deletions | additions       diff --git a/Early RGB.tex b/Early RGB.tex  index 1b909b3..b16701c 100644  --- a/Early RGB.tex  +++ b/Early RGB.tex 
...
\subsection{Early RGB}  %We show in Fig.~\ref{omega} the early red giant evolution of the core angular velocity in rotating $1.5\mso$ models %with different assumptions for the internal angular momentum transport. The models are initially rotating with a surface %velocity of $50\kms$. %$150\kms$   %The plot shows how the contraction of the core leads to different rates of spin up for the different models. On the other %hand the expanding envelope slows down substantially, following the expected $P_{\rm rot} \propto R^2$ scaling from %homologous expansion and angular momentum conservation (no substantial mass is lost in this phase). First we note that %models including angular momentum transport due to rotation couple very weakly core and envelope, resulting in very %similar core rotation rates than in models with homologous contraction and angular momentum conservation (no angular %momentum transport).   %Our calculations confirm the results of \citet{Eggenberger:2012}. Models that only include angular momentum transport due %to rotational instabilities and circulations clearly fail to reproduce the splittings observed in the RGB star $\KIC$ %(which, using the relation $P_{c} \simeq (2\delta\nu_{\rm max})^{-1}$, results in the estimate for the core rotation shown %as a star symbol in Fig.~\ref{period} and \ref{period_evolution}). The resulting cores are so rapidly rotating that 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 (Fig.~\ref{splitting}). This is in perfect agreement with the %calculations of \citet{Eggenberger:2012}, despite the implementation of the physics of rotation being 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 rotating $1.5\mso$ models with different assumptions for the internal angular momentum transport is shown in Fig.~\ref{period}. The models are initially rotating with a surface velocity of $50\kms$. 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}).  The contraction of the core leads to different rates of spin up for the different angular momentum transport mechanisms considered. At the same time he expanding envelope slows down substantially, following the expected $P_{\rm rot} \propto R^2$ scaling from homologous expansion and angular momentum conservation (no substantial mass is lost in this phase).  Overall in the calculations the cores seem to rotate about 1 to 3 orders of magnitude faster than the values inferred by asteroseismology. Note that during this phase the mass of the core increases only slightly (See Fig.~\ref{kipp}).  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 ($P_{\rm c} \propto R^{-0.58}$ for models including ST and $P_{\rm c} \propto R^{-1.32}$ for models only including angular momentum transport by rotational instabilities\footnote{These exponents have been calculated for the range R/$\rso$=[3,15]}), 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 to carefully compare the rotation rates in the models with the ones derived from the observations one has to calculate the relevant eigenfunctions for the mixed modes. This allows to calculate the rotational kernels $K_{n,\ell}(r)$ and compute the predicted splittings (See Fig.~\ref{kernels} and Sec.~\ref{splittings}). The results of a detailed comparison for the predicted splittings of $\KIC$ as function of different initial rotational velocities and physics of angular momentum transfer is shown in Fig.~\ref{splitting}. The splittings have been calculated using the ADIPLS code and the MESA background structure at the location that matches the asteroseismic properties of the star.  Our calculations confirm the results of \citet{Eggenberger:2012}. Models that only include angular momentum transport due to rotational instabilities and circulations clearly fail to reproduce the splittings observed in the RGB star $\KIC$ (which, using the relation $P_{c} \simeq (2\delta\nu_{\rm max})^{-1}$, results in the estimate for the core rotation shown as a star symbol in Fig.~\ref{period} and \ref{period_evolution}). The resulting cores are so rapidly rotating that 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 (Fig.~\ref{splitting}). This is in perfect agreement with the calculations of \citet{Eggenberger:2012}, even if despite  the implementation of the physics of rotation is being  quite different in the GENEVA code compared to MESA \citep[See Sec.6 in ][and references therein]{Paxton:2013}. 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.