The CoRoT target HD 49933: 1- Role of the metal abundance
R. Samadi, H.-G. Ludwig, K. Belkacem, M.J. Goupil, M.-A. Dupret
aa r X i v : . [ a s t r o - ph . S R ] N ov Astronomy & Astrophysics manuscript no. 11867 c (cid:13)
ESO 2018June 27, 2018
The CoRoT ⋆ target HD 49933: R. Samadi , H.-G. Ludwig , K. Belkacem , , M.J. Goupil , and M.-A. Dupret , 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´ee du 6 Aoˆut 17 – B 4000 Li`ege, BelgiumJune 27, 2018
ABSTRACT
Context.
Solar-like oscillations are stochastically excited by turbulent convection at the surface layers of the stars.
Aims.
We study the role of the surface metal abundance on the efficiency of the stochastic driving in the case of theCoRoT target HD 49933.
Methods.
We compute two 3D hydrodynamical simulations representative – in effective temperature and gravity – ofthe surface layers of the CoRoT target HD 49933, a star that is rather metal poor and significantly hotter than theSun. One 3D simulation has a solar metal abundance, and the other has a surface iron-to-hydrogen, [Fe/H], abundanceten times smaller. For each 3D simulation we match an associated global 1D model, and we compute the associatedacoustic modes using a theoretical model of stochastic excitation validated in the case of the Sun and α Cen A.
Results.
The rate at which energy is supplied per unit time into the acoustic modes associated with the 3D simulationwith [Fe/H]=-1 is found to be about three times smaller than those associated with the 3D simulation with [Fe/H]=0.As shown here, these differences are related to the fact that low metallicity implies surface layers with a higher meandensity. In turn, a higher mean density favors smaller convective velocities and hence less efficient driving of the acousticmodes.
Conclusions.
Our result shows the importance of taking the surface metal abundance into account in the modeling ofthe mode driving by turbulent convection. A comparison with observational data is presented in a companion paperusing seismic data obtained for the CoRoT target HD 49933.
Key words. convection - turbulence - atmosphere - Stars: oscillations - Stars: individual: HD 49933 - Sun: oscillations
1. Introduction
Using the measured linewidths and the amplitudes of thesolar acoustic modes, it has been possible to infer the rate atwhich energy is supplied per unit time into the solar acous-tic modes. Using these constraints, different models of modeexcitation by turbulent convection have been extensivelytested in the case of the Sun (see e.g. recent reviews bySamadi et al. (2008b) and Houdek (2006)). Among the dif-ferent approaches, we can distinguish pure theoretical ap-proaches (e.g. Samadi & Goupil 2001; Chaplin et al. 2005),semi-analytical approaches (e.g. Samadi et al. 2003b,a) andpure numerical approaches (e.g. Nordlund & Stein 2001;Stein et al. 2004; Jacoutot et al. 2008). The advantage ofa theoretical approach is that it easily allows massive com-putation of the mode excitation rates for a wide varietyof stars with different fundamental parameters (e.g. effec-tive temperature, gravity) and different surface metal abun-dance. However, pure theoretical approaches are based oncrude or simplified descriptions of turbulent convection.On the other hand, a semi-analytical approach is gener- ⋆ 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] ally more realistic since the quantities related to turbulentconvection are obtained from 3D hydrodynamical simula-tion. 3D hydrodynamical simulations are at this point intime too time consuming, so that a fine grid of 3D modelswith a sufficient resolution in effective temperature ( T eff ),gravity (log g ) and surface metal abundance ( Z ) is not yetavailable. In the present paper, we study and provide a pro-cedure to interpolate for any value of Z the mode excitationrates P between two 3D simulations with different Z butthe same T eff and log g . With such interpolation procedureit is no longer required to have at our disposal a fine gridin Z of 3D simulations.The semi-analytical mode that we consider here is basedon Samadi & Goupil (2001)’s theoretical model with theimprovements proposed by Belkacem et al. (2006a). Thissemi-analytical model satisfactorily reproduces the solarseismic data (Samadi et al. 2003a; Belkacem et al. 2006b).Recently, the seismic constraints obtained for α Cen A(HD 128620) have provided an additional validation ofthe basic physical assumptions of this theoretical model(Samadi et al. 2008a). The star α Cen A has a sur-face gravity (log g = 4 . g ⊙ = 4.438), but its effective temperature ( T eff =5810 K) does not significantly differ from that of the Sun( T eff , ⊙ = 5780 K). The higher T eff , the more vigorous the R. Samadi et al.: The CoRoT target HD 49933: 1 - Role of the metal abundance convective velocity at the surface and the stronger the driv-ing by turbulent convection (see e.g. Houdek et al. 1999).For main sequence stars with a mass M . . M ⊙ , an in-crease of the convective velocity is expected to be associatedwith a larger turbulent Mach number, M t (Houdek et al.1999). However, the theoretical models of stochastic exci-tation are strictly valid in a medium where M t is – as in theSun and α Cen A – rather small. Hence, the higher M t , themore questionable the different approximations and the as-sumptions involved in the theory (see e.g. Samadi & Goupil2001). It is therefore important to test the theory with an-other star characterized by a T eff significantly higher thanin the Sun.Furthermore, the star α Cen A has an iron-to-hydrogenabundance slightly larger than the Sun, namely [Fe/H]=0.2 (see Neuforge-Verheecke & Magain 1997). However, themodeling performed by Samadi et al. (2008a) for α Cen Aassumes a solar iron abundance ([Fe/H]=0). According toHoudek et al. (1999), the mode amplitudes are expected tochange with the metal abundance. However, Houdek et al.(1999)’s result was obtained on the basis of a mixing-lengthapproach involving several free parameters and by using atheoretical model of stochastic excitation in which a freemultiplicative factor is introduced in order to reproduce themaximum of the solar mode excitation rates. Therefore, it isimportant to extend Houdek et al. (1999)’s study by usinga more realistic modeling based on 3D hydrodynamical sim-ulation of the surface layers of stars and a theoretical modelof mode driving that reproduces – without the introductionof free parameters – the available seismic constraints.To this end, the star HD 49933 is an interesting casefor three reasons: First, this star has T eff = 6780 ± g ≃ . ± .
13 (Bruntt et al.2008) and [Fe/H] ≃ − .
37 dex (Solano et al. 2005;Gillon & Magain 2006). The properties of its surface lay-ers are thus significantly different from those of the Sunand α Cen A. Second, HD 49933 was observed in Dopplervelocity with the HARPS spectrograph. A seismic analysisof these data performed by Mosser et al. (2005) has pro-vided the maximum of the mode surface velocity ( V max ).Third, the star was more recently observed continuouslyin intensity by CoRoT during 62 days. Apart from ob-servations for the Sun, this is the longest seismic observa-tion ever peformed both from the ground and from space.This long term and continuous observation provides a veryhigh frequency resolution ( ∼ . µ Hz). The seismic analy-sis of these observations undertaken by Appourchaux et al.(2008) or more recently by Benomar et al. (2009) have pro-vided the direct measurements of the mode amplitudesand the mode linewidths with an accuracy not previouslyachieved for a star other than the Sun.We consider two 3D hydrodynamical simulations rep-resentative – in effective temperature and gravity – ofthe surface layers of HD 49933. One 3D simulation has[Fe/H]=0, while the second has [Fe/H]= -1. For each 3Dsimulation, we match an associated global 1D model andcompute the associated acoustic modes and mode excita-tion rates, P . This permits us to quantify the variation of P induced by a change of the surface metal abundance Z .From these two sets of calculation, we then deduce P forHD 49933 by taking into account the observed iron abun-dance of the star (i.e. [Fe/H]=-0.37). In a companion paper(Samadi et al. 2009, hereafter Paper II), we will use thesetheoretical calculations of P and the mode linewidths ob- tained from the seismic analysis of HD 49933 performedwith the CoRoT data to derive the expected mode ampli-tudes in HD 49933. These computed mode amplitudes willthen be compared with the observed ones. This compari-son will then constitute a test of the stochastic excitationmodel with a star significantly different from the Sun and α Cen A. It will also constitute a test of the procedure pro-posed here for deriving P for any value of Z between two3D simulations with different Z .The present paper is organised as follows: we first de-scribe in Sect. 2 the method to compute the theoreticalmode excitation rates associated with the two 3D hydro-dynamical simulations. Next, the effects on P of a differentsurface metal abundance are presented in Sect. 3. Then, bytaking into account the actual iron abundance of HD 49933,we derive theoretical values of P expected for HD 49933.Finally, Sect. 5 is dedicated to our conclusions.
2. Calculation of mode excitation rates
The energy injected into a mode per unit time P is givenby the relation (see Samadi & Goupil 2001; Belkacem et al.2006b): P = 18 I (cid:0) C R + C S (cid:1) , (1)where C R and C S are the turbulent Reynolds stress andentropy contributions, respectively, and I = Z M dm | ξ r | (2)is the mode inertia, ξ r is the adiabatic radial mode dis-placement and M is the mass of the star. The expressionsfor C R and C S are given for a radial mode with frequency ω osc by C R = 64 π Z dm ¯ ρ ˜ u k ω K w f r S R ( r, ω osc ) , (3) C S = 16 π ω Z dm ( α s ˜ s ˜ u ) ¯ ρ k ω g r S s ( r, ω osc ) (4)where we have defined the “source functions”: S R ( r, ω osc ) = k ω ˜ u Z d kk E ( k ) × Z d ω χ k ( ω + ω osc ) χ k ( ω ) (5) S s ( r, ω osc ) = k ω ˜ u ˜ s Z d kk E ( k ) E s ( k ) × Z d ω χ k ( ω + ω osc ) χ k ( ω ) (6)where P is the gas pressure, ρ the density, s the entropy,¯ ρ the equilibrium density profile, α s ≡ ( ∂P/∂s ) ρ , f r ≡ (d ξ r / d r ) and g r are two functions that involve the firstand second derivatives of ξ r respectively, k is the wavenum-ber, E ( k ) is the turbulent kinetic energy spectrum, E s ( k )is the spectrum associated with the entropy fluctuations( s ), ˜ s is the rms of s , χ k is the time-correlation functionassociated with the velocity, ˜ u is a characteristic velocitydefined in a way that 3 ˜ u = h u i , h . i refers to horizontal . Samadi et al.: The CoRoT target HD 49933: 1 - Role of the metal abundance 3 and time average, u is the turbulent velocity field, and fi-nally K w ≡ h u z i / h u z i is the Kurtosis (see Belkacem et al.2006a,b, for details). Furthermore, we have introduced forconvenience the characteristic frequency ω and the char-acteristic wavenumber k : ω ≡ k ˜ u (7) k ≡ π Λ (8)where Λ is a characteristic size derived from E ( k ) as ex-plained in Samadi et al. (2003b). Note that the introduc-tion of the term k ω ˜ u − in the RHS of Eq. (5) and theterm k ω ˜ u − ˜ s − in the RHS of Eq. (6) ensure dimension-less source functions.The kinetic spectrum E ( k ) is derived from the 3D sim-ulation as detailled in Samadi et al. (2003b). As shown bySamadi et al. (2003b), the k -dependence of E s ( k ) is similarto that of the E ( k ). Accordingly, we assume E s ∝ E .In Samadi et al. (2008a), two different analytical func-tions for χ k ( ω ) have been considered, namely a Lorentzianfunction and a Gaussian one. In the present study we willin addition derive χ k ( ω ) directly from the 3D simulationsas detailled in Samadi et al. (2003a). Once χ k ( ω ) is derivedfrom the 3D simulation, it is implemented in Eq. (5) andEq. (6).We compute the mode excitation as detailled inSamadi et al. (2008a): all required quantities – except ξ r , I and ω osc – are obtained directly from two 3D hydro-dynamical simulations representative of the outer layers ofHD 49933, whose characteristics are described in Sect. 2.2below.The quantities related to the modes ( ω osc , I and ξ r ) are calculated using the adiabatic pulsationcode ADIPLS (Christensen-Dalsgaard & Berthomieu 1991)from 1D global models. The outer layers of these 1D modelsare derived from the 3D simulation as described in Sect. 2.3. We computed two 3D radiation-hydrodynamical model at-mospheres with the code CO BOLD (Freytag et al. 2002;Wedemeyer et al. 2004). One 3D simulation had a solariron-to-hydrogen [Fe/H]=0.0 while the other had [Fe/H]=-1.0. The 3D model with [Fe/H]=0 (resp. [Fe/H]=-1) willbe hereafter referred to as model S0 (resp. S1). The as-sumed chemical composition is similar (in particular forthe CNO elements) to that of the solar chemical composi-tion proposed by Asplund et al. (2005). The abundances ofthe α -elements in model S1 were assumed to be enhancedby 0.4 dex. For S0 we obtain Z/X = 0.01830 and Y=0.249,and for S1
Z/X = 0.0036765 and Y =0.252. Both 3D sim-ulations have exactly the same gravity (log g =4.25) andare very close in effective temperature ( T eff ). Both modelsemploy a spatial mesh with 140 × ×
150 grid points, anda physical extent of the computational box of 16 . × . × . . The equation of state takes into account theionisation of hydrogen and helium as well as the formationof H molecules according to the Saha-Boltzmann statis-tics. The wavelength dependence of the radiative transferis treated by the opacity binning method (Nordlund 1982;Ludwig 1992; V¨ogler et al. 2004) using five wavelength binsfor model S0 and six for model S1. Detailed wavelength-dependent opacities were obtained from the MARCS model atmosphere package (Gustafsson et al. 2008). Table 1 sum-marizes the characteristics of the 3D models. The effectivetemperature and surface gravity correspond to the param-eters of HD 49933 within the observational uncertainties,while the two metallicities bracket the observed value.For each 3D simulation, two time series were built. Onehas a long duration (38h and 20h for S0 and S1, respec-tively) and a low sampling frequency (10 mn). This timeseries is used to compute time averaged quantities (¯ ρ , E ( k ),etc.). The second time series is shorter (8.8h and 6.8h forS0 and S1, respectively), but has a high sampling frequency(1 mn). Such high sampling frequency is required for thecalculation of χ k ( ω ). Indeed, the modes we are looking atlie between ν ≈ .
25 mHz and ν ≈ . Label [Fe/H]
Y Z Z/X T eff [K]S0 0 0.249 13 . − ± .
74 10 − ± Table 1.
Characteristics of the 3D simulations.The two 3D simulations extend up to T = 100 000 K.However, for T &
30 000 K, the 3D simulations are notcompletely realistic. First of all, the MARCS-based opaci-ties are provided only up to a temperature of 30 000 K; forhigher temperatures the value at 30 000 K is assumed. Notethat we refer to the opacity per unit mass here. For the ra-diative transfer the opacity per unit volume is the relevantquantity, i.e. the product of opacity per mass unit and den-sity. Since in the simulation the opacity is still multiplied ateach position with the correct local density, the actual errorwe make when extrapolating the opacity is acceptable.Another limitation of the simulations is the restrictedsize of the computational box which does not allow for afull development of the largest flow structures, again in thelayers above T ≃
30 000 K. Two hints make us believe thatthe size of the computational domain is not fully sufficient:i) in the deepest layers of the simulations there is a ten-dency that structures align with the computational grid;ii) the spatial spectral power P of scalar fields in a hor-izontal layer does not tend towards the expected asymp-totic behaviour P × k for low spatial wavenumber k . Wenoticed this shortcoming only after the completion of thesimulation runs. To mitigate its effect in our analysis, wewill later by default integrate the mode excitation ratesup to T = 30 000 K. However, for comparison purposes,some computations have been extended down to the bot-tom of the 3D simulations. For S0, the layers located below T ≃
30 000 K contribute only by .
10 % to the excitation ofthe modes lying in the frequency range where modes havethe most chance to be detected ( ν ≃ . − . ∼ BOLD on the one hand side implicitely by thenumerical scheme (Roe-type approximate Riemann solver),and on the other hand explicitely by a sub-grid model ac-cording to the classical Smagorinsky (1963) formulation.Jacoutot et al. (2008) found that computed mode excita-tion rates significantly depend on the adopted sub-gridmodel. Samadi et al. (2007) have found that solar mode ex-
R. Samadi et al.: The CoRoT target HD 49933: 1 - Role of the metal abundance citation rates computed in the manner of Nordlund & Stein(2001), i.e., using data directly from the 3D simulation, de-crease as the spatial resolution of the solar 3D simulationdecreases. As a conclusion the spatial resolution or the sub-grid model can influence computed mode excitation rates(see a discussion in Samadi et al. 2008a). However, concern-ing the spatial resolution and according to Samadi et al.(2007)’s results, the present spatial resolution (1/140 ofthe horizontal size of the box and about 1/150 of the ver-tical extent of the simulation box) is high enough to ob-tain accurate computed energy rates. The increased spa-tial resolution of our models in comparison to the work ofJacoutot et al. (2008) reduces the impact of the unresolvedscales.
For each 3D model we compute an associated 1D globalmodel. The models are built in the manner of Trampedach(1997) as detailled in Samadi et al. (2008a) in such waythat their outer layers are replaced by the averaged3D simulations described in Sect. 2.2. The interior ofthe models are obtained with the CESAM code assum-ing standard physics: Convection is described accord-ing to B¨ohm-Vitense (1958)’s local mixing-length theoryof convection (MLT), and turbulent pressure is ignored.Microscopic diffusion is not included. The OPAL equationof state is assumed. The chemical mixture of the heavy ele-ments is similar to that of Asplund et al. (2005)’s mixture.As in Samadi et al. (2008a), we will refer to these modelsas “patched” models hereafter.The two models have the effective temperature and thegravity of the 3D simulations. One model is matched withS0 and has [Fe/H]=0, while the second is matched with S1and has [Fe/H]=-1. The 1D models have the same chemicalmixture as their associated 3D simulations. The parametersof the 1D patched models are given in Table 2. The stratifi-cation in density and temperature of the patched 1D modelsare shown in Fig. (1). At any given temperature the den-sity is larger in S1 as a consequence of its lower metal abun-dance. Indeed, the lower the metal abundance, the lower theopacity ; then, at a given optical depth ( τ ), the density islarger in S1 compared to S0. The photosphere correspondsto the optical depth τ = 2 /
3. Since the two 3D simulationshave approximatively the same effective temperature, thedensity in S1 is larger at optical depth τ = 2 /
3. Since thedensity in S1 increases with depth even more rapidly thanin S0, the density in S1 remains larger for τ > /
3. Effects of the metal abundance on excitationrates
The mode excitation rates ( P ) are computed for the two3D simulations according to Eqs. (1)-(6). The integrationis performed from the top of the simulated domains down to T = 30 000 K (see Sect. 2.2). In the following, P (resp. P )corresponds to the mode excitation rates associated withthe 3D model with [Fe/H]=-1 (resp. [Fe/H]=0) Fig. 1.
Mean density ¯ ρ as a function of temperature, T .The solid line corresponds to the 3D model with the metalabundance (S0) and the dashed line to metal poor 3D model(S1). The filled dots show the location where the 1D modelshave been matched to the associated 3D simulation. Figure 2 shows the effect of the assumed metal abundance ofthe stellar model on the mode excitation rates. P is foundto be three times smaller than P , i.e. p modes associatedwith the metal poor 3D model (S1) receive approximativelythree times less energy per unit time than those associatedwith the 3D model with the solar metal abundance (S0).For both 3D models, the dominant part of the driving isensured by the Reynolds stresses. The entropy fluctuationscontribute by only ∼
30 % of the total power for both S0and S1. By comparison, in the case of the Sun and α Cen Ait contributes by only ∼
15 %. Furthermore, we find thatthe contribution of the entropy source term is – as for theReynolds stress term – about three times smaller in S1 thanin S0. We conclude that the effect of the metal abundanceon the excitation rates is almost the same for the Reynoldsstress contribution and the entropy source term.
From Eqs. (1), (2), (3), (7) and (8) we show that at a givenlayer the power supplied to the modes – per unit mass – bythe
Reynolds stress is proportional to F kin Λ S R / M , where F kin is the flux of the kinetic energy, which is proportionalto ¯ ρ ˜ u , Λ is a characteristic length (see Sect. 2.1) and M isthe mode mass defined as: M = Iξ r (9)where ξ r is the mode displacement evaluated at the layerin the atmosphere where the mode is measured.The power supplied to the modes – per unit mass –by the entropy source term is proportional to ¯ ρ ˜ u Λ R S s where ω osc is the mode frequency, R ∝ F conv /F kin , where F conv ∝ w α s ˜ s is the convective flux, and finally ˜ s is therms of the entropy fluctuations (see Samadi et al. 2006). Werecall that the higher R , the higher the relative contributionof the entropy source to the excitation. We study below therole of M , F kin , Λ, S R , S s and R : . Samadi et al.: The CoRoT target HD 49933: 1 - Role of the metal abundance 5[Fe/H] Y Z T eff [K] R/R ⊙ M/M ⊙ α . − .
74 10 − Table 2.
Characteristics of the 1D “patched” models. α is the mixing-length parameter. Fig. 2.
Mode excitation rates P as a function of the modefrequency, ν . The solid line corresponds to the 3D modelwith the canonical metal abundance (S0) and the dashedline to the metal poor 3D model (S1). The dot-dashed linecorresponds to the mode excitation rates derived for thespecific case of HD 49933 as explained in Appendix A. Mode mass ( M ): The frequency domain, where modesare strongly excited, ranges between ν ≈ . ν ≈ . M associated with S0 are quite similar to those associated withS1 (not shown). Consequently the differences between P and P do not arise from the (small) differences in M . Kinetic energy flux ( F kin ): The larger F kin , the largerthe driving by the Reynolds stress. However, we find thatthe two 3D models have very similar F kin . This is not sur-prising since the two 3D models have very similar effectivetemperatures. This means that the differences between P and P do not arise from the (small) differences in F kin . Characteristic length ( Λ ): In the manner ofSamadi et al. (2003b) we derive from the kinetic energyspectra E ( k ) of the two 3D simulations the characteristiclength Λ (Λ = 2 π/k , see Eq. (8)) for each layer of thesimulated domain. We find that the differences in Λbetween the two 3D simulations is small and does notplay a significant role in the differences in P . This canbe understood by the fact that S0 and S1 have the samegravity. Indeed, as shown by Samadi et al. (2008a) – ata fixed effective temperature – Λ scales as the inverse of g . We conclude that the differences between P and P donot originate from the (small) differences in Λ. Source functions ( S R and S s ): The dimensionless sourcefunctions S R and S s are defined in Eqs. (5) and (6) re-spectively. Both source functions involve the eddy time-correlation function χ k ( ω ). We define ω k as the frequency width of χ k ( ω ). As shown by Samadi et al. (2003a) andas verified in the present case, ω k can be evaluated as theproduct k u k where u k is given by the relation (Stein 1967): u k = Z kk d k E ( k ) (10)where E ( k ) is normalised as: Z + ∞ d k E ( k ) = h u i ≡ ˜ u . (11)According to Eqs. (10) and (11), u k is directly proportionalto ˜ u . At a fixed k/k , we then have ω k ∝ ˜ u k = ω .As seen above, ω controls ω k , the frequency width of χ k . Then, at fixed ω osc , we can easly see from Eqs. (5) and(6) that the smaller ω , the smaller S R ( ω osc ) and S s ( ω osc ).Since ω = ˜ u k = 2 π ˜ u/ Λ and since both 3D simulationshave approximately the same Λ, smaller ˜ u results directlyin smaller ω and hence in smaller source functions.We have plotted in Fig. 3 the characteristic velocity ˜ u .This quantity is found to be up to 15 % smaller for S1 com-pared with S0. In other words, the metal poor 3D model ischaracterized by lower convective velocities. Consequently,the source functions are smaller for S1 compared to S0.Although the convective velocities differ between S0 andS1 by only 15 %, the excitation rates differ by a factor ∼ u , decrease very rapidly with ˜ u . Thisis the consequence of the behavior of the eddy-time correla-tion χ k . Indeed, this function varies with the ratio ω osc /ω k approximately as a Lorentzian function. This is why χ k varies rapidly with ˜ u (we recall that ω k ∝ ˜ u k ).In conclusion, the differences between P and P aremainly due to differences in the characteristic velocity ˜ u .In turn, the low convective velocity in S1 is a consequenceof the larger density compared to S0. Indeed, as shownin Fig. 1, the density is systematically higher in S1. Atthe layer where the modes are the most excited (i.e. at T ∼ K ), the density is ∼
50 % higher. Since the two3D models have a similar kinetic energy flux (see above),it follows that a larger density for S1 then implies lowerconvective velocities.
Relative contribution of the entropy source term ( R ): The convective flux F conv in S1 is almost identical to that ofS0. This is due to the fact that the two 3D simulations havealmost the same effective temperature. Furthermore, aspointed out above, the differences in F kin between S1 and S0are small. As a consequence, the ratio R ∝ F conv /F kin doesnot differ between the two 3D simulations. Accordingly, asfor the Reynolds contribution, the variation of the excita-tion rates with the metal abundance is only due to thesource term S S . The latter varies with ω in the samemanner as S R , which is turn the reason for As a conse-quence, the contribution of the entropy fluctuations to showthe same trend with the metal abundance as the Reynoldsstress term. R. Samadi et al.: The CoRoT target HD 49933: 1 - Role of the metal abundance
Fig. 3.
Characteristic velocity ˜ u defined in Eq. (7) as afunction of temperature, T . The solid and dashed lines havethe same meaning as in Fig. 2. The dot-dashed line corre-sponds to the solid line multiplied by γ , where γ ( T ) ≡ (¯ ρ / ¯ ρ ) / and ¯ ρ (resp. ¯ ρ ) is the mean density stratifica-tion of S0 (resp. S1)(see Appendix A).
4. Theoretical calculation of P for HD 49933 We derive the mode excitation rates P for HD 49933.According to Gillon & Magain (2006), HD 49933 has[Fe/H]=-0.37 ± P between S0 andS1 are a direct consequence of the differences in the sourcefunctions S R and S S . It follows that in order to derive P for HD 49933, we only have to derive the expected val-ues for S R and S S . As seen in Sect. 3.2, differences in S R (or in S S ) between S0 and S1 are related to the surfacemetal abundance through the surface densities that impactthe convective velocities (˜ u ). The determination of the HD49933 convective velocities allows us to determine its sourcefunction. To this end, we use the fact that the kinetic fluxis almost unchanged between S1 and S0 (see Sect. 3.2) toderive the profile of ˜ u ( T ), expected at the surface layers ofHD 49933. This is performed by interpolating in Z betweenS0 and S1, the surface density stratification representativeof the surface layers of HD 49933. The whole procedure isdescribed in Appendix A.In order to compute P for HD 49933, we then needto know Z for this star. Since we do not know its sur-face helium abundance, we will assume by default thesolar value for Y : Y = 0 . ± Z/X ) ⊙ = 0 . ± Z = 5 . − ± . − for HD 49933. Note that as-suming Grevesse & Noels (1993)’s chemical mixture yields Z = 7 . − ± . − .The result of the calculation is shown in Fig. 2.The maximum P is 1.08 ± J/s whenAsplund et al. (2005)’s chemical composition is assumed (see Appendix A). This is about 30 times larger than inthe Sun and about 14 times larger than in α Cen A.When Grevesse & Noels (1993)’s chemical mixture is as-sumed, the maximum in P is in that case equal to1.27 ± J/s, that is about 30 % larger than withAsplund et al. (2005)’s solar chemical mixture.We note that the uncertainties in the knowledge of[Fe/H] set uncertainties on P which are the on order of10 % in the frequency domain of interest.
5. Conclusion
We have built two 3D hydrodynamical simulations repre-sentative in effective temperature ( T eff ) and gravity ( g )of the surface layers of an F type star on the main se-quence. One model has a solar iron-to-hydrogen abundance([Fe/H]=0) and the other has [Fe/H]=-1. Both models havethe same T eff and g . For each 3D simulation, we have com-puted an associated “patched” 1D full model. Finally, wehave computed the mode excitation rates P associated withthe two “patched” 1D models.Mode excitation rates associated with the metal poor3D simulation are found to be about three times smallerthan those associated with the 3D simulation which has asolar surface metal abundance. This is explained by the fol-lowing connections: the lower the metallicity, the lower theopacity. At fixed effective temperature and surface gravity,the lower the opacity, the denser the medium at a given op-tical depth. The higher the density, the smaller are the con-vective velocities to transport the same amount of energyby convection. Finally, smaller convective velocities resultin a less efficient driving. On the other hand, a surface metalabundance higher than the solar metal abundance will re-sult in a lower surface density, which in turn will result in ahigher convective velocity and then in a more efficient driv-ing. Our result can then be qualitatively generalised for anysurface metal abundance.By taking into account the observed surface metal abun-dance of the star HD 49933 (i.e. [Fe/H]=-0.37), we have de-rived, using two 3D simulations and the interpolation pro-cedure developed here, the rates at which acoustic modesare expected to be excited by turbulent convection in thecase of HD 49933. These excitation rates P are found to beabout two times smaller than for a model built assuming asolar metal abundance. These theoretical mode excitationrates will be used in Paper II to derive the expected modeamplitudes from measured mode linewidths. We will thenbe able to compare these amplitudes with those derived forHD 49933 from different seismic data. This will constitutean indirect test of our procedure which permits us to in-terpolate for any value of Z the mode excitation rates P between two 3D simulations with different Z but the same T eff and log g . We must stress that a more direct valida-tion of this interpolation procedure will be to compute athird 3D model with the surface metal abundance of thestar HD 49933 and to compare finally the mode excitationrates obtained here with the interpolation procedure withthat obtained with this third 3D model. This represents along term work since several months (about three to fourmonths) are required for the calculation of this additional3D model, which is in progress. Acknowledgements.
We thank C. Catala for useful discussions con-cerning the spectrometric properties of HD 49933. We are indebted to . Samadi et al.: The CoRoT target HD 49933: 1 - Role of the metal abundance 7
J. Leibacher for his careful reading of the manuscript. K.B. acknowl-edged financial support from Li`ege University through the SubsideF´ed´eral pour la Recherche 2009.
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Online Material . Samadi et al.: The CoRoT target HD 49933: 1 - Role of the metal abundance , Online Material p 2
Appendix A: Theoretical calculation of the modeexcitation rates for HD 49933
The mode excitation rate P is inversely proportional to themode mass M (see Eqs. (9), (2) and (2)). This is why wecan derive M and M P separately in order to derive P forHD 49933. A.1. Derivation of
M P
As pointed out in Sect. 3.2, the kinetic flux F kin = ¯ ρ ˜ u is almost unchanged between S1 and S0 because both 3Dmodels have the same T eff . This has also to be the casefor HD 49933 (same T eff and same log g than S0 and S1).Therefore, the calculation of M P for HD 49933 relies onlyon the evaluation of the values reached – at a fixed modefrequency – by the source functions S R and S S .As seen in Sect. 3.2, ω = k ˜ u controls the width of χ k in a way that the source functions S R ( ω osc ) and S S ( ω osc )can be seen as functions of the dimensionless ratio ω osc /ω .The variation of E with k as well as the variation of χ k with ω/ω are shown to be similar in the two 3D simula-tions. Furthermore, S0 and S1 have approximately the samecharacteristic length Λ and hence approximately the same k ≡ π/ Λ. Therefore, the source function S R (resp. S S )associated with S0 only differs from that of S1 by the char-acteristic velocity ˜ u . This must then also be the case forHD 49933. Further, in order to evaluate the source func-tions in the case of HD 49933, we only need to know thefactor γ by which ˜ u is modified in HD 49933 with respectto S1 or S0. According to Eq. (5) (resp. Eq. (6)), multipling˜ u by γ is equivalent to replace S R ( ω osc ) (reps. S S ( ω osc )) by γ S R ( ω osc /γ ) (resp. γ S s ( ω osc /γ )).Since the kinetic flux F kin in HD 49933 must be the samefor S0 or S1, the characteristic velocity ˜ u can be derivedfor HD 49933 according to ˜ u ∗ ( T ) = ˜ u γ ∗ with γ ∗ ( T ) ≡ (¯ ρ / ¯ ρ ∗ ) / where ¯ ρ ( T ) is the mean density stratification ofS1, ˜ u ( T ) the characteritic velocity of S1 and ¯ ρ ∗ the meandensity of HD 49933. Once γ ∗ and then ˜ u ∗ are derived forHD 49933, we then compute the source functions associatedwith HD 49933. Finally, we compute M P by keeping F kin constant. We now turn to the derivation of the factor γ ∗ . A.2. Derivation of γ ∗ To derive γ ∗ at a given T , we need to know how the meandensity ¯ ρ varies with the metal abundance Z . In order tothis we consider five “standard” 1D models with five differ-ent values of Z . These 1D models are built using the samephysics as described in Sect. 2.3. Two of these models havethe same abundance as S0 and S1. All of the 1D modelshave approximately the same gravity (log g ≃ .
25) andthe same effective temperature ( T eff ≃ ρ varies with Z ratherlinearly. In order to derive ¯ ρ for HD 49933, we apply – atfixed T and between S0 and S1 – a linear interpolation of¯ ρ ( T ) with respect to Z . A.3. Derivation of M As shown in Sect. 3.2 above in the frequency domain wheremodes are detected in HD 49933, M does not change sig- nificantly between S0 and S1. This suggests that the modemasses associated with a patched 1D model with the metalabundance expected for HD 49933 would be very similarto those associated with S0 or S1. Consequently we willassume for the case of HD 49933 the same mode massesas those associated with S1, since this 3D model has a Z abundance closer to that of HD 49933. A.4. Derivation of P Before deriving P for HD 49933, we check that, from S0 andthe knowledge of P , we can approximately reproduce P ,the mode excitation rates, associated with S1 following theprocedure described above. Let γ ≡ (¯ ρ / ¯ ρ ) / . As seen inFig. 3, when we multiply ˜ u by γ ( T ) we matche ˜ u . Then,using γ ( T ) and following the procedure described above,we derive P , the mode excitation rates associated withS1 but derived from S0. The result is shown in Fig. A.1. P matches P rather well. However, there are differencesremaining in particular in the frequency domain ν =1.2-1.5 mHz. Nevertheless, the differences between P and P are in any case not significant compared to the accuracy atwhich the mode amplitudes are measured with the CoRoTdata (see Paper II). This validates the procedure, at leastat the level of the current seismic precisions. Fig. A.1.
Mode excitation rates P as a function of themode frequency ν . The thin dot-dashed line correspondsto P , the mode excitation rates derived for S1 from S0(see Appendix A4). The other lines have the same meaningas in Fig. 2.Since the metal abundance Z of HD 49933 is closer tothat of S1 than that of S0, we derive the mode excitationrates P associated with HD 49933 from S1 following theprocedure detailled above. The result is shown in Fig. 2.As expected, the mode excitation rates P associated withHD 49933 lie between those of S0 and S1, while remainingcloser to S1 than to S0. Note that the differences between P and the excitation rates derived for HD 49933 ( P ) are of thesame order as the differences seen locally between P and P . These differences remain small compared to the cur-rent seismic precisions. On the other hand the differences . Samadi et al.: The CoRoT target HD 49933: 1 - Role of the metal abundance , Online Material p 3 between P and P0