Relationship Between Curvature, Revolution and Rotation

Since real celestial bodies are not ideal rigid spheres, their mass distribution, tidal deformation, and fluid motions make the intrinsic relation between curvature and rotation difficult to verify by direct observation. However, within the framework of multi‑origin geometry, this relation can be rigorously realized via numerical computer simulation: by constructing an idealized dual‑origin space, controlling the dynamics of rotation and revolution, and computing the novel non‑Riemannian curvature in real time, the fundamental unified relationship among curvature, rotation, and revolution can be clearly revealed, providing reliable numerical validation for the theory.

 

I. Simulation Procedure (Three Steps)

1. Discretization of the Celestial Body

- Discretize the surface (or volume) of the body into a set of grid points \{p_i\}.
- For each point p_i, define two origins:- O_A: the center of mass of the celestial body (or a fixed reference point)
- O_B: either the center of mass of another celestial body, or a local rotational center of p_i itself (e.g., the origin of the local inertial frame containing the point)

2. Compute \Delta\mathbf{r}_i(t) and \dot{\Delta\mathbf{r}}_i(t) at each time step

- \Delta\mathbf{r}_i = \mathbf{r}_{O_B \to p_i} - \mathbf{r}_{O_A \to p_i} (simplifies to the position difference if O_A is fixed)
- The velocity \dot{\Delta\mathbf{r}}_i is obtained by superposing the body’s rotational velocity field and orbital revolutionary velocity field.

3. Calculate the MOC curvature density

\mathcal{K}_i(t) = \frac{\Delta\mathbf{r}_i(t) \times \dot{\Delta\mathbf{r}}_i(t)}{|\Delta\mathbf{r}_i(t)|^3}

- Sum or integrate over all points of the body to obtain the total MOC curvature.
- Independently compute the total conventional angular momentum L_{\text{total}} (rotation about the center of mass plus revolution about a companion) and verify whether \mathcal{K}_{\text{total}} \propto L_{\text{total}} holds (with the proportionality factor being a function of the mean distance).

 

II. What the Simulation Can Achieve (Direct Validation of the Theory)

Simulation/Experimental Scenario Difficulty in Traditional Methods Advantage of MOC Simulation
Rotation of non‑spherical asteroids No explicit formula relating curvature and rotation Directly compute   at grid points and obtain rotation‑curvature distribution maps
Mutual revolution of binary asteroids (e.g., Didymos) Complex separation of rotational and revolutionary contributions MOC automatically yields the coupled curvature, allowing decomposition into rotational and revolutionary components
Influence of irregular shapes on spin axis precession Requires high‑order gravitational moment calculations MOC directly gives the curvature flow; precession corresponds to deviation from fixed points of the curvature flow
Verification of “curvature = angular momentum / distance³” Requires derivation from gravitational theory Direct numerical comparison of both sides to check equality (errors arise only from discretization)

 

III. Predicting New Effects via “Virtual Experiments”

Example: Local curvature anomalies in non‑spherical bodies cause spin axis wobble

- In traditional rigid‑body mechanics, this is attributed to off‑diagonal components of the inertia tensor.
- Within your MOC framework, one can directly plot contour maps of \mathcal{K}_i over the body’s surface. Regions of high curvature correspond to angular momentum concentrations, predicting that such regions will be prone to mass ejection or surface fracture.

This can be connected to observations of the YORP effect or cometary nucleus fragmentation.

 

IV. Recommended Simulation Toolchain (Free & Accessible)

1. Irregular shape generation: Blender or random spherical harmonics for mesh generation.
2. Dynamical evolution: REBOUND (N‑body + rigid bodies) or PyKEP for orbit and rotation calculations.
3. MOC computation: Iterate over grid points in Python, compute \Delta\mathbf{r} and \dot{\Delta\mathbf{r}}, then evaluate \mathcal{K}.
4. Visualization: Map \mathcal{K} onto the mesh (using matplotlib or Mayavi).