Asteroseismic data obtained by the Kepler spacecraft have led to the recent detection and characterization of rotational frequency splittings of mixed modes in red-giant stars. This has opened the way to the determination of the core rotation rates for these stars, which is of prime importance to progress in our understanding of internal angular momentum transport. In this contribution, we discuss which constraints can be brought by these asteroseismic measurements on the modelling of angular momentum transport in stellar radiative zone

Eggenberger, P.

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New windows on massive stars: asteroseismology, interferometry, andspectropolarimetryProceedings IAU Symposium No. 307, 2014G. Meynet, C. Georgy, J. H. Groh & Ph. Stee, eds.c© International Astronomical Union 2015doi:10.1017/S1743921314006656Asteroseismology of red giants to constrainangular momentum transportP. EggenbergerGeneva Observatory, University of Geneva, Maillettes 51, 1290, Sauverny, Switzerlandemail: patrick.eggenberger@unige.chAbstract. Asteroseismic data obtained by the Kepler spacecraft have led to the recent detectionand characterization of rotational frequency splittings of mixed modes in red-giant stars. This hasopened the way to the determination of the core rotation rates for these stars, which is of primeimportance to progress in our understanding of internal angular momentum transport. In thiscontribution, we discuss which constraints can be brought by these asteroseismic measurementson the modelling of angular momentum transport in stellar radiative zones.Keywords. stars: evolution, stars: interiors, stars: oscillations, stars: rotation1. IntroductionThe inclusion of rotational eﬀects basically changes all outputs of stellar models (e.g.Maeder 2009). A coherent physical description of rotational eﬀects has to take into ac-count simultaneously the transport of chemical elements and angular momentum. Theeﬃciency of angular momentum transport is then found to have an important impact onthe properties of rotating models. To progress in our understanding of internal angularmomentum transport, one needs direct observational constraints on the internal rotationof stars, which are now available thanks to seismic data.Helioseismic measurements indicate a nearly ﬂat rotation proﬁle in the radiative zone ofthe Sun (Brown et al. 1989; Elsworth et al. 1995; Kosovichev et al. 1997; Couvidat et al.2003; Garc´ıa et al. 2007), while solar models including only shellular rotation predicta rapidly rotating core (Pinsonneault et al. 1989; Chaboyer et al. 1995; Eggenbergeret al. 2005; Turck-Chie`ze et al. 2010). This is a ﬁrst indication that an undeterminedmechanism for the angular momentum transport is at work in the solar interior.In the continuity of helioseismic measurements, solar-like oscillations have been ob-served for a large number of stars. In particular, rotational frequency splittings havebeen recently detected and characterized for red giants (Beck et al. 2012; Deheuvelset al. 2012; Mosser et al. 2012; Deheuvels et al. 2014). In this contribution, we brieﬂy dis-cuss the implications of these new observational constraints on the modelling of angularmomentum transport in stellar radiative zones.2. Mixed oscillation modes in the red giant KIC 8366239The ﬁrst detection and characterization of rotational frequency splittings of mixedmodes has been obtained for the red giant KIC 8366239 (Beck et al. 2012). These mea-surements show that oscillation modes at the center of the dipole forests have lowervalues of rotational splitting than modes in the wings of the dipole forests. This directlyindicates that diﬀerential rotation is present in the interior of this red giant, with a corerotating more rapidly than the surface (Beck et al. 2012). At ﬁrst sight, this seems tobe in good agreement with stellar models, which predict diﬀerential rotation during the165at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921314006656Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:21:07, subject to the Cambridge Core terms of use, available166 P. Eggenbergerred-giant phase because of the rapid contraction of the central layers after the main se-quence (e.g Palacios et al. 2006; Eggenberger et al. 2010). However, by computing modelsincluding rotation for this target and comparing the theoretical values for the rotationalsplittings with the observed ones, we conclude that rotating models predict too steepinternal rotation proﬁles compared to asteroseismic measurements (Eggenberger et al.2012; Marques et al. 2013). This clearly indicates that an additional mechanism for theinternal angular momentum transport is at work during the post-main sequence phase.In order to study how this unknown transport mechanism can be constrained by as-teroseismic data, Eggenberger et al. (2012) introduced an additional viscosity νadd cor-responding to this process in the equation describing the internal transport of angularmomentum. By computing models of KIC 8366239 for diﬀerent values of νadd and of theinitial velocity on the zero-age main sequence (ZAMS), one ﬁnds that the mean valuefor the eﬃciency of this unknown transport mechanism is strongly constrained by aster-oseismic measurements to the value of νadd = 3 · 104 cm2 s−1 for this 1.5M red giant(Eggenberger et al. 2012).3. KIC 7341231: mixed modes in a low-mass red giantIn the preceding section, we have seen that an additional mechanism for the internaltransport of angular momentum is needed during the post-main sequence evolution of a1.5M. This star exhibits a convective core during the main sequence and one can thenwonder whether such a mechanism is also needed for a low-mass star with a radiative coreduring the main sequence. Rotational splittings have been obtained for KIC 7341231, ared giant with a mass of about 0.84M (Deheuvels et al. 2012). For this star witha radiative core during the main sequence, one also ﬁnds that an eﬃcient additionalmechanism for the internal transport of angular momentum is needed in order to correctlyreproduce the asteroseismic data (Ceillier et al. 2012, 2013).As for the more massive red giant KIC 8366239, the eﬃciency of this unknown trans-port process can be determined for KIC 7341231 by computing models including anadditional viscosity νadd . In this case, we ﬁnd a lower value of about 3 · 103 cm2 s−1for the additional viscosity. The model of KIC 7341231 has a convective enveloppe dur-ing the main sequence and the initial velocity on the ZAMS is thus found to dependon the adopted eﬃciency for the associated surface magnetic braking during the mainsequence. Interestingly, the value of the additional viscosity νadd can be precisely deter-mined independently from the uncertainties on the adopted magnetic braking during themain sequence. The diﬀerences in νadd between KIC 8366239 and KIC 7341231 suggeststhat the eﬃciency of the undetermined additional process for the transport of angularmomentum decreases when the mass of the star decreases.4. Rotational splittings for a sample of six subgiants and young redgiantsAsteroseismic measurements have been recently obtained for a sample of six Kepleryoung red giants (Deheuvels et al. 2014). Estimates of the core and surface rotation rateshave then been deduced for these stars. These results suggest that the degree of diﬀer-ential rotation increases when evolution proceeds, with an increase of the core rotationrate and a decrease of the surface rotation rate when gravity decreases (Deheuvels et al.2014).at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921314006656Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:21:07, subject to the Cambridge Core terms of use, availableAsteroseismology of red giants 167Interestingly, rotational splittings suggest that for two of these targets a discontinuousinternal rotation proﬁle is favored compared to a smooth rotation proﬁle. The determi-nation of the location of such a discontinuity suggests that it corresponds to the locationof the hydrogen-burning shell. These results seem to indicate that the eﬃciency of theadditional mechanism of internal angular momentum transport should strongly decreasein a region of large chemical gradients. Such a sharp discontinuity in the rotation proﬁleseems to be diﬃcult to reproduce with fossil magnetic ﬁelds and could favor diﬀusiveprocesses with a reduced eﬃciency related to the presence of steep chemical gradients.By comparing the deduced core and surface rotation rates with rotating models includ-ing an additional viscosity associated to the unkown transport process, one can preciselydetermined the eﬃciency of this process for the six targets. This also suggests an in-crease of νadd with the mass in good agreement with the preceding result obtained bycomparing KIC 8366239 and KIC 7341231. Interestingly, the simultaneous determina-tion of the core and surface rotation rates for these stars can also constrain the eﬃciencyof the magnetic braking associated to the presence of a convective envelope during themain-sequence phase. One also ﬁnds that the change of the eﬃciency of the transport ofangular momentum in the radiative zone has a negligible impact on the surface rotationrates during the red-giant phase (see dashed lines in Fig. 1). This is obvious in viewof the diﬀerence in the moment of inertia between the core and the enveloppe of a redgiant. It is however interesting to recall this point in the context of red giants exhibitinga rapid surface rotation velocity, which origin is more related to binary interaction orplanet engulfment than to an eﬃcient transport of angular momentum in the radiativeinterior.5. Core rotation rates for a large sample of red giantsMosser et al. (2012) have determined mean core rotation rates from rotational splittingsfor a large sample of red giants observed with the Kepler spacecraft. Interestingly, thisstudy suggests that core rotation rates slightly decrease during the evolution on thered-giant branch.As shown in Fig. 1, a model including only shellular rotation (dotted line) predictsa very rapidly rotating core during the red giant phase. Moreover, it predicts an in-crease of the core rotation rate when the star evolves on the red-giant branch, which isalso in contradiction with asteroseismic data. When an additional constant viscosity of3 · 104 cm2 s−1 is included (continuous line in Fig. 1), the model correctly reproduces theasteroseismic data available for the red giant KIC 8366239 and also predicts a slight in-crease in the core rotation rates of red giants with a radius lower than about 5R. Thisis in good agreement with core rotation rates deduced from asteroseismic data. How-ever, for higher values of the radius, this model predicts an increase of the core rotationrates when evolution proceeds. This discrepancy with the trend deduced from asteroseis-mic measurements indicates that the eﬃciency of the unknown transport mechanism ofangular momentum must increase when the star evolves along the red-giant branch.6. Comparison with mechanisms already included in stellar evolutioncodesIn the preceding sections, we have seen that according to asteroseismic data an un-known mechanism is needed in addition to meridional circulation and the shear instabilityfor the transport of angular momentum in the radiative zone of red giants. Asteroseismicdata also bring valuable constraints on the eﬃciency of such a process for red giants ofat https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921314006656Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:21:07, subject to the Cambridge Core terms of use, available168 P. EggenbergerFigure 1. Core (continuous and dotted lines) and surface (dashed lines) rotation periods for tworotating models of 1.5M with an initial velocity of 20 km s−1 on the ZAMS. The dotted line(blue) indicates a model computed with shellular rotation only. The continuous line (red) corre-sponds to the same model except for the inclusion of an additional viscosity of 3× 104 cm2 s−1 .diﬀerent masses and at diﬀerent evolutionary stages. One can then wonder whether theseseismic constraints can be correctly reproduced by mechanisms already included in someevolution codes such as magnetic ﬁelds and internal gravity waves.Internal gravity waves are found to be able to eﬃciently transport angular momentumin the radiative zone of solar-type stars (see e.g. Zahn et al. 1997; Talon & Charbonnel2003; Charbonnel & Talon 2005; Mathis et al. 2013). It would then be interesting to seewhether rotating models of red giants including the transport of angular momentum byinternal gravity waves are able to correclty reproduce the asteroseismic constraints nowavailable for the internal rotation of red giants.Concerning magnetic ﬁelds, Cantiello et al. (2014) recently compared rotating modelsof red giants including the Tayler-Spruit dynamo (Spruit 1999, 2002) to asteroseismicmeasurements. They ﬁnd that models including the transport of angular momentum bythe Tayler-Spruit dynamo are in better agreement with asteroseismic determinations ofcore rotation rates of red giants than models including only rotational eﬀects. However,models including the Tayler-Spruit dynamo still predict core rotation rates which are atleast ten times higher than the observed values.7. SummaryAsteroseismic constraints on the internal rotation of red giants are now available tohelp us progress in the modelling of the diﬀerent transport processes at work in stellarradiative zones. The goal is now to use these new valuable constraints to obtain a coherentglobal physical description of transport processes in radiatives zones that should be validfor low and massive stars, from the pre-main sequence to advanced stages of evolution.Preliminary results of comparison of these asteroseismic measurements with stellarmodels clearly show that an undetermined mechanism for the internal transport of an-gular momentum is needed during the post-main sequence evolution. This is the case forat https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921314006656Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:21:07, subject to the Cambridge Core terms of use, availableAsteroseismology of red giants 169low-mass red giants with a radiative core during the main sequence as well as for moremassive stars with a convective core during the main sequence. Thanks to seismic data,the eﬃciency of this unknown transport process can be precisely determined. Asteroseis-mic measurements suggest an increase of the eﬃciency of this process when the massincreases as well as when the star evolves on the red-giant branch.Concerning the physical nature of this unknown process, it will be interesting to com-pare the rotation proﬁles of red-giant models including angular momentum transport byinternal gravity waves to the asteroseismic constraints now available. Diﬀerent clues maysuggest that transport of angular momentum by magnetic instabilities could be a promis-ing candidate. While a ﬁrst study of the impact of the Tayler-Spruit dynamo indicatesthat the eﬃciency of this process is not suﬃcient to account for the low core rotationrates deduced from seismic data (Cantiello et al. 2014), it will be particularly interestingto consider the eﬀects of the azimuthal magnetorotational instability in this context (e.g.Ru¨diger et al. 2014, and see the contribution by M. Gellert in this volume).ReferencesBeck, P. G., Montalban, J., Kallinger, T., et al. 2012, Nature 481, 55Brown, T. M., Christensen-Dalsgaard, J., Dziembowski, W. A., et al. 1989, ApJ 343, 526Cantiello, M., Mankovich, C., Bildsten, L., Christensen-Dalsgaard, J., & Paxton, B. 2014, ApJ788, 93Ceillier, T., Eggenberger, P., Garc´ıa, R. A., & Mathis, S. 2012, Astronomische Nachrichten 333,971Ceillier, T., Eggenberger, P., Garc´ıa, R. A., & Mathis, S. 2013, A&A 555, A54Chaboyer, B., Demarque, P., & Pinsonneault, M. H. 1995, ApJ 441, 865Charbonnel, C. & Talon, S. 2005, Science 309, 2189Couvidat, S., Garc´ıa, R. A., Turck-Chie`ze, S., et al. 2003, ApJ (Letters) 597, L77Deheuvels, S., Dog˘an, G., Goupil, M. J., et al. 2014, A&A 564, A27Deheuvels, S., Garc´ıa, R. A., Chaplin, W. J., et al. 2012, ApJ 756, 19Eggenberger, P., Maeder, A., & Meynet, G. 2005, A&A 440, L9Eggenberger, P., Miglio, A., Montalban, J., et al. 2010, A&A 509, A72Eggenberger, P., Montalba´n, J., & Miglio, A. 2012, A&A 544, L4Elsworth, Y., Howe, R., Isaak, G. R., et al. 1995, Nature 376, 669Garc´ıa, R. A., Turck-Chie`ze, S., Jime´nez-Reyes, S. J., et al. 2007, Science 316, 1591Kosovichev, A. G., Schou, J., Scherrer, P. H., et al. 1997, Sol. Phys. 170, 43Maeder, A. 2009, Physics, Formation and Evolution of Rotating Stars, Springer BerlinHeidelbergMarques, J. P., Goupil, M. J., Lebreton, Y., et al. 2013, A&A 549, A74Mathis, S., Decressin, T., Eggenberger, P., & Charbonnel, C. 2013, A&A 558, A11Mosser, B., Goupil, M. J., Belkacem, K., et al. 2012, A&A 548, A10Palacios, A., Charbonnel, C., Talon, S., & Siess, L. 2006, A&A 453, 261Pinsonneault, M. H., Kawaler, S. D., Soﬁa, S., & Demarque, P. 1989, ApJ 338, 424Ru¨diger, G., Gellert, M., Schultz, M., Hollerbach, R., & Stefani, F. 2014, MNRAS 438, 271Spruit, H. C. 1999, A&A 349, 189Spruit, H. C. 2002, A&A 381, 923Talon, S. & Charbonnel, C. 2003, A&A 405, 1025Turck-Chie`ze, S., Palacios, A., Marques, J. P., & Nghiem, P. A. P. 2010, ApJ 715, 1539Zahn, J.-P., Talon, S., & Matias, J. 1997, A&A 322, 320DiscussionMoravveji: Can we take the lessons learned about viscosity on RGB phase to otherstallar evolution phases?at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921314006656Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 30 May 2017 at 17:21:07, subject to the Cambridge Core terms of use, available170 P. EggenbergerEggenberger: Yes, the idea is to use all available observational constraints to progressin the modelling of transport processes in stellar radiative zones. In this context, theasteroseismic data obtained for red giants are particularly valuable. A coherent physicaldescription of these processes should then be valid for low-mass stars as well as for massivestars and from the pre-main sequence to advanced stages of evolution.Hirschi: Can you conﬁrm that a process transporting angular momentum all the wayto the centre of the star is needed to reproduce observations?Eggenberger: Yes, core rotation rates of red giants deduced from asteroseismic mea-surements clearly show that an eﬃcient process for the transport of angular momentum isneeded in addition to meridional circulation and the shear instability to extract angularmomentum from the central layers.at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921314006656Downloaded from https:/www.cambridge.org/core. 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Asteroseismology of red giants to constrain angular momentum transport

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Asteroseismology of red giants to constrain angular momentum transport

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