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The aim here is to show how the synodic periods and orbits of these three planets align with the so-called Grand Synod, a period of about 4628 years which has 27 Uranus-Neptune conjunctions and almost 233 Jupiter-Saturn conjunctions. Its half-period is sometimes referred to as the Hallstatt cycle (2314 years +/- a variable margin).

**1. U-N ‘long period’**

1420 Uranus-Neptune conjunctions = 1477 Neptune orbits

(for calculations, see Footnote)

1477 – 1420 = 57

Uranus-Neptune 360 degrees return is 1420/57 U-N = 24.91228 U-N long period = 4270.119 years

**2. GS : U-N ratio**

Grand Synod = 27 U-N = 4627.967 years (= ~233 Jupiter-Saturn conjunctions)

27 / 24.91228 = 1.0838028

1.0838028 * 12 = 13.005633

Therefore the ratio of 4627.967:4270.119 is almost exactly 13:12 (> 99.956% true)

**3. Orbital data**

Turning to the orbit periods nearest to the Grand Synod:

28 Neptune = 4614.157y

55 Uranus = 4620.927y

(Data: https://ssd.jpl.nasa.gov/?planet_phys_par )

**4. Factor of 12**

These periods fall slightly short of the 27 U-N Grand Synod (~4628 years).

However, multiplying by 12 and adding one orbit to each, gives:

28*12,+1 (337) Neptune = 55534.67y

55*12,+1 (661) Uranus = 55535.14y

27*12 (661 – 337) U-N = 55535.61y

Now the numbers match to within a year +/- 55535 years.

Also, the period is 12 Grand Synods (12*4628 = 55536y), or 13 U-N ‘long’ periods.

**5. Pluto data**

Pluto’s orbit period is 247.92065 years.

55535 / 247.92065y = 224.003

So 224 Pluto orbits also equate to 12 Grand Synods.

Therefore, a U-N-P synodic chart can be created for that period of time.

**6. Neptune:Pluto orbits**

Neptune has one more orbit in the period than an exact 3:2 ratio with Pluto – a planetary resonance.

224 P = 112*2

337 N = 112*3, +1

113 N-P = 112, +1

**7. Phi factor**

Uranus and Neptune both have one more orbit than this ratio:

660:336 = (55*12):(21*16)

55/21 = Phi²

12/16 = 3/4

Therefore the U:N ratio is almost (3/4 of Phi²):1

The U-N-P chart should repeat every 12 Grand synods i.e. every 55,535 years or so.

– – –

**Footnote**

360 / Neptune orbit (164.79132) = 2.184581

2.184581 * U-N conjunction (171.40619) = 374.4507

374.4507 – 360 = 14.4507

Obtain nearest multiple of 360 degrees:

1420 * 14.4507 = 20519.9994

20520 / 360 = 57

1420 + 57 = 1477

1420 U-N = 1477 Neptune orbits

1420 + 1477 = 2897 Uranus orbits

**Update (Dec. 2020):** the number of occurrences of the Uranus-Neptune conjunction precession in the period is 337 – 324 = 13. Therefore the ratio of that period to the Grand Synod (GS) is 13:12, because the GS is 27 U-N and 324 = 27*12.