34-EFT.WP.Astro.Acceleration v1.0 | Chapter 8 Escape, Transport & Losses

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Chapter 8 Escape, Transport & Losses

I. Abstract & Scope
This chapter provides a unified system for particle escape, transport, and energy losses in a medium, forming the minimal S52-* closure to compute D(E), tau_esc(E), A_loss(E), and the spatial distribution n(E, r), coupled with Chapter 7’s spectrum-formation equation to yield observables Phi(E). All symbols use English notation with backticks; SI units; composite expressions are parenthesized. Any time-of-arrival (ToA) quantity carries explicit path gamma(ell) and measure d ell.

II. Dependencies & References

• Unified symbols & do-not-confuse: Chapter 2 Tab. 2-1 and P12-*.
• Kinematics & channel composition: Chapter 3 S20-* (A_acc(E), tau_acc(E)); reconnection & shear: Chapter 4 S30-, Chapter 5 S40-; comparators: Chapter 6 S45-*.
• Spectrum formation & observables mapping: Chapter 7 S50-* (N(E), Phi(E)).
• Media ontology & metrology: Core.Density v1.0, Core.Tension v1.0, Core.Metrology v1.0.

III. Normative Anchors (added in this chapter, S52-*)

• S52-0 (State & Flux): local spectral density n(E, r) and transport flux
  J(E, r) = - D(E, r) * ∇ n(E, r) + u_adv(r) * n(E, r), with diffusion coefficient D(E, r) and advective speed u_adv(r).
• S52-1 (Steady-State Transport Master Equation):
  0 = ∇ · J(E, r) + ∂/∂E ( b_loss(E, r) * n(E, r) ) - n(E, r)/tau_frag(E, r) + q(E, r),
  where b_loss(E, r) = | dE/dt |_loss > 0, tau_frag is fragmentation/dissipation timescale, and q is the source density.
• S52-2 (Volume Integration & Spectral Quantity): N(E) = ∫_V n(E, r) d^3r. In a one-zone effective model, N(E) is governed by effective tau_esc(E) and A_loss(E).
• S52-3 (Escape Timescale): for a connected region with characteristic scale L_esc,
  tau_esc(E) = L_esc^2 / ( kappa_esc * D(E) ), with geometric factor kappa_esc ∈ (0,∞).
• S52-4 (Loss Rate & Composition): A_loss(E) = b_loss(E) / E, with
  A_loss(E) = A_rad(E) + A_ad(E) + A_coll(E) + A_ion(E) + … (radiative, adiabatic, collisional, ionization; units s^-1).
• S52-5 (Typical Loss Terms): radiative b_rad(E) = k_rad(r) * E^2 (depends on B_vec, radiation field), adiabatic b_ad(E) = ( 1/3 ) * ( ∇ · u_adv ) * E, collisional/ionization b_coll(E) = k_coll(r) * f_med(E).
• S52-6 (Boundary Conditions): Dirichlet: n|_{∂Ω} = 0; Neumann: ( ∇ n · n_hat )|_{∂Ω} = 0; mixed: J · n_hat + κ_b * n = 0. Boundary choice fixes the kappa_esc aperture.
• S52-7 (Observed-Flux Mapping):
  Phi_obs(E) = C_geom * ∫_{gamma(ell)} R(E, r) * n(E, r) d ell, with response/attenuation R(E, r) and geometric factor C_geom.
• S52-8 (Two ToA Forms, Record Both):
  T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ) and T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ), carrying explicit gamma(ell) and d ell, and recording delta_form.
• S52-9 (Coupling to Spectrum Formation): feed D(E), tau_esc(E), A_loss(E) into Chapter 7 S50-* to obtain joint solutions for alpha_loc(E), E_br, E_max.
• S52-10 (Multi-Zone Equivalents): for partitions {Ω_i}, combine via effective series/parallel timescales: tau_esc^{-1} ≈ Σ w_i / tau_esc,i, A_loss ≈ Σ w_i * A_loss,i, with path or volume weights w_i.

IV. Body Structure

I. Transport Modeling & Geometry

• Regions & scales: a connected domain Ω with characteristic {L_esc, L_grad} defines the propagation environment; write D(E, r) as an isotropic dominant term plus anisotropy corrections.
• Advection & shear: u_adv(r) comes from large-scale flows; in strong-shear zones, u_adv and the Chapter 5 tensor S together set adiabatic/geometric terms.
• One-zone vs multi-zone: in one-zone, n(E, r) is volume-averaged and governed by tau_esc(E) and A_loss(E); for multi-zone, use S52-10 equivalents.

II. Key Equations & Derivations (S-series)

• S52-1 (Steady Transport): 0 = ∇ · ( - D ∇ n + u_adv n ) + ∂/∂E ( b_loss n ) - n/tau_frag + q.
• S52-3 (Escape Timescale): tau_esc = L_esc^2 / ( kappa_esc D ).
• S52-4 (Loss Rate): A_loss = b_loss / E = A_rad + A_ad + A_coll + A_ion + ….
• S52-5 (Typical Terms): b_rad = k_rad E^2, b_ad = ( 1/3 ) ( ∇ · u_adv ) E, b_coll = k_coll f_med(E).
• S52-7 (Observed Mapping): Phi_obs(E) = C_geom * ∫_{gamma(ell)} R(E, r) n(E, r) d ell.
• S52-8 (Two ToA Forms): record both and delta_form.
• S52-9 (Spectral Coupling): {D, tau_esc, A_loss} → Chapter 7 S50-* to produce N(E) and Phi(E).

III. Methods & Flows (M-series)

• M52-1 (Region & Boundary Setup): input geometry and medium parameters; choose boundary type (Dirichlet/Neumann/mixed); infer kappa_esc and L_esc.
• M52-2 (Coefficient Calibration): invert from data/simulations to obtain D(E), u_adv(r), k_rad(r), k_coll(r); return priors/posteriors.
• M52-3 (Numerical Solver): finite-volume/finite-difference steady solver for S52-1 to yield n(E, r) and derived {tau_esc(E), A_loss(E)}; in one-zone, use algebraic closure instead of PDE.
• M52-4 (Path Integration & Observable Synthesis): line-of-sight integration per S52-7 over gamma(ell) to synthesize Phi_obs(E); for ToA, use both forms in parallel and record delta_form.
• M52-5 (Uncertainty Propagation): perturb {D, u_adv, k_rad, k_coll, L_esc, kappa_esc} and propagate to {tau_esc, A_loss, Phi_obs} and Chapter 7 {alpha_loc, E_br, E_max}.
• M52-6 (Channel Masking & Harmonization): combine Chapter 6 dominance bands eta_dom(E) to mask channel weights in A_acc(E); harmonize with {tau_esc, A_loss} before passing to Chapter 7.

IV. Cross-References within/beyond this Volume

• Kinematics & channels: Chapter 3 S20-; Chapter 4 S30-; Chapter 5 S40-; Chapter 6 S45-.
• Spectrum formation: Chapter 7 S50-* (solutions/diagnostics for b(E), N(E), Phi(E)).
• Metrology & media: Core.Metrology v1.0 (dimensional checks), Core.Density v1.0 (medium fields), Core.Tension v1.0.

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • Spatial gradients of Phi_obs(E) track the energy scaling of D(E) (stronger diffusion → smoother distributions, softer spectra).
   • With adiabatic term present, high-energy alpha_loc(E) increases with ∇ · u_adv.
   • Path-integral synthesis per S52-7 consistently explains Phi_obs(E) and the frequency dependence of T_arr.
2. Negative criteria:
   • If turning off the dominant term in D(E) or A_loss(E) does not degrade fit quality, the corresponding mechanism is nonessential.
   • Changing boundary aperture (Dirichlet ↔ Neumann) without lowering evidence indicates inconsistencies in kappa_esc or geometry.
3. Contrasts:
   • With fixed A_acc(E), compare {diffusion only, diffusion+advection, diffusion+advection+adiabatic, full} for impacts on Phi_obs(E).
   • Fit ToA with both forms in parallel and compare evidence gaps via delta_form.

VI. Summary & Handoff
This chapter establishes the S52-* minimal closure and numerical workflow for escape, transport, and losses, delivering D(E), tau_esc(E), A_loss(E), and n(E, r), and harmonizing with Chapter 7’s spectrum-formation equation to synthesize Phi(E) and diagnostics. Subsequent chapters apply this closure to source classes and benchmarks.

V. Figures & Tables (this chapter)

• Tab. 8-1 Local Symbol Table (this chapter)

• Tab. 8-2 Boundary Conditions & Apertures

• Tab. 8-3 Loss-Term Packaging

• Tab. 8-4 Solution & Fitting Workflow (summary)