parameter β calibrated with helioseismic data. We show here that de-spite the deficiency of the quasi-adiabatic approximation,it nevertheless provides the correct scaling, at least at lowfrequency and at the level of the present seismic precisions.As an alternative approach, comparing the spectrum ob-tained from the 3D models in intensity with that obtainedin velocity can provide valuable information concerning theintensity-velocity relation, in particular concerning the de-parture from the adiabatic approximation and the sensi-tivity to the surface metal abundance. We have started tocarry out such a study. For the velocity, the (few) acous-tic modes trapped in the simulated boxes can be extractedand their properties measured. But this was impossible todo for the intensity with the simulations at our disposalbecause the computed spectrum for the intensity is domi-nated by the granulation background. As a consequence itis not possible to extract the mode amplitudes in intensitywith sufficient accuracy . A comparison between the spec-tra obtained from the 3D models requires a much longertime series (work in progress). R. Samadi et al.: The CoRoT target HD 49933: 2 - Comparison with observations
Sensitivity to the metal abundance:
We have shown in this study how the mode amplitudesin the velocity are sensitive to the surface metal abun-dance. An open question is how sensitive is the intensity-velocity relation in general to the metal abundance? A the-oretical answer to this question would require a realisticand validated non-adiabatic treatment. The pure numeri-cal approach mentionned above can also in principle pro-vide some answers to this question. However, as discussedabove, this approach is not applicable with the time se-ries at our disposal. Concerning the quasi-adiabatic rela-tion of Eq. (5): a change of the metal abundance has adirect effect on Γ and an indirect effect on the propertiesof the (radial) eigen-displacement ξ r . However, the com-parison between the metal-poor 3D model (S1) and the 3Dmodel with the solar abundances (S0) shows that – at afixed frequency ω osc – the ratio ( δL/L ) /v , which is equalto 4 β (Γ −
1) (d ξ r / d r ) / ( ξ r ω osc ), is almost unchanged be-tween S0 and S1 (the differences are less than ∼ β . As seen in this study, the surface metal abundance has apronounced effect on the mode excitation rates. One maythen wonder about the previous validation of the theoret-ical model of stochastic excitation in the case of the Sun(Belkacem et al. 2006; Samadi et al. 2008b). Indeed, thisvalidation was carried out with the use of a solar 3D modelbased on an ”old” solar chemical mixture (namely thoseproposed by Anders & Grevesse 1989) while the ”new”chemical mixture by Asplund et al. (2005) is characterizedby a significantly lower metal abundance.In order to adress this issue, we have first considered twoglobal 1D solar models. One model has an ”old” solar abun-dance (Grevesse & Noels 1993, model M old hereafter) whilethe second one has the ”new” abundances (Asplund et al.2005, model M new hereafter). At the surface where the ex-citation occurs, the density of the solar model M new is only ∼ old . According tothe arguments developed in Paper I, this difference in thedensity must imply a difference in the convective velocities(˜ u ) of the order of ∼ ( ρ old /ρ new ) / , where ρ old (resp. ρ new )is the surface density associated with M old (resp. M new ).Accordingly, ˜ u is expected to be ∼ new compared to M old .The next question is what is the change in the solarmode excitation rates induced by the above difference in ˜ u ?We have computed the solar mode excitation rates exactlyin the same manner as for HD 49333 by using a solar 3Dsimulations based on the ”old” abundances. We obtained arather good agreement with the different helioseismic data(see the result in Samadi et al. 2008b). To derive the solarmode excitations expected with the ”new” solar abudance,we have proceeded in a similar way as the one done in PaperI: we have increased the convective velocity ˜ u derived fromthe solar 3D model by 2 % while keeping the kinetic fluxconstant (see details in Paper I). This increase of ∼ u results in an increase of ∼
10 % of the mode excitationrates. This increase is significantly lower than the current uncertainties associated with the different helioseismic data(Baudin et al. 2005; Samadi et al. 2008b).
The discrepancy betwen theoretical calculations and ob-servations is particularly pronounced at high frequency.This discrepancy may be attributed to a canceling be-tween the entropy and the Reynolds stress contributions(see Sect. 4.5.1) or the ”scale length separation” assump-tion (see Sect. 4.5.2).
The relative contribution of the entropy fluctuations to theexcitation is found to be about 30 % of the total excitation.This is two times larger than in the case of the Sun ( ∼
15 %).This can be explained by the fact HD 49933 is significantlyhotter than the Sun and, as pointed-out by Samadi et al.(2007), the larger (
L/M ) ∝ T /g , the more important therelative contribution of the entropy. Although more impor-tant than in the Sun, the contribution of the entropy fluctu-ations remains relatively smaller than the uncertainties as-sociated with the current seismic data. This is illustrated inFig. 3: the difference between theoretical mode amplitudeswhich take into account only the Reynolds stress contribu-tion ( C R , see Eq. (3) of Paper I) and those that includeboth contributions (entropy and Reynolds stress) is lowerthan σ v . In terms of amplitudes, the entropy fluctuationscontribute only ∼
15 % of the global amplitude. This issignificantly smaller than the uncertainties associated withthe current seismic measurements. Seismic data of a betterquality are then needed to constrain the entropy contribu-tion and its possible canceling with the Reynolds stress.Numerical simulations show some cancellation betweenthe entropy source term and the one due to the Reynoldsstress (Stein et al. 2004). However, in the present theoreti-cal model of stochastic excitation, the cross terms betweenthe entropy fluctuations and the Reynolds stresses vanish(see Samadi & Goupil 2001). This is a consequence of thedifferent assumptions concerning the entropy fluctuations(see Samadi & Goupil 2001, see also the recent discussionin Samadi et al. (2008b)). Accordingly, the entropy sourceterm is included as a source independent from the Reynoldsstress contribution. As suggested by Houdek (2006), a par-tial canceling between the entropy fluctuations and theReynolds stress can decrease the mode amplitudes of F-type stars and reduce the discrepancy between the theoret-ical calculations and the observations.There is currently no theoretical description of these in-terferences. In order to have an upper limit of the interfer-ences, we assume that both contributions locally and fully interfer. This assumption leads to the computation of theexcitation rates per unit mass as: d P dm = (cid:18) d P dm (cid:19) RS + (cid:18) d P dm (cid:19) E − s(cid:18) d P dm (cid:19) RS (cid:18) d P dm (cid:19) E (8)where ( d P /dm ) RS and ( d P /dm ) E are the contributionsper unit mass of the Reynolds stress and entropy respec-tively. The result is presented in Fig. 3 in terms of velocity(top pannel) and in terms of intensity (bottom pannel). . Samadi et al.: The CoRoT target HD 49933: 2 - Comparison with observations 7 The mode amplitudes are decreased by up to ∼
55 %. Inthat case, ( δL/L ) CoRoT is systematically under-estimated.Obviously, a partial canceling between the entropy contri-bution and the Reynolds stress would result in a smallerdecrease.We have assumed here that the cancellation betweenthe two terms is independent of the mode frequency (seeEq. (8)). However, according to Stein et al. (2004), the levelof the cancellation depends on the frequency (see theirFig. 8). In particular, for F-type stars, the cancellation isexpected to be more important around and above the peakfrequency.As a conclusion, the existence of a partial cancelingbetween the entropy fluctuations and the Reynolds stresscan decrease the mode amplitude and could improve theagreement with the seismic observations at high frequency.However, there is currently no theoretical modeling of theinterference between theses two terms. Further theoreticaldevelopements are required.
The ”scale length separation” assumption (see the reviewby Samadi et al. 2008b) consists of the assumption thatthe eddies contributing effectively to the driving have acharacteristic length scale smaller than the mode wave-length. This assumption is justified for a low Mach num-ber ( M t ). However, this approximation is less valid in thesuper-adiabatic region where M t reaches a maximum (forthe Sun M t is up to 0.3) and accordingly affects the high-frequency modes more. This approximation is then ex-pected to be even more questionable for stars hotter thanthe Sun, since M t increases with T eff . This spatial separa-tion can be avoided, however if the kinetic energy spectrumassociated with the turbulent elements ( E ( k )) is properlycoupled with the spatial dependence of the modes (workin progress). In that case, we expect a more rapid decreaseof the driving efficiency with increasing frequency than inthe present formalism where the spatial dependence of themodes is totally decoupled from E ( k ) (i.e. ”scale lengthseparation”).
5. Conclusion
From the mode linewidths measured by CoRoT and theo-retical mode excitation rates derived for HD 49933, we havederived the expected mode surface velocities v s which wehave compared with v HARPS , the mode velocity spectrumderived from the seismic observations obtained with theHARPS spectrograph (Mosser et al. 2005). Except at highfrequency ( ν & v s and v HARPS is within the 1- σ domain associated withthe seismic data from the HARPS spectrograph. However,there is a clear tendency to overestimate v HARPS above ν ∼ calibrated quasi-adiabatic approximation to re-late the mode velocity to the mode amplitude in inten-sity (Eq. 5), we have derived for the case of HD 49933 theexpected mode amplitudes in intensity. Computed modeintensity fluctuations, δL/L , are within 1- σ in agreementwith the seismic constraints derived from the CoRoT data(Benomar et al. 2009). However, as for the velocity, there isa clear tendency at high frequency ( ν & δL/L compared to the CoRoT observations. Fig. 3. Top:
Same as Fig. 1. The thin dashed line corre-sponds to a calculation that takes only the contribution ofthe Reynolds stress into account. The dot-dashed line corre-sponds to a calculation in which we have assumed that thecontribution of the Reynolds stress interferes totally withthat of the entropy fluctuations (see text). The thick solidline has the same meaning as in Fig 1.
Bottom:
Same astop for δL/L . The triangles and associated error bars havethe same meaning as in Fig. 2Calculations that assume a solar surface metal abun-dance result, both in velocity and in intensity, in ampli-tudes larger by ∼
35 % around the peak frequency ( ν max ≃ R. Samadi et al.: The CoRoT target HD 49933: 2 - Comparison with observations test the validity of the calibrated quasi-adiabatic relation(Eq. (5)). Our comparison shows that this relation providesthe correct scaling, at least at the level of the present seis-mic precisions, .Both in terms of surface velocity and of intensity, thedifferences between predicted and observed mode ampli-tudes are within the 1- σ uncertainty domain, except athigh frequency. This result then validates for low frequencymodes the basic underlying physical assumptions includedin the theoretical model of stochastic excitation for a starsignificantly different in effective temperature, surface grav-ity, turbulent Mach number ( M t ) and metallicity comparedto the Sun or α Cen A.As discussed in Sect. 4, the clear discrepancy betweenpredicted and observed mode amplitudes seen at high fre-quency may have two possible origins: First, a cancelingbetween the entropy contribution and the Reynolds stressis expected to occur and to be important around and abovethe frequency of the maximum of the mode excitation rates(see Sect. 4.5.1). Second, the assumption called the “scalelength separation” (Samadi et al. 2008b) may also resultin an over-estimation of the mode amplitudes at high fre-quency (see Sect. 4.5.2). These issues will be investigatedin a forthcoming paper.
Acknowledgements.
The CoRoT space mission, launched onDecember 27 2006, has been developed and is operated by CNES,with the contribution of Austria, Belgium, Brasil, ESA, Germanyand Spain.We are grateful to the referee for his pertinent comments. We areindebted to J. Leibacher for his careful reading of the manuscript.K.B. acknowledged financial support from Li`ege University throughthe Subside F´ed´eral pour la Recherche 2009.