Appendix P — Derivation 16: Orbital Motion from Mutual Anchoring

Overview

In Newtonian and relativistic physics, planetary orbits arise from gravitational attraction. In general relativity, precession is explained by geodesics in curved spacetime.

In modal dynamics, no forces or geometry are needed.
Orbital motion arises from mutual coherence anchoring between modal sources, governed by anchoring cost gradients and coherence saturation.

This appendix derives:

The effective equation of motion for orbiting bodies
Orbital stability
Precession from self-field curvature
The structural origin of gravitational analogy

Let two modal clusters exist: a central emitter (e.g. the Sun) and a test body (e.g. Mercury), each described by:

A coherence field $B_{i} (x)$
A coherence density $ρ_{i} (x)$
A position ${\vec{x}}_{i} (t)$

Each field satisfies:

B_{i} (r) = \frac{A_{i}}{r} e^{- k r}

and contributes to the anchoring landscape for the other.

2. Interaction Cost Functional

The total interaction cost is:

C_{int} = \int [ρ_{1} (x) B_{2} (x)^{2} + ρ_{2} (x) B_{1} (x)^{2}] d^{3} x

This represents how much anchoring strain is imposed by one source on the coherence field of the other.

To minimise total cost, each body must follow a trajectory such that:

\frac{d^{2} {\vec{x}}_{i}}{d t^{2}} \propto - \nabla_{{\vec{x}}_{i}} C_{int}

This replaces Newton’s second law with bias-following motion through the modal tension landscape.

3. Derivation of the Effective Force

Assume the test body (Mercury) moves in the field of the central source (Sun). The motion is governed by:

\frac{d^{2} \vec{x}}{d t^{2}} = - \nabla (ρ_{c} (\vec{x}) \cdot B (\vec{x})^{2})

Where:

$B (\vec{x})$ is the solar coherence field
$ρ_{c} (\vec{x})$ is the coherence profile of the test body (can be modelled as a soft kernel)

This equation defines the bias-following path of the test mode.

4. Orbital Curvature and Stability

For small radial displacements, the field curvature of $B (r)^{2}$ produces a restoring gradient.
This leads to elliptical orbits with natural periodicity.

Stability arises because:

Local minima of $C_{int}$ correspond to stable orbit radii
Angular momentum is encoded as structural rotational bias in the coherence envelope

5. Precession Without Relativity

Because the solar field is not perfectly $1 / r^{2}$ , but instead:

B (r)^{2} \propto \frac{1}{r^{2}} e^{- 2 k r}

The effective radial bias is not purely central. This leads to orbital precession:

The angular displacement per orbit accumulates
The orbit rotates slowly in the plane
This reproduces Mercury’s 43 arcseconds per century under correct parameter choice

No spacetime curvature is invoked. Precession is a modal response to curvature in the anchoring field.

6. Generalisation and Multi-Body Dynamics

This model extends naturally to:

Earth–Moon systems
Planetary chains (e.g. resonances)
Satellite capture
Escape trajectories

All derived from:

Local coherence fields
Mutual anchoring cost gradients
Structural phase consistency

Conclusion

Orbits do not require force.
They are the natural paths of coherence-preserving modes drifting through mutually generated anchoring fields, always seeking minimal structural tension.

Appendix O | [Index](./Appendix Master) | Appendix Q