Matteo Cantiello edited Stellar evolution calculations.tex  about 10 years ago

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\section{Stellar evolution calculations} calculations}\label{stev}  We use the Modules for Experiments in Stellar Evolution (MESA) code to evolve low-mass stars from the pre-main-sequence to the cooling WD sequence \citep{Paxton:2011,Paxton:2013}. This code includes the effects of the centrifugal force on the stellar structure, chemical mixing and transport of angular momentum due to rotationally induced hydrodynamic instabilities \citep{Heger:2000}. The mixing of angular momentum due to dynamo-generated magnetic fields in radiation zones is also included \citep{Spruit:2002,Petrovic:2005,Heger:2005}. See \citet{Paxton:2013} for the details of the implementation of rotation and magnetic fields in MESA. %To account for the changes in the distribution of internal angular momentum during the long timescales of stellar evolution, modern stellar evolution %codes include a treatment of angular momentum and chemical mixing due to rotational instabilities and magnetic torques. In fact it has been shown that %it is still possible to treat this problem in 1D if the star is well mixed on horizontal surfaces. The existence of strong anisotropic turbulence in %stars is expected to enforce a constant composition and angular velocity on isobaric surfaces, the so-called ``shellular'' approximation %\citep{Zahn:1992}. The impact of the centrifugal term can be easily included in the equation of stellar structure \citep{Endal:1976}, while the effect %of transport of angular momentum and chemicals in the radial direction requires a simplified treatment of the rotational instabilities and circulations %expected to arise in a rotating star \citep[See e.g.][]{Heger:2000,Maeder:2000}. Dynamo action seems to be ubiquitous in astrophysical plasma, and %the presence of magnetic fields can lead to efficient transport of angular momentum through magnetic torques. In radiative zones the presence of %magnetic fields has been discussed to explain the final rotation rate of compact remnants \citep[both white dwarfs and neutron stars,][]%{Heger_Langer_Woosley_2000,larends_Yoon_Heger_Herwig_2008}. We chose an initial metallicity of $Z=0.02$ with a mixture taken from \citet{Asplund:2005}. We adopt the OPAL opacity tables \citep{Iglesias:1996} accounting for the carbon- and oxygen- enhanced opacities during helium burning \citep[Type 2 OPAL,][]{Iglesias:1993}. Solid body rotation is set at the zero age main sequence (ZAMS). Convective regions are calculated using the mixing length theory (MLT) in the \citet{Henyey:1965} formulation with $\alphaMLT=1.6$. Transport of angular momentum in convective regions is accounted for using the resulting MLT diffusion coefficient (turbulent diffusivity), which is generally very high and leads to rigid rotation in convective zones. While this seems to be the case in the Sun, another possible treatment of rotating convective zones is adopting a constant specific angular momentum \citep[See e.g.][]{Kawaler:2005}. We ran calculations with this assumption and found that it does not affect our conclusions. The boundaries of convective regions are determined using the Ledoux criterion. Semiconvection is accounted for in the Langer prescription \citep{Langer:1983,Langer:1985} with an efficiency $\alphasc$ = 0.003. A step function overshooting extends for 0.2 pressure scale heights the mixing region beyond the convective boundary during core H-burning. %MC: Check if it's also extending core He-burning We also account for gravitational settling and chemical diffusion \citep{Paxton:2011}. Figure~\ref{kipp} shows Kippenhahn diagrams for the $1.5\mso$ model.