Convective shells in the interior of Cepheid variable stars: Overshooting models based on hydrodynamic simulations

¹ Lawrence Livermore National Laboratory, 7000 East Ave, Livermore, CA 94550, USA
² Department of Physics and Astronomy, Georgia State University, Atlanta, GA 30303, USA
³ Department of Physics and Astronomy, University of Exeter, EX4 4QL Exeter, United Kingdom
⁴ École Normale Supérieure de Lyon, CRAL (UMR CNRS 5574), Université de Lyon 1, 69007 Lyon, France
⁵ Los Alamos National Laboratory, Los Alamos, NM 87545, USA
⁶ Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
⁷ Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, Gif-sur-Yvette F-91191, France

^⋆ Corresponding author.

Received: 15 April 2025
Accepted: 6 May 2025

Abstract

Context. Because Cepheid variable stars have long been used as a cosmic benchmark for scaling distances in our Galaxy and beyond, the accuracy of stellar evolution models for Cepheids have wide-reaching effects. However, our understanding of the dynamics in the interiors of these physically complex stars is limited.

Aims. Our goal is to provide a detailed multi-dimensional picture of hydrodynamic convection and convective boundary mixing in the interior of Cepheids.

Methods. Using the Modules for Experiments in Stellar Astrophysics (MESA), we studied the structure of intermediate-mass stars that cross the instability strip. Then, we performed two-dimensional hydrodynamic simulations of six stars with the fully compressible Multidimensional Stellar Implicit Code (MUSIC). Our simulations did not model the radial pulsations but focused on the interior structure of this family of stars. We developed and applied a new statistical analysis to examine convection and convective boundary mixing in the interior of these stellar simulations.

Results. Based on a grid of MESA models, we demonstrated that a common structure for intermediate mass Cepheids includes an interior convective shell as well as a thin outer convective envelope. Using the extreme value theory approach to analyze our MUSIC simulation data, we found that overshooting above the convective shell fills the space between these convectively unstable layers. We developed a new statistical analysis that provides a clearer picture of how overshooting fills this layer; it also allowed us to formulate a detailed comparison between overshooting above and below the convective shell. Our analysis effectively decomposes the overshooting layer into two layers: a weak overshooting layer and a strong overshooting layer. Statistically, this is accomplished by decomposing the strongly non-Gaussian probability density function into a mixture of gamma distributions. Using our mixture model, we showed that the ratio of overshooting lengths above and below the convective shell depends directly on the radial extent of the convective shell as well as its depth in the star. We proposed a new form for the diffusion coefficient that addresses the need for overlapping overshooting layers between convective shells. We introduced the idea of a “super-mixing layer” where overshooting from both the convective shell and the convective envelope results in efficient mixing and could be viewed as merging the two adjacent convective zones.