Earth’s Clock

The model reads every long cycle of Earth’s orbit and spin against one clock: the mean lunisolar precession period, the time Earth’s spin axis takes to trace one circle on the sky. The clock is not a constant of nature. It is composed from the Sun’s and the Moon’s pull on Earth’s equatorial bulge, and both of those change on geological timescales, so the clock lengthens — and every period read against it is a reading of date, not a fixed number.

This page establishes:

• What the clock is made of, and its J2000 value, 25,771.6 years — What the clock is made of.
• Earth’s five long periods as ratios to it, three of them frame arithmetic once two are known — Earth’s five long periods.
• How the clock lengthens through geological time, and which periods follow it and which do not — The clock through geological time.
• What the model no longer carries — the fitted anchor, its counters and correction series, the relations read off them and the integer framing — and why — Retired machinery.

What the clock is made of

Earth’s spin axis precesses because the Sun and the Moon pull on its equatorial bulge. The rate is a composition of the two torques on the spin:

ψ̇(t) = [ω(t)/ω₀] · p₀ · [f_S + (1 − f_S) · (a₀/a_M(t))³]

where ω is Earth’s spin rate, p₀ the J2000 rate, f_S the Sun’s share of the J2000 torque, and a_M the Moon’s distance. At J2000:

• Rate: 50.2880″ per year — the IAU 2006 general precession p_A, taken as an input. The Sun’s torque supplies 31.6 % of it and the Moon’s the rest.
• Period: 25,771.6 years — one full circle at that rate, the model’s one J2000 precession reading.

The J2000 rate is cited, not fitted, and every face of the model — the simulator’s panels, the published packages, this site — reads the same value. The year lengths follow from it: the J2000 tropical year, 365.242190469 days, is derived from the IAU sidereal year and this rate (Days & Years).

Two things change the clock through time: the tides slow the spin (ω falls), and the Moon recedes (its torque weakens as the inverse cube of its distance). Both are measured histories, not fitted constants — the same recession history that carries the length-of-day record. Expanding Resonance is the time-evolution layer.

Earth’s five long periods, against the clock

Earth’s orbit and spin carry five long periods. Each is the model’s own dynamical value — read from its N-body chain and its composed spin rate — and each is best understood as a ratio to the clock.

Three of these are frame arithmetic once the other two are known. The perihelion-of-date period is the beat of the clock with the apsidal period (1/T_peri = 1/T_p + 1/T_aps): the equinox precesses one way, the perihelion drifts the other, and the two rates add. The obliquity beat is the spin axis’s own inertial precession (ψ̇ᵢₙ, 0.097 ″/yr faster than the equinox clock — the planetary precession) against the dominant nodal mode of Earth’s orbit (2π/(ψ̇ᵢₙ − |s₃|)): the tilt oscillates at the rate the spin axis gains on the orbit plane. The two motions themselves — what moves, in which direction, and how each is observed — are the subject of Precession; the obliquity beat and the orbit plane are Obliquity & Inclination.

The clock through geological time

The period is not a constant even within the modern era: it oscillates by about two thousand years as the orbit’s own plane moves, and Vondrák’s expression — drawn here across the span it is valid for — follows the same oscillation. Within the ±26,000-year window the axial period wanders between 24,802 and 26,559 years as the orbit’s eccentricity and inclination modulate the torques, and the perihelion-of-date period between 11,303 and 26,699 years. What changes at geological time is the mean the oscillation rides on.

Because the spin slows and the Moon recedes, the clock lengthens:

The precession period shortens monotonically into the past. The equinoxes completed one circuit in ~16,584 years at 1 Gyr ago against ~~25,772 yr today, and the period collapses to a few years at the end of the cited distance history (the Earth–Moon genesis, 4.429 Ga, the Moon at 6.19 Earth radii) — the composition law at the edge of the input’s domain, not a period the model reads off. Two things follow from a clock of years. The tropical year of date is the sidereal year beaten with the clock, 1/T_trop = 1/T_sid + 1/T_p — Hipparchus’s relation, the one by which precession was found, and its deep-time coupling to the day and the Moon’s distance is Berger, Loutre and Dehant’s (1989) and Lambeck’s (1980); the model’s part is to hold it on the same axis as every other period — so at the edge it is 327.8 SI days against a sidereal year of 364.96 — 1,429 solar days of 5.51 hours, a ratio of 0.8975 to today’s year. And the edge value is a floor, not the Earth’s: the law’s lunar term assumes the Moon’s orbit near the ecliptic, while inside the Laplace radius (about 10 to 17 Earth radii, the earliest part of the history) the Moon’s orbit follows the equator and its torque on the equatorial bulge averages toward zero — on the solar torque alone the period there would be of order ten thousand years. Modelling that transition is an open item; until then every value inside it is reproducible, not converged, and says so where it is shown. The classical observable known since Hipparchus is not a fixed astronomical constant but a now-snapshot of a slowly-evolving clock.

The obliquity beat shortens faster than the clock itself, because the nodal mode s₃ is an orbital quantity that does not follow Earth’s spin: as the precession rate rises into the past, the beat 2π/(ψ̇ − |s₃|) closes in. A dated record sees the beat of the pole rate’s mean over the record’s window, not the J2000 instant: over the LR04 record (0–5.32 Ma) that mean is 50.519 ″/yr and the realized obliquity period ~40,924 yr, measured on the model’s own obliquity history (~40,916 yr) and on La2004’s (~40,900 yr) alike (Supporting Evidence §7). This split — spin quantities ride the clock, orbital quantities do not — is the model’s pre-registered deep-time prediction, tested against Precambrian cyclostratigraphy at 1.4 and 2.46 Gyr. The climate formula’s precession-band lines ride the clock; its eccentricity-band lines, planetary beats, do not (Climate Formula).

At today’s rate the clock lengthens by ~0.025 % per million years, about 6 yr per Myr at the modern epoch — small on human timescales but in principle observable in high-precision IAU precession-rate measurements over decades; distinguishing it from mainstream tidal-LOD contributions and planetary-precession evolution is the open observational question. The strongest evidence comes from deep time, where the cumulative shift is large: paleontological day counts and cyclostratigraphic period drift across the Phanerozoic (Supporting Evidence §11 for the Wells 1963 Devonian-coral validation table).

Retired machinery

The model carries no fitted timing anchor, no unit built on one, no counters and no correction combs. Every time-domain quantity rides the clock T_p(t) and the N-body series: Earth’s eccentricity is the series, the perihelion of date is the series’ own ϖ, the obliquity is the integrated spin axis precessing about the series’ orbit plane, and the year and day lengths are one family of date built on the clock and the IAU year inputs. The 1246 AD perihelion–December-solstice alignment is a dated event of the IAU mean elements (the model’s own series crosses at 1246.1 AD), not the phase origin of anything; the perihelion reaches the June solstice half a perihelion-of-date cycle later.

This section is the one record of what left the model and why.

• The fitted anchor and its correction series. A fitted interval, calibrated on the 1246 AD alignment, served as the unit of small Fourier corrections on its integer fractions — on the perihelion, the obliquity, the year lengths, the cardinal points, the Sun’s longitude and a right-ascension day offset. All of them are deleted, with the counter that measured time in the anchor’s units. The quantities they corrected are now read directly from the clock and the series. The “Earth Rate Deviation” (the perihelion series’ derivative) and the “measured solar day” built on the RA day offset (an instrument’s window term, not a motion) went with them.
• Relations that held only on the counter. Two relations were properties of the deleted counter’s own definition, not of the dynamics, and are withdrawn: the day-count near-invariant (the counter times the days per year held near-constant) and the Lunar Precession Invariant (the Moon’s apsidal period times the counter held constant, the lunar periods scaling as its square). The Moon’s of-date perigee and node are compositions of its inertial chain, and at deep time they part from the retired law. The deep-time lunar prediction built on the invariant is withdrawn and stays listed as such on Predictions.
• The integer framing. The anchor was once read as Earth’s master cycle, the five long periods as its integer divisors, and a “Solar System Resonance Cycle” of eight such intervals as the common period every planet’s long periods divided into, with a “System Reset” at its end. On the model’s own dynamics none of it holds: the five periods sit off the integer values and the ratios wander far from them across the ±26,000-year window (table above), and the planets’ N-body eigenfrequencies land on the eight-unit grid no more often than chance. The planets’ periods are the N-body chain’s own values (Moon & Planets, N-body Exploration); the full record is doc 109 in the 3d repository .

Continue to Precession for the two motions behind the clock and how each is observed, Obliquity & Inclination for the obliquity beat and the orbit plane, and Expanding Resonance for the clock through geological time.