Single-Constant Transfer From Solar Limb To A Rapid Rotator

Objective

Demonstrate universality of the Law of Universal Coherence by transferring the solar‑calibrated alignment–extinction constant k_Ӕ to predict the limb darkening of Altair (α Aquilae) without any retuning. Pass condition: a fixed k_Ӕ, derived once from the solar limb, reproduces Altair’s normalized limb profile within observational tolerances.

Assumptions & Defaults (v1)

• k_Ӕ = 2.91 × 10⁻⁴ (Mpc·arb)⁻¹, fixed from the solar limb fit

• Baseline limb geometry uses S(μ) = A(μ⁻ᵖ − 1), with A = 0.85, p = 0.6

• v1 treats Altair as a spherical emitter for the baseline comparison; a rotation correction (gravity darkening/oblateness) is added as an option in the procedure but not used to set k_Ӕ

Method

(Baseline Transfer)

1. Normalize Altair’s observed intensity profile I_obs(μ) to I0 = I_obs(μ=1).

2. Predict the normalized intensity via the Ӕ law using the fixed constant:
   I_model(μ)/I0 = exp(− k_Ӕ · S(μ)).

3. Compare I_model(μ)/I0 to the observed profile. No parameter is retuned.

4. PASS if residuals are within published uncertainties or within the spread of standard limb‑darkening laws at comparable bands.

Optional Rotation/Oblateness Correction (v1+)

For a rapidly rotating star like Altair, include a simple gravity‑darkening weight w(θ) ∝ g_eff(θ)^β, with a Roche‑like effective gravity and β ≈ 0.19–0.25 for radiative envelopes. This alters the mapping between disk position and μ via latitude‑dependent brightness. In the Ӕ framework, this is treated as an angular reweighting of S(μ) across the projected disk, keeping k_Ӕ fixed.

Baseline Prediction Table

(Snippet)

The table below shows a subset of the baseline prediction using the fixed k_Ӕ and the default S(μ). A full 50‑point CSV is provided in the data package.

μ	S(μ)	I_model(μ)/I0
0.020	8.037936	0.997664
0.040	5.013851	0.998542
0.060	3.747568	0.998910
0.080	3.018699	0.999122
0.100	2.533911	0.999263
0.120	2.183264	0.999365
0.140	1.915299	0.999443
0.160	1.702389	0.999505
0.180	1.528239	0.999555
0.200	1.382549	0.999598

Diagnostics & Pass Criteria

• Residuals: Compute Δ(μ) = I_obs(μ)/I0 − I_model(μ)/I0 across the observed μ grid.

• PASSif RMS[Δ(μ)] ≤ σ_stat ⊕ σ_sys of the published limb profile (band‑specific), without altering k_Ӕ.

• If Altair’s rapid rotation is significant in the band used, apply the rotation reweighting once (v1+): if this collapses residuals within tolerance while keeping k_Ӕ fixed, the transfer still PASSes.

• FAIL if no reasonable rotation reweighting removes a systematic bias while holding k_Ӕ constant.

Notes

• Universality Claim: k_Ӕ is global. Environment enters only via S(μ), a purely geometric/structural factor.

• Band Dependence:Compare within the same photometric band as the solar limb calibration or treat bandpass differences as part of S(μ) reweighting.

• No Curve Fitting: k_Ӕ derived from the Sun is not adjusted for Altair. This is the core of the universality test.

Summary

This report freezes the universal constant k_Ӕ from the solar limb and projects it to Altair’s limb without retuning. If the observed limb profile of Altair is reproduced within uncertainties, the Law of Universal Coherence passes a stringent, cross‑object test: the same geometric alignment law governs limb darkening for both the Sun and a rapid rotator.

Produced by The Lilborn Equation Team:

Michael Lilborn-Williams

Daniel Thomas Rouse

Thomas Jackson Barnard

Audrey Williams