Heartbeat Stars: Tidal Dynamics in Binaries

• Heartbeat Stars (HBSs) are eccentric binary systems that display distinctive, short-lived photometric pulses at periastron due to tidal distortions and a mix of equilibrium and dynamical effects.
• Modeling techniques range from analytic equilibrium-tide fits with MCMC to comprehensive binary-star light-curve and radial-velocity analyses, enabling precise orbital and stellar parameter determination.
• Surveys from Kepler, OGLE, TESS, and Gaia reveal a diverse population across various periods, eccentricities, and stellar types, expanding our understanding of tidal interactions and binary evolution.

Heartbeat stars (HBSs) are eccentric binary systems in which strong tidal forcing at periastron produces a characteristic, short-lived photometric “heartbeat” and, in many systems, tidally excited oscillations (TEOs) at exact integer multiples of the orbital frequency. In the Kepler-era usage, they were often identified as short-period eccentric binaries with prominent periastron distortions, but subsequent OGLE, TESS, and Gaia-linked surveys established a much broader phenomenology spanning detached, non-eclipsing systems, eclipsing systems, massive binaries, and long-period red-giant binaries. Their importance lies in the fact that the observable light-curve morphology, pulsation spectrum, rotation state, and radial-velocity orbit all respond directly to tidal forcing, making HBSs a laboratory for equilibrium tides, dynamical tides, synchronization, circularization, apsidal motion, and resonance locking (Zimmerman et al., 2017, Wrona et al., 2021, Shporer et al., 2016).

1. Defining phenomenology and observable signature

The defining observational feature of an HBS is a sharp flux variation near periastron that resembles an electrocardiogram “heartbeat.” In the standard analytic description used for OGLE and many TESS analyses, the fractional flux perturbation is written as

$$\frac{\Delta F(t)}{F} = S\,\frac{1 - 3\sin^2 i\,\sin^2\!\bigl[\varphi(t)+\omega\bigr]}{\bigl[R(t)/a\bigr]^3}+C,$$

where $i$ is the orbital inclination, $\omega$ the argument of periastron, $a$ the semimajor axis, and $R(t)=a(1-e^2)/(1+e\cos f(t))$ the instantaneous separation. The $1/R^3$ scaling makes the signal sharply concentrated around periastron, while the angular factor sets whether the waveform appears as a central brightening flanked by dips or the reverse (Wrona et al., 2021, Wrona et al., 2021).

The photometric heartbeat is not produced by a single mechanism. The data consistently attribute it to some combination of equilibrium tidal distortion, heating or reflection, and Doppler boosting; detailed binary modeling with ELLC likewise includes gravity darkening, limb darkening, Doppler boosting, and reflection in addition to the tidal distortion itself (Shporer et al., 2016, Cheng et al., 2020). In many systems, especially those with high-quality Kepler photometry, the periastron pulse is accompanied by TEOs that persist through the orbit and appear at exact orbital harmonics (Fuller, 2017).

In Fourier space, the heartbeat pulse and the oscillatory components are separable in a diagnostically useful way. The non-sinusoidal orbital heartbeat appears as a comb of equally spaced, very narrow peaks at exact harmonics of the orbital frequency, whereas rotational modulation and some stellar variability appear as broader, more sinusoidal groups of peaks. This distinction was exploited both for rotation-period measurements in Kepler systems and for automated recognition methods based on the first 100 orbital harmonics (Zimmerman et al., 2017, Li et al., 7 May 2025).

A common misconception is that HBSs are merely non-eclipsing ellipsoidal variables. The larger catalogs show that the class is broader: some systems are eclipsing, some contain red giants, some show strong intrinsic pulsations unrelated to orbital harmonics, and some require full binary-star modeling rather than the simplest analytic equilibrium-tide approximation (Wrona et al., 2021, Kołaczek-Szymański et al., 2020).

2. Population structure and parameter space

Early Kepler work emphasized short-period, highly eccentric systems. A radial-velocity campaign for Kepler heartbeat stars measured 19 orbits with periods from 7 to 90 d and eccentricities from 0.2 to 0.9, and showed that heartbeat stars draw the upper envelope of the eccentricity-period distribution (Shporer et al., 2016). Spectroscopic solutions for six Kepler systems yielded $P=7$–20 days and $e>0.34$, confirming that the photometric class corresponds to genuine eccentric binaries rather than an artifact of light-curve morphology alone (Smullen et al., 2015).

OGLE transformed the demographic picture. Its catalog contains 991 HBS candidates: 512 toward the Galactic bulge, 439 in the Large Magellanic Cloud, and 40 in the Small Magellanic Cloud. That sample separates into two large groups with different evolutionary states: about 100 systems with a hot main-sequence or Hertzsprung-gap primary, and about 900 with a red giant primary. The main-sequence/Hertzsprung-gap systems occupy roughly $P\approx2$–100 d with amplitudes $\sim0.001$–0.005 mag, whereas the red-giant systems occupy roughly $i$0–1500 d with amplitudes $i$1–0.05 mag (Wrona et al., 2021).

The observed range of HBS parameter space is therefore much broader than the Kepler archetype suggests. In the OGLE modeling, the full sample spans $i$2 d with median $i$3 d, $i$4, inclinations from $i$5 to $i$6, and an approximately uniform distribution of $i$7 over $i$8 modulo the analytic model’s symmetry (Wrona et al., 2021, Wrona et al., 2021).

Survey or study	Sample	Salient result
OGLE collection (Wrona et al., 2021)	991 candidates	Two dominant groups: hot MS/HG and RG primaries
Kepler RV monitoring (Shporer et al., 2016)	19 measured orbits	$i$9–90 d, $\omega$0–0.9, upper $\omega$1–$\omega$2 envelope
Massive TESS sample (Kołaczek-Szymański et al., 2020)	20 systems	Seven show TEOs; six are eclipsing
Gaia + TESS search (Callahan et al., 17 Jun 2025)	112 new systems	Non-giant HBSs have evolved off the main sequence

TESS-based expansion of the class has reinforced both the astrophysical diversity of HBSs and the presence of survey selection effects. One TESS study reported 23 new systems with periods from 2.7 to 20 days and eccentricities from 0.08 to 0.70, and argued that the survey is biased toward massive HBSs with shorter orbital periods, higher temperatures, and higher luminosities (Li et al., 2024). A later catalog of 42 new TESS HBSs similarly concluded that TESS HBSs have higher temperatures and greater luminosities than Kepler HBSs (Li et al., 2 Aug 2025). A plausible implication is that the current observational census remains incomplete at low amplitude, long period, and low luminosity.

3. Modelling frameworks and orbital inference

The principal modeling approaches fall into two categories: analytic equilibrium-tide fits and full binary-star light-curve plus radial-velocity models. The analytic approach descends from Kumar et al. (1995) and is widely used in OGLE and TESS work, often in a corrected “K95$\omega$3” form with MCMC inference of $\omega$4, $\omega$5, $\omega$6, $\omega$7, $\omega$8, and nuisance scaling parameters (Wrona et al., 2021, Li et al., 2024, Li et al., 2024). In OGLE, cleaned and detrended $\omega$9-band light curves were fit with emcee using flat priors on $a$0, $a$1, and $a$2, plus Gaussian priors on $a$3 and $a$4 (Wrona et al., 2021).

More detailed forward modeling is used when eclipses, spectroscopy, or TEO subtraction demand higher fidelity. ELLC models each star as a Roche-potential equipotential perturbed by rotation and tides and includes gravity darkening, limb darkening, Doppler boosting, and reflection; it was used together with MCMC to fit Kepler light curves and radial velocities for KIC 6117415, KIC 11494130, and KIC 5790807 (Cheng et al., 2020). PHOEBE was modified to include tidally induced pulsations and Doppler boosting in the analysis of KIC 3749404, and later used in Gaia-TESS joint modeling of two SB2 heartbeat stars (Hambleton et al., 2016, Callahan et al., 17 Jun 2025). ELISa was used for the eclipsing heartbeat system V680 Mon, where the fitted parameters included $a$5, $a$6, $a$7, $a$8, surface potentials, and $a$9 (Paunzen et al., 2021).

Radial velocities remain essential for establishing dynamical orbits and masses. In the Kepler HIRES campaign, the orbit was fit with the standard Keplerian model

$R(t)=a(1-e^2)/(1+e\cos f(t))$0

with $R(t)=a(1-e^2)/(1+e\cos f(t))$1, $R(t)=a(1-e^2)/(1+e\cos f(t))$2, $R(t)=a(1-e^2)/(1+e\cos f(t))$3, $R(t)=a(1-e^2)/(1+e\cos f(t))$4, and $R(t)=a(1-e^2)/(1+e\cos f(t))$5 as free parameters and the photometric period as a prior. From $R(t)=a(1-e^2)/(1+e\cos f(t))$6, $R(t)=a(1-e^2)/(1+e\cos f(t))$7, and $R(t)=a(1-e^2)/(1+e\cos f(t))$8 one derives the spectroscopic mass function

$R(t)=a(1-e^2)/(1+e\cos f(t))$9

These measurements yielded the largest sample of heartbeat stars with orbits measured using a single instrument and roughly doubled the number of heartbeat stars with an RV-measured orbit at the time (Shporer et al., 2016).

The empirical status of the analytic heartbeat model is mixed rather than uniform. For six Kepler systems, spectroscopic orbital elements agreed well with those derived from photometric heartbeat modeling, supporting the view that photometric data are sufficient to derive reliable orbital parameters in some regimes (Smullen et al., 2015). By contrast, the TESS survey of 20 massive systems concluded that Kumar’s model does not provide reliable parameters when compared with detailed light-curve modeling, attributing the discrepancy to neglected irradiation, Doppler beaming, and higher-order multipoles (Kołaczek-Szymański et al., 2020). This tension is not a contradiction so much as a reminder that the validity of analytic inference depends on system architecture, data quality, and the degree to which non-tidal contributions contaminate the periastron waveform.

4. Tidally excited oscillations and mode diagnostics

TEOs are the observational signature of the dynamical tide. After subtraction of the equilibrium-tide or full binary model, they appear at frequencies

$1/R^3$0

and are interpreted mainly as forced stellar pulsations, usually g-modes (Cheng et al., 2020). Their detectability is strongly temperature dependent: theoretical work argued that TEOs are more visible in hot stars with surface temperatures $1/R^3$1 because cool stars’ thick surface convection zones isolate g-mode eigenfunctions from the photosphere, reducing the observable luminosity perturbation (Fuller, 2017). Kepler ensemble work recovered the same trend empirically, with almost every HBS hosting detectable TEOs lying above $1/R^3$2 K (Li et al., 2023).

Large-sample studies have made TEOs a statistical property of the class rather than an exceptional feature. In the OGLE sample, 52 systems, about 5% of the catalog, show at least one TEO, for a total of 78 distinct modes with orbital harmonics $1/R^3$3–79 and typical amplitudes 0.5–6 mmag (Wrona et al., 2021). A reanalysis of Kepler HBSs identified TEOs in 21 previously unstudied systems, of which 12 show prominent harmonics with $1/R^3$4; it also found positive correlations between orbital eccentricity and TEO harmonic number and between orbital period and harmonic number (Li et al., 2023). In the massive TESS sample, seven of 20 systems show TEOs, but at lower orbital harmonics, $1/R^3$5–36 with median value 9, lower than in known Kepler systems with TEOs (Kołaczek-Szymański et al., 2020).

Phase diagnostics provide a powerful route to mode identification. For aligned systems with adiabatic, standing-wave oscillations, the expected pulsation phase can be written as

$1/R^3$6

On this basis, most pulsation phases in most systems can be explained by dominant $1/R^3$7, $1/R^3$8, or $1/R^3$9 modes, assuming aligned spin and orbital axes (Li et al., 2024). Earlier phase work on eight Kepler systems reached the same qualitative conclusion, while noting significant deviations in some cases, especially the misaligned system KIC 8164262 (Guo et al., 2019).

Not all TEOs are consistent with the simplest standing-wave picture. Fourteen-system Kepler phase analysis identified two misaligned systems, several partially consistent systems, and harmonics in KIC 4377638, KIC 5090937, and KIC 11403032 that are expected to be travelling waves rather than standing waves (Li et al., 2024). TESS work on five new HBSs with TEOs likewise reported one system whose phase deviates by $P=7$0 from adiabatic expectations and is therefore expected to host a traveling wave (Li et al., 2024).

The central theoretical distinction is between chance resonances and resonance locking. Detailed ELLC plus MESA plus GYRE modeling of three Kepler HBSs concluded that resonance locking is likely occurring in KIC 11494130, but not in KIC 6117415 or KIC 5790807 (Cheng et al., 2020). The broader theoretical framework for forced amplitudes, non-adiabatic luminosity perturbations, traditional-approximation rotation, and spin-orbit misalignment was laid out in the formalism of “Heartbeat Stars, Tidally Excited Oscillations, and Resonance Locking,” which also derived a statistical theory for deciding whether an observed TEO is consistent with a chance resonance or requires resonance locking (Fuller, 2017).

5. Spin, synchronization, circularization, and secular evolution

The spin state of HBS components is commonly discussed in terms of Hut’s pseudosynchronization theory. In the equilibrium-tide, constant-time-lag model, the pseudosynchronous angular rate is

$P=7$1

with corresponding pseudosynchronization period $P=7$2 (Zimmerman et al., 2017). Kepler rotation measurements for 24 heartbeat systems showed that only a few stars are actually pseudosynchronized: only two rotate faster than their Hut-predicted pseudosynchronous rate, four rotate more slowly, and the vast majority, about 12 out of 18 systems with reliable eccentricities, cluster around $P=7$3 (Zimmerman et al., 2017). Four systems showed two distinct rotation clusters, naturally interpreted as the two stars’ individual spin signals. This systematic offset suggests that the equilibrium-tide prediction is incomplete for real HBSs.

Several mechanisms have been proposed for the offset from Hut’s prediction. The list explicitly includes dynamical tides, magnetic braking in stars with convective envelopes, differential rotation and spot latitude effects, and third bodies or secular effects such as rapid apsidal motion (Zimmerman et al., 2017). A plausible implication is that “pseudosynchronization” in observed HBSs is often a torque balance that includes more than the equilibrium tide alone.

In orbital-ensemble terms, HBSs are closely connected to high-eccentricity tidal migration. The Kepler RV sample showed that heartbeat stars occupy the upper envelope of the eccentricity-period distribution, and that this envelope can be approximated by

$P=7$4

for $P=7$5–10 d under constant orbital angular momentum (Shporer et al., 2016). The same interpretation recurs in later TESS catalogs, which report a positive eccentricity-period correlation together with evidence for orbital circularization (Li et al., 2024, Li et al., 2 Aug 2025). The Kepler HIRES study further noted that HBS periastron distances and tidal force ratios occupy a narrow range, suggesting that systems with smaller periastron distances have already circularized while wider systems produce photometric signals too small to be flagged as heartbeat stars (Shporer et al., 2016).

Secular evolution can be unusually rapid in individual systems. KIC 3749404 has $P=7$6 d and $P=7$7, and its observed apsidal advance, $P=7$8, exceeds the classical plus general-relativistic theoretical rate, $P=7$9, by roughly two orders of magnitude (Hambleton et al., 2016). The analysis concluded that tidally induced pulsations alone cannot explain the effect and hypothesized a third body in the system. Such cases show that HBSs probe not only tides internal to the binary, but also hierarchical dynamics.

6. Population astrophysics, discovery pipelines, and notable systems

Population-level analysis of the Magellanic Cloud heartbeat stars has linked observed amplitudes to stellar structure and tidal dissipation. Using 479 OGLE HBSs in the LMC and SMC with broadband SED fitting, one study found two clusters in the H-R diagram: hot systems with $e>0.34$0 K near or just off the main sequence, and cool systems with $e>0.34$1 K occupying the red-giant region (Macleod et al., 21 Mar 2025). The observed amplitude distribution was interpreted through a simplified model that couples post-main-sequence radius growth to orbital circularization through linear tidal dissipation. In that framework, cool stars imply $e>0.34$2, while hot stars imply $e>0.34$3, qualitatively consistent with efficient dissipation in convective envelopes and weaker dissipation in radiative envelopes (Macleod et al., 21 Mar 2025). The same study argued that hot stars approach, but do not exceed, the threshold for nonlinear tidal wave breaking, suggesting saturation of tidal amplitudes at that threshold.

The growth of HBS catalogs has also made automated recognition feasible. A recurrent-neural-network method based on the first 100 Fourier harmonic amplitudes of a light curve trained on 52,000 synthetic binaries generated with ELLC, using shallow GRU and LSTM architectures to predict eccentricity. On synthetic test data both networks achieved 95% accuracy under the criterion $e>0.34$4, and on real OGLE, Kepler, TESS, and eccentric eclipsing-binary samples they achieved an average accuracy of 88% under the criterion $e>0.34$5 (Li et al., 7 May 2025). Applied to 2,623 Kepler eclipsing binaries not already identified as HBSs, the method recovered four new heartbeat stars: KIC 4940438, 6794131, 7601633, and 9243795 (Li et al., 7 May 2025).

A complementary search strategy starts from spectroscopic binaries rather than photometric morphology alone. The Gaia-plus-TESS search identified 112 new heartbeat star systems and found that periods and eccentricities derived from the TESS heartbeat fit agree with Gaia solutions for 85% of single-line spectroscopic binaries but only 20% of double-line spectroscopic binaries (Callahan et al., 17 Jun 2025). For the two SB2 systems with consistent orbits, joint PHOEBE analysis yielded direct masses and radii, illustrating the dynamical payoff of combining phase-folded heartbeat morphology with spectroscopic velocity semi-amplitudes (Callahan et al., 17 Jun 2025).

Several individual systems mark the extremes of the class. MACHO 80.7443.1718 is a young, massive LMC binary composed of a B0 Iae supergiant and an O9.5V secondary on an eccentric orbit with $e>0.34$6 d and $e>0.34$7. It shows the largest variability amplitude among known heartbeat stars, with a peak-to-peak periastron brightening of about 40%, TEOs at the $e>0.34$8 and $e>0.34$9 harmonics, and a circumstellar disk that dissipates at periastron and re-emerges shortly afterward (Jayasinghe et al., 2021). V680 Mon is the first known heartbeat binary containing a mercury-manganese star and only the fifth known eclipsing CP3 star; its primary is a very young, zero-age main-sequence HgMn star in an eccentric, tidally distorted configuration (Paunzen et al., 2021). These systems demonstrate that HBSs are not confined to the A/F-type detached-binary regime emphasized by early Kepler discoveries.

Taken together, the observational and theoretical literature shows that heartbeat stars are not a narrowly defined photometric curiosity but a heterogeneous tidal-binary population. They include hot main-sequence systems, evolved red-giant binaries, massive stars, eclipsing and non-eclipsing configurations, systems with standing-wave TEOs, systems with travelling waves, and systems whose secular evolution appears to be influenced by third bodies or resonance locking. This suggests that the unifying property of the class is not any single period, mass, or morphology, but the fact that periastron tides are directly observable in both the orbital light curve and the stellar oscillation spectrum (Fuller, 2017, Wrona et al., 2021, Macleod et al., 21 Mar 2025).