Parabolic Stellar Flybys

• Parabolic stellar flybys are unbound, zero-energy encounters that significantly disturb circumstellar and protoplanetary discs in young stellar clusters.
• They generate distinct spiral arms, truncate and warp discs, and promote dust trapping that can trigger streaming instability and planetesimal formation.
• Flybys play a key role in ejecting planetesimals into interstellar space, thereby influencing outer solar system architecture and disc evolution.

A parabolic stellar flyby is a dynamical event in which a star (the perturber) passes another star (the host) on a parabolic orbit—characterized by zero specific orbital energy ($E = 0$) and eccentricity $e=1$—with a closest approach (periastron) that can strongly perturb any circumstellar or protoplanetary disc around the host. Parabolic flybys play a fundamental role in the evolution of planet-forming discs, the birth of planetesimals, the shaping of stellar systems' outer architectures, and the ejection of interstellar objects. Their prevalence and dynamical consequences are especially pronounced in young, dense stellar clusters.

1. Dynamical Framework of Parabolic Flybys

A parabolic encounter is defined by its zero specific energy condition

$$\varepsilon = \frac{v^2}{2} - \frac{G(M_1+M_2)}{r} = 0,$$

where $M_1$ and $M_2$ are the masses of the host and the perturber, $v$ is the relative speed at distance $r$. This yields a trajectory

$r(\theta) = \frac{2 r_p}{1 + \cos\theta},$

with $r_p$ the periastron distance and $\theta$ the true anomaly ($e=1$0 at periastron). The velocity as a function of separation becomes $e=1$1. The time since periastron is given by Barker’s equation: $e=1$2 Such encounters are unbound, with the perturber entering from and departing to infinity at zero velocity.

The orbital angular momentum per unit reduced mass is $e=1$3. The key parameters governing the interaction are the mass ratio $e=1$4, the periastron $e=1$5, and the inclination $e=1$6 of the flyby orbit relative to the disc. Inclined and retrograde parabolic flybys produce significantly different outcomes from coplanar prograde ones.

2. Gas and Dust Morphology: Spirals, Truncation, and Warps

Parabolic flybys excite distinctive dynamical structures within planet-forming discs, primarily:

• Spiral arms: Two-armed, large-scale spiral patterns emerge promptly as the perturber nears periastron (Cuello et al., 2018, Smallwood et al., 2023). Gas spirals possess initial pitch angles of $e=1$7–$e=1$8 near periastron, winding down to $e=1$9–$$\varepsilon = \frac{v^2}{2} - \frac{G(M_1+M_2)}{r} = 0,$$0 over $$\varepsilon = \frac{v^2}{2} - \frac{G(M_1+M_2)}{r} = 0,$$1 years, with lifetime set by several orbital periods at the relevant disc radii (e.g., $$\varepsilon = \frac{v^2}{2} - \frac{G(M_1+M_2)}{r} = 0,$$2 years at $$\varepsilon = \frac{v^2}{2} - \frac{G(M_1+M_2)}{r} = 0,$$3 au) (Cuello et al., 2018, Smallwood et al., 2023). Dust spirals are sharper and more tightly wound, often spatially offset from gas spirals especially for grains with high Stokes numbers (St$$\varepsilon = \frac{v^2}{2} - \frac{G(M_1+M_2)}{r} = 0,$$4), leading to rapid decoupling and ring formation (Su et al., 24 Jan 2026, Prasad et al., 15 Sep 2025).
• Disc truncation: Tidal forces strip and truncate the outer disc. For coplanar, prograde flybys, the final gas disc radius can be approximated as $$\varepsilon = \frac{v^2}{2} - \frac{G(M_1+M_2)}{r} = 0,$$5 au (Cuello et al., 2018). Prograde coplanar encounters truncate more severely (up to 20% reduction) than polar or retrograde orbits; dust discs are left even more compact due to radial drift and truncation, with differences $$\varepsilon = \frac{v^2}{2} - \frac{G(M_1+M_2)}{r} = 0,$$610 au for St$$\varepsilon = \frac{v^2}{2} - \frac{G(M_1+M_2)}{r} = 0,$$71 grains (Cuello et al., 2018). Analytical estimates yield the tidal truncation radius:

$$\varepsilon = \frac{v^2}{2} - \frac{G(M_1+M_2)}{r} = 0,$$8

(Cuello et al., 2022).

• Warps and misalignments: Inclined encounters warp the disc; for example, an inclined retrograde flyby ($$\varepsilon = \frac{v^2}{2} - \frac{G(M_1+M_2)}{r} = 0,$$9) can tilt the inner disc by $M_1$0–$M_1$1 and twist it by $M_1$2–$M_1$3 (Cuello et al., 2018). Retrograde, highly inclined flybys can induce warps and broken discs with misalignments up to $M_1$4 (Xiang-Gruess, 2015).
• Disc eccentricity: Massive prograde flybys induce significant eccentricity growth (up to $M_1$5–$M_1$6 for $M_1$7) that recircularizes with a damping timescale of $M_1$8 yr (Smallwood et al., 2023).

3. Dust Trapping, Streaming Instability, and Planetesimal Formation

Parabolic flybys are efficient in creating long-lived dust traps and triggering planetesimal formation:

• Dust spirals, rings, and trapping: Strong tidal spirals in the gas lead to corresponding dust structures. For grains with large Stokes numbers (cm-sized, St$M_1$910–100), dust quickly decouples from gas spirals, forming kinematic ring-like overdensities at $M_2$050–80 au, depending on the flyby geometry (Su et al., 24 Jan 2026, Prasad et al., 15 Sep 2025). These rings are sites of strongly enhanced dust-to-gas ratios, $M_2$1, reaching $M_2$2–1 (up to 100× enhancement over background) (Prasad et al., 15 Sep 2025, Su et al., 24 Jan 2026).
• Streaming instability (SI): Dust overdensities in flyby-induced spirals often satisfy $M_2$3, where for St$M_2$410,

$M_2$5

(Prasad et al., 15 Sep 2025). For flyby-induced rings, $M_2$6 can robustly exceed $M_2$7, allowing local streaming instability on $M_2$8 timescales, resulting in gravitationally bound clumps (planetesimals, $M_2$9–$v$0 km) (Prasad et al., 15 Sep 2025, Su et al., 24 Jan 2026).

• Dependence on perturber mass: Equal-mass flybys sharply truncate the disc, form tightly wound ring-like spirals, and enhance dust concentration; low-mass flybys, in contrast, can suppress SI by diluting the local solid abundance (Su et al., 24 Jan 2026). The spatial offset between gas and dust spirals in the weak coupling regime further promotes dust trapping at pressure maxima.

4. Disc–Disc and Multistar Encounters

For interactions involving two star–disc systems on parabolic trajectories:

• Linear (tidal) regime: If the periastron exceeds approximately twice the disc radius ($v$1), only mild truncation and angular momentum exchange occurs, with

$v$2

(Muñoz et al., 2014). For $v$3, periastron distances in this regime leave discs largely intact.

• Nonlinear (colliding/grazing) regime: For $v$4, violent hydrodynamic interactions occur, leading to shock heating and rapid orbital energy loss. If the shock-heated gas cools rapidly ($v$5), the two stars can be captured into a bound binary; in rare cases, a circumbinary disc forms (Muñoz et al., 2014).
• Parameter sensitivity: Outcomes depend sensitively on mass ratio, disc mass, periastron, and orientation. For $v$6, even wider encounters can transfer sufficient angular momentum for capture.

5. Ejection of Planetesimals and Production of Interstellar Objects (ISOs)

Parabolic flybys are a primary mechanism for ejecting planetesimals into interstellar space:

• Tidal stripping and ISO yields: The tidal truncation radius $v$7 defines the region inside which disc material remains bound; outside, planetesimals are readily ejected (Pfalzner et al., 2021). The fraction of the disc unbound per encounter is $v$8; mean ejection velocities for unbound bodies are $v$9–$r$0 km/s (distinct from the higher velocities expected from planet–planet scattering or binary ejection) (Pfalzner et al., 2021).
• Cluster statistics: In typical young clusters, $r$1 of ISOs are ejected per solar-type star (ONC-like), up to $r$2 in extremely dense clusters (NGC 3603), with the Solar System likely losing $r$3–$r$4 in planetesimals to similar encounters (Pfalzner et al., 2021).
• Composition of ISOs: In "normal" clusters, ISOs are drawn from the volatile-rich outer disc and are thus predominantly comet-like; in extremely dense clusters with truncation radii interior to the CO ice line, ISOs may be volatile-depleted (Pfalzner et al., 2021).

6. Observable Signatures and Empirical Systems

Parabolic flybys imprint a characteristic suite of morphological and kinematic signatures:

• Spiral morphology: Two-armed grand-design spirals in scattered light images (μm dust), and sharper structures in mm continuum (cm-sized dust), with the two arms typically exhibiting distinct pitch angles (Cuello et al., 2018, Smallwood et al., 2023). The arms wind up and become more tightly wound binaries disappear after several local orbits ($r$5–$r$6 yr) (Smallwood et al., 2023).
• Differential truncation: The gas disc is radially larger than the dust disc by 10–20% due to the combined effect of truncation and dust radial drift (Cuello et al., 2018).
• Warps: Inclined and retrograde parabolic flybys induce warps and kinematic twists, resulting in misalignment between inner and outer disc planes of $r$710–$r$8, and up to $r$9 for strong retrograde, inclined encounters (Cuello et al., 2018, Xiang-Gruess, 2015).
• Accretion outbursts: Prograde flybys can trigger accretion bursts on $r(\theta) = \frac{2 r_p}{1 + \cos\theta},$0 yr timescales, with the accretion rate increasing by up to an order of magnitude, analogous to FU Orionis events (Cuello et al., 2018, Dong et al., 2022). In observed systems such as Z CMa, coincidence of streamers in dust and gas, alignment with a distant point-like source (the perturber), and temporally correlated bursts provide compelling evidence of a parabolic flyby "in action" (Dong et al., 2022).
• Candidate systems: Multiple discs display direct evidence for recent parabolic flybys, including UX Tau, RW Aur, FU Ori, ISO-Oph 2, and Z CMa—each showing bridges, warps, spiral arms, or misaligned discs consistent with flyby models (Cuello et al., 2022, Dong et al., 2022).

7. Solar System Case Study and Long-Term Implications

A parabolic stellar flyby offers a unified scenario for the structure of the outer solar system:

• Solar flyby parameters: The observed orbits of trans-Neptunian objects (TNOs) are best reproduced by a parabolic encounter involving a $r(\theta) = \frac{2 r_p}{1 + \cos\theta},$1 star passing at $r(\theta) = \frac{2 r_p}{1 + \cos\theta},$2 au, inclined by $r(\theta) = \frac{2 r_p}{1 + \cos\theta},$3 (Pfalzner et al., 2024). This matches simultaneously the cold classical Kuiper belt, Sedna-like objects, high-inclination/extreme TNOs, and the observed retrograde TNO population.
• Numerical modeling: Large suites of three-body integrations demonstrate that a single encounter of these parameters can populate all dynamical classes of distant TNOs, leaving the inner planetary system unperturbed inside $r(\theta) = \frac{2 r_p}{1 + \cos\theta},$4–$r(\theta) = \frac{2 r_p}{1 + \cos\theta},$5 au (Pfalzner et al., 2024). The predicted fraction of Sedna-like and retrograde TNOs increases sharply as survey depth improves.
• Stability and secular evolution: The TNO population resulting from such a flyby remains stable for over a Gyr, preserving dynamical structure, while planetary orbits remain unchanged.
• Prevalence: At least $r(\theta) = \frac{2 r_p}{1 + \cos\theta},$6 solar-type stars in the Milky Way are likely to have experienced a similar close parabolic flyby (Pfalzner et al., 2024).

Table: Key Outcomes of Parabolic Flybys

Phenomenon	Mass/Geometry Required	Observational/Physical Effect
Two-armed spirals	Any $r(\theta) = \frac{2 r_p}{1 + \cos\theta},$7, $r(\theta) = \frac{2 r_p}{1 + \cos\theta},$8 disc	Large-scale spiral patterns (dust/gas)
Disc truncation	Any, strongest for prograde	Outer disc radius reduced; gas$r(\theta) = \frac{2 r_p}{1 + \cos\theta},$9dust size
Dust rings/traps	St$r_p$01–100 grains, $r_p$1	Local $r_p$2, SI triggered
Warping/misalignment	Inclined, retrograde flyby	Disc tilt up to $r_p$3
Planetesimal ejection (ISOs)	$r_p$4, $r_p$5	0.5–2 km/s ejection vel.; Earth-mass in ISOs
Accretion outburst	Prograde, $r_p$6	$r_p$7 increase by $r_p$8 (FUor-like)

• (Cuello et al., 2018) — Flybys in protoplanetary discs: I. Gas and dust dynamics
• (Prasad et al., 15 Sep 2025) — Dust trapping in protoplanetary discs after stellar flybys
• (Pfalzner et al., 2021) — Significant interstellar object production by close stellar flybys
• (Smallwood et al., 2023) — Exciting spiral arms in protoplanetary discs from flybys
• (Cuello et al., 2022) — Close encounters: How stellar flybys shape planet-forming discs
• (Su et al., 24 Jan 2026) — Evolution of dust in a protoplanetary disc driven by stellar flybys: implications for the streaming instability
• (Dong et al., 2022) — A likely flyby of binary protostar Z CMa caught in action
• (Pfalzner et al., 2024) — Trajectory of the stellar flyby that shaped the outer solar system
• (Xiang-Gruess, 2015) — Generation of highly inclined protoplanetary discs through single stellar flybys
• (Muñoz et al., 2014) — Stellar orbit evolution in close circumstellar disc encounters

Parabolic stellar flybys constitute a robust mechanism for sculpting the structure, dynamics, and planet-forming potential of circumstellar discs. Their signatures are imprinted on observed disc substructure, outer solar system architecture, the population of interstellar objects, and the dynamical pathways available in dense stellar environments.