Angular momentum redistribution by mixed modes in evolved lowmass stars
II. Spindown of the core of red giants induced by mixed modes
^{1}
LESIA, Observatoire de Paris, PSL Research University, CNRS, Université
Pierre et Marie Curie, Université Denis Diderot,
92195
Meudon,
France
email:
kevin.belkacem@obspm.fr
^{2}
Institut d’Astrophysique Spatiale, CNRS, Université Paris
XI, 91405
Orsay Cedex,
France
^{3}
Stellar Astrophysics Centre, Department of Physics and Astronomy,
Aarhus University, Ny Munkegade
120, 8000
Aarhus C,
Denmark
^{4}
Institut d’Astrophysique et de Géophysique, Université de
Liège, Allée du 6 Août
17, 4000
Liège,
Belgium
^{5}
Laboratoire AIM ParisSaclay, CEA/DSMCNRSUniversité Paris
Diderot; IRFU /SAp, Centre de
Saclay, 91191
GifsurYvette Cedex,
France
Received: 6 March 2015
Accepted: 19 May 2015
The detection of mixed modes in subgiants and red giants by the CoRoT and Kepler spaceborne missions allows us to investigate the internal structure of evolved lowmass stars, from the end of the main sequence to the central heliumburning phase. In particular, the measurement of the mean core rotation rate as a function of the evolution places stringent constraints on the physical mechanisms responsible for the angular momentum redistribution in stars. It showed that the current stellar evolution codes including the modelling of rotation fail to reproduce the observations. An additional physical process that efficiently extracts angular momentum from the core is thus necessary. Our aim is to assess the ability of mixed modes to do this. To this end, we developed a formalism that provides a modelling of the wave fluxes in both the mean angular momentum and the mean energy equations in a companion paper. In this article, mode amplitudes are modelled based on recent asteroseismic observations, and a quantitative estimate of the angular momentum transfer is obtained. This is performed for a benchmark model of 1.3 M_{⊙} at three evolutionary stages, representative of the evolved pulsating stars observed by CoRoT and Kepler. We show that mixed modes extract angular momentum from the innermost regions of subgiants and red giants. However, this transport of angular momentum from the core is unlikely to counterbalance the effect of the core contraction in subgiants and early red giants. In contrast, for more evolved red giants, mixed modes are found efficient enough to balance and exceed the effect of the core contraction, in particular in the hydrogenburning shell. Our results thus indicate that mixed modes are a promising candidate to explain the observed spindown of the core of evolved red giants, but that an other mechanism is to be invoked for subgiants and early red giants.
Key words: waves / stars: evolution / stars: oscillations / stars: interiors / stars: rotation
© ESO, 2015
1. Introduction
The CoRoT (Baglin et al. 2006a,b; Michel et al. 2008) and Kepler (Borucki et al. 2010) spaceborne missions provided a wealth of observed stars exhibiting solarlike oscillations, from the mainsequence to the redgiant phases (see Chaplin & Miglio 2013, for a review). A large number of those stars are lowmass evolved stars showing a rich spectrum of mixed modes, which behave as acoustic modes in the stellar envelope and as gravity modes in the core. They are therefore detectable at the stellar surface while yielding information on the innermost regions (e.g., Dziembowski 1971; Scuflaire 1974; Aizenman et al. 1977; Dziembowski et al. 2001; Dupret et al. 2009).
While mixed modes allowed unveiling the structure of the red giants core (Bedding et al. 2011; Mosser et al. 2011, 2014), the detection of associated rotational splittings enabled measuring the mean core rotation of subgiant and red giant stars (Beck et al. 2012; Deheuvels et al. 2012, 2014; Mosser et al. 2012b). It turns out that the core of lowmass red giant stars slows down during evolution. Since local conservation of angular momentum would imply a spinup of the core, these observations indicate a strong transfer of angular momentum from the inner to the outer layers of these stars. However, current stellar evolutionary rotating models that take the transport of angular momentum into account fail to reproduce the observations. Meridional circulation or shear instabilities are not efficient enough to slow down the redgiant cores (e.g., Eggenberger et al. 2012; Marques et al. 2013; Ceillier et al. 2013), and recent attempts to include the effect of propagative internal gravity waves (Fuller et al. 2014) or magnetic fields using the TaylorSpruit dynamo formalism (Cantiello et al. 2014) failed to solve the problem. Nonetheless, for subgiants and early red giants, Rüdiger et al. (2015) have recently shown that magnetorotational instabilities of a toroidal magnetic field could explain the angular momentum redistribution in subgiants and early red giants (see also Maeder & Meynet 2014).
Hence, an additional physical mechanism that efficiently extracts angular momentum from the core of red giants seems to be needed. Nonradial modes in subgiants and red giants are potential candidates. Indeed, prograde and retrograde mixed modes are differentially damped in the presence of rotation, allowing a net transport of angular momentum (e.g., Ando 1986; Lee & Saio 1993). Our aim is thus to investigate the ability of mixed modes to transfer angular momentum from the inner radiative interior to the outer convective layers.
In the first article of this series (Belkacem et al. 2015, hereafter Paper I), we used the transformed Eulerian mean (TEM) formalism to account for the effect of wave driving through both the wave momentum flux in the mean angular momentum equation and the wave heat flux in the mean energy equation. The wave field was then modelled using the asymptotic, quasiadiabatic, and slowrotation approximations, valid in the dense radiative layers of evolved stars. This allowed us to obtain an explicit expression of the wave fluxes appearing in the mean angular momentum and energy equations. This article, the second of this series, is dedicated to a quantitative determination of the angular momentum transported by mixed modes, which requires the knowledge of mode amplitudes. Recent CoRoT and Kepler observations (Baudin et al. 2011; Mosser et al. 2012a; Samadi et al. 2012) allow us to determine mixedmode amplitudes and subsequently to determine the efficiency of angular momentum transport by mixed modes.
The paper is organized as follows: Sect. 2 recalls the formalism developed in Paper I, while Sect. 3 presents the modelling of the equilibrium models and related oscillations. In Sect. 4, based on the observations, we describe the modelling of mode amplitudes. Finally, the estimate of the angular momentum transport efficiency is presented in Sect. 5, and Sect. 6 is dedicated to discussions and conclusions.
2. Theoretical formalism
In this section, we recall the formalism developed in Paper I as well as the related assumptions for both the mean flow and the wave fluxes appearing in the mean equations.
2.1. Mean flow equations
Following Paper I, we adopt the transformed Eulerian mean formalism (TEM). In addition, we assume shellular rotation (see Maeder 2009, for an extensive discussion) so that at dominant order the mean angular momentum equation is (1)where Ω_{0} is the rotation angular frequency that depends only on the radius, ρ is the density, the overbar and the symbol ⟨⟩ denotes the horizontal average (i.e. azimuthal and meridional). The fluxes are defined by where ϖ = rsinθ, the overbar is the azimuthal average, is the residual meridional velocity, ν_{v} is the radial eddy viscosity, s is the specific entropy, and the prime denotes perturbations associated with the nonradial oscillations, so that , , are the azimuthal, radial, and meridional component of the wave velocity field, and s′ the wave Eulerian perturbation of entropy. We have also introduced the Lagrangian derivative d/dt = ∂/∂t + ṙ∂/∂r.
Under the same assumptions as for Eq. (1), the mean entropy equation is (5)with (6)and (7)where T is the temperature, ε is the nuclear energy generation rate and χ is the thermal conductivity (see, e.g., Sect. 6 of Mathis & Zahn 2004, and also Maeder & Zahn 1998). This expresses energy conservation on level surfaces.
To quantify the effect of the waves on the rotation profile, we further need to provide a modelling of the wave field^{1}.
2.2. Wave flux modelling
Our aim is to determine the angular momentum redistribution induced by mixed modes in the radiative region of lowmass evolved stars. Therefore, the wave fluxes in Eqs. (1) and (5) must be modelled. This has been done in Paper I, so that for the momentum equation we have (8)where the coefficients are given by Eqs. (A.25) to (A.28) of Paper I, and (with σ_{R} the modal frequency). The amplitude a_{ℓ,m} corresponds to the amplitude of a mode of a given angular degree, ℓ, and azimuthal order, m. We note that the amplitude is considered statistically constant in time since we deal with solarlike oscillations that result from a balance between mode driving and damping (see Samadi 2011, for details).
For the entropy equation, we obtained (9)where (10)with L the luminosity, N the buoyancy frequency, ∇ and ∇_{ad} the actual and adiabatic temperature gradients, respectively, and the radial component of the eigenfunction for a mode of angular degree ℓ and azimuthal order m. The overbars and angular brackets have been dropped for ease of notation.
We also recall that in the course of deriving Eqs. (8) and (9) several simplifying approximations were performed, namely:

1.
The quasiadiabatic approach: it consists of neglecting the difference between adiabatic and nonadiabatic eigenfunctions in the full wave equations. This approximation is valid provided the local thermal timescale is much longer than the modal period. This is justified in the radiative region of evolved lowmass stars.

2.
The lowrotation limit: we consider the lowrotation limit in the radiative regions or, more precisely, σ_{R} ≫ Ω_{0}. This is justified by recent inferences of the rotation rate in the core of subgiants (Deheuvels et al. 2012, 2014) and red giants (Mosser et al. 2012b) using seismic constraints from Kepler.

3.
The asymptotic limit: we used an asymptotic description for gravity modes (e.g., Dziembowski et al. 2001; Godart et al. 2009), which is valid for mixed modes in the inner radiative region of subgiants and red giants (e.g., Goupil et al. 2013).
Consequently, our formalism is valid in the dense radiative core of evolved slowly rotating stars.
3. Computing selected stellar models
We have computed an evolutionary sequence of a benchmark model of a 1.3 M_{⊙} evolved lowmass star representative of CoRoT and Kepler observations. We selected three M = 1.3 M_{⊙} models at three postmain sequence stages of this evolutionary sequence, as discussed below. The location of the models in the HertzsprungRussell (HR) diagram is shown in Fig. 1, and their characteristics are summarised in Table 1.
Fig. 1 Evolutionary track on the HertzsprungRussell (HR) diagram of a 1.3 M_{⊙} model, showing the location of the selected models (see Sect. 3 for details). 

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3.1. Equilibrium models and related eigenfunctions and eigenfrequencies
We used the stellar evolution code CESTAM (Marques et al. 2013) to compute the equilibrium models. The atmosphere was computed assuming a grey Eddington approximation. Convection was included according to Canuto et al. (1996), with a mixinglength parameter α = 0.67. The initial chemical composition follows Asplund et al. (2005), with a helium mass fraction of Y = 0.261 and a metallicity of Z = 0.0138. We used the OPAL equation of state (Rogers et al. 1996) and opacities (Iglesias & Rogers 1996), complemented, at T< 10^{4} K, by the Alexander & Ferguson (1994) opacities. We adopted the NACRE nuclear reaction rates from Angulo et al. (1999) except for the 14N + p reaction, where we used the reaction rates given in Imbriani et al. (2004).
Finally, we used the ADIPLS code to compute adiabatic oscillations (ChristensenDalsgaard 2008, 2011), that is, to obtain the eigenfunctions and eigenfrequencies.
3.2. Rotation profiles
We consider that rotation has negligible effects on the equilibrium structure of the star and that it can be considered as a perturbation for the oscillations. This approximation is consistent with the formalism developed in Paper I.
The rotation profiles are therefore determined by computing synthetic profiles that match the observations. The synthetic profiles should have a core that rotates faster than the envelope, with a transition region located at the Hburning shell in the case of red giant stars (models M1 and M2). The width of the transition region is related to the width of the Hburning shell. We consequently used the following profile (11)where Ω_{S} and Ω_{C} are the surface and central angular velocities, respectively, and r_{t} and w are the radius and width of the burning shell. Ω_{C} was chosen to reproduce the mean core rotation rate of evolved lowmass stars as derived from Kepler observations (see Mosser et al. 2012b; Deheuvels et al. 2014, and the surface rotation rate (Ω_{S}) was obtained so as to conserve the total angular momentum from the subgiant stage (model M0).
For the subgiant model (model M0), we also used Eq. (11), but with both r_{t} and w given by the radius of the central convective zone at the end of the main sequence (just before it disappears). In this case, the central and surface rotation (Ω_{C} and Ω_{S}, respectively) were chosen to reproduce the results of Deheuvels et al. (2012). The values of the parameters for models M0, M1, and M2 are summarised in Table 2, and the resulting rotation profiles are displayed in Fig. 2.
Fig. 2 Synthetic rotation profiles, computed as described in Sect. 3.2, for models M0, M1, and M2 as a function of the radius normalised by the total radius. We used logarithmic scales. 

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4. Modelling mode amplitudes
There are mainly two approaches to compute mode amplitudes. The first is based on a full nonadiabatic computation including a timedependent treatment of convection. This procedure is timeconsuming, however, and still suffers from theoretical uncertainties (see Dupret et al. 2009; Grosjean et al. 2014, for details). The second is based on recent CoRoT and Kepler observations that allowed us to measure the amplitudes of solarlike modes in a wide variety of stars in different evolutionary stages. This wealth of observations made it possible to establish scaling relations that provide mode amplitudes versus global stellar parameters (e.g., Mosser et al. 2012a; Samadi et al. 2012). In this paper, we adopt the latter approach.
4.1. Radial mode amplitudes
We first express the amplitudes of radial modes. The surface velocity of the radial modes, at the frequency of maximum height (ν_{max}) in the oscillation power spectrum, follows the scaling relation (see Belkacem & Samadi 2013, for details) (12)where T_{eff} is the effective temperature, Δν is the large separation, and the ⊙ symbols stand for the solar values. Here we adopt the values T_{⊙ ,eff} = 5777 K, ν_{⊙ ,max} = 3050 μHz, and Δν_{⊙} = 135 μHz.
To account for the frequency dependence of radial mode amplitudes, we follow the work of Mosser et al. (2010) , who used a Gaussian envelope centred on ν_{max} with a fullwidth at halfmaximum δν_{env}, that is, (13)with . Based on Kepler observations, we take δν_{env} = 150 μHz for M0, δν_{env} = 70 μHz for M1, and δν_{env} = 30 μHz for M2 for the following.
4.2. Nonradial mode amplitudes
The next step is to consider the amplitudes of nonradial modes. The ratio between radial and nonradial mode amplitudes of comparable frequencies can be written as (see Benomar et al. 2014, for details) (14)where is the mode mass, ξ is the eigenfunction, m is the mass profile, R is the total radius, and Γ_{ℓ} is the mode linewidth that is related to the mode damping rate η_{ℓ} by Γ_{ℓ} = η_{ℓ}/π.
For high angular degrees, the radiative work of the oscillations in the innermost regions dominantes the work done in the uppermost layers (see Grosjean et al. 2014, for details). It follows that the mode amplitudes of high angular degrees become very small due to strong radiative damping. We thus define a cutoff angular degree, ℓ_{max}, below which we assume that the work of nonradial modes equals the work of radial modes (15)where the product between the mode mass and the mode damping rate equals the work integral (e.g., Dupret et al. 2009; Benomar et al. 2014; Grosjean et al. 2014). Now, using Eq. (15) together with Eq. (14) leads to (16)For ℓ>ℓ_{max}, we assume that the radiative damping in the innermost region becomes dominant over the damping of the upper layers so that mode amplitudes become negligible.
The cutoff angular degree is obtained by assuming that the work done over an oscillation period through the effect of radiative losses in the innermost layers becomes of the same order of magnitude as the work done in the uppermost layers. We obtain ℓ_{max} = 6 for M0, ℓ_{max} = 4 for M1, and ℓ_{max} = 2 for M2 (see Appendix A for details). Note that this approach for the modelling of mode amplitudes provides a lower limit to the amount of angular momentum transported by mixed modes since we neglect the contribution of angular degrees higher than ℓ_{max}. However, this is enough to determine whether mixed modes are able to slow down the rotating core of red giants.
Finally, we assume that the relation between the surface mode velocity and the mode amplitude is given by (see Samadi 2011). The results obtained for mode amplitudes of angular degrees ℓ = { 0,1,2 } are presented in Fig. 3 for the three models M0, M1, and M2, described in Sect. 3.
Fig. 3 Mode amplitudes versus mode frequencies for models M0 (top panel), M1 (middle panel), and M2 (bottom panel). The amplitudes are computed as described in Sect. 4. 

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5. Quantitative estimate of angular momentum extracted by mixed modes
Our aim is to estimate the rate of angular momentum transported by mixed modes for the benchmark models M0, M1, and M2. To this end, as in Paper I, we introduce the notation (17)where the righthand side is given by Eq. (8). Using the approach presented in Sect. 4 to derive the amplitudes, we compute the wave fluxes in the righthand side of Eqs. (1) and (5) using Eqs. (8) and (9).
Fig. 4 Rate of temporal variation of the mean specific angular momentum ( as defined by Eq. (17)) as a function of the star radius normalised by the radius of the base of the convective envelope. Red corresponds to and blue to . 

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The contribution of the mixed modes to the mean angular momentum equation is shown in Fig. 4 for models M0, M1, and M2. The main effect of mixed modes is to slow down the core of the stars in the region near the maximum of the buoyancy frequency that also corresponds to the edge of the fast rotating core. For the redgiant models, this region is the hydrogenburning shell. The decrease shown towards the upper radiative layers is the result of both a lower rotation rate and a lower value of the buoyancy frequency. This behaviour is similar in the three models. In addition, the amount of angular momentum extracted by mixed modes hardly depends on the rotation gradient between the core and the envelope (the sensitivity of the results to the parameter w in Eq. (11) remains weak).
The amount of angular momentum extracted by mixed modes increases with the evolutionary stage of the star. This effect is the result of several factors. First, as shown in Fig. 3, mode amplitudes increase from models M0 to M2 and thus more energy is available to transport angular momentum. Moreover, the buoyancy frequency also increases and so does the radial wave number in the gravitymode cavity. As the energy exchanges between modes and the background is proportional to the radial wave number, the amount of angular momentum extracted also increases. Finally, the number of mixed modes between two pdominated modes significantly increases between M0 and M2.
Furthermore, the angular momentum is mainly carried away by gdominated modes of the highest angular degrees. Indeed, these modes are very efficiently trapped in the core and undergo a larger radiative damping (due to the larger radial wave number). The predominance of high angular degree modes implies that is sensitive to our determination of ℓ_{max}. As mentioned before, the estimate of the amount of angular momentum extracted in the radiative regions is thus a lower limit. In other words, adding the contribution of angular degrees higher than ℓ_{max} would strengthen these trends.
To proceed and assess the efficiency of the transport of angular momentum for our benchmark models, we compare two timescales, namely:

the timescale associated with the efficiency of the transport of angular momentum by mixed modes, defined as (18)

the timescale associated with the contraction of the star, defined as (19)where ṙ = dr/ dt.
The two timescales are shown in Fig. 5 (top panel). In all the radiative regions of models M0 and M1, the extraction of angular momentum remains negligible compared to the star contraction. In contrast, for model M2, the timescale of angular momentum extraction is of the same order of magnitude as the contraction timescale and even lower in the hydrogenshellburning region, for which the contraction is maximal (see Fig. 5, bottom panel). However, in the very centre or in the upper radiative region, the mixed mode extraction of angular momentum is again negligible.
Therefore, we conclude that the extraction of angular momentum by mixed modes is negligible in subgiants and early red giants, but becomes strong in the hydrogenburning shell in stars higher on the redgiant branch. In these cases, mixed modes are able to counterbalance the spinup due to the star contraction and can thus enforce a spindown in those layers. We note, however, that the exact location in the HR diagram of the transition between inefficient and efficient extraction by mixed modes is likely to depend on the mass and internal physics of the models.
Fig. 5 Top panel: timescales versus normalised radius (i.e., normalised by the radius of the base of the convective envelope) for models M0, M1, and M2. The solid lines correspond to the timescale associated with the transport of angular momentum by mixed modes (see Eq. (18)) and the dotted lines correspond to the timescale associated with the contraction of the star (see Eq. (19)). Bottom panel: same as the top panel, except that only a zoom for model M2 is shown. 

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6. Concluding remarks
Based on the theoretical formalism developed in Paper I, we computed the amount of angular momentum extracted by mixed modes in evolved lowmass stars. To this end, we considered 1.3 M_{⊙} benchmark models of a subgiant (model M0), an early red giant (model M1), and a more evolved red giant (model M2) for which we computed realistic mode amplitudes based on recent CoRoT and Kepler observations. We found that mixed modes extract angular momentum in the innermost region of subgiants and red giants. This extraction of angular momentum is found negligible for the subgiant and lower red giant branch models, whereas it is efficient enough to counterbalance the effect of the stellar contraction for the most evolved model.
For this evolved star, mixed modes efficiently transport the angular momentum in the hydrogenburning shell (that corresponds to the maximum of the buoyancy frequency) but becomes less efficient in the very central layers. We note, however, that mixed modes can be considered as an external torque for meridional circulation (e.g., Mathis et al. 2013). Therefore, such a localized spindown in the hydrogenburning shell is likely to feed a strong redistribution of angular momentum in the central layers by meridional circulation. This would nevertheless deserve a quantitative estimate.
Indeed, evolutionary calculations including the effect of mixed modes coupled with meridional circulation and shear instabilities are the next step for a definitive conclusion on the ability of mixed modes to slow down the cores of evolved red giants. Modelling of mode amplitudes would require further investigation since we assumed that the contribution of highangular degree modes would be negligible due to their high radiative damping. Therefore, we believe that our calculations provide a lower limit. Full nonadiabatic computations would provide more accurate results, however.
We conclude that, for subgiants and early red giants, our results indicate that extraction of angular momentum by mixed modes cannot explain the current observations (Deheuvels et al. 2014). Other physical mechanisms such as internal gravity waves (Talon & Charbonnel 2008; Fuller et al. 2014) or magnetic fields (Rüdiger et al. 2015) should also be present in these stars in order to extract enough angular momentum from the core. In contrast, as demonstrated in this work, the transport of angular momentum by mixed modes is a serious explanation for the observed spin down of the core of evolved stars on the redgiant branch. At least, given the order of magnitude of the induced angular momentum extraction, it seems difficult to neglect this mechanism in the future.
Finally, we note that an extension of this work to central helium burning stars would be desirable in the future. Indeed, even if the mode amplitudes are similar for stars on the vertical branch and on the clump for the same ν_{max} (e.g., Mosser et al. 2012a), the maximum of buoyancy frequency is lower and shifted toward higher radius for clump stars (e.g., Montalbán et al. 2013). Therefore, we can expect a lower efficiency of the transport by mixed modes but this needs to be definitively quantified.
Acknowledgments
We acknowledge the ANR (Agence Nationale de la Recherche, France) program IDEE (N° ANR12BS050008) “Interaction Des Étoiles et des Exoplanètes” as well as financial support from “Programme National de Physique Stellaire” (PNPS) of CNRS/INSU, France. R.M.O. aknowledges funding for the Stellar Astrophysics Centre, provided by The Danish National Research Foundation, and funding for the ASTERISK project (ASTERoseismic Investigations with SONG and Kepler) provided by the European Research Council (Grant agreement No. 267864).
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Appendix A: Cutoff angular degrees
The cutoff angular degree, ℓ_{max} as introduced in Sect. 4.2, can be defined as the angular degree for which the work performed over an oscillation period through the effect of radiative losses in the innermost layers becomes of the same order of magnitude as for the work performed in the uppermost layers. More precisely, ℓ_{max} is obtained at ν = ν_{max} through the relation (A.1)The lefthand side of Eq. (A.1) can be identified with a radial mode of comparable frequency for which the mode mass is computed as described in Sect. 3 and the associated damping rate is obtained using the scaling relation as provided by Belkacem et al. (2012; see also Baudin et al. 2011; Appourchaux et al. 2012). It gives η (ℓ = 0,ν = ν_{max}) = 2.2 μHz for M0, η (ℓ = 0,ν = ν_{max}) = 0.95 μHz for M1, and η (ℓ = 0,ν = ν_{max}) = 0.06 μHz for M2.
The righthand side of Eq. (A.1) can be identified with a gravity dominated mode for which the mode mass is computed following Sect. 3 and the damping rate is obtained using the asymptotic relation (e.g. Dziembowski et al. 2001; Godart et al. 2009) (A.2)where P is the pressure, L is the luminosity, R_{c} is the base of the convective region, σ_{R} = 2πν, ∇ is the temperature gradient and ∇_{ad} its adiabatic counterpart.
Finally, using Eqs. (A.1), and (A.2), one obtains ℓ_{max} = 6 for M0, ℓ_{max} = 4 for M1, and ℓ_{max} = 2 for M2.
All Tables
All Figures
Fig. 1 Evolutionary track on the HertzsprungRussell (HR) diagram of a 1.3 M_{⊙} model, showing the location of the selected models (see Sect. 3 for details). 

Open with DEXTER  
In the text 
Fig. 2 Synthetic rotation profiles, computed as described in Sect. 3.2, for models M0, M1, and M2 as a function of the radius normalised by the total radius. We used logarithmic scales. 

Open with DEXTER  
In the text 
Fig. 3 Mode amplitudes versus mode frequencies for models M0 (top panel), M1 (middle panel), and M2 (bottom panel). The amplitudes are computed as described in Sect. 4. 

Open with DEXTER  
In the text 
Fig. 4 Rate of temporal variation of the mean specific angular momentum ( as defined by Eq. (17)) as a function of the star radius normalised by the radius of the base of the convective envelope. Red corresponds to and blue to . 

Open with DEXTER  
In the text 
Fig. 5 Top panel: timescales versus normalised radius (i.e., normalised by the radius of the base of the convective envelope) for models M0, M1, and M2. The solid lines correspond to the timescale associated with the transport of angular momentum by mixed modes (see Eq. (18)) and the dotted lines correspond to the timescale associated with the contraction of the star (see Eq. (19)). Bottom panel: same as the top panel, except that only a zoom for model M2 is shown. 

Open with DEXTER  
In the text 