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ReferenceFormulas

Calculation Formulas

This page provides formulas for calculating key values at any given year in the Holistic Universe Model. Sections 1–6 include Excel-compatible formulas (single input: MODEL_YEAR). Section 7 (planetary precession) uses Python due to the ~2,400-term complexity.

These formulas are derived from measurements. Every formula on this page was derived from values measured directly in the 3D SimulationΒ  β€” not the other way around. The simulation produces raw data (year lengths, day lengths, precession periods, orbital parameters) using objective measurement functions; the analytical formulas here were then derived to reproduce that data. See Analysis & Export Tools for how measurements are taken.

How to use these formulas: Replace MODEL_YEAR with your target year (e.g., 2000, -10000, 50000). Negative years represent BCE dates. You can use an Excel translatorΒ  to translate formulas into your own language. Or try the Interactive Calculator to compute all values instantly in your browser.

MODEL_YEAR vs calendar year: MODEL_YEAR counts at the model’s mean tropical year (~365.2422 days). Versus the Gregorian calendar (365.2425 days/year, used post-1582 AD) drift is negligible (~0.0003 d/yr). Versus the proleptic Julian calendar (365.25 days/year, used for pre-1582 dates) drift is ~0.0078 d/yr β€” small per year, but accumulating over millennia. Example: MODEL_YEAR βˆ’302,652 β‰ˆ Julian calendar year βˆ’302,646 (~6 years over ~300,000 years). All formulas expect MODEL_YEAR.

Contents

  1. Obliquity (Axial Tilt)
  2. Eccentricity
  3. Inclination to Invariable Plane
  4. Longitude of Perihelion β€” includes ERD definition
  5. Length of Days & Years β€” solar year, sidereal year, day length, solar year (seconds), sidereal day, stellar day
  6. Precession Durations β€” axial, perihelion, inclination (coin rotation paradox), anomalistic year (3-step formula)
  7. Planetary Precession Fluctuation β€” all 7 planets, observed & predictive formulas
  8. How the Formulas Connect
  9. Verification
  10. Dashboard

1. Obliquity (Axial Tilt)

Calculates Earth’s axial tilt in degrees at any given year.

Scientific Notation:

Ξ΅(t)=Ξ΅0βˆ’Acos⁑(2Ο€tTI)+Acos⁑(2Ο€tTO)\varepsilon(t) = \varepsilon_0 - A \cos\left(\frac{2\pi t}{T_I}\right) + A \cos\left(\frac{2\pi t}{T_O}\right)

Where:

  • Ξ΅0\varepsilon_0 = 23.41354Β° β€” Mean obliquity
  • AA = 0.63603Β° β€” Amplitude of variation
  • TI=T_I = ~111,772 years β€” Inclination precession period (H/3)
  • TO=T_O = ~41,915 years β€” Obliquity cycle period (H/8)
  • tt = MODEL_YEAR + 302,635 β€” Time relative to anchor year (302,635 BC)

The conceptual 2-cosine form above captures the dominant behaviour (Β±A\pm A at H/3 vs H/8). The calculator uses the full 16-harmonic Fourier fit β€” every harmonic from H/2 to H/32 β€” with an additive constant Ξ΅Λ‰=\bar{\varepsilon} = 23.453Β°. This pythagoreanMeanObliquity is the zero-frequency component of the 16-harmonic decomposition and differs slightly from meanObliquity (used in the simpler 2-cosine form) because the harmonic fit absorbs additional structure from the higher-order terms.

EXCEL FORMULA OBLIQUITY IN DEGREES (16-harmonic Fourier fit)
=23.453262-0.000003*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/2))-0.000062*COS(2*PI()*(MODEL_YEAR+302635)/(335317/2))+0.032098*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/3))-0.634890*COS(2*PI()*(MODEL_YEAR+302635)/(335317/3))-0.000078*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/5))-0.008146*COS(2*PI()*(MODEL_YEAR+302635)/(335317/5))+0.000449*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/6))-0.004044*COS(2*PI()*(MODEL_YEAR+302635)/(335317/6))-0.032069*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/8))+0.634911*COS(2*PI()*(MODEL_YEAR+302635)/(335317/8))+0.000009*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/9))-0.000055*COS(2*PI()*(MODEL_YEAR+302635)/(335317/9))-0.000897*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/11))+0.008090*COS(2*PI()*(MODEL_YEAR+302635)/(335317/11))-0.000002*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/13))+0.000042*COS(2*PI()*(MODEL_YEAR+302635)/(335317/13))-0.000027*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/14))+0.000164*COS(2*PI()*(MODEL_YEAR+302635)/(335317/14))+0.000448*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/16))-0.004045*COS(2*PI()*(MODEL_YEAR+302635)/(335317/16))-0.000001*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/17))+0.000004*COS(2*PI()*(MODEL_YEAR+302635)/(335317/17))+0.000027*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/19))-0.000164*COS(2*PI()*(MODEL_YEAR+302635)/(335317/19))+0.000001*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/22))-0.000005*COS(2*PI()*(MODEL_YEAR+302635)/(335317/22))-0.000009*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/24))+0.000055*COS(2*PI()*(MODEL_YEAR+302635)/(335317/24))-0.000001*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/27))+0.000004*COS(2*PI()*(MODEL_YEAR+302635)/(335317/27))+0.000000*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/32))-0.000001*COS(2*PI()*(MODEL_YEAR+302635)/(335317/32))

Components:

  • pythagoreanMeanObliquity β€” Additive constant (23.453Β°), the mean over one Earth Fundamental Cycle
  • Dominant harmonics: H/3 and H/8 with ∣cos⁑∣|\cos| coefficient β‰ˆ 0.635 (amplitude matches earthInclAmplitude = 0.63603Β°)
  • Anchor offset (302,635) β€” added to MODEL_YEAR in the Excel formula to align time t with the start of the current Earth Fundamental Cycle (302,635 BC)

Example Values:

MODEL_YEARObliquity
2000 AD23.4393Β°
-10000 (10,000 BCE)24.5293Β°
+10000 (10,000 AD)22.6182Β°

Obliquity for Other Planets

The same two-component structure applies to every planet with an obliquity cycle. The formula uses the planet’s ICRF perihelion period (inclination component) and obliquity cycle period (obliquity component), with amplitude A=ψ/(dΓ—m)A = \psi / (d \times \sqrt{m}) from Law 2:

Ξ΅p(t)=Ξ΅p,J2000βˆ’Ap[cos⁑(2Ο€tTICRF)βˆ’cos⁑(2Ο€t2000TICRF)]+Ap[cos⁑(2Ο€tTObliq)βˆ’cos⁑(2Ο€t2000TObliq)]\varepsilon_p(t) = \varepsilon_{p,J2000} - A_p \left[\cos\left(\frac{2\pi t}{T_{\text{ICRF}}}\right) - \cos\left(\frac{2\pi t_{2000}}{T_{\text{ICRF}}}\right)\right] + A_p \left[\cos\left(\frac{2\pi t}{T_{\text{Obliq}}}\right) - \cos\left(\frac{2\pi t_{2000}}{T_{\text{Obliq}}}\right)\right]

The anchoring terms ensure Ρp(2000)=Ρp,J2000\varepsilon_p(2000) = \varepsilon_{p,J2000} exactly. Venus and Neptune have TObliq=∣TICRF∣T_{\text{Obliq}} = |T_{\text{ICRF}}| (auto-derived from their ecliptic periods; tidally damped), so the two cosine terms cancel exactly and the formula returns the static J2000 tilt.

PlanetTICRFT_{\text{ICRF}}TObliqT_{\text{Obliq}}Amplitude AAJ2000 tilt
Mercuryβˆ’28,844 yr894,179 yrΒ±\pm0.386477Β°0.03Β°
Marsβˆ’38,877 yr127,740 yrΒ±\pm1.164214Β°25.19Β°
Jupiterβˆ’41,915 yr167,659 yrΒ±\pm0.021404Β°3.13Β°
Saturnβˆ’15,967 yr111,772 yrΒ±\pm0.065192Β°26.73Β°
Uranusβˆ’33,532 yr167,659 yrΒ±\pm0.023831Β°82.23Β°

Mean Obliquity (analytical, averaged over 8H)

The mean obliquity over the Solar System Resonance Cycle is computed analytically from the anchoring offsets (cosine terms average to zero over 8H):

Ξ΅Λ‰p=Ξ΅p,J2000+Apcos⁑(2Ο€t2000TICRF)βˆ’Apcos⁑(2Ο€t2000TObliq)\bar{\varepsilon}_p = \varepsilon_{p,J2000} + A_p \cos\left(\frac{2\pi t_{2000}}{T_{\text{ICRF}}}\right) - A_p \cos\left(\frac{2\pi t_{2000}}{T_{\text{Obliq}}}\right)
PlanetJ2000Mean (8H)
Mercury0.03Β°0.01Β°
Mars25.19Β°25.40Β°
Jupiter3.13Β°3.12Β°
Saturn26.73Β°26.80Β°
Uranus82.23Β°82.24Β°

2. Eccentricity

Calculates Earth’s orbital eccentricity at any given year.

Scientific Notation (law of cosines β€” Earth at distance ebasee_{\text{base}} from the wobble centre; perihelion point at distance AA from Earth; eccentricity e(t)e(t) is the distance between them):

e(t)=ebase 2+A2βˆ’2 ebase A cos⁑ϕe(t) = \sqrt{e_{\text{base}}^{\,2} + A^2 - 2\,e_{\text{base}}\,A\,\cos\phi}

Where:

  • Ο•=2Ο€tTP\phi = \frac{2\pi t}{T_P} β€” Phase angle
  • ebasee_{\text{base}} = 0.015386 β€” orbital radius of Earth’s perihelion point around the Sun
  • AA = 0.001356 β€” Earth’s orbital radius around its wobble center
  • TP=T_P = ~20,957 years β€” Perihelion precession period (H/16)
  • tt = MODEL_YEAR + 302,635 β€” Time relative to anchor year (302,635 BC)

Only 2 free parameters (ebasee_{\text{base}} and AA). Minimum (Ο•=0\phi = 0, both points on the same side) and maximum (Ο•=Ο€\phi = \pi, opposite sides) eccentricities are exactly ∣ebaseβˆ’A∣|e_{\text{base}} - A| and ebase+Ae_{\text{base}} + A. The derived mean e0=ebase2+A2e_0 = \sqrt{e_{\text{base}}^2 + A^2} = 0.0154454 appears at Ο•=Ο€/2\phi = \pi/2.

EXCEL FORMULA ORBITAL ECCENTRICITY (Change the MODEL_YEAR value)
=SQRT(0.015386^2+0.001356^2-2*0.015386*0.001356*COS(RADIANS(((MODEL_YEAR--302635)/(335317/16))*360)))

Components:

  • eccentricityBase β€” orbital radius of Earth’s perihelion point (0.015386)
  • eccentricityAmplitude β€” Earth’s orbital radius around its wobble center (0.001356)
  • SQRT(eccentricityBaseΒ²+eccentricityAmplitudeΒ²) β€” Derived mean eccentricity at Ο•=Ο€/2\phi = \pi/2 (0.0154454)
  • H/16 β€” Perihelion precession cycle (H/16 = ~20,957 years)

Example Values:

MODEL_YEAREccentricity
2000 AD0.01671022
Minimum~0.0140
Maximum~0.0167

3. Inclination to Invariable Plane

Calculates Earth’s orbital inclination relative to the solar system’s invariable plane in degrees.

Scientific Notation:

I(t)=I0βˆ’Acos⁑(2Ο€tTI)I(t) = I_0 - A \cos\left(\frac{2\pi t}{T_I}\right)

Where:

  • I0I_0 = 1.48113Β° β€” Mean inclination
  • AA = 0.63603Β° β€” Amplitude of variation (same as obliquity)
  • TI=T_I = ~111,772 years β€” Inclination precession period (H/3)
  • tt = MODEL_YEAR + 302,635 β€” Time relative to anchor year (302,635 BC)

Note: The inclination and obliquity share the same amplitude (AA = 0.63603Β°) because they are geometrically coupled β€” as Earth’s orbital plane tilts relative to the invariable plane, Earth’s axis maintains its orientation in inertial space, causing the obliquity (angle between axis and orbit normal) to change by the same amount.

EXCEL FORMULA INCLINATION TO THE INVARIABLE PLANE IN DEGREES (Change the MODEL_YEAR value)
=1.48113+(-0.63603*(COS(RADIANS(((MODEL_YEAR--302635)/(335317/3))*360))))

Components:

  • earthInclMean β€” Mean inclination (1.48113Β°)
  • earthInclAmplitude β€” Amplitude of variation (0.63603Β°)
  • H/3 β€” Inclination precession cycle (H/3 = ~111,772 years)

Example Values:

MODEL_YEARInclination
2000 AD~1.57869Β°
Maximum~2.117Β°
Minimum~0.845Β°

4. Longitude of Perihelion

Calculates the longitude of perihelion (direction of Earth’s closest approach to the Sun) in degrees. Analysis of Earth perihelion data revealed multiple harmonic terms that improve accuracy.

Scientific Notation:

Ο–(t)=Ο–0+Ο‰Pβ‹…t+βˆ‘i[aisin⁑(2Ο€tTi)+bicos⁑(2Ο€tTi)]+c0\varpi(t) = \varpi_0 + \omega_P \cdot t + \sum_{i} \left[ a_i \sin\left(\frac{2\pi t}{T_i}\right) + b_i \cos\left(\frac{2\pi t}{T_i}\right) \right] + c_0

Note: The Excel formula below uses the full 25-harmonic Fourier fit β€” identical coefficients and periods to the Python/TypeScript implementation used by the Orbital Calculator.

EXCEL FORMULA LONGITUDE OF PERIHELION (25-harmonic Fourier fit)
=MOD(270+(MODEL_YEAR+302635)*(360/(335317/16))-0.131811*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/3))+0.007264*COS(2*PI()*(MODEL_YEAR+302635)/(335317/3))-0.003219*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/5))+0.000012*COS(2*PI()*(MODEL_YEAR+302635)/(335317/5))+0.120192*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/8))-0.007799*COS(2*PI()*(MODEL_YEAR+302635)/(335317/8))+0.011620*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/13))+0.000536*COS(2*PI()*(MODEL_YEAR+302635)/(335317/13))+4.835748*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/16))-0.021962*COS(2*PI()*(MODEL_YEAR+302635)/(335317/16))+0.000499*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/19))+0.000023*COS(2*PI()*(MODEL_YEAR+302635)/(335317/19))-0.003468*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/21))+0.000048*COS(2*PI()*(MODEL_YEAR+302635)/(335317/21))+0.130382*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/24))+0.006004*COS(2*PI()*(MODEL_YEAR+302635)/(335317/24))+0.001754*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/26))-0.000024*COS(2*PI()*(MODEL_YEAR+302635)/(335317/26))-0.130892*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/29))-0.006027*COS(2*PI()*(MODEL_YEAR+302635)/(335317/29))+2.659030*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/32))+0.247035*COS(2*PI()*(MODEL_YEAR+302635)/(335317/32))-0.005621*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/35))-0.000259*COS(2*PI()*(MODEL_YEAR+302635)/(335317/35))-0.000681*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/37))+0.000000*COS(2*PI()*(MODEL_YEAR+302635)/(335317/37))+0.016174*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/40))+0.000644*COS(2*PI()*(MODEL_YEAR+302635)/(335317/40))-0.010562*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/45))-0.000385*COS(2*PI()*(MODEL_YEAR+302635)/(335317/45))+0.218927*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/48))+0.019922*COS(2*PI()*(MODEL_YEAR+302635)/(335317/48))-0.000502*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/53))-0.000040*COS(2*PI()*(MODEL_YEAR+302635)/(335317/53))+0.006926*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/56))+0.000872*COS(2*PI()*(MODEL_YEAR+302635)/(335317/56))-0.006476*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/61))-0.000855*COS(2*PI()*(MODEL_YEAR+302635)/(335317/61))+0.070333*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/64))+0.012269*COS(2*PI()*(MODEL_YEAR+302635)/(335317/64))+0.001276*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/72))+0.000169*COS(2*PI()*(MODEL_YEAR+302635)/(335317/72))-0.000999*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/77))-0.000132*COS(2*PI()*(MODEL_YEAR+302635)/(335317/77))+0.010342*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/80))+0.001892*COS(2*PI()*(MODEL_YEAR+302635)/(335317/80))+0.002705*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/96))+0.000684*COS(2*PI()*(MODEL_YEAR+302635)/(335317/96))+0.000507*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/112))+0.000139*COS(2*PI()*(MODEL_YEAR+302635)/(335317/112))-0.259775,360)

Earth Rate Deviation (ERD)

The ERD is the derivative of the oscillation terms β€” it measures how Earth’s perihelion rate deviates from the mean rate. ERD is essential for accurate planetary precession calculations.

Scientific Notation:

ERD=dΞΈEdtβˆ’Ο‰0\text{ERD} = \frac{d\theta_E}{dt} - \omega_0

Where:

  • dΞΈEdt\frac{d\theta_E}{dt} = Instantaneous Earth perihelion rate (Β°/year)
  • Ο‰0\omega_0 = 360Β°/~20,957 = 0.017178Β°/year β€” Mean rate

Excel Formula (numerical derivative):

This formula calculates ERD from consecutive Longitude of Perihelion values, with angle wraparound correction at Β±180Β°:

EXCEL FORMULA ERD (requires two consecutive years)
=IF(ΞΈ_E_current - ΞΈ_E_previous < -180,
    (ΞΈ_E_current - ΞΈ_E_previous + 360) / (MODEL_YEAR_current - MODEL_YEAR_previous),
  IF(ΞΈ_E_current - ΞΈ_E_previous > 180,
    (ΞΈ_E_current - ΞΈ_E_previous - 360) / (MODEL_YEAR_current - MODEL_YEAR_previous),
    (ΞΈ_E_current - ΞΈ_E_previous) / (MODEL_YEAR_current - MODEL_YEAR_previous)
  )
) - 360/20957

Analytical ERD Formula (recommended):

The true derivative of the perihelion harmonics gives the mathematically correct instantaneous rate. For each harmonic term asin⁑(Ο‰t)+bcos⁑(Ο‰t)a \sin(\omega t) + b \cos(\omega t), the derivative is aΟ‰cos⁑(Ο‰t)βˆ’bΟ‰sin⁑(Ο‰t)a \omega \cos(\omega t) - b \omega \sin(\omega t).

EXCEL FORMULA ERD (Analytical, 25-harmonic)
=-0.131811*(2*PI()/(335317/3))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/3))-0.007264*(2*PI()/(335317/3))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/3))-0.003219*(2*PI()/(335317/5))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/5))-0.000012*(2*PI()/(335317/5))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/5))+0.120192*(2*PI()/(335317/8))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/8))+0.007799*(2*PI()/(335317/8))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/8))+0.011620*(2*PI()/(335317/13))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/13))-0.000536*(2*PI()/(335317/13))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/13))+4.835748*(2*PI()/(335317/16))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/16))+0.021962*(2*PI()/(335317/16))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/16))+0.000499*(2*PI()/(335317/19))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/19))-0.000023*(2*PI()/(335317/19))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/19))-0.003468*(2*PI()/(335317/21))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/21))-0.000048*(2*PI()/(335317/21))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/21))+0.130382*(2*PI()/(335317/24))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/24))-0.006004*(2*PI()/(335317/24))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/24))+0.001754*(2*PI()/(335317/26))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/26))+0.000024*(2*PI()/(335317/26))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/26))-0.130892*(2*PI()/(335317/29))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/29))+0.006027*(2*PI()/(335317/29))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/29))+2.659030*(2*PI()/(335317/32))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/32))-0.247035*(2*PI()/(335317/32))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/32))-0.005621*(2*PI()/(335317/35))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/35))+0.000259*(2*PI()/(335317/35))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/35))-0.000681*(2*PI()/(335317/37))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/37))+0.000000*(2*PI()/(335317/37))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/37))+0.016174*(2*PI()/(335317/40))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/40))-0.000644*(2*PI()/(335317/40))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/40))-0.010562*(2*PI()/(335317/45))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/45))+0.000385*(2*PI()/(335317/45))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/45))+0.218927*(2*PI()/(335317/48))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/48))-0.019922*(2*PI()/(335317/48))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/48))-0.000502*(2*PI()/(335317/53))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/53))+0.000040*(2*PI()/(335317/53))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/53))+0.006926*(2*PI()/(335317/56))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/56))-0.000872*(2*PI()/(335317/56))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/56))-0.006476*(2*PI()/(335317/61))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/61))+0.000855*(2*PI()/(335317/61))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/61))+0.070333*(2*PI()/(335317/64))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/64))-0.012269*(2*PI()/(335317/64))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/64))+0.001276*(2*PI()/(335317/72))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/72))-0.000169*(2*PI()/(335317/72))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/72))-0.000999*(2*PI()/(335317/77))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/77))+0.000132*(2*PI()/(335317/77))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/77))+0.010342*(2*PI()/(335317/80))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/80))-0.001892*(2*PI()/(335317/80))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/80))+0.002705*(2*PI()/(335317/96))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/96))-0.000684*(2*PI()/(335317/96))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/96))+0.000507*(2*PI()/(335317/112))*COS(2*PI()*(MODEL_YEAR+302635)/(335317/112))-0.000139*(2*PI()/(335317/112))*SIN(2*PI()*(MODEL_YEAR+302635)/(335317/112))

This returns ERD in Β°/year (typically ~Β±0.0002 Β°/year).

Key Values:

MODEL_YEARLongitude
1246 AD90Β° (aligned with December solstice)
2000 AD102.947Β°
22,203 AD90Β° (next December solstice alignment)

ICRF Perihelion Longitude (J2000 epoch)

The ICRF perihelion longitude is the longitude measured against the fixed International Celestial Reference Frame, rather than the moving ecliptic of date. It precesses at a slower rate because the general precession (H/13H/13) is removed.

For Earth, the ICRF perihelion is the ecliptic longitude minus the general precession:

Ο–ICRF,Β Earth(t)=Ο–ecl,Β Earth(t)βˆ’360Β°H/13β‹…(tβˆ’2000)\varpi_{\text{ICRF, Earth}}(t) = \varpi_{\text{ecl, Earth}}(t) - \frac{360Β°}{H/13} \cdot (t - 2000)

For other planets, the ICRF rate is computed from the ecliptic perihelion period Tp,eclT_{p,\text{ecl}} minus the general precession:

1TICRF=1Tp,eclβˆ’1H/13\frac{1}{T_{\text{ICRF}}} = \frac{1}{T_{p,\text{ecl}}} - \frac{1}{H/13} Ο–ICRF,p(t)=Ο–p,J2000+360Β°TICRFβ‹…(tβˆ’2000)\varpi_{\text{ICRF}, p}(t) = \varpi_{p,J2000} + \frac{360Β°}{T_{\text{ICRF}}} \cdot (t - 2000)

Earth is the only planet with prograde ICRF perihelion precession (period H/3=H/3 = ~111,772 years). All other planets have retrograde ICRF rates because their ecliptic rates are slower than the general precession.

Earth example: At J2000, Ο–ICRF,Β Earth=\varpi_{\text{ICRF, Earth}} = 102.947Β°. After one full H/3H/3 cycle (~111,772 years), it returns to the same value.

5. Length of Days & Years

Year-length variations are modelled as Fourier harmonic series around derived means. The means are computed from two model constants (the input mean tropical year and H); only the harmonic coefficients are empirical (fitted from 491 data points spanning Β±25,000 years).

Fourier Harmonic Model

The sidereal and anomalistic year types follow this pattern directly:

Y(t)=YΛ‰+βˆ‘i[sisin⁑ ⁣(2Ο€tTi)+cicos⁑ ⁣(2Ο€tTi)]Y(t) = \bar{Y} + \sum_i \left[ s_i \sin\!\left(\frac{2\pi t}{T_i}\right) + c_i \cos\!\left(\frac{2\pi t}{T_i}\right) \right]

Where t=MODEL_YEARβˆ’AnchorYeart = \text{MODEL\_YEAR} - \text{AnchorYear}, with AnchorYear = -302,635.

The tropical (solar) year is technically computed by the calculator as the mean of 4 cardinal-point year lengths (each a 24-harmonic derivative of its Julian-Day harmonic series). For Excel use, the 12-harmonic Fourier approximation shown below agrees with the calculator’s 4-CP method to within ~0.04 seconds across the full Β±100,000-year range β€” counterintuitively, the peak disagreement (~0.04 s) sits near J2000 where harmonic phases align unfavourably, while at Β±100,000 years it falls to ~0.02 s. Either way: sub-second precision over the full Earth Fundamental Cycle.

Derived Means

All three mean year lengths derive from inputmeanlengthsolaryearindays = 365.2422 and H = 335,317:

Year typeFormulaMean (days)
Tropical (Yˉtrop\bar{Y}_{\text{trop}})ROUND(input × H/16) / (H/16)365.242203646102
Sidereal (YΛ‰sid\bar{Y}_{\text{sid}})YΛ‰tropΓ—HHβˆ’13\bar{Y}_{\text{trop}} \times \frac{H}{H-13}365.2563644
Anomalistic (YΛ‰anom\bar{Y}_{\text{anom}})YΛ‰tropΓ—HHβˆ’16\bar{Y}_{\text{trop}} \times \frac{H}{H-16}365.2596324

Solar Year (in days)

Dominant period: H/8 (obliquity cycle, ~41,915 years). Amplitude: Β±1.8 seconds.

The Excel formula uses 12 harmonics; the table below shows the 3 dominant ones (H/8, H/3, H/16). The remaining 9 (H/5, H/6, H/9, H/11, H/14, H/19, H/22, H/24, H/32) each contribute < 0.1 second and are included in the formula for completeness.

HarmonicPeriodsin coeff (sis_i)cos coeff (cic_i)Amplitude
H/8~41,915 yrβˆ’1.299452eβˆ’05+2.227963eβˆ’041.93 s
H/3~111,772 yr+4.963656eβˆ’06βˆ’8.354141eβˆ’050.72 s
H/16~20,957 yr+6.560490eβˆ’07βˆ’6.416442eβˆ’060.56 s
EXCEL FORMULA LENGTH OF SOLAR YEAR IN DAYS (Fourier)
=LET(t, MODEL_YEAR+302635, H, 335317, mean, 365.242203646102, mean + (4.963655515994E-06)*SIN(2*PI()*t/(H/3)) + (-8.354140858289E-05)*COS(2*PI()*t/(H/3)) + (-1.645990068981E-08)*SIN(2*PI()*t/(H/5)) + (3.346788536693E-07)*COS(2*PI()*t/(H/5)) + (2.433753464246E-07)*SIN(2*PI()*t/(H/6)) + (-2.329695678249E-06)*COS(2*PI()*t/(H/6)) + (-1.299451587802E-05)*SIN(2*PI()*t/(H/8)) + (2.227963456434E-04)*COS(2*PI()*t/(H/8)) + (9.743882426511E-09)*SIN(2*PI()*t/(H/9)) + (-6.177071904623E-08)*COS(2*PI()*t/(H/9)) + (-8.831015131162E-07)*SIN(2*PI()*t/(H/11)) + (8.543316063918E-06)*COS(2*PI()*t/(H/11)) + (-4.515837982944E-08)*SIN(2*PI()*t/(H/14)) + (2.883165423683E-07)*COS(2*PI()*t/(H/14)) + (6.560490474823E-07)*SIN(2*PI()*t/(H/16)) + (-6.416442022713E-06)*COS(2*PI()*t/(H/16)) + (6.126554369035E-08)*SIN(2*PI()*t/(H/19)) + (-3.916922799144E-07)*COS(2*PI()*t/(H/19)) + (3.783127036528E-09)*SIN(2*PI()*t/(H/22)) + (-1.802048457974E-08)*COS(2*PI()*t/(H/22)) + (-2.612611973930E-08)*SIN(2*PI()*t/(H/24)) + (1.653905514523E-07)*COS(2*PI()*t/(H/24)) + (-2.964708323237E-09)*SIN(2*PI()*t/(H/32)) + (3.095269447075E-08)*COS(2*PI()*t/(H/32)))

Sidereal Year (in days)

Variations are 15Γ— smaller than tropical β€” the orbital period is nearly constant.

The Excel formula uses 5 harmonics (H/3, H/5, H/8, H/16, H/32) matching the calculator’s SIDEREAL_YEAR_HARMONICS exactly. The table shows the 2 dominant ones.

HarmonicPeriodsin coeff (sis_i)cos coeff (cic_i)Amplitude
H/8~41,915 yrβˆ’1.059417eβˆ’06+1.713869eβˆ’090.092 s
H/3~111,772 yr+3.973165eβˆ’07βˆ’2.397135eβˆ’100.034 s
EXCEL FORMULA LENGTH OF SIDEREAL YEAR IN DAYS (Fourier)
=LET(t, MODEL_YEAR+302635, H, 335317, mean, 365.242203646102*(H/13)/((H/13)-1), mean + (3.973164944361E-07)*SIN(2*PI()*t/(H/3)) + (-2.397135263624E-10)*COS(2*PI()*t/(H/3)) + (3.745546829271E-10)*SIN(2*PI()*t/(H/5)) + (3.355989685870E-07)*COS(2*PI()*t/(H/5)) + (-1.059417380507E-06)*SIN(2*PI()*t/(H/8)) + (1.713869484534E-09)*COS(2*PI()*t/(H/8)) + (1.991940272716E-08)*SIN(2*PI()*t/(H/16)) + (-3.841448696173E-07)*COS(2*PI()*t/(H/16)) + (-3.885222338193E-09)*SIN(2*PI()*t/(H/32)) + (3.535868885074E-08)*COS(2*PI()*t/(H/32)))

Sidereal Year (in seconds)

The sidereal year in SI seconds is fixed β€” it is the orbital period:

Ysid(s)Y_{\text{sid}}^{\text{(s)}} = 31,558,149.76 SI seconds

Day Length (in SI seconds)

D=Ysid(s)Ysid(days)D = \frac{Y_{\text{sid}}^{\text{(s)}}}{Y_{\text{sid}}^{\text{(days)}}}
EXCEL FORMULA LENGTH OF DAY IN SI SECONDS
=LET(t, MODEL_YEAR+302635, H, 335317, sid_days, 365.242203646102*(H/13)/((H/13)-1) + (3.973164944361E-07)*SIN(2*PI()*t/(H/3)) + (-2.397135263624E-10)*COS(2*PI()*t/(H/3)) + (3.745546829271E-10)*SIN(2*PI()*t/(H/5)) + (3.355989685870E-07)*COS(2*PI()*t/(H/5)) + (-1.059417380507E-06)*SIN(2*PI()*t/(H/8)) + (1.713869484534E-09)*COS(2*PI()*t/(H/8)) + (1.991940272716E-08)*SIN(2*PI()*t/(H/16)) + (-3.841448696173E-07)*COS(2*PI()*t/(H/16)) + (-3.885222338193E-09)*SIN(2*PI()*t/(H/32)) + (3.535868885074E-08)*COS(2*PI()*t/(H/32)), 31558149.76/sid_days)

Solar Year (in seconds)

Ysolar(s)=Ysolar(days)Γ—D(s)Y_{\text{solar}}(s) = Y_{\text{solar}}^{\text{(days)}} \times D(s)
EXCEL FORMULA SOLAR YEAR IN SI SECONDS
=LET(t, MODEL_YEAR+302635, H, 335317, sol_days, 365.242203646102 + (4.963655515994E-06)*SIN(2*PI()*t/(H/3)) + (-8.354140858289E-05)*COS(2*PI()*t/(H/3)) + (-1.645990068981E-08)*SIN(2*PI()*t/(H/5)) + (3.346788536693E-07)*COS(2*PI()*t/(H/5)) + (2.433753464246E-07)*SIN(2*PI()*t/(H/6)) + (-2.329695678249E-06)*COS(2*PI()*t/(H/6)) + (-1.299451587802E-05)*SIN(2*PI()*t/(H/8)) + (2.227963456434E-04)*COS(2*PI()*t/(H/8)) + (9.743882426511E-09)*SIN(2*PI()*t/(H/9)) + (-6.177071904623E-08)*COS(2*PI()*t/(H/9)) + (-8.831015131162E-07)*SIN(2*PI()*t/(H/11)) + (8.543316063918E-06)*COS(2*PI()*t/(H/11)) + (-4.515837982944E-08)*SIN(2*PI()*t/(H/14)) + (2.883165423683E-07)*COS(2*PI()*t/(H/14)) + (6.560490474823E-07)*SIN(2*PI()*t/(H/16)) + (-6.416442022713E-06)*COS(2*PI()*t/(H/16)) + (6.126554369035E-08)*SIN(2*PI()*t/(H/19)) + (-3.916922799144E-07)*COS(2*PI()*t/(H/19)) + (3.783127036528E-09)*SIN(2*PI()*t/(H/22)) + (-1.802048457974E-08)*COS(2*PI()*t/(H/22)) + (-2.612611973930E-08)*SIN(2*PI()*t/(H/24)) + (1.653905514523E-07)*COS(2*PI()*t/(H/24)) + (-2.964708323237E-09)*SIN(2*PI()*t/(H/32)) + (3.095269447075E-08)*COS(2*PI()*t/(H/32)), sid_days, 365.242203646102*(H/13)/((H/13)-1) + (3.973164944361E-07)*SIN(2*PI()*t/(H/3)) + (-2.397135263624E-10)*COS(2*PI()*t/(H/3)) + (3.745546829271E-10)*SIN(2*PI()*t/(H/5)) + (3.355989685870E-07)*COS(2*PI()*t/(H/5)) + (-1.059417380507E-06)*SIN(2*PI()*t/(H/8)) + (1.713869484534E-09)*COS(2*PI()*t/(H/8)) + (1.991940272716E-08)*SIN(2*PI()*t/(H/16)) + (-3.841448696173E-07)*COS(2*PI()*t/(H/16)) + (-3.885222338193E-09)*SIN(2*PI()*t/(H/32)) + (3.535868885074E-08)*COS(2*PI()*t/(H/32)), sol_days*31558149.76/sid_days)

Sidereal Day (in SI seconds)

Dsid=Ysolar(s)Ysolar(s)86400+1D_{\text{sid}} = \frac{Y_{\text{solar}}(s)}{\frac{Y_{\text{solar}}(s)}{86400} + 1}

The sidereal day is the time for one full rotation relative to the vernal equinox β€” slightly shorter than the solar day because Earth must rotate an extra ~1Β° per day to compensate for its orbital motion.

EXCEL FORMULA SIDEREAL DAY IN SI SECONDS
=LET(t, MODEL_YEAR+302635, H, 335317, sol_days, 365.242203646102 + (4.963655515994E-06)*SIN(2*PI()*t/(H/3)) + (-8.354140858289E-05)*COS(2*PI()*t/(H/3)) + (-1.645990068981E-08)*SIN(2*PI()*t/(H/5)) + (3.346788536693E-07)*COS(2*PI()*t/(H/5)) + (2.433753464246E-07)*SIN(2*PI()*t/(H/6)) + (-2.329695678249E-06)*COS(2*PI()*t/(H/6)) + (-1.299451587802E-05)*SIN(2*PI()*t/(H/8)) + (2.227963456434E-04)*COS(2*PI()*t/(H/8)) + (9.743882426511E-09)*SIN(2*PI()*t/(H/9)) + (-6.177071904623E-08)*COS(2*PI()*t/(H/9)) + (-8.831015131162E-07)*SIN(2*PI()*t/(H/11)) + (8.543316063918E-06)*COS(2*PI()*t/(H/11)) + (-4.515837982944E-08)*SIN(2*PI()*t/(H/14)) + (2.883165423683E-07)*COS(2*PI()*t/(H/14)) + (6.560490474823E-07)*SIN(2*PI()*t/(H/16)) + (-6.416442022713E-06)*COS(2*PI()*t/(H/16)) + (6.126554369035E-08)*SIN(2*PI()*t/(H/19)) + (-3.916922799144E-07)*COS(2*PI()*t/(H/19)) + (3.783127036528E-09)*SIN(2*PI()*t/(H/22)) + (-1.802048457974E-08)*COS(2*PI()*t/(H/22)) + (-2.612611973930E-08)*SIN(2*PI()*t/(H/24)) + (1.653905514523E-07)*COS(2*PI()*t/(H/24)) + (-2.964708323237E-09)*SIN(2*PI()*t/(H/32)) + (3.095269447075E-08)*COS(2*PI()*t/(H/32)), sid_days, 365.242203646102*(H/13)/((H/13)-1) + (3.973164944361E-07)*SIN(2*PI()*t/(H/3)) + (-2.397135263624E-10)*COS(2*PI()*t/(H/3)) + (3.745546829271E-10)*SIN(2*PI()*t/(H/5)) + (3.355989685870E-07)*COS(2*PI()*t/(H/5)) + (-1.059417380507E-06)*SIN(2*PI()*t/(H/8)) + (1.713869484534E-09)*COS(2*PI()*t/(H/8)) + (1.991940272716E-08)*SIN(2*PI()*t/(H/16)) + (-3.841448696173E-07)*COS(2*PI()*t/(H/16)) + (-3.885222338193E-09)*SIN(2*PI()*t/(H/32)) + (3.535868885074E-08)*COS(2*PI()*t/(H/32)), sol_s, sol_days*31558149.76/sid_days, sol_s/(sol_s/86400+1))

Stellar Day (in SI seconds)

Dstar=DsidΓ—(1+1PAΓ—R)D_{\text{star}} = D_{\text{sid}} \times \left(1 + \frac{1}{P_A \times R}\right)

Where PAP_A is the axial precession period and R=Ysolar(s)/DsidR = Y_{\text{solar}}(s) / D_{\text{sid}} is rotations per year. The stellar day is ~9.12 ms longer than the sidereal day because the vernal equinox precesses westward, so Earth must rotate slightly more to reach the same fixed star.

EXCEL FORMULA STELLAR DAY IN SI SECONDS
=LET(t, MODEL_YEAR+302635, H, 335317, sol_days, 365.242203646102 + (4.963655515994E-06)*SIN(2*PI()*t/(H/3)) + (-8.354140858289E-05)*COS(2*PI()*t/(H/3)) + (-1.645990068981E-08)*SIN(2*PI()*t/(H/5)) + (3.346788536693E-07)*COS(2*PI()*t/(H/5)) + (2.433753464246E-07)*SIN(2*PI()*t/(H/6)) + (-2.329695678249E-06)*COS(2*PI()*t/(H/6)) + (-1.299451587802E-05)*SIN(2*PI()*t/(H/8)) + (2.227963456434E-04)*COS(2*PI()*t/(H/8)) + (9.743882426511E-09)*SIN(2*PI()*t/(H/9)) + (-6.177071904623E-08)*COS(2*PI()*t/(H/9)) + (-8.831015131162E-07)*SIN(2*PI()*t/(H/11)) + (8.543316063918E-06)*COS(2*PI()*t/(H/11)) + (-4.515837982944E-08)*SIN(2*PI()*t/(H/14)) + (2.883165423683E-07)*COS(2*PI()*t/(H/14)) + (6.560490474823E-07)*SIN(2*PI()*t/(H/16)) + (-6.416442022713E-06)*COS(2*PI()*t/(H/16)) + (6.126554369035E-08)*SIN(2*PI()*t/(H/19)) + (-3.916922799144E-07)*COS(2*PI()*t/(H/19)) + (3.783127036528E-09)*SIN(2*PI()*t/(H/22)) + (-1.802048457974E-08)*COS(2*PI()*t/(H/22)) + (-2.612611973930E-08)*SIN(2*PI()*t/(H/24)) + (1.653905514523E-07)*COS(2*PI()*t/(H/24)) + (-2.964708323237E-09)*SIN(2*PI()*t/(H/32)) + (3.095269447075E-08)*COS(2*PI()*t/(H/32)), sid_days, 365.242203646102*(H/13)/((H/13)-1) + (3.973164944361E-07)*SIN(2*PI()*t/(H/3)) + (-2.397135263624E-10)*COS(2*PI()*t/(H/3)) + (3.745546829271E-10)*SIN(2*PI()*t/(H/5)) + (3.355989685870E-07)*COS(2*PI()*t/(H/5)) + (-1.059417380507E-06)*SIN(2*PI()*t/(H/8)) + (1.713869484534E-09)*COS(2*PI()*t/(H/8)) + (1.991940272716E-08)*SIN(2*PI()*t/(H/16)) + (-3.841448696173E-07)*COS(2*PI()*t/(H/16)) + (-3.885222338193E-09)*SIN(2*PI()*t/(H/32)) + (3.535868885074E-08)*COS(2*PI()*t/(H/32)), sol_s, sol_days*31558149.76/sid_days, sid_day, sol_s/(sol_s/86400+1), prec, sid_days/(sid_days-sol_days), rot, sol_s/sid_day, sid_day*(1+1/(prec*rot)))

6. Precession Durations

With the help of the above formulas and the coin rotation paradox, we can calculate all precession durations.

Variables used throughout this section:

  • YsidY_{\text{sid}} = Sidereal year (days or seconds, context-dependent)
  • YsolarY_{\text{solar}} = Solar (tropical) year (days or seconds, context-dependent)
  • YanomY_{\text{anom}} = Anomalistic year (days or seconds, context-dependent) β€” defined below
  • D(s)D(s) = Day length in SI seconds
  • PAP_A = Axial precession period (years)
  • PPP_P = Perihelion precession period (years)
  • PIP_I = Inclination precession period (years)
  • H=H = 335,317 years = Earth Fundamental Cycle

The Axial precession only needs the sidereal and solar year lengths (Section 5). Perihelion and Inclination precession also need the Anomalistic year β€” so that one is defined first (in days, then in seconds), before the three precession-duration formulas.

Anomalistic Year (in days β€” Fourier)

The anomalistic year uses 8 harmonics, with H/24 and H/13 as the dominant terms:

HarmonicPeriodsin coeff (sis_i)cos coeff (cic_i)Amplitude
H/2413,972 yrβˆ’2.801670eβˆ’07+2.357807eβˆ’090.024 s
H/1917,648 yr+2.216796eβˆ’07βˆ’1.497201eβˆ’090.019 s
H/13~25,794 yr+1.517396eβˆ’07βˆ’6.519321eβˆ’100.013 s
H/16~20,957 yr+5.329737eβˆ’10+9.467577eβˆ’080.008 s
H/8~41,915 yrβˆ’9.335198eβˆ’08+2.725305eβˆ’100.008 s
H/1130,483 yrβˆ’1.906973eβˆ’10βˆ’6.507288eβˆ’080.006 s
H/5~67,063 yrβˆ’5.388601eβˆ’11βˆ’2.957320eβˆ’080.003 s
H/3~111,772 yrβˆ’1.036667eβˆ’12+2.896582eβˆ’12< 10⁻⁸ s (structural, negligible)
EXCEL FORMULA LENGTH OF ANOMALISTIC YEAR IN DAYS (Fourier)
=LET(t, MODEL_YEAR+302635, H, 335317, mean, 365.242203646102*(H/16)/((H/16)-1), mean + (-1.036666643334E-12)*SIN(2*PI()*t/(H/3)) + (2.896582452374E-12)*COS(2*PI()*t/(H/3)) + (-5.388600561351E-11)*SIN(2*PI()*t/(H/5)) + (-2.957320128182E-08)*COS(2*PI()*t/(H/5)) + (-9.335197976099E-08)*SIN(2*PI()*t/(H/8)) + (2.725304683341E-10)*COS(2*PI()*t/(H/8)) + (-1.906973375443E-10)*SIN(2*PI()*t/(H/11)) + (-6.507288400912E-08)*COS(2*PI()*t/(H/11)) + (1.517395717327E-07)*SIN(2*PI()*t/(H/13)) + (-6.519320976697E-10)*COS(2*PI()*t/(H/13)) + (5.329736788609E-10)*SIN(2*PI()*t/(H/16)) + (9.467576939646E-08)*COS(2*PI()*t/(H/16)) + (2.216796076052E-07)*SIN(2*PI()*t/(H/19)) + (-1.497200952707E-09)*COS(2*PI()*t/(H/19)) + (-2.801669607452E-07)*SIN(2*PI()*t/(H/24)) + (2.357806548736E-09)*COS(2*PI()*t/(H/24)))

Anomalistic Year (in seconds)

The anomalistic year in seconds is derived by multiplying the Fourier result (in days) by the current day length:

Yanom(s)=Yanom(days)Γ—D(s)Y_{\text{anom}}(s) = Y_{\text{anom}}^{\text{(days)}} \times D(s)
EXCEL FORMULA ANOMALISTIC YEAR IN SI SECONDS
=LET(t, MODEL_YEAR+302635, H, 335317, sid_days, 365.242203646102*(H/13)/((H/13)-1) + (3.973164944361E-07)*SIN(2*PI()*t/(H/3)) + (-2.397135263624E-10)*COS(2*PI()*t/(H/3)) + (3.745546829271E-10)*SIN(2*PI()*t/(H/5)) + (3.355989685870E-07)*COS(2*PI()*t/(H/5)) + (-1.059417380507E-06)*SIN(2*PI()*t/(H/8)) + (1.713869484534E-09)*COS(2*PI()*t/(H/8)) + (1.991940272716E-08)*SIN(2*PI()*t/(H/16)) + (-3.841448696173E-07)*COS(2*PI()*t/(H/16)) + (-3.885222338193E-09)*SIN(2*PI()*t/(H/32)) + (3.535868885074E-08)*COS(2*PI()*t/(H/32)), anom_days, 365.242203646102*(H/16)/((H/16)-1) + (-1.036666643334E-12)*SIN(2*PI()*t/(H/3)) + (2.896582452374E-12)*COS(2*PI()*t/(H/3)) + (-5.388600561351E-11)*SIN(2*PI()*t/(H/5)) + (-2.957320128182E-08)*COS(2*PI()*t/(H/5)) + (-9.335197976099E-08)*SIN(2*PI()*t/(H/8)) + (2.725304683341E-10)*COS(2*PI()*t/(H/8)) + (-1.906973375443E-10)*SIN(2*PI()*t/(H/11)) + (-6.507288400912E-08)*COS(2*PI()*t/(H/11)) + (1.517395717327E-07)*SIN(2*PI()*t/(H/13)) + (-6.519320976697E-10)*COS(2*PI()*t/(H/13)) + (5.329736788609E-10)*SIN(2*PI()*t/(H/16)) + (9.467576939646E-08)*COS(2*PI()*t/(H/16)) + (2.216796076052E-07)*SIN(2*PI()*t/(H/19)) + (-1.497200952707E-09)*COS(2*PI()*t/(H/19)) + (-2.801669607452E-07)*SIN(2*PI()*t/(H/24)) + (2.357806548736E-09)*COS(2*PI()*t/(H/24)), anom_days*31558149.76/sid_days)

Axial Precession

Scientific Notation:

PA=YsidYsidβˆ’YsolarP_A = \frac{Y_{\text{sid}}}{Y_{\text{sid}} - Y_{\text{solar}}}

This is the β€œcoin rotation paradox” β€” the precession period equals the sidereal year divided by the difference between sidereal and solar years.

Mean value: PA=H13=P_A = \frac{H}{13} = ~25,794 years (range over one H cycle: ~25,226 – 26,334 yr; currently 25,771 yr at J2000)

Example Values:

MODEL_YEARAxial Precession (yr)
2000 AD25,771
Minimum25,226
Maximum26,334
EXCEL FORMULA LENGTH OF AXIAL PRECESSION IN YEARS
=LET(t, MODEL_YEAR+302635, H, 335317, sol_days, 365.242203646102 + (4.963655515994E-06)*SIN(2*PI()*t/(H/3)) + (-8.354140858289E-05)*COS(2*PI()*t/(H/3)) + (-1.645990068981E-08)*SIN(2*PI()*t/(H/5)) + (3.346788536693E-07)*COS(2*PI()*t/(H/5)) + (2.433753464246E-07)*SIN(2*PI()*t/(H/6)) + (-2.329695678249E-06)*COS(2*PI()*t/(H/6)) + (-1.299451587802E-05)*SIN(2*PI()*t/(H/8)) + (2.227963456434E-04)*COS(2*PI()*t/(H/8)) + (9.743882426511E-09)*SIN(2*PI()*t/(H/9)) + (-6.177071904623E-08)*COS(2*PI()*t/(H/9)) + (-8.831015131162E-07)*SIN(2*PI()*t/(H/11)) + (8.543316063918E-06)*COS(2*PI()*t/(H/11)) + (-4.515837982944E-08)*SIN(2*PI()*t/(H/14)) + (2.883165423683E-07)*COS(2*PI()*t/(H/14)) + (6.560490474823E-07)*SIN(2*PI()*t/(H/16)) + (-6.416442022713E-06)*COS(2*PI()*t/(H/16)) + (6.126554369035E-08)*SIN(2*PI()*t/(H/19)) + (-3.916922799144E-07)*COS(2*PI()*t/(H/19)) + (3.783127036528E-09)*SIN(2*PI()*t/(H/22)) + (-1.802048457974E-08)*COS(2*PI()*t/(H/22)) + (-2.612611973930E-08)*SIN(2*PI()*t/(H/24)) + (1.653905514523E-07)*COS(2*PI()*t/(H/24)) + (-2.964708323237E-09)*SIN(2*PI()*t/(H/32)) + (3.095269447075E-08)*COS(2*PI()*t/(H/32)), sid_days, 365.242203646102*(H/13)/((H/13)-1) + (3.973164944361E-07)*SIN(2*PI()*t/(H/3)) + (-2.397135263624E-10)*COS(2*PI()*t/(H/3)) + (3.745546829271E-10)*SIN(2*PI()*t/(H/5)) + (3.355989685870E-07)*COS(2*PI()*t/(H/5)) + (-1.059417380507E-06)*SIN(2*PI()*t/(H/8)) + (1.713869484534E-09)*COS(2*PI()*t/(H/8)) + (1.991940272716E-08)*SIN(2*PI()*t/(H/16)) + (-3.841448696173E-07)*COS(2*PI()*t/(H/16)) + (-3.885222338193E-09)*SIN(2*PI()*t/(H/32)) + (3.535868885074E-08)*COS(2*PI()*t/(H/32)), sid_days/(sid_days-sol_days))

Perihelion Precession

Scientific Notation (coin rotation paradox):

PP=Yanom(s)Yanom(s)βˆ’Ysolar(s)P_P = \frac{Y_{\text{anom}}(s)}{Y_{\text{anom}}(s) - Y_{\text{solar}}(s)}

Where YanomY_{\text{anom}}(s) and YsolarY_{\text{solar}}(s) are the anomalistic and solar years in seconds. This is the coin rotation paradox applied to the anomalistic and solar years β€” the perihelion precession period equals the anomalistic year (in seconds) divided by the difference between the anomalistic and solar years (in seconds).

Mean value: PP=H16=P_P = \frac{H}{16} = ~20,957 years (equivalent to PAβ‹…1316P_A \cdot \frac{13}{16}; range over one H cycle: ~20,580 – 21,312 yr; currently ~20,941 yr at J2000)

Example Values:

MODEL_YEARPerihelion Precession (yr)
2000 AD20,941
Minimum20,580
Maximum21,312
EXCEL FORMULA LENGTH OF PERIHELION PRECESSION IN YEARS (time-varying)
=LET(t, MODEL_YEAR+302635, H, 335317, sol_days, 365.242203646102 + (4.963655515994E-06)*SIN(2*PI()*t/(H/3)) + (-8.354140858289E-05)*COS(2*PI()*t/(H/3)) + (-1.645990068981E-08)*SIN(2*PI()*t/(H/5)) + (3.346788536693E-07)*COS(2*PI()*t/(H/5)) + (2.433753464246E-07)*SIN(2*PI()*t/(H/6)) + (-2.329695678249E-06)*COS(2*PI()*t/(H/6)) + (-1.299451587802E-05)*SIN(2*PI()*t/(H/8)) + (2.227963456434E-04)*COS(2*PI()*t/(H/8)) + (9.743882426511E-09)*SIN(2*PI()*t/(H/9)) + (-6.177071904623E-08)*COS(2*PI()*t/(H/9)) + (-8.831015131162E-07)*SIN(2*PI()*t/(H/11)) + (8.543316063918E-06)*COS(2*PI()*t/(H/11)) + (-4.515837982944E-08)*SIN(2*PI()*t/(H/14)) + (2.883165423683E-07)*COS(2*PI()*t/(H/14)) + (6.560490474823E-07)*SIN(2*PI()*t/(H/16)) + (-6.416442022713E-06)*COS(2*PI()*t/(H/16)) + (6.126554369035E-08)*SIN(2*PI()*t/(H/19)) + (-3.916922799144E-07)*COS(2*PI()*t/(H/19)) + (3.783127036528E-09)*SIN(2*PI()*t/(H/22)) + (-1.802048457974E-08)*COS(2*PI()*t/(H/22)) + (-2.612611973930E-08)*SIN(2*PI()*t/(H/24)) + (1.653905514523E-07)*COS(2*PI()*t/(H/24)) + (-2.964708323237E-09)*SIN(2*PI()*t/(H/32)) + (3.095269447075E-08)*COS(2*PI()*t/(H/32)), sid_days, 365.242203646102*(H/13)/((H/13)-1) + (3.973164944361E-07)*SIN(2*PI()*t/(H/3)) + (-2.397135263624E-10)*COS(2*PI()*t/(H/3)) + (3.745546829271E-10)*SIN(2*PI()*t/(H/5)) + (3.355989685870E-07)*COS(2*PI()*t/(H/5)) + (-1.059417380507E-06)*SIN(2*PI()*t/(H/8)) + (1.713869484534E-09)*COS(2*PI()*t/(H/8)) + (1.991940272716E-08)*SIN(2*PI()*t/(H/16)) + (-3.841448696173E-07)*COS(2*PI()*t/(H/16)) + (-3.885222338193E-09)*SIN(2*PI()*t/(H/32)) + (3.535868885074E-08)*COS(2*PI()*t/(H/32)), anom_days, 365.242203646102*(H/16)/((H/16)-1) + (-1.036666643334E-12)*SIN(2*PI()*t/(H/3)) + (2.896582452374E-12)*COS(2*PI()*t/(H/3)) + (-5.388600561351E-11)*SIN(2*PI()*t/(H/5)) + (-2.957320128182E-08)*COS(2*PI()*t/(H/5)) + (-9.335197976099E-08)*SIN(2*PI()*t/(H/8)) + (2.725304683341E-10)*COS(2*PI()*t/(H/8)) + (-1.906973375443E-10)*SIN(2*PI()*t/(H/11)) + (-6.507288400912E-08)*COS(2*PI()*t/(H/11)) + (1.517395717327E-07)*SIN(2*PI()*t/(H/13)) + (-6.519320976697E-10)*COS(2*PI()*t/(H/13)) + (5.329736788609E-10)*SIN(2*PI()*t/(H/16)) + (9.467576939646E-08)*COS(2*PI()*t/(H/16)) + (2.216796076052E-07)*SIN(2*PI()*t/(H/19)) + (-1.497200952707E-09)*COS(2*PI()*t/(H/19)) + (-2.801669607452E-07)*SIN(2*PI()*t/(H/24)) + (2.357806548736E-09)*COS(2*PI()*t/(H/24)), day_s, 31558149.76/sid_days, sol_s, sol_days*day_s, anom_s, anom_days*day_s, anom_s/(anom_s-sol_s))

Inclination Precession

Scientific Notation (coin rotation paradox):

PI=Yanom(s)Yanom(s)βˆ’Ysid(s)P_I = \frac{Y_{\text{anom}}(s)}{Y_{\text{anom}}(s) - Y_{\text{sid}}(s)}

Where YanomY_{\text{anom}}(s) and YsidY_{\text{sid}}(s) are the anomalistic and sidereal years in seconds. This is the coin rotation paradox applied to the anomalistic and sidereal years β€” the inclination precession period equals the anomalistic year (in seconds) divided by the difference between the anomalistic and sidereal years (in seconds).

Mean value: PI=H3=P_I = \frac{H}{3} = ~111,772 years (equivalent to PAβ‹…133P_A \cdot \frac{13}{3}; range over one H cycle: ~111,717 – 111,835 yr β€” exceptionally narrow since the sidereal year is almost constant; currently ~111,717 yr at J2000)

Example Values:

MODEL_YEARInclination Precession (yr)
2000 AD111,717
Minimum111,717
Maximum111,835
EXCEL FORMULA LENGTH OF INCLINATION PRECESSION IN YEARS (time-varying)
=LET(t, MODEL_YEAR+302635, H, 335317, sid_days, 365.242203646102*(H/13)/((H/13)-1) + (3.973164944361E-07)*SIN(2*PI()*t/(H/3)) + (-2.397135263624E-10)*COS(2*PI()*t/(H/3)) + (3.745546829271E-10)*SIN(2*PI()*t/(H/5)) + (3.355989685870E-07)*COS(2*PI()*t/(H/5)) + (-1.059417380507E-06)*SIN(2*PI()*t/(H/8)) + (1.713869484534E-09)*COS(2*PI()*t/(H/8)) + (1.991940272716E-08)*SIN(2*PI()*t/(H/16)) + (-3.841448696173E-07)*COS(2*PI()*t/(H/16)) + (-3.885222338193E-09)*SIN(2*PI()*t/(H/32)) + (3.535868885074E-08)*COS(2*PI()*t/(H/32)), anom_days, 365.242203646102*(H/16)/((H/16)-1) + (-1.036666643334E-12)*SIN(2*PI()*t/(H/3)) + (2.896582452374E-12)*COS(2*PI()*t/(H/3)) + (-5.388600561351E-11)*SIN(2*PI()*t/(H/5)) + (-2.957320128182E-08)*COS(2*PI()*t/(H/5)) + (-9.335197976099E-08)*SIN(2*PI()*t/(H/8)) + (2.725304683341E-10)*COS(2*PI()*t/(H/8)) + (-1.906973375443E-10)*SIN(2*PI()*t/(H/11)) + (-6.507288400912E-08)*COS(2*PI()*t/(H/11)) + (1.517395717327E-07)*SIN(2*PI()*t/(H/13)) + (-6.519320976697E-10)*COS(2*PI()*t/(H/13)) + (5.329736788609E-10)*SIN(2*PI()*t/(H/16)) + (9.467576939646E-08)*COS(2*PI()*t/(H/16)) + (2.216796076052E-07)*SIN(2*PI()*t/(H/19)) + (-1.497200952707E-09)*COS(2*PI()*t/(H/19)) + (-2.801669607452E-07)*SIN(2*PI()*t/(H/24)) + (2.357806548736E-09)*COS(2*PI()*t/(H/24)), anom_s, anom_days*31558149.76/sid_days, anom_s/(anom_s-31558149.76))

7. Planetary Precession Fluctuation

Calculates the perihelion precession β€œfluctuation” for any planet β€” the difference between observed precession and Newtonian planetary perturbation predictions. In the Holistic Universe Model, this fluctuation arises from Earth’s changing reference frame, not from relativistic effects.

What this formula calculates: The β€œanomaly” traditionally attributed to General Relativity. The model interprets it as a geometric effect from the interaction between Earth’s effective perihelion cycle (~20,957 years) and each planet’s own perihelion cycle.

Key inputs used in this section:

  • ERD (Earth Rate Deviation) β€” measures how Earth’s perihelion rate deviates from its mean rate. Calculated as the analytical derivative of the 25-harmonic perihelion formula in Section 4: Longitude of Perihelion.

Generic Formula Structure

Simplified Approximation: This two-term formula shows the conceptual structure of planetary precession fluctuation. The actual unified system uses ~2,400 terms for all 7 planets (planet-specific counts 2,393–2,435), accounting for:

  • Frequency mixing between Earth and planetary cycles
  • ERD (Earth Rate Deviation) corrections
  • Triple interactions (ERD Γ— periodic Γ— angle terms)
  • Higher harmonics (2ΞΈ, 3ΞΈ, 4ΞΈ patterns)
  • Time-varying obliquity/eccentricity (critical for Saturn)

This generic formula illustrates the basic concept. See the Predictive Formulas section below for the complete implementations for all planets.

For any planet P, the fluctuation formula has two components:

fluct_P = A_E Γ— cos(Ο‰_E Γ— (MODEL_YEAR + 302635) + Ο†_E)
        + A_P Γ— cos(n Γ— Ο‰_P Γ— (MODEL_YEAR + 302635) + Ο†_P)
        + C

Where:

  • Ο‰_E = 360 / ~20,957 = 0.017178Β°/year β€” Earth’s effective perihelion rate (same for all planets)
  • Ο‰_P = 360 / T_P β€” Planet’s perihelion precession rate
  • n = angle multiplier (typically 2 for double-angle pattern)
  • A_E = Earth term amplitude (planet-specific)
  • A_P = Planet term amplitude (planet-specific)
  • Ο†_E, Ο†_P = Phase offsets (planet-specific)
  • C = Offset constant (planet-specific)
  • 302,635 = Anchor year (Balanced Year, 302,635 BC)

Planetary Parameters

Planete (J2000)a (AU) †T_P (years)Ο‰_P (Β°/year)Direction
Mercury0.205640.3871243,8670.001476Prograde
Venus0.006780.7233447,089-0.000805Ecliptic-retrograde
Earth0.016711.0000ICRF ~111,772; effective ~20,957*0.017178Prograde
Mars0.093391.523776,6440.004697Prograde
Jupiter0.048395.199667,0630.005368Prograde
Saturn0.053869.531041,915-0.008589Ecliptic-retrograde
Uranus0.0472619.1408111,7720.003221Prograde
Neptune0.0085929.9282670,6340.000537Prograde

*Earth has true perihelion period ~111,772 years, but effective period ~20,957 years due to axial precession interaction. † Semi-major axis derived from Kepler’s 3rd law (a = (T_planet / T_earth)^(2/3)) using the model’s orbital period and MEAN_SOLAR_YEAR_DAYS as the Earth reference. Earth = 1.0 AU by definition.

Amplitude Scaling

The amplitudes scale with each planet’s orbital characteristics:

A_E = 329 Γ— (e / a)    [Earth term amplitude, arcsec/century]
A_P = 219 Γ— e          [Planet term amplitude, arcsec/century]

Observed-Angle Formulas (All 7 Planets)

The observed-angle formulas were developed during model fitting and require observational inputs (Earth perihelion, planetary perihelion, obliquity, eccentricity, ERD). They achieve RΒ² β‰ˆ 1.0000 for all 7 planets.

For predictions using only year as input (no observations required), see the Predictive Formulas below.

Predictive Formulas (Year-Only Input)

A key validation of the Holistic Model is that planetary precession can be predicted using only the year as input - no observations required. All parameters (Earth perihelion, obliquity, eccentricity, ERD, and the planet’s predicted perihelion) are computed from their respective formulas.

Why This Matters: Standard formulas require observed data (Earth perihelion position, planetary perihelion position, obliquity, eccentricity). The predictive formulas demonstrate that all these quantities are deterministic functions of time. Given only a year, we can compute all the necessary inputs from formulas and predict the precession fluctuation. This validates the model’s claim that observed precession β€œanomalies” are reference frame effects, not gravitational perturbations.

Predictive Perihelion Calculation

ΞΈP(t)=ΞΈ0+360Β°TPΓ—(tβˆ’2000)\theta_P(t) = \theta_0 + \frac{360Β°}{T_P} \times (t - 2000)

Where:

  • ΞΈ0\theta_0 = J2000 perihelion longitude (degrees)
  • TPT_P = Planet’s precession period
  • tt = MODEL_YEAR
PlanetPeriod (years)ΞΈβ‚€ (J2000)
Mercury243,86777.457Β°
Venus447,089131.577Β°
Mars76,644336.065Β°
Jupiter67,06314.707Β°
Saturn41,91592.128Β°
Uranus111,772170.731Β°
Neptune670,63445.801Β°

All planets use the same formula structure with their J2000 perihelion longitude as the reference point. For accuracy metrics (RΒ², RMSE) of all 7 planets, see the Summary table below.

Mercury

Uses predicted Mercury perihelion: ΞΈ_M = 77.457Β° + (360Β°/243,867) Γ— (MODEL_YEAR βˆ’ 2000)

Baseline precession: 531.4β€³/century = 1,296,000β€³ Γ· 243,867 years Γ— 100

Mercury Precession (Python)
from predict_precession import predict, predict_total
 
year = 2026
fluctuation = predict(year, 'mercury')  # Returns fluctuation only
total = predict_total(year, 'mercury')  # Returns baseline + fluctuation
print(f"Mercury {year}: {total:+.2f} arcsec/century (fluctuation: {fluctuation:+.2f})")

Python Result: RΒ² = 0.999999, RMSE = 0.0974β€³/century (2,421 unified terms with full precision coefficients)

Venus

Uses predicted Venus perihelion: ΞΈ_V = 131.577Β° + (360Β°/447,089) Γ— (MODEL_YEAR βˆ’ 2000)

Venus Precession (Python)
from predict_precession import predict, predict_total
 
# Calculate Venus precession for any year
year = 2026
fluctuation = predict(year, 'venus')  # Returns fluctuation only
total = predict_total(year, 'venus')  # Returns baseline + fluctuation
print(f"Venus {year}: {total:+.2f} arcsec/century (fluctuation: {fluctuation:+.2f})")

Result: RΒ² = 0.999997, RMSE = 0.9160β€³/century (2,421 unified terms, analytical ERD)

Outer Planets (Mars, Jupiter, Saturn, Uranus, Neptune)

Outer Planet Precession (Python)
from predict_precession import predict_total, get_all_predictions
 
year = 2026
 
# Calculate total observed precession for each planet (baseline + fluctuation)
print(f"Mars {year}:    {predict_total(year, 'mars'):+.2f} arcsec/century")
print(f"Jupiter {year}: {predict_total(year, 'jupiter'):+.2f} arcsec/century")
print(f"Saturn {year}:  {predict_total(year, 'saturn'):+.2f} arcsec/century")
print(f"Uranus {year}:  {predict_total(year, 'uranus'):+.2f} arcsec/century")
print(f"Neptune {year}: {predict_total(year, 'neptune'):+.2f} arcsec/century")
 
# Or get all predictions at once:
results = get_all_predictions(year)
for planet, data in results.items():
    print(f"{data['name']}: {data['total']:+.2f} arcsec/century")

Expected output at J2000:

PlanetTotal Precession (β€³/century)BaselineFluctuation
Mercury~+569+531.4~+38
Venus~+-65-289.9~+225
Mars~+1,574+1,690.9~-117
Jupiter~+1,780+1,932.5~-153
Saturn~-3,372-3,092.0~-280
Uranus~+1,116+1,159.5~-44
Neptune~+196+193.2~+3

Unified Feature Matrix Structure (~2,400 terms for all planets; planet-specific counts 2,393–2,435):

The unified feature matrix is organized into 25 groups:

GroupTermsDescription
118Angle terms (Ξ΄, Ξ£, harmonics)
26Obliquity/Eccentricity terms
312ERD terms (linear, quadratic, cubic)
422Periodic terms (11 periods Γ— 2)
522ERD Γ— Periodic
612ERDΒ² Γ— Periodic
724Periodic Γ— Angle
816ERD Γ— Periodic Γ— Angle
916Periodic Γ— 2Ξ΄
1012Periodic Γ— Periodic
111Constant
124ERD Γ— Sum-angle
1360Extended harmonics (3Ξ΄, beats)
1416Venus periodic terms
1512Time-varying obliq/ecc (critical for Saturn)
1620Venus fine-tuning
1714Greedy-selected periods (H/9, H/45, H/60, H/41, H/56, H/75, H/110)
1810Greedy-selected periods (H/39, H/37, H/96, H/112, H/7)
196Greedy-selected periods (H/18, H/38, H/31)
204Greedy-selected periods (H/35, H/128)
212Greedy-selected period (H/42)
2214ERD Γ— Greedy periods
2314Angle Γ— Greedy periods
2418Saturn high-frequency harmonics H/(8Γ—34Γ—n), n=1..9
2522Venus high-frequency triplets (carrier Β±16 sidebands)

Where:

  • Ξ΄ = ΞΈ_E βˆ’ ΞΈ_P (difference angle between Earth and planet perihelia)
  • Ξ£ = ΞΈ_E + ΞΈ_P (sum angle)
  • ΞΈ_E = Earth perihelion from 25-harmonic formula
  • ΞΈ_P = planet perihelion calculated from J2000 value and period

Saturn: Critical Obliquity/Eccentricity Coupling

Saturn’s perihelion period (H/8 = 41,915 years) equals the obliquity cycle, creating strong coupling with Earth’s obliquity/eccentricity variations. The GROUP 15 terms capture this coupling:

Ξ΅osc(t)=cos⁑(2Ο€tTH/8),eosc(t)=cos⁑(2Ο€tTH/8)\varepsilon_{\text{osc}}(t) = \cos\left(\frac{2\pi t}{T_{H/8}}\right), \quad e_{\text{osc}}(t) = \cos\left(\frac{2\pi t}{T_{H/8}}\right)

where TH/8T_{H/8} = 41,915 years (Saturn’s perihelion period = obliquity cycle).

Saturn: The Obliquity Driver

Saturn is the only planet among all seven that requires the GROUP 15 time-varying obliquity/eccentricity terms to achieve accurate predictive modeling. Without these terms, Saturn accuracy degrades significantly. The GROUP 24 terms (H/(8Γ—34Γ—n) Saturn-Fibonacci harmonics) are a key driver of Saturn’s final accuracy (0.0977β€³/century).

This mathematical requirement strongly suggests that Saturn drives Earth’s obliquity cycle. The exact period match (Saturn precession = obliquity cycle = 41,915 years = H/8) is not coincidenceβ€”the model cannot achieve RΒ² = 1.0000 for Saturn without explicitly including this coupling.

See Scientific Background: Milankovitch Theory for the physical interpretation.

Venus: Fine-Tuning for Large Fluctuations

Venus has very large fluctuations (up to ~1000 arcsec/century) compared to other planets. GROUPs 14, 16, and 25 provide Venus-specific fine-tuning β€” GROUP 25 adds high-frequency triplets (carrier Β±16 sidebands at H/124–H/158) from Earth perihelion modulation, improving Venus accuracy to RΒ² = 0.999997, RMSE = 0.9160β€³/century.

Summary: All Predictive Formulas (Unified 2400-Term System)

PlanetRΒ²RMSE (β€³/century)TermsBaseline (β€³/century)Implementation
Mercury0.9999990.09742,421531.4Python
Venus0.9999970.91602,421-289.9Python
Mars0.9999990.09802,4351,690.9Python
Jupiter0.9999990.09752,4071,932.5Python
Saturn1.0000000.09772,407-3,092.0Python
Uranus0.9999980.10272,4071,159.5Python
Neptune0.9999850.09782,393193.2Python

Notes:

  • All 7 planets share the same unified ~2,400-term feature matrix (planet-specific term counts 2,393–2,435) with planet-specific coefficients
  • All formulas output total observed precession (baseline + fluctuation)
  • Saturn has negative baseline (ecliptic-retrograde precession)
  • All RΒ² values > 0.99998

The predictive formulas require no observations whatsoever - only the year is needed as input. All parameters (Earth perihelion, obliquity, eccentricity, ERD) are computed from their respective formulas. This demonstrates that planetary precession patterns are deterministic consequences of Earth’s orbital parameters and time, validating the Holistic Model’s framework.

For the physical basis and complete derivation, see Scientific Background: Mercury Perihelion.

8. How the Formulas Connect

All formulas are interconnected through the Earth Fundamental Cycle (335,317 years) and its Fibonacci divisions:

CycleDivisorDuration (years)Used In
Earth Fundamental Cycle1335,317All calculations
Inclination PrecessionΓ·3~111,772Obliquity, Inclination
Obliquity CycleΓ·8~41,915Obliquity
Axial PrecessionΓ·13~25,794Longitude of perihelion
Perihelion PrecessionΓ·16~20,957Eccentricity, Day/Year, Planetary Fluctuation
Mercury PerihelionΓ—8/11243,867Mercury Precession Fluctuation
Venus Perihelionβˆ’8/6447,089 (ecliptic-retrograde)Venus Precession Fluctuation
Mars PerihelionΓ—8/3576,644Mars Precession Fluctuation
Jupiter PerihelionΓ·567,063Jupiter Precession Fluctuation
Saturn Perihelionβˆ’1/841,915 (ecliptic-retrograde)Saturn Precession Fluctuation
Uranus PerihelionΓ·3111,772Uranus Precession Fluctuation
Neptune PerihelionΓ—2670,634Neptune Precession Fluctuation

Note: All planet perihelion periods are Fibonacci-fraction multiples of the Earth Fundamental Cycle. Saturn’s period coincides with the obliquity cycle (H/8) and is ecliptic-retrograde; Venus is also ecliptic-retrograde (βˆ’8H/6). Uranus’s period coincides with inclination precession (H/3). Neptune is at 2H. Earth’s effective perihelion period (~20,957 years = H/16) is the common component for calculating fluctuations of ALL planets.

β”Œβ”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”
β”‚                    HOLISTIC-YEAR (H years)                       β”‚
β”œβ”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€
β”‚                              β”‚                                  β”‚
β”‚         Γ·3 = H/3             β”‚         Γ·13 = H/13               β”‚
β”‚    (Inclination Precession)  β”‚     (Axial Precession)           β”‚
β”‚              β”‚               β”‚              β”‚                   β”‚
β”‚              β–Ό               β”‚              β–Ό                   β”‚
β”‚         Inclination          β”‚      Longitude of                β”‚
β”‚           Formula            β”‚       Perihelion                 β”‚
β”‚              β”‚               β”‚              β”‚                   β”‚
β”‚              β–Ό               β”‚              β”‚                   β”‚
β”‚         Γ·8 = H/8       ◄─────┼───────────────                   β”‚
β”‚       (Obliquity Cycle)      β”‚              β”‚                   β”‚
β”‚              β”‚               β”‚              β–Ό                   β”‚
β”‚              β–Ό               β”‚       Γ·16 = H/16                 β”‚
β”‚          Obliquity           β”‚  (Earth Effective Perihelion)    β”‚
β”‚           Formula            β”‚              β”‚                   β”‚
β”‚                              β”‚              β”œβ”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β–Ί β”‚
β”‚                              β”‚              β”‚   ALL PLANETS     β”‚
β”‚                              β”‚              β”‚   Fluctuation     β”‚
β”‚                              β”‚              β”‚   (Earth Term)    β”‚
β”‚                              β”‚              β–Ό                   β”‚
β”‚                              β”‚    Day Length, Year Lengths      β”‚
β”‚                              β”‚    (Fourier harmonics at H/8,    β”‚
β”‚                              β”‚     H/3, H/16, H/24)            β”‚
β”‚                              β”‚                                  β”‚
β”‚   Planet-specific terms: ────┼──────────────────────────────────│
β”‚   Mercury: HΓ—8/11            β”‚   Venus: -8H/6 (retrograde)      β”‚
β”‚   Mars: HΓ—8/35               β”‚   Jupiter: H/5                   β”‚
β”‚   Saturn: -H/8 (retrograde)  β”‚   Uranus: H/3                    β”‚
β”‚   Neptune: HΓ—2               β”‚                                  β”‚
β””β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”˜

9. Verification

To verify these formulas, compare results against:

  1. J2000 Values (Year = 2000):

    • Obliquity: 23.4393Β° (IAU: 23.439291Β°) βœ“
    • Eccentricity: 0.01671 (NASA: 0.01671022) βœ“
    • Longitude of perihelion: 102.947Β° (NASA: 102.94719Β°) βœ“

  2. Historical Values:

    • Obliquity at -10,000: 24.5293Β° (La2004: 24.1592Β°, Chapront: 24.3053Β°) β€” model agrees within ~0.37Β° at this epoch; near-term values match to within 0.001Β°. See Model vs La2004/Chapront comparison table for the full series.
    • Perihelion aligned with December solstice at 1246 AD βœ“

  3. Mercury Precession Fluctuation (verified against 3D simulation output):

    β€œModel Data” in this table refers to values computed by the Holistic Model’s 3D simulation, which calculates planetary positions using the model’s geometric framework. The formula results are compared against these simulation outputs.

    MODEL_YEARFormula Result3D SimulationError
    191244.6β€³/century42.94β€³/century+1.7
    202340.1β€³/century38.36β€³/century+1.7
    26894.4β€³/century4.18β€³/century+0.2
    3244-25.3β€³/century-26.10β€³/century+0.8
    • Model baseline: 531.4β€³/century (Newtonian prediction from period)
    • At year 2023: fluctuation β‰ˆ +38β€³/century
    • Observed total (Park et al. 2017): 575.31 Β± 0.0015β€³/century

  4. Interactive Calculator:

    • Use the Orbital Calculator to compute all values for any year instantly β€” it shows J2000 reference values side-by-side with model output

  5. 3D Simulation:

    • Use the Interactive 3D Simulation to verify values visually
    • Console tests (F12) compare model values against IAU reference data

10. Dashboard

All formulas and calculated values are available in the Data Explorer dashboardΒ . Click the PURPLE Data Button at the top of any page to open it.

Formula Derivation

For the detailed derivation of these formulas β€” including the Fibonacci hierarchy, resonance analysis, coefficient breakdowns per planet, and observed-angle formulas β€” see the Formula Derivation documentationΒ  in the technical GitHub repository.

Return to Mathematical Foundations or explore the Appendix for reference data.

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