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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: YEAR). Section 7 (planetary precession) uses Python due to the 429-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 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.

Quick Reference

TermValueMeaning
Holistic-Year (H)335,317 yearsMaster cycle from which all periods derive via Fibonacci fractions
Anchor Year-302,635 (302,635 BC)Year zero of the current Holistic cycle; all formulas use Year + <V k="anchorYearOffset" />
ERDEarth Rate DeviationHow much Earth’s perihelion rate differs from its mean (Β°/year); see Section 4

New to the Holistic Model? Start with the Model Overview for the conceptual framework, or see Scientific Background for the physical basis of these formulas.

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 = Year + 302,635 β€” Time relative to anchor year (302,635 BC)
EXCEL FORMULA OBLIQUITY IN DEGREES (Change the YEAR value)
=23.41354+(-0.63603*(COS(RADIANS(((YEAR--302635)/(335317/3))* 360))))+(0.63603*(COS(RADIANS(((YEAR--302635)/(335317/8))*360))))

Components:

  • meanObliquity β€” Mean obliquity (23.41354Β°)
  • earthInclAmplitude β€” Amplitude of variation (0.63603Β°)
  • H/3 β€” Inclination precession cycle (H/3 = 111,772 years)
  • H/8 β€” Obliquity cycle (H/8 = 41,915 years)
  • -balancedYearOffset β€” Anchor year (-302,635 = 302,635 BC, start of current Holistic-Year cycle)

Example Values:

YearObliquity
2000 AD23.439Β°
-10000 (10,000 BCE)24.23Β°
+10000 (10,000 AD)22.41Β°

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 Ξ΅p(2000)=Ξ΅p,J2000\varepsilon_p(2000) = \varepsilon_{p,J2000} exactly. Venus and Neptune have TObliq=TICRFT_{\text{Obliq}} = T_{\text{ICRF}} (both 8H/100), 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’28,844-28{,}844 yr894,179 yrΒ±\pm0.386477Β°0.03Β°
Marsβˆ’38,877-38{,}877 yr125,744 yrΒ±\pm1.164217Β°25.19Β°
Jupiterβˆ’41,915-41{,}915 yr167,659 yrΒ±\pm0.021404Β°3.13Β°
Saturnβˆ’15,967-15{,}967 yr111,772 yrΒ±\pm0.065192Β°26.73Β°
Uranusβˆ’33,532-33{,}532 yr167,659 yrΒ±\pm0.023831Β°82.23Β°

Mean Obliquity (analytical, averaged over 8H)

The mean obliquity over the Grand Holistic Octave 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:

e(t)=e0+(βˆ’Aβˆ’(e0βˆ’ebase)β‹…cos⁑ϕ)β‹…cos⁑ϕe(t) = e_0 + \left( -A - (e_0 - e_{\text{base}}) \cdot \cos\phi \right) \cdot \cos\phi

Where:

  • Ο•=2Ο€tTP\phi = \frac{2\pi t}{T_P} β€” Phase angle
  • ebasee_{\text{base}} = 0.015386 β€” Arithmetic midpoint of data range: (max + min) / 2
  • AA = 0.001356 β€” Half-range amplitude: (max - min) / 2
  • e0e_0 = √(ebasee_{\text{base}}Β² + AAΒ²) = 0.0154454 β€” Derived mean eccentricity
  • TP=T_P = 20,957 years β€” Perihelion precession period (H/16)
  • tt = Year + 302,635 β€” Time relative to anchor year (302,635 BC)

Only 2 free parameters (ebasee_{\text{base}} and AA), both derived directly from the data extremes. The extremes are exactly ebasee_{\text{base}} Β± AA. The mean e0e_0 and the cosΒ² coefficient (e0e_0 βˆ’ ebasee_{\text{base}}) are both derived from these two inputs. RΒ² = 0.9999968 against model data.

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

Components:

  • eccentricityBase β€” Arithmetic midpoint (0.015386): (max + min) / 2
  • eccentricityAmplitude β€” Half-range amplitude (0.001356): (max - min) / 2
  • SQRT(eccentricityBaseΒ²+eccentricityAmplitudeΒ²) β€” Derived mean eccentricity (0.0154454)
  • H/16 β€” Perihelion precession cycle (H/16 = 20,957 years)

Example Values:

YearEccentricity
2000 AD0.016710
Minimum0.013898
Maximum0.016744

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 = 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 YEAR value)
=1.48113+(-0.63603*(COS(RADIANS(((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:

YearInclination
2000 AD~1.58Β°
Maximum~2.115Β°
Minimum~0.848Β°

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

Harmonic Terms (all 25 periods derived from H):

PeriodH Divisionsin coeffcos coeffPhysical Origin
20,957H/16+4.835748Β°βˆ’0.021962Β°Primary perihelion
10,479H/32+2.659030Β°+0.247035Β°2nd harmonic
6,986H/48+0.218927Β°+0.019922Β°3rd harmonic
5,239H/64+0.070333Β°+0.012269Β°4th harmonic
111,772H/3βˆ’0.131811Β°+0.007264Β°Inclination coupling
11,563H/29βˆ’0.130892Β°βˆ’0.006027Β°Residual correction
13,972H/24+0.130382Β°+0.006004Β°Residual correction
41,915H/8+0.120192Β°βˆ’0.007799Β°Obliquity coupling
8,383H/40+0.016174Β°+0.000644Β°Fine correction
25,794H/13+0.011620Β°+0.000536Β°Axial precession coupling
7,451H/45βˆ’0.010562Β°βˆ’0.000385Β°Beat frequency
4,191H/80+0.010342Β°+0.001892Β°Venus fine-tuning
5,988H/56+0.006926Β°+0.000872Β°Beat frequency
5,497H/61βˆ’0.006476Β°βˆ’0.000855Β°Beat frequency
9,580H/35βˆ’0.005621Β°βˆ’0.000259Β°Beat frequency
15,967H/21βˆ’0.003468Β°+0.000048Β°Venus coupling
67,063H/5βˆ’0.003219Β°+0.000012Β°Jupiter period
3,493H/96+0.002705Β°+0.000684Β°Fine correction
17,648H/19+0.000499Β°+0.000023Β°Fine correction
12,897H/26+0.001754Β°βˆ’0.000024Β°Beat frequency
9,063H/37βˆ’0.000681Β°βˆ’0.000000Β°Beat frequency
4,657H/72+0.001276Β°+0.000169Β°Fine correction
4,355H/77βˆ’0.000999Β°βˆ’0.000132Β°Fine correction
2,994H/112+0.000507Β°+0.000139Β°Fine correction

Constant offset: c0=βˆ’0.2598Β°c_0 = -0.2598Β°

Accuracy: RMSE = 0.0006Β° vs actual orbital data, J2000 error = 0.0000Β° (25-term Fourier fit)

Note: The Excel formula below uses the 12 dominant harmonics (out of 25) β€” this captures ~99.99% of the variation. J2000 reproduction is exact to 0.0001Β°. For full 25-term precision, use the Python/TypeScript implementation or the Orbital Calculator.

EXCEL FORMULA LONGITUDE OF PERIHELION (12 dominant harmonics)
=MOD(270+(YEAR+302635)*(360/(335317/16))+4.835748*SIN(2*PI()*(YEAR+302635)/(335317/16))-0.021962*COS(2*PI()*(YEAR+302635)/(335317/16))+2.659030*SIN(2*PI()*(YEAR+302635)/(335317/32))+0.247035*COS(2*PI()*(YEAR+302635)/(335317/32))+0.218927*SIN(2*PI()*(YEAR+302635)/(335317/48))+0.019922*COS(2*PI()*(YEAR+302635)/(335317/48))-0.131811*SIN(2*PI()*(YEAR+302635)/(335317/3))+0.007264*COS(2*PI()*(YEAR+302635)/(335317/3))-0.130892*SIN(2*PI()*(YEAR+302635)/(335317/29))-0.006027*COS(2*PI()*(YEAR+302635)/(335317/29))+0.130382*SIN(2*PI()*(YEAR+302635)/(335317/24))+0.006004*COS(2*PI()*(YEAR+302635)/(335317/24))+0.120192*SIN(2*PI()*(YEAR+302635)/(335317/8))-0.007799*COS(2*PI()*(YEAR+302635)/(335317/8))+0.070333*SIN(2*PI()*(YEAR+302635)/(335317/64))+0.012269*COS(2*PI()*(YEAR+302635)/(335317/64))+0.016174*SIN(2*PI()*(YEAR+302635)/(335317/40))+0.000644*COS(2*PI()*(YEAR+302635)/(335317/40))+0.011620*SIN(2*PI()*(YEAR+302635)/(335317/13))+0.000536*COS(2*PI()*(YEAR+302635)/(335317/13))-0.010562*SIN(2*PI()*(YEAR+302635)/(335317/45))-0.000385*COS(2*PI()*(YEAR+302635)/(335317/45))+0.010342*SIN(2*PI()*(YEAR+302635)/(335317/80))+0.001892*COS(2*PI()*(YEAR+302635)/(335317/80))-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=360Β°<Vk="periPrecYears"/>=0.01718Β°\omega_0 = \frac{360Β°}{<V k="periPrecYears" />} = 0.01718Β°/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) / (Year_current - Year_previous),
  IF(ΞΈ_E_current - ΞΈ_E_previous > 180,
    (ΞΈ_E_current - ΞΈ_E_previous - 360) / (Year_current - Year_previous),
    (ΞΈ_E_current - ΞΈ_E_previous) / (Year_current - 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, 12 dominant harmonics)
=4.835748*(2*PI()/(335317/16))*COS(2*PI()*(YEAR+302635)/(335317/16))+0.021962*(2*PI()/(335317/16))*SIN(2*PI()*(YEAR+302635)/(335317/16))+2.659030*(2*PI()/(335317/32))*COS(2*PI()*(YEAR+302635)/(335317/32))-0.247035*(2*PI()/(335317/32))*SIN(2*PI()*(YEAR+302635)/(335317/32))+0.218927*(2*PI()/(335317/48))*COS(2*PI()*(YEAR+302635)/(335317/48))-0.019922*(2*PI()/(335317/48))*SIN(2*PI()*(YEAR+302635)/(335317/48))-0.131811*(2*PI()/(335317/3))*COS(2*PI()*(YEAR+302635)/(335317/3))-0.007264*(2*PI()/(335317/3))*SIN(2*PI()*(YEAR+302635)/(335317/3))-0.130892*(2*PI()/(335317/29))*COS(2*PI()*(YEAR+302635)/(335317/29))+0.006027*(2*PI()/(335317/29))*SIN(2*PI()*(YEAR+302635)/(335317/29))+0.130382*(2*PI()/(335317/24))*COS(2*PI()*(YEAR+302635)/(335317/24))-0.006004*(2*PI()/(335317/24))*SIN(2*PI()*(YEAR+302635)/(335317/24))+0.120192*(2*PI()/(335317/8))*COS(2*PI()*(YEAR+302635)/(335317/8))+0.007799*(2*PI()/(335317/8))*SIN(2*PI()*(YEAR+302635)/(335317/8))+0.070333*(2*PI()/(335317/64))*COS(2*PI()*(YEAR+302635)/(335317/64))-0.012269*(2*PI()/(335317/64))*SIN(2*PI()*(YEAR+302635)/(335317/64))+0.016174*(2*PI()/(335317/40))*COS(2*PI()*(YEAR+302635)/(335317/40))-0.000644*(2*PI()/(335317/40))*SIN(2*PI()*(YEAR+302635)/(335317/40))+0.011620*(2*PI()/(335317/13))*COS(2*PI()*(YEAR+302635)/(335317/13))-0.000536*(2*PI()/(335317/13))*SIN(2*PI()*(YEAR+302635)/(335317/13))-0.010562*(2*PI()/(335317/45))*COS(2*PI()*(YEAR+302635)/(335317/45))+0.000385*(2*PI()/(335317/45))*SIN(2*PI()*(YEAR+302635)/(335317/45))+0.010342*(2*PI()/(335317/80))*COS(2*PI()*(YEAR+302635)/(335317/80))-0.001892*(2*PI()/(335317/80))*SIN(2*PI()*(YEAR+302635)/(335317/80))

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

Key Values:

YearLongitude
1246 AD90Β° (aligned with December solstice)
2000 AD~102.95Β°
22,203.344 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=102.947Β°\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 a single input constant; only the harmonic coefficients are empirical (fitted from 491 data points spanning Β±25,000 years).

Fourier Harmonic Model

All three year types follow the same pattern:

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=YEARβˆ’AnchorYeart = \text{YEAR} - \text{AnchorYear}, with AnchorYear = -302,635.

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.

HarmonicPeriodsin coeff (sis_i)cos coeff (cic_i)Amplitude
H/841,915 yrβˆ’1.299452eβˆ’05+2.227963eβˆ’041.93 s
H/3111,772 yr+4.963656eβˆ’06βˆ’8.354141eβˆ’050.72 s
H/1620,957 yr+6.560490eβˆ’07βˆ’6.416442eβˆ’060.56 s
EXCEL FORMULA LENGTH OF SOLAR YEAR IN DAYS (Fourier)
=LET(t, 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.

HarmonicPeriodsin coeff (sis_i)cos coeff (cic_i)Amplitude
H/841,915 yrβˆ’1.059417eβˆ’06+1.713869eβˆ’090.092 s
H/3111,772 yr+3.973165eβˆ’07βˆ’2.397135eβˆ’100.034 s
EXCEL FORMULA LENGTH OF SIDEREAL YEAR IN DAYS (Fourier)
=LET(t, 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)=<VΒ k="siderealYearSeconds"Β />Β SIΒ secondsY_{\text{sid}}^{\text{(s)}} = \text{<V k="siderealYearSeconds" />} \text{ 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, 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, 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, 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, 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.

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

EXCEL FORMULA LENGTH OF AXIAL PRECESSION IN YEARS
=LET(t, 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})

EXCEL FORMULA LENGTH OF PERIHELION PRECESSION IN YEARS (time-varying)
=LET(t, 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})

EXCEL FORMULA LENGTH OF INCLINATION PRECESSION IN YEARS (time-varying)
=LET(t, 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))

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/1325,794 yr+1.517396eβˆ’07βˆ’6.519321eβˆ’100.013 s
H/1620,957 yr+5.329737eβˆ’10+9.467577eβˆ’080.008 s
H/841,915 yrβˆ’9.335198eβˆ’08+2.725305eβˆ’100.008 s
H/1130,483 yrβˆ’1.906973eβˆ’10βˆ’6.507288eβˆ’080.006 s
H/567,063 yrβˆ’5.388601eβˆ’11βˆ’2.957320eβˆ’080.003 s
H/3111,772 yrβˆ’1.036667eβˆ’12+2.896582eβˆ’120.000 s
EXCEL FORMULA LENGTH OF ANOMALISTIC YEAR IN DAYS (Fourier)
=LET(t, 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, 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)

Where:

  • YsidY_{\text{sid}} = Sidereal year in days
  • YsolarY_{\text{solar}} = Solar year in days
  • D(s)D(s) = Day length in SI seconds
  • PAP_A = Axial precession period in years
  • PPP_P = Perihelion precession period in years
  • H=H = 335,317 years = Holistic-Year

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 21-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 429 terms for all 7 planets, 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— (YEAR + 302635) + Ο†_E)
        + A_P Γ— cos(n Γ— Ο‰_P Γ— (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

Planetea (AU)e Γ— a (AU)T_P (years)Ο‰_P (Β°/year)Direction
Mercury0.205640.38710.0796243,8670.001476Prograde
Venus0.006780.72330.0049670,6340.000537Prograde
Earth0.016711.00000.0167111,772 / 20,957*0.017178Prograde
Mars0.093391.52370.142376,6440.004697Prograde
Jupiter0.048395.19970.251667,0630.005368Prograde
Saturn0.053869.53010.513341,915-0.008589Ecliptic-retrograde
Uranus0.0472619.13800.9044111,7720.003221Prograde
Neptune0.0085929.97030.2574670,6340.000537Prograde

*Earth has true perihelion period 111,772 years, but effective period 20,957 years due to axial precession interaction.

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 = 1.0000 for all 7 planets using 225 terms per planet (328 for Venus).

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 = Year
PlanetPeriod (years)ΞΈβ‚€ (J2000)
Mercury243,86777.457Β°
Venus670,634131.577Β°
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— (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 = 2022
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 = 0.9999, RMSE = 0.75β€³/century (429 unified terms with full precision coefficients)

Venus

Uses predicted Venus perihelion: ΞΈ_V = 131.577Β° + (360Β°/670,634) Γ— (Year βˆ’ 2000)

Venus Precession (Python)
from predict_precession import predict, predict_total
 
# Calculate Venus precession for any year
year = 2022
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 = 1.0000, RMSE = 3.47β€³/century (429 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 = 2022
 
# 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~+361+193.2~+168
Mars~+1,591+1,690.9~-100
Jupiter~+1,780+1,932.5~-153
Saturn~-3,372-3,092.0~-280
Uranus~+1,072+1,159.5~-87
Neptune~+201+193.2~+8

Unified Feature Matrix Structure (429 terms for all planets):

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 21-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€t41915),eosc(t)=cos⁑(2Ο€t41915)\varepsilon_{\text{osc}}(t) = \cos\left(\frac{2\pi t}{41915}\right), \quad e_{\text{osc}}(t) = \cos\left(\frac{2\pi t}{41915}\right)

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) further improve Saturn from RMSE 11.59 to 3.72β€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 = 1.0000, RMSE = 3.47β€³/century.

Summary: All Predictive Formulas (Unified 429-Term System)

PlanetRΒ²RMSE (β€³/century)TermsBaseline (β€³/century)Implementation
Mercury0.99990.75429531.4Python
Venus1.00003.47429193.2Python
Mars0.99962.034291,690.9Python
Jupiter0.99962.334291,932.5Python
Saturn0.99963.72429-3,092.0Python
Uranus0.99961.404291,159.5Python
Neptune0.99990.25429193.2Python

Notes:

  • All 7 planets use the same unified 429-term feature matrix with planet-specific coefficients
  • All formulas output total observed precession (baseline + fluctuation)
  • Saturn has negative baseline (ecliptic-retrograde precession)
  • All RΒ² values > 0.999

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 Holistic-Year (335,317 years) and its Fibonacci divisions:

CycleDivisorDuration (years)Used In
Holistic-Year1335,317All calculations
Inclination PrecessionΓ·3111,772Obliquity, Inclination
Obliquity CycleΓ·841,915Obliquity
Axial PrecessionΓ·1325,793.62Longitude of perihelion
Perihelion PrecessionΓ·1620,957Eccentricity, Day/Year, Planetary Fluctuation
Mercury PerihelionΓ—8/11243,867Mercury Precession Fluctuation
Venus PerihelionΓ—2670,634Venus Precession Fluctuation
Mars PerihelionΓ—8/3576,644Mars Precession Fluctuation
Jupiter PerihelionΓ·567,063Jupiter Precession Fluctuation
Saturn PerihelionΓ·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 Holistic-Year. Saturn’s period coincides with the obliquity cycle (H/8) and is ecliptic-retrograde. Venus and Neptune share the same period (HΓ—2). Uranus’s period coincides with inclination precession (H/3). Earth’s effective perihelion period (20,957 years = H/16) is the common component for calculating fluctuations of ALL planets.

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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: HΓ—2                     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: 23.439Β° (IAU: 23.439291Β°) βœ“
    • Eccentricity: 0.01671 (NASA: 0.01671022) βœ“
    • Longitude of perihelion: 102.95Β° (NASA: 102.94719Β°) βœ“

  2. Historical Values:

    • Obliquity at -10,000: 24.23Β° (Laskar: 24.23Β°) βœ“
    • 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.

    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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