Tidal Asteroseismology in Binary Stars

• Tidal asteroseismology is the study of stellar oscillations in stars perturbed by gravitational tides, primarily in close binaries.
• It integrates classical asteroseismic techniques with tidal theory to analyze resonant oscillation modes, non-linear coupling, and energy transfer.
• Observations from missions like Kepler and TESS detect tidally excited modes in heartbeat and Algol-type binaries, advancing our understanding of binary evolution.

Tidal asteroseismology is the paper of stellar oscillations in stars perturbed by tidal forces, typically within close binaries, with an emphasis on extracting constraints on stellar interiors and binary properties by analyzing oscillation modes that are excited, modified, or revealed by tidal interactions. This field synthesizes classical asteroseismic diagnostics with the theory of dynamical tides, non-linear mode coupling, and angular momentum exchange, enabling the investigation of stellar interiors and their response to external potential perturbations across a diverse set of astrophysical environments.

1. Theoretical Framework of Tidal Asteroseismology

Tidal asteroseismology extends classical asteroseismology by focusing on stars in close binaries where the gravitational potential of a companion induces static (equilibrium) and dynamical tides within the stellar envelope and interior. The interaction excites oscillation modes—most importantly gravity (g) and pressure (p) modes—with tidal forcing frequencies given by integer harmonics of the orbital frequency: $$\nu_{\rm tid} = k\,\nu_{\rm orb}, \quad k\in\mathbb{Z}$$ (Kurtz, 2022). For eccentric binaries, the decomposition of the time-dependent tidal potential via Hansen coefficients results in a broad spectrum of orbital harmonics (Burkart et al., 2011).

The dynamical aspect arises when the forcing frequency nears the eigenfrequency of a stellar oscillation mode, leading to resonant excitation with an amplitude scaling as a Lorentzian function of the detuning parameter: $$A_{nlmk} \propto \frac{Q_{nl} X_{lm}^k}{(\omega_{nl}^2 - (k-m)^2) - 2i\gamma_{nl}(k-m)}$$ where $Q_{nl}$ is the spatial overlap integral, $X_{lm}^k$ is the Hansen coefficient, $\omega_{nl}$ the eigenfrequency, $k$ the orbital harmonic, $m$ the azimuthal number, and $\gamma_{nl}$ the damping rate (Burkart et al., 2011).

Non-linear phenomena, such as three-mode coupling, arise when strongly forced modes exceed an instability threshold, redistributing energy into daughter modes whose frequencies sum to a parent mode's frequency, $\omega_a \approx \omega_b + \omega_c$ [(Burkart et al., 2011); (Guo, 2020)].

2. Observational Signatures and Diagnostics

The observational landscape is dominated by the discovery of heartbeat stars and Algol-type binaries exhibiting tidally excited oscillations (TEOs) and tidally perturbed modes. In these systems:

• Pulsations appear at integer multiples of the orbital frequency due to linear tidal excitation, while additional anharmonic (nonharmonic) modes—frequencies offset from exact harmonics—signal non-linear mode coupling or perturbations to the pulsation cavity [(Burkart et al., 2011); (Bowman et al., 2019)].
• High-cadence, long-baseline photometry from Kepler and TESS enables extraction of rich asteroseismic frequency spectra, including both p-modes and g-modes, with sufficient resolution to disentangle mode amplitudes, phases, lifetimes, and possible tidal modulation (Kurtz, 2022).
• Free modes in close binaries, when not strictly at harmonic frequencies, may reveal subtle shifts due to tidal deformation or asynchronous rotation. Cases such as U Gru demonstrate longitudinal series of modes offset from orbital harmonics, serving as a direct observational signature of tidal perturbation (Bowman et al., 2019).

Key asteroseismic parameters such as the large frequency separation $\Delta\nu$ and the period spacing $\Delta\Pi$ retain their interpretive power (e.g., for mean density and core properties), but tidally induced, shifted, or phase-modulated modes provide additional constraints on tidal dissipation, angular momentum exchange, and interior structure [(Christensen-Dalsgaard et al., 2011); (Guo et al., 2022)].

3. Physical Mechanisms: Excitation, Damping, and Non-linear Interactions

Tidal excitation occurs via coupling between the time-dependent external potential and the star's normal modes, with the efficiency shaped by:

• The spatial overlap between the tidal potential and mode eigenfunction.
• The proximity of the forcing frequency to natural mode frequency (resonance), enhanced when the Doppler-shifted frequency is matched via stellar rotation.
• Mode damping, set by radiative or convective dissipation, which limits amplitude growth and controls the observable pulsation lifetime [(Burkart et al., 2011); (Nance et al., 2018)].

Non-linear three-mode coupling has emerged as a critical process. In this paradigm:

• A parent mode, driven above a threshold, transfers energy to two daughter modes with $m_a = m_b + m_c$ and adherence to angular momentum selection rules, often stabilizing into an equilibrium state with constant amplitude and phase over long timescales (Guo, 2020).
• The equilibrium state is quantitatively expressed as: $$(A_b)^2 / (A_c)^2 = (\omega_b / \gamma_b)/(\omega_c / \gamma_c)$$ and the phase detuning satisfies: $\tan(2\delta) = -2\mu / (1 - \mu^2)$ where $\mu$ parametrizes the frequency detuning relative to combined damping rates (Guo, 2020).

The role of stellar rotation is captured via the traditional approximation, which replaces the $l(l+1)$ angular term with a rotation-dependent eigenvalue $\lambda(q)$, thereby modifying the spatial pattern and resonance conditions (Burkart et al., 2011).

4. Methodological Developments and Modeling Techniques

Tidal asteroseismology requires forward-modeling frameworks that integrate:

• Linear nonadiabatic oscillation equations, often including the Coriolis force, radiative damping, and complex boundary conditions at the photosphere (Burkart et al., 2011).
• Expansion of the tidal potential in spherical harmonics and Hansen coefficients, enabling time-dependent forcing in eccentric binaries.
• Non-linear mode coupling amplitude equations for simulating the transfer of energy between forced and daughter modes (Guo, 2020).

Analytic models for ellipsoidal variation and irradiation have been developed to reproduce static brightness modulation, employing expansions in low-$l$ spherical harmonics with terms for limb darkening and geometric projection (Burkart et al., 2011).

Advanced asteroseismic modeling utilizes seismic observations (e.g., period spacing sequences, mixed mode patterns) to fit stellar evolution and pulsation models, as implemented via MESA or GYRE. For instance, period spacings of nonlinearly-excited g modes in KOI-54 directly constrain mass, radius, and metallicity (Guo et al., 2022).

In gravitational-wave asteroseismology, the imprint of tidal deformability and resonant f-mode excitation is modeled via parameters such as the Love number $k_2$, compactness $C$, and tidal deformability $\Lambda$, with direct consequences for the GW phase evolution (Pratten et al., 2019, Ng et al., 2020).

5. Astrophysical Applications and Notable Case Studies

Heartbeat systems such as KOI-54 serve as archetypes, displaying tens of orbital-harmonic pulsations, periastron brightening from ellipsoidal distortion, and non-harmonic signals attributable to nonlinear coupling [(Burkart et al., 2011); (Guo et al., 2022)]. Period spacing patterns of non-linearly excited g-modes provide direct seismological constraints and, uniquely, allow for mode identification not possible in linear models.

Eccentric red-giant binaries like KOI-3890 (Kuszlewicz et al., 2019) employ global oscillation parameters and rotational splitting to derive mass, radius, age, inclination, and obliquity, yielding system alignment measurements and constraining M-dwarf companions' radii against theoretical predictions.

TESS discoveries in Algol-type binaries (e.g., U Gru (Bowman et al., 2019)) showcase asteroseismic mapping of mass-accreting, tidally perturbed pulsators, essential for constraining stellar evolution and the role of tides in angular momentum transport and mixing.

Compact object binaries studied via gravitational waves provide a new frontier. For binary neutron stars, tidal asteroseismology allows measurement of the f-mode frequency during inspiral and merger, yielding direct constraints on the nuclear equation of state and tidal deformability (Pratten et al., 2019, Ng et al., 2020, Flores et al., 19 Feb 2024).

6. Observational and Computational Challenges

Key challenges in tidal asteroseismology include:

• The need for extremely long, precise photometric baselines to resolve closely spaced modes, damped modes with high inertia, and subtle variability signatures (Christensen-Dalsgaard et al., 2011).
• The complexity of seismic inversions in the presence of tides, rotation, and possibly magnetic fields, which all break spherical symmetry and complicate the mode geometry (Dupret, 2019).
• The high sensitivity of TEO amplitudes to frequency detuning, which necessitates dense model grids and robust non-linear modeling to interpret observed frequency spectra (Guo et al., 2022).
• The overlap of tidally induced variability with convective or spot-driven signals, especially in evolved stars or mass-transfer binaries, which can hinder mode extraction and identification (Hey et al., 12 Mar 2025).

7. Future Prospects and Open Questions

The field is positioned for rapid progress driven by:

• Expanding observational datasets from current and upcoming missions (e.g., TESS, PLATO) increasing the sample of tidally interacting binaries and enabling population studies of TEOs, resonance locking, and tidal mixings (Huber et al., 2019, Kurtz, 2022).
• Improved integration of asteroseismic data (e.g., phase modulation and frequency modulation techniques) with binary orbital modeling to extract masses, radii, and rotation rates without reliance on spectroscopy (Kurtz, 2022).
• Enhanced modeling of non-linear mode coupling, secular amplitude evolution (including observed multi-year TEO amplitude drifts), and tidal dissipation processes (Guo, 2020, Guo, 2020).
• Application of gravitational-wave asteroseismology, where constraints on the f-mode spectrum, damping times, and tidal deformability may probe both the nuclear equation of state and dark sector physics in compact stars (Pratten et al., 2019, Ng et al., 2020, Flores et al., 19 Feb 2024).

A major ongoing challenge remains the full integration of tidal, rotational, and magnetic effects in theoretical models and seismic inversions. The synergy between classical photometric asteroseismology and gravitational-wave observations promises transformative progress in understanding the interplay between stellar interiors and dynamic tidal interactions.

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Key Equations in Tidal Asteroseismology

Equation	Description
$\nu_{\rm tid} = k\,\nu_{\rm orb}$	Tidal forcing frequency as harmonics of orbital frequency (Kurtz, 2022)
$$A_{nlmk} \propto \frac{Q_{nl} X_{lm}^k}{(\omega_{nl}^2 - (k-m)^2) - 2i\gamma_{nl}(k-m)}$$	Lorentzian amplitude of forced oscillation (Burkart et al., 2011)
$$\Delta\nu = \left(2\int_0^R \frac{dr}{c}\right)^{-1}$$	Large frequency separation for p-modes (Christensen-Dalsgaard et al., 2011)
$$\Delta\Pi = \frac{2\pi^2}{\sqrt{l(l+1)}\left(\int_{r_1}^{r_2} \frac{N}{r}dr\right)^{-1}}$$	Period spacing for g-modes (Christensen-Dalsgaard et al., 2011)
$\Lambda = \frac{2k_2}{3C^5}$	Dimensionless tidal deformability (Pratten et al., 2019)
$$(A_b)^2 / (A_c)^2 = (\omega_b / \gamma_b)/(\omega_c / \gamma_c)$$	Three-mode coupling equilibrium (Guo, 2020)

The ongoing revolution in high-precision photometry, spectroscopy, and gravitational-wave astronomy ensures that tidal asteroseismology will remain a forefront discipline for probing stellar structure and binary evolution across the mass spectrum.