Asteroseismology and Buoyancy Glitch Inversion with Fourier Spectra of Gravity Mode Period Spacings

TL;DR

The paper develops a robust framework to diagnose buoyancy glitches in stellar interiors by Fourier analyzing gravity-mode period spacings. By linking $FT(\Delta P_k)$ to the derivative of the glitch profile $\delta N/N$ and $FT(\delta P/P)$ to the glitch amplitude, it enables direct inversion of the Brunt–Väisälä profile and precise age/mixing constraints from g-mode data. The method is validated with MESA/GYRE models and applied to real SPB and $\gamma$ Dor stars, including a rotating SPB, demonstrating that $u_\mu$ (the dominant Fourier frequency) tracks central hydrogen abundance $X_c$ and thus stellar age with weak mass dependence. This approach supports fast ensemble asteroseismology of g-modes and promises extensions to other pulsator classes and tidal studies, offering a practical pathway to connect internal structure with evolution and rotation.

Abstract

We investigate the small, quasi-periodic modulations seen in the gravity-mode period spacings of pulsating stars. These ``wiggles'' are produced by buoyancy glitches -- sharp features in the buoyancy frequency ($N$) caused by composition transitions and the convective-radiative interface. Our method takes the Fourier transform of the period-spacing series, $FT(ΔP_k)$ as a function of radial order $k$. We show that $FT(ΔP_k)$ traces the radial derivative of the normalized glitch profile $δN/N$ with respect to the normalized buoyancy radius; peaks in $FT(ΔP_k)$ therefore pinpoint jump/drop locations in $N$ and measure their sharpness. We also note that the Fourier transform of relative period perturbations (deviations from asymptotic values), $FT(δP/P)$, directly recovers the absolute value of the glitch profile $|δN/N|$, enabling a straightforward inversion for the internal structure. The dominant $FT(ΔP_k)$ frequency correlates tightly with central hydrogen abundance ($X_c$) and thus with stellar age for slowly pulsating B-stars, with only weak mass dependence. Applying the technique to MESA stellar models and to observed slowly pulsating B-stars and $γ$ Dor pulsators, we find typical glitch amplitudes $δN/N \lesssim 0.01$ and derivative magnitudes $\lesssim 0.1$, concentrated at chemical gradients and the convective boundary. This approach enables fast, ensemble asteroseismology of g-mode pulsators, constrains internal mixing and ages, and can be extended to other classes of pulsators, with potential links to tidal interactions in binaries.

Figures (4)

• Figure 1: Brunt--Väisälä (buoyancy) profiles in the left and middle columns are plotted as a function of radial coordinate $x=r/R$ (left) and the buoyancy coordinate $u$ (middle), for an $M=3.2M_{\odot}$ MESA stellar model with solar metallicity and convective overshooting $f_{ov}=0.015$. The hydrogen mass fraction $X$ is shown by green dashed lines (multiplied by 10 for clarity). The gray-shaded regions denote the receding convective core. The dipole gravity-mode period spacings ($\Delta P_k$) are shown in the right panel, and their Fourier amplitude spectra $FT(\Delta P)$ are overplotted in the middle panels (red lines + super-Nyquist gray parts). The dominant Fourier spectra peak aligns with the sharp drop points of the Brunt profile and is labeled as $u_{\mu}$ in red.
• Figure 2: (a): Brunt profile decomposed to a smooth part $N(u)$ and a glitch part $\delta N(u)$. (b): Pulsation periods vs radial order $k$ ($l=1$ g modes). (c): Comparison between $FT(\Delta P)$ and the derivative of $FT(\delta P/P)$. (d): Period spacings $\Delta P_k$. (e): relative period spacings $\Delta P_k$ with respect to the asymptotic value $\Pi_l$. (g): ($\delta P_k$) Deviations (perturbations) of pulsation period from asymptotic values. (h): relative perturbations of pulsation periods ($\delta P_k/P_k$). (f): Comparison of the Fourier spectrum of $\Delta P_k$ and the buoyancy glitch derivatives. (i): Comparison of the Fourier spectrum of $\delta P_k/P_k$ and the buoyancy glitch profile $\delta N/N$. All the plots are based on a middle-main-sequence ($X_c=0.4$), $M=4.5M_{\odot}$ stellar model and its dipole g modes.
• Figure 3: Panel (b): Observed g-mode period spacings $\Delta P$ and periods for KIC 10526294, KIC 11145123, KIC 7760680. The Fourier spectra of the former two stars are shown in panel (a) and (d), respectively. The dominant peaks are labeled by the filled circles with corresponding frequency $u_{\mu}$. Panel (c) and (f) presents the g-mode $\Delta P$ variational frequency $u_{\mu}$ as a function of central hydrogen mass fraction $X_c$ for MESA models of masses $M=6.0$ (blue), $3.2$ (red) and $1.6M_{\odot}$ (purple). Panel (c) shows models with different convective-core overshooting $f_{ov}$ and (f) displays the effect of the envelope mixing $D_{min}$. Panel (e) demonstrates the amplitudes of FT($\Delta P$) for MESA models with different masses and $D_{min}$.
• Figure 4: Left column: Gravity-mode period spacing of KIC7760680 in the inertial frame ($\Delta P_{iner}$ vs $P_{iner}$, upper) and co-rotating frame ($\Delta P_{co}$ vs $P_{co}$, lower panel). Right column: Fourier spectra of normalized $\Delta P_{co}$ (upper) and relative period perturbation $\delta P_{co}/P_{co}$ (lower). The two spectra are linked by a relation $FT[(\Delta P -\tilde{\Pi}_l)/\tilde{\Pi}_l]=d FT(\delta P/P)/d\ln u$, and compared in the upper panel.