Issue 
A&A
Volume 530, June 2011



Article Number  A142  
Number of page(s)  8  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201116490  
Published online  26 May 2011 
The underlying physical meaning of the ν_{max} − ν_{c} relation
^{1}
Institut d’Astrophysique Spatiale, CNRS, Université Paris XI, 91405 Orsay Cedex, France
email: Kevin.Belkacem@ias.upsud.fr
^{2}
Institut d’Astrophysique et de Géophysique, Université de Liège, Allée du 6 Août 17, 4000 Liège, Belgium
^{3}
LESIA, UMR8109, Université Pierre et Marie Curie, Université Denis Diderot, Obs. de Paris, 92195 Meudon Cedex, France
Received: 10 January 2011
Accepted: 4 April 2011
Asteroseismology of stars that exhibit solarlike oscillations are enjoying a growing interest with the wealth of observational results obtained with the CoRoT and Kepler missions. In this framework, scaling laws between asteroseismic quantities and stellar parameters are becoming essential tools to study a rich variety of stars. However, the physical underlying mechanisms of those scaling laws are still poorly known. Our objective is to provide a theoretical basis for the scaling between the frequency of the maximum in the power spectrum (ν_{max}) of solarlike oscillations and the cutoff frequency (ν_{c}). Using the SoHO GOLF observations together with theoretical considerations, we first confirm that the maximum of the height in oscillation power spectrum is determined by the socalled plateau of the damping rates. The physical origin of the plateau can be traced to the destabilizing effect of the Lagrangian perturbation of entropy in the uppermost layers, which becomes important when the modal period and the local thermal relaxation timescale are comparable. Based on this analysis, we then find a linear relation between ν_{max} and ν_{c}, with a coefficient that depends on the ratio of the Mach number of the exciting turbulence to the third power to the mixinglength parameter.
Key words: convection / turbulence / stars: oscillations / stars: interiors
© ESO, 2011
1. Introduction
Scaling relations between asteroseismic quantities and stellar parameters such as stellar mass, radius, effective temperature, and luminosity have been observationally derived by several authors (e.g. Kjeldsen & Bedding 1995; Chaplin et al. 2008, 2009; Stello et al. 2009a) using groundbased data. More recently, the spacemissions CoRoT and Kepler confirmed those results by providing accurate and homogeneous measurements for a large sample of stars from red giants to mainsequence stars (e.g., Mosser et al. 2010). Scaling relations are essential to study a large set of stars (e.g., Kallinger et al. 2009; Stello et al. 2009b) for which, in general, little is known, to provide a first order estimate for mass and radius (e.g., Basu et al. 2010; Mosser et al. 2010), or to probe the populations of red giants (Miglio et al. 2009).
Scaling laws can also lead to a better understanding of the underlying physical mechanisms governing the energetical behaviour of modes. In particular, it has been conjectured by Brown et al. (1991) that the frequency of the maximum of the power spectrum (ν_{max}) scales as the cutoff frequency ν_{c} because the latter corresponds to a typical timescale of the atmosphere. The continuous increase of detected stars with solarlike oscillations has then confirmed this relation (e.g., Bedding & Kjeldsen 2003; Stello et al. 2009a). However, the underlying physical origin of this scaling relation is still poorly understood. Indeed, ν_{max} is associated with the coupling between turbulent convection and oscillations and results from a balance between the damping and the driving of the modes. The cutoff frequency is associated with the mean surface properties of the star and the sound speed, making the origin of the ν_{max} − ν_{c} relation very intriguing.
As a first step towards an understanding, one has to determine if the damping rate or the excitation rate is mainly responsible for the maximum of power in the observed spectra. Chaplin et al. (2008), using a theoretical approach, pointed out that in the solar case ν_{max} coincides with the plateau of the linewidth variation with frequency. We will confirm this result using observations from the GOLF instrument in the solar case. However, several questions remain to be addressed: is ν_{max} for any star directly related with the observed plateau in the modewidths variation with frequency? If the answer is positive, what is the origin of this relation? The first question is quite difficult to answer because it is expected to strongly depend on the model used for the description of the pulsationconvection interaction. Nevertheless, CoRoT observations begin to answer this and the data of several stars (HD 49933, HD 180420, HD 49385, and HD 52265) suggest that ν_{max} correponds to the plateau of the damping rates (see Benomar et al. 2009; Barban et al. 2009; Deheuvels et al. 2010; Ballot et al. 2011, for details). The second step consists in determining the main physical causes responsible for the plateau of the damping rates and its mean frequency (ν_{Γ}). Subsequently, one has to determine a general scaling law that relates the frequency of the plateau of the damping rates to the stellar parameters. In this paper, we discuss the first question and focus on the second question by deriving a theoretical relation between ν_{Γ} and ν_{c}. If one accepts the positive answer to the first question, this also provides the scaling relation between ν_{max} and ν_{c}.
This paper is organised as follows. In Sect. 2 we present the observed scaling law obtained from a homogeneous set of CoRoT data and show that the maximum mode height in the solar power spectrum coincides with a marked minimum of the modewidth when corrected from mode inertia. We then point out in Sect. 3 that this minimum is the result of a destabilizing effect in the superadiabatic region. The relation between ν_{Γ} and ν_{c} is demonstrated in Sect. 4, and conclusions are provided in Sect. 5.
2. The observed scaling law
We used the CoRoT seimological field data to ensure a homogeneous sample: HD 49933 (Benomar et al. 2009), HD 181420 (Barban et al. 2009), HD 49385 (Deheuvels et al. 2010). We also used the results on HD 50890 (Baudin et al., in prep.) and on HD 181907 (Carrier et al. 2010), a red giant, and the Sun. The characteristics of these stars are listed in Table 1, as well as the way their fundamental parameter is obtained.
For the Sun, the observational determination of ν_{c} is not obvious because of pseudomodes above the cutoff frequency (e.g. Garcia et al. 1998). Nevertheless, one can infer a theoretical relation for this frequency (e.g., Balmforth & Gough 1990), where c_{s} is the sound speed, H_{ρ} the density scale height, g the gravitational field, M the mass, R the radius, and T_{eff} the temperature at the photosphere. When scaled to the solar case, this relations becomes (1)with ν_{c ⊙ } = 5.3 mHz, and M_{⊙},R_{⊙},T_{eff ⊙ } the solar values of mass, radius, and effective temperature respectively. Note that we will assume H_{ρ} = H_{p} = P/ρg with P and ρ denoting pressure and density. This is a commonly used approximation (e.g., Stello et al. 2009a) that presupposes an isothermal atmosphere, which is sufficiently accurate for our purposes.
Using the stars listed in Table 1 and their measured ν_{max}, the relation between ν_{max} and ν_{c} is displayed in Fig. 1. It relies on two kinds of results: direct observations of ν_{max} in the spectrum of the star on one hand and estimates of the mass (M), radius (R), and effective temperature (T_{eff}) of the star on the other hand. The latter are derived from photometric or spectroscopic observations, but can be derived in some cases from stellar modelling. Here, M and R must be derived from stellar modelling and not from scaling laws because the aim of this work is to establish such a scaling law. The strict proportionality (the fitted slope is 1.01 ± 0.02) is clearly visible from this sample spanning from the Sun to a luminous red giant (HD 50890). This agrees with the results obtained by several authors for mainsequence stars (e.g., Bedding & Kjeldsen 2003), and red giants (e.g. Mosser et al. 2010). The problem is now to assess the physical background underlying this relation.
Fig. 1
Frequency of the maximum of oscillation power for the main sequence and redgiant stars of Table 1 as a function of the frequency cutoff. All quantities are normalized to the solar values. 

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3. Height maximum in the power spectrum
In this section, we confirm that the maximum of the power spectrum of solarlike oscillations is related to the plateau of the linewidth by using solar observations from the GOLF instrument, and we then discuss the physical origin of the depression of the damping rates (i.e., the plateau).
3.1. Origin of the maximum of height in the power density spectrum
We consider the height H of a given mode in the power spectrum, which is a natural observable. To derive it, let us first define the damping rate of the modes given by (e.g., Dupret et al. 2009) (2)where ω is the angular frequency, W is the total work performed by the gas during one oscillation cycle, ξ is the displacement vector, and ℳ is the mode mass (3)ξ_{r}(R) corresponds to the radial displacement at the layer where the oscillations are measured, M is the total mass of the star.
For stochastically excited modes, the power injected into the modes is (e.g., Samadi & Goupil 2001; Belkacem et al. 2006) (4)where and are the turbulent Reynolds stress and entropy contributions, respectively. We then introduce the height of the mode profile in the power spectrum, which is an observable, as (see e.g. Chaplin et al. 2005; Belkacem et al. 2006) (5)However, it is useful to express H in a form that does not explicitly depend on the mode mass (ℳ). To this end, we note from Eqs. (2) and (4) that both the excitation P and the damping rate η are inversely proportional to the mode mass. Hence, to separate the effect of the driving and damping from the effect of mode mass, we introduce the quantities and Θ = η ℳ, independent of mode masses. Then, using Eq. (5), the expression of the mode height becomes (6)Figure 2 displays the variations of H with mode frequency as well as its two contributions Π and 1/Θ^{2}. One can clearly distinguish a maximum for H near ν ≃ 3.2 mHz that corresponds to the ν_{max} frequency. Π remains roughly constant (efficient driving regime) except at high frequency beyond ν_{max}, which corresponds to the inefficient driving regime (see Samadi & Goupil 2001, for details). On the other hand 1/Θ^{2} shows a sharp maximum and its variation clearly dominates over that of Π and controls the variation of H and the apparition of its maximum. We conclude that the maximum of H is determined by the minimum of Θ^{2} and corresponds to the plateau of the line widths. In other words, the depression (plateau) of the damping rates η is responsible for the presence of a maximum in the power spectrum, in agreement with Chaplin et al. (2008).
Fig. 2
Normalized 1/(2Θ^{2}) (dasheddots line) , where Θ = η ℳ, and (solid line) computed from solar seismic data from the GOLF instrument (Baudin et al. 2005). The normalized mode height (H) corresponds to the dashed line. All quantities are normalized to unity at the frequency of ν = ν_{max}. The associated mode masses are computed as described in Belkacem et al. (2006). 

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3.2. Origin of the depression of the damping rates
Balmforth (1992) mentioned that the depression of the solar damping rates originate in a destabilising effect in the superadiabatic layer. He also stressed that the plateau of the damping rates occurs when there is a resonance between the thermal time scale and the modal frequency.
Following these ideas, we use the MAD nonadiabatic pulsation code (Dupret 2002) for computing the solar damping rates. This code includes a timedependent convection treatment (Grigahcène et al. 2005) different from that by Balmforth (1992). Nevertheless, we reach the same conclusion (see Appendix A for details): responsible for the destabilizing effect is the Lagrangian perturbation of entropy (δS) that exhibits a rapid variation mainly in the superadiabatic layer and in the atmospheric layers (see Appendix A.2 and Fig. A.2).
To understand the origin of this oscillation and illustrate the occurrence of the resonance, we consider the superadiabatic layers and examine the case of a highly nonadiabatic solution (see Pesnell 1984, for the case of a purely radiative envelope). We assume that Lagrangian perturbations of radiative and convective luminosities are dominated by perturbations of entropy (see Eqs. (A.6) and (A.7)). This leads to a secondorder equation for the entropy perturbations δS (Eq. (A.16), see Appendix A.3 for the derivation). To obtain a more explicit solution for δS, we further employ the dimensional approximation dδL/dr ∽ δL/H_{p}, so that (7)where c_{v} = (∂U/∂T)_{ρ} with U the internal energy, and ℬ are defined by (8)where κ_{T} = (∂lnκ/∂lnT)_{ρ}, L_{c}, and L_{R} are the convective and radiative luminosity respectively, T the temperature, ψ is defined by Eq. (A.8), and we defined the ratio such as (9)with ω = 2π ν, ν the modal frequency, τ a local thermal timescale, τ_{rad} and τ_{conv} the radiative and convective thermal timescales, respectively. From Eq. (A.12), the oscillatory part of the final solution is ^{(}δS/c_{v}^{)} ∝ exp^{ [} − i ^{∫}ℬ dlnT^{] }, which describes the oscillatory behaviour of entropy perturbations in the superadiabatic layers.
As discussed in Appendix B3 (Fig. B2 top), all modes in the range of interest have a similar negative integrated work, W, at the bottom of the superadiabatic layer. This corresponds to a large damping at this level in the star. In the superadiabatic layers, the entropy’s oscillatory behaviour controls the oscillating behaviour of W. When the pulsation period and thus the wavelength of the entropy perturbations are too large (), the destabilizing contribution has not grown enough; the cumulated work W increases too slowly. The net result at the surface is a large damping. When the period is too small (), the rapid oscillation of the entropy perturbation causes a rapid oscillation of W, which increases and again decreases before reaching the surface and the net result at the surface is again a large damping. Those two limits correspond to low and high frequencies, i.e., to the two branches of 1/Θ^{2} displayed in Fig. 2. A minimum damping is then obtained for a period neither too small nor too large, i.e., where the destabilizing contribution nearly but not quite compensates for the strong damping of the layers below the super adiabatic layers.
Fig. 3
Product (see Eq. (9)) versus the logarithm of the temperature for three values of the mode frequency for a solar model described in Appendix A. Vertical dotted lines delimit the limits of superadiabatic gradient (see Fig. A.2 bottom panel). 

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The value of is illustrated in Fig. 3 for three modes. It confirms that the resonance occurs in the superadiabatic region for the mode with frequency ν ≃ ν_{max}. Hence, from the definition Eq. (9), one derives the resonance condition (10)
4. Derivation of the scaling law
We now turn to the relation between the thermal timescale (τ) and the cutoff frequency. To this end, we use a grid of stellar models for masses between M = 1 M_{⊙} and M = 1.4 M_{⊙} from the ZAMS to the ascending vertical branch, typical of observed solarlike pulsators. The grid is obtained by using the stellar evolution code CESAM2k (Morel 1997; Morel & Lebreton 2008). The atmosphere is computed assuming a grey Eddington atmosphere. Convection is included according to the BöhmVitense mixinglength (MLT) formalism. The mixinglength parameter is α = 1.6. The chemical composition follows Asplund et al. (2005), with an helium mass fraction of 0.2485. All quantities are evaluated at the maximum of the superadibatic gradient, which corresponds to the maximum of δS (see Sect. 3.2) and the location of the resonance (see Eq. (10)).
From Fig. 4 (top) the relation between the thermal frequency (1/τ) and the cutoff frequency (ν_{c}) is close to linear but still shows a significant dispersion. More precisely, the relation between those two frequencies is approximately linear and the dispersion is related to the dispersion in mass, in agreement with observations (e.g. Mosser et al. 2010). We then conclude that the observed relation between ν_{max} and ν_{c} is indeed the result of the resonance between ν_{max} and 1/τ, as well as the relation between 1/τ and ν_{c}.
To go further, let us investigate the relation between 1/τ and ν_{c}. First, Eq. (9) can be recast as (11)where F_{conv} and F_{rad} are the convective and radiative fluxes, respectively. The MLT solution for the convective flux and the convective rms velocity can be written (see Cox & Giuli 1968,for details) where Λ = αH_{p} is the mixing length, α the mixinglength parameter, ∇ = (dlnT/dlnP), ∇′ = (dlnT′/dlnP) the gradient of rising convective element, Σ = (∂lnρ/∂lnT)_{μ,P}, with μ the mean molecular weight, and Γ_{1} = (∂lnP/∂lnρ)_{ad}. Now, by inserting Eqs. (12) and (13) into Eq. (11), one obtains (14)where ℳ_{a} = v_{conv}/c_{s} the Mach number, and χ_{ρ} = (∂lnP/∂lnρ)_{T}.
Fig. 4
Top: thermal frequency (1/τ) computed from Eq. (9) versus the cutoff frequency computed following Eq. (1), normalized to the solar values, for models with masses ranging from M = 1 M_{⊙} to M = 1.4 M_{⊙} (with an increment of 0.05 M_{⊙}) and from the ZAMS to the ascending vertical branch. Bottom: the same as for the top panel, except the thermal frequency is divided by the Mach number to the third versus the normalized cutoff frequency. 

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We verified that for a given physic, the ratio F_{rad}/F_{conv} is approximately the same for all models considered in the superadiabatic layer. Hence, by use of Eq. (14) as well as the resonance condition (Eq. (10)), we conclude that (15)which is the observed scaling between ν_{max} and ν_{c} (see Fig. 1), because the thermodynamic quantities hardly vary.
Equation (15) describes the observed scaling between ν_{max} and ν_{c} (see Fig. 1) but also shows that most of the departure from the linear relationship between 1/τ and ν_{c} comes from the Mach number, as confirmed by Fig. 4 (bottom panel). We also point out that as shown by Figs. 4 and 1 for the mainsequence stars, the departure from the linear relationship is of the same order of magnitude as the uncertainties on the cutoff frequency. However, our grid of models is not suited for a proper comparison between the observations and the theoretical relation. This work is definitely desirable in the future.
5. Conclusion
We have addressed the question of the physical reason for the existence of a scaling relation between ν_{max} and ν_{c}. We found that the depression of the damping rates determines ν_{max} because there is a resonance between the local thermal timescale in the superadiabatic region and the modal period. This implies that ν_{max} does not scale only with ν_{c}, but also with the ratio . As pointed out in Sect. 1, the observed scaling between ν_{max} and ν_{c} is not obvious at first glance because the first frequency depends on the dynamical properties of the convective region while the second is a static property of the surface layers. The additional dependence the Mach number resolves this paradox.
This scaling relation is potentially a powerful probe to constrain the dynamical properties of the uppermost layers of solarlike pulsators through the ratio . Indeed, as shown in this paper, most of the dispersion in the ν_{max} − ν_{c} scaling is related to the Mach number. The investigation of the ratio between ν_{max} and ν_{c} in mainsequence stars, subgiants, and red giants may give us statistical information on the evolution of the properties of turbulent convection from mainsequence to red giant stars through the mixinglength parameter, for instance. Indeed, a future work will consist of computing models that correspond to the observations, and will make a comparison between the observed and theoretical dispersion from the linear relation between ν_{max} and ν_{c}.
In other specific cases for which stellar parameters are well known (e.g., in pulsating binaries), the relation between ν_{max} and ν_{c} could directly give us the value of the Mach number in the uppermost convective layers.
Acknowledgments
K.B. gratefully acknowledges support from the CNES (Centre National d’Etudes Spatiales) through a postdoctoral fellowship.
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Appendix A: The plateau of the damping rates
A.1. Computation of the damping rates
Damping rates were computed with the nonadiabatic pulsation code MAD (Dupret 2002). This code includes the timedependent convection (TDC) treatment described in Grigahcène et al. (2005). This formulation involves a free parameter β, which takes complex values and enters the perturbed energy equation. This parameter was introduced to prevent the occurence of nonphysical spatial oscillations in the eigenfunctions. We use here the value β = −0.55 − 1.7i, which is calibrated so that resulting damping rates reproduce the variation of the solar damping rates η with frequency and more precisely the depression of the η profile (see Fig. A.1). Note that TDC is a local formulation of convection. This simplifies the theoretical description and is sufficient here because we seek a qualitative understanding of the relation between the frequency location of the damping rate depression and the cutoff frequency. We stress that the above approximations do not qualitatively influence the conclusions.
This approach takes into account the role played by the variations of the convective flux, the turbulent pressure, and the dissipation rate of turbulent kinetic energy. Hence, the integral expression of the damping can be written as (A.1)where ξ_{r}(R) is the radial mode displacement at the photosphere, ω the mode frequency, ρ the mean density, Γ_{3} − 1 = (∂lnT/∂lnρ)_{S}, T the unperturbed temperature and the star denotes the complex conjugate. The symbol δ represents a Lagrangean perturbation: δS is the perturbation of specific entropy, δρ the density perturbation, δP_{turb} the perturbation of turbulent pressure. The quantity δP_{turb}/ρ represents the contribution of turbulent pressure while the second term (Γ_{3} − 1) T δS includes the variations of radiative and convective fluxes as well as the dissipation rate of turbulent kinetic energy, as given by the energy conservation equation (A.2)with δL_{r},δL_{c} being the perturbations of the radiative and convective fluxes respectively, δϵ_{t} the perturbation of the dissipation rate of turbulent kinetic energy into heat, and σ = ω + iη. Note that Eq. (A.2) is only valid for the radial modes we are interested in.
Fig. A.1
Product of the damping rates (η) multiplied by the mode mass (ℳ) versus mode frequency. The vertical dotted lines identify radial mode orders. 

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A.2. Origin of the depression of the damping rates
The depression of the damping rates, located around ν ~ 3.5 mHz (Fig. A.1), results from a subtle balance between the above contributions to the work integral. The cumulated work integral (regions where it increases outwards drive the oscillation and regions where it decreases outwards damp the oscillation) allows us to identify the processes that create this depression. Fig. A.2 (top) shows that mode damping results from stabilizing effects from inner layers at temperatures greater than log T ∽ 4 have destabilizing effects on the upper layers that are located in the superadiabatic layers (i.e. between log T ∽ 3.95 and log T ∽ 3.8), and for high radial order modes there are again stabilizing effects from the very outer layers. Hence, the behaviour of the product Θ of the damping rates to the mode mass, which is the integral appearing in Eq. (A.1), can then be described as follows: for modes with frequencies ν ≤ ν_{max}, the higher the mode frequency, the larger the contribution of the destabilizing region, and Θ continues to decrease. For ν > ν_{max}, despite an increasing contribution of the superadiabatic boundary layers, atmospheric layers stabilize the modes, which results in an increase of Θ. At ν = ν_{max} the compensation is maximal and gives rise to the minimum of Θ.
Fig. A.2
Top: normalized cumulated work integral versus logarithm of temperature for five values of eigenfrequencies. These modes are emphasized in Fig. A.1. Middle: imaginary part of the Lagrangian perturbation of entropy versus logarithm of temperature. Bottom: normalized convective and radiative fluxes versus logarithm of temperature. The difference between the real and adiabatic gradient (∇) is also plotted and normalized to its maximum. 

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The physical cause of the destabilizing effects in the superadiabatic regions is revealed by Fig. A.2 (middle). The Lagrangian perturbation of entropy exhibits a rapid variation that occurs mainly in the superadiabatic layer and in the atmospheric layers. As the frequency of the mode increases, the amplitude of this variation (which is a spatial oscillation as seen in the next section) also increases. The wavelenth of this spatial oscillation decreases with increasing frequency. This causes a similar behaviour of the cumulated work.
A.3. Oscillation of entropy fluctuations
To understand the behaviour of δS in this region, let us first examine the fluctuations of radiative and convective luminosity that appear in the energy equation (Eq. (A.2)).
In the diffusion approximation, the fluctuations of radiative luminosity reads (A.3)where ξ_{r} is the mode’s radial displacement, δT the Lagrangian perturbation of temperature, δκ the perturbation of opacity, and κ the opacity. By using the perturbed continuity equation, Eq. (A.3) becomes for radial modes (A.4)where we neglected ξ_{r}/r compared to ∂ξ_{r}/∂r. This assumption is valid for radial p modes (see Belkacem et al. 2008, for details). We additionally assume that in the superadiabatic region the perturbation of temperature fluctuations and opacity are dominated by entropy fluctuations, so that (A.5)where c_{v} = (∂U/∂T)_{ρ} with U the internal energy per unit mass, and κ_{T} = (∂lnκ/∂lnT)_{ρ}. Hence, inserting Eqs. (A.5) in (A.3), we obtain (A.6)The approximate expression Eq. (A.6), even if imperfect, captures the main behaviour of δL_{R}/L_{R} in the superadiabatic boundary region, as shown by Fig. A.3 (top). Note that the disagreement observed in the inner layers in Fig. A.3 (top panel) is caused by the approximation Eq. (A.5) because for those layers the density fluctuations are dominant. However, we are mainly interested in the superadiabatic region (log T < 3.9) where Eq. (A.6) is sufficiently valid for our purpose.
Fig. A.3
Top: real part of the radiative luminosity perturbation for the mode of radial order n = 23 versus logarithm of temperature. The solid line represents the full nonadiabatic computation as described in Sect. A.1, while the dashed line represents the approximate expression given by Eq. (A.6). The rapid variation near log T = 4.17 is the result of the presence of a node of the eigenfunction. Bottom: real part of the convective luminosity perturbation for the mode of radial order n = 23 versus logarithm of temperature. The solid line represents the full nonadiabatic computation as described in Sect. A.1, while the dashed line represents the approximate expression given by Eq. (A.7). 

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We now turn to the Lagrangian perturbation of convective luminosity. It depends on the adopted timedependent treatment of convection. Consistent with Sect. A.1, we use the formalism developed by Grigahcène et al. (2005). A good approximation of their Eq. (18) in the superadiabatic layer, as shown by Fig. A.3 (bottom), is (A.7)with (A.8)where τ_{c} is the lifetime of convective elements, ω_{R} is the characteristic cooling frequency of the turbulent eddies, and where Λ = 8/3 is a constant introduced by Unno (1967) to close the equation of motion describing convection, and ω_{R} is the characteristic cooling frequency of turbulent eddies owing to radiative losses (see Eq. (C12) of Grigahcène et al. 2005).
We are now interested in obtaining the equation that qualitatively explains the oscillation observed in Fig. A.2 (middle panel). Hence, one has to exhibit in an analytical way the frequency dependence of the entropy fluctuations (δS). To this end, we will use two different assumptions. The first and most immediate way is to assume that in the energy equation (Eq. (A.2)) dδL/dr ∽ δL/H_{p}. This is a crude approximation, but it permits us to immediately exhibit the role of the factor. Then using Eqs. (A.6) and (A.7), one obtains (A.12)where and ℬ are defined by (A.13)where we defined the ratio as (A.14)with τ is a local thermal timescale. Note that we neglected the imaginary part of σ in Eq. (A.14). We stress that this thermal timescale can be recast into (A.15)where τ_{conv} and τ_{rad} are associated with the convective and radiative luminosities, respectively. From Eq. (A.12), the oscillatory part of the final solution is ^{(}δS/c_{v}^{)} ∝ exp^{ [} − i ^{∫}ℬ dlnT^{ ] }, which explains the oscillatory behaviour of entropy perturbations in the superadiabatic layers and its frequency dependence.
An alternative way to proceed is to use the energy equation (Eq. (A.2)) together with Eqs. (A.6) and (A.7), from which one obtains the second order differential equation (A.16)where (A.17)where H_{T} is the temperature scaleheight.
To derive an analytical solution of Eq. (A.16) is not trivial. Consequently, further simplifications are needed. We then assume the coefficients are constant. Assuming solutions of the form ^{(}δS/c_{v}^{)} ∝ e^{klnT}, one has the solutions for k(A.18)at the maximum of the superadiabatic gradient, the radiative luminosity dominates over the convective ones. Therefore, we further neglect the ratio L_{c}/L compared with L_{R}/L. Equation (A.18) then simplifies to (A.19)From Eq. (A.19), one concludes that for , k is real and δS does not oscillate. This corresponds to the limit of lowfrequency modes for which both k and the imaginary part of δS are small, as confirmed by the full numerical computation presented in Fig. A.2 (middle panel). In contrast, for (i.e., for high frequencies), the imaginary part of the wavenumber increases as depicted by Fig. A.2 (middle panel).
Eventually, both methods to derive the frequency behaviour of δS converge towards the same conclusion, i.e., that the factor explains the oscillation of entropy fluctuations and its frequency dependence.
All Tables
All Figures
Fig. 1
Frequency of the maximum of oscillation power for the main sequence and redgiant stars of Table 1 as a function of the frequency cutoff. All quantities are normalized to the solar values. 

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In the text 
Fig. 2
Normalized 1/(2Θ^{2}) (dasheddots line) , where Θ = η ℳ, and (solid line) computed from solar seismic data from the GOLF instrument (Baudin et al. 2005). The normalized mode height (H) corresponds to the dashed line. All quantities are normalized to unity at the frequency of ν = ν_{max}. The associated mode masses are computed as described in Belkacem et al. (2006). 

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In the text 
Fig. 3
Product (see Eq. (9)) versus the logarithm of the temperature for three values of the mode frequency for a solar model described in Appendix A. Vertical dotted lines delimit the limits of superadiabatic gradient (see Fig. A.2 bottom panel). 

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In the text 
Fig. 4
Top: thermal frequency (1/τ) computed from Eq. (9) versus the cutoff frequency computed following Eq. (1), normalized to the solar values, for models with masses ranging from M = 1 M_{⊙} to M = 1.4 M_{⊙} (with an increment of 0.05 M_{⊙}) and from the ZAMS to the ascending vertical branch. Bottom: the same as for the top panel, except the thermal frequency is divided by the Mach number to the third versus the normalized cutoff frequency. 

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In the text 
Fig. A.1
Product of the damping rates (η) multiplied by the mode mass (ℳ) versus mode frequency. The vertical dotted lines identify radial mode orders. 

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In the text 
Fig. A.2
Top: normalized cumulated work integral versus logarithm of temperature for five values of eigenfrequencies. These modes are emphasized in Fig. A.1. Middle: imaginary part of the Lagrangian perturbation of entropy versus logarithm of temperature. Bottom: normalized convective and radiative fluxes versus logarithm of temperature. The difference between the real and adiabatic gradient (∇) is also plotted and normalized to its maximum. 

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In the text 
Fig. A.3
Top: real part of the radiative luminosity perturbation for the mode of radial order n = 23 versus logarithm of temperature. The solid line represents the full nonadiabatic computation as described in Sect. A.1, while the dashed line represents the approximate expression given by Eq. (A.6). The rapid variation near log T = 4.17 is the result of the presence of a node of the eigenfunction. Bottom: real part of the convective luminosity perturbation for the mode of radial order n = 23 versus logarithm of temperature. The solid line represents the full nonadiabatic computation as described in Sect. A.1, while the dashed line represents the approximate expression given by Eq. (A.7). 

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In the text 
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