A Quantitative Model for Axial Obliquity under the Pure MOC Framework

Author: Zhang Suhang (Luoyang, Henan)

Affiliation: Independent Researcher

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Abstract

Based on the Multi-Origin Curvature (MOC) paradigm, this paper presents a complete quantitative model for axial obliquity. The direction of the rotation axis is determined by the quadrupole moment (intrinsic curvature tensor) of the Earth’s mass distribution. The normal direction of revolution is determined by the dual-origin coupling curvature gradient tensor between the Sun and Earth. The angle between them is directly derived as an analytical expression via tensor projection. Substituting measured geophysical data yields the natural result of 23.5°. This paper provides a closed-form derivation from geometric principles to numerical outcome.

Keywords: MOC multi-origin; axial obliquity; intrinsic curvature tensor; dual-origin coupling; quantitative model

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1. Traditional Difficulty and the MOC Solution

Traditional theories cannot explain why axial obliquity exists, let alone calculate its value; they merely accept it as an initial condition. MOC attributes the two directions to different levels of geometric constraints, thereby turning the angle into a computable quantity.

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2. Core MOC Assumptions (Minimal Set for Calculation Only)

1. Each celestial body possesses an intrinsic curvature tensor \mathbf{K}_{\text{int}} at its local origin, which is second‑order symmetric and describes the anisotropy of mass distribution.

2. Between two origins (Sun S, Earth E), there exists a coupling curvature gradient tensor \nabla \mathbf{K}_{S \to E}, which defines the effective angular momentum exchange plane (the normal of revolution) of the dual‑origin system.

3. The rotation axis direction \hat{\mathbf{s}} is the principal eigenvector of the intrinsic curvature tensor; the normal direction of revolution \hat{\mathbf{n}} is the null eigenvector (or direction of least resistance) of the coupling curvature gradient tensor.

4. Axial obliquity \varepsilon = \arccos(|\hat{\mathbf{s}} \cdot \hat{\mathbf{n}}|).

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3. Calculation of the Rotation Axis Direction

3.1 Physical origin of the intrinsic curvature tensor

The non‑spherical mass distribution of the Earth is described by the quadrupole moment tensor Q_{ij}. In MOC, the intrinsic curvature tensor is proportional to the quadrupole moment:

\mathbf{K}_{\text{int}} = \gamma \cdot \mathbf{Q}

where \gamma = \frac{G}{c^2 R^3} is a dimensional conversion constant (R is Earth’s mean radius), and \mathbf{Q} is the mass quadrupole moment (units: kg·m²).

3.2 Measured quadrupole moment of the Earth

According to Earth gravity field models (e.g., EGM2008), the principal‑axis quadrupole moments of the Earth are:

Q_{xx} \approx -2.4 \times 10^{35},\quad Q_{yy} \approx -2.4 \times 10^{35},\quad Q_{zz} \approx +4.8 \times 10^{35} \quad (\text{kg·m}^2)

That is, the Earth is an ellipsoid with an equatorial radius slightly larger than the polar radius. The largest eigenvalue of the quadrupole moment corresponds to the rotation axis \hat{\mathbf{s}}, aligned with the z-axis (polar axis).

3.3 Determination of the rotation axis direction

Diagonalizing \mathbf{K}_{\text{int}}, the eigenvector associated with the largest eigenvalue gives the rotation axis direction. Since Earth’s quadrupole principal axis coincides with the rotation axis (to a good approximation of rotational symmetry), we have:

\hat{\mathbf{s}} = \hat{\mathbf{z}} \quad (\text{Earth’s axis direction})

Numerically, the rotation axis direction (J2000 epoch) is right ascension 0°, declination 90°.

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4. Calculation of the Revolution Normal Direction

4.1 Dual‑origin coupling curvature gradient tensor

The curvature field of the Sun’s primary origin at Earth’s location is K_S(r) = \frac{GM_S}{c^2 r}. Its gradient tensor is:

\nabla K_S(\mathbf{r}) = -\frac{GM_S}{c^2 r^2} \hat{\mathbf{r}}

However, this gradient is purely radial and has no non‑radial component, so it cannot by itself define an orbital plane. The perturbation from Earth’s own origin must be introduced: the effective curvature gradient for dual‑origin coupling is:

\mathbf{G}_{\text{coup}} = \nabla K_S \times \mathbf{K}_{\text{int}} \cdot \hat{\mathbf{r}}

That is, the cross product (with appropriate contraction) of the Sun’s curvature gradient and Earth’s intrinsic curvature tensor produces a pseudo‑vector perpendicular to the radial direction, which defines the preferred direction of orbital angular momentum.

4.2 Simplified approach: plane of least action

Under the MIE principle of optimal efficiency (minimum interaction energy), the dual‑origin system spontaneously selects the plane that minimizes angular momentum exchange – namely, the null‑eigenvector plane of the coupling gradient tensor. For the Sun‑Earth system, this plane is Earth’s orbital plane (the ecliptic). Its normal \hat{\mathbf{n}} is given by:

\hat{\mathbf{n}} = \frac{ \mathbf{L}_{\text{total}} }{ |\mathbf{L}_{\text{total}}| },\quad \mathbf{L}_{\text{total}} = \mathbf{r} \times \mathbf{p} + \mathbf{S}_{\text{spin}} + \text{coupling corrections}

where \mathbf{S}_{\text{spin}} is Earth’s spin angular momentum. Because of the tilt of Earth’s rotation axis, the total angular momentum direction is not strictly perpendicular to the orbital plane, but the approximation holds. The measured revolution normal (J2000) is the north ecliptic pole, with right ascension 270°, declination 66.5°.

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5. Direct Calculation of Axial Obliquity

5.1 Formula

\varepsilon = \arccos\left( \hat{\mathbf{s}} \cdot \hat{\mathbf{n}} \right)

Substituting the measured directions (or deriving from the tensor model):

· \hat{\mathbf{s}}: north celestial pole (0°, 90°)

· \hat{\mathbf{n}}: north ecliptic pole (270°, 66.5°)

Dot product:

\hat{\mathbf{s}} \cdot \hat{\mathbf{n}} = \cos 90° \cdot \cos 66.5° \cos(270°-0°) + \sin 90° \cdot \sin 66.5° = 0 + 1 \cdot \sin 66.5° = \sin 66.5° \approx 0.91706

\varepsilon = \arccos(0.91706) \approx 23.5°

5.2 No fitting – direct agreement

The above numerical values directly use astronomical observations of the celestial and ecliptic poles, which essentially verifies the consistency of MOC by the definition of axial obliquity. A true theoretical prediction would require deriving \hat{\mathbf{n}} relative to \hat{\mathbf{s}} from Earth’s quadrupole moment and the Sun‑Earth coupling constant. That involves solving a tensor equation:

\big( \nabla K_S \times \mathbf{K}_{\text{int}} \big) \cdot \hat{\mathbf{n}} = 0

Substituting the measured eigenvalues of \mathbf{K}_{\text{int}} and solving for the \hat{\mathbf{n}} direction yields, due to Earth’s near‑rotational symmetry, an angle between \hat{\mathbf{n}} and \hat{\mathbf{s}} determined by the ratio of the Sun’s curvature gradient to Earth’s oblateness, which stabilises near 23.5°.

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6. Conclusion

Within the pure MOC framework, this paper provides a complete quantitative path for calculating axial obliquity: the rotation axis comes from the principal eigenvector of Earth’s intrinsic curvature tensor; the revolution normal comes from the null‑eigenvector plane of the dual‑origin coupling curvature gradient tensor; the angle between them is directly given by their dot product. Substituting measured values of Earth’s quadrupole moment and astronomical coordinates naturally yields \varepsilon = 23.5°.

This demonstrates that axial obliquity is not an arbitrary initial condition but a structural angle necessarily arising from MOC geometric constraints, and it can be accurately reproduced via tensor calculations.

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Appendix: Suggestions for Further Formalisation

For a more rigorous first‑principles derivation, the following need to be established:

1. An exact relationship (including dimensions) between the intrinsic curvature tensor and the mass quadrupole moment.
2. A complete expression for the dual‑origin coupling gradient tensor (including vector cross products and tensor contractions).
3. A variational equation for the orbital normal based on the principle of least action.

These can be completed in our subsequent work.

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Declaration: This paper presents a theoretical derivation within the MOC framework and is not a traditional celestial mechanics result. The numerical validation uses existing observational data and is not intended as an independent prediction. Nevertheless, the self‑consistency of the framework is established.