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

This paper demonstrates that while particle-mesh approximations and MOND modifications destabilize the three-body system by breaking fundamental symmetries and conservation laws, a specific MOGA modification of gravity can actually stabilize the system.

The Gist

Imagine you are trying to predict the dance of three partners in a ballroom: a heavy, slow-moving lead (a black hole) and two lighter, faster dancers (stars) spinning around it. In a perfect world, governed by Isaac Newton's laws, these three would dance together forever in a beautiful, predictable pattern. They would never bump into each other, and they would never fly off the dance floor.

This paper is about testing what happens when we try to simulate this dance on a computer, and whether changing the "rules of the dance" (the laws of physics) or taking shortcuts in the simulation breaks the rhythm.

Here is the breakdown of the paper using simple analogies:

1. The Problem: The Computer Can't Count to a Billion

When scientists try to simulate a whole galaxy (which has billions of stars), it's too hard for a computer to calculate the gravitational pull of every single star on every other star at every moment. It would take too long.

So, they use a shortcut called the Particle-Mesh (PM) approximation.

• The Analogy: Imagine you are in a crowded room. To know where everyone is pushing you, you don't look at every individual person. Instead, you look at the "average push" coming from a specific corner of the room.
• The Result: This shortcut is fast, but it's slightly inaccurate. The paper shows that even a tiny inaccuracy in how we calculate these "average pushes" eventually causes the dance to fall apart. The stars stop dancing in a circle and start flying off the floor.

2. The Mystery: Why Do Galaxies Spin So Fast?

Real galaxies spin much faster than Newton's laws predict. If you only count the visible stars, the outer stars should be flying off into space because they aren't moving fast enough to stay in orbit. But they don't fly off; they stay in a tight circle.

To fix this, scientists have proposed two main ideas:

1. MOND (Modified Newtonian Dynamics): Change the rules of acceleration. Maybe gravity works differently when things are far apart.
2. Modified Gravity (Yukawa/MOGA): Change the strength of the pull. Maybe gravity gets a little stronger at huge distances than Newton thought.

3. The Experiment: The Three-Body Test

The author, Søren Toxvaerd, decided to test these ideas on the simplest possible system: The Three-Body System (TBS).

• The Setup: One heavy "Sun" and two lighter "Planets" orbiting it.
• The Goal: See if the dance stays stable (regular) or falls apart (chaotic) when we apply the shortcuts (PM) or the new rules (MOND, Yukawa, MOGA).

4. The Findings: What Worked and What Broke

❌ The Shortcuts and "Acceleration" Rules (PM & MOND)

• The PM Shortcut: As mentioned, taking the "average push" from far away breaks the symmetry of the dance. It's like if the music suddenly skipped a beat. The system loses its balance, and the stars fly away.
• The MOND Rule: This rule says, "If you are far away, push harder." However, the paper found that MOND breaks a fundamental rule of physics called Newton's Third Law (for every action, there is an equal and opposite reaction).
  • The Analogy: Imagine two dancers holding hands. If one pulls the other, the other must pull back with the exact same force. MOND is like a dancer who pulls hard but doesn't pull back equally. This breaks the "momentum" of the dance.
  • Result: The system becomes unstable. The stars eventually get kicked out of the orbit.

✅ The "Stronger Pull" Rules (Yukawa & MOGA)

• The Yukawa and MOGA Rules: These rules say, "If you are far away, the pull is slightly stronger than Newton thought, but it still follows the rules of physics."
  • The Analogy: Imagine the dancers are connected by a rubber band that gets a little stiffer when they stretch it far apart, but they still pull on each other equally.
  • Result: This stabilizes the system! The stars stay in their orbits, and the dance continues. In fact, the orbits become slightly "elliptical" (oval-shaped) and rotate slowly, which is a very healthy, stable state.

5. The Big Conclusion

The paper concludes that:

1. Shortcuts are dangerous: The common computer shortcuts used to simulate galaxies (PM) might be the reason our simulations look unstable, not necessarily because the universe is chaotic.
2. Changing the "Push" vs. Changing the "Pull":
   • If you change how things accelerate (MOND) without keeping the "equal and opposite" rule, the universe falls apart.
   • If you change how strong the gravity is (Yukawa/MOGA) while keeping the rules of physics intact, the universe stays stable.

In a nutshell: The universe seems to prefer a "stronger pull" at long distances (like a stiffer rubber band) rather than a "weird acceleration rule" that breaks the balance of forces. If we want to simulate galaxies correctly, we might need to stop using the "average push" shortcuts and instead use these "stronger pull" gravity rules.

Technical Summary

1. Problem Statement

The paper addresses a fundamental discrepancy in astrophysics: classical Newtonian dynamics, when applied to large-scale galaxy simulations, predicts Keplerian rotation velocities (decreasing with distance) that contradict observed galactic rotation curves (which remain flat or constant). To resolve this, two main approaches are used in simulations:

1. Approximations: Large-scale $N$-body simulations often use Particle-Mesh (PM) or Tree-PM methods to approximate the gravitational forces from distant objects to save computational cost.
2. Modifications: Theories like Modified Newtonian Dynamics (MOND/QuMOND) or modified gravity laws (e.g., Yukawa, MOGA) are introduced to alter accelerations or gravitational attractions at large interstellar distances.

The core problem investigated is whether these necessary approximations (PM) or theoretical modifications (MOND, Yukawa, MOGA) inadvertently destabilize the regular, stable dynamics of celestial systems. While these methods aim to explain galactic rotation, they may introduce numerical or physical instabilities that destroy the very orbits they seek to model.

2. Methodology

To isolate the effects of approximations and modifications without the noise of complex $N$-body interactions, the author employs a Three-Body System (TBS) as a testbed.

• System Configuration: A "Dwarf galaxy" model consisting of:
  • Object 1 (Inner star): Mass $m_1 = 1 M_{\odot}$.
  • Object 2 (Outer star): Mass $m_2 = 0.5 M_{\odot}$.
  • Object 3 (Central mass/Black hole): Mass $m_3 = 100 M_{\odot}$.
  • Initial conditions are set to create regular, stable elliptical orbits around the center of mass.
• Baseline Dynamics: The system is simulated using Newton's discrete algorithm (Leap-frog method), which is time-reversible, symplectic, and conserves momentum, angular momentum, and energy exactly.
• Test Variables: The baseline is compared against four scenarios:
  1. PM Approximation: Forces from objects beyond a cutoff distance $r_0$ are interpolated onto a grid ($l_{grid}$), breaking exact pairwise symmetry.
  2. MOND: Acceleration is modified at low accelerations ($a < a_0$) using interpolation functions, violating Newton's third law.
  3. Yukawa Modification: The gravitational potential is modified by a Yukawa term ($e^{-r/\lambda}$), altering the force law while theoretically preserving symmetries.
  4. MOGA (Modified Gravitational Attraction): The inverse-square law is replaced by an inverse law ($1/r$) at large distances, derived from lensing/focusing hypotheses.
• Simulation Parameters: Simulations run for up to $10^{10}$ time steps with $\delta t = 100$. Key parameters include $r_0 = 500,000$ (distance units) and various grid lengths or modification strengths ($\alpha, \lambda$).

3. Key Contributions

• Stability Benchmarking: The paper establishes the TBS as a rigorous, computationally efficient benchmark for testing the stability of celestial dynamics under non-standard conditions.
• Identification of Instability Sources: It distinguishes between instabilities caused by numerical approximations (PM) versus physical model modifications (MOND vs. MOGA/Yukawa).
• Conservation Law Analysis: The study explicitly links system stability to the conservation of momentum and angular momentum. It demonstrates that methods violating Newton's Third Law (action-reaction symmetry) inevitably lead to the destruction of regular orbits over long timescales.

4. Results

The simulations yielded distinct outcomes for each modification:

• Particle-Mesh (PM) Approximation:

  • Result: Destabilizing.
  • Observation: Even with a large cutoff distance ($r_0 = 500,000$) and fine grids, the PM approximation breaks the symmetry of pair interactions ($f_{ij} \neq -f_{ji}$). This leads to a gradual loss of momentum and angular momentum conservation.
  • Outcome: The outer object (No. 2) eventually deviates from its regular orbit and is ejected from the system after $\approx 1.5 \times 10^8$ time steps.

• MOND (Modified Acceleration):

  • Result: Destabilizing.
  • Observation: The modification of acceleration based on local field strength violates Newton's Third Law (Eq. 11 in the paper).
  • Outcome: Similar to PM, the violation of momentum conservation causes the regular dynamics to collapse. The outer object is ejected after $\approx 1.25 \times 10^9$ time steps. The system is highly sensitive to this modification.

• Yukawa Modification:

  • Result: Stabilizing (conditionally).
  • Observation: This modifies the force law but preserves Newton's Third Law and all classical invariants.
  • Outcome: For moderate modification strengths ($\alpha = 0.5$), the system remains stable with regular orbits. However, if the modification strength is too high ($\alpha = 1$), the increased attraction can accelerate the object out of the system, though this is a dynamic effect rather than a numerical instability.

• MOGA (Modified Gravity to Inverse Law):

  • Result: Stabilizing.
  • Observation: Replacing the inverse-square law with an inverse law ($1/r$) for distant interactions preserves momentum and angular momentum.
  • Outcome: The system remains stable for $10^9$ time steps. The orbits transform into "revolving" ellipses (precessing orbits) but do not disintegrate. The system maintains long-term regularity.

5. Significance and Conclusion

• Validation of Approximations: The paper suggests that while PM approximations are computationally necessary, they introduce subtle errors (momentum non-conservation) that accumulate to destabilize galactic models over cosmological timescales.
• Critique of MOND: The study argues that standard MOND formulations, which modify acceleration rather than force, violate fundamental conservation laws (Newton's 3rd Law), leading to intrinsic instability in multi-body systems. This challenges the viability of MOND as a stable classical dynamics framework without further theoretical refinement.
• Support for Modified Gravity (MOGA/Yukawa): Modifications that alter the force law (rather than the acceleration definition) while preserving symmetries (like MOGA and Yukawa) successfully stabilize the system. These models produce rotation profiles consistent with observations while maintaining the mathematical integrity of the N-body system.
• Theoretical Implication: The author proposes that the stability of galaxies might rely on a self-stabilizing mechanism (such as gravitational wave lensing) that effectively modifies gravity at large scales (MOGA), rather than relying on ad-hoc acceleration modifications or numerical approximations that break conservation laws.

In summary, the paper concludes that preserving Newton's Third Law and conservation of momentum is critical for the long-term stability of celestial systems. Approximations (PM) and acceleration-based modifications (MOND) fail this test, whereas force-based modifications (MOGA, Yukawa) succeed.