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 and coefficients below 0.15″ — and even that term is now decomposed by measurement, in one sign convention — the patch is Meeus − JPL: −1.13″ is named series truncation (Meeus’s three-term compression of ELP’s ~14,000-term planetary series), and the remaining −4.05″ is the offset between analytic lunar theory and the JPL DE441 numerical ephemeris. That second half is not a step between analytic lineages — ELP-2000/82B and ELP/MPP02 agree to 0.03″ on this term, so JPL sits ~4″ from both — which means it has no series-term decomposition in any analytic theory (see §4). 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.

Epoch. Every period, rate and angle below is a J2000 value — J2000 (2000 January 1.5 TT) is the epoch at which the framework is anchored, and it is the epoch at which all the comparisons in this document are made. These are not constants of nature: under the deep-time layer (§6) the periods and rates evolve, while the integer structure that organises them does not. Only the genuinely universal quantities in the second group — the speed of light and the mass ratios — are epoch-free.

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 deep-time branch carries this same content as a bounded carrier rather than a polynomial (§6), and its empirical size is confirmed against JPL’s DE441 ephemeris: a 300-epoch fit across 27 centuries reads the certified tail at ×1.0 (2c = +0.11 ± 0.13″/cy²).

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 number every reference gives for the Moon — 5.145° — is worth pausing on, because the Moon’s orbit does not hold a fixed inclination: it swings between roughly 4.98° and 5.30° over an 18.6-year node cycle, driven by the Sun. So “the inclination” can only ever mean some particular average or normalization, and the two in use here are different ones. The catalog ”5.1453964°” is not the mean tilt of the orbit plane — it is Brown/ELP’s theory constant, the normalization of the latitude coefficient family. The mean of the actual oscillation, 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 — derived. The Moon’s spin axis obeys Cassini’s laws (spin axis, orbit normal, and ecliptic pole coplanar, co-precessing with the node), holding a measured 1.5424° obliquity to the ecliptic. This number is now derived: averaging the Earth’s gravity-gradient torque over the locked triaxial figure (built from three documented GRAIL/LLR constants — J₂ = 203.305 × 10⁻⁶, C₂₂ = 22.4261 × 10⁻⁶, C/MR² = 0.392728) and balancing it against the framework’s own node-regression rate yields an equilibrium obliquity of 1.5528° — 100.7% of the measured value. The Earth-only mass fraction M_E/(M_E+M_M) in the torque is the 1.2% term naive closed forms miss. Averaging the torque over the Moon’s real (solar-perturbed) orbit instead of the Keplerian ellipse — the classical ELP-2000/82B series, integrated over a full 18.6-year node cycle in the frame co-rotating with the mean node — gives 1.5551° (100.83%), no closer to the measurement. Nor could any named correction close the remaining ~0.8%: the fluid core’s direct pressure torque is ~176× too small, the Sun-coherent orbit oscillations are worth only −6.5″, and no input can absorb it (C/MR² is known to 3×10⁻⁵, J₂ to 10⁻⁹).

That every candidate fell short by 5–10× was the clue — and it pointed at the method, not at the Moon. An averaged balance describes a body spinning uniformly about a fixed axis, whereas the real Moon obeys the coupled Euler equations, in which the pole orientation and the physical librations evolve together and feed back on each other. No average can contain that coupling. This is well understood in the field: modern lunar rotation theory integrates those equations numerically rather than using averaged forms, for precisely this reason. Doing the same here — three rotational degrees of freedom, torque from Earth and the Sun over the real orbit, no averaging anywhere — gives 1.5470° (100.30% of the measured value). The libration–pole coupling is worth −29.3″, 64% of the discrepancy, and the residual falls to 16.5″ (0.30%) — the size of the known small channels not modelled here (elastic tidal modification of the effective moments, degree-3 gravity, a two-layer fluid core), rather than many times larger than all of them.

The integration is verified two ways: two different starting obliquities converge to the same forced state within 0.6″, and the result is unchanged at half the step size.

What is and isn’t claimed. The Cassini-state physics is classical — Colombo (1966) and Peale (1969) established it, and the numerical treatment is standard practice. Nothing here is new lunar theory. The claim is narrower and is the same one this document makes throughout: the framework’s own constants — its sidereal month, its node rate, its mass ratio — together with three published gravity coefficients and Newtonian gravity, reproduce a lunar constant the model had previously adopted, to 0.30%, with nothing fitted. The remaining 16″ is named rather than absorbed; closing it would mean modelling elasticity and the fluid core, which is a question for lunar interior physics, not for this framework.

The catalog “tilt to orbit” of 6.687° remains 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. The simulation now applies the composition in its own convention (dynamical inclination + measured obliquity), so the rendered spin-to-ecliptic obliquity equals the measured 1.5424°.

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 plus coefficients under 0.15″ — and the 5.1″ term is decomposed by direct measurement against the full 37,863-term ELP-2000/82B series: −1.13″ is named truncation content (almost entirely the planetary family — the abridged tables compress ELP’s planetary series into 3 additive terms; the 60-term main-problem cut itself contributes only −0.04″, a testament to Meeus’s selection), and −4.05″ is the offset between analytic lunar theory and the JPL DE441 numerical ephemeris the correction was fitted against. Measuring the MPP02 series directly refutes the idea that this half is an ELP-82→MPP02 step: the two analytic theories agree to 0.03″ on this term and to 0.30″ RMS in longitude over 2000–2050, so both sit together while JPL sits ~4″ from each. The gap is analytic-vs-numerical, flat rather than growing across the window, and therefore has no series-term decomposition in any analytic theory. Attributed by cause rather than by term: documented model lineage, not free physics.

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 T³/T⁴ tail rides the secular-ë completion carrier — the integral of the curvature difference between the secular Taylor (§0 inputs) and the H/3 line, under a cos² taper on the H/12 quarter period: exact in the historical window, saturating to a frozen offset at deep time instead of diverging;
• the of-date rates ride the dynamical axial precession — the beat of the sidereal and solar year evaluators (the same identity the simulator’s precession panel displays: ~25 771 yr at J2000, evolving with the model’s own year lengths at any epoch), integrated against the kinematic lattice mean, which is naturally bounded because the beat oscillates about that mean; the residual chain-rate offsets are self-measured at build against the certified J2000 rates and applied under the same taper;
• 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, by measurement, to the lineage of the tables: one part named truncation, five parts the generation gap between the classical series and the modern numerical ephemeris.

8. Addendum — the derived Delaunay tail (DLT-1.1)

After the DLT-1 freeze, a dedicated accuracy campaign asked how far the derived series could be pushed toward the analytic limit. Two instruments — a term-by-term budget attribution of the residual against JPL, and an alias-breaker that separates near-degenerate Delaunay arguments by frequency clustering (a naive joint fit had exploded a near-degenerate triplet to compensating 150,000″ amplitudes — the instrument now refuses that class) — identified five further longitude terms in the classical Delaunay families. All five are derived-class: standard main-problem arguments whose amplitudes come from the three-body laboratory, not from fitting.

With the tail in place the shipped series stands at λ 2.84″ / β 0.35″ RMS against JPL (all-phase, modern window). Two further options were measured and rejected:

• Declared-fitted head amplitudes (adjusting the three biggest Meeus amplitudes by ~5×10⁻⁵ toward their DE-fitted values, worth 0.85″): rejected by owner decision — the chain keeps every constant derived.
• Lifting the MPP02 series wholesale: rejected for the same reason; the framework derives, it does not adopt.

What remains is the ~4″ analytic-vs-DE441 representational floor already characterised in §4 — the flat offset every analytic lunar theory (ELP-2000/82B and MPP02 agree with each other to 0.03″) holds against the numerical ephemeris. Going below it would mean leaving analytic theory altogether.

DLT-1.2 — the node family and the deep dust

Two further derived rounds followed, and both began by finding that a “floor” was a blind spot.

The J2 node family. A slow ~17-year band in the residual had been written off as beyond-three-body content. A 200-year probe showed its period to be 16.9 years, not the 18.6-year node or the 19.9-year Jupiter–Saturn beat, and named it: the family of arguments built on the lunar node Ω (Mp ± Ω, 2F + Ω, 2D + Ω, F − Ω), which no integer Delaunay catalog contains. The three-body laboratory could not produce it — until Earth’s oblateness was switched on. With the J2 torque in the integration every member appears at the amplitude JPL shows, and two of the classical theory’s documented J2 terms (the +1962e-6 and −2235e-6 additives in the Meeus tables) are re-derived at 1.03 and 0.97 as internal controls. Five node-family terms shipped, all derived; the latitude went from 0.65″ to 0.61″.

The deep dust. What was left was thought to be thousands of sub-0.1″ terms. A census against the ELP/MPP02 catalog — used strictly to find candidates, never to take amplitudes — showed the leading “dust” to be ordinary main-problem terms sitting just below the Meeus tables’ 0.4″ cutoff, in argument classes no earlier extraction had enumerated (three times the elongation, three times the solar anomaly, four times the argument of latitude). Every candidate was confirmed real against dense JPL positions, and the laboratory then derived 43 longitude and 33 latitude terms at the census amplitudes; the four it could not reproduce were left out, on the rule that the chain derives or does not ship. The series now stands at λ 2.84″ / β 0.35″ — the latitude at the floor of what a comparison with JPL can even resolve, since the reference theory itself agrees with JPL to only ~0.33″ there.

One eccentricity law. The E-factor above rides the framework’s H/3 eccentricity line. Since these rounds the eclipse Sun’s equation of centre rides the same line (The Derived Sun §4): eccentricity is frame-invariant and may carry only fixed-frame lattice periods, and giving Sun and Moon one law was measured to remove the Sun’s entire eccentricity-rate error against JPL.

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.