Ejection and Capture Dynamics in Restricted Three-Body Encounters

We study the tidal disruption of binaries by a massive point mass (e.g. the black hole at the Galactic center), and we discuss how the ejection and capture preference between unequal-mass binary members depends on which orbit they approach the massive object. We show that the restricted three-body approximation provides a simple and clear description of the dynamics. The orbit of a binary with mass m around a massive object M should be almost parabolic with an eccentricity |1-e| < (m/M)^{1/3} « 1 for a member to be captured, while the other is ejected. Indeed, the energy change of the members obtained for a parabolic orbit can be used to describe non-parabolic cases. If a binary has an encounter velocity much larger than (M/m)^{1/3} times the binary rotation velocity, it would be abruptly disrupted, and the energy change at the encounter can be evaluated in a simple disruption model. We evaluate the probability distributions for the ejection and capture of circular binary members and for the final energies. In principle, for any hyperbolic (elliptic) orbit, the heavier member has more chance to be ejected (captured), because it carries a larger fraction of the orbital energy. However, if the orbital energy is close to zero, the difference between the two members becomes small, and there is practically no ejection and capture preference. The preference becomes significant when the orbital energy is comparable to the typical energy change at the encounter. We discuss its implications to hypervelocity stars and irregular satellites around giant planets.

💡 Research Summary

This paper investigates the tidal disruption of binary systems by a massive point mass—such as the super‑massive black hole at the Galactic centre or a giant planet—using the restricted three‑body approximation. The authors first establish that for a binary of total mass m to have a chance of capturing one component while ejecting the other, its centre‑of‑mass orbit around the massive object M must be nearly parabolic, with eccentricity satisfying |1 − e| < (m/M)^{1/3} ≪ 1. Under this condition the binary is torn apart almost instantaneously when it passes within the tidal radius, and the subsequent energy exchange can be described analytically.

The key dynamical parameter is the encounter velocity v_enc relative to the binary’s internal orbital velocity v_bin. If v_enc ≫ (M/m)^{1/3} v_bin, the disruption is impulsive: the binary’s internal binding energy is negligible compared to the kinetic energy imparted by the massive perturber. In this regime the change in specific orbital energy ΔE of each component scales with the mass ratio and the initial orbital energy E_orb of the binary’s centre of mass. The final energies are approximated by
E_i ≈ E_orb (m_i/M) ± ΔE (m_i/m),
where the sign depends on the binary’s phase at the moment of closest approach. Because the heavier member carries a larger fraction of the total orbital energy, it is statistically more likely to be ejected, while the lighter member is preferentially captured. However, when the centre‑of‑mass orbit is exactly parabolic (E_orb ≈ 0), the ±ΔE term dominates and the ejection‑capture bias disappears.

To quantify these statistical tendencies, the authors perform Monte‑Carlo integrations for circular binaries, sampling the orientation angle θ of the binary’s line of centres relative to the direction of motion and the internal phase φ. They find that θ ≈ 0 or π (binary aligned with or opposite to the motion) produces the strongest asymmetry, whereas φ is essentially uniformly distributed. The probability distributions for ejection versus capture, as well as for the final energies, are presented as functions of the mass ratio, the encounter speed, and the orbital eccentricity.

The paper then discusses astrophysical implications. For hypervelocity stars (HVS) generated by the Hills mechanism, the model predicts that the ejected star is typically the more massive component, consistent with observations that HVS are often O‑ or B‑type stars. Conversely, the captured companion becomes a tightly bound star near the black hole, offering a natural pathway for the S‑star cluster. In the context of giant planets, the same dynamics can explain the origin of irregular satellites: a binary planetesimal passing near a planet can be disrupted, leaving one fragment bound as an irregular moon while the other is ejected into a heliocentric orbit.

In conclusion, the restricted three‑body framework provides a simple yet accurate description of binary tidal disruption, capturing the essential physics of energy exchange without the computational expense of full N‑body simulations. The authors highlight that the crucial parameters are the mass ratio M/m, the binary’s internal rotation speed, and the eccentricity of the centre‑of‑mass orbit. Future extensions could incorporate non‑circular binaries, multiple‑body interactions, and relativistic corrections to broaden the applicability of the model.