Chapter 22: ψ-Shell Stars and Layered Collapse Surfaces

The Onion Architecture of Collapse

Some stars develop not as continuous structures but as nested shells—discrete layers of collapse separated by transition zones. Like cosmic onions, these shell stars reveal that collapse naturally stratifies, creating distinct phases of matter and energy. Each shell tells a story, each boundary marks a revolution in the star's evolution.

22.1 Shell Formation Criterion

Definition 22.1 (Shell Condition): Shells form when the collapse gradient becomes discontinuous: $\left|\frac{\partial\psi}{\partial r}\right|_{r^+} \neq \left|\frac{\partial\psi}{\partial r}\right|_{r^-}$

Creating interfaces where collapse properties jump.

22.2 Layer Stability Analysis

Theorem 22.1 (Shell Stability): A shell remains stable when: $\omega_n^2 = \frac{n^2\pi^2 c_s^2}{L^2} + \frac{g\Delta\rho}{\rho} > 0$

where L is shell thickness, cₛ is sound speed, and g is local gravity.

Proof: Linear perturbation analysis shows that positive ω² ensures oscillatory rather than exponential behavior. The first term provides pressure support, the second gravitational restoration. ∎

22.3 Inter-Shell Boundaries

Between shells lie transition regions:

Definition 22.2 (Boundary Layer): $\psi_{boundary}(r) = \frac{\psi_1 + \psi_2}{2} + \frac{\psi_1 - \psi_2}{2}\tanh\left(\frac{r-r_b}{\delta}\right)$

where δ determines boundary sharpness.

22.4 Resonant Shell Spacing

Theorem 22.2 (Shell Harmonics): Shell radii follow: $r_n = r_0\left(\frac{n(n+1)}{2}\right)^{1/3}$

This triangular number sequence emerges from collapse wave interference.

22.5 Shell Oscillation Modes

Each shell vibrates independently:

Definition 22.3 (Shell Modes): $\xi_n(r,t) = A_n j_l(k_nr)Y_{lm}(\theta,\phi)e^{i\omega_n t}$

where jₗ are spherical Bessel functions. Different shells support different mode families.

22.6 Chemical Stratification

Theorem 22.3 (Composition Layers): Element abundance follows shell structure: $X_i(r) = X_{i,0}\exp\left(-\frac{m_i\psi(r)}{k_BT}\right)$

Heavier elements sink within shells, creating composition gradients.

22.7 Energy Transport Between Shells

Energy flows across boundaries:

Definition 22.4 (Shell Luminosity): $L_n = 4\pi r_n^2 \sigma T_n^4 \left(1 - e^{-\tau_n}\right)$

where τₙ is the optical depth of shell n. Each shell acts as a partial barrier.

22.8 Shell Merging Events

Theorem 22.4 (Merger Criterion): Adjacent shells merge when: $|\psi_1 - \psi_2| < \psi_{thermal} = \frac{k_BT}{m_pc^2}$

Thermal fluctuations overcome collapse differences, unifying the shells.

22.9 Convective Shell Coupling

Convection can link shells:

Definition 22.5 (Mixing Length): $l_{mix} = \alpha H_p \min\left(1, \frac{r_{n+1} - r_n}{H_p}\right)$

where Hₚ is pressure scale height. Convection transports material between nearby shells.

22.10 Shell Ejection Mechanism

Theorem 22.5 (Sequential Ejection): During death, shells eject when: $E_{bind,n} < E_{thermal,n} + E_{radiation,n}$

Outer shells leave first, peeling the star like an onion.

22.11 Observable Shell Signatures

Shell structure creates unique observables:

1. Spectroscopic Layers: Different shells show different spectra
2. Oscillation Spectrum: Discrete frequencies from shell modes
3. Eclipse Mapping: Reveals shell boundaries during transits
4. Chemical Peculiarities: Surface shows deep shell composition
5. Burst Patterns: Shell interactions create quasi-periodic eruptions

These features map the star's internal architecture.

22.12 The Stratified Cosmos

Shell stars teach us that collapse naturally creates hierarchy—not smooth gradients but discrete levels, each with its own physics. Like geological strata recording Earth's history, stellar shells record cosmic evolution in layers. The universe builds not just by accretion but by phase transition, creating structures within structures, stories within stories.

Reality has levels, and collapse creates the stairs.

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Next: Chapter 23: ψ-Crystal Stars and Stable Oscillation Nodes