Modelling $δ$ Scuti pulsations: A new grid of p, g, and f modes across pre-main-sequence to post-main-sequence evolution

TL;DR

This work delivers a public, comprehensive grid of 25 million δ Sct pulsation models spanning pre-MS to post-MS evolution, computed with MESA and GYRE for $M=1.4$–$2.5\,M_\odot$, $Z=0.001$–0.026, and rotation up to $\Omega/\Omega_{\rm crit}=0.3$. It includes adiabatic frequencies for $\ell=0$–$3$ (p, g, f modes) and accounts for rotation and mode coupling via avoided crossings, providing detailed asteroseismic diagnostics ($\Delta\nu$, $\varepsilon$, and the $p_{n_1\ell_0}$–$\Delta\nu$ relation). Key findings show robust $\Delta\nu$–$\bar{\rho}$ scaling with $\Delta\nu/\Delta\nu_\odot = 0.857\,(\bar{\rho}/\bar{\rho}_\odot)^{0.507}$ and a tight, rotation-aware linear relation between the fundamental radial mode and $\Delta\nu$, enabling efficient mode identification across evolutionary stages. The grid reveals significant f- and low-order g-mode visibility during late pre-MS to early MS, and demonstrates that including p, g, and f modes alongside their rotational splittings substantially improves age and structural inferences for young A- and F-type stars.

Abstract

Space-based photometry reveals regular high-frequency patterns in many young $δ$ Scuti stars. These pulsations provide a powerful means of inferring stellar properties, particularly ages, for young $δ$ Scuti stars for which traditional age-dating methods are poorly constrained. Realising this potential requires theoretical models that capture the complexities of stellar structure and evolution. We present a comprehensive grid of 25 million stellar pulsation models, computed using the mesa stellar evolution code and the gyre stellar oscillation code, tailored to $δ$ Scuti stars. The grid spans a wide range of masses, metallicities and rotation velocities, and covers evolutionary phases from the early pre-main-sequence through the main sequence and into the post-main sequence contraction phase. For each model, we computed adiabatic pulsation frequencies for degrees $\ell$ = 0 to 3, capturing p modes, g modes, f modes and their interactions through avoided crossings. We find that f and low-order g modes have mode inertias comparable to or lower than the fundamental radial mode during the late pre-MS and early MS, implying that these modes should be observable. We revisit $δ$ Scuti scaling relations and map asteroseismic observables, including the large frequency separation ($Δν$) and phase offset parameter ($\varepsilon$), across age, mass, metallicity, and rotation. This new model grid, which is publicly available, improves upon previous such model grids by facilitating interpretation of $δ$ Scuti pulsations, allowing for more reliable age estimates and tighter constraints on stellar evolutionary pathways, and planet formation in A- and F-type stars.

Figures (18)

• Figure 1: Phases of evolution of a typical 1.7-M$_\odot$, solar-metallicity $\delta$ Sct star on the Hertzsprung-Russell diagram.
• Figure 2: Evolution of the large frequency separation ($\Delta\nu$) for a typical 1.7 solar mass star with solar metallicity. The dashed line shows the ZAMS stabilization point, where both the stellar density (hence $\Delta\nu$) and nuclear burning have reached equilibrium.
• Figure 3: Kippenhahn diagrams for 12 stars from our model grid, illustrating how variations in stellar mass and metallicity influence core structure and evolution. These factors affect the size of the convective core, the extent of near-core overshooting, and the timing of the transition to stable hydrogen burning at the ZAMS (indicated by black dashed lines). Shading represents $\log_{10}$ of the net energy generation rate, defined as nuclear energy production minus neutrino losses, in units of erg g$^{-1}$s$^{-1}$. Mixing occurs within convective zones and the adjacent overshoot regions, as indicated by the shaded areas and legend. Stellar mass increases from top to bottom, and metallicity increases from left to right.
• Figure 4: Evolution of a 1.7-M$_\odot$ star with solar metallicity. Figure \ref{['fig:phases1']} shows the pre-MS evolution up to the ZAMS with three panels: the top panel displays the evolutionary track on the HR diagram coloured by $\log_{10}$ of the surface gravity (cm s$^{-2}$), the bottom left panel zooms in to show the ZAMS, while the bottom right panel presents Kippenhahn diagram corresponding to the top panel. Figure \ref{['fig:phases2']} shows the MS and post-MS evolution and contains two panels. The top panel shows the HR diagram evolution from the ZAMS through TAMS up to the end of the subgiant phase, while the bottom panel presents the same evolution as a Kippenhahn diagram, in which shading represents $\log_{10}$ of the net energy generation rate, defined as nuclear energy production minus neutrino losses, in units of erg g$^{-1}$s$^{-1}$. Shaded areas show mixing regions as indicated in the legend. The stellar track shown here was computed at 5$\times$ higher temporal resolution during the MS and post-MS than the grid models to more clearly illustrate phase transitions.
• Figure 5: Percentage of the main-sequence lifetime spent in each evolutionary phase for stellar models across a range of four masses and three metallicities. Each bar represents a star with a given mass (x-axis) and metallicity (panel title), subdivided into pre-main-sequence (Pre-MS), main-sequence (MS), post-MS contraction, and subgiant branch phases. The hatched portion of each bar indicates the time spent within the classical instability strip dupret_theoretical_2004 during that phase.