34-EFT.WP.Astro.Acceleration v1.0 | Chapter 7 Energy Gain & Spectrum Formation

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Chapter 7 Energy Gain & Spectrum Formation

I. Abstract & Scope
This chapter derives steady and quasi-steady spectrum-formation equations under unified A_acc(E), tau_acc(E), tau_loss(E), and tau_esc(E), defines the local spectral index alpha_loc(E), break and cutoff scales, and maps to observables N(E) and Phi(E). Equations use English notation in backticks; SI units; composite expressions are parenthesized.

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)), Chapter 4 S30-* (reconnection), Chapter 5 S40-* (shear), Chapter 6 S45-* (comparators/boundaries).
• Transport & losses: Chapter 8 S52-*.
• Inference & reproducibility: Methods.Inference v1.0, Methods.Repro v1.0.

III. Normative Anchors (added in this chapter, S50-*)

• S50-0 (Net Energy Change): b(E) = dE/dt = E * ( A_acc(E) - A_loss(E) ).
• S50-1 (Master Equation, Steady State): 0 = d/dE ( b(E) * N(E) ) + N(E)/tau_esc(E) + N(E)/tau_loss(E) - Q(E), with source term Q(E).
• S50-2 (Local Spectral Index): alpha_loc(E) = - d ln N(E) / d ln E.
• S50-3 (Closed Form in a Locally Constant Band): if A_acc, tau_esc, tau_loss are approximately constant over an energy band, alpha_loc(E) ≈ 1 + ( 1/tau_esc + 1/tau_loss ) / A_acc.
• S50-4 (Semi-Closed Solution, General Case): with B(E) = b(E) and Λ(E) = ( 1/tau_esc(E) + 1/tau_loss(E) ) / B(E),
  N(E) = ( 1 / |B(E)| ) * exp( - ∫_{E_0}^{E} Λ(E') dE' ) * ∫_{E_0}^{E} Q(E'') * exp( ∫_{E_0}^{E''} Λ(E') dE' ) dE''.
• S50-5 (Escape-Time Packaging): diffusive escape tau_esc(E) = L_esc^2 / ( kappa_esc * D(E) ), with geometric factor kappa_esc ∈ (0,∞) and diffusion coefficient D(E).
• S50-6 (Loss-Time Packaging): tau_loss(E) = E / | dE/dt |_loss, with mechanism composition in Chapter 8.
• S50-7 (Breaks & Upper Limits): E_br solves tau_acc(E_br) = tau_loss(E_br) or tau_acc(E_br) = tau_esc(E_br);
  E_max = max{ E : A_acc(E) > ( 1/tau_esc(E) + 1/tau_loss(E) ) }.
• S50-8 (Channel Composition): A_acc(E) = A_rec(E) + A_shear(E) (Ch.3–5); comparators per Ch.6.
• S50-9 (Observables Mapping): differential flux Phi(E) = C_geom * C_prop * N(E); any ToA-related terms must record both forms with explicit path and measure and delta_form:
  T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ) and T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ).
• S50-10 (Index–Timescale Check): with weak energy dependence, alpha_spec ≈ 1 + tau_esc / tau_acc; when losses dominate, alpha_spec increases with tau_loss / tau_acc.

IV. Body Structure

I. Master Equation & Dimensional Audit

• Use S50-1; [N(E)] and [Q(E)] are consistent; [b(E)] = J·s^-1; all tau_* in s.
• b(E) via channel composition A_acc(E) = A_rec(E) + A_shear(E); A_loss(E) matches Chapter 8 mechanisms.
• alpha_loc is dimensionless; Phi(E) carries steradian and energy dimensions (e.g., m^-2·s^-1·sr^-1·eV^-1).

II. Approximations & Boundaries

• Locally constant band: use S50-3 for segmented power-law fits.
• Weak energy dependence: first-order expand A_acc(E), tau_esc(E), tau_loss(E) into S50-4, keeping at most one kernel integral in the exponent.
• Loss-dominated: if tau_loss(E) << tau_esc(E), alpha_loc(E) steepens above E_br.
• Escape-dominated: if tau_esc(E) << tau_loss(E), alpha_loc(E) is set mainly by geometry and diffusion scaling.
• Bounds: E_0 from injection/bandpass; E_max from S50-7.

III. Breaks & Cutoffs

• Break criterion: E_br where tau_acc = tau_loss or tau_acc = tau_esc; for multiple mechanisms, take the lesser root.
• Soft cutoff: apply f_cut(E) = ( 1 + ( E / E_cut )^{p_cut} )^{-1} to N(E) or A_acc(E), with p_cut > 0.
• Hard cutoff: N(E ≥ E_max) → 0; E_max per S50-7 and consistent with dominant-channel bands (Ch.6).
• Multi-breaks: differing scalings among {tau_acc, tau_loss, tau_esc} allow segmented power laws with continuous breaks; fit per band and compare evidences.

IV. Observational Mapping & Diagnostics

• Flux: Phi(E) = C_geom * C_prop * N(E) with geometry/radiation in C_geom and propagation/absorption in C_prop.
• Local index: alpha_loc(E) = - d ln N / d ln E, evaluated in sliding windows and cross-diagnosed against energy dependences of tau_*.
• Polarization linkage: reconnection-dominated bands show rapid co-variations of alpha_loc(E) with Pi; shear-dominated bands show smoother co-variations.
• ToA: record both forms with explicit gamma(ell) and d ell and delta_form as in S50-9.

V. Key Equations & Derivations (S-series)

• S50-1: 0 = d/dE ( E * ( A_acc - A_loss ) * N ) + N/tau_esc + N/tau_loss - Q.
• S50-2: alpha_loc = - d ln N / d ln E.
• S50-3: alpha_loc ≈ 1 + ( 1/tau_esc + 1/tau_loss ) / A_acc.
• S50-4: semi-closed N(E) as exponential–convolution form above.
• S50-5: breaks and upper limits (E_br, E_max) per S50-7.
• S50-6: A_acc = A_rec + A_shear; dominant-band logic via eta_dom(E) (Ch.6).

VI. Methods & Flows (M-series)

• M50-1 (Timescale Aggregation): from Ch.4–5 obtain {A_rec(E), A_shear(E)} to form A_acc(E); from Ch.8 obtain {tau_loss(E), D(E)} and compute tau_esc(E).
• M50-2 (Spectral Solution & Binning): compute N(E) on an energy grid via S50-4; output alpha_loc(E) and segmented power-law parameters.
• M50-3 (Break Detection): seed with intersections of tau_acc(E) and {tau_loss, tau_esc}; refine with curvature of alpha_loc(E).
• M50-4 (Multiband Fitting): minimize L = L_spec + L_timing + L_polar; return {posterior, evidence}.
• M50-5 (Uncertainty Propagation): perturb {A_rec, A_shear, tau_loss, tau_esc} and propagate to {alpha_loc, E_br, E_max} with 68%/95% intervals.
• M50-6 (ToA Recording): apply both ToA forms in parallel and record delta_form.

IV. Cross-References within/beyond this Volume

• Kinematics & channels: Chapter 3 S20-; Chapter 4 S30-; Chapter 5 S40-; Chapter 6 S45-.
• Transport & losses: Chapter 8 S52-* (D(E), A_loss(E), boundary conditions).
• Inference & falsification: Methods.Inference v1.0, Methods.Falsification v1.0.

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • An energy band with tau_acc(E) < min{ tau_loss(E), tau_esc(E) } and decreasing alpha_loc(E) (hardening).
   • Systematic shifts of {E_br} under modulation of {A_rec, A_shear}.
   • Phi(E) synthesized via S50-3/S50-4 matches multi-band data with consistent indices over the same window.
2. Negative criteria:
   • If k_STG → 0, beta_TPR → 0, or gamma_Path → 0 without degraded fit quality, the corresponding mechanism is falsified.
   • If removing the dominant term in tau_loss or tau_esc does not lower evidence, the associated mechanism is nonessential.
3. Contrasts: with fixed sources/boundaries, compare {A_rec-only}, {A_shear-only}, and {A_rec + A_shear} and report differences in {alpha_loc(E), E_br, E_max} and Phi(E).

V. Figures & Tables (this chapter)

4. Fig. 7-1 Schematic of energy gain–timescale–spectral index relations (A_acc, tau_*, alpha_loc, breaks).
5. Tab. 7-1 Local Symbol Table (this chapter)

1. Tab. 7-2 Index–Timescale Relations (locally constant band)

1. Tab. 7-3 Break & Upper-Limit Extraction