The Derived Moon

DLT-1 — a framework-native lunar theory in the form of Meeus, with every constant derived, attributed, or anchored by design.

Every practical lunar ephemeris of the last century descends from one lineage: Hansen and Delaunay, then E. W. Brown’s Theory of the Motion of the Moon, then the Éphéméride Lunaire Parisienne (ELP2000), distilled for working astronomers by Jean Meeus in Chapter 47 of Astronomical Algorithms. Those tables are magnificent — and, as presented, they are numbers without origins: polynomial coefficients, periodic amplitudes, and secular terms adopted from fits and inherited conventions.

This document presents the same theory in Meeus’s own form, with one difference: every number now carries its origin. Nothing here is a new observation and nothing is a re-fit. It is a derivation of the existing baseline from the constants of the Holistic Universe Model — the Earth Fundamental Cycle (H = 335,317 years), a small set of physical constants and J2000 anchors (the complete input list is §0), and Newtonian gravity — with the framework’s own three-body laboratory (a first-principles numerical integrator built from those constants alone) as the derivation instrument.

Every value belongs to exactly one of four origin classes:

The remaining fitted content of the entire theory is a single 5.1″ residual term (a series-truncation artifact of the abridged tables) and coefficients below 0.15″. The complete technical derivation record — every experiment, budget, and gate — lives in the repository’s technical documentation (doc 66) .

0. The inputs — the complete list

Everything in this document is computed from the closed set below plus Newtonian gravity. Nothing else enters. The set has four groups:

Framework constants (the model’s own structure — shared with every other part of the Holistic Universe Model, not lunar-specific):

Physical constants (standard observed values — IAU/JPL/LLR class, cited, none fitted here):

Lunar inputs (the Moon’s own numbers — and this is the point of §3: most of them are not free):

Epoch phases (initial conditions, required by any theory including Brown’s and ELP): the five J2000 argument constants (L′₀, D₀, M₀, M′₀, F₀), the scene’s three start positions, and the Sun-side J2000 anchors (mean longitude 280.46646°, perihelion 102.947°).

That is the entire list. Given these inputs, gravity produces the periodic tables of §2, the emergent periods of §3, the secular budget of §1 and §5, and the deep-time behaviour of §6 — with one 5.1″ fitted residual remaining in the whole construction.

1. The five fundamental arguments

Meeus Ch. 47 builds the theory on five polynomial arguments. Their J2000 constants are epoch anchors (by design); their rates and higher coefficients are now derived or attributed:

L′ = 218.3164477 + 481267.88123421·T − 0.0015786·T² + T³/538841 − T⁴/65194000
D  = 297.8501921 + 445267.1114034·T  − 0.0018819·T² + T³/545868  − T⁴/113065000
M  = 357.5291092 + 35999.0502909·T   − 0.0001536·T² + T³/24490000
M′ = 134.9633964 + 477198.8675055·T  + 0.0087414·T² + T³/69699   − T⁴/14712000
F  = 93.2720950  + 483202.0175233·T  − 0.0036539·T² − T³/3526000 + T⁴/863310000

The rates. Every rate is an equinox-of-date quantity: inertial content plus the general precession p = 360·13/H of the Earth Fundamental Cycle. Two of the composite argument rates reduce to exact lattice identities: the Moon’s sidereal mean longitude rate equals L′_tropical − p to 0.003 ppm, and the Jupiter-perigee argument rate (2L′_trop − M′ − 2n_Jupiter) closes to 0.19 ppm. At geological time the arguments do not ride these frozen polynomials: they are chain-integrated through the framework’s evolving months, precession, and solar mass loss.

The L′ secular terms — the crown of the derivation. The T² coefficient (−5.68296″/cy²) decomposes against primary sources with zero free parameters, closing to 0.08″:

The T³ term (1/538841 °/cy³) — historically an unexplained tail — is derived at 98.8%: it is the Adams–Laplace channel’s own curvature (the secular ë of Earth’s eccentricity passed through the channel, +104.2%) plus the obliquity carrier’s second order (−6.5%) and the frame’s ṗ_A T³ (+1.2%). The T⁴ is 41% channel, remainder documented (0.004° at 584 BCE).

The perigee and node (M′, F). Their secular T² coefficients follow the solar-perturbation channel Ẍ/Ẋ = s·3eė/(1−e²) along the framework’s own H/3 eccentricity line. The shipped exponents (s_ϖ = 2.407, s_Ω = 1.018) are frame-effective values that reproduce Meeus exactly; removing the equinox-of-date frame acceleration gives the physical exponents 2.479 and 0.867 — which the three-body laboratory reproduces from pure gravity at 100.3% and 101.5%.

The channel in full, checkable form — the eccentricity line, the perturbation strength, and the rate law:

e_E(t) = base · (1 + cos θ(t) / 2)          θ(t) = 3 · (t − balancedYear)/H · 360° − 180°
                                            (balancedYear = −302,635 — the System Reset epoch)
g(e)   = (1 − e²)^(−3/2)                    (solar perturbation strength)

ϖ(T)   = ϖ₀ + ϖ̇₀ · [ T + ∫₀ᵀ ( (g(e_E(t))/g₀)^s_ϖ − 1 ) dt ]     (node Ω: same with s_Ω)

J2000 Taylor check:  T² = s · Ẋ₀ · κ / 2,   κ = 3eė/(1−e²)
  s_ϖ = 2.407  →  −0.010318 °/cy²   (Meeus ϖ: −0.010320)
  s_Ω = 1.018  →  +0.0020752 °/cy²  (Meeus Ω: +0.0020753)

The line’s inputs are §0’s base, the balanced year, and H — nothing lunar; the observed J2000 eccentricity (−0.9%) and its rate (+1.7%) are predictions of this line, not inputs.

2. The periodic series, derived

The heart of Meeus Ch. 47 is its tables of periodic terms with argument multipliers (D, M, M′, F). The framework’s three-body laboratory — built only from the framework’s constants (GM of the Earth–Moon system, the mass ratio, the AU, the sidereal year, and the lunar orbit inputs) — reproduces them from gravity alone. Coefficients in Meeus’s unit of 0.000001°:

Longitude (ΣL), top of the table:

Latitude (ΣB), the full main family:

The classical small-parameter scalings behind the table — every named term traces to one product of the §0 inputs (m = n_Sun/n_Moon ≈ 1/13.37):

equation of center  =  2·e_M − e_M³/4  =  6.288774°     (exact identity, verified at 2 ppm)
evection            ∝  m · e_M         →  1.274027°
variation           ∝  m²              →  0.658314°
annual equation     ∝  m · e_S         →  0.185116°     (carries the E-factor = e_E(t)/e_E(0))
parallactic ineq.   ∝  m · (a/a′)      →  0.034720°

Three structural identities inside these tables:

• The equation of center is exact framework algebra: the main term 6.288774° = 2e − e³/4 at the framework’s lunar eccentricity, an identity verified at 2 ppm.
• Meeus’s E-factor (the eccentricity damping of M-bearing terms, 1 − 0.002516T − 0.0000074T²) is e_Earth(t)/e₀ — the framework replaces the unbounded polynomial with its bounded H/3 eccentricity line.
• The three additive corrections: A2’s amplitude is derived from gravity at 101 ± 8% (322 vs Meeus’s 318, in 10⁻⁶ ° — and the historical “window-growing multiplet” around it dissolved into a measurement artifact); the “Lp − F” flattening term (1962 × 10⁻⁶ °) is derived from Earth’s oblateness at 99.8%; A1’s amplitude is gravity-sized while its rate is observationally defined — the framework proved it sits on the 18V − 16E − M′ near-resonance where parts-per-million in planetary years move the beat by degrees per century, so no finite theory can pin it: nature does.

3. The periods that emerge

Brown needed the lunar theory’s free constants; ELP adopted them; the framework derives two of the three lunar inputs from the third plus gravity:

The catalog periods and the dynamical ones differ by exactly the frame: node and equinox are both retrograde (the equinox chases the node), the apsis is prograde against it. The framework’s H/13-frame bookkeeping, the of-date observables, and the star-referenced dynamics are three consistent views of one motion:

frame conversion:   N_of-date = N_star ∓ 13 cycles per H     (the equinox regresses once per H/13)
emergence law:      T_precession ∝ T_year² / T_month          (Brouwer–Clemence scaling, ± the e_E channel)
deep-time invariant: T_apsidal × H = H₀²/N = constant

At J2000 the invariant products are T_apsidal × H = 2,966,728 yr² and T_nodal × H = 6,241,369 yr² (37,899.5 and 18,014.875 cycles per H — eighth-integers, because the lattice counts are integers per 8H) — held exact at every epoch by the (H/H₀)² scaling.

The inclination. The catalog ”5.1453964°” is not the Moon’s dynamical tilt — it is Brown/ELP’s theory constant, the normalization of the latitude coefficient family. The Moon’s true mean osculating inclination, measured directly from the theory itself (angular-momentum vector over two node cycles), is 5.1573°, and the framework’s laboratory reproduces the compression between the two views (sinF coefficient / dynamical tilt = 0.9944) at 0.01% from pure gravity. A curiosity worth recording: the compact formula “1 − m²” reproduces this compression to 1″ at the real Moon — and the framework’s parameter scans disproved it as a theorem: it is a coincidence of our Moon’s particular eccentricity and tilt ((3/2)e² + sin²i/8 ≈ m² to 1%). The honest statement — a measured map, not a closed form — is part of this theory’s discipline.

The axial tilt. The Moon’s spin axis obeys Cassini’s laws (spin axis, orbit normal, and ecliptic pole coplanar), holding a measured 1.5424° obliquity to the ecliptic. The catalog “tilt to orbit” of 6.687° is a convention-composed number: 5.1453964° + 1.5424° = 6.6878°, built on the Brown/ELP inclination constant — with the dynamical tilt (5.1573°) the same sum reads 6.700°, so the two catalog numbers must be read in one convention together. Deriving the 1.5424° Cassini obliquity from the framework’s nodal rate and the lunar gravity moments is registered future work — it would remove one more input and settle which plane convention the composition should carry.

4. What the fitted patch turned out to be

The model’s Moon historically carried a small fitted RA/Dec correction against JPL. Decomposing it revealed that 98–102% of the patch was annual aberration — the model’s frames carry apparent-Sun content while the reference was astrometric. The correction is now computed analytically from the framework’s speed of light and Sun vector (itself free of hardcoded coefficients: rates from the framework year and H/16 perihelion cycle, equation-of-center from the Kepler identity):

u′ = normalize( u − v_Earth / c )        (astrometric direction from the apparent one;
                                          |v_Earth|/c = κ = 20.5″, the aberration constant)

What remains fitted in the entire visual pipeline is one 5.1″ term — the truncation difference between the abridged 60-term tables and the full numerical ephemerides — plus coefficients under 0.15″.

5. The precession acceleration, composed

The deepest frame quantity in the budget — the acceleration ṗ_A of general precession (+1.1054″/cy², IAU 2006) — is composed from first principles:

• the ecliptic-of-date motion comes from the framework’s eight-body laboratory, pure gravity: π̇ = 47.5″/cy (observed: 47.0);
• the obliquity history ε(T) is the framework’s own H/3 + H/8 cycle line;
• the equator is a luni-solar precession cone about the moving ecliptic pole with the classical cos ε torque law;
• one rate anchor pins the composed J2000 rate — and independently lands the luni-solar rate at 5039.2″/cy, within 0.013% of the IAU value that was never an input.

Composed result: +1.15″/cy² = 104% of the observed acceleration. En route, the framework’s long-standing “remaining 38% of the equinox acceleration” was identified exactly: it is the planetary χ-channel — the equinox sliding along the equator because the ecliptic tilts underneath it.

6. Bounded at deep time

Polynomial ephemerides explode outside their fit windows: Meeus’s T² parabola for the mean longitude reaches 7 892° of spurious longitude at +200 000 years, and his E-factor reaches E = −40. The framework replaces every secular polynomial with a bounded carrier riding a physical cycle:

• the planetary term rides the bounded H/3 eccentricity line (staying ≤ ~220° where the parabola reached 7 892°);
• the figure + frame terms ride the bounded obliquity cycle;
• the tidal evolution runs through the framework’s Earth-rotation chain, valid across ±500 million years;
• the lunar precession periods obey the deep-time invariant T_apsidal × H = constant.

The complete factored deep-time law for the precession periods — invariant mean × bounded modulation, implemented identically in the simulator and the website’s deep-time engine:

T(t) = T₀ · (T_year(t)/T_year,0)² · (T_month,0/T_month(t))  /  (g(e_E(t))/g₀)^s

The same construction that reproduces Meeus in the historical window degrades gracefully into deep time instead of diverging.

7. Verification

The derivation is validated once, against every record class at hand:

The last line of the ledger is the discipline that produced it: the framework derives; the classical values confirm. Where a number could not be derived, this document says so — the A1 rate belongs to nature, the epoch phases belong to the calendar, and 5.1 arcseconds belong, for now, to the truncation of the tables.

Complete technical record: experiments, budgets, certification gates, and the laboratory source are in the repository documentation . Values in this document are frozen as DLT-1.