Eccentricity Tides in Astrophysics

• Eccentricity tides are tidal forces generated by non-circular orbits that yield enhanced, time-varying potentials, driving internal heating and orbital evolution.
• They are modeled using Fourier and equilibrium tide frameworks, with energy dissipation scaling quadratically with eccentricity and marked harmonic signatures.
• Observational signatures include photometric heartbeat features, gravitational wave phase shifts, and variations in spin–orbit coupling, crucial for deciphering system dynamics.

Eccentricity tides refer to tidal interactions in astrophysical systems that are uniquely driven or amplified by orbital eccentricity. Distinct from tides in circular orbits, eccentricity tides produce time-varying tidal potentials with enhanced harmonic content, leading to richer dynamical, thermal, and observational phenomena across exoplanets, binaries, and triple systems. Their physical consequences include internal heating, orbital evolution, spin–orbit coupling, and altered gravitational wave signals, with manifestations ranging from photometric "heartbeat" features to accelerated eccentricity damping by inertial waves.

1. Principles and Formalism of Eccentricity Tides

The foundation of eccentricity tides lies in the periodic variation of star–companion separation and orbital velocity inherent to eccentric orbits. The leading (quadrupolar) component of the tidal potential varies as

$U(t) \propto \frac{1}{[a(1-e\cos E)]^3},$

where $a$ is the semi-major axis, $e$ orbital eccentricity, and $E$ the eccentric anomaly. This yields an instantaneous tidal forcing much greater at pericenter.

The dissipative power from equilibrium eccentricity tides is given in the weak friction limit by

$$P_{\text{tide}} = \frac{21}{2}\frac{k_2}{Q} \frac{G M_*^2 R_p^5}{a^6} e^2 n,$$

for a planet of radius $R_p$, host mass $M_*$, eccentricity $e$, mean motion $n$, Love number $k_2$, and tidal quality factor $Q$ (Yee et al., 11 Nov 2025, Husnoo et al., 2012). The quadratic scaling with $e$ reflects the growing role of pericenter passages in energy injection.

The time-varying tidal potential can be expanded in Hansen or Fourier coefficients, and the resulting forced response in the secondary (planet or star) displays harmonics at integer multiples of $n$, each with amplitude scaling as powers of $e$ (Penoyre et al., 2018, Auclair-Desrotour et al., 2019). Dynamical tides, in contrast, couple resonantly to normal modes of the body, whose excitation depends sensitively on both the frequency content set by $e$ and the body's internal structure.

2. Eccentricity Tides in Exoplanets and Binaries: Heating, Damping, and Spin States

Internal Heating and Inflation of Exoplanets

Eccentricity tides have been invoked as a potential energy source for the radii inflation observed in low-density exoplanets. However, precise radial velocity measurements (e.g., of WASP-107 b, TOI-1173 b, HAT-P-18 b) indicate that for these "popcorn planets," upper limits on eccentricity ($e < 0.03\text{--}0.05$) combined with standard $Q' \sim 10^5$ yield insufficient $P_{\text{tide}}$, typically at least two orders of magnitude below the $10^{26}$–$10^{27}$ W heating budget necessary for inflation (Yee et al., 11 Nov 2025). Attaining the required heating would demand tidal dissipation efficiencies $Q' \lesssim 10^2$, which is not only inconsistent with gas-giant structure but would also circularize the orbits on $\lesssim 10^7$ yr timescales, much shorter than system ages.

Eccentricity Damping and Spin Evolution

The classical equilibrium-tide model predicts eccentricity damping timescales scaling as

$$\tau_{e,p} = \frac{Q_p}{21/2\, k_{2,p}} \frac{M_p}{M_*} \left( \frac{a}{R_p} \right)^5 n^{-1}$$

with $\dot{e}/e \propto -e$, yielding exponential damping (Husnoo et al., 2012). In binaries with convective envelopes, inertial wave excitation at frequencies up to $2\Omega$ (with $\Omega$ the spin) can greatly accelerate damping, yielding $\dot{e}/e \propto -e^{-2}$ and enhancing circularization rates by orders of magnitude at modest $e$ (Dewberry, 29 Oct 2025).

Tidal torques also drive spin synchronization and pseudosynchronization. Eccentricity tides introduce multiple harmonic torques that, when resonant with oceanic or inertial modes, allow for asynchronous rotation equilibria. In planetary cases with oceans, critical $e$ for asynchrony can be suppressed from $\sim$0.3 (dry) to 0.01–0.06 (with resonant ocean modes) (Auclair-Desrotour et al., 2019).

3. Eccentricity Tides in Multi-Body Systems

Hierarchical Triples: Tertiary Tides

In hierarchical triples, time-varying quadrupole forcing by the inner binary can efficiently siphon energy from the inner orbit into the outer ("tertiary") star or into the outer orbit itself, greatly accelerating orbital evolution. Dissipation rates are sensitive to the radii of the tertiary, the proximity to Roche filling, and especially the inner eccentricity $e_1$: $$\frac{dE_1}{dt} \propto (1-e_1^2), \qquad \frac{dE_2}{dt} \propto e_1^2$$ where $E_1, E_2$ are the inner and outer orbit energies, illustrating the strong modulation by eccentricity (Gao et al., 1 Sep 2025). Eccentricities in the outer orbit are rapidly erased (trivial), whereas nonzero $e_1$ leads to an additional drain from $E_2$. The regime of interest for significant "tertiary tides" is when the tertiary is nearly Roche-filling in a system with weak primary binary tidal interactions.

Resonant Interactions and Migration

Eccentricity tides can drive near-resonant planet pairs apart ("resonant repulsion"), explaining ~2:1 mean-motion-resonance offsets, but in Kepler systems, required initial $e$ vastly exceed dynamical stability thresholds, especially when considering the stunting effect of "resonant tugging" where the resonance links the orbital evolution of both planets (Silburt et al., 2015). Thus, pure eccentricity tides alone cannot reproduce observed distributions without invoking additional migration or excitation from protoplanetary disks and other perturbers.

4. Observational Manifestations: Photometry, Spectroscopy, and Gravitational Waves

Photometric and Spectroscopic Signatures

Eccentricity tides produce time-varying, non-sinusoidal tidal bulges in host stars, yielding photometric variations ("heartbeat" features) and radial velocity signals with harmonics at integer multiples of the orbital frequency (Penoyre et al., 2018). Power spectra show strong $n \geq 2$ harmonics, with amplitude scaling steeply as $(1-e)^{-3}$ at high $e$. Such signals are distinct from reflection, beaming, or thermal emission, which tend to favor $n=1$ harmonics.

These tidal features are observable at all inclinations and directly constrain $e$, viewing geometry, and planetary mass. Notably, high-precision light curves can detect orbital eccentricities $e \sim 0.05$ (e.g., HAT-P-16 b: $e=0.034 \pm 0.003$) (Husnoo et al., 2012).

Gravitational Wave Effects

In eccentric compact-object binaries (white dwarf, neutron star, black hole), dynamical tides can resonantly excite modes, leading to measurable modifications of the gravitational waveform:

• Periastron Precession: Resonant excitation causes precession rates to spike in narrow bands, covering up to $\sim10\%$ of the frequency space at $e\gtrsim0.5$, with dynamical tides contributing up to 20% of the total shift even off-resonance (Lau et al., 2022). For high-mass systems, LISA will be able to distinguish these effects from equilibrium tides and relativistic precession.
• Stochastic Phase Shifts: In highly eccentric neutron star binaries, consecutive pericenter passages stochastically excite the $f$-mode, transferring orbital energy into internal oscillations and producing cumulative gravitational wave phase shifts, detectable if $r_{\rm p} \lesssim 70$ km and $e \to 1$ (Takátsy et al., 24 Jul 2024).
• EOS Constraints: The frequency and amplitude of these phase shifts encode both tidal deformability and $f$-mode frequencies, providing a unique probe of compact object interiors and possible nuclear phase transitions.

5. Eccentricity Tides and Orbital/Thermal Evolution in Planetary Contexts

Oceanic and Surface Effects

Planets orbiting low-mass stars near the habitable zone with moderate-to-large $e$ experience extreme ocean tides, with global maximum elevations of $\mathcal{O}(10^3)$ m and mean dissipation orders of magnitude greater than on Earth. Models show that for $e=0.3$, mean dissipation reaches $\sim100$ W m$^{-2}$, tidal amplitudes can approach or exceed a kilometer, and mean flow speeds $>1$ m s$^{-1}$ (Shi et al., 5 Jul 2025). Such dissipation can circularize orbits on Gyr timescales—1–2 orders of magnitude faster than for rocky planets without oceans.

These effects are further modulated by ocean depth, topography, and rotation. Resonant excitation of ocean internal modes at frequencies determined by $e$ and rotation state can dramatically amplify both tidal torque and energy dissipation (Auclair-Desrotour et al., 2019).

Planetary Spin-Orbit Outcomes

Resonant enhancement of certain tidal harmonics due to eccentricity enables stable asynchronous states distinct from classical 1:1 locks. For instance, with moderate oceanic Rayleigh drag ($\gamma$) and resonance, the critical eccentricity for an asynchronous equilibrium (e.g., 3:2 spin-orbit) can be $e_c \sim 0.01$–0.06 in typical terrestrial-ocean planet cases, well below $e_c \sim 0.3$ for dry bodies (Auclair-Desrotour et al., 2019). This fundamentally alters climate and day-night cycles on exoplanets, with implications for surface habitability.

6. Mechanism-Dependent Outcomes: Inflation, Circularization, and Excitation

Limitations of Eccentricity Tides for Planet Inflation

New constraints, such as those from MAROON-X spectroscopy, exclude significant contribution of eccentricity tides to inflating the radii of certain large exoplanets unless $Q'$ values are extremely low and inconsistent with maxima observed for gas giants (Yee et al., 11 Nov 2025). Alternative mechanisms now favored include obliquity tides (where excitation can reach the inflated values for $P_{\rm obliq}$ at high obliquity) and Ohmic dissipation driven by atmospheric flows and magnetic fields.

Eccentricity Maintenance and Pumping

Secular gravitational forcing by external companions can, under certain circumstances, pump eccentricity against tidal decay, particularly when periastron precession matches the timescale of forced oscillations. The net criterion for secular growth is achieved if the companion-induced forcing exceeds the dissipative loss, determined by the balance of $$A e_{\rm out} \gtrsim (21/2) (k_2/Q) (M_*/m_p)(R_p/a)^5 n e$$ (Correia et al., 2011). This explains why some moderately close-in exoplanets retain significant $e$ despite apparently short tidal damping times, provided suitable outer companions exist.

Disk-Driven Eccentricity in Binaries

Post-main-sequence binaries can in principle acquire high $e$ via angular momentum transfer to a massive circumbinary disk, mediated by disk–binary tide resonances. However, once realistic viscous spreading, disk reaccretion, and dust formation are accounted for, the net eccentricity growth is greatly suppressed. Only very massive, long-lived, and non-reaccreting disks can reach $e\gtrsim0.3$ (Rafikov, 2016).

7. Current Frontiers, Robustness, and Astrophysical Implications

Robustness Across Models and Observable Predictions

The efficiency of eccentricity tide dissipation is highly sensitive to internal rheology, presence of oceans or convective envelopes, and resonant locking phenomena (e.g., inertial modes). Inertial-wave dissipation introduces a torque balance sequence and discrete spin–orbit commensurabilities, providing a strong test in rotational period/orbital period distributions observed in eclipsing binaries (Dewberry, 29 Oct 2025).

LISA, aLIGO, and JWST promise to resolve or rule out eccentricity tide effects across multiple domains: GW phase shifts, periastron precession, thermal light curve inflation, and planetary atmospheric behavior.

Summary Table: Key Roles and Regimes of Eccentricity Tides

Context	Effect Driven/Enhanced by Eccentricity	Dominant Outcomes
Hot (and cool) Jupiters	Apastron/periastron power, harmonics	Rapid circularization, inflation (ruled out in “popcorn planets”), “heartbeat” signals
Terrestrial/ocean worlds	Ocean resonance, frequency content	Asynchronous spin equilibria, kilometer-scale tides, rapid tidal dissipation
Hierarchical triples	Quadrupolar inner-binary modulation	Efficient shrinkage or chaos, modulation of inner and outer energy, ejection or GW merger acceleration
Compact binaries	Resonant mode excitation, phase shifts	Periastron precession, GW signal distortion, eccentricity-tide phase coupling ($\sim e^2\Lambda$), stochastic energy transfer
Multi-planet systems	Resonant tugging, migration	Suppressed MMR migration, need for hybrid disk/migration scenarios

Astrophysical consequences are diverse, and the interplay between eccentricity amplitude, tidal quality factor, internal structure, and external perturbation determines the dominance or suppression of eccentricity tides in a given system. In several domains, their effects are measurable, testable, and—where ruled out—demand further investigation of alternative dissipation or heating mechanisms.