Calculation Formulas

This page provides Excel and Python formulas for calculating key values at any given year in the Holistic Universe Model. Sections 1–6 include Excel formulas (single input: YEAR). Section 7 (planetary precession fluctuation) uses Python only.

Quick Reference

Contents

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

1. Obliquity (Axial Tilt)

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

Scientific Notation:

Where:

• $\varepsilon_0 = 23.414°$ — Mean obliquity
• $A = 0.634°$ — Amplitude of variation
• $T_I = 111,296$ years — Inclination precession period (H/3)
• $T_O = 41,736$ years — Obliquity cycle period (H/8)
• $t$ = Year + 301,340 — Time relative to anchor year

=23.41398+(-0.633849*(COS(RADIANS(((YEAR--301340)/(333888/3))* 360))))+(0.633849*(COS(RADIANS(((YEAR--301340)/(333888/8))*360))))

Components:

• 23.41398 — Mean obliquity (degrees)
• 0.633849 — Amplitude of variation (degrees)
• 333888/3 — Inclination precession cycle (111,296 years)
• 333888/8 — Obliquity cycle (41,736 years)
• -301340 — Anchor year (start of current Holistic-Year cycle)

Example Values:

2. Eccentricity

Calculates Earth’s orbital eccentricity at any given year.

Scientific Notation:

Where:

• $\phi = \frac{2\pi t}{T_P}$ — Phase angle
• $e_{\text{base}}$ = 0.015321 — Arithmetic midpoint of data range: (max + min) / 2
• $A = 0.0014226$ — Half-range amplitude: (max - min) / 2
• $e_0$ = √($e_{\text{base}}$² + $A$²) = 0.0153869 — Derived mean eccentricity
• $T_P = 20,868$ years — Perihelion precession period (H/16)
• $t$ = Year + 301,340 — Time relative to anchor year

Only 2 free parameters ($e_{\text{base}}$ and $A$), both derived directly from the data extremes. The extremes are exactly $e_{\text{base}}$ ± $A$. The mean $e_0$ and the cos² coefficient ($e_0$ − $e_{\text{base}}$) are both derived from these two inputs. R² = 0.9999968 against model data.

=(SQRT(0.015321^2+0.0014226^2)+((-0.0014226-(SQRT(0.015321^2+0.0014226^2)-0.015321)*(COS(RADIANS(((YEAR--301340)/(333888/16))*360)))))*(COS(RADIANS(((YEAR--301340)/(333888/16))*360))))

Components:

• 0.015321 — Arithmetic midpoint: (max + min) / 2
• 0.0014226 — Half-range amplitude: (max - min) / 2
• SQRT(0.015321^2+0.0014226^2) — Derived mean eccentricity (0.0153869)
• 333888/16 — Perihelion precession cycle (20,868 years)

Example Values:

3. Inclination to Invariable Plane

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

Scientific Notation:

Where:

• $I_0 = 1.4816°$ — Mean inclination
• $A = 0.634°$ — Amplitude of variation (same as obliquity)
• $T_I = 111,296$ years — Inclination precession period (H/3)
• $t$ = Year + 301,340 — Time relative to anchor year

Note: The inclination and obliquity share the same amplitude ($A = 0.634°$) because they are geometrically coupled — 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.

=1.481592+(-0.633849*(COS(RADIANS(((YEAR--301340)/(333888/3))*360))))

Components:

• 1.481592 — Mean inclination (degrees)
• 0.633849 — Amplitude of variation (degrees)
• 333888/3 — Inclination precession cycle (111,296 years)

Example Values:

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:

Harmonic Terms (all 12 periods derived from H):

Constant offset: $c_0 = -0.3071°$

Accuracy: RMSE = 0.042° vs actual orbital data (improved from 0.14° with 7 terms)

=MOD(270+(YEAR+301340)*(360/20868)+5.05*SIN(2*PI()*(YEAR+301340)/20868)+2.46*SIN(2*PI()*(YEAR+301340)/10434)+0.2206*SIN(2*PI()*(YEAR+301340)/10434)+0.2439*COS(2*PI()*(YEAR+301340)/10434)+0.231*SIN(2*PI()*(YEAR+301340)/6956)+0.0205*COS(2*PI()*(YEAR+301340)/6956)+0.0715*SIN(2*PI()*(YEAR+301340)/5217)+0.0127*COS(2*PI()*(YEAR+301340)/5217)-0.1445*SIN(2*PI()*(YEAR+301340)/111296)+0.0072*COS(2*PI()*(YEAR+301340)/111296)+0.115*SIN(2*PI()*(YEAR+301340)/41736)-0.007*COS(2*PI()*(YEAR+301340)/41736)-0.1305*SIN(2*PI()*(YEAR+301340)/11513)-0.0052*COS(2*PI()*(YEAR+301340)/11513)+0.1279*SIN(2*PI()*(YEAR+301340)/13912)+0.0059*COS(2*PI()*(YEAR+301340)/13912)-0.0392*SIN(2*PI()*(YEAR+301340)/333888)-0.0002*COS(2*PI()*(YEAR+301340)/333888)-0.0196*SIN(2*PI()*(YEAR+301340)/166944)+0.0154*SIN(2*PI()*(YEAR+301340)/8347)+0.0006*COS(2*PI()*(YEAR+301340)/8347)-0.3071,360)

Earth Rate Deviation (ERD)

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

Scientific Notation:

Where:

• $\frac{d\theta_E}{dt}$ = Instantaneous Earth perihelion rate (°/year)
• $\omega_0 = \frac{360°}{20,868} = 0.01725°$/year — Mean rate

Excel Formula (numerical derivative):

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

=IF(θ_E_current - θ_E_previous < -180,
    (θ_E_current - θ_E_previous + 360) / (Year_current - Year_previous),
  IF(θ_E_current - θ_E_previous > 180,
    (θ_E_current - θ_E_previous - 360) / (Year_current - Year_previous),
    (θ_E_current - θ_E_previous) / (Year_current - Year_previous)
  )
) - 360/20868

Simplified (for consecutive years with Δt = 1):

=IF(AI2-AI1<-180,(AI2-AI1+360),IF(AI2-AI1>180,(AI2-AI1-360),(AI2-AI1)))-360/20868

Where AI = Longitude of Perihelion column (row 2 = current year, row 1 = previous year).

Analytical ERD Formula (recommended):

The true derivative of the perihelion harmonics gives the mathematically correct instantaneous rate. For each harmonic term $a \sin(\omega t) + b \cos(\omega t)$, the derivative is $a \omega \cos(\omega t) - b \omega \sin(\omega t)$.

=5.05*(2*PI()/20868)*COS(2*PI()*(YEAR+301340)/20868)+2.46*(2*PI()/10434)*COS(2*PI()*(YEAR+301340)/10434)+0.2206*(2*PI()/10434)*COS(2*PI()*(YEAR+301340)/10434)-0.2439*(2*PI()/10434)*SIN(2*PI()*(YEAR+301340)/10434)+0.231*(2*PI()/6956)*COS(2*PI()*(YEAR+301340)/6956)-0.0205*(2*PI()/6956)*SIN(2*PI()*(YEAR+301340)/6956)+0.0715*(2*PI()/5217)*COS(2*PI()*(YEAR+301340)/5217)-0.0127*(2*PI()/5217)*SIN(2*PI()*(YEAR+301340)/5217)-0.1445*(2*PI()/111296)*COS(2*PI()*(YEAR+301340)/111296)-0.0072*(2*PI()/111296)*SIN(2*PI()*(YEAR+301340)/111296)+0.115*(2*PI()/41736)*COS(2*PI()*(YEAR+301340)/41736)+0.007*(2*PI()/41736)*SIN(2*PI()*(YEAR+301340)/41736)-0.1305*(2*PI()/11513)*COS(2*PI()*(YEAR+301340)/11513)+0.0052*(2*PI()/11513)*SIN(2*PI()*(YEAR+301340)/11513)+0.1279*(2*PI()/13912)*COS(2*PI()*(YEAR+301340)/13912)-0.0059*(2*PI()/13912)*SIN(2*PI()*(YEAR+301340)/13912)-0.0392*(2*PI()/333888)*COS(2*PI()*(YEAR+301340)/333888)+0.0002*(2*PI()/333888)*SIN(2*PI()*(YEAR+301340)/333888)-0.0196*(2*PI()/166944)*COS(2*PI()*(YEAR+301340)/166944)+0.0154*(2*PI()/8347)*COS(2*PI()*(YEAR+301340)/8347)-0.0006*(2*PI()/8347)*SIN(2*PI()*(YEAR+301340)/8347)

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

Key Values:

5. Length of Days & Years

These formulas explain the connection between all day and year lengths.

Solar Year (in days)

Scientific Notation:

Where:

• $Y_0 = 365.2422$ days — Mean solar year
• $k_\varepsilon = \frac{2.29}{D_{\text{mean}}}$ days/degree — Obliquity sensitivity
• $D_{\text{mean}} = \frac{31558149.724}{365.256363} = 86399.9887$ SI seconds — Mean day length
• $\varepsilon$ = Current obliquity (degrees)
• $\varepsilon_0 = 23.414°$ — Mean obliquity

=365.242188997508-(2.29/86399.9886961896)*(<Current obliquity>-23.41398)

Where <Current obliquity> is calculated using the obliquity formula above.

Sidereal Year (in days)

Scientific Notation:

Where:

• $Y_0 = 365.2422$ days — Mean solar year
• $P_A = 25,684$ years — Axial precession period (H/13)
• $k_e = \frac{-3208}{D_{\text{mean}}}$ days/unit eccentricity — Eccentricity sensitivity
• $D_{\text{mean}} = 86,399.9887$ SI seconds — Mean day length ($Y_{\text{sid}}(\text{s}) / Y_{\text{sid}}(\text{days})$)
• $e$ = Current eccentricity
• $e_0 = 0.015387$ — Eccentricity derived mean ($e_{\text{mean}} = \sqrt{e_{\text{base}}^2 + e_{\text{amp}}^2}$)

=(365.242188997508*(333888/13)/((333888/13)-1)) + ((-3208 / 86399.9886961896) * (<Current eccentricity>-SQRT(0.015321^2+0.0014226^2)))

Where <Current eccentricity> is calculated using the eccentricity formula above.

Sidereal Year (in seconds)

The sidereal year in SI seconds is fixed:

Day Length (in SI seconds)

Scientific Notation:

Where:

• $Y_{\text{sid}}^{\text{(sec)}}$ = 31,558,149.724 seconds — Fixed sidereal year
• $Y_{\text{sid}}^{\text{(days)}}$ = Sidereal year in days (from formula above)

=31558149.724/<Current sidereal year in days>

Solar Year (in seconds)

Scientific Notation:

Where:

• $Y_{\text{solar}}^{\text{(days)}}$ = Solar year in days (from formula above)
• $D(s)$ = Day length in SI seconds (from formula above)

=<Solar year (days)>*<Day length (SI seconds)>

Sidereal Day (in SI seconds)

Scientific Notation:

The sidereal day is the time for one full rotation relative to the stars — slightly shorter than the solar day because Earth must rotate an extra ~1° per day to compensate for its orbital motion. Based on 86400-second solar days.

=<Solar year (SI seconds)>/(<Solar year (SI seconds)>/86400+1)

Stellar Day (in SI seconds)

Scientific Notation:

The stellar day accounts for the additional precession correction beyond the sidereal day. Based on 86400-second solar days. $H/13 = 25,684$ years is the axial precession period.

=((<Solar year (SI seconds)>/(<Solar year (days)>+1))/(333888/13))/(<Solar year (days)>+1)+<Sidereal day (SI seconds)>

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:

This is the “coin rotation paradox” — the precession period equals the sidereal year divided by the difference between sidereal and solar years.

Mean value: $P_A = \frac{H}{13} = 25,684$ years

=<Sidereal year (days)>/(<Sidereal year (days)>-<Solar year (days)>)

Perihelion Precession

Scientific Notation (coin rotation paradox):

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

Mean value: $P_P = \frac{H}{16} = 20,868$ years (equivalent to $P_A \cdot \frac{13}{16}$)

=<Anomalistic year (SI seconds)>/(<Anomalistic year (SI seconds)>-<Solar year (SI seconds)>)

Inclination Precession

Scientific Notation (coin rotation paradox):

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

Mean value: $P_I = \frac{H}{3} = 111,296$ years (equivalent to $P_A \cdot \frac{13}{3}$)

=<Anomalistic year (SI seconds)>/(<Anomalistic year (SI seconds)>-31558149.724)

Anomalistic Year (in days — simplified)

Scientific Notation (mean value approximation):

The anomalistic year (perihelion to perihelion) is slightly longer than the solar year because Earth must “catch up” to its advancing perihelion point.

=((<Perihelion precession in years>*<Solar year (days)>)/(<Perihelion precession in years>-1))

Anomalistic Year (in seconds — full formula)

The full anomalistic year calculation is a 3-step process that accounts for eccentricity variation and apsidal correction:

Step 1 — Raw anomalistic year in days (eccentricity variation):

Where:

• $Y_{\text{anom}}^0 = Y_0 \cdot \frac{H/16}{H/16 - 1}$ — Mean anomalistic year in days (~365.2597)
• $k_a = -6$ seconds/unit eccentricity — Anomalistic eccentricity amplitude
• $D_0$ = Mean day length in SI seconds (~86400.0)
• $e_0 = 0.015387$ — Eccentricity mean (EARTH_ECC_MEAN)

Step 2 — Convert to seconds:

Step 3 — Apsidal correction (seconds experienced on Earth):

The apsidal correction uses the factors $\frac{13}{3}$ and $\frac{16}{3}$ (derived from the H-harmonic ratios) to convert from raw orbital seconds to seconds as experienced on Earth with its precessing apsidal line.

=LET(mean_anom_days, 365.242188997508*(333888/16)/((333888/16)-1), mean_day, 31558149.724/(365.242188997508*(333888/13)/((333888/13)-1)), raw_days, mean_anom_days - (-6/mean_day)*(<Current eccentricity>-SQRT(0.015321^2+0.0014226^2)), raw_s, raw_days*<Day length (SI seconds)>, raw_s - ((raw_s - mean_anom_days*mean_day)/(13/3))*(16/3))

Anomalistic year in days (from seconds):

=<Anomalistic year (SI seconds)>/86400

Where:

• $Y_{\text{sid}}$ = Sidereal year in days
• $Y_{\text{solar}}$ = Solar year in days
• $D(s)$ = Day length in SI seconds
• $P_A$ = Axial precession period in years
• $P_P$ = Perihelion precession period in years
• $H = 333,888$ years = Holistic-Year

7. Planetary Precession Fluctuation

Calculates the perihelion precession “fluctuation” for any planet — 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.

Generic Formula Structure

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

fluct_P = A_E × cos(ω_E × (YEAR + 301340) + φ_E)
        + A_P × cos(n × ω_P × (YEAR + 301340) + φ_P)
        + C

Where:

• ω_E = 360 / 20,868 = 0.017252°/year — Earth’s effective perihelion rate (same for all planets)
• ω_P = 360 / T_P — 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)
• φ_E, φ_P = Phase offsets (planet-specific)
• C = Offset constant (planet-specific)
• 301340 = Anchor year (Balanced Year, 301,340 BC)

Planetary Parameters

*Earth has true perihelion period 111,296 years, but effective period 20,868 years due to axial precession interaction.

Amplitude Scaling

The amplitudes scale with each planet’s orbital characteristics:

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

For calculated amplitudes per planet, see Formula Derivation.

Observed-Angle Formulas (All 7 Planets)

For the complete observed-angle formulas including:

• ERD (Earth Rate Deviation) helper — analytical derivative of 12-harmonic perihelion formula
• Fluctuation Formulas — 225 terms per planet (328 for Venus), R² = 1.0000 for all 7 planets
• Mercury — 225 terms, R² = 1.0000, RMSE = 0.08″/century
• Venus — 328 terms, R² = 1.0000, RMSE = 0.27″/century
• Mars — 225 terms, R² = 1.0000, RMSE = 0.02″/century
• Jupiter — 225 terms, R² = 1.0000, RMSE = 0.03″/century
• Saturn — 225 terms, R² = 1.0000, RMSE = 0.03″/century
• Uranus — 225 terms, R² = 1.0000, RMSE = 0.01″/century
• Neptune — 225 terms, R² = 1.0000, RMSE = 0.01″/century

See Formula Derivation: Observed-Angle Formulas.

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.

Predictive Perihelion Calculation

Where:

• $\theta_0$ = J2000 perihelion longitude (degrees)
• $T_P$ = Planet’s precession period
• $t$ = Year

All planets use the same formula structure with their J2000 perihelion longitude as the reference point. For accuracy metrics (R², RMSE) of all 7 planets, see the Summary table below.

Mercury Predictive Excel Formula (Total Observed Precession)

Uses predicted Mercury perihelion: θ_M = 77.457° + (360°/242,828) × (Year − 2000)

This formula is 100% predictive - all inputs come from other formulas, not from observations. The formula outputs total observed precession (baseline + fluctuation), not just the fluctuation component.

Baseline precession: 533.7″/century = 1,296,000″ ÷ 242,828 years × 100

Cell References (from master spreadsheet, all formula-derived):

The formulas use cell references from the analysis spreadsheet. To use these formulas, either:

• Replace the references with the actual formula outputs, or
• Set up helper columns with the same structure

=-60.72*ABS(SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180))))*COS(((AE2738*PI()/180)+((74.199+360*A2738/242828)*PI()/180)))-785.3*COS(((AE2738*PI()/180)+((74.199+360*A2738/242828)*PI()/180)))-12.51*SIN(((AE2738*PI()/180)+((74.199+360*A2738/242828)*PI()/180)))-44.39*COS(2*((74.199+360*A2738/242828)*PI()/180))+0.61*SIN(2*((74.199+360*A2738/242828)*PI()/180))+198.3*COS(2*(AE2738*PI()/180))+5.11*SIN(2*(AE2738*PI()/180))-0.73*COS(2*((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+1.09*SIN(2*((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+2.92*COS(3*((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+2.98*SIN(3*((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+0.54*COS(4*((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-104.3*(S2738-23.414)+434.2*(E2738-0.015354)+8.71*(S2738-23.414)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+101.5*(S2738-23.414)*SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-28.87*(E2738-0.015354)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-26.26*DT2738+4.24*DT2738*DT2738+9022*DT2738*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+34.18*DT2738*SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-212.2*DT2738*COS(2*((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+5773*DT2738*COS(((AE2738*PI()/180)+((74.199+360*A2738/242828)*PI()/180)))+1725*DT2738*SIN(((AE2738*PI()/180)+((74.199+360*A2738/242828)*PI()/180)))-9539*DT2738*(S2738-23.414)-4.62*DT2738*(E2738-0.015354)+2040*DT2738*(S2738-23.414)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-16.40*DT2738*(E2738-0.015354)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-59.31*SIN(2*PI()*(A2738+301340)/242828)+51.55*COS(2*PI()*(A2738+301340)/242828)-0.91*SIN(2*PI()*(A2738+301340)/333888)+6.72*COS(2*PI()*(A2738+301340)/333888)+6.46*SIN(2*PI()*(A2738+301340)/20868)-21.71*COS(2*PI()*(A2738+301340)/20868)-1.99*SIN(2*PI()*(A2738+301340)/10434)+202.9*COS(2*PI()*(A2738+301340)/10434)-21.34*SIN(2*PI()*(A2738+301340)/6956)+12.82*COS(2*PI()*(A2738+301340)/6956)-11.13*SIN(2*PI()*(A2738+301340)/5217)+8.39*COS(2*PI()*(A2738+301340)/5217)-1.13*SIN(2*PI()*(A2738+301340)/41736)+72.58*COS(2*PI()*(A2738+301340)/41736)+4.75*SIN(2*PI()*(A2738+301340)/111296)-73.43*COS(2*PI()*(A2738+301340)/111296)+399.9*SIN(2*PI()*(A2738+301340)/7161)+93.71*COS(2*PI()*(A2738+301340)/7161)+1136*SIN(2*PI()*(A2738+301340)/19217)-449.9*COS(2*PI()*(A2738+301340)/19217)+12320*DT2738*SIN(2*PI()*(A2738+301340)/242828)+30.87*DT2738*COS(2*PI()*(A2738+301340)/242828)+473.0*DT2738*SIN(2*PI()*(A2738+301340)/333888)-1595*DT2738*COS(2*PI()*(A2738+301340)/333888)+16632*DT2738*SIN(2*PI()*(A2738+301340)/20868)+2468*DT2738*COS(2*PI()*(A2738+301340)/20868)+12993*DT2738*SIN(2*PI()*(A2738+301340)/10434)+502.8*DT2738*COS(2*PI()*(A2738+301340)/10434)-2179*DT2738*SIN(2*PI()*(A2738+301340)/6956)+1230*DT2738*COS(2*PI()*(A2738+301340)/6956)-652.1*DT2738*SIN(2*PI()*(A2738+301340)/5217)+595.4*DT2738*COS(2*PI()*(A2738+301340)/5217)-268.2*DT2738*SIN(2*PI()*(A2738+301340)/41736)+6578*DT2738*COS(2*PI()*(A2738+301340)/41736)+54.07*DT2738*SIN(2*PI()*(A2738+301340)/111296)-6199*DT2738*COS(2*PI()*(A2738+301340)/111296)+15923*DT2738*SIN(2*PI()*(A2738+301340)/7161)+2785*DT2738*COS(2*PI()*(A2738+301340)/7161)+6469*DT2738*SIN(2*PI()*(A2738+301340)/19217)+1327*DT2738*COS(2*PI()*(A2738+301340)/19217)-28.89*DT2738*DT2738*SIN(2*PI()*(A2738+301340)/242828)-95.19*DT2738*DT2738*COS(2*PI()*(A2738+301340)/242828)+431.7*DT2738*DT2738*SIN(2*PI()*(A2738+301340)/333888)-4.83*DT2738*DT2738*COS(2*PI()*(A2738+301340)/333888)-5.99*DT2738*DT2738*SIN(2*PI()*(A2738+301340)/20868)-4.34*DT2738*DT2738*COS(2*PI()*(A2738+301340)/20868)+491.3*DT2738*DT2738*SIN(2*PI()*(A2738+301340)/41736)+529.7*DT2738*DT2738*COS(2*PI()*(A2738+301340)/41736)+477.6*DT2738*DT2738*SIN(2*PI()*(A2738+301340)/111296)+758.5*DT2738*DT2738*COS(2*PI()*(A2738+301340)/111296)-46.82*SIN(2*PI()*(A2738+301340)/20868)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-24.70*SIN(2*PI()*(A2738+301340)/20868)*SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-80.80*COS(2*PI()*(A2738+301340)/20868)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+53.85*COS(2*PI()*(A2738+301340)/20868)*SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+182.1*SIN(2*PI()*(A2738+301340)/10434)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+668.5*SIN(2*PI()*(A2738+301340)/10434)*SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-14.78*COS(2*PI()*(A2738+301340)/10434)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-172.3*COS(2*PI()*(A2738+301340)/10434)*SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+0.52*SIN(2*PI()*(A2738+301340)/41736)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+3.80*SIN(2*PI()*(A2738+301340)/41736)*SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-6.73*COS(2*PI()*(A2738+301340)/41736)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-65.75*COS(2*PI()*(A2738+301340)/41736)*SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-1.97*SIN(2*PI()*(A2738+301340)/111296)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+2.07*SIN(2*PI()*(A2738+301340)/111296)*SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+7.76*COS(2*PI()*(A2738+301340)/111296)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+57.12*COS(2*PI()*(A2738+301340)/111296)*SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+3030*DT2738*SIN(2*PI()*(A2738+301340)/20868)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-21069*DT2738*SIN(2*PI()*(A2738+301340)/20868)*SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+12237*DT2738*COS(2*PI()*(A2738+301340)/20868)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-71.89*DT2738*COS(2*PI()*(A2738+301340)/20868)*SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+2083*DT2738*SIN(2*PI()*(A2738+301340)/10434)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-235.0*DT2738*SIN(2*PI()*(A2738+301340)/10434)*SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-5196*DT2738*COS(2*PI()*(A2738+301340)/10434)*COS(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))-5369*DT2738*COS(2*PI()*(A2738+301340)/10434)*SIN(((AE2738*PI()/180)-((74.199+360*A2738/242828)*PI()/180)))+529.1+955.3*DT2738*SIN(2*PI()*(A2738+301340)/28117)

Result: R² = 0.9896, RMSE = 7.72″/century (6,456 characters, 106 terms, includes 533.7″ baseline)

Note: This legacy Excel formula uses the old Mercury perihelion reference (74.199° from year 0) and old eccentricity mean (0.015354), with only 106 terms. The Python implementation uses J2000 references for all planets (Mercury θ₀ = 77.457°), achieves R² = 0.9990 with 273 unified terms and the updated eccentricity mean $e_0 = \sqrt{0.015321^2 + 0.0014226^2} = 0.015387$. The Excel formula is preserved here for reference but the Python version is the current implementation.

Python Alternative: For easier use, the unified Python implementation provides higher accuracy:

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: R² = 0.9990, RMSE = 2.44″/century (273 unified terms with full precision coefficients)

Venus Predictive Excel Formula

Uses predicted Venus perihelion: θ_V = 131.577° + (360°/667,776) × (Year − 2000)

The legacy Venus Excel formula used 163 terms with analytical ERD (true derivative of perihelion harmonics). The current Python implementation uses 273 unified terms (see below). The legacy formula structure includes:

• Angle terms (δ, 2δ, 3δ, 4δ where δ = θ_E − θ_V)
• ERD linear and quadratic terms
• Periodic terms for 9 key periods (all derived from H)
• Triple interactions (ERD × Periodic × Angle)

Cell References (from master spreadsheet, all formula-derived):

Python Implementation: predict_precession.py on GitHub

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: R² = 0.9983, RMSE = 21.64″/century (273 unified terms, analytical ERD)

Note: Venus accuracy was improved from R² = 0.9716 to R² = 0.9983 by adding H/78, H/94, H/77, H/55 periods to the unified feature matrix (GROUP 16 fine-tuning terms).

Outer Planet Predictive Formulas (Mars, Jupiter, Saturn, Uranus, Neptune)

All planets (including outer planets) now use the same unified 273-term feature matrix with planet-specific coefficients trained via ridge regression.

Python Implementation:

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 year 2022:

Unified Feature Matrix Structure (273 terms for all planets):

The unified feature matrix is organized into 16 groups:

Where:

• δ = θ_E − θ_P (difference angle between Earth and planet perihelia)
• Σ = θ_E + θ_P (sum angle)
• θ_E = Earth perihelion from 12-harmonic formula
• θ_P = planet perihelion calculated from J2000 value and period

Saturn: Critical Obliquity/Eccentricity Coupling

Saturn’s perihelion period (H/8 = 41,736 years) equals the obliquity cycle, creating strong coupling with Earth’s obliquity/eccentricity variations. The GROUP 15 terms capture this coupling:

Venus: Fine-Tuning for Large Fluctuations

Venus has very large fluctuations (up to ~1000 arcsec/century) compared to other planets. GROUP 14 and GROUP 16 terms provide Venus-specific fine-tuning with periods H/78, H/94, H/77, and H/55, improving Venus accuracy from R² = 0.9716 to R² = 0.9983.

Summary: All Predictive Formulas (Unified 273-Term System)

Notes:

• All 7 planets use the same unified 273-term feature matrix with planet-specific coefficients
• All formulas output total observed precession (baseline + fluctuation)
• Saturn has negative baseline (ecliptic-retrograde precession)
• Mercury is the only planet with an Excel formula due to formula length limits
• All R² values > 0.998, demonstrating excellent fit across all planets

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 (333,888 years) and its Fibonacci divisions:

*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 (H×2). Uranus’s period coincides with inclination precession (H/3). Earth’s effective perihelion period (20,868 years = H/16) is the common component for calculating fluctuations of ALL planets.

┌─────────────────────────────────────────────────────────────────┐
│                HOLISTIC-YEAR (333,888 years)                    │
├─────────────────────────────────────────────────────────────────┤
│                              │                                  │
│         ÷3 = 111,296         │         ÷13 = 25,684             │
│    (Inclination Precession)  │     (Axial Precession)           │
│              │               │              │                   │
│              ▼               │              ▼                   │
│         Inclination          │      Longitude of                │
│           Formula            │       Perihelion                 │
│              │               │              │                   │
│              ▼               │              │                   │
│         ÷8 = 41,736    ◄─────┼──────────────┤                   │
│       (Obliquity Cycle)      │              │                   │
│              │               │              ▼                   │
│              ▼               │       ÷16 = 20,868               │
│          Obliquity           │  (Earth Effective Perihelion)    │
│           Formula            │              │                   │
│                              │              ├─────────────────► │
│                              │              │   ALL PLANETS     │
│                              │              │   Fluctuation     │
│                              │              │   (Earth Term)    │
│                              │              ▼                   │
│                              │    Eccentricity, Day Length      │
│                              │      Year Lengths                │
│                              │                                  │
│   Planet-specific terms: ────┼──────────────────────────────────│
│   Mercury: ×8/11 = 242,828   │   Venus: 667,776 years           │
│   Mars: ×3/13 = 77,051       │   Jupiter: 66,778 years          │
│   Saturn: 41,736 years (ret) │   Uranus: 111,296 years          │
│   Neptune: 667,776 years     │                                  │
└─────────────────────────────────────────────────────────────────┘

9. Verification

To verify these formulas, compare results against:

1. J2000 Values (Year = 2000):

   • Obliquity: 23.439° (IAU: 23.439291°) ✓
   • Eccentricity: 0.01671 (NASA: 0.01671022) ✓
   • Longitude of perihelion: 102.95° (NASA: 102.94719°) ✓

2. Historical Values:

   • Obliquity at -10,000: 24.23° (Laskar: 24.23°) ✓
   • Perihelion aligned with December solstice at 1246 AD ✓

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

   “Model 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.

   Year	Formula Result	3D Simulation	Error
   1912	44.6″/century	42.94″/century	+1.7
   2023	40.1″/century	38.36″/century	+1.7
   2689	4.4″/century	4.18″/century	+0.2
   3244	-25.3″/century	-26.10″/century	+0.8

   • Model baseline: 533.7″/century (Newtonian prediction from period)
   • At year 2023: fluctuation ≈ +38″/century
   • Observed total (Park et al. 2017): 575.31 ± 0.0015″/century

4. Interactive Calculator:

   • Use the Orbital Calculator to compute all values for any year instantly — 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. Excel Spreadsheet

The spreadsheet with all formulas and interactive calculations is available. Click the PURPLE Data Button at the top of any page to access a simplified view.

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