Tidally Excited Oscillations (TEOs)

• TEOs are coherent stellar pulsations driven by time-varying tidal forces in eccentric binaries, commonly appearing as gravity modes at integer orbital harmonics.
• The methodology involves decomposing tidal potentials into spherical harmonics and Fourier components to match resonant conditions, enabling precise identification using phase diagnostics and amplitude modeling.
• Observations, particularly in systems like KOI-54, demonstrate that resonance locking and nonlinear tidal coupling can significantly influence the pulsation amplitudes, frequencies, and evolutionary dynamics.

Tidally excited oscillations (TEOs) are coherent stellar pulsations driven by the time-varying tidal potential of a companion, most prominently in eccentric binaries whose periastron passages produce strongly non-sinusoidal forcing. In heartbeat stars, the equilibrium tide shapes the broad periastron brightening, while the dynamical tide excites oscillatory responses that appear as narrow Fourier peaks, usually at exact integer multiples of the orbital frequency. In the binary-star literature, TEOs are usually gravity modes, especially low-degree quadrupole modes, and they have become a central observable for tidal asteroseismology because their frequencies, amplitudes, and phases encode orbital geometry, rotation, damping, and internal mode structure (Fuller et al., 2011, Fuller, 2017).

1. Dynamical-tide framework

The standard description of TEOs begins by expanding the tidal potential into spatial spherical-harmonic components and temporal orbital harmonics. In the formulation developed for eccentric binaries, the forcing can be written as

$$\left(\frac{a}{D}\right)^{l+1} e^{-im\Phi} = \sum_{N=-\infty}^{\infty} F_{Nm} e^{-iN\Omega t},$$

so each harmonic $N\Omega$ can drive a stellar mode whose inertial-frame frequency

$\sigma_\alpha = \omega_\alpha + m\Omega_s$

lies near the forcing frequency. The resonant condition is therefore

$\sigma_\alpha \approx N\Omega,$

or, in the rotating-frame notation frequently used for eccentric heartbeat stars,

$\omega_{Nm} = N\Omega - m\Omega_s.$

This Fourier-harmonic structure is the direct reason that TEOs appear at exact orbital harmonics in photometric Fourier spectra (Fuller et al., 2011, Fuller, 2017).

For a single near-resonant mode, the steady-state response has the usual resonant denominator. One form given for the mode energy is

$$E_\alpha \simeq \frac{\Delta E_\alpha}{4\sin^2(\sigma_\alpha P/2)+(\gamma_\alpha P)^2},$$

while a commonly used luminosity-amplitude expression is

$$A_N = \epsilon_\ell X_{Nm} V_{\ell m} |Q_\alpha L_\alpha| \frac{\omega_{Nm}}{\sqrt{(\omega_\alpha - \omega_{Nm})^2 + \gamma_\alpha^2}}.$$

In these expressions, the amplitude depends on tidal forcing strength, mode visibility, overlap integrals, non-adiabatic luminosity response, damping, and above all the detuning between forcing and eigenfrequency. This extreme sensitivity to detuning is a recurring feature of the TEO literature and motivates both statistical amplitude modeling and resonance-locking interpretations (Cheng et al., 2020, Fuller, 2017).

Quadrupole tides dominate. The binary-star surveys and theoretical treatments consistently emphasize $l=2$ modes, with $m=0$ and $m=\pm2$ as the principal observable components. One theoretical survey of eccentric ellipsoidal variables restricted its analysis to $N\Omega$0, $N\Omega$1 because the quadrupole tide dominates, higher $N\Omega$2 is suppressed by $N\Omega$3, and $N\Omega$4 is about 2–3 orders of magnitude smaller than $N\Omega$5 or $N\Omega$6 (Kołaczek-Szymański et al., 2023). Rotation modifies the frequencies through splitting such as

$N\Omega$7

and many modern treatments incorporate rotation with the traditional approximation, non-adiabatic luminosity perturbations, and the coherent sum of many modal contributions at the same harmonic (Fuller et al., 2011, Fuller, 2017).

2. Observational diagnostics: harmonics, amplitudes, and phases

The defining observational signature of a TEO is a coherent peak at or extremely close to an integer orbital harmonic,

$N\Omega$8

This by itself is not always sufficient, because imperfect subtraction of binary light variations can also leave harmonic residuals. Consequently, recent survey work treats TEO identification as a combined frequency-and-phase problem, with additional checks in original and residual light-curve spectra (Li et al., 2023, Li et al., 2 Aug 2025).

Pulsation phase is the main mode-identification diagnostic. Under the assumptions that the spin, orbit, and pulsation axes are aligned, the oscillations are adiabatic standing waves, and the dominant visible modes are $N\Omega$9 with $\sigma_\alpha = \omega_\alpha + m\Omega_s$0, the expected phase relative to periastron is written as

$\sigma_\alpha = \omega_\alpha + m\Omega_s$1

Equivalent forms appear across the Kepler and TESS heartbeat-star literature, including

$\sigma_\alpha = \omega_\alpha + m\Omega_s$2

These relations place $\sigma_\alpha = \omega_\alpha + m\Omega_s$3 and $\sigma_\alpha = \omega_\alpha + m\Omega_s$4 modes on narrow phase strips, so observed Fourier phases can be used to assign likely azimuthal order or to reject dubious harmonic candidates (Guo et al., 2019, Li et al., 2024).

Inclination strongly affects visibility. In the KOI-54 analysis, axisymmetric $\sigma_\alpha = \omega_\alpha + m\Omega_s$5 modes can generate larger flux variations than $\sigma_\alpha = \omega_\alpha + m\Omega_s$6 modes for small spin inclination, with the visibility scaling for $\sigma_\alpha = \omega_\alpha + m\Omega_s$7 more like $\sigma_\alpha = \omega_\alpha + m\Omega_s$8, while for $\sigma_\alpha = \omega_\alpha + m\Omega_s$9 the geometrical factor includes

$\sigma_\alpha \approx N\Omega,$0

A related amplitude-ratio expression derived for heartbeat binaries,

$\sigma_\alpha \approx N\Omega,$1

implies that low-inclination systems favor $\sigma_\alpha \approx N\Omega,$2, whereas medium and high inclinations can show both $\sigma_\alpha \approx N\Omega,$3 and $\sigma_\alpha \approx N\Omega,$4 (Fuller et al., 2011, Guo et al., 2019). Survey papers explicitly use this geometric trend: low inclinations favor $\sigma_\alpha \approx N\Omega,$5, while medium and high inclinations can show both $\sigma_\alpha \approx N\Omega,$6 and $\sigma_\alpha \approx N\Omega,$7 (Li et al., 2024).

A recurrent misconception is that every peak near an orbital harmonic is physically meaningful as a tidal mode. The survey literature rejects this. Phase inconsistency, low amplitude, sensitivity to detrending, or persistence of periastron residuals can all indicate that a nominal harmonic is an artifact or a different form of variability (Li et al., 2023, Li et al., 2 Aug 2025).

3. KOI-54 and the prototype of tidal asteroseismology

KOI-54 established the modern observational paradigm for TEOs. Kepler revealed a sharp periodic brightening of about 0.7% every 41.8 days together with a $\sigma_\alpha \approx N\Omega,$8 beat pattern of pulsations phase-locked to those brightenings. The system is a highly eccentric, nearly face-on binary of nearly identical A stars with $\sigma_\alpha \approx N\Omega,$9 d, $\omega_{Nm} = N\Omega - m\Omega_s.$0, and $\omega_{Nm} = N\Omega - m\Omega_s.$1. After subtraction of the binary brightening component, the power spectrum showed a large set of coherent peaks, with 30 pulsations of signal-to-noise ratio $\omega_{Nm} = N\Omega - m\Omega_s.$2; 23 of the 30 strongest peaks are close to exact orbital harmonics. The two dominant modes are the 90th and 91st harmonics of the orbital frequency, and their frequency separation produces a beat period essentially equal to the orbital period (Welsh et al., 2011).

KOI-54 was immediately interpreted as an unusually clean dynamical-tide laboratory. One analysis argued that the strongest observed oscillations at 90 and 91 times the orbital frequency are likely due to prograde $\omega_{Nm} = N\Omega - m\Omega_s.$3 modes locked in resonance with the orbit, whereas many of the remaining harmonic peaks are likely nearly resonant $\omega_{Nm} = N\Omega - m\Omega_s.$4 g-modes. In that interpretation, the system can evolve into a state in which at least one $\omega_{Nm} = N\Omega - m\Omega_s.$5 mode is resonance locked under the combined effects of dynamical tides on the stellar spin and orbit and intrinsic stellar spindown (Fuller et al., 2011).

The resonance-locking condition is written as

$\omega_{Nm} = N\Omega - m\Omega_s.$6

and the KOI-54 analysis introduced a critical harmonic number

$\omega_{Nm} = N\Omega - m\Omega_s.$7

such that locking is possible only for $\omega_{Nm} = N\Omega - m\Omega_s.$8. This was used to explain why strong oscillations are observed only up to about $\omega_{Nm} = N\Omega - m\Omega_s.$9–$$E_\alpha \simeq \frac{\Delta E_\alpha}{4\sin^2(\sigma_\alpha P/2)+(\gamma_\alpha P)^2},$$0: above that threshold, the system sweeps through resonance too rapidly for the mode frequency to keep pace with orbital evolution (Fuller et al., 2011).

KOI-54 also became the main case for nonlinear tidal phenomena. The 2011 theoretical treatment found evidence from the published Kepler result that three-mode nonlinear coupling occurs in the system and suggested that non-harmonic oscillations may arise through parametric resonance satisfying

$$E_\alpha \simeq \frac{\Delta E_\alpha}{4\sin^2(\sigma_\alpha P/2)+(\gamma_\alpha P)^2},$$1

A later reanalysis argued that the anharmonic, non-orbital-harmonic TEOs are genuine stellar eigenmodes excited nonlinearly, and identified a quadrupole g-mode period-spacing pattern with

$$E_\alpha \simeq \frac{\Delta E_\alpha}{4\sin^2(\sigma_\alpha P/2)+(\gamma_\alpha P)^2},$$2

at a detection significance level of $$E_\alpha \simeq \frac{\Delta E_\alpha}{4\sin^2(\sigma_\alpha P/2)+(\gamma_\alpha P)^2},$$3. That study identified 16 candidate $$E_\alpha \simeq \frac{\Delta E_\alpha}{4\sin^2(\sigma_\alpha P/2)+(\gamma_\alpha P)^2},$$4 eigenfrequencies and concluded that the dominant harmonic peaks at $$E_\alpha \simeq \frac{\Delta E_\alpha}{4\sin^2(\sigma_\alpha P/2)+(\gamma_\alpha P)^2},$$5 and $$E_\alpha \simeq \frac{\Delta E_\alpha}{4\sin^2(\sigma_\alpha P/2)+(\gamma_\alpha P)^2},$$6 are very close to resonance with $$E_\alpha \simeq \frac{\Delta E_\alpha}{4\sin^2(\sigma_\alpha P/2)+(\gamma_\alpha P)^2},$$7 eigenmodes, likely arising from different stars in the binary (Guo et al., 2022). Taken together, these results made KOI-54 the prototype system in which orbital harmonics, resonance locking, nonlinear coupling, and direct seismic inference could all be studied simultaneously.

4. Detection methodology in heartbeat-star surveys

The practical identification of TEOs generally begins by modeling and subtracting the heartbeat signal, because the equilibrium tide itself contributes orbital-harmonic power. Survey work based on Kepler and TESS uses either analytic heartbeat prescriptions derived from Kumar et al. or full binary light-curve synthesis such as PHOEBE and ELLC. A corrected Kumar-type model frequently appears as

$$E_\alpha \simeq \frac{\Delta E_\alpha}{4\sin^2(\sigma_\alpha P/2)+(\gamma_\alpha P)^2},$$8

or, in a closely related notation,

$$E_\alpha \simeq \frac{\Delta E_\alpha}{4\sin^2(\sigma_\alpha P/2)+(\gamma_\alpha P)^2},$$9

These analytic models are used as baseline tidal-geometry descriptions for heartbeat subtraction, while PHOEBE or ELLC is adopted when eclipses and additional proximity effects must be modeled more explicitly (Li et al., 2 Aug 2025, Kołaczek-Szymański et al., 2020, Cheng et al., 2020).

Residual time series are then searched with Fourier tools such as FNPEAKS, Period04, or Lomb–Scargle periodograms, usually after detrending and iterative prewhitening. A standard threshold is $$A_N = \epsilon_\ell X_{Nm} V_{\ell m} |Q_\alpha L_\alpha| \frac{\omega_{Nm}}{\sqrt{(\omega_\alpha - \omega_{Nm})^2 + \gamma_\alpha^2}}.$$0, and harmonic candidacy is commonly defined by

$$A_N = \epsilon_\ell X_{Nm} V_{\ell m} |Q_\alpha L_\alpha| \frac{\omega_{Nm}}{\sqrt{(\omega_\alpha - \omega_{Nm})^2 + \gamma_\alpha^2}}.$$1

or

$$A_N = \epsilon_\ell X_{Nm} V_{\ell m} |Q_\alpha L_\alpha| \frac{\omega_{Nm}}{\sqrt{(\omega_\alpha - \omega_{Nm})^2 + \gamma_\alpha^2}}.$$2

with $$A_N = \epsilon_\ell X_{Nm} V_{\ell m} |Q_\alpha L_\alpha| \frac{\omega_{Nm}}{\sqrt{(\omega_\alpha - \omega_{Nm})^2 + \gamma_\alpha^2}}.$$3 propagated from the uncertainties in $$A_N = \epsilon_\ell X_{Nm} V_{\ell m} |Q_\alpha L_\alpha| \frac{\omega_{Nm}}{\sqrt{(\omega_\alpha - \omega_{Nm})^2 + \gamma_\alpha^2}}.$$4 and $$A_N = \epsilon_\ell X_{Nm} V_{\ell m} |Q_\alpha L_\alpha| \frac{\omega_{Nm}}{\sqrt{(\omega_\alpha - \omega_{Nm})^2 + \gamma_\alpha^2}}.$$5. Kepler survey work further requires that a robust harmonic TEO be present in both the original and residual spectra, and, if necessary, in spectra recomputed after removing data near periastron to suppress subtraction artifacts (Li et al., 2023, Li et al., 2 Aug 2025).

Large-sample surveys have established the population context. A Kepler study of 146 heartbeat stars found 21 systems exhibiting TEOs, of which 12 have prominent TEOs defined by $$A_N = \epsilon_\ell X_{Nm} V_{\ell m} |Q_\alpha L_\alpha| \frac{\omega_{Nm}}{\sqrt{(\omega_\alpha - \omega_{Nm})^2 + \gamma_\alpha^2}}.$$6; that work also reported harmonic numbers as high as $$A_N = \epsilon_\ell X_{Nm} V_{\ell m} |Q_\alpha L_\alpha| \frac{\omega_{Nm}}{\sqrt{(\omega_\alpha - \omega_{Nm})^2 + \gamma_\alpha^2}}.$$7 in KIC 5006817 and $$A_N = \epsilon_\ell X_{Nm} V_{\ell m} |Q_\alpha L_\alpha| \frac{\omega_{Nm}}{\sqrt{(\omega_\alpha - \omega_{Nm})^2 + \gamma_\alpha^2}}.$$8 in KIC 8459354 (Li et al., 2023). A TESS catalog of 42 new heartbeat stars found TEOs in 10 systems, with most pulsation phases explicable by dominant $$A_N = \epsilon_\ell X_{Nm} V_{\ell m} |Q_\alpha L_\alpha| \frac{\omega_{Nm}}{\sqrt{(\omega_\alpha - \omega_{Nm})^2 + \gamma_\alpha^2}}.$$9, $l=2$0, or $l=2$1 spherical harmonics (Li et al., 2 Aug 2025). A search of massive heartbeat stars in TESS sectors 1–16 found 20 massive heartbeat systems, of which seven show TEOs; the harmonic numbers span $l=2$2 to $l=2$3, with a median $l=2$4, lower than the corresponding median of about 28 quoted for known Kepler systems with TEOs (Kołaczek-Szymański et al., 2020).

The same methodology also clarifies what TEOs are not. TIC 92828790, for example, shows no TEOs but does show a significant non-harmonic $l=2$5 Doradus pulsation. In that system the observed frequency is far from the pseudo-synchronous rotation frequency computed from Hut’s formula, and the authors therefore identify it as an intrinsic $l=2$6 Dor pulsation rather than a tidal harmonic (Li et al., 2 Aug 2025). Such cases demonstrate that heartbeat morphology and gravity-mode variability do not automatically imply TEOs.

5. Resonance locking, nonlinear coupling, and coexistence with intrinsic pulsation

Individual systems beyond KOI-54 show that TEOs occupy a continuum between ordinary near-resonant forcing, long-lived resonance locking, and nonlinear mode coupling. In the detailed characterization of KIC 6117415, KIC 11494130, and KIC 5790807, the authors used binary models, MESA, and GYRE to compare observed TEO amplitudes with both statistical chance-resonance expectations and resonance-locking predictions. KIC 11494130 has a prominent $l=2$7 TEO much stronger than the chance-resonance prediction and is therefore interpreted as a likely resonance lock, whereas KIC 6117415 and KIC 5790807 are consistent with ordinary near-resonant forcing (Cheng et al., 2020).

KIC 4142768 provides a different benchmark: an eclipsing eccentric binary containing two evolved A-type stars, hybrid $l=2$8 Dor/$l=2$9 Sct pulsations, and a set of low-frequency orbital harmonics that are exact multiples of

$m=0$0

Its most convincing TEOs have $m=0$1 and lie between about $m=0$2 and $m=0$3. Their amplitudes and phases agree with predictions from linear tidal theory for $m=0$4 prograde g-modes, and the same system also shows self-excited $m=0$5 Dor g modes with nearly regular period spacings around $m=0$6–2 days and $m=0$7 s. This coexistence makes KIC 4142768 a direct example of a low-frequency spectrum containing both orbital-harmonic TEOs and free g modes (Guo et al., 2019).

Evidence for nonlinear tides is explicit in several systems. FX UMa shows 17 orbital harmonics from $m=0$8 to $m=0$9, with 10 harmonics of $m=\pm2$0 treated as high-probability linearly excited TEOs. The same residual spectrum contains four anharmonic frequencies that pair up so that

$m=\pm2$1

which is interpreted as non-linear tidal mode coupling and daughter-mode generation (Wang et al., 2022). KIC 4544587 similarly exhibits eight g-mode frequencies at orbital harmonics and an echelle diagram in the p-mode region modulo the orbital frequency, leading to the conclusion that tides are also influencing the p modes (Hambleton et al., 2013).

Massive systems add temporal evolution to this picture. In MACHO 80.7443.1718, TEOs at $m=\pm2$2 and 230 change amplitude on months-to-years timescales. For the $m=\pm2$3 mode, amplitude and frequency changes are related in a way interpreted as growing detuning from exact resonance, and the system also shows an orbital period decrease of about $m=\pm2$4 (Kołaczek-Szymański et al., 2021). This suggests that in some massive heartbeat stars, TEOs are not merely passive tracers of tidal forcing but part of an actively evolving tidal-orbital system.

6. Deviations from the simplest picture, population trends, and broader domains

The aligned, adiabatic, standing-wave $m=\pm2$5 framework is effective but not universal. A substantial part of the recent literature is devoted to explaining phase deviations from the standard strips. In a study of 14 Kepler heartbeat stars, most pulsation phases could be explained by dominant $m=\pm2$6, $m=\pm2$7, or $m=\pm2$8 spherical harmonics, but the largest deviations exceeded $m=\pm2$9 in KIC 8459354, where spin-orbit misalignment was proposed, and KIC 5877364 showed a similar scenario. In KIC 4377638, KIC 5090937, and KIC 11403032, harmonics with large deviations $N\Omega$00 were suggested to be traveling waves rather than standing waves; the same study also proposed apsidal motion as a cause of systematic phase offsets (Li et al., 2024).

The phase-based approach was anticipated by earlier work on eight heartbeat systems, which found that simple adiabatic $N\Omega$01, $N\Omega$02, or $N\Omega$03 geometry can account for more than half the sample if spin and orbit are aligned, but that systems such as the misaligned KIC 8164262 deviate strongly because the aligned-axis phase formula breaks down and resonance locking can make the phase effectively arbitrary (Guo et al., 2019). TESS discoveries reinforce the same point. In five newly identified TESS heartbeat stars with TEOs, four have phases consistent with the dominant $N\Omega$04, $N\Omega$05, or $N\Omega$06 spherical harmonic, while TIC 447927324 shows a $N\Omega$07 phase deviation and is interpreted as a likely traveling wave rather than a standing wave (Li et al., 2024). The 42-system TESS catalog also highlights TIC 156846634, whose $N\Omega$08 harmonic deviates by more than $N\Omega$09 and may be a traveling wave or nonadiabatic response, and TIC 184413651, where the $N\Omega$10 harmonic is argued not to be a reliable TEO candidate (Li et al., 2 Aug 2025).

Population studies show clear trends. In the Kepler heartbeat-star sample, the harmonic number of TEOs has a positive correlation with orbital eccentricity and also with orbital period; prominent TEO systems cluster at surface effective temperatures

$N\Omega$11

supporting the idea that hot stars are more favorable for detection (Li et al., 2023). This agrees with the theoretical statement that TEOs are more visible in hot stars with $N\Omega$12, because in cool stars a thick surface convection zone traps g modes below the surface, whereas in hot stars the modes reach the photosphere more easily and produce larger luminosity perturbations (Fuller, 2017). TESS heartbeat stars occupy higher-temperature and higher-luminosity regions than the Kepler sample, which significantly enhances the detectability of massive heartbeat stars and those containing TEOs (Li et al., 2 Aug 2025).

Theoretical population synthesis of 20,000 eccentric ellipsoidal variables extends these empirical trends. That study found resonance histories ranging from resonance-free intervals to exceptionally long resonances of $N\Omega$13–$N\Omega$14 years, average resonance rates of about $N\Omega$15 for the most massive stars versus $N\Omega$16 for intermediate-mass stars, and a strong tendency for both components to experience increased resonance rates near TAMS. The conclusion was that TEOs should be more frequent and more pronounced in massive systems, especially when one component is close to TAMS, and that they may play an important role in angular-momentum transport within massive and intermediate-mass stars (Kołaczek-Szymański et al., 2023).

The concept of TEOs also extends beyond ordinary eccentric binaries. In compact hierarchical triples, the inner binary produces a genuinely three-body tidal forcing spectrum in the outer star, with inertial-frame forcing frequencies such as

$N\Omega$17

allowing excitation of high-frequency p-modes even in circular, aligned, synchronized configurations. HD 181068 is interpreted in exactly this way (Fuller et al., 2012). Hot helium white dwarfs provide another extension: for $N\Omega$18 kK and $N\Omega$19–$N\Omega$20, the dynamical tide may induce fractional flux changes $N\Omega$21, whereas cold white dwarfs with $N\Omega$22 kK are unlikely to show observable dynamical-tide pulsations. In the traveling-wave limit, the flux variation is typically reduced to 0.1%–1% and the excess phase is likely to be 90 degrees (Yu et al., 2020).

Across these domains, TEOs remain defined by the same core physics: periodic tidal forcing excites real stellar oscillation modes rather than a purely static bulge. What differs from system to system is the forcing spectrum, the mode cavity, the degree of resonance, and the extent to which nonlinear coupling, misalignment, nonadiabaticity, and secular evolution must be included in the description.