Days & Years

A sidereal day and a stellar day differ by just ~9.12 milliseconds. That tiny gap — caused by axial precession — connects the definition of a “day” to the definition of a “year” and ultimately to the ~25,794-year precession cycle.

Three types of days

A solar day is ~4 minutes longer than a sidereal day because Earth moves along its orbit while rotating: after one full rotation relative to the stars Earth must rotate a little more for the Sun to return to the same position.

The ~9.12 ms difference between stellar and sidereal day has been debated for decades without official consensus. The model attributes it to axial precession: as Earth’s axis precesses over the mean axial precession cycle (~25,794 years), the reference point for the sidereal day (the precessing equinox) shifts slightly each day relative to the fixed stars. The slip accumulates with precise consequences shown in The precession accumulation below.

Three types of years

The sidereal year in seconds is the model’s J2000 anchor — Earth’s orbit measured against an unchanging reference frame, held constant within the modern-era scope. Every other year-length value derives from it. The solar year fluctuates because the equinox shifts with axial precession; the anomalistic year fluctuates because perihelion shifts with perihelion precession.

Solar vs sidereal year

The solar year is 1,224.6 seconds shorter than the sidereal year. This is the same axial-precession slip seen at the day level, scaled up:

Every year the Sun appears 1,224.6 seconds “behind” its previous position relative to the fixed stars when measured at the equinox. Over ~25,771 years this accumulates to a full 360° shift — one complete precession cycle:

1,224.6 seconds/year × ~25,771 years = 31,558,149.76 seconds ≈ 1 sidereal year

This is why the equinoxes “precess” through the zodiac constellations.

The coin rotation paradox

The coin rotation paradox  is the key to the day/year hierarchy:

When a coin rolls around another coin of equal size it rotates twice — once for the orbit, plus once for its own rotation.

At the day level, in one year Earth rotates:

• ~365.25 solar days (rotations relative to the Sun)
• ~366.25 sidereal days (rotations relative to the stars)

The difference is exactly 1 extra rotation — Earth’s orbital motion adds one rotation per year.

At the year level, the paradox works in reverse because the two motions run in opposite directions: Earth orbits its wobble center clockwise over ~25,794 years, while Earth’s perihelion point orbits the Sun counter-clockwise over ~111,772 years.

Just as there is exactly 1 more sidereal day than solar days per year, there is exactly 1 fewer sidereal year than solar years per axial precession cycle.

The precession accumulation

The coin rotation paradox is not just a counting trick — it verifies quantitatively at both levels. Because the precessing equinox completes one full loop over the mean axial precession cycle, the accumulated slip between precessing and fixed references must equal exactly one full rotation (day level) or one full orbit (year level):

Day level:   9.12 ms/sidereal day × 366.25 sidereal days/year × ~~25,794 years
           = ~86,164.09 seconds = 1 sidereal day → 1 sidereal day less per axial precession cycle

Year level:  1,223.37 s/year (mean solar–sidereal year difference) × ~~25,794 years
           = 31,558,149.76 seconds = 1 sidereal year → 1 sidereal year less per axial precession cycle

The 9.12 ms/day is the stellar–sidereal day difference introduced above. The 1,223.37 s/year is the mean difference between the solar year and the sidereal year. Both use different epoch values but produce the same result: the product always equals the sidereal year (J2000 anchor), because a faster precession rate means a smaller annual difference and vice versa.

How the years connect geometrically

Starting from Earth’s perspective at Position 0 (Sun and Earth aligned at the start):

1. After 1 solar year (~365.2421898 days) the Sun returns to the same seasonal position (Position A).
2. After 1 sidereal year (~365.25636297 days) the Sun aligns with the same fixed star again (Position B).

The angular difference between A and B is the annual precession shift (~50.29 arcsec/yr at J2000).

The anomalistic year

The anomalistic year measures perihelion to perihelion:

The anomalistic year is longer than the solar year because perihelion shifts forward in time due to perihelion precession.

What each year type depends on

The sidereal year in days varies because day length changes:

Sidereal Year (days) = Sidereal Year (seconds) / Day Length (seconds)

As orbital elements change over millennia the sidereal year in days changes (via H/8 and H/3 harmonics), which changes how many days fit into the J2000-anchor number of seconds. Day length itself derives from the same anchor:

Day Length = Sidereal Year (seconds) / Sidereal Year (days)
         = 31,558,149.76 s / 365.25636297 days
         = 86,400.001798 seconds

Cardinal point variation

The tropical year length depends on which cardinal point is used to measure it. At the current epoch (perihelion in early January), Earth moves faster near perihelion (Kepler’s 2nd law) and slower near aphelion:

Total spread ~91 seconds. The pattern reverses when perihelion precesses to July (~11,725 AD): winter solstice becomes shortest and summer solstice longest. The mean of all four cardinal points cancels this effect and gives the true mean tropical year.

Physical vs kinematic LOD (the H/5 correction)

The model maintains two distinct mean-LOD values because they answer two different physical questions:

The ~3.527 ms H/5 correction represents Earth’s need to rotate slightly more per solar day to catch the Sun on the meridian, because the Sun’s apparent motion follows the ecliptic — which precesses at H/5 (the ecliptic precession cycle, ~~67,063 years). Over one solar day the ecliptic advances by 1/((H/5)·mSY) revolutions — requiring that many extra revolutions of Earth rotation:

δ_rev = 1 / ((H/5) × mSY) revolutions per day
raw H/5 kinematic  = LOD_mean + LOD_mean × δ_rev    ≈ 86,399.999676 s + 3.527 ms
                 = 86,400.003535 s          (overshoots USNO by ~1.74 ms)
LOD_real (Layer 3) = raw H/5 kinematic + Σ calibrated ΔT cycles
                 = 86,400.001798 s at J2000     (matches USNO anchor exactly)

Why other precession cycles don’t need their own correction:

• H/13 axial precession is already implicit in LOD_mean via the H/(H−13) denominator (over H tropical years the sidereal frame counts H−13 years — the missing 13 IS the axial precession). Adding an explicit H/13 correction would double-count.
• H/8 obliquity is an oscillation, not a precession — its time-averaged contribution to LOD is zero.
• H/3 inclination precession applies to the invariable-plane frame, not to the Sun’s apparent motion — using it as the LOD correction would give the wrong reference frame.
• H/16 perihelion motion contributes to the anomalistic year, not to the tropical-day counting relative to the Sun.

Only H/5 (ecliptic precession) gives the correct reference for the Sun’s apparent motion and therefore the correct solar-day correction.

Where each LOD is used:

See Timekeeping & Delta-T for how each LOD enters the ΔT calculation and the two zoom-outs visualising LOD_real(t) across ±12 kyr (short-term ripples on the tidal trend) and −248 to +102 kyr (Milankovitch coupling to glacial cycles), and the Formulas reference for closed-form expressions.

Current vs mean values

Each measurement has a mean value over the full precession cycles; current values fluctuate around that mean.

LOD_mean and LOD_real differ by the H/5 correction plus the calibrated cyclic ΔT stack contribution (~+3.527 ms raw H/5, net ~+1.79 ms at J2000 after the ΔT stack). Both are mean quantities — they have no separate “current at J2000” value distinct from the mean, unlike the sidereal/stellar day and the year lengths which do fluctuate through the year (equation of time) and through the precession cycles. Both fluctuate slowly with H(t) across geological time. See Physical vs kinematic LOD above for the derivation.

How everything connects

┌─────────────────────────────────────────────────────────────────┐
│                    COIN ROTATION PARADOX                        │
├─────────────────────────────────────────────────────────────────┤
│  366.25 sidereal days = 365.25 solar days (1 more rotation)     │
│  25,793 sidereal years = ~25,794 solar years (1 fewer orbit)     │
└─────────────────────────────────────────────────────────────────┘
                            │
                            ▼
┌─────────────────────────────────────────────────────────────────┐
│                    DAY-YEAR CONNECTIONS                         │
├─────────────────────────────────────────────────────────────────┤
│  Stellar Day ────────► Sidereal Year (J2000 anchor)             │
│       │                      │                                  │
│       ▼                      ▼                                  │
│  ~9.12ms difference    ~1,223.37s difference                   │
│       │                      │                                  │
│       ▼                      ▼                                  │
│  Sidereal Day ───────► Solar Year                               │
│       │                      │                                  │
│       ▼                      ▼                                  │
│  Axial Precession     Perihelion Precession                     │
│  (~~25,794 years)      (~~20,957 years)                           │
│                              │                                  │
│                              ▼                                  │
│                       Anomalistic Year                          │
└─────────────────────────────────────────────────────────────────┘

Compute day & year lengths at any year

Complete closed-form expressions for solar year, sidereal year, day length, and precession durations at arbitrary year: Formulas.

Continue to Timekeeping & Delta-T for how Earth’s rotation cycles affect time measurement.