Timekeeping & Delta-T

Earth’s rotation is not constant — it speeds up and slows down in cycles, causing the Length of Day (LOD) to vary. This variation creates the difference between atomic time and observed time, known as Delta-T (ΔT). This page describes the millennial-scale LOD cycle, derived closed-form from the J2000-anchor value H = 335,317 yr. The long-term secular evolution of LOD across geological time (Driver 1 — Earth-Moon tidal recession, the very-long-term H expansion) is the subject of Expanding Resonance; the cycle described here sits on top of that slow background. This page builds on Days & Years.

Two timekeeping systems

TT is a uniform timescale calibrated to the mean solar day around 1820 AD — independent of Earth’s actual rotation, used for astronomical predictions. UT represents Earth’s observed rotation angle relative to an inertial reference frame — time as actually experienced on Earth.

What is Delta-T?

Delta-T (ΔT) is the difference between the two systems:

ΔT = TT − UT

• When Earth rotates slower (LOD > 86,400 s): UT falls behind TT, ΔT increases.
• When Earth rotates faster (LOD < 86,400 s): UT gains on TT, ΔT decreases.

The IERS maintains the official ΔT values. Leap seconds are occasionally added to UTC to keep it within 0.9 s of UT1; per the 2022 CGPM resolution, leap seconds are scheduled to be phased out (or have their tolerance significantly relaxed) by 2035.

The Delta-T V-shape

The historical Delta-T curve from 1650 AD has a distinctive V-shape:

1. ~1820 AD: SI second was calibrated to the mean solar day → ΔT = 0 by definition.
2. ~1820–1900 AD: ΔT decreased slightly into negative values as LOD stayed near (and briefly below) 86,400 s.
3. ~1900 AD: ΔT reached its actual minimum (~−3 s) when LOD finally exceeded 86,400 s and the trend reversed.
4. After ~1900 AD: LOD has stayed above 86,400 s, so TT pulls ahead of UT (ΔT increasing).

The underlying trend is that LOD has been slowly increasing over centuries, crossing the 86,400-second mark around 1900 AD.

The model’s interpretation: a millennial cycle

In the Holistic model, LOD varies cyclically over millennia, driven by obliquity and eccentricity:

• LOD slightly increases until ~30,000 AD
• LOD then decreases until ~2,000 AD
• Short-term fluctuations are superimposed on this long-term trend

This millennial cycle is itself superimposed on the slow secular LOD evolution captured by the proper-physics two-layer formula (Expanding Resonance): Earth-Moon tidal coupling makes the very-long-term LOD trend monotonically upward (H expands by ~0.02 % per Myr at the current rate), but on millennial timescales the LOD oscillates around that secular background. The two effects act on very different timescales and add: the long secular tidal background is what Wells 1963 measured as ~400 days/year in the Devonian; the millennial cycle described here is what’s measured in century-scale Delta-T records.

The model does NOT claim tidal friction doesn’t exist. Established physics is well-documented: lunar laser ranging measures the Moon receding at ~3.8 cm/yr (angular momentum transferred from Earth); Earth’s rotation slows at ~+1.7 to +2.3 ms/century from the combined tidal-friction signal and post-glacial rebound; eclipse records over 2,700 years confirm the long-term slowing (Stephenson, Morrison & Hohenkerk 2016 ); since ~2000, ice melt in Greenland and Antarctica has added ~+1.33 ms/century, projected to reach ~+2.62 ms/century by 2100 (Adhikari & Ivins 2016 ).

The model’s claim is that a millennial-scale variation is superimposed on the tidal slowing. Standard theory holds that tidal slowing dominates monotonically over all timescales. This is the testable difference: can a millennial cycle exist alongside (and partially counteract) tidal slowing? The model predicts yes — continued precision measurement over centuries can distinguish the two pictures.

Two pieces of independent evidence are consistent with cyclical LOD behaviour and are treated canonically at Supporting Evidence:

• The 2020–present rotation speedup: Earth has been rotating faster against the long-term tidal slowing. 2020 produced 28 shortest days since atomic clocks were invented; July 5, 2024 set the all-time record at 1.66 ms under 24 hours; 2025 saw three notably short days (July 9, July 22, August 5) without breaking the 2024 record. The IERS acknowledges difficulty predicting LOD beyond ~6–12 months. Qualitatively consistent with the model’s cyclical prediction; not yet evidence of the long-term trend reversal itself. See Supporting Evidence §3.
• The 1-billion-year day-length stall: Mitchell & Kirscher (2023, Nature Geoscience 16, 567) showed Earth’s day length stalled at ~19 hours for roughly 1 billion years in the mid-Proterozoic (2.0–1.0 Ga) — atmospheric thermal tides balanced lunar tidal drag. Complex, non-monotonic LOD dynamics are not unprecedented. See Supporting Evidence §4.

Empirical validation against the historical eclipse record

The closed-form ΔT formula on this page has been tested directly against the documented historical record. 19 well-localized solar eclipses spanning -762 BCE (Bur-Sagale, Nineveh) to 1654 CE (European total) were used to compare the model’s pure-tidal ΔT against the empirical Stephenson-Morrison 2004 fit. Both ΔT models were evaluated against the same observation sites and the same Moon polynomial:

Pure-tidal wins both — the broad visibility count and the per-event mean residual. The Stephenson fit fails on the two Cairo events of Ibn Yunus 979 May 28 and 1004 Jan 24, where its ΔT is too low and pushes sub-solar east of the observation site. The model’s higher pure-tidal ΔT recovers both.

The longstanding consensus that “Stephenson’s empirical fit is preferred because non-tidal Earth-rotation speedup is real” turns out, on this cleaner methodology, not to be required by the data. A pure-tidal Farhat-based ΔT — with no fitted parameters and no non-tidal component — fits the eclipse record at least as well as Stephenson’s polynomial.

The full validation methodology, the underlying Moon polynomial cross-check against NASA’s Five Millennium Catalog (±15 min over 2,500 years), and the open question about the Thales eclipse date are treated canonically at Historical Eclipse Validation.

Deep-time LOD — the secular background

The closed-form formulas on this page describe the millennial cycle around the modern J2000 anchor. Across the much longer timescales of paleoclimate and the Earth-Moon system’s full history, LOD has evolved monotonically with Earth-Moon tidal recession:

These values come from a proper-physics two-layer LOD formula: a Moon-distance polynomial calibrated against Farhat 2022’s deep-time tidal-evolution data, combined with angular-momentum conservation. The Devonian prediction matches Wells 1963’s coral growth-ring count to within 1 % (Supporting Evidence §14). Full derivation and the Hadean Moon-at-Roche-limit self-validation are at Expanding Resonance.

Compute ΔT and LOD at any year

The complete closed-form expressions for ΔT and LOD are in Formulas.

Continue to Fibonacci Laws to see how a single timescale generates all planetary precession periods, orbital tilts, and eccentricities.