Approximations and modifications of celestial dynamics tested on the three-body system

TL;DR

The study addresses the stability of celestial dynamics under common large-scale simulation approximations. It assesses particle-mesh ($PM$) schemes and gravity modifications, notably MOND and MOGA, using the simple three-body system (TBS) governed by the inverse-square law $F \propto 1/r^2$, with Newton's discrete algorithm as the reference. The results show that $PM$ and MOND perturb regular TBS orbits and violate conservation of momentum and angular momentum, while the MOGA modification preserves dynamical invariances and stabilizes the system, producing revolving yet bounded orbits. The work highlights the sensitivity of regular celestial dynamics to modeling choices, cautions against naive extrapolation from a three-body test to galactic scales, and suggests that MOGA-like approaches may reconcile observed rotation features with stability, albeit at significant computational and theoretical cost.

Abstract

Large-scale simulations of celestial systems are based on approximations or modifications of classical dynamics. The approximations are with ``particle-mesh'' (PM) substitutions of the attractions from objects far away, or one modify the Newtonian accelerations (MOND) or the gravities (MOGA). The PM approximation and MOND modification of classical dynamics break the invariances of classical dynamics. The simple three-body system (TBS) is the simplest system to test the approximations and modifications of celestial dynamics, and it is easy to implement on a computer. Simulations of the TBS show that the PM approximation and MOND destabilize TBS. In contrast, the MOGA modification of gravity by replacing Newton's inverse square attraction with an inverse attraction for far-away interactions stabilizes the system. The PM approximation and the MOND modification of classical dynamics do not preserve the momentum and angular momentum of a conservative system exactly, and PM does not obey Newton's third law. Although the errors and shortcomings of these PM approximations and MOND modifications are small, they cause the instability of the regular dynamics.

Figures (6)

• Figure 1: The regular orbits in a TBS system with two light objects around a heavy object, and for $10^7$ discrete timesteps. The $\approx$ 45 orbits in green is for the object No. 1 with start position at its aphelion (green dot), and with blue is the corresponding $\approx$ four orbits for the object No. 2 with start position at its perihelion (blue dot). The red "dot" is the orbits of the heavy object No. 3 with start position at the center of mass (origin), and the inset shows the $\approx$ four orbits of the object.
• Figure 2: The distances $r_{12}(t)$ (green), $r_{13}(t)$ (red), and $r_{23}(t)$ (blue) between the three objects as a function of time for the regular orbits, shown in Figure 1. The thick black straight line is the borderline for the PM approximation, $r_0$=500000, and the thinn black lines are $r_0 \pm l_{grid}$ with $l_{grid}$= 10000, used in the PM approximation.
• Figure 3: The dynamics of TBS with the PM approximation with $r_0=500000$ and $l_{grid}$=10000. The orbits are for $10^7$ timesteps. The blue orbits are for No. 2. The thick blue ellipse is without PM approximation (also shown in Figure 1), and the orbits with thin blue are with the PM approximation. No. 1's orbits are in green with or without PM (the differences are not visible on the figures).
• Figure 4: The dynamics of TBS, also shown in Figure 3, but for 1.49899436$\times 10^8$ timesteps with PM, and when No. 2 was released from the TBS.
• Figure 5: The MOND modification for $a_0=4 \times 10^{-12 }$ (corresponding to $r_0=500000$ in MOGA and PM). The orbits with MOND dynamics are shown in thin blue for No. 2, and in thin black for No. 1. The position of No. 2 at $t=1.2576 \times 10^9$ at its release is marked with a blue sphere. The ellipses with thick blue and green (from Fig. 1) are without MOND modification.