Matteo Cantiello edited Results2.tex  over 10 years ago

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In fact our calculations confirm the results of \citet{Eggenberger:2012}. Models that only include angular momentum transport due to rotational instabilities and circulations fail to reproduce the splittings observed in the RGB star $\KIC$ (which result in the estimate for the core rotation shown as a dot in Fig.~\ref{omega}). The resulting cores are so rapidly rotating that the the perturbative approach to the splitting calculation is no more justified. Even models with an extremely slow initial rotation of $1 \kms$ result in rotational splittings one order of magnitude larger than the observed ones, which clearly shows this class of models can not explain the observations. This is in perfect agreement with the calculations of \citet{Eggenberger:2012}, even if the implementation of the physics of rotation is quite different in the GENEVA code compared to MESA \citep[See Sec.6 in ][and references therein]{Paxton:2013}.   The evolution of core rotation during the early RGB for our models is shown in Fig.~\ref{period}. Here the value shown for $P_{\rm c}$ is a mass average of the rotational period in the region below the maximum of the energy generation $\epsilon_{\rm nuc}$ in the H-burning shell (see e.g. Fig.~\ref{kernels}). This is already showing that our cores seem to rotate about 1 to 3 orders of magnitude faster than the values inferred by asteroseismology.   In particular the work of \citet{Mosser:2012} reveals that the cores of stars in the mass range 1.2...1.5$\mso$ {\bf spin down} ascending the early RGB as $P_{\rm c} \propto R^{0.7\pm0.3}$, while our stellar evolution calculations show spin up with different slopes, slopes ($P_{\rm c} \propto R^{-0.6}$ for models including ST and $P_{\rm c} \propto R^{-1.7}$ for models only including rotational angular momentum transport),  depending on the assumptions for angular momentum transport (Fig.~\ref{period}). This is clearly showing that the amount of torque between core and envelope during the RGB evolution is underestimated in the models. Of course it could still be that the inferred rotation rates are much lower due to the contribution of the p-dominated part of the mode (which mostly lives in the slowly rotating envelope). However detailed calculations of the mixed modes and their splittings using ADIPLS confirm that our models can not reproduce the observed splittings (see Fig.~\ref{kernels} and Fig.~\ref{splittings}).   Overall we found that the models including angular momentum transport due to Spruit-Tayler magnetic fields \citep[which are generated by differential rotation in radiative regions][]{Spruit:2002} do a much better job, but still result in rotational splittings on the order of $1\mu$Hz (Fig.~\ref{splitting}) which overestimate the core rotation rates by a factor of $\sim10$ (See Fig.~\ref{omega} and Fig.~\ref{period}). We explored if an increase in the efficiency of the ST mechanism could reconcile the models with the observations. However even increasing the efficiency of the diffusion coefficient resulting from the magnetic torques by a factor 100 results in a very small change in the overall coupling. This is due to the self-regulating nature of the Spruit-Tayler dynamo. The poloidal component of the magnetic field $B_r$ is generated by the Tayler instability that occurs in the toroidal component $B_{\phi}$ of the field. However the toroidal component is amplified by the differential rotation, which is in turn suppressed by the torque $\propto B_r B_{\phi}$. It is not too surprising then to observe that the system tend to relax around some average differential rotation state which is weakly dependent on a possible increase of the efficiency of the Tayler-Spruit dynamo loop.