Seismic diagnostics for transport of angular momentum in stars

Seismic diagnostics for transport of angular momentum in stars

1. Rotational splittings from the PMS to the RGB
J. P. Marques Georg-August-Universität Göttingen, Institut für Astrophysik, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany Observatoire de Paris, LESIA, CNRS UMR 8109, F-92195 Meudon, France    M.J. Goupil Observatoire de Paris, LESIA, CNRS UMR 8109, F-92195 Meudon, France    Y. Lebreton Observatoire de Paris, GEPI, CNRS UMR 8111, F-92195 Meudon, France Institut de Physique de Rennes, Université de Rennes 1, CNRS UMR 6251, 35042, Rennes, France    S. Talon Département de Physique, Université de Montréal, Montréal PQ H3C 3J7, Canada    A. Palacios LUPM - UM2/CNRS UMR 5299 - Place Eugène Bataillon cc72 - F- 34095 Montpellier, France    K. Belkacem Observatoire de Paris, LESIA, CNRS UMR 8109, F-92195 Meudon, France    R.-M. Ouazzani Institut d’Astrophysique, Géophysique et Océanographie de l’Université de Liège, Allée du 6 Août 17, 4000 Liège, Belgium Observatoire de Paris, LESIA, CNRS UMR 8109, F-92195 Meudon, France    B. Mosser Observatoire de Paris, LESIA, CNRS UMR 8109, F-92195 Meudon, France   
A. Moya
Departamento de Astrofísica, Centro de Astrobiología (INTA-CSIC), PO Box 78, 28691 Villanueva de la Cañada, Madrid, Spain
   P. Morel Laboratoire Lagrange, UMR 7293, CNRS, Observatoire de la Côte d’Azur, Université de Nice Sophia-Antipolis, Nice, France.    B. Pichon Laboratoire Lagrange, UMR 7293, CNRS, Observatoire de la Côte d’Azur, Université de Nice Sophia-Antipolis, Nice, France.    S. Mathis Laboratoire AIM Paris-Saclay, CEA/DSM-CNRS-Université Paris Diderot, IRFU/SAp Centre de Saclay, 91191 Gif-sur-Yvette, France Observatoire de Paris, LESIA, CNRS UMR 8109, F-92195 Meudon, France    J.-P. Zahn Observatoire de Paris, LUTH, CNRS UMR 8102, F-92195 Meudon, France    S. Turck-Chièze Laboratoire AIM Paris-Saclay, CEA/DSM-CNRS-Université Paris Diderot, IRFU/SAp Centre de Saclay, 91191 Gif-sur-Yvette, France    P A. P. Nghiem Laboratoire AIM Paris-Saclay, CEA/DSM-CNRS-Université Paris Diderot, IRFU/SAp Centre de Saclay, 91191 Gif-sur-Yvette, France
 today
Key Words.:
star: evolution – stars: interiors – stars: rotation – stars: oscillations
offprints: J.P. Marques
Abstract

Context:Rotational splittings are currently measured for several main sequence stars and a large number of red giants with the space mission Kepler. This will provide stringent constraints on rotation profiles.

Aims: Our aim is to obtain seismic constraints on the internal transport and surface loss of angular momentum of oscillating solar-like stars. To this end, we study the evolution of rotational splittings from the pre-main sequence to the red-giant branch for stochastically excited oscillation modes.

Methods:We modified the evolutionary code CESAM2K to take rotationally induced transport in radiative zones into account. Linear rotational splittings were computed for a sequence of models. Rotation profiles were derived from our evolutionary models and eigenfunctions from linear adiabatic oscillation calculations.

Results: We find that transport by meridional circulation and shear turbulence yields far too high a core rotation rate for red-giant models compared with recent seismic observations. We discuss several uncertainties in the physical description of stars that could have an impact on the rotation profiles. For instance, we find that the Goldreich-Schubert-Fricke instability does not extract enough angular momentum from the core to account for the discrepancy. In contrast, an increase of the horizontal turbulent viscosity by 2 orders of magnitude is able to significantly decrease the central rotation rate on the red-giant branch.

Conclusions:Our results indicate that it is possible that the prescription for the horizontal turbulent viscosity largely underestimates its actual value or else a mechanism not included in current stellar models of low mass stars is needed to slow down the rotation in the radiative core of red-giant stars.

1 Introduction

Stars rotate, and this rotation has important consequences on their evolution. On the one hand, centrifugal acceleration reduces local gravity, mimicking a lower mass. On the other hand, rotation induces meridional circulation (1925Obs....48...73E; 1953MNRAS.113..716M) and shear and baroclinic instabilities (2004A&A...425..243M), which contribute to the mixing of chemical elements.

The problem of transport of angular momentum inside stars has not yet been fully understood. Several mechanisms seem to be active, the most commonly invoked being diffusion by turbulent viscosity, transport by meridional circulation (1992A&A...265..115Z; 1998A&A...334.1000M), torques due to magnetic fields (2004A&A...422..225M; 2005A&A...440..653M; 2011A&A...532A..34S), and transport by gravity waves (2005A&A...440..981T). For stars with significant convective envelopes (), the problem is complicated further by the magnetic braking by stellar winds (1988ApJ...333..236K). Another problem concerns the initial rotation state, which can have an influence on the rotation profile during the pre-main sequence (PMS) and early main sequence (MS). The initial rotation state depends on the presence and lifetime of a circumstellar disk during the first stages of the PMS; magnetic fields seem to effectively lock the stellar surface to the disk (1994ApJ...429..781S). 2009pfer.book.....M’s 2009pfer.book.....M textbook contains an extensive description of these processes.

Several stellar evolutionary codes already include transport of angular momentum and associated chemical element mixing (1995ApJ...441..865C; 1997A&A...322..209T; 2003A&A...399..603P; 2008Ap&SS.316...43E). They have been developed first with a focus on studying consequences on the evolution of massive stars or evolved stars (see, e. g., 2000ARA&A..38..143M; 2000ApJ...540..489S). The Sun and solar-like stars have been studied as above (e. g., 1995ApJ...441..865C; 1997A&A...322..320Z; 1997PhDT........18T), and one important application was the lithium surface depletion (e. g., 1998A&A...335..959T; 1999A&A...351..635C).

One way to study the internal transport and evolution of angular momentum is to obtain seismic information on the internal rotation profile of stars in the low and intermediate mass range at different stages of evolution. The SoHO satellite for the Sun (2010ApJ...715.1539T) and the ultra high precision photometry (UHP) asteroseismic space missions CoRoT (Baglin:2003) and Kepler (2010Sci...327..977B) offer such an opportunity. Seismic information has being obtained for PMS and MS stars (e. g., 2010AN....331..956B; 2012A&A...543A..96E), and now we also have seismic observational constraints on the internal rotation of red-giant stars (e. g., 2012Natur.481...55B; 2012arXiv1209.3336M; 2012ApJ...756...19D).

Our goal is to use seismic diagnostics to test the description of transport of angular momentum processes in 1-D stellar models of low-mass stars, from the PMS to the red-giant branch (RGB). Rotational splittings (differences between nonaxisymmetric oscillation modes) are such diagnostics. When the rotation is slow, the relation between rotational splittings and rotation rate is linear and therefore relatively easy to interpret.

The information that rotational splittings provide on the internal rotation, however, depends on the physical nature of the stochastically excited modes, which in turn depend on the structure of the star, hence its age. In addition, some rotation gradient is expected to develop during the evolution; the core rotation rate is expected to become higher than the surface, as the core contracts and the envelope expands. If the rotation rate of some layers becomes too high, the associated distortion of the star and/or the Coriolis acceleration invalidates the linear approximation that provides the usual (linear) rotational splittings. In such cases, nonperturbative methods must be used (such as those developed in 2006A&A...455..607L; 2006A&A...455..621R; Ouazzani2012). It is therefore necessary to determine if linear splitting are valid for an evolutionary stage in order to interpret correctly the observations.

We have thus started a series of papers devoted to establish seismically validated processes of transport of angular momentum that play an essential role in shaping the rotation profile of stars. The present paper is the first of this series. The second and the third will focus specifically on the case of red-giant stars. They will investigate the seismic diagnostics of rotation for slowly rotating red-giant stars for which linear rotational splittings are valid, and then rapidly rotating red-giant stars with nonperturbative methods.

In the present paper, we compute the rotation profiles and their evolution with time from the PMS to the RGB with an evolutionary code where rotationally induced mixing using the prescription of 1992A&A...265..115Z as refined by 1998A&A...334.1000M has been implemented. We then follow the evolution of the linear rotational splittings calculated with the rotation profiles obtained from our evolutionary models and discuss their validity.

The paper is organized as follows: in Section 2 we describe the physical inputs that are implemented in our evolutionary code, with special emphasis on the transport of angular momentum and rotation induced mixing. We developed a version CESAM2K of the code CESAM (Morel:1997; 2008Ap&SS.316...61M) by implementing rotation-induced transport in radiative zones with careful attention to conservation and transport of angular momentum. The numerical scheme for the rotationally induced transport had to be modified compared to the general scheme used in CESAM2K, and some differences in the physical inputs also exist. For these reasons, the modified version will be referred to as the CESTAM code hereafter. This will avoid any possible confusion with results from other versions of CESAM2K used in the international community. The acronym CESAM stands for Code d’Evolution Stellaire Adaptatif et Modulaire, and the extra T in CESTAM stands for transport. A validated standard version of CESTAM will be made freely available.

The numerical implementation is described in Section 3. We then show comparisons with other codes for validation (Section 4). During the implementation of the code, we encountered some difficulties already present but not clearly explained in previous work. We chose to describe them carefully. In Sect. 5 we follow the evolution of theoretical linear rotational splittings for a 1.3 sequence of stellar models evolved from the PMS to the RGB. The modes are computed with the ADIPLS code (2008Ap&SS.316..113C) in a frequency range where the modes are expected to be stochastically excited. We discuss the behavior of the splittings with frequency that can be observed during the various phases of evolution. For red-giant models, we find rotational splittings that are much higher than observed. In Section 6 we consider possible reasons for this disagreement. Finally, conclusions and some perspectives are given in Sect. 7.

2 Physical input

The original version of the CESAM2K code (Morel:1997; 2008Ap&SS.316...61M) consists of a set of routines that calculates 1-D quasi-hydrostatic stellar evolution including microscopic diffusion of chemical species. The solution of the quasi-static equilibrium is performed by a collocation method based on piecewise polynomial approximations projected on a B-spline basis. For models without diffusion, the evolution of the chemical composition is solved by stiffly stable schemes of orders up to four; in the convection zones mixing and evolution of chemicals are simultaneous. The solution of the diffusion equation employs the Galerkin finite elements scheme, and the mixing of chemicals in convective zones is then performed by a strong turbulent diffusion.

The code CESAM2K allows the choice of several options for the physics. The microscopic input physics is updated regularly. The opacity tables are presently the OPAL95 data (Iglesias/Rogers:1996) complemented at low temperatures by the Wichita opacity data (2005ApJ...623..585F). Several sets of opacity tables are provided that correspond to various mixtures of chemical elements, for instance the 1993PhST...47..133G or 2005ASPC..336...25A; 2009ARA&A..47..481A solar mixtures, as well as -element enhanced mixtures. We included Pothekhin’s updated conductive opacities (see e. g. 2007ApJ...661.1094C). Several options are possible for the equation of state, the most commonly used being the OPAL2005 EoS (Rogersetal:1996; Rogers/Nayfonov:2002). Several networks of nuclear reactions (and corresponding updated nuclear reaction rates) were implemented, allowing the evolution of stars to be calculated from the PMS up to helium burning. The microscopic diffusion of chemical elements includes gravitational settling, thermal, and concentration diffusion terms, but no radiative accelerations. Two formalisms are available for microscopic diffusion transport (Burgers:1969; Michaud/Proffitt:1993). Also, two options are available for treating convection, the classical MLT theory of Bohm-Vitense:1958 or the Canutoetal:1996 so-called full spectrum of turbulence. The atmospheric boundary condition is derived either from gray model atmospheres or from Kurucz ATLAS9 models (2005MSAIS...8...14K). Overshooting is an option.

Mass loss can be considered with different prescriptions. In the models presented hereafter we use the empirical mass loss rates of 1975MSRSL...8..369R scaled with metallicity according to 1992A&AS...96..269S.

Recently, the CESAM2K code has been involved in the ESTA activities undertaken to prepare the interpretation of the seismic observations of CoRoT. Stellar models were calculated for a range of mass, chemical composition, and the evolutionary stages corresponding to CoRoT main targets and have been compared with the results of several other evolutionary codes showing a very good general agreement (2008Ap&SS.316..187L; 2008Ap&SS.316....1L; 2008Ap&SS.316..219M).

CESAM2K is freely available for download, with the details in 2008Ap&SS.316...61M.

2.1 Transport of angular momentum

In convective zones, although there is differential rotation in latitude, the mean rotation rate at a given radius weakly depends on the radius. Therefore, it is often assumed in 1-D stellar evolution codes that convective zones rotate as solid bodies (e. g. 1997A&A...322..209T; Meynet/Maeder:2000; 2003A&A...399..603P). In extended convective zones, however, comparisons with 3-D numerical simulations suggest instead a prescription of uniform specific angular momentum (e. g. 2000MNRAS.316..395D; 2006A&A...453..261P). Both options can be used in CESTAM.

In a radiative zone, we used the formalism of 1992A&A...265..115Z, refined in 1998A&A...334.1000M, to model the transport of angular momentum and chemical species by meridional circulation and shear-induced turbulence. In what follows, we sketch the model to make it clear exactly which equations we adopted, as several slightly different versions have been used (e. g. 1992A&A...265..115Z; 1997A&A...322..209T; 1998A&A...334.1000M; 2000MNRAS.316..395D; 2003A&A...399..603P; 2004A&A...425..229M).

Turbulence is expected to be highly anisotropic owing to the stable stratification in radiative zones. Turbulence would then be much stronger in the horizontal than in the vertical direction. Thus, the hypothesis of “shellular rotation” can be used: as differential rotation is presumably weak along isobars, it is treated as a perturbation. All variables can be split into a mean value over an isobar and a perturbation (as in 1992A&A...265..115Z):

(1)

where is the second-order Legendre polynomial and is the pressure. Higher order effects, included in the formalism developed by 2004A&A...425..229M, are not considered here for the moment.

The velocity of meridional circulation can be written in a spherical coordinate system as

(2)

where is the mean radius of the isobar and the colatitude. The vertical component is given in Appendix A, following 1998A&A...334.1000M. The equation of continuity in the anelastic approximation gives then the horizontal component :

(3)

where is the density.

The transport of angular momentum obeys an advection-diffusion equation:

(4)

where is the vertical component of the turbulent viscosity and represents the Lagrangian time derivative.

The relative horizontal variation of the density, , and the mean molecular weight, , obey

(5)
(6)

where is the horizontal component of turbulent diffusion discussed in Appendix B. The evolution of , Eq. (6), depends on the competition between the advection of a mean molecular weight gradient and its destruction by the horizontal turbulent diffusion.

2.2 Evolution of the chemical composition

The vertical advection of chemicals due to the large-scale meridional circulation coupled with a strong horizontal turbulent diffusion results in a vertical diffusion process (see, e. g. 1992A&A...253..173C). The equation of the chemical composition evolution can then be written as

(7)

where is the abundance by mass of the -th nuclear species and and the vertical diffusivity and the diffusion coefficient associated with the meridional circulation; and are discussed in Appendix B.

A necessary condition for shear instability is the Richardson criterion as given by 1997A&A...317..749T. Another condition is that the turbulent viscosity must be greater than the molecular viscosity, , as expressed by the Reynolds criterion

(8)

where is the critical Reynolds number (2000A&A...364..876S). When condition (8) is not satisfied we use .

2.3 Initial conditions

Stars are fully convective when they start their PMS evolution on the Hayashi track. Assuming that convective zones rotate like solid bodies, the star should have uniform angular velocity at the beginning of its evolution.

Several facts complicate this simple picture. First, Palla/Stahler:1991 showed that stars that are more massive than about are no longer fully convective when they appear on the PMS (the exact mass depends mainly on the protostellar accretion rate and deuterium abundance). Second, if during the PMS the only process slowing down stellar rotation were the magnetic braking by stellar winds mentioned above, stars would rotate much more rapidly than observed on the ZAMS. The PMS is too short for this process to slow down the star significantly. An additional process is needed during the PMS, most likely disk locking (e. g., 1997A&A...326.1023B).

Young stars are most often surrounded by a circumstellar disk left over after the main accretion phase is over. The magnetic coupling between the star and the disk slows the star down (see, e. g., 1994ApJ...429..781S), and this effect is often modeled by assuming that the stellar surface corotates with the disk at a constant angular velocity (see 1997A&A...326.1023B) as long as the disk exists. Once the disk has disappeared, the star surface rotation evolves freely. This scenario is implemented in CESTAM with the disk lifetime and period as free parameters of the model.

2.4 Magnetic braking

Stars that are less massive than about have significant outer convective zones. A solar-type dynamo operates there and generates a magnetic field. The coupling between the magnetic field and the plasma in the stellar wind strongly brakes the rotation of the star. 1988ApJ...333..236K proposed the following law for the loss of angular momentum :

(9)

where is a saturation angular velocity, above which magnetic field generation seems to saturate. The value of is often set at (as in 1997A&A...326.1023B). There are indications, however, that varies with stellar mass. (see, e. g. 1997ApJ...480..303K; 2003ApJ...582..358A).

The parameter in Eq. (9) is usually calibrated by requiring that calibrated solar models have . The precise value of needed to spin down the Sun to its current period depends on the prescription for the transport of angular momentum adopted (see section 4.2 below). The parameter should also depend on stellar mass (see, e. g. 1997ApJ...480..303K).

Recently, 2012ApJ...746...43R have criticized the approach used in K88. Specifically, K88 supposed that the surface magnetic flux goes as some power of the angular velocity, whereas RM12 suggest instead that it is the magnetic field strength that obeys such a law. As a consequence, they obtain the following law:

(10)

The value of does not depend on the stellar mass, and . Both prescriptions, Eqs. (9) and (10), are implemented in CESTAM and tested below.

3 Numerical procedure

3.1 The overall problem

Stellar evolution depends on two interconnected problems: the problem of stellar structure (solving the stellar structure equations), and the problem of the evolution of the chemical composition. It has proved very difficult to solve the two problems simultaneously; different stellar evolution codes employ several techniques to overcome this difficulty. Some codes (e. g., 2008Ap&SS.316...25D) compute the solutions of the structure equations and chemical composition in an independent way, where they use the structure calculated at the previous time step to evolve the chemical composition to the current time step, and then use this composition to calculate the structure at the current time step (or the other way around: first the structure, then the chemical composition). A second kind of code computes the chemical composition between each iteration of the structure problem, as in 2008Ap&SS.316...83S. And finally, Eggleton’s code (1971MNRAS.151..351E) solves the two problems simultaneously. 2006MNRAS.370.1817S has shown that differences between the three kinds of codes are only significant on the AGB.

The CESAM2K code (Morel:1997; 2008Ap&SS.316...61M) belongs to the second category above. It begins a time step by updating the chemical composition. With the new chemical composition, a new structure is obtained by performing one iteration of the algorithm used to solve the structure equations. This structure is used to update the chemical composition again, before a new iteration on the structure is performed. The procedure is repeated until convergence of the structure algorithm. CESTAM keeps this iterative scheme.

Rotation with angular momentum transport introduces a new problem interconnected with the previous two. The evolution of the chemical composition depends on the turbulent mixing induced by differential rotation, whereas the structure equations are changed by the inclusion of the centrifugal acceleration. In CESTAM, we inserted the resolution of the angular momentum transport equations at the beginning of the cycle, because the rotation profile is required to calculate the turbulent diffusion coefficients that are needed to update the chemical composition. Our cycle, then, is as follows: we update the rotation profile, then the chemical composition, and finally we iterate on the structure. The cycle is repeated until convergence.

In the absence of external torques, total angular momentum is conserved. The rotation profile obtained at this stage does not enforce angular momentum conservation, however, because the profile was computed with the stellar structure obtained in the iteration before convergence. The total angular momentum is and the function changes between iterations. We need to compute the rotation profile one more time after convergence to make sure that it is consistent with the stellar structure, so that angular momentum is numerically conserved.

3.2 The rotation profile

Equation (4) (with Eq. (LABEL:eq:u2), Eq. (5) and Eq. (6)), is a fourth-order differential equation in . To solve it, we split it into four first-order differential equations, and solve the system using the well known relaxation method (Henyeyatal:1964; numrecipes). Equation (6) is solved simultaneously with Eq. (4), so that we have a total of five finite difference equations. We chose this method for its simplicity in dealing with the complex relations between variables expressed in Eq. (4), taking Eqs. (LABEL:eq:u2), (5), and (6) into account. Methods that require that the solutions are approximated by a linear combination of known functions (collocation methods, spectral methods) are therefore difficult to implement. In practice, the relaxation method we employed proved to be efficient, robust, and fairly stable.

The scheme can be fully implicit or semi-implicit in time (to ensure higher order accuracy in time). The time step can be subdivided if needed, a useful feature for future developments involving faster processes.

Four boundary conditions are needed. At the top of a radiative zone, we impose conservation of angular momentum and no differential rotation:

(11)

where is the mass inside the top of the radiative zone. If the convective zone above it is at the surface, is the torque applied at the surface of the star, Eq. (9), otherwise . At the bottom of a radiative zone, similarly,

(12)

where is the mass inside the bottom of the radiative zone. If the center of the star is radiative (), Eq. (12) is replaced by , the requirement that there is no mass flow out of the center.

Intermediate convective regions (regions that are neither at the center nor at the surface) are treated as if they were special points. The equations between the beginning and the end of an intermediate convective zone (between and ) are

(13)
(14)
(15)
(16)
(17)

Equation (13) guarantees conservation of total angular momentum, Eqs. (14-15) impose no shear at the borders of the convective zone, and Eqs. (16-17) result from the absence of -gradients in a convective zone.

The four equations resulting from Eq. (4) are written in finite-difference form between pairs of points, say, at and . Equation (6) requires a special treatment, because it is not a differential equation in (or ). We write Eq. (6) as an algebraic equation at point , because otherwise we would be solving for (the average between and instead of the values at and ). Indeed, only appears as in the other equations, since it has no space derivatives. A fifth boundary condition is needed: at (the top of the radiative zone) as in Eqs. (16-17).

Case (cgs)
GN93, no b. 0.7031 0.2792 0.02101 0
GN93, b. 0.6842 0.2760 0.02019
AGS05, no b. 0.6702 0.2607 0.01371 0
AGS05, b. 0.6502 0.2566 0.01307
111 is the mixing length, , the initial helium and metallicity abundances, and the braking parameter entering Eq. (9). No b. indicates no braking, b. braking.
Table 1: Parameters of calibrated solar models.
Case
GN93, no braking 0.2511 0.01950 0.7123
GN93, braking 0.2613 0.01950 0.7166
AGS05, no braking 0.2319 0.01260 0.7314
AGS05, braking 0.2425 0.01260 0.7364
222 and are the surface helium abundance and metallicity, respectively, and is the normalized radius at the base of the convective envelope.
Table 2: Characteristics of calibrated solar models.

4 Comparison with results from other evolutionary codes

The complicated nature of the equations describing the evolution of the rotation profile (and particularly the evolution of ) made it important to validate our approach. To do that we compared our results with those obtained with other implementations of rotational mixing in stellar evolution codes, namely STAREVOL (2003A&A...399..603P) for the solar case (see also 2010ApJ...715.1539T) and the Geneva stellar evolution code (1997A&A...322..209T) for the case of and stars.

4.1 Standard physics

In these comparisons, we used the OPAL equation of state (Rogersetal:1996) and opacities (Iglesias/Rogers:1996), complemented at by the Alexander/Ferguson:1994 opacities. We used the NACRE nuclear reaction rates of Anguloetal:1999 except for the reaction, for which we used the reaction rates given in 2004A&A...420..625I. The solar compositions of 1993PhST...47..133G and 2005ASPC..336...25A were used. The Schwarzschild criterion was used to determine convective instability. Convective core overshoot fully mixes the chemical composition to a distance from the border of the convective core (where is the radius of the core determined by the Schwarzschild criterion). The temperature gradient in the overshoot zone is . The centrifugal acceleration is taken into account by adding the average centrifugal acceleration to gravity in the hydrostatic equilibrium equation. The atmosphere is computed in the gray approximation and integrated up to an optical depth of .

Figure 1: Differences in sound speed profiles between the seismic models of 2001ApJ...555L..69T and CESTAM.

4.2 The Sun

We computed calibrated rotating solar models using the compositions of GN93 and AGS05, with and without magnetic braking according to the magnetic braking law, Eq. (9). Models were calibrated to within in luminosity, radius, and surface metallicity ( for GN93, for AGS05). Initial rotational velocities (and parameter for the models with magnetic braking) were chosen so that models have an equatorial velocity km s at the solar age (4.6 Gyrs).

The temperature gradient in convective zones was computed using the convection model of Canutoetal:1996 with . All models include microscopic diffusion and settling using the approximations proposed by Michaud/Proffitt:1993. The initial rotational velocities for the cases with braking were chosen in order to have km s at the ZAMS.

Table 1 shows the values of parameters resulting from the solar calibration in , the initial helium, and metallicity abundances ( and ). The greatest differences between the cases with and without magnetic braking concern the initial abundances. This is because models with braking rotate much faster at the center (as shown in 2010ApJ...715.1539T) and thus have a higher -gradient, leading to a much higher turbulent diffusion. Turbulent diffusion tends to homogenize the radiative zone to a large extent, partially erasing the composition gradient created by gravitational settling.

Table 2 shows some characteristics of the calibrated solar models. The higher turbulent diffusion partially stops the settling of helium, causing a higher helium abundance in the convective zone. However, the strong -gradient predicted by models with braking in the radiative zone is in direct contradiction with helioseismic results, which indicate a flat rotation profile to within of the center (see, e. g. 1999MNRAS.308..405C; 2003ApJ...597L..77C; 2008ApJ...679.1636E). A new physical mechanism is needed to explain the discrepancy, such as transport of angular momentum by internal gravity waves (see 2005A&A...440..981T) and/or magnetic stresses (2004A&A...422..225M; 2005A&A...440..653M)

Figure 1 shows differences between the sound speed profiles of the seismic Sun of 2001ApJ...555L..69T using results from SoHO (GOLF-MDI) and our calibrated solar models (see also 2011JPhCS.271a2031G, for intermediate opacities AGS09). The seismic Sun of 2001ApJ...555L..69T agrees with 2009ApJ...699.1403B using results from BiSON and MDI. A good agreement at the level of 0.3 % is seen when using the old mixture of GN93 with only small discrepancies below the convection zone and in the central region. On the other hand, severe discrepancies occur when more recent mixtures are used, such as AGS05. These results are quite similar to what is found in the literature (see, e. g., 1997MNRAS.292..243B; 2008PhR...457..217B; 2011ApJ...731L..29T, for models with no rotation induced transport).

We compared our results for calibrated solar models including rotation-induced transport of Type I with the results obtained in 2010ApJ...715.1539T with the STAREVOL code. We found good agreement between the profiles of the angular velocity, , and the two components of turbulent viscosity. Several improvements have been implemented in the code CESTAM compared to the CESAM2K version used in 2010ApJ...715.1539T, but with no consequence for the solar case. We confirm that rotation induced transport of Type I does not help remove the discrepancies, as found by 2010ApJ...715.1539T.

4.3 Higher mass, main sequence stellar models

Figure 2: Profiles of the vertical component of the meridional circulation as a function of normalized radius () for 5 stellar models computed with CESTAM (continuous line) and the Geneva code (dashed line) when the central hydrogen abundance is (see text). The left panel shows the central regions, the right panel the surface.

We compared results for models with higher mass stars. These stars have thin convective or fully radiative surface layers and are therefore not expected to undergo magnetic braking. Calculations here are carried out assuming global conservation of angular momentum, which is obtained at the precision level of .

The first comparison concerns the evolution of a model computed with CESTAM and Geneva codes (as in 1997A&A...322..209T), using and (and no magnetic braking). These models were computed without microscopic diffusion and settling. Figure 2 shows the profile of at the middle of the MS (when ). Curves for the same quantities superimpose, showing excellent agreement between the results obtained with the two codes. There are small differences that we attribute to the different microphysics used in the codes. The rotation profiles, not shown, also agree during the course of the evolution.


Figure 3: Evolutionary tracks on the HRD for a model calculated with CESTAM code assuming either no extra mixing (dashed) or rotation with an initial equatorial velocity  km s at the ZAMS (continuous line) or without rotation but with overshooting and (dot-dashed line) and (dotted line).

We also compared the effects of rotational mixing on the evolutionary tracks on the Herzsprung-Russel diagram (HRD) between CESTAM models and those of 2010A&A...509A..72E computed with the Geneva code (2008Ap&SS.316...43E). Figure 3 shows evolutionary tracks on the HRD for a model calculated without extra mixing (microscopic diffusion, convective overshoot, rotation), with rotational mixing only and with overshooting only.

We considered two cases for the models with overshooting, and . The model with rotational mixing has  km s at the ZAMS. As in 2010A&A...509A..72E, the evolution of the rotating model closely resembles the evolution with . The difference between the rotating models and the nonrotating models at the ZAMS is due to the centrifugal acceleration. The decrease in the effective gravity (gravity minus centrifugal force) replicates a nonrotating star with a lower mass.

The evolutionary tracks shown in Fig. 3 are identical to those in Fig. 2 of 2010A&A...509A..72E.

4.4 Evolved low-mass models

The rotation profiles that we obtain for our red-giant models are similar to those computed by 2010A&A...509A..72E. Examples for are given in the section below.

5 Evolution of rotational splittings from PMS to RGB

Figure 4: Evolutionary track for a sequence of models from PMS to RGB. Dots indicate models for which rotational splittings are computed. The numbering of the selected models stars with the first dot on the PMS and ends with number 25 as the highest dot on the red-giant branch

Rotational splittings are useful as seismic diagnostics for measuring the stellar internal rotational profile. For slow rotation, a first order perturbation description provides rotational splittings that are linearly dependent on the rotation profile. They are therefore convenient tools, easy to compute from stellar models and theoretical oscillation codes for comparison with observations. The first order approximation provides the following expression for the linear rotational splittings (1991sia..book..401C, and references therein):

(18)

where is the normalized radius, is the angular rotation (rad/s) and the rotational kernel takes the form

(19)

where is the mode inertia

(20)

The quantities entering the equations above are the fluid vertical and horizontal displacement eigenfunctions, and respectively, and the density . For solid-body rotation, the rotational splittings become:

(21)

with

(22)

where are the Ledoux coefficients.

For asymptotic pure p-modes, and . For pure g-modes, and , i. e. for modes.

Figure 5: Evolution of the central (blue line) and surface (green line) rotation rates for a sequence of models from the PMS to the RGB. The insert shows the central and surface rotation rates along the red-giant branch.

To investigate the evolution of rotational splittings with stellar age, we computed a 1.3 evolutionary sequence including rotationally induced transport as described in Section 2 above. We used the same input physics as in the case AGS05 with braking described in Section 4.2, but without microscopic diffusion and settling, as we found that microscopic diffusion and settling do not affect the rotation profiles of our models. We chose because it is a typical seismic mass obtained for the Kepler and CoRoT red-giant stars.

Figure 4 shows an evolutionary track for a sequence of stellar models in a HR diagram computed with CESTAM. In order to interpret the rotational splittings, we plot the evolution of the central and surface rotation rates with age in Fig. 5.

We then computed oscillation frequencies and rotational splittings for several models spanning the track (shown in Fig. 4) using the freely available ADIPLS adiabatic oscillation code (2008Ap&SS.316..113C). We computed frequencies of axisymmetric modes and rotational splittings for modes.

The models have a convective envelope from PMS to the RGB, hence we expect stochastically excited modes all along the sequence. Thus, we chose the frequency range that spans an interval of a few radial orders below and above , the radial order corresponding to the frequency at maximum power. It is estimated as , where the frequency at maximum power spectrum and the mean large separation are given by the usual scalings relations

(23)
(24)

with , Hz and Hz for the Sun. Indeed, it has been conjectured, then shown observationally, that these relations predict well the location of the excited frequency range of stochastically excited modes (e g. 1991ApJ...368..599B; 1995A&A...293...87K; 2010A&A...509A..77K)

5.1 Validity of linear approximation for the rotation splittings

Figure 6: The evolution of the parameters , Eq. (25) (top) and , Eq. (28) (bottom), with the effective temperature of the model along its evolution from the PMS to the red-giant branch. To be conservative, the layer where is maximum is considered for and the layer where the gradient is maximum is chosen for computing . For , we used , since is higher for prograde modes, while for we considered as no difference is found for due to the fact that where is maximum.

The validity of the linear approximation is estimated by comparing the rotation rate to the oscillation frequency. In a perturbative approach, the parameter

(25)

is thus assumed smaller than unity. We evaluate this parameter at the center of each selected equilibrium model of the evolutionary sequence, where is largest, and for a range of frequency spanning the radial order interval .

Sharp rotation gradients develop in the central layers and can also cause some departure from the linear approximation. In the oscillation equations (see, e. g. Eqs. (34.7-34.12) in 1989nos..book.....U), the rotation gradient term appears with a factor

(26)

Thus, we consider that the linear approximation is valid if

(27)

where the first inequality follows from assuming .

We then evaluate the quantity

(28)

at the radius where the rotation gradient is greatest and for . Fig. 6 shows both quantities, and , as functions of the effective temperature of the models. Both parameters remain much smaller than unity until the model reaches the base of the red-giant branch. Therefore, the use of the linear approximation for the rotational splittings can be safely used for stochastically excited solar-type modes of low mass stars except for the fast rotating cores of red giants.

5.2 Evolution of rotational splittings of stochastically excited modes along an evolutionary track

Figure 7: Top panel: Rotation profiles as a function of the normalized mass for PMS models # 1 (black), # 2 (cyan), # 3 (red), # 4 (green), # 5 (blue) and #6 (magenta) shown in Fig. 4. Second panel: Rotational splittings for modes as a function of the normalized frequency for models # 1,# 3,# 5 and #6 (same colors as above). Third panel: for the same models. Bottom panel: for the same models. The large separation goes from Hz (model # 1) to Hz (model # 6).

In the following, we discuss the information that can be retrieved from the average value throughout the star weighted by the mode. This definition is given at fixed , and depends on the radial order .

  • The PMS regime: On the PMS, the central and surface rotation rates are equal as long as the model is fully convective. The rotation rate remains constant in time for the duration of disk locking. When the radiative core appears, the central rotation starts to increase due to the contraction of the central layers. When disk locking stops, the surface rotation rate starts increasing as the star contracts. Fig. 7 shows the rotation profiles of the selected PMS models shown in Fig. 4. The surface rotation rate evolves from (model # 1) to (model # 5) while the central rotation rate increases from to . The rotational splittings for modes are computed for models # 1 to # 6 according to Eq. (18) and are also shown in Fig. 7. The splittings are nearly independent on radial order and increase steadily with the surface rotation rate as the model evolves along the PMS towards the ZAMS. The value of the Ledoux constant almost vanishes, and , because the modes are essentially pressure modes in the frequency range where they are expected to be stochastically excited. These modes have no amplitudes in the inner layers and cannot probe the rotation there. Thus the mean rotation rate corresponds essentially to the rotation averaged over the outer layers. As the rotation increases inwards in these layers, we find that for Model # 1, close to - but slightly larger than - its surface rotation rate. Model # 6 is already on the MS. Its surface rotation is faster than that of younger models but its central rotation is slower. This is not seen in the corresponding splittings and the averaged rotation remains close to the surface value. Results are similar for modes (not shown).

    Figure 8: Top panel: Rotation profiles as a function of the normalized mass for MS models # 6 (magenta), # 7 (blue), # 9 (red), #10 (green) and # 11 (black) shown in Fig. 4. Second panel: Rotational splittings for modes as a function of the normalized frequency for models # 7,# 9,# 10 and # 11 (same colors as above). Third panel: for the same models. Bottom panel: for the same models. The large separation goes from Hz (model # 7) to Hz (model # 11).
  • The MS regime: On the MS, the surface rotation rate decreases with age due to both braking at the surface and an increase of the stellar radius. The central rotation rate also decreases with time due to transport of angular momentum from the core to the surface. The rotation in the central regions first decreases from Model #6 to #9 and increases by roughly 60 % from model #9 to #11 (Fig. 8). The excited modes are still in the frequency domain of high order p-modes and their rotational splittings reflect the surface behavior only. The splittings decrease with the surface angular velocity. At the end of the MS (model #11), the central rotation rate has already reached 15.4 Hz, while the surface rotates at a rate of 0.6 Hz: the core is rotating roughly 26 times faster than the surface. As can be seen in Fig. 8, the rotational splittings for this model yield a mean rotation , that is 1.22 higher than the surface rotation rate but much lower than the central rotation rate.

    Figure 9: Top panel: Rotation profiles (logarithmic scale) as a function of the normalized mass for subgiant models # 12 (blue solid) to # 17 (black solid) as indicated in Fig. 4. Second panel: Rotational splittings for modes as a function of the normalized frequency for model #12,#14,#15 and #16. Third panel: for the same models. Bottom panel: for the same models. The large separation goes from Hz (model # 12) to Hz (model # 16).
  • The subgiant regime: When the model cools along the subgiant branch, its rotation keeps evolving differently in the inner and outer parts, slowing down at the surface and accelerating in the inner regions (Fig. 9). Substantial changes appear when the model leaves the MS and evolves as a subgiant. The Brunt-Väisälä frequency of red-giant stars is very high in the radiative interior, and the frequencies of g-modes enter the frequency domain of p-modes in the frequency range where stochastically excited modes are expected (2011arXiv1110.5012C). During the subgiant phase, avoided crossings between p- and g-modes appear. As observationally shown (2010A&A...515A..87D; 2012A&A...540A.143M), some p-modes become mixed modes. These modes have amplitudes both in the inner regions, where they behave as gravity modes, and in the surface layers, where they behave as acoustic modes (1991A&A...248L..11D). They are very interesting as they allow probing regions deep inside the star. The number of such mixed modes is small at the beginning of the subgiant branch and increases as the star evolves toward the red-giant branch. During this phase, modes will eventually undergo avoided crossings, exchanging nature from p to g.

    Figure 10: Rotation profiles as a function of the normalized mass for models # 18 to # 21 and # 23, # 25 (from blue to black) on the red giant branch as indicated in Fig. 4.

    Figure 9 shows that Model #12 is not evolved enough so that no mixed modes are present in the frequency domain shown. The splittings remain nearly frequency independent and provide a mean rotation rate close to its surface value. The variation in the splittings with frequency for the more evolved models # 13 to # 17 clearly shows the presence of mixed modes. With amplitudes in the dense interior, mixed modes have a much stronger inertia than the neighboring p-modes (2009A&A...506...57D). For the same reason, the rotational splittings exhibit the same behavior as the mode inertia although in a more pronounced way. They vary from one mode to the next. They are also larger than for pure p-modes when the rotation increases inward within the model. For subgiant stars, only one mixed mode exists between two consecutive radial modes. This explains the saw-like aspect of the variation in the splittings with frequency. This is confirmed by the same variation in with frequency: p-modes have close to 1 while mixed modes have close to 1/2. Mixed modes enter the frequency domain by the low part and one can see that is closer to 1 at high frequency where the g nature of the mixed modes is less pronounced. Because of the weighting by the eigenfunction, the values of the splittings yield a mean rotation that is no longer dominated by the outer layer contribution, since the mixed modes have amplitudes in the central region where the rotation rate is larger. As a result, one obtains a rotation rate that is higher than the surface value but still much lower than the rotation rate of the central regions.

  • The red-giant branch: When the outer convective region progresses inward, its uniform rotation extends deeper as well. The surface rotation rate decreases with time essentially due to an increase in the stellar radius. The core accelerates, and in the intermediate region, where the H-burning shell lies, a sharp rotation gradient develops (see 2006A&A...453..261P). The frequency spectrum of red giants is composed of g-dominated mixed modes and mixed modes with nearly equal p and g character. When the star evolves up the red-giant branch, the number of g-dominated modes increases and largely outnumbers the p-g mixed modes (2009A&A...506...57D). These modes mostly probe the core rotation (2012Natur.481...55B; 2012ApJ...756...19D; 2012arXiv1209.3336M). For model # 25 ( the last shown in Fig. 4), the core rotation amounts to Hz and the expected excited frequencies are about Hz. This indicates a ratio (Eq. 25).

    At such rotation rates, the first-order perturbation is not relevant to computing the rotational splittings. Nonperturbative methods must then be used.

    However, observations show that several RGB stars have core rotation rates that are much lower than predicted by our models. (2012arXiv1209.3336M). Indeed the observed values amount to a few hundred nHz, whereas here we find several dozen to a few hundred Hz. This leads to the conclusion that the central regions of our models rotate too fast and that rotationally induced transport of the kind included in our models is not efficient enough to slow down the central rotation of red-giant stars.

6 Seismic tests of transport of angular momentum: slowing down the rotating core of red-giant stars

As in the solar case, one can wonder whether another mechanism, such as internal gravity waves or magnetic fields, operates more efficiently to shape the rotation profile, either at the red-giant phase or before. However, several issues must be discussed before we can reach such a conclusion.

6.1 Uncertainties on stellar modeling and transport of angular momentum

A possible reason for this discrepancy is some inaccuracy in the current physical description of stellar models. Indeed several assumptions enter the rotationally induced transport as prescribed by 1992A&A...265..115Z and 1998A&A...334.1000M. Besides, several other uncertainties on the input physics of stellar models can also affect the resulting rotation profile to some extent. In what follows, we discuss the sensitivity of the rotation profile of red-giant models to several uncertainties.

6.1.1 Effect of convective overshoot

A star has a convective core during the MS. Overshoot shifts the tracks on the HR diagram, imitating tracks with higher masses. As a result, at a given radius in the RGB, the core of the model with overshoot has had less time to spin up, since it started its contraction later in the evolution. An overshoot of reduces the core rotation rate by about 32 % when .

On the other hand, we need to reduce the mass of the model with overshoot by to reproduce the RGB at the same location in the HR diagram. In that case, the model with overshoot and has a central rotation rate (again, when ) that is similar to the central rotation rate of the model without overshoot and .

6.1.2 Effect of the initial rotation state

It is well known that, after the early stages on the MS, the surface rotation rate does not depend on the initial conditions; it only depends on the magnetic braking law (Eqs. 9 and 10; see, e. g. 2009pfer.book.....M and references therein). We found that it is indeed the case for our models. The internal rotation profile down to the center is also independent of the initial conditions after the early stages of the MS.

6.1.3 Impact of the magnetic braking law

We computed an evolution with the braking law of K88 with a value of twice the solar calibrated value to reduce the rotation rate. We found that the surface rotation rate is slowed down, as expected, but the central rotation rate changes only slightly. For instance, at the TAMS, the surface rotation rate changes by 20 %, and by 3 % at the base of the RGB, while the central rotation rate changes by 2 % at the TAMS, 3 % at the base of the RGB.

We calculated another evolutionary sequence with the RM12 magnetic braking law. With a coefficient calibrated so as to yield the solar rotation rate in a solar model, our is much more efficiently slowed down compared to the evolution computed with K88. The central rotation rate, however, changes by at most 2 %.

6.1.4 Stability of the rotation profile

The changes on the surface rotation rate caused by the mechanisms described above do not change the central rotation rate significantly because the extraction of angular momentum from the core is not efficient enough. The core and the surface are not sufficiently coupled. One potential way of coupling the core to the surface is through instabilities that might arise because of rotation gradient becoming very steep (as seen in Fig. 10). The Rayleigh criterion requires that increases outwards. For our model, the rotation profile becomes unstable according to the Rayleigh criterion in two places, at the shell source and just below the convective envelope. However, the stabilizing effect of the density stratification overcomes this instability, as expressed by the Solberg-Hoiland criterion:

(29)

where is the Rayleigh frequency in a rotating medium, given at the equator by

(30)
Figure 11: Top: Rotation profiles for models at the end of the MS calculated assuming standard viscosity coefficients (standard), a vertical turbulent viscosity computed with (), a horizontal viscosity coefficient 100 times the standard value (), and all of the above (All). Bottom: The same for models at the base of the RGB when ().

According to the Solberg-Hoiland criterion, the rotation profiles we obtain are stable throughout the evolution. However, the Goldreich-Schubert-Fricke instability (GSF, see 1967ApJ...150..571G; 1968ZA.....68..317F) may occur in stellar conditions. 2010A&A...519A..16H have studied its effects on pre-supernova models and find that if the GSF instability is always present regardless of the and gradients. We implemented their prescription in CESTAM. Where , turbulent transport by the GSF instability operates with a viscosity coefficient given by the solution of their Eq. (20),

(31)

where is the thermal diffusivity given by Eq. (38). The viscosity coefficient associated to the GSF instability is then given by .

Figure 12: Top: HR diagram showing the evolutionary track of a model computed with an overshoot of , a vertical turbulent viscosity computed with , a increased by a factor (red line, dashed), and a model computed without overshooting and with the “standard” viscosities (blue line, full). Dots indicate the location of models with . Bottom: Rotation profiles for the models above.
Figure 13: Top: Rotational splittings for modes as a function of the normalized frequency associated with the rotation profile shown in Fig. 12 for the model. Middle: Corresponding for modes as a function of the normalized frequency . Bottom: The same for modes.

We found that in two regions from the end of the MS, just below the border of the outer CZ and at the shell source. At the shell source, the gradient is high because of the contraction of the layers left behind by the movement of the shell source, while the descending CZ lowers just above its border. In our models, the GSF turbulent viscosity is on the order of in the unstable regions. The turbulent transport by the GSF instability reduces the gradient in the unstable region at the shell source, reducing the central rotation rate by 3 % in models at the base of the RGB. The turbulent transport by the GSF instability does not have time to change the gradient below the CZ before the convective instability sweeps over this region.

6.1.5 Uncertainties on the turbulent viscosity coefficients

To extract more angular momentum, we can either increase the meridional circulation or increase the vertical turbulent viscosity. To increase the meridional circulation, one option is to decrease the inhibiting effect of (see the expression for the -currents, Eq. 36). According to Eq. (6), an increase in will lead to a decrease in . The prescription for the horizontal coefficient of turbulence is still open to discussion. Prescriptions derived by 1992A&A...265..115Z, 2003A&A...399..263M, and 2004A&A...425..243M can differ by as much as two orders of magnitude. We adopted, as a test, a value of that is higher than that previously used (Eq. 44).

The value of the critical Richardson number in Eq. (46) is usually assumed to lie between 1/4 and 1/6. 2001MNRAS.320...73B and 2002A&A...384.1119C indicate, however, that we can expect instability for . To increase the efficiency of the vertical turbulent viscosity in transporting angular momentum from the core, we computed an evolutionary sequence using to test the effect of the uncertainties. Figure 11 shows profiles of the rotation rate computed with different assumptions of the diffusion coefficients, at the TAMS and at the base of the RGB when . Models have the same mass and radius, hence the same . As expected, with an increased value of from 1/6 to 1, the central rotation rate decreases by 18% at the TAMS and by 30% at the base of the RGB. With a increased by two orders of magnitude (and ), the central rotation rate decreases by 84% at the TAMS and the base of the RGB. When both changes are included (an increase of and as above), the central rotation rate is decreased by an order of magnitude, while we need a decrease of two orders of magnitude to reproduce the observed rotational splittings.

6.1.6 A slowly rotating red-giant model

We eventually computed an evolutionary sequence where we combined all effects causing a decrease in the rotation rate in the central region; an overshoot of and a vertical turbulent viscosity computed with , together with increased by a factor . This model was computed with because during the RGB the evolutionary track on the HRD of a computed with all these effects lies close to the evolutionary track of the models used as comparison. The resulting rotation profile for models at the base of the RGB with same is significantly decreased in the central region with the central value Hz and is displayed in Fig. 12.

Calculations of linear rotational splittings shows that the maximum splittings amounts to 5-6 Hz for model #20 at the base of the RGB (Fig. 13). This corresponds to a maximum value for of Hz. The rotation gradient and the value of the central rotation are now too low for nonperturbative calculations to give rise to significant corrections to the values of the linear rotational splittings. Such rotation splittings lead to a core rotation rate that is closer to the observations but still too high by almost one order of magnitude compared with what is seen in recent observations (2012Natur.481...55B; 2012ApJ...756...19D; 2012arXiv1209.3336M).

The seismology of red-giant stars emphasizes that an additional mechanism is needed to achieve a slower core rotation of red-giant stars.

7 Conclusions and perspectives

We have computed stellar evolution models of low-mass stars, taking the transport of angular momentum by turbulent viscosity and meridional circulation in radiative zones into account. As found by 2006A&A...453..261P and 2010A&A...509A..72E, the core accelerates very rapidly in our models during the subgiant and giant phases. We showed that the linear approximation for computing rotational splittings remains nevertheless valid for stochastically excited solar-type modes, for all evolutionary stages until the RGB.

For red-giant stars, interpretations of recent Kepler observations lead to a ratio between the core and surface rotation rates of five to ten, far lower than in our models where the ratio between the core and surface rotation rates approaches . Similar conclusions were reached by 2012A&A...544L...4E, which was published during the submission process of the present paper. Nonperturbative calculations lead to complex frequency spectra, with in particular nonsymmetric multiplets (see for instance without Coriolis force 2006A&A...455..607L; 2006A&A...455..621R; Ouazzani2012, with both centrifugal and Coriolis force). Observations show that several red-giant stars do indeed have such complex spectra. Their core rotation is fast enough to have entangled rotational splittings and mixed-mode spacings. In such cases, rotation may have to be studied with nonperturbative methods. On the other hand, other red-giant stars (2012Natur.481...55B) show frequency spectra where the rotation splittings are easily identified as symmetric patterns around axisymmetric modes. The values of the corresponding splittings are quite low, close to or smaller than Hz, and using the linear approximation to derive the rotation splittings from stellar models is fully justified.

We have computed the evolution of a stellar model including rotationally induced transport taking uncertainties in the values of the parameters entering such a description into account. We achieved a decrease in the central rotation rate of a red-giant star by an order of magnitude at most, mostly due to an increase in by two orders of magnitude. The modification of the parameters were chosen so as to enhance the transport of angular momentum so that the core rotation is slowed down as much as possible. The models in the red-giant phase were then slowed enough that the calculations of linear rotation splittings are valid. They were found to be one order of magnitude larger still than observed. This indicates that extraction of angular momentum from the core in our models is not efficient enough.

We need to decrease it by another order of magnitude to reproduce the observations. A similar situation is encountered in the case of the Sun. Several possibilities have been proposed, the main candidates being magnetic fields and internal gravity waves. We showed that such mechanisms must operate or keep operating during the whole subgiant and red-giant phase. In this context, 2008A&A...482..597T show that transport of angular momentum by internal waves generated by the convective envelope can play an important role from the subgiant phase to the base of the RGB. They also find that this transport seems to have no major impact on stars ascending the RGB itself and later on. In the RGB phase, another mechanism must be called for to explain the observations. This will be investigated further in future work. On the observational side, measurements of rotational splittings for subgiant stars are crucially needed to constrain the efficiency for their central layers to slow down.

A standard, validated version of CESTAM will be available for download soon, together with grids of stellar models computed with several rotational velocities and adiabatic oscillation frequencies for selected models.

Acknowledgements.
JPM acknowledges financial support through a 3-year CDD contract with the CNES. The authors also acknowledge financial support from the French National Research Agency (ANR) for the project ANR-07-BLAN-0226 SIROCO (SeIsmology, ROtation and COnvection with the CoRoT satellite). A.M., acknowledges the support of the Direccion General de Investigacion under project ESP2004-03855-C03-01. He also acknowledges a stay of two years at the Observatoire de Paris-Meudon.

Appendix A Meridional circulation

The vertical component of the meridional circulation is given by

where , and have their usual meanings (as defined in Kippenhahn/Weigert:1991). In Eq. (LABEL:eq:u2) above, is the reduced mass,

(33)

where and are the mass and mean density inside an isobar, respectively, and and denote the so-called - and -currents (following 2003A&A...399..603P). The quantities and are obtained from the equation of state, and are defined as in Kippenhahn/Weigert:1991,

(34)

The - and -currents are given by

(35)
(36)

In Eqs. (35) and (36) we used the following quantities: and are the nuclear and gravitational energy generation rates, the mean energy production inside a sphere of radius , and the mean density. The thermal conductivity is given by

(37)

and the thermal diffusivity is

(38)

We also used

(39)

the temperature scale height;

(40)

the logarithmic derivatives of ;

(41)

the logarithmic derivatives of ; and

(42)

Appendix B Turbulent coefficients of diffusivity

The coefficient is given by (1992A&A...253..173C)

(43)

We use the prescription of 2004A&A...425..243M for the horizontal component of turbulent diffusivity :

(44)

where

(45)

We take the coefficient (1999A&A...347..734R).

For the vertical component , we use the prescription of 1997A&A...317..749T,

(46)

where is the critical Richardson number, the thermal diffusivity, and the chemical and thermal parts of the Brunt-Va̋isa̋la̋ frequency, :

(47)

References

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