Merger Dynamics of N+N Co-planar Particles in Newton Gravitation

Abstract

We model the inspiral and merger dynamics of two co-planar rings in Newtonian mechanics with GR motivated corrections and illustrate their similarity with those of black hole binary systems on the orbital plane. Our simulation reveals a banana-shape deformation of the ``black holes'' involved, and a typhoon-like spiral structure in the merger product. Using an eXact One-Body approach, we compute the full gravitational waveform of this process and qualitatively reproduce results consistent with those of numerical relativity. Our simulation offers a transparent link between the feature of gravitational waveforms and the internal structure of black holes, thus a complementary interpretation of physics behind numerical relativity.

Figures (6)

• Figure 1: Evolution of the ring structures during binary merger under varying values of the parameter $K$. A higher $K$ indicates stronger GW back-reaction, leading to faster orbital decay and more pronounced deformation.
• Figure 2: Influence of particle number $n$ on the deformation of ring profile. As $n$ increases, the system evolves with greater spatial resolution but reduced numerical stability, shortening the duration of coherent evolution. The trend of the four sub-figure reflects the finite-N nature of the toy model; it is not intended to represent a continuum limit, but to illustrate the qualitative patterns. Fig. 2-4 are represent 10 uniformly spaced time snapshots between $t=0$ and $t=T$, shown with a purple–yellow gradient indicating time evolution.
• Figure 3: Time evolution of the ring configuration for varying values of $L$ with fixed particle number $n=8$. Asymmetric banana-shape deformations emerge due to uneven GW back-reaction and internal gravitational interactions.
• Figure 4: Deformation of the system for varying initial radius ratios $r_0/a_0$ and evolution times $T$. Top row: comparison of different $r_0/a_0$ at fixed times, showing stronger orbit contraction for smaller ratios. Bottom row: time evolution at fixed $r_0/a_0$, where the system gradually contracts into a spiral structure before stabilizing.
• Figure 5: Fitting results of circumference of the ring over time. The coefficient of determination is $R^2=0.999549$.