Close Hyperbolic Encounters (CHEs)

• Close Hyperbolic Encounters (CHEs) are unbound two-body flybys with eccentricity >1 that produce observable dynamical effects, including gravitational-wave bursts and orbital changes.
• CHEs generate unique gravitational-wave signatures, with burst spectra scaling as f² or f^(2/3) depending on the environment, and serve as probes in both compact-object and primordial black hole studies.
• Methodologies to study CHEs include precise orbital parameterization, numerical simulations, and cataloging stellar and planetary encounters to assess rates and gravitational-wave background contributions.

Close hyperbolic encounters (CHEs) are close two-body passages on unbound orbits, conventionally identified by eccentricity $e>1$, a nonzero asymptotic speed at infinity, and a periapsis small enough for the encounter to have an observable dynamical or radiative consequence. In the compact-object literature, CHEs are single-scattering events that emit gravitational-wave bremsstrahlung concentrated near closest approach; in primordial-black-hole (PBH) scenarios they contribute both burst signals and a stochastic gravitational-wave background (SGWB); in planetary and stellar dynamics they describe single close passages that can eject small bodies, strip planetary mantles, or define catalogs of rare unbound flybys in the Solar neighborhood (Garcia-Bellido et al., 2017, García-Bellido et al., 2021, Gòmez-Aguilar et al., 23 Sep 2025, Monk et al., 4 Feb 2026, Deng, 2019, Hansen, 2021).

1. Orbital definition and kinematic parametrization

Across the literature, CHEs are parameterized by the total mass $M\equiv m_1+m_2$, the relative speed at infinity $v_\infty$ or $v_0$, the impact parameter $b$, and the periapsis distance $r_p$ or $r_{\min}$. In Newtonian two-body dynamics, the hyperbolic eccentricity is

$e=\sqrt{1+\frac{b^2 v_\infty^4}{G^2 M^2}}>1,$

while gravitational focusing gives

$$b^2=r_p^2\left(1+\frac{2GM}{r_p v_\infty^2}\right).$$

Equivalent forms used in the compact-object literature include

$$r_p=a(e-1)=b\sqrt{\frac{e-1}{e+1}}, \qquad a=-\frac{GM}{v_\infty^2},$$

and the scattering cross section

$M\equiv m_1+m_2$0

These relations appear with minor notational differences in PBH and compact-object treatments of CHEs (Garcia-Bellido et al., 2017, García-Bellido et al., 2021, Gòmez-Aguilar et al., 23 Sep 2025).

The same orbital language is used in stellar and planetary applications. For unbound stellar pairs, the standard relations are often written in terms of the specific orbital energy $M\equiv m_1+m_2$1, specific angular momentum $M\equiv m_1+m_2$2, eccentricity $M\equiv m_1+m_2$3, and perihelion distance $M\equiv m_1+m_2$4, with the escape condition $M\equiv m_1+m_2$5 (Hansen, 2021). In the Venus–proto-Mercury simulations, CHEs are further characterized by pericenter distances $M\equiv m_1+m_2$6 and hyperbolic excess speeds $M\equiv m_1+m_2$7 (Deng, 2019).

This suggests that the qualifier “close” is application-dependent. In the Solar-neighborhood stellar catalog, it denotes perihelion distances of order $M\equiv m_1+m_2$8 AU or less as the operative scale for identifying notable unbound flybys (Hansen, 2021). In planetary mantle-stripping and compact-object GW problems, it denotes passages only a few body radii or a few Schwarzschild radii from the primary interaction scale (Deng, 2019, Jaraba et al., 2021).

2. Gravitational-wave emission from a single encounter

In the compact-object and PBH literature, a CHE is distinguished from GW capture by the comparison between radiated energy and the kinetic energy at infinity. If the energy radiated during the fly-by satisfies

$M\equiv m_1+m_2$9

the objects remain unbound and the event is a CHE; if the radiative loss is larger, the pair can become bound and form an eccentric binary black hole (García-Bellido et al., 2021).

At quadrupole order, the total GW energy radiated in a single hyperbolic encounter is

$v_\infty$0

where $v_\infty$1 is an eccentricity-dependent factor. The burst is sharply peaked near periastron, and the frequency-domain power can be written as

$v_\infty$2

with $v_\infty$3 and $v_\infty$4. An approximate peak frequency is

$v_\infty$5

which encodes the impact parameter and eccentricity dependence of the burst spectrum (Garcia-Bellido et al., 2017).

A source-frame spectral representation used in SGWB calculations is

$v_\infty$6

where $v_\infty$7 controls the spectral shape (García-Bellido et al., 2021). In the time domain, the signal is a burst event with the majority of the released energy occurring during the closest approach, and in the PBH-burst literature it is described as having a “chirp” then “anti-chirp” structure (Garcia-Bellido et al., 2017).

Several corrections beyond the leading Newtonian quadrupole have been analyzed. Orbital precession can be incorporated through

$v_\infty$8

which produces a slight chirping and asymmetry of the burst near periastron (Caldarola et al., 2023). The same work gives a linear memory contribution

$v_\infty$9

for the non-precessing case (Caldarola et al., 2023). At next-to-leading multipole order, the mass octupole, current quadrupole, and 1PN quadrupole correction can contribute at the $v_0$0 level for close encounters with small periapses and high mass ratios; numerical examples in the literature identify $v_0$1, $v_0$2, and $v_0$3 as a regime where these corrections can be important (Roskill et al., 2023).

3. Population synthesis and the stochastic background

For unresolved populations of CHEs, the GW observable is the SGWB energy density

$v_0$4

with $v_0$5 and $v_0$6 the comoving event rate (García-Bellido et al., 2021). In dense PBH clusters, the low-frequency tail depends on the redshift evolution $v_0$7, giving

$v_0$8

whereas the binary-black-hole background retains $v_0$9 independently of $b$0 (García-Bellido et al., 2021).

The dwarf-galaxy PBH calculation extends this framework to dense cores in which both hierarchical binary black hole mergers and CHEs are present. In that model, the core is divided into ten concentric shells, the encounter rate is summed shell by shell, and the calculation is repeated across four redshift epochs between $b$1 and $b$2, with masses, densities, and velocities updated after each merger generation (Gòmez-Aguilar et al., 23 Sep 2025). The resulting CHE background follows $b$3 up to a turnover at $b$4 Hz, with amplitude at $b$5 Hz of order few $b$6 (Gòmez-Aguilar et al., 23 Sep 2025).

In the same calculation, CHEs occur earlier, provide the first GW signals, and contribute a continuous though subdominant background that becomes relatively more significant once the initial PBH population is depleted and binary formation is suppressed (Gòmez-Aguilar et al., 23 Sep 2025).

Channel	Spectral scaling	Relative role in the dwarf-galaxy PBH model
CHE	$b$7 up to $b$8 Hz	Earlier; first GW signals; roughly an order of magnitude below BBH at the peaks; dominates at the highest frequencies just below $b$9Hz
BBH mergers	$r_p$0	Dominates the total emission

The contrast between $r_p$1 and $r_p$2 is therefore not merely formal; it is the principal spectral discriminator used in CHE-background studies of PBH populations (García-Bellido et al., 2021, Gòmez-Aguilar et al., 23 Sep 2025).

4. Dynamical outcomes in compact-object, stellar, and planetary systems

CHEs are not limited to GW burst phenomenology. In dense black-hole clusters, close flybys can transfer angular momentum and induce spin. Numerical-relativity simulations with the Einstein Toolkit find that for equal masses the maximum induced spin is $r_p$3, while large mass ratios can yield $r_p$4, with the highest spin induced on the more massive black hole; for small induced spins, analytic expressions depend on the relative velocity and impact parameter (Jaraba et al., 2021).

For neutron stars, close passages can resonantly excite crust-core interface modes and produce shattering flares. The criterion for crust failure is

$r_p$5

and the electromagnetic counterpart is effectively prompt with respect to the GW burst, with $r_p$6 ms of periapse (Tsang, 2013). The same work provides updated encounter-rate estimates in dense stellar environments and argues that triggered GW searches using hard X-ray or gamma-ray flashes are relevant for this source class (Tsang, 2013).

In planetary dynamics, a single hyperbolic encounter can map a bound orbit to an unbound one. In the coplanar restricted three-body treatment, the deflection satisfies

$r_p$7

and the ejection criterion can be written compactly as

$r_p$8

The same analysis identifies a minimum-e threshold $r_p$9 for efficient ejection in a single close encounter, and numerical experiments show agreement at the $r_{\min}$0 level except for grazing or multi-encounter cases (Monk et al., 4 Feb 2026).

A different planetary application is the Venus–proto-Mercury scenario. Smoothed-particle-hydrodynamics and N-body modeling indicate that tidal disruption of proto-Mercury always removes part of its silicate mantle while the iron core remains intact; in favorable cases, four close encounters with fast spinning projectiles can lead to the present-day Mercury iron fraction (Deng, 2019).

CHEs also arise in observational stellar dynamics. A Gaia EDR3-based catalog of unbound stellar pairs within $r_{\min}$1 pc identifies $r_{\min}$2 independent CHEs after duplicate removal, with relative velocities at infinity typically $r_{\min}$3 and times of closest approach ranging from $r_{\min}$4 kyr to $r_{\min}$5 yr (Hansen, 2021). In that context, CHEs have been discussed both as tracers of rare local stellar dynamics and as a finite target list for SETI-style searches under a migration-during-flyby hypothesis (Hansen, 2021).

5. Searches and observational prospects

The first dedicated LVK search for compact-object hyperbolic encounters in O3b used a model-informed machine-learning-enhanced Coherent WaveBurst pipeline. No significant event was identified in addition to known detections of compact binary coalescences (Bini et al., 2023). The injections employed a non-spinning third Post-Newtonian hyperbolic model with component masses in $r_{\min}$6, impact parameter in $r_{\min}$7, and eccentricity in $r_{\min}$8 (Bini et al., 2023).

For O3b, the best sensitivity was obtained in the $r_{\min}$9 mass bin, where the sensitive spacetime volume reached

$e=\sqrt{1+\frac{b^2 v_\infty^4}{G^2 M^2}}>1,$0

corresponding to a rate density upper limit

$e=\sqrt{1+\frac{b^2 v_\infty^4}{G^2 M^2}}>1,$1

Projected sensitivities for the same mass bin improve to $e=\sqrt{1+\frac{b^2 v_\infty^4}{G^2 M^2}}>1,$2 for O4 and $e=\sqrt{1+\frac{b^2 v_\infty^4}{G^2 M^2}}>1,$3 for O5 (Bini et al., 2023).

A complementary model-independent study uses BayesWave with exponential shapelets to reconstruct binary-black-hole hyperbolic-encounter waveforms in simulated detector noise. For a typical hyperbolic orbit with total mass $e=\sqrt{1+\frac{b^2 v_\infty^4}{G^2 M^2}}>1,$4, the detectable luminosity distance is $e=\sqrt{1+\frac{b^2 v_\infty^4}{G^2 M^2}}>1,$5 Mpc; the same study forecasts ranges of $e=\sqrt{1+\frac{b^2 v_\infty^4}{G^2 M^2}}>1,$6 Mpc for Cosmic Explorer and $e=\sqrt{1+\frac{b^2 v_\infty^4}{G^2 M^2}}>1,$7 Mpc for the Einstein Telescope, depending on mass ratio and periapsis (Lott et al., 1 Dec 2025).

For unresolved backgrounds, the detector landscape is frequency-dependent. In the dwarf-galaxy PBH model, CHEs might be marginally accessible to DECIGO around $e=\sqrt{1+\frac{b^2 v_\infty^4}{G^2 M^2}}>1,$8 Hz, whereas BBHs are within reach of ET and in part LISA at low frequency (Gòmez-Aguilar et al., 23 Sep 2025). In dense PBH cluster models, CHE backgrounds of order few $e=\sqrt{1+\frac{b^2 v_\infty^4}{G^2 M^2}}>1,$9 at $$b^2=r_p^2\left(1+\frac{2GM}{r_p v_\infty^2}\right).$$0 Hz are identified as targets for third-generation ground-based detectors such as Einstein Telescope and Cosmic Explorer (García-Bellido et al., 2021).

6. Delimiting criteria, approximations, and theoretical extensions

A recurring misconception is to treat every close unbound passage as equivalent to a capture precursor or merger. The PBH-scattering literature is explicit that CHEs are the subset of close flybys for which the radiated energy is insufficient to bind the pair; once $$b^2=r_p^2\left(1+\frac{2GM}{r_p v_\infty^2}\right).$$1, the outcome is an eccentric binary rather than a CHE (García-Bellido et al., 2021). This distinction matters because CHEs and BBH mergers generate different rate equations, different temporal behavior, and different SGWB slopes (García-Bellido et al., 2021, Gòmez-Aguilar et al., 23 Sep 2025).

The underlying calculations also rely on strong approximations that vary by subfield. The dwarf-galaxy PBH SGWB model assumes monochromatic PBH masses, a spherical Plummer profile for the core, a two-body single-encounter approximation, no GW-recoil ejection or binary disruption, and nonrelativistic quadrupole estimates for the CHE cross section and $$b^2=r_p^2\left(1+\frac{2GM}{r_p v_\infty^2}\right).$$2; it further notes that the per-event spectral shape $$b^2=r_p^2\left(1+\frac{2GM}{r_p v_\infty^2}\right).$$3 is not derived in detail, and that the $$b^2=r_p^2\left(1+\frac{2GM}{r_p v_\infty^2}\right).$$4 scaling enters through analytic approximations (Gòmez-Aguilar et al., 23 Sep 2025). In the planetary ejection problem, the treatment assumes the restricted three-body problem, coplanar orbits, an instantaneous kick with equal incoming and outgoing planet-centric speed, and validity mainly for encounters inside $$b^2=r_p^2\left(1+\frac{2GM}{r_p v_\infty^2}\right).$$5 with $$b^2=r_p^2\left(1+\frac{2GM}{r_p v_\infty^2}\right).$$6 (Monk et al., 4 Feb 2026). In the Solar-neighborhood stellar catalog, many line-of-sight velocities are unknown, so closest approaches are inferred statistically through Monte Carlo sampling and straight-line motion during the brief flyby (Hansen, 2021).

CHEs have also been used as probes of modified gravity. In metric $$b^2=r_p^2\left(1+\frac{2GM}{r_p v_\infty^2}\right).$$7 gravity, linearization yields the usual tensor sector together with a scalar mode obeying

$$b^2=r_p^2\left(1+\frac{2GM}{r_p v_\infty^2}\right).$$8

The corresponding hyperbolic-encounter waveform contains a scalar breathing component $$b^2=r_p^2\left(1+\frac{2GM}{r_p v_\infty^2}\right).$$9, and the paper identifies a scalar-to-tensor amplitude ratio $$r_p=a(e-1)=b\sqrt{\frac{e-1}{e+1}}, \qquad a=-\frac{GM}{v_\infty^2},$$0 for $$r_p=a(e-1)=b\sqrt{\frac{e-1}{e+1}}, \qquad a=-\frac{GM}{v_\infty^2},$$1 together with an inter-mode delay $$r_p=a(e-1)=b\sqrt{\frac{e-1}{e+1}}, \qquad a=-\frac{GM}{v_\infty^2},$$2 s for $$r_p=a(e-1)=b\sqrt{\frac{e-1}{e+1}}, \qquad a=-\frac{GM}{v_\infty^2},$$3 kpc, $$r_p=a(e-1)=b\sqrt{\frac{e-1}{e+1}}, \qquad a=-\frac{GM}{v_\infty^2},$$4 Hz, and $$r_p=a(e-1)=b\sqrt{\frac{e-1}{e+1}}, \qquad a=-\frac{GM}{v_\infty^2},$$5 eV (Bruton et al., 25 Jun 2025). In this extension, CHEs are treated as burst-like tests of extra GW polarizations rather than only as sources of tensor bremsstrahlung.

Taken together, the literature presents CHEs as a common dynamical motif rather than a single specialized phenomenon: unbound close passages with $$r_p=a(e-1)=b\sqrt{\frac{e-1}{e+1}}, \qquad a=-\frac{GM}{v_\infty^2},$$6 that, depending on scale and environment, generate GW bursts, SGWB contributions, orbital ejection, tidal stripping, stellar near-miss catalogs, or tests of non-GR radiation sectors (Garcia-Bellido et al., 2017, Gòmez-Aguilar et al., 23 Sep 2025, Monk et al., 4 Feb 2026, Hansen, 2021, Bruton et al., 25 Jun 2025).