\newcommand{\ensuremath}{}
%\newcommand{\mesa}{\code{MESA}}
\newcommand{\MESA}{\mesa}
\newcommand{\ga}{\gt}
\newcommand{\varv}{v}
\newcommand{\vca}{v_{\mathrm {c}}}
\newcommand{\mso}{\,\mathrm{M}_\odot}
\newcommand{\rso}{\,{\rm R}_\odot}
\newcommand{\lso}{\,{\rm L}_\odot}
\newcommand{\hrd}{HR diagram}
\newcommand{\kms}{\, {\rm km}\, {\rm s}^{-1}}
\newcommand{\egs}{\, {\rm erg}\, {\rm s}^{-1}}
\newcommand{\teff}{\log\, T_{\rm }\,}
\newcommand{\llso}{\log\, L/ L_\odot \,}
\newcommand{\simle}{\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}}
\newcommand{\simgr}{\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}}
\newcommand{\grad}{\nabla}
\newcommand{\adrad}{\nabla_{\mathrm{\!rad}}}
\newcommand{\adgrad}{\nabla_{\mathrm{\!ad}}}
\newcommand{\mugrad}{\nabla_{\!\mu}}
\newcommand{\cp}{c_{\!P}}
\newcommand{\hp}{H_{\mathrm{P}}}
% macros for naming codes, plus shortcuts for some common ones
%\newcommand{\code}[1]{\texttt{#1}}
\newcommand{\code}{}
\newcommand{\flash}{FLASH}
\newcommand{\kepler}{\textit{Kepler} }
\newcommand{\Kepler}{\textit{Kepler}}
\newcommand{\KEPLER}{\textit{Kepler}}
\newcommand{\nonsmoker}{NON-SMOKER}
\newcommand{\mesa}{\code{MESA}}
\newcommand{\MESA}{\mesa}
\newcommand{\STERN}{STERN}
\newcommand{\ADIPLS}{\code{ADIPLS}}
\newcommand{\DSEP}{DSEP}
\newcommand{\KIC}{\rm{KIC}8366239}
\newcommand{\KID}{\rm{KIC}5006817}
% modules for MESA, from the original instrument paper
\newcommand{\alert}{\code{alert}}
\newcommand{\utils}{\code{utils}}
\newcommand{\const}{\code{const}}
\newcommand{\chem}{\code{chem}}
\newcommand{\diffusion}{\code{diffusion}}
\newcommand{\mtx}{\code{mtx}}
\newcommand{\mesastar}{\mesa~\code{star}}
\newcommand{\MESAstar}{\mesastar}
\newcommand{\num}{\code{num}}
\newcommand{\nuc}{\code{nuc}}
\newcommand{\kap}{\code{kap}}
\newcommand{\eos}{\code{eos}}
\newcommand{\intone}{\code{interp\_1d}}
\newcommand{\inttwo}{\code{interp\_2d}}
\newcommand{\atm}{\code{atm}}
\newcommand{\diff}{\code{diffusion}}
\newcommand{\mlt}{\code{mlt}}
\newcommand{\rates}{\code{rates}}
\newcommand{\net}{\code{net}}
\newcommand{\neu}{\code{neu}}
\newcommand{\weak}{\code{weaklib}}
\newcommand{\screen}{\code{screen}}
\newcommand{\ioniz}{\code{ionization}}
\newcommand{\adipls}{\code{adipls}}
\newcommand{\colors}{\code{colors}}
\newcommand{\reaclib}{\code{reaclib}}
\newcommand{\karo}{\code{karo}}
\newcommand{\astero}{\code{astero}}
\newcommand{\sdk}{\code{SDK}}
\newcommand{\SDK}{\sdk}

%derivatives
\newcommand{\dif}{\ensuremath{\mathrm{d}}}
\newcommand{\D}{{\mathrm d}}
\newcommand{\DD}{{\,\D\!\!\;}}
\newcommand{\ddt}[1]{\frac{\partial #1}{\partial t}} %partial time derivative 
\newcommand{\DDt}[1]{\frac{\dif #1}{\dif t}} %total time derivative
\newcommand{\ddx}[1]{\frac{\partial #1}{\partial x}} %partial derivative wrt x 
\newcommand{\ddy}[1]{\frac{\partial #1}{\partial y}} %partial derivative wrt y 
\newcommand{\DDy}[1]{\frac{\dif #1}{\dif y}} %total derivative wrt y
\newcommand{\ddz}[1]{\frac{\partial #1}{\partial z}} %partial derivative wrt z 
\newcommand{\ppl}[2]{\left(\frac{\partial\ln #1}{\partial\ln
      #2}\right)_{\rho,T}}
\newcommand{\ppll}[2]{\left(\frac{\partial\ln #1}{\partial\ln
      #2}\right)_{\rho,T,\{X_{j\neq i}\}}}
\newcommand{\ddl}[2]{\frac{{\rm d}\ln #1}{{\rm d}\ln #2}}
\newcommand{\DxDy}[2]{{\frac{\D{#1}}{\D{#2}}}}
\newcommand{\dxdy}[2]{{\frac{\partial{#1}}{\partial{#2}}}}
\newcommand{\dxdyind}[3]{{\Brak{\frac{\D{#1}}{\D{#2}}}_{{#3}}}}
\newcommand{\dxdycz}[3]{{\Brak{\frac{\partial{#1}}{\partial{#2}}}_{{#3}}}}

% nuclides
% base command
\newcommand{\nuclei}[2]{\ensuremath{\mathrm{^{#1}#2}}}
% nuclides, with most highest abundance or longest half-life as default
% for example, \carbon produces ^{12}C, \carbon[13] produces ^{13}C
\newcommand{\neutron}{\ensuremath{n}}
\newcommand{\nt}{\neutron}
\newcommand{\proton}{\ensuremath{p}}
\newcommand{\pt}{\proton}
\newcommand{\photon}{\ensuremath{\gamma}}
\newcommand{\hydrogen}[1][1]{\nuclei{#1}{H}}
\newcommand{\helium}[1][4]{\nuclei{#1}{He}}
\newcommand{\lithium}[1][7]{\nuclei{#1}{Li}}
\newcommand{\beryllium}[1][9]{\nuclei{#1}{Be}}
\newcommand{\boron}[1][11]{\nuclei{#1}{B}}
\newcommand{\carbon}[1][12]{\nuclei{#1}{C}}
\newcommand{\nitrogen}[1][14]{\nuclei{#1}{N}}
\newcommand{\oxygen}[1][16]{\nuclei{#1}{O}}
\newcommand{\fluorine}[1][19]{\nuclei{#1}{F}}
\newcommand{\neon}[1][20]{\nuclei{#1}{Ne}}
\newcommand{\sodium}[1][23]{\nuclei{#1}{Na}}
\newcommand{\magnesium}[1][24]{\nuclei{#1}{Mg}}
\newcommand{\aluminum}[1][27]{\nuclei{#1}{Al}}
\newcommand{\silicon}[1][28]{\nuclei{#1}{Si}}
\newcommand{\phosphorus}[1][31]{\nuclei{#1}{P}}
\newcommand{\sulfur}[1][32]{\nuclei{#1}{S}}
\newcommand{\chlorine}[1][35]{\nuclei{#1}{Cl}}
\newcommand{\argon}[1][36]{\nuclei{#1}{Ar}}
\newcommand{\potassium}[1][39]{\nuclei{#1}{K}}
\newcommand{\calcium}[1][40]{\nuclei{#1}{Ca}}
\newcommand{\scandium}[1][45]{\nuclei{#1}{Sc}}
\newcommand{\titanium}[1][48]{\nuclei{#1}{Ti}}
\newcommand{\vanadium}[1][51]{\nuclei{#1}{V}}
\newcommand{\chromium}[1][52]{\nuclei{#1}{Cr}}
\newcommand{\manganese}[1][55]{\nuclei{#1}{Mn}}
\newcommand{\iron}[1][56]{\nuclei{#1}{Fe}}
\newcommand{\cobalt}[1][59]{\nuclei{#1}{Co}}
\newcommand{\nickel}[1][58]{\nuclei{#1}{Ni}}
\newcommand{\copernicum}[1][285]{\nuclei{#1}{Cn}}

% typesetting of units
% basic unit typesetteing
\newcommand{\unitspace}{\ensuremath{\,}}
\newcommand{\usp}{\unitspace}
\newcommand{\numberspace}{\ensuremath{\;}}
\newcommand{\nsp}{\numberspace}
\newcommand{\unitstyle}[1]{\ensuremath{\mathrm{#1}}}
\newcommand{\power}[2]{\ensuremath{{#1}^{#2}}}
\newcommand{\ee}[1]{\ensuremath{\times 10^{#1}}}

% prefixes
\newcommand{\nano}{\unitstyle{n}}
\newcommand{\milli}{\unitstyle{m}}
\newcommand{\centi}{\unitstyle{c}}
\newcommand{\kilo}{\unitstyle{k}}
\newcommand{\Mega}{\unitstyle{M}}
\newcommand{\Giga}{\unitstyle{G}}

% base units, mks
\newcommand{\meter}{\unitstyle{m}}
\newcommand{\kilogram}{\kilo\gram}
\newcommand{\second}{\unitstyle{s}}

\newcommand{\Kelvin}{\unitstyle{K}}
\newcommand{\K}{\Kelvin}  %degrees Kelvin

% base units, cgs
\newcommand{\cm}{\centi\meter}
\newcommand{\gram}{\unitstyle{g}}

% derived units
\newcommand{\grampercc}{\gram\usp\power{\cm}{-3}} %mass density
\newcommand{\grampersquarecm}{\gram\usp\power{\cm}{-2}} %column depth
\newcommand{\squarecmpergram}{\power{\cm}{2}\usp\power{\gram}{-1}} %opacity
\newcommand{\GramPerCc}{\grampercc}
\newcommand{\GramPerSc}{\grampersquarecm}
\newcommand{\GramSc}{\gram\usp\power{\cm}{2}}
\newcommand{\columnunit}{\grampersquarecm}
\newcommand{\dyne}{\unitstyle{dyn}} %dyne
\newcommand{\erg}{\unitstyle{ergs}} %ergs
\newcommand{\ergs}{\erg}
\newcommand{\gauss}{\unitstyle{G}} %gauss
\newcommand{\ergspersecond}{\erg\unitspace\power{\second}{-1}}
\newcommand{\ergspergram}{\erg\unitspace\power{\gram}{-1}}
\newcommand{\ergspergs}{\erg\unitspace\power{\gram}{-1}\unitspace\power{\second}{-1}} %angular momentum
\newcommand{\ergssecond}{\erg\unitspace\second}
\newcommand{\cgsflux}{\erg\unitspace\power{\cm}{-2}\usp\power{\second}{-1}}

% Nuclear and atomic units
\newcommand{\amu}{\unitstyle{u}} %atomic mass unit
\newcommand{\angstrom}{\mbox{\AA}} %Angstrom
\newcommand{\fermi}{\unitstyle{fm}} %fermi
\newcommand{\eV}{\unitstyle{eV}}        %eV
\newcommand{\keV}{\kilo\eV} %Kev
\newcommand{\MeV}{\Mega\eV} %MeV

% solar and astronomical units
\newcommand{\Msun}{\ensuremath{\unitstyle{M}_\odot}}
\newcommand{\Lsun}{\ensuremath{\unitstyle{L}_{\odot}}}
\newcommand{\Rsun}{\ensuremath{\unitstyle{R}_{\odot}}}
\newcommand{\Zsun}{\ensuremath{Z_{\odot}}}
\newcommand{\Myr}{\Mega\yr}
\newcommand{\Gyr}{\Giga\yr}
\newcommand{\parsec}{\unitstyle{pc}}
\newcommand{\kpc}{\kilo\parsec} %kiloparsec
\newcommand{\mJy}{\unitstyle{\mu Jy}} %micro Jansky
\newcommand{\Msunyr}{\Msun\,\power{\yr}{-1}}
\newcommand{\MJ}{\ensuremath{\mathrm{M_J}}}
\newcommand{\RJ}{\ensuremath{\mathrm{R_J}}}
\newcommand{\AU}{\unitstyle{AU}}

% misc. units
\newcommand{\minute}{\unitstyle{min}} %minute
\newcommand{\hour}{\unitstyle{hr}} %hour
\newcommand{\yr}{\unitstyle{yr}}        %year
\newcommand{\km}{\kilo\meter}   %kilometers
\newcommand{\Hz}{\unitstyle{Hz}}        %Hertz
\newcommand{\ksec}{\kilo\second} %kilosecond
\newcommand{\kms}{\ensuremath{\mathrm{km}\,\second^{-1}}}
\newcommand{\vcrit}{{\varv_{\mathrm{crit}}}}
\newcommand{\vkep}{{\varv_{\mathrm{kep}}}}
\newcommand{\vsurf}{{\varv_{\mathrm{surf}}}}
\newcommand{\mol}{\unitstyle{mol}}% mole
\newcommand{\barn}{\unitstyle{b}} %barn
\newcommand{\gcms}{\, {\rm g}\, {\rm cm}^{2}\, {\rm s}^{-1}}
\newcommand{\cms}{\,  {\rm cm}^{2}\, {\rm s}^{-1}}
% command to include values
\newcommand{\unit}[2]{\ensuremath{#1\numberspace\mathrm{#2}}}

\newcommand{\bvec}[1]{\ensuremath{\boldsymbol{#1}}} %boldface vector style
\newcommand{\grad}{\bvec{\nabla}} %gradient
\newcommand{\divr}{\nabla \cdot} %divergence
\newcommand{\curl}{\bvec{\nabla \times}} %curl
\newcommand{\lap}{\ensuremath{\nabla^2}} %Laplacian
\newcommand{\btens}[1]{\ensuremath{\boldsymbol{\mathsf{#1}}}}
\newcommand{\vcross}{\bvec{\times}}
\newcommand{\vdot}{\bvec{\cdot}}

% formatting.tex --- Formatting Macros
%References
\newcommand{\lSect}[1]{{\label{sec:#1}}}
\newcommand{\lFig}[1]{{\label{fig:#1}}}
\newcommand{\lEq}[1]{{\label{eq:#1}}}
\newcommand{\lTab}[1]{{\label{tab:#1}}}
\newcommand{\pFig}[1]{{\placefigure{fig:#1}}}
\newcommand{\pTab}[1]{{\placetable{tab:#1}}}
\newcommand{\Tabff}[1]{{\ref{tab:#1}}}
\newcommand{\Tab}[1]{{Table~\Tabff{#1}}}
\newcommand{\Tabs}[1]{{Tables~\Tabff{#1}}}
\newcommand{\TabRange}[2]{{Tables~\Tabff{#1} - \Tabff{#2}}}
\newcommand{\TabTwo}[2]{{Tables~\Tabff{#1} and \Tabff{#2}}}
\newcommand{\pan}[1]{{\textit{#1}}}
\newcommand{\Pan}[1]{{Panel~\pan{#1}}}
\newcommand{\Pans}[1]{{Panels~\pan{#1}}}
\newcommand{\FIGFF}[2]{{\ref{fig:#2}\pan{#1}}}
\newcommand{\Figff}[1]{{\FIGFF{}{#1}}}
\newcommand{\FIG}[2]{{Fig.~\FIGFF{#1}{#2}}}
\newcommand{\Fig}[1]{{\FIG{}{#1}}}
\newcommand{\FigTwo}[2]{{\FIGS{}{#1} and \FIGFF{}{#2}}}
\newcommand{\FIGS}[2]{{Figs.~\FIGFF{#1}{#2}}}
\newcommand{\Figs}[1]{{\FIGS{}{#1}}}
\newcommand{\Figure}[1]{{Figure~\FIGFF{}{#1}}}
\newcommand{\Figures}[1]{{Figures~\FIGFF{}{#1}}}
\newcommand{\Sectff}[1]{{\ref{sec:#1}}}
\newcommand{\Sect}[1]{{\S\Sectff{#1}}}
\newcommand{\Appendix}[1]{{Appendix~\Sectff{#1}}}
\newcommand{\Appendices}[1]{{Appendices~\Sectff{#1}}}
\newcommand{\Sects}[1]{{\S\S\Sectff{#1}}}
\newcommand{\Eqref}[1]{{\ref{eq:#1}}}
\newcommand{\Eqff}[1]{{(\Eqref{#1})}}
\newcommand{\EQ}[1]{{Equation~\Eqff{#1}}}
\newcommand{\Equation}[1]{{Equation~\Eqff{#1}}}
\newcommand{\Eq}[1]{{Eq.~\Eqff{#1}}}
\newcommand{\Eqs}[1]{{Eqs.~\Eqff{#1}}}
\newcommand{\Leg}[1]{{\textit{#1}}}

% formatting macros
\newcommand{\isofont}[1]{{\mathrm{#1}}}
\newcommand{\isomass}[1]{{\ensuremath{\isofont{^{#1}}}}}
\newcommand{\isocharge}[1]{{\ensuremath{\isofont{_{#1}}}}}
\newcommand{\isotope}[3]{{\ensuremath{\isocharge{#1}\isomass{#2}\isofont{#3}}}}
\newcommand{\Solar}[1]{{\ensuremath{\left[\right.}#1\ensuremath{\right.\left]}}}
\newcommand{\I}[2]{{\isotope{}{#1}{#2}}}
\newcommand{\El}[1]{{\I{}{#1}}}
\newcommand{\Rat}[4]{{\I{#1}{#2}/\I{#3}{#4}}}
\newcommand{\SolRatI}[4]{{\Solar{\Rat{#1}{#2}{#3}{#4}}}}
\newcommand{\SolI}[2]{{\Solar{\I{#1}{#2}}}} \newcommand{\SolEl}[1]{{\Solar{\El{#1}}}} 
\newcommand{\Ep}[1]{{\ensuremath{10^{#1}}}}
\newcommand{\E}[1]{{\ensuremath{\powersep\Ep{#1}}}}
\newcommand{\EE}[2]{{\ensuremath{\powersep\Ep{#1#2}}}}
\newcommand{\powersep}{\times}
\newcommand{\Brak}[1]{{\left({#1}\right)}}
\newcommand{\SBrak}[1]{{\left[{#1}\right]}}

% symbols for commonly used expressions in the paper.  The point is to define these for 
% consistent notation.  Before defining or using an expression, check that someone hasn't defined it already.
\newcommand{\epsnuc}{\ensuremath{\epsilon_{\mathrm{nuc}}}}	% nuclear heating rate
\newcommand{\epsgrav}{\ensuremath{\epsilon_{\mathrm{grav}}}} % gravitational heating rate
\newcommand{\Teff}{\ensuremath{T_{\!\mathrm{eff}}}}	% effective temperature
\newcommand{\teff}{\Teff}
\newcommand{\Ledd}{\ensuremath{L_{\mathrm{Edd}}}} % Eddington Luminosity
\newcommand{\logg}{\ensuremath{\log g}}	% log surface gravity
\newcommand{\Tc}{\ensuremath{T_{\mathrm{\!c}}}} % central temperature
\newcommand{\Pc}{\ensuremath{P_{\mathrm{\!c}}}} % central pressure
\newcommand{\rhoc}{\ensuremath{\rho_{\mathrm{c}}}} % central density
\newcommand{\CP}{\ensuremath{C_{\!P}}} % specific heat at constant pressure
\newcommand{\Mdot}{\ensuremath{\dot{M}}} % mass-loss rate
\newcommand{\Mc}{\ensuremath{M_{\rm c}}} % core mass
\newcommand{\Mm}{\ensuremath{M_{\rm m}}} % modeled mass
\newcommand{\Rc}{\ensuremath{R_{\rm c}}} % core radius
\newcommand{\Lc}{\ensuremath{L_{\rm c}}} % core luminosity
\newcommand{\Lacc}{\ensuremath{L_{\rm acc}}} % accretion luminosity

% opacity stuff
\newcommand{\kappath}{\ensuremath{\kappa_{\mathrm{th}}}} % opacity for thermal radiation orig.\ in planet
\newcommand{\kappav}{\ensuremath{\kappa_{\mathrm{v}}}} % opacity for irradiation from star

% for correction between baryon densities and mass-energy densities
\newcommand{\nB}{\ensuremath{n_{\mathrm{B}}}}	% baryon density
%
% symbols from the first instrument paper
\newcommand{\alphaMLT}{\ensuremath{\alpha_{\mathrm{MLT}}}}	% mixing length parameter
\newcommand{\chirho}{\ensuremath{\chi_{\rho}}}	% $(\partial\ln P/\partial\ln\rho)_T$
\newcommand{\chiT}{\ensuremath{\chi_{\raisebox{-2pt}{$\scriptstyle T$}}}}	% $(\partial\ln P/\partial\ln T)_{\rho}$
\newcommand{\Gammaone}{\ensuremath{\Gamma_{\!1}}} % $ (\partial\ln P/\partial \ln\rho)_S$
\newcommand{\Dov}{\ensuremath{D_{\mathrm{ov}}}}	% overshoot diffusion coefficient
\newcommand{\nablaad}{\ensuremath{\nabla_{\!\mathrm{ad}}}}	% adiabatic temperature gradient
\newcommand{\nablarad}{\ensuremath{\nabla_{\!\mathrm{rad}}}}	% radiative temperature gradient
\newcommand{\nablaT}{\ensuremath{\nabla_{\!T}}}	% actual temperature gradient
\newcommand{\nablaL}{\ensuremath{\nabla_{\mathrm{\!L}}}}	% Ledoux criterion
\newcommand{\scaleheight}{\ensuremath{\lambda_P}}	% pressure scale height
\newcommand{\Pgas}{\ensuremath{P_{\!\!\mathrm{gas}}}}	% gas pressure
\newcommand{\timestep}{\ensuremath{\delta t}} % numerical timestep
%
% more symbols for radiation and gas pressures
\newcommand{\Prad}{\ensuremath{P_{\!\!\mathrm{rad}}}}	% radiation pressure
\newcommand{\Lrad}{\ensuremath{L_{\mathrm{rad}}}}	% radiative luminosity
\newcommand{\tderiv}[3]{\ensuremath{\left(\frac{\partial #1}{\partial #2}\right)_{#3}}} %thermodynamic derivative
\newcommand{\Lrho}{\ensuremath{L_{\mathrm{inv}}}}	% luminosity at which a density inversion occurs
\newcommand{\Lonset}{\ensuremath{L_{\mathrm{onset}}}}	% luminosity at which the onset of convection occurs
\newcommand{\Fconv}{\ensuremath{F_{\!\mathrm{conv}}}}		% convective flux
\newcommand{\Frad}{\ensuremath{F_{\!\mathrm{rad}}}}	% radiative flux
\newcommand{\supernab}{\ensuremath{\delta_\nabla}}  % superadiabaticity, $\nablaT-\nablaad$
\newcommand{\superthresh}{\ensuremath{\delta_{\nabla,\mathrm{thresh}}}}  % controls when MLT++ is applied
\newcommand{\fsuper}{\ensuremath{f_\nabla}} % reduction factor for $\supernab$
\newcommand{\asuper}{\ensuremath{\alpha_\nabla}}  % smoothing parameter for MLT++
\newcommand{\asupert}{\ensuremath{\widetilde{\asuper}}} % MLT++ parameter used in construction of \asuper
\newcommand{\lambdamax}{\ensuremath{\lambda_{\max}}} % $ \max(\Lrad/\Ledd)$
\newcommand{\betamin}{\ensuremath{\beta_{\min}}} % $ \min(P/\Pgas)$

% mixing symbols
\newcommand{\alphasc}{\ensuremath{\alpha_{\mathrm{sc}}}} % semiconvection efficiency parameter
\newcommand{\alphath}{\ensuremath{\alpha_{\mathrm{th}}}} % thermohaline efficiency parameter
\newcommand{\Dth}{\ensuremath{D_{\mathrm{th}}}} % thermohaline diffusion coefficient

\newcommand{\EFc}{\ensuremath{E_{\mathrm{F,c}}}}  % Fermi energy at center
%
% physical constants
\newcommand{\kB}{\ensuremath{k_\mathrm{B}}} % Boltzmann constant
\newcommand{\NA}{\ensuremath{N_\mathrm{\!A}}} % Avogadro number
\newcommand{\mb}{\ensuremath{m_\mathrm{u}}} % atomic mass unit
\newcommand{\sigmaSB}{\ensuremath{\sigma_\mathrm{\!SB}}} % Stefan-Boltzmann constant

% rotation
\newcommand{\veq}{\ensuremath{\varv_{\mathrm{eq}}}} % equatorial velocity
\newcommand{\veqi}{\ensuremath{\varv_{\mathrm{eq,ini}}}}
\newcommand{\Om}{\ensuremath{\Omega}}  % surface angular velocity
\newcommand{\Omc}{\ensuremath{\Om_{\mathrm{crit}}}} % surface critical angular velocity
\newcommand{\om}{\ensuremath{\omega}}  %  angular velocity
\newcommand{\tkh}{\ensuremath{\tau_{\mathrm{KH}}}} % thermal (Kelvin-Helmholtz) timescale

% asteroseismology
\newcommand{\numax}{\ensuremath{\nu_{\mathrm{max}}}} % frequency of maximum power
\newcommand{\dnu}{\ensuremath{\Delta\nu}}  % large frequency separation of pulsation modes
\newcommand{\fov}{\ensuremath{f_{\mathrm{ov}}}} % convective overshoot parameter
\newcommand{\cs}{\ensuremath{c_{\rm s}}} % adiabatic sound speed
\newcommand{\Slamb}{\ensuremath{S_{\!\ell}}} % Lamb frequency

% misc. abbreviations
\newcommand{\mesaone}{Paper~I} % the first mesa paper
\newcommand{\bvfreq}{Brunt-V\"ais\"al\"a frequency}
\newcommand{\bvv}{Brunt-V\"ais\"al\"a}
\newcommand{\muhz}{$\mu$Hz}
\newcommand{\mso}{\Msun}
\newcommand{\rso}{\Rsun}
\newcommand{\lso}{\Lsun}