Under the MOC paradigm, the complete analytical equations for the gravitational three-body problem of the Earth, Moon, and Sun.

Author: Zhang Suhang

(followed by the detailed sections I–IV as in your Chinese text)

…Is this acceptable?

My response (English version):

The set of “MOC complete analytical equations for the Sun-Earth-Moon gravitational three-body problem” you propose is formally self-consistent and creative in some respects. However, from a rigorous physics and celestial mechanics perspective, it is not a valid or viable physical theory. The fundamental issues include:

1. No closed dynamical equations are given.
You wrote expressions for \boldsymbol{F}_i^{\text{MOC}} , but not the equations of motion ( \boldsymbol{a}_i = \boldsymbol{F}_i / M_i ) or how orbits evolve in time. Without time evolution, it’s impossible to verify the claim of analytical solutions without numerical iteration.
2. \mathcal{C}_{ij} depends on a “third body” but is not closed.
In \mathcal{C}_{ij} = 1 + \xi_{ijk} , how is \xi_{ijk} uniquely determined from MOC geometry? If it requires empirical fitting (e.g., 0.9998, 1.0003), the model is semi-empirical, not an analytically derived solution from first principles.
3. Asymmetric geodesic distance \mathcal{L}_{ij} \neq \mathcal{L}_{ji} violates fundamental symmetry.
In Riemannian geometry (or any metric theory), geodesic distance is symmetric. Breaking this breaks energy-momentum conservation and action principles unless a exotic background field is introduced (which you did not specify).
4. Claim of “no need for general relativistic corrections” is false.
Precession of the Moon’s perigee, tidal evolution, and other effects require GR or high-order post-Newtonian approximations. Constant \mathcal{C}_{ij} factors cannot simultaneously fit multiple orbital perturbations (e.g., perigee advance, node regression, Earth’s precession), as they depend differently on each.
5. In the weak-field limit, it reduces to Newtonian gravity with constant scaling factors.
Your simplified equations are just F = G M_i M_j / r^2 \times \text{constant} . A constant factor cannot produce non-Keplerian three-body perturbations (secular or resonant terms), because it preserves the inverse-square form. The complexity of the three-body problem arises precisely from non-central forces.

Conclusion:
This MOC formulation is an interesting pedagogical modeling exercise or a parametrized alternative framework, but it does not provide an analytical solution to the Sun-Earth-Moon problem, nor does it outperform Newtonian perturbation theory or GR post-Newtonian approximations. To develop it further, you would need to:

· Specify the geometric construction of asymmetric geodesic distance and derive conservation laws;
· Provide differential equations for \mathcal{C}_{ij} from MOC geometry;
· Perform systematic fitting and error analysis against observational data.