High-order Gravity-mode Period Spacing Patterns of Intermediate-mass (1.5 M ⊙ M M ⊙ ) Main-sequence Stars. I. Perturbative Analysis

Abstract Theoretical study of the high-order gravity-mode period spacing (Δ P g ) pattern is relevant to a better understanding of the internal properties of intermediate-mass (1.5 M ⊙ < M < 8 M ⊙ ) main-sequence g -mode pulsators. In this paper, we carry out first-order perturbative analysis to evaluate effects of a sharp, though not discontinuous, transition in the Brunt–Väisälä (BV) frequency on the Δ 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 ⊙ < M < 3 M ⊙ ) 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 for the Δ P g pattern. The analytical expression shows that the amplitude of the oscillatory Δ 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 (∼2 M ⊙ ) main-sequence stellar models show that the analytical expression can reproduce the numerically computed Δ P g patterns reasonably well. The results of our perturbative analysis will be useful, for example, in improving semi-analytical expressions for the Δ P g pattern, thereby enabling investigations of Δ P g patterns in slowly pulsating B-type stars and γ Dor stars to infer their chemical composition profiles and rotation rates.