High-order Gravity-mode Period Spacing Patterns of Intermediate-mass ($1.5 \, M_\odot < M < 3 \, M_{\odot}$) Main-sequence Stars I. Perturbative Analysis

Abstract

Theoretical study of high-order gravity-mode period spacing ($\Delta P_g$) pattern is relevant for the better understanding of internal properties of intermediate-mass ($1.5 \, M_\odot < M < 8 \, M_{\odot}$) main-sequence g-mode pulsators. In this paper, we carry out the first-order perturbative analysis to evaluate effects of a sharp, though not discontinuous, transition in the Brunt-V\"{a}is\"{a}l\"{a} (BV) frequency on the $\Delta P_g$ pattern. Such a finite-width transition in the BV frequency, whose scale height can be comparable to the local wavelength of gravity waves, is expected to develop in relatively low-mass ($1.5 \, M_\odot < M < 3 \, M_{\odot}$) main-sequence stars, causing a bump in the second derivative of the BV frequency. Inspired by Unno et al.'s formulation, we treat the bump in the second derivative of the BV frequency as a small perturbation, which allows us to derive an analytical expression of the $\Delta P_g$ pattern. The analytical expression shows that the amplitude of the oscillatory $\Delta P_g$ pattern is determined by a weighted average of the bump in the second derivative of the BV frequency where the weighting function is given by the g-mode eigenfunction. Tests with low-mass ($\sim 2 \, M_\odot$) main-sequence stellar models show that the analytical expression can reproduce the $\Delta P_g$ patterns numerically computed reasonably well. The results of our perturbative analysis will be useful for, e.g., improving semi-analytical expressions of the $\Delta P_g$ pattern, which would enable us to investigate $\Delta P_g$ patterns of SPB stars and $\gamma$ Dor stars for inferring chemical composition profile and rotation rates.