The Derived Sun

DST-1 — the apparent solar longitude assembled from the framework’s own structure, with zero fitted solar constants.

Every eclipse computation stands or falls with the Sun. Classical practice takes the apparent solar longitude from a fitted polynomial-plus-periodic series — Meeus Chapter 25, VSOP87, or a numerical ephemeris — in which the coefficients are adopted numbers. The Holistic Universe Model’s certified Sun is different in kind: it is assembled, not fitted. Its mean motion is the framework’s tropical rate, its drift is the integral of the model’s own year-length physics, and its equation of centre runs on an eccentricity law that is itself derived. The result is verified at 0.95″ RMS against JPL over the modern window — where the Meeus Chapter 25 reference itself reads 1.28″ — and it is the Sun inside every eclipse result this site reports, and (since the scene-wheel unification) the Sun the visible simulator displays.

The companion page The Derived Moon documents the lunar half of the eclipse chain; this page documents the solar half. The complete derivation record lives in the repository’s technical documentation — doc 99  and doc 65 .

Every element belongs to one of three origin classes:

There is no fourth class. No solar coefficient anywhere in the chain is fitted to observations.

1. The assembly

λ(t) = L(t) + EoC( e(t), M(t) )                      (apparent longitude)

L(t) = L₀ + n_trop · (t − t₀) + D(t)                 (mean longitude)

• L₀ is the J2000 anchor 280.46646°.
• n_trop is the mean tropical rate — 360° per mean tropical year of 365.2422036 days, itself the IAU input snapped to the framework’s H/8 grid.
• D(t) is the drift term: the accumulated difference between the actual year length at each epoch and the mean — the integral of the model’s year-length physics (§2–§4).
• EoC is the classical Kepler equation of centre, evaluated with the derived e(t) of §4 — no fitted equation-of-centre coefficients.

Everything below is the anatomy of D(t) and e(t).

2. The year-length harmonics — the amplitude is derived

The tropical year is not constant: it oscillates by tens of seconds over the Earth Fundamental Cycle (H = 335,317 years). The framework models this as 12 Fourier terms on H-lattice divisors (the TROPICAL_YEAR_HARMONICS), dominated by two lines:

The two amplitudes sit in an exact 8:3 ratio — one oscillation per equinox cycle — and the common amplitude factor is not fitted: it is A·cot ε, where A = 0.63605° is Earth’s invariable-plane inclination amplitude (itself pinned by the IAU obliquity rate dε/dt = -46.836769″/cy) and ε = 23.41353° is the mean obliquity. A year-length spectrum, derived from an inclination amplitude and an obliquity — nothing solar was fitted to produce it.

Integrated over time, these oscillations displace the Sun’s mean longitude by whole arcminutes at historical epochs. That integral is D(t)‘s periodic part.

3. The obliquity-torque term

The luni-solar precession rate p depends on the obliquity: the classical torque law gives δp = −p₀ · tan ε · δε. As the obliquity cycles (H/8) and the inclination precesses (H/3), the precession rate breathes, and that breathing feeds back into the tropical-year clock.

Integrated on the lattice, the effect scales the two drift harmonics of §2 by

scale(div) = 1 + p₀ · tan²ε · H / (2π · div)
           = 1.306  for the H/8 line
           = 1.815  for the H/3 line

with p₀ = 360°·13/H, the framework’s general precession. This term was nearly dismissed — a quick estimate suggested a ~3% contribution — until the full derivation showed that both mechanisms lengthen the year at obliquity maximum, so they add. The measured record then confirmed the structure independently: the ancient eclipse corpus preferred the term’s per-divisor scaling before the derivation existed (see §6). The full episode is recorded in doc 99.

4. The closed form and the derived e(t)

The drift D(t) is evaluated analytically — no lookup table and no fit window. Each harmonic contributes its exact antiderivative on the integrated lattice phase, plus a sin-saturated rebase ramp (H/2πk)·sin(Δφₖ) that anchors the drift and its slope to zero at J2000 by construction. (Extending the ramp linearly instead would be precisely the fitted-linear-slope trap the framework’s hard rules forbid: a linear extension is exact in the fit window and tens of minutes wrong at deep time; the saturated form is exact where it must be and bounded everywhere else.) The smooth deep-time part rides the framework’s H(t) evolution, so the same expression that reproduces the modern window degrades gracefully at ±3,000 years and beyond instead of diverging.

The eccentricity e(t) entering the equation of centre is two derived pieces:

e(t) = e_H16(t)  −  (e_base / 2) · ( cos φ₃(t) − cos φ₃(2000) )

• e_H16 is the framework’s H/16 perihelion-cycle eccentricity law (period 20,957 years), built from the Law-5 base eccentricity 0.015386 — the same law the cardinal-point braid and the climate work use.
• The second piece is the derived H/3 inclination coupling on the integrated lattice phase φ₃ — the factor e_base/2 comes from the same geometry as the framework’s derived-eccentricity channel. No coefficient in it is fitted.

5. One Sun everywhere — the eclipse chain and the visible wheel

The assembled longitude ships in the published @essrt/physics package and is consumed identically in all three runtimes: the package eclipse chain (tier finders and the Besselian umbra engine), the Node verification engine, and the browser simulator. There are not two Suns to drift apart.

The visible scene wheel keeps its legacy display stack (linear tropical rate + Kepler equation of centre + the retired fitted harmonics) and adds one term on top:

δ(t) = λ_certified(t) − λ_twin(t)

— the difference between the certified Sun and the wheel’s own twin evaluation — applied inside a clock-convention window: full weight within 3,000 years of J2000 (where eclipse truth lives, on the TT clock), tapering by cos² to zero at 20,000 years (where the deliberately-UT deep-time scene would clash with a TT-clock Sun). Both endpoints are derived from the model’s own conventions; only the cos² easing shape is a convention, and there is nothing physical for it to derive from. Measured effect on the wheel: within the window the display agrees with the certified Sun to a few arcseconds; at −135 (the Babylonian eclipse epoch) the wheel error fell from 1,138″ to 10″, and at −3000 from 5,953″ to 164″.

Replacing the wheel’s legacy stack outright was tried and rejected by measurement — it double-counts the wheel’s geometric-split ellipse and moves the planets by a degree. The δ overlay is the honest architecture: the certified physics owns the longitude; the legacy stack remains as scaffolding whose fit quality no longer matters.

6. Verification

The doctrine behind this page: long-term ephemerides are not truth — the eclipses are. The Sun above was never tuned to any of the rows in this table; the rows are what the assembled structure produces.

The derivation experiments, the rejected alternatives (full equation-of-centre replacement, corrections on the wheel’s tropical axis, UT-assembled δ), and the certification gates are recorded in the repository’s doc 99 §“The framework-native Sun”  and doc 65 . Values in this document are frozen as DST-1.