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 framework’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 1.03″ RMS against JPL over the modern window — where the Meeus Chapter 25 reference itself reads 1.02″ — 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 era device’s integer-day grid (Formulas §4).
• 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 anchor interval (335,317 years, the correction bases’ unit). The framework models this as 12 Fourier terms on the anchor’s fixed divisors — a bounded harmonic basis (the TROPICAL_YEAR_HARMONICS) — dominated by two lines:

The two amplitudes sit in an exact 8:3 ratio — one oscillation per equinox cycle — and their common factor is A·cot ε, carried over from the Earth side rather than fitted here: A = 0.63607° is the device’s invariable-plane inclination amplitude (pinned against the IAU obliquity rate dε/dt = -46.836769″/cy) and ε = 23.41353° its mean obliquity. Both are device constants set against the observed obliquity range and accounted in Ledger 3 (Mathematical Foundations) — so the claim here is the narrower one: a year-length spectrum built from an inclination amplitude and an obliquity, with nothing solar 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 (the obliquity line) and the inclination precesses (the apsidal line), the precession rate breathes, and that breathing feeds back into the tropical-year clock.

Integrated on the device’s bases, the effect scales the two drift harmonics of §2 by

scale(line) = 1 + p₀ · tan²ε · T_line / (2π)
            = 1.306  for the obliquity line
            = 1.815  for the apsidal line

with T_line the line’s period and p₀ the device’s general-precession reading (360° per its anchor’s precession period). 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 mean longitude is the integral of the tropical year of date — L(t) = L0 + 360°·∫dt/T(t), with T(t) the model’s one tropical-year family (the sidereal year from the N-body drift and the solar-mass law, reduced by the equinox rate the same movement supplies), accumulated as a cumulative table from J2000 in both directions and never as a rate multiplied by a span. No fitted drift term enters. This replaced an earlier closed-form drift built on the frozen era clock’s year-length harmonics, whose tropical year carried no secular drift at all, so the earlier Sun wandered twelve minutes around 500 AD and diverged after 2100, outside every eclipse and JPL check of the time. Checked against JPL Horizons’ apparent Sun on a ten-day grid over ±3000 years (219,152 instants), the model’s Sun sits within 3.6″ of Horizons at its worst millennium and within about one arcsecond in 1000–3000 (part of the ancient offset is Horizons’ own precession theory, IAU 1976, which parts from the long-term Vondrák series by several arcseconds there; carried into the fixed J2000 frame against DE441 the model’s Sun sits within two arcseconds over −1500 to +3000), with 0.78″ scatter in the modern window; the equinox and solstice instants the model publishes are the apparent crossings of this Sun (aberration and nutation derived, no fitted cardinal constants) and sit −0.71 minutes from Horizons’ own crossings on average, within a minute in 1000–3000.

The eccentricity e(t) and the perihelion ϖ(t) entering the equation of centre are the model’s own banked N-body series — the secular orbit of Earth from the eight-body integration of the J2000 state (its J2000 element coincides with the Laskar secular element) — plus one derived constant per element, the mean-element offset of the era:

e_Sun(t) = e_series(t) + Δe        ϖ_Sun(t) = ϖ_series(t) + Δϖ

• Why an offset at all. A secular series removes the short-period Venus and Jupiter terms; their average over an era does not vanish, and the Sun’s equation of centre needs the mean elements that carry it. The fast part of that channel is already in the planetary completion’s direct terms — adding it a second time was measured to double-count.
• How it is derived, not fitted. The same J2000 seed vectors are integrated as a nine-body system; the osculating Earth–Moon-barycentre e and ϖ are sampled daily over 1890–2110, the secular series is subtracted, and the window mean is taken: Δϖ = +70.08″, Δe = +5.81e-6. No ephemeris and no eclipse enters the derivation.
• What it reproduces unread. The resulting Sun perihelion at J2000 matches the classical mean perihelion of the era (Simon et al. 1994, 102.937°) to 0.2″. Against JPL over the modern window the Sun residual equals that of the former J2000-anchored law it replaces, and the ancient Babylonian anchor improves by two minutes.

The published Earth surface — the panel’s e and ϖ of date — stays the secular series; the offset belongs to the Sun’s equation of centre alone. The Moon’s E-factor keeps its own J2000-anchored eccentricity line (see The Derived Moon); the cardinal-point braid takes the Sun’s e(t) above, and the climate work keeps its own of-date quantities.

Nothing fitted remains in the Sun chain. The planetary completion’s last declared-fitted term — a +1.42″ semiannual correction inherited from the finder’s early calibration — turned out to be the semiannual nutation term in disguise; the chain keeps Sun and Moon on the mean equinox of date, where nutation cancels in eclipse geometry, so a Sun-only nutation term was a frame inconsistency. It is gone: the syzygy elongation fleet improved from 3.88″ to 3.76″ and the centerline mean from 2.4″ to 2.2″ on its removal alone.

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 runs its own longitude stack — linear tropical rate + full Kepler equation of centre, exact by derivation, with no fitted display-path correction — 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 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, and the wheel underneath now realizes exact Kepler on the framework’s own laws to sub-arcsecond annual agreement with its analytic twin, so δ compares the certified Sun against an exact structure with no fitted absorber in the loop.

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.