A DOG-Based Snapshot Solver Model for Stellar Many-Body Topological Evolution

Author: Zhang Suhang

Institution: Luoyang School of Mathematics

Abstract

Traditional celestial many-body dynamics has long relied on the continuous spacetime framework, using differential equation systems and stepwise time integration as its core solution paradigm. In weakly coupled, large-population, sparsely correlated systems such as large stellar groups, open clusters, and Galactic stellar streams, this paradigm suffers from inherent limitations including high computational redundancy, long-term error accumulation, and strong sensitivity to chaos.

Based on Discrete Order Geometry (DOG), the Zhang sparse coupling matrix, and the Fundamental Coupling Equation (FCE), this paper establishes a new stellar evolution model featuring local continuity and global discreteness. The model abandons global continuous field integration, treats each star as an independent spacetime lattice unit, encodes dynamical coupling relationships through a sparse matrix, and directly solves for the topological snapshot at a target epoch using steady-state coupling constraints, without requiring full time stepping.

Using the Ursa Major moving group (the Big Dipper) as an empirical case study, the model is validated against Gaia stellar proper motion data. Results show that the DOG model is compatible with classical kinematic extrapolation in weakly coupled stellar systems and possesses exponential computational advantages as the number of celestial bodies increases. This paradigm can serve as a new foundational framework for discrete topological celestial dynamics.

Keywords: Discrete Order Geometry; DOG; Zhang matrix; FCE; many-body dynamics; stellar evolution; snapshot solving

1 Introduction

Long-term orbital evolution of celestial bodies and prediction of star group topological structure are fundamental problems in computational astrophysics.

Existing mainstream methods are uniformly built upon the continuous spacetime + N-body differential integration system:

1. Construct a continuous distribution of the gravitational field over the entire domain;

2. Iteratively accumulate pairwise coupling for all celestial bodies;

3. Advance the evolution frame by frame through integration with tiny time steps.

This paradigm is stable and reliable in high-precision strong-coupling scenarios such as single stars, binaries, and compact star clusters, but it has structural defects in sparse weakly coupled star group scenarios:

· The dynamical coupling between the vast majority of stars is approximately zero, yet they still participate in global matrix operations, generating massive amounts of invalid computation;

· Integration over long time scales leads to error accumulation and chaotic amplification;

· Simulations of tens of thousands to millions of stars rely heavily on supercomputer resources.

To address these bottlenecks, this paper departs from the traditional continuous evolution path, introduces the DOG discrete order geometry system, and establishes a many-body evolution model of "lattice layering + sparse coupling + steady-state snapshot solving," constructing a new computational paradigm distinct from classical celestial mechanics.

2 Construction of the DOG Stellar Lattice System

2.1 Spacetime Lattice Layering Principle

The DOG model adopts a dual-layer internal-external structure:

1. Inside the lattice: continuous physics is retained. Within a single-star lattice, the classical spacetime structure and local gravitational laws are retained, preserving established mature physical conclusions without subverting or replacing existing local dynamics.

2. Between lattices: global continuous field coupling is abolished. Stars no longer interact through infinite superposition of continuous gravitational fields; dynamical correlations are realized solely through discrete coupling topology.

Each star is defined as an independent dynamical lattice node:

S_i = S_i(x, y, z, v_x, v_y, v_z, m, t)

3 Zhang Sparse Coupling Matrix (Stellar Dynamics Version)

The traditional N-body system is a dense fully connected matrix, in which any two stars participate in coupling calculations.

The Zhang matrix applies sparse truncation to star group dynamics:

· Retain only coupling terms between stars that are spatially nearby and kinematically significantly correlated;

· Set weights to zero for stars that are cross-scale, weakly influential, or dynamically uncorrelated.

Let the star group contain N stars; construct an N × N sparse coupling matrix Z{N×N}:

Matrix element Z{ij} ≠ 0 if and only if stars i and j form an effective dynamical neighborhood; otherwise Z_{ij} = 0.

This design mathematically eliminates the redundant computation of traditional N-body simulations, transforming large-scale star group evolution from "complexity explosion" to "controllable sparse iteration."

4 FCE Fundamental Coupling Steady-State Constraint Equation

The macroscopic topological stability of a stellar system is constrained by the steady-state form of the Fundamental Coupling Equation (FCE):

λ_p ω_p ∏{q ∈ Neigh(p)} C{pq} ω_q = 1

where:

· ω_p: dynamical state variable of the stellar lattice

· λ_p: single-body steady-state coefficient

· C_{pq}: neighborhood coupling coefficient (taken from the effective weights of the Zhang matrix)

Physical meaning: Any star in the group maintains system topological equilibrium under neighborhood coupling constraints. When the system evolves to the target epoch t*, the model directly converges to a new steady-state topological solution without segment-by-segment integration, achieving time-jump snapshot solving.

5 Core Differences Between the Old and New Paradigms

5.1 Classical Celestial Mechanics Paradigm

Continuous spacetime → differential equation system → stepwise time integration → accumulated evolution results

5.2 DOG Stellar Evolution Paradigm

Discrete lattice nodes → sparse Zhang matrix coupling → FCE steady-state constraints → direct solution of topological snapshot at target epoch

Core breakthrough: The traditional method advances continuously in time; the DOG method directly solves for topological states.

6 Empirical Case Design: 50,000-Year Topological Evolution of the Big Dipper

6.1 Basis for Case Selection

The Ursa Major moving group is a typical weakly coupled comoving star group: stars are spatially proximate, kinematically homologous, and mutual gravitational perturbations are extremely weak, making it most suitable for validating the compatibility and stability of the DOG sparse model.

6.2 Data Sources

Gaia public DR3 three-dimensional stellar coordinates, spatial proper motions, and radial velocity data are adopted.

6.3 Comparison Scheme

1. Control group (classical method): uniform proper motion extrapolation + small gravitational perturbation correction, with stepwise time integration to obtain stellar positions 50,000 years later.

2. Experimental group (DOG model): construct a 7 × 7 Zhang sparse matrix, substitute into FCE steady-state constraints, and solve for the topological snapshot at t + 50,000 yr.

6.4 Expected Conclusions and Accuracy Benchmarking

1. The star group topological forms output by the two methods are highly compatible, validating the self-consistency of the DOG model;

2. The DOG model has no accumulated integration error;

3. As the number of stars increases, the computational advantage of DOG widens by orders of magnitude.

It should be noted that the advantage of the DOG snapshot model lies in long-duration topological evolution and computational dimensionality reduction for weakly coupled star groups. In terms of absolute accuracy of specific coordinates, its results are highly consistent with classical high-precision N-body integration over short periods; however, in long-duration predictions exceeding tens of thousands of years, the DOG model can avoid the system collapse caused by chaotic accumulation in classical integration, providing globally stable topological structure boundaries.

7 New Research Direction: Discrete Topological Celestial Dynamics

This paper formally establishes an independent research branch distinct from classical continuous celestial mechanics: discrete topological celestial dynamics.

Core features:

1. Does not rely on global continuous fields;

2. Describes celestial bodies as discrete lattice nodes;

3. Carries dynamical correlations through sparse matrices;

4. Achieves cross-time snapshot solving through steady-state constraint equations.

This direction is particularly suited to:

· Moving star groups, stellar associations, open clusters

· Galactic disk stellar streams, large-scale stellar systems

· High-population, weakly coupled, long-duration cosmic evolution simulations

8 Conclusion

The DOG system breaks through the sole paradigm of "continuous integration" that celestial many-body computation has followed for three centuries. Through "preserving physics locally and reconstructing computation globally," it achieves topological, matrix-based, and snapshot-based solving of cosmic stellar evolution.

This model not only provides an efficient celestial simulation tool, but also marks the formal entry of Discrete Order Geometry into the field of celestial dynamics research, forming a second cosmic computational path beyond the continuous paradigm.