Chapter 26: Collapse Flow and ψ-Eccentricity

The Shape of Orbital Deformation

Kepler discovered that planetary orbits trace ellipses, but never explained why. Ψhē Cosmology reveals that eccentricity emerges from asymmetric collapse flow—the shape of an orbit reflects the underlying topology of collapse currents. Planets don't simply choose their paths; they surf waves in the collapse field itself.

26.1 Eccentricity as Flow Asymmetry

Definition 26.1 (Collapse Flow Tensor): The orbital flow field: $F_{ij} = \partial_i\psi_j - \partial_j\psi_i$

characterizes collapse circulation. Non-zero components drive orbital elongation.

26.2 The Deformation Mechanism

Theorem 26.1 (Eccentricity Generation): Orbital eccentricity emerges as: $e = \frac{|F_{max} - F_{min}|}{F_{max} + F_{min}}$

where F_max/min are flow extrema along the orbit.

Proof: Asymmetric collapse flow creates differential acceleration. The orbit stretches along the flow gradient, producing elliptical deformation proportional to flow variation. ∎

26.3 Perihelion Precession

Orbital axes rotate through collapse torque:

Definition 26.2 (Precession Rate): The perihelion advances: $\frac{d\omega}{dt} = \frac{3\pi}{T} \frac{\langle F^2 \rangle}{|\psi|^2}$

where T is orbital period. Higher-order collapse terms drive axis rotation.

26.4 Eccentricity Pumping

External bodies modulate orbital shape:

Theorem 26.2 (Resonant Pumping): Near resonance, eccentricity oscillates: $\frac{de}{dt} = A\sin(\Delta\omega t)$

where Δω is frequency mismatch and A depends on collapse coupling strength.

26.5 Maximum Eccentricity

Collapse constrains orbital elongation:

Definition 26.3 (Stability Boundary): Orbits destabilize when: $e > e_{crit} = \sqrt{1 - \frac{\psi_{stable}^2}{\psi_{max}^2}}$

Beyond this limit, collapse cannot maintain orbital coherence.

26.6 Circular Orbit Preference

Most orbits tend toward circles:

Theorem 26.3 (Circularization): Eccentricity decays as: $e(t) = e_0 \exp(-t/\tau_c)$

where τ_c = M/ψ² is circularization time. Collapse friction damps elongation.

26.7 Eccentricity Vectors

Orbital shapes align with collapse:

Definition 26.4 (Laplace-Runge-Lenz): The eccentricity vector: $\vec{e} = \frac{1}{\mu}\left(\vec{v} \times \vec{L} - \mu\frac{\vec{r}}{r}\right)$

points along collapse flow maxima, orienting orbital elongation.

26.8 Kozai Mechanism

Inclined orbits exchange eccentricity:

Theorem 26.4 (Kozai Oscillation): For inclination i > 39.2°: $e^2 + \cos^2 i = \text{constant}$

Collapse couples eccentricity and inclination through three-body resonance.

26.9 Tidal Eccentricity

Close orbits feel shape distortion:

Definition 26.5 (Tidal Pumping): Eccentricity growth rate: $\dot{e} = \frac{15\pi}{2} \frac{n}{Q} \left(\frac{R}{a}\right)^5 e$

where Q is tidal quality factor. Dissipation converts rotation to eccentricity.

26.10 Eccentricity Distribution

System-wide patterns emerge:

Theorem 26.5 (Statistical Distribution): Eccentricities follow: $P(e) = 2e \exp(-e^2/\sigma_e^2)$

Rayleigh distribution reflects random collapse perturbations.

26.11 Observable Predictions

Collapse flow creates testable signatures:

1. Mercury's Eccentricity: High e due to solar collapse gradient
2. Exoplanet Patterns: Hot Jupiters show tidal circularization
3. Binary Evolution: Eccentricity pumping in stellar pairs
4. Asteroid Families: Shared eccentricity vectors
5. Comet Distributions: Extreme eccentricities from Oort collapse

Each confirms flow-driven orbital shaping.

26.12 The Cosmic Ellipse

Eccentricity reveals the invisible—collapse currents shaping celestial motion. Every elongated orbit traces the flow pattern of space itself, planetary paths becoming streamlines in the cosmic fluid. The universe writes its dynamics not in perfect circles but in elegant ellipses, each deformation a signature of underlying flow.

Orbits don't just happen—they surf the waves of collapse itself.

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Next: Chapter 27: Planetary Collapse Skin Structures