On the proper use of the Schwarzschild and Ledoux criteria in stellar evolution computations

TL;DR

This paper clarifies the correct method for locating convective zone boundaries in stellar models, highlighting common misuses and their impact on stellar evolution predictions, and proposes improved practices for accurate modeling.

Contribution

It establishes the proper way to locate convective boundaries in stellar models and discusses the consequences of common misapplications, improving the accuracy of stellar evolution computations.

Findings

Correct boundary location involves interpolating within the convective zone.

Misuse of boundary conditions can lead to incorrect convective core sizes.

Using a double mesh point improves boundary accuracy.

Abstract

The era of detailed asteroseismic analyses opened by space missions such as CoRoT and $\textit{Kepler}$ has highlighted the need for stellar models devoid of numerical inaccuracies, in order to be able to diagnose which physical aspects are being ignored or poorly treated in standard stellar modeling. We tackle here the important problem of fixing convective zones boundaries in the frame of the local mixing length theory. First we show that the only correct way to locate a convective zone boundary is to find, at each iteration step, through interpolations or extrapolations from points $\textit{within the convective zone}$, the mass where the radiative luminosity is equal to the total one. We then discuss two misuses of the boundary condition and the way they affect stellar modeling and stellar evolution. The first one consists in applying the neutrality condition for convective instability on the $\textit{radiative}$ side of the convective boundary. The second way of misusing the boundary condition comes from the process of fixing the convective boundary through the search for a change of sign of a possibly \textit{discontinuous} function. We show that these misuses can lead to completely wrong estimates of convective core sizes with important consequences for the following evolutionary phases. We point out the advantages of using a double mesh point at each convective zone boundaries. The specific problem of a convective shell is discussed and some remarks concerning overshooting are given.