A Two-Dimensional Analytic Solution for the Generation of Hyperbolic Trajectories Via A Single Close Encounter with Applications To Interstellar Objects

TL;DR

The paper develops a closed-form, Opik-style analytic framework to map pre-encounter orbital elements to post-encounter, ejected trajectories in a restricted three-body setup, focusing on single close encounters with a massive perturber. A key result is a simple ejection predictor $\beta=U^2+2U\cos(\gamma-\delta)$, with ejection corresponding to $\beta\ge1$ (i.e., $e'>1$), and an emphasis on the y-component of the planet-centric velocity as the main driver of ejection. The methodology is validated against numerical simulations and applied to Solar System analogues and exoplanetary systems (β Pictoris, HR 8799), illustrating where ejection is most efficient and how initial eccentricity $e$ above ~0.4 enhances ejection probability. The framework offers a fast, transparent tool for identifying ejection-prone reservoirs in planetary systems, aiding interpretations of interstellar object production and guiding observational expectations, while acknowledging planar limitations and the need for full 3D extensions for complex architectures.

Abstract

The discovery of interstellar interlopers such as 1I/`Oumuamua, 2I/Borisov, and 3I/ATLAS have highlighted the necessity of understanding the dynamical pathways that eject small bodies from planetary systems into hyperbolic trajectories. In this paper we examine the orbital elements of particles in the restricted three-body problem prior to and post scattering onto hyperbolic trajectories by massive perturbers. Building on previous work, we calculate closed-form -- but approximate -- analytic criteria that map pre- to post-encounter orbital elements. An application of these equations demonstrates that ejection occurs most efficiently when the orbital eccentricity of the massless test particle exceeds a minimum threshold, $e\gtrsim0.4$. The primary driver of the final eccentricity is the component of the perturber-centric velocity projected along the direction of motion of the perturber. These analytic criteria are then benchmarked and validated against numerical simulations which demonstrate that they provide a reasonably good zeroth-order approximation for ejection behavior. However, system-specific cases will generally require numerical simulations in addition to this analytic construction. The methodology is applied to (i) the solar system and exoplanetary systems (ii) $β$ Pictoris and (iii) HR 8799 to evaluate the pre-scattering orbits of ejected particles. This method provides a transparent and computationally efficient tool for identifying orbits within a given system from which interstellar objects are efficiently ejected via a single scattering event from a massive perturber.

Figures (10)

• Figure 1: Post-encounter eccentricity of a massless test particle that scatters off of a Jupiter analog. Color indicates the final eccentricity as a function of initial eccentricity and semimajor axis for interactions with a Jupiter-mass perturber at an impact parameter of 10 $R_J$. The blank sections of the parameter space correspond to non-crossing orbits, where $U_x$ is undefined by Equation \ref{['eq:U_component_matrix']}.
• Figure 2: Each subpanel is similar to that shown in Figure \ref{['fig:jupiter_ejections']} for a range of impact parameters between 10-400 $R_J$. The black, dashed lines follow $\beta=1$. $e'$ decreases for more distant interactions. At 400 $R_J$, almost no orbits with $a\in[0, 40]$ au are ejected
• Figure 3: Higher eccentricity initial trajectories are more efficiently ejected. The color indicates the total fraction of orbits that are ejected for impact parameters ranging from $R_p$ to $R_H$.
• Figure 4: The distribution of particles ejected from the suite of simulations of massless test particles interacting with a Jupiter analogue perturber. 1072 particles were ejected and 66 particles on highly eccentric initial orbits, $e>0.97$, were excluded. The numerical simulations provide nominal agreement with the analytic theory shown by the $\beta=1$ line. The overplotted $\beta=1$ contours are calculated from a range of $b$-values logarithmically sampled from [$R_J$,$R_H$].
• Figure 5: Evolution of the orbit of a massless test particle ejected by a Jupiter analogue. (top left) Top down view of the star, planet, and particle. The particle starts on a more distant orbit that leaves the frame of the plot, before being redirected onto a much closer orbit and is then ejected. The arrows indicate the direction of the orbits. (top right) Top down view centered on the massive perturber. The Hill Sphere of the perturber is shown as a dotted line, and the path of the particle is shown in red. Circled numbers indicate the incoming and outgoing paths of the first and second close encounters. (bottom left) The temporal evolution of semimajor axis and eccentricity of the ejected particle in Earth years. Both heliocentric and barycentric coordinates are shown. (bottom right) Zoomed out version of the top down orbits of the star and particle. An animated version of this figure is available online.