34-EFT.WP.Astro.Acceleration v1.0 | Chapter 4 Reconnection Acceleration (STG/TBN)

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Chapter 4 Reconnection Acceleration (STG/TBN)

I. Abstract & Scope
This chapter specifies reconnection geometry and the reconnection rate R_rec, constructs the minimal S30-* set for tension-release–driven energy injection and particle energy gain, and provides M30-* workflows for simulation and fitting to connect with spectrum formation (Ch.7) and transport/losses (Ch.8). Equations use English notation with backticks; SI units are used; composite expressions are parenthesized.

II. Dependencies & References

• Terms & symbols: Chapter 2 Tab. 2-1 and P12-*.
• Kinematic baseline: Chapter 3 S20-* (A_acc(E), tau_acc(E), g_cycle(E), tau_cycle(E)).
• Ontology & dynamics: EFT.WP.Core.Threads v1.0, EFT.WP.STG.Dynamics v1.0.
• Simulation & reproducibility: EFT.WP.Methods.SimStack v1.0, EFT.WP.Methods.Repro v1.0.
• Metrology: EFT.WP.Core.Metrology v1.0 Ch.1–3.

III. Normative Anchors (added in this chapter)

• S30-0 (Geometry & Frames): The sheet is described by length L_sheet, thickness delta_sheet, openness chi_open ∈ (0,1], inflow/outflow speeds u_in / u_out, and tension scale T_fil. Geometry is evaluated in F_sheet; rates and timescales are mapped back to F_flow.
• S30-1 (Reconnection Rate): R_rec = ( xi_rate * u_in ) / L_sheet, with xi_rate ∈ (0,1].
• S30-2 (Cycle Time): tau_rec(E) = tau_adv_rec + tau_sc_rec(E), with tau_adv_rec ≈ ( L_cross / u_in ) and L_cross ≤ L_sheet.
• S30-3 (Per-Cycle Gain): g_rec(E) = eta_acc(E) * Phi_geom, where Phi_geom = chi_open * f_out(u_out / c_ref) * Theta_T^{q_rec}, Theta_T = T_fil / T_ref.
• S30-4 (Acceleration-Rate Closure): A_rec(E) = g_rec(E) / tau_rec(E).
• S30-5 (Energy-Dependent Efficiency): eta_acc(E) = ( 1 + ( E / E_cut_rec )^{p_eta} )^{-1}, with E_cut_rec > 0, p_eta > 0.
• S30-6 (Velocity Ceiling & Constraints): u_out ≤ u_max, u_in ≤ u_out; if a medium-imposed limit u_lim exists, set u_out = min( u_out, u_lim ).
• S30-7 (Composition with Total Acceleration): A_acc(E) = A_rec(E) + A_shear(E) (see Chapter 5 for A_shear(E)).

IV. Body Structure

I. Topology & Geometry

• Sheet definition in F_sheet: neutral plane centered layer with thickness delta_sheet (normal), length L_sheet (tangential); chi_open captures topological openness of magnetic/Thread structures.
• Kinematic mapping: u_in (normal inflow), u_out (tangential outflow). The F_sheet → F_flow mapping is used for timescale and gain evaluation.
• Tension scale: the dimensionless ratio Theta_T = T_fil / T_ref enters the geometric flux factor Phi_geom.

II. Key Equations & Derivations (S-series)

• S30-1: R_rec = ( xi_rate * u_in ) / L_sheet.
• S30-2: tau_rec(E) = ( L_cross / u_in ) + tau_sc_rec(E).
• S30-3: g_rec(E) = eta_acc(E) * chi_open * f_out(u_out / c_ref) * Theta_T^{q_rec}, with f_out(·) monotone non-decreasing, f_out(0)=0, f_out(1)≤1.
• S30-4: A_rec(E) = g_rec(E) / ( ( L_cross / u_in ) + tau_sc_rec(E) ).
• S30-5: eta_acc(E) = ( 1 + ( E / E_cut_rec )^{p_eta} )^{-1}.
• S30-6: A_acc(E) = A_rec(E) + A_shear(E) (consistent with Chapter 3 S20-2).
• Heuristic mapping (lossless limit, for Ch.7 use): alpha_spec(E) ≈ 1 + tau_esc(E) / tau_acc(E), where tau_acc(E) = 1 / A_acc(E); exact treatment in Chapter 7 S50-*.

III. Methods & Flows (M-series)

• M30-1 (Geometry Inversion): Input {u_in, u_out, L_sheet, delta_sheet} or observational proxies (e.g., timing, polarization structures); output {chi_open, L_cross, xi_rate} with uncertainties.
• M30-2 (Rate & Gain Computation): Compute {R_rec, tau_rec(E), g_rec(E), A_rec(E)} per S30-1…S30-4, with energy-binned outputs.
• M30-3 (Multiband Fitting): Jointly with Ch.7/Ch.8 minimize L = L_spec + L_timing + L_polar, returning {posterior, evidence}. Any ToA terms must use both forms T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ) and T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ), explicitly carrying gamma(ell) and d ell, and recording delta_form.
• M30-4 (Simulation-Stack Interface): Use SimStack to generate sheet geometry and inflow/outflow fields; output synthetic spectra, lepton–hadron components, and polarization sequences for benchmarks and regression tests.
• M30-5 (Sensitivity & Power): Perturb {chi_open, xi_rate, E_cut_rec, p_eta} and report sensitivity of A_rec(E) and observables.

IV. Cross-References within/beyond this Volume

• Kinematics: Chapter 3 S20-* (composition of A_acc, timescales).
• Shear: Chapter 5 S40-* (A_shear(E)) for channel separation.
• Spectrum formation & transport: Chapter 7 S50-, Chapter 8 S52-.
• ToA & paths: any integral involving T_arr must carry gamma(ell) and d ell; provide both ToA forms and record delta_form.

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • A_rec(E) increases monotonically with u_out / c_ref, chi_open, and Theta_T.
   • An energy band exists with tau_rec(E) < tau_loss(E) and spectrum hardening in that band.
   • Perturbations of L_sheet or delta_sheet drive coordinated responses in timing and polarization.
2. Negative criteria:
   • As xi_rate → 0 or chi_open → 0, if fit quality does not degrade, the reconnection channel is falsified.
   • Abnormal sensitivity of A_rec(E) to the F_sheet → F_flow mapping indicates geometric or frame inconsistency.
3. Contrasts: Hold A_loss(E) and A_shear(E) fixed; compare A_rec(E)=0 vs A_rec(E)>0 to localize observational differences.

VI. Summary & Handoff
This chapter provides the S30-* closure linking reconnection geometry, rate, and per-cycle gain, plus M30-* computation and fitting flows for Chapters 7–8. Chapter 5 specifies the statistical closure for shear acceleration and establishes channel discrimination against this chapter.

V. Figures & Tables (this chapter)

• Fig. 4-1 Reconnection-sheet geometry and frames (F_sheet ↔ F_flow, u_in/u_out, L_sheet/delta_sheet, chi_open).
• Tab. 4-1 Local Symbol Table (this chapter)

• Tab. 4-2 Parameters & Default Priors (examples)

• Tab. 4-3 Observational Criteria