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Astronomy & Astrophysicsmanuscript no. aa˙2015˙27591˙spada˙ea c© ESO 2016March 4, 2016

Angular momentum transport efficiencyin post-main sequence low-mass stars

F. Spada1, M. Gellert1, R. Arlt1 and S. Deheuvels2,3

1 Leibniz-Institut fur Astrophysik Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germanye-mail:fspada@aip.de2 Universite de Toulouse, UPS-OMP, IRAP, 31028 Toulouse, France3 CNRS, IRAP, 14 avenue Edouard Belin, 31400 Toulouse, France

Received; accepted

ABSTRACT

Context. Using asteroseismic techniques, it has recently become possible to probe the internal rotation profile of low-mass (≈ 1.1–1.5M⊙) subgiant and red giant stars. Under the assumption of localangular momentum conservation, the core contraction andenvelope expansion occurring at the end of the main sequencewould result in a much larger internal differential rotation than observed.This suggests that angular momentum redistribution must betaking place in the interior of these stars.Aims. We investigate the physical nature of the angular momentum redistribution mechanisms operating in stellar interiors byconstraining the efficiency of post-main sequence rotational coupling.Methods. We model the rotational evolution of a1.25 M⊙ star using the Yale Rotational stellar Evolution Code. Our models takeinto account the magnetic wind braking occurring at the surface of the star and the angular momentum transport in the interior, withan efficiency dependent on the degree of internal differential rotation.Results. We find that models including a dependence of the angular momentum transport efficiency on the radial rotational shearreproduce very well the observations. The best fit of the datais obtained with an angular momentum transport coefficient scaling withthe ratio of the rotation rate of the radiative interior overthat of the convective envelope of the star as a power law of exponent≈ 3.This scaling is consistent with the predictions of recent numerical simulations of the Azimuthal Magneto-Rotational Instability.Conclusions. We show that an angular momentum transport process whose efficiency varies during the stellar evolution through adependence on the level of internal differential rotation is required to explain the observed post-main sequence rotational evolution oflow-mass stars.

Key words. Asteroseismology – Magnetohydrodynamics (MHD) – Stars:rotation – Stars: solar-type – Stars: magnetic fields – Stars:interiors

1. Introduction

Rotation is a very important property of stars, as it can signifi-cantly affect stellar structure and evolution in a variety of ways(e.g., directly, Sills et al. 2000; by enhancing element mixing,Pinsonneault 1997; by powering dynamo action and magneticactivity, Noyes et al. 1984).

Solar-like stars show a significant rotational evolution dur-ing their pre-main sequence (PMS) and main sequence (MS)lifetime and beyond, as their magnetized stellar winds effec-tively drain angular momentum from their surfaces (Schatzman,1962; Kraft, 1967; Skumanich, 1972). Simple rotational evolu-tion models can reproduce the basic features of this spin-down(see, e.g., Gallet & Bouvier, 2013), when also taking into ac-count the occurrence of structural readjustments (e.g., the grad-ual development of an inner radiative zone during PMS contrac-tion), and of angular momentum transport within the interior,typically ensuring an efficient rotational coupling withina timescale of a few hundreds of Myr (see also MacGregor & Brenner,1991; Denissenkov et al., 2010; Spada et al., 2011).

Although it has been possible to measure the surface rota-tion of stars for a long time (see, e.g., Kraft, 1970, and refer-ences therein), much less is known about the rotational stateof stellar interiors. Through helioseimology, the solar rotation

profile has been mapped from the surface down to≈ 0.2 so-lar radii (Schou et al., 1998; Howe, 2009), showing a remark-ably low radial differential rotation; similarly, using asteroseis-mic techniques, it has recently become possible to place con-straints on the radial differential rotation in the interior of otherstars. The analysis of the rotational splittings carried out on sixsolar-like stars by Nielsen et al. (2014) ruled out strong radialrotational gradients. Similarly, Benomar et al. (2015) reportedratios of interior to surface rotation rates smaller than a fac-tor of 2 in a sample of stars of mass between1.1 and1.6M⊙,independently of their age. Quantitative constraints on the de-gree of internal differential rotation can also be placed for starssufficiently evolved to show mixed modes (Dupret et al., 2009;Beck et al., 2012; Deheuvels et al., 2012; Mosser et al., 2012;Deheuvels et al., 2014).

Observations thus strongly support an efficient angu-lar momentum redistribution in the interior of low- andintermediate-mass stars, leading to a state of almost uni-form rotation within the mature stages of their MS life-time. The physical nature of the processes responsible forthis rotational coupling, however, remains elusive. Both in-ternal gravity waves (see Mathis, 2013, for a review),and magnetic fields (e.g., Charbonneau & MacGregor, 1993;Rudiger & Kitchatinov, 1996) have been proposed as viable

1

F. Spada, M. Gellert, R. Arlt and S. Deheuvels: Angular momentum transport efficiency in post-main sequence low-mass stars

mechanisms to redistribute angular momentum in the interior ofMS stars.

The rotational evolution beyond the TAMS is equally non-trivial. As the stellar core contracts and the envelope expands atthe end of the hydrogen burning phase, local angular momentumconservation would require the development of a strong differen-tial rotation. The moderate differential rotation observed in KIC7341231 (Deheuvels et al., 2012) or KIC8366239 (Beck et al.,2012) is incompatible with this scenario, suggesting that effi-cient angular momentum transport is taking place during thepost-main sequence (poMS) phase as well. Purely hydrodynam-ical effects, such as meridional circulation and the shear instabil-ity, have been shown to be insufficient to reconcile the theoreticalmodels with this observed behavior (Eggenberger et al., 2012;Ceillier et al., 2013; Marques et al., 2013). Models including an-gular momentum transport mediated by the so-called Tayler-Spruit Dynamo (Cantiello et al., 2014) and by gravity waves(Fuller et al., 2014) also produce too fast core rotation in sub-giants in comparison with the observations.

The stars in the samples of Deheuvels et al. (2014, hereafterD+14) and Mosser et al. (2012, M+12 in the following), com-bined together, offer the unique opportunity to trace the evolu-tion of core–envelope differential rotation from immediately af-ter the TAMS to the giant branch stage.

In this work, we focus on modeling the poMS rota-tional evolution of stars of approximately solar mass (M .1.5M⊙), using the Yale Rotational stellar Evolution Code(YREC; Pinsonneault et al., 1989; Demarque et al., 2008). Inparticular, motivated by the results of recent numerical simula-tions of the Azimuthal Magneto-Rotational Instability (AMRI;Rudiger et al. 2015, Gellert et al., in preparation), we wish to testthe dependence of the angular momentum transport efficiencyonthe degree of internal differential rotation.

The AMRI is a destabilization of hydrodynamically sta-ble differential rotation by current-free toroidal magnetic fields(Rudiger et al., 2007). In contrast to the magneto-rotational in-stability of an axial field (see Velikhov, 1959), it is naturally non-axisymmetric. The instability extracts its energy from thediffer-ential rotation, thus working more effectively for steeperrotationprofiles, and disappearing in the presence of solid-body rotation.The turbulent viscosity generated by the AMRI can be very ef-fective at transporting angular momentum (Rudiger et al.,2015),with the magnetic contribution due to Maxwell stresses stronglydominating over the kinetic component. As a consequence, mix-ing is much less enhanced by the AMRI than angular momen-tum transport, and the Schmidt number (the ratio of turbulentviscosity and turbulent element diffusion coefficient) is largeenough not to speed up stellar evolution significantly Schatzman(1977); Lebreton & Maeder (1987); Brott et al. (2008). A dis-tinctive property of the AMRI-induced viscosity is its depen-dence on the angular velocity shear present in the region wherethe instability develops. As a consequence, the time scale for thequenching of the differential rotation is not constant in time, butincreases as the rotation profile becomes flatter. This is a specificprediction that can be tested using our formulation.

The paper is organized as follows: in Section 2 we introducethe data used to constrain our models; in Section 3 we describeour models and the implementation of the turbulent angular mo-mentum diffusion dependent on the internal differential rotation.Our results are presented in Section 4 and discussed in Section5. We summarize our conclusions in Section 6.

Fig. 1. Asteroseismic constraints on the internal differential ro-tation of evolved low mass stars stars used in this work. Thebroken line is a fit of the red giants in the M+12 sample.

2. Asteroseismic constraints on poMS rotationalcoupling

We consider asteroseismic constraints on the poMS rotationalevolution of low-mass stars from the following two studies (seeFigure 1).

The D+14 sample contains six stars of mass between1.1 and1.5 M⊙ and approximately solar composition. For these stars,the surface gravity,log g, and the core and envelope angular ve-locities,Ωg andΩp (filled and empty circles, respectively), havebeen determined from individual asteroseismic modelling (herewe loosely refer to the rotation rate averaged over theg-modeand thep-mode cavities, respectively, as “core” and “envelope”rotation rates; see section6 of D+14 for more details). Thesestars apparently possess some degree of internal differential ro-tation, with a ratioΩg/Ωp in the range2.5–20, which increaseswith decreasinglog g, i.e., while the stars evolve away from theMS towards the red giant branch.

For the stars in the M+12 sample, values oflog g andΩg ob-tained from the ensemble asteroseismology technique (see theirpaper for details) are available. This sample contains stars whoseevolutionary stage ranges from the early red giant to the redclump. While the red giants (diamond symbols in Figure 1) areroughly in the same mass range as those of the D+14 sample,the clump stars (triangle symbols) are much less homogeneousin mass, containing both low- and intermediate mass stars. Forthis reason, only the red giants of the M+12 sample will be con-sidered here.

These two samples offer the possibility of a quantitativecomparison between observed and modeledΩg/Ωp as a func-tion of log g, to constrain the efficiency of internal angular mo-mentum transport through the poMS evolution. As can be seenfrom Figure 1, after the TAMS a transition from the core spin-up of the subgiants to core spin-down among the red giants isobserved.

In the following, we will compare the observedΩg andΩp

with the average rotation rates over the radiative interiorand theconvective envelope,Ωrad andΩenv respectively, extracted fromour models. It should be noted, however, that theg-mode cavitycovers only part of the radiative interior, and thereforeΩg does

2

F. Spada, M. Gellert, R. Arlt and S. Deheuvels: Angular momentum transport efficiency in post-main sequence low-mass stars

not exactly translate toΩrad. We will neglect this effect fromnow on (see the discussion in section 6 of D+14 for details).

3. Modeling poMS rotational evolution oflow-mass stars

3.1. The stellar evolution code

The models discussed here were constructed using the YRECstellar evolution code (see Demarque et al. 2008, for a descrip-tion of its non-rotational configuration, and Endal & Sofia 1976;Pinsonneault et al. 1989, for the treatment of rotation-relatedphysics).

We use the OPAL 2005 Equation of State(Rogers & Nayfonov, 2002), and the OPAL Rosseland opacities(Iglesias & Rogers, 1996), complemented by the Ferguson et al.(2005) opacities at low temperatures; the nuclear energy gener-ation rates are those recommended by Adelberger et al. (2011).The surface boundary conditions are based on the classicalEddington grayT–τ relationship. Convection is described withthe mixing length theory (Bohm-Vitense, 1958). We adoptthe Grevesse & Noels (1993) value of the solar metallicity,(Z/X)⊙ = 0.0245. The resulting solar-calibrated value ofthe mixing length parameter (the ratio of the mixing lengthover the pressure scale height) isα = 1.832, which is adoptedthroughout in our modeling. To keep the number of parametersto a minimum, the effect of elements diffusion and convectivecore overshooting are ignored (see Marques et al., 2013, fora discussion of the effects of microphysics on the angularmomentum transport).

In rotating models, the effect of rotation on the stellar struc-ture, the angular momentum loss from the surface (if present),and the internal redistribution of angular momentum must alsobe taken into account.

In the treatment of the structural effects of rotation (i.e., in-crease in effective temperature, decrease in luminosity, increaseof the MS lifetimes) we follow the standard YREC implemen-tation. These effects are quite small in low-mass stars (see, e.g.,figures4 and5 of Sills et al. 2000).

For the wind braking we adopt the parametrization ofKawaler (1988):

(

dJ

dt

)

wind

= KW

(

R∗/R⊙

M∗/M⊙

)1/2

Ω3

∗, (1)

whereR∗, M∗, Ω∗ are the radius, mass, and surface rotationrate of the star, and the overall scaling factorKW is an ad-justable parameter. The dependence onΩ3

∗ of equation (1) re-sults in an asymptotic rotational evolution that follows the em-pirically well-supportedProt ∝ t1/2 relation, whereProt is thesurface rotation period of the star andt its age (Skumanich,1972). The second factor to the right hand side of equation (1),containingR∗ andM∗, has been shown not to capture the fullmass dependence of the wind braking phenomenon as observedin young open clusters (see Barnes & Kim, 2010; Barnes, 2010;Meibom et al., 2015; Lanzafame & Spada, 2015, for details). Tocompensate for the incorrect mass dependence in equation (1),Chaboyer et al. (1995a,b) introduced a “saturation phase”,ofmass-dependent duration (during whichJwind ∝ Ω∗). However,since for a star of mass& 1.1M⊙ the saturation phase only lastsfor the first few Myr of the MS rotational evolution, to keep thenumber of parameters to a minimum we absorb both the satu-ration effect and the residual mass dependence of equation (1)within KW , and retain it as the only freely adjustable parameter.

In the following, to keepKW of the order of unity for conve-nience, we scale it over1.13 · 1047 g cm2s, the value recom-mended by Kawaler (1988).

We describe the radial transport of angular momentum in thestellar interiors as a diffusion process:

ρr4∂Ω

∂t=

∂

∂r

[

ρr4D∂Ω

∂r

]

, (2)

whereΩ(r, t) is the angular velocity,ρ is the density, andD isthe angular momentum diffusion coefficient. Note that the valuesof D adopted in the following are in any case representative of aturbulent process (i.e., much stronger than those resulting fromjust the molecular viscosity). Solid-body rotation is enforced atall times within convective zones, as their typical mixing timescales are much faster than the processes discussed here. Thegeneration of latitudinal shear in convection zones does not af-fect their total angular momentum, and is therefore not relevantto our model.

In the standard version of YREC, the coefficientD is cal-culated at each evolutionary time step by taking into accountseveral hydrodynamical instabilities (Pinsonneault et al., 1989).Purely hydrodynamic instabilities, however, have alreadybeenshown not to be sufficiently effective at transporting angu-lar momentum to explain the internal rotation of red giants(Eggenberger et al., 2012; Ceillier et al., 2013; Marques etal.,2013). For this reason, here we follow a different approach.Weexplore a simple, power law dependence ofD on the internal dif-ferential rotation, as measured by the ratio of the average angularvelocity of the radiative interior,Ωrad, over that of the convectiveenvelope,Ωenv:

D = D0

(

Ωrad

Ωenv

)α

. (3)

In the equation above,D0 andα are parameters to be determinedthat set the overall scale factor and the sensitivity of the depen-dence on the internal differential rotation, respectively. A pos-sible physical interpretation of such a dependence will be pro-posed in Section 5.

By determining the value ofα that leads to the best agree-ment of the models with the data, we can obtain useful clueson the physical nature of the rotational coupling mechanisms inpoMS stars.

3.2. The solar benchmark

In order to set baseline values of the constantsKW andD0, weapply the constantD prescription, i.e.,α = 0 in equation (3), tothe MS rotational evolution of the Sun.

From the full solution forΩ(r, t) obtained for a1M⊙ model,we extract the average rotation rate of the radiative and convec-tive zones,Ωrad(t) andΩenv(t), as a function of time. Simple,unweighted averages over the mass shells belonging to the ra-diative interior and to the convective envelope, respectively, areused.

The initial conditions are chosen such that the period at1Myr is 8 days. This choice roughly coincides with the medianof the observed period distribution in the Orion Nebula Cluster(ONC; Rebull, 2001). We adjust the values ofKW andD0 inorder to satisfy the following constraints, which are meanttorepresent the present solar rotation (e.g., Schou et al., 1998):

Ωenv(t⊙) = 1.0Ω⊙ ; Ωrad(t⊙)/Ωenv(t⊙) = 1.02,

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F. Spada, M. Gellert, R. Arlt and S. Deheuvels: Angular momentum transport efficiency in post-main sequence low-mass stars

Fig. 2. PoMS rotational evolution of a1.25M⊙ model with con-stant angular momentum diffusion coefficient, compared withthe subgiant stars of D+14 (open circles:Ωp; filled circles:Ωg)and the red giants from M+12 (diamonds). The broken line is alinear fit (in the variableslogΩ/2π vs.log g) of the core rotationof the red giants. Values ofD0 in the legend are in cm2s−1.

wheret⊙ = 4.57 Gyr andΩ⊙ = 2.78 · 10−6 s−1. The valueadopted forΩrad(t⊙)/Ωenv(t⊙) was estimated via numerical in-tegration, using the fit of the solar rotation profile given byequa-tions (14) and (15) of Roxburgh (2001). We thus obtain:

KW,⊙ = 3.2; D0,⊙ = 2.5 · 105 cm2s−1.

The parameterKW is mostly determined by the condition onΩenv(t⊙), but it is also moderately sensitive to the behavior ofinternal differential rotation. For comparison, solid-body rota-tion is enforced at all times during the evolution ifD0 & 106

cm2 s−1 (see also Denissenkov et al., 2010), in which case oursolar calibration givesKW,⊙ = 2.9.

Our solar-calibrated D0,⊙ is in good agreementwith previous studies (e.g. Rudiger & Kitchatinov, 1996;Denissenkov et al., 2010; Spada et al., 2010). Another term ofcomparison is the value of3 · 104 cm2 s−1, which was found byEggenberger et al. (2012) to reproduce the observed rotationalsplittings of KIC8366239.

4. Results

We model the poMS rotational evolution of a solar composition,1.25M⊙ star. This choice of parameters is assumed to be repre-sentative of both the D+14 sample and of the red giants in theM+12 sample. Modelling the red clump stars of M+12 is out-side the aims of this paper; more massive stars, such as thoseinthe sample of Deheuvels et al. (2015), will be the subject of asubsequent investigation.

4.1. Evolution of Ωenv and calibration of Kw

The rotational evolution is an initial value problem, requiringsuitable initial conditions. We evolve our1.25M⊙ modelstarting from the early PMS (age. 1 Myr). This choice hasseveral advantages: since the initial model is fully convective,we can assume it has a solid-body rotation profile; moreover,

Fig. 3. PoMS rotational evolution models with angular momen-tum diffusion coefficient scaling with the ratioΩrad/Ωenv ac-cording to equation (3). A solid-body interior rotation profile isenforced until the TAMS. Upper panel: as in Figure 2, but show-ing various non-zeroα models as well. For each value ofα, thescaling parameterD0 is calibrated to match the most evolved starin the D+14 sample. Lower panel: evolution ofD as a functionof log g. The values ofD0 shown in the legend are in cm2s−1.

due to the strong convergence properties of theΩ dependencein equation (1), the subsequent evolution is not too sensitiveto the details of the initial conditions. Similarly to the solarbenchmark model discussed in Section 3.2, the initial periodis assigned so that the period at the age of1 Myr is 8 days.This is still compatible with the observed rotation perioddistribution in ONC (Rebull, 2001), at least within the level ofaccuracy with which its mass dependence is currently known.Furthermore, rigid rotation within the model is enforced untilthe end of the MS (i.e., until the central hydrogen abundancedrops below≈ 10−4). Indeed, theoretical models of stars ofmass& 1M⊙ predict that they attain a quasi-rigid rotationstate by the time they have reached their mature MS (i.e.,once they are older than≈ 1 Gyr: see Gallet & Bouvier, 2013;Lanzafame & Spada, 2015). This is also observed in the Sun andin other solar-like stars (Nielsen et al., 2014; Benomar et al.,2015). Our poMS rotational evolution modeling thus begins

4

F. Spada, M. Gellert, R. Arlt and S. Deheuvels: Angular momentum transport efficiency in post-main sequence low-mass stars

from a (realistic) state of negligible internal differential rotation.At the same time, the problem at hand has been effectivelydecoupled from the MS rotational evolution problem, whereangular momentum redistribution is a highly debated, currentlyunsettled issue in itself (e.g. Charbonneau & MacGregor,1993; Rudiger & Kitchatinov, 1996; Mathis et al., 2008;Garaud & Garaud, 2008; Denissenkov et al., 2010).

The parameterKW can be fixed by requiring that the evolu-tion ofΩenv(t) extracted from the models matches that ofΩp ofthe subgiants in thelog g-Ω plane. As can be seen from Figure 2,adjusting the single parameterKW produces a remarkably goodfit of all theΩp in the D+14 sample. This implies that the surfacespin-down of these subgiants is still in the Skumanich regime(i.e.,P ∝ t1/2).

The calibration ofKW depends on the mass of the model, butis insensitive to the value ofD0, as can be seen from the overlapof theΩenv evolutions plotted in Figure 2, or even to the pre-scription used for the angular momentum diffusion coefficient(compare Figure 2 with Figures 3 and 4). ForM∗ = 1.25M⊙

we obtainKW = 0.8. This value ofKW is kept fixed in all thefollowing calculations.

4.2. Constant angular momentum diffusion coefficient

We first discuss models with a diffusion coefficient independentof differential rotation (i.e.,α = 0 in equation 3). The resultingpoMS rotational evolution for several values ofD0 is shown inFigure 2.

From the Figure we can draw two main conclusions: first,although the overall behavior is not captured at all, valuesofD0 ≈ 104 cm2s−1 roughly match the order of magnitude ofboth the subgiants and the red giants core rotation (for com-parison, the typical value of the molecular viscosity in thera-diative zone is2–20 cm2s−1). This is significantly smaller thanwhat is required during the MS evolution of the Sun to matchthe helioseismic constraints (by a factor of25 according to ourestimate of Section 3.2), but comparable to the value found byEggenberger et al. (2012) for KIC8366239 (3 · 104 cm2s−1). Inother words, the poMS angular momentum transport is, over-all, less efficient than during the MS (see also the discussion inTayar & Pinsonneault, 2013).

Secondly, as was already noted by Cantiello et al. (2014),with a constant angular momentum diffusion efficiency it is im-possible to satisfactorily reproduce both the subgiant core spin-up and the subsequent spin-down occurring during the red giantphase. The small value ofD0 required to allow the developmentof internal differential rotation shortly after the TAMS resultsin a strong monotonic increase of the core rotation rate at latertimes, at odds with the trend observed in the M+12 red giants.

4.3. Angular momentum diffusion dependent ondifferential rotation

We now discuss the effect of a dependence of the angular mo-mentum diffusion coefficient on the degree of internal differen-tial rotation, in the form of equation (3). The resulting poMSrotational evolutions forα = 0, 1, 2, 3, and4 are shown in theupper panel of Figure 3. At first, we do not attempt to fit the ro-tational evolution of the subgiants in detail, but rather the overallspin-up of the core during the subgiant phase, as represented bythe most evolved star of the D+14 sample, and the subsequentcore spin-down. This calibration fixes the value ofD0, givenα. A more satisfactory fit of all the stars in the D+14 sample

Fig. 4. As in Figure 3, but showing models kept in a state ofrigid rotation until approximately1 Gyr after the TAMS. Upperpanel: poMS rotational evolution for various values ofα. Foreachα, D0 has been calibrated to obtain an optimal fit of boththe subgiants and the red giants. Lower panel: the evolutionofthe angular momentum diffusion coefficient according to equa-tion (3).

will be presented in Section 4.4. The lower panel of the Figureshows the corresponding evolution ofD according to equation(3). Note that, as a result of the calibration ofD0 chosen, the dif-fusion coefficient has approximately the same value (≈ 4.5 · 103

cm2s−1) aroundlog g = 3.8 in all the models.The rotational evolution from the early red giant phase on-

wards (log g . 3.5) is very sensitive toα. Qualitatively, theslope of the fit to the red giants trend (shown as a broken linein the Figure) is best reproduced withα ≈ 3.

4.4. Prolonged post-TAMS solid-body rotation phase

Introducing a dependence of the angular momentum diffusioncoefficient on the internal differential rotation according to equa-tion (3) allows the models to reproduce quite well the core spin-down trend on the red giant branch. The agreement with the sub-giant data is, however, not equally satisfactory. From Figures 2and 3 it is apparent that, as soon as the solid-body rotation con-

5

F. Spada, M. Gellert, R. Arlt and S. Deheuvels: Angular momentum transport efficiency in post-main sequence low-mass stars

Fig. 5. Kippenhahn diagram (run of interior structure quantitiesas a function of mass coordinate and age) for a1.25M⊙ model.The main region of hydrogen burning (i.e., whereεnucl > 10 ergs−1g−1) is hatched; “cloudy” areas indicate convection.

straint is relaxed,Ωrad andΩenv decouple very quickly, and thevalues ofD0 required by the overall fit allow the ratioΩrad/Ωenv

to reach too large values in comparison with the youngest star inthe D+14 sample.

In the models shown so far, a solid-body rotation profile isartificially enforced until the TAMS. In this Section, we explorethe possibility of a prolonged rigid rotation phase extendingbeyond the TAMS, of adjustable duration. Indeed, as was dis-cussed in the Introduction, there are good theoretical and obser-vational reasons (Gallet & Bouvier, 2013; Lanzafame & Spada,2015; Nielsen et al., 2014; Benomar et al., 2015) to assume thatstars of mass& 1M⊙ attain a rotationally coupled state on theirmature main sequence. There are, however, no constraints onwhether this state should cease immediately at the end of corehydrogen burning or last longer.

In Figure 4 we plot the results obtained with a prolongedrigid rotation phase. The best agreement with the D+14 sub-giants is found when enforcing solid-body rotation until the ageof ≈ 4.8 Gyr. This roughly corresponds to the age at which shellhydrogen burning has become well established (see Figure 5).For future reference, this occurs when the stellar radius has ex-panded to≈ 2.16R⊙, or about1 Gyr after the end of core hydro-gen burning. Bothα = 0 andα = 1, 2, 3, 4 models are shown inFigure 4. This time, for each value ofα, D0 has been calibratedto optimize the overall fit. The lower panel of Figure 4 shows thecorresponding evolution of the angular momentum diffusionco-efficientD. Note that the tuned values ofD0 reported in Figure4 are smaller by a factor of≈ 3 than those in Figure 3. This isdue to the prolonged rigid rotation phase that results in a weakerdifferential rotation to be quenched afterwards.

A remarkable agreement with the data, from the subgiants allthe way to the red giant branch, is achieved forα ≈ 3, D0 ≈ 1cm2s−1.

5. Discussion

The results of the previous section show that, during the poMSevolution of low-mass stars, the rotational coupling efficiency re-quired to explain the observations is compatible with some tur-

bulent process (i.e., it is enhanced compared to the molecularviscosity) and it is not constant in time. The models including apower law dependence of the angular momentum diffusion co-efficient on the ratioΩrad/Ωenv (see equation 3) can reproducethis behavior withD0 ≈ 1 andα ≈ 3. As the lower panelsof Figures 3 and 4 show,D becomes essentially independent ofD0 andα during the red giant evolution forα & 2. This is aconsequence of the differential rotation feedback introduced byequation (3), which allows the angular momentum transport ef-ficiency to regulate itself and to reach a quasi-stationary state.

Various angular momentum transport processes in stellar in-teriors depend on the internal shear: for example, purely hydro-dynamic instabilities, such as the dynamical and secular shearinstability (e.g., Zahn, 1974), or magnetohydrodynamic instabil-ities, such as the Tayler instability (Tayler, 1957, 1973) or theAMRI (Rudiger et al., 2007, 2015).

For the AMRI, in particular, a scaling of the turbulent vis-cosity in terms of dimensionless numbers has been establishedthrough direct numerical simulations performed in a Taylor-Couette cylindrical setup, taking into account the thermalstrat-ification of the background fluid (Gellert et al., in preparation).According to these simulations, the enhancement of the turbu-lent viscosity with respect to its molecular value,νT /ν, is givenby:

νTν

∝

√

Pm

Ra

Re

µ2

Ω

, (4)

wherePm ≡ ν/η is the Prandtl number,Re ≡ ΩiRi(Ro −Ri)/ν the Reynolds number,Ra ≡ Q g (To−Ti) (Ro−Ri)

3/νχthe Rayleigh number, andµΩ ≡ Ωo/Ωi. In the previous defini-tions,g is the acceleration of gravity,η the magnetic diffusivity,χ the thermal conductivity, andQ the thermal expansion coef-ficient of the fluid;Ri, Ti, Ωi andRo, To, Ωo are the radius,temperature, and angular velocity at the inner and outer bound-ary of the simulation domain, respectively. The dependenceofνT /ν on rotation only is thus:

νTν

∝Ωi

(Ωo/Ωi)2=

Ω3i

Ω2o

. (5)

Since, as was shown in Section 4, our best-fitting model requiresD ∝ (Ωrad/Ωenv)

α with α ≈ 3, andD becomes almost inde-pendent ofα on the red giant branch forα & 2 (see Figure 4),we suggest the loose identificationΩi ≈ Ωrad andΩo ≈ Ωenv.

It must be emphasized, however, that, although the effec-tiveness of angular momentum transport by the AMRI in achemically homogeneous fluid has been shown by Rudiger et al.(2015), it is possible that the strong molecular weight gradientthat develops in the core of a poMS star suppresses the insta-bility. To assess the importance of this effect, the profile of themean molecular weightµ ≈ (2X + 3

4Y + 1

2Z)−1 in the interior

of our reference1.25M⊙ model at three different evolutionarystages is plotted in Figure 6. The profiles shown in the Figureroughly correspond to models right after the TAMS, and in thesubgiant and red giant stages. As expected, the mean molecularweight ranges fromµ ≈ 1.33 to µ ≈ 0.6 within the He coreand in the convection zone, respectively. The strongest gradientis found at the transition between the pure helium core and therest of the star. Clearly, more extensive numerical and modelingwork, taking into account the effect ofµ gradients directly, isrequired to put the identification suggested above between ourheuristic scaling (3) and equation (4) on firmer ground.

A satisfactory agreement of our models with both the sub-giants in the D+14 sample and the red giants in the M+12 sam-

6

F. Spada, M. Gellert, R. Arlt and S. Deheuvels: Angular momentum transport efficiency in post-main sequence low-mass stars

Fig. 6. Mean molecular weight profile as a function of the frac-tional mass in the interior of1.25M⊙ models at three differentevolutionary stages (cf. Figure 1): early post-TAMS (red, dia-monds), subgiant (green, squares) and red giant (blue, circles).The approximate formulaµ ≈ (2X + 3

4Y + 1

2Z)−1, valid for

a fully ionized gas, has been used. Ages in Gyr andlog g areshown on the side. The filled and open symbols on each curvemark the He core outer boundary and the bottom of the outerconvection zone, respectively. Dotted lines indicate the pure he-lium µ = 4/3 and pristine compositionµ ≈ 0.6.

ple can only be obtained by assuming that the solid-body rota-tion regime achieved during the mature MS continues to holduntil the nuclear energy source of the model has shifted fromcore to shell hydrogen burning. This occurs about1 Gyr after theTAMS in our 1.25M⊙ model. Since there are currently no ob-servational constraints on this issue, we can only propose sometheoretical arguments. The two leading explanations for the cou-pling in the interior of low-mass MS stars are magnetic fields(e.g., Rudiger & Kitchatinov, 1996) and internal gravity waves(e.g., Mathis et al., 2008). In the case of the former, it is plausi-ble that the action of magnetic fields continues beyond the end ofthe hydrogen burning phase, and becomes ineffective only whenthe star has undergone significant structural changes evolvingtowards the red giant branch. For the latter, Fuller et al. (2014)have shown that internal gravity waves can affect internal rota-tion on a short (≈ 100 Myr) time scale on the MS, but that thistime scale progressively increases, eventually leading toa de-coupling between the stellar core and envelope. They estimatedthat, for low-mass stars (M . 1.5M⊙), this occurs when thestellar radius has increased to about1.75 times the MS radius.This is in very good agreement with our prolonged solid-bodyrotation scenario discussed in Section 4.4, where the strong cou-pling regime was assumed to last untilR∗ ≈ 2.16R⊙ (note thatour 1.25M⊙ model has a MS radius of about1.2R⊙). Eitherway, we could speculate that differential rotation begins to de-velop at some point after the TAMS, during the subgiant/earlyred giant phase, until angular momentum redistribution is takenover by some other, dominant process during the poMS.

Finally, we note that the overshooting at the bottom of thesurface convection zone, which has been ignored in our calcu-lations, could have a significant impact on our results if it canbring the bottom of the outer convection zone close enough tothe hydrogen-burningshell, bridging the gap of the region where

most of the shear develops1. As a crude estimate, we find thaton a typical evolution, and considering an overshooting of0.20pressure scale heights at the bottom of the outer convectionzone,the overshooting layer is less than≈ 20% of the radial extensionof the shear region. This is not enough to change the qualitativepicture of the evolution from the subgiant through the red giantstages.

6. Conclusions

We have discussed poMS rotational evolution models for a solarcomposition,1.25M⊙ star, representative of the low-mass com-ponent (1.0 . M/M⊙ . 1.5) of the asteroseismic analyses ofDeheuvels et al. (2014) and Mosser et al. (2012).

The models include a standard parametrization of the brak-ing of the stellar surface due to the magnetized stellar winds.Angular momentum transport in the interior is treated as a dif-fusion process, implementing a simple formulation for a depen-dence of the diffusion coefficient on the internal differential ro-tation.

Our main conclusions are the following:

1. Angular momentum transport in the early poMS is less ef-ficient, by more than one order of magnitude, than that re-quired for the PMS and MS evolution of the Sun to matchthe helioseismic constraints;

2. Assuming an angular momentum diffusion coefficient con-stant in time results in a monotonic spin-up of the stellarcore from the subgiant phase onwards, which is incompat-ible with the available observational constraints;

3. An angular momentum diffusion coefficient dependent onthe internal shear can establish a quasi-stationary core spin-down regime during the red giant phase, and lead to a rota-tional evolution in agreement with the observations;

4. Full agreement between models and data from the TAMS allthe way to giant branch evolution (before the red clump) canbe achieved assuming that stars remain in a rigid rotationstate until the shell hydrogen burning phase (≈ 1 Gyr afterthe TAMS for a1.25M⊙ star).

Acknowledgements. FS acknowledges support from the Leibniz Institute forAstrophysics Potsdam (AIP) through the Karl SchwarzschildPostdoctoralFellowship. MG would like to acknowledge support by the Helmholtz AllianceLIMTECH.

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