[Paper Review] Chaotic motion and ballistic ejection of gravitating shells

Maxim V. Barkov, Vladimir Belinski+1 more|arXiv (Cornell University)|Jul 3, 2001

Cosmology and Gravitation Theories7 citations

TL;DR

This paper models ballistic ejection of gravitating spherical shells in Newtonian gravity, showing that energy transfer during shell intersections can expel one shell at subrelativistic speeds up to $ v_{\text{exp}} \approx 0.25c $, with maximum ejection efficiency at mass ratio $ m/M \approx 1 $. The system exhibits chaotic dynamics due to sensitive dependence on initial conditions and strong energy exchange between shells.

ABSTRACT

It is shown that during the motion of two initially gravitationally bound spherical shells, consisting of point particles moving along ballistic trajectories, one of the shell may be expelled to infinity at subrelativistic speed $v_{exp}\leq 0.25 c$. The problem is solved in Newtonian gravity. Motion of two intersecting shells in the case when they do not runaway shows a chaotic behaviour. We hope that this toy and oversimplyfied model can give nevertheless a qualitative idea on the nature of the mechanism of matter outbursts from the dense stellar clusters.

Motivation & Objective

• To investigate whether gravitational interactions between two massive, gravitationally bound spherical shells can lead to ballistic ejection of one shell to infinity.
• To quantify the maximum ejection velocity achievable via pure gravitational energy transfer in a simplified Newtonian model.
• To explore the emergence of chaotic dynamics in the motion of intersecting gravitating shells.
• To assess the relevance of this mechanism as a possible origin for high-velocity stars and intergalactic matter ejection in dense stellar systems.

Proposed method

• Model two spherical shells of point particles with equal specific angular momentum and energy, moving ballistically in a central gravitational potential.
• Use Newtonian equations of motion for spherical shells, incorporating gravitational interactions between shells and a central supermassive black hole (SBH).
• Apply energy and angular momentum conservation to analyze energy transfer during shell intersections.
• Introduce a minimum radius cutoff $ r_m \sim 3r_g $ to prevent collapse into the SBH, modeling realistic physical limits.
• Perform numerical simulations of shell trajectories to observe ejection and chaotic behavior under varying mass ratios.
• Analyze sensitivity to initial conditions to assess chaotic dynamics, including perturbations in integrator precision and angular momentum.

Experimental results

Research questions

• RQ1 Can pure gravitational interactions between two gravitating shells lead to ballistic ejection of one shell at subrelativistic speeds?
• RQ2 What is the maximum ejection velocity achievable via gravitational energy transfer in this system?
• RQ3 How does the mass ratio between the shells and the central SBH affect the ejection efficiency and energy transfer?
• RQ4 Under what conditions does the system exhibit chaotic motion due to repeated shell intersections?
• RQ5 How sensitive is the system's dynamics to small changes in initial conditions, indicating chaos?

Key findings

• The maximum ejection velocity reaches $ v_{\text{max}} \approx 0.3547v_p $ when the shell mass equals the central black hole mass ($ m/M = 1 $), corresponding to $ v_{\text{exp}} \approx 0.25c $.
• For equal-mass shells, the ejection velocity peaks at approximately 0.25 times the speed of light, indicating significant energy transfer is possible via gravitational interaction alone.
• Chaotic motion emerges when shells have comparable mass, leading to strong, unpredictable energy exchange and trajectory changes after each intersection.
• The system exhibits extreme sensitivity to initial conditions: changing integrator precision from $ 10^{-6} $ to $ 3 \times 10^{-7} $ or angular momentum by 0.7% leads to drastically different long-term behavior.
• When shell masses are large relative to the central mass, the motion becomes fully chaotic, with shells exchanging positions and orbits becoming highly irregular, as seen in simulations with $ m/M = 0.08 $.
• In the absence of a central mass, two self-gravitating shells still exhibit chaotic motion, demonstrating that chaos arises from mutual gravitational interactions alone.