From Tallbloke’s Talkshop

By oldbrew

Here we show a period of nearly 2500 years when the conjunction periods of Jupiter and Mars sum to an exact number of Earth years. Using this result, and referring to previous Talkshop findings, we determine how Jupiter, Mars and Earth are related to each other in terms of orbits and conjunctions (joint alignments to the Sun). We note that Mars has a more eccentric orbit than (for example) Earth, at least partly due to its far larger ‘neighbour’, Jupiter. This can lead to slight variability in its conjunctions with both Jupiter and Earth (and other planets), typically +/- a few days or maybe a few weeks. However over long enough periods these variations can balance out, so we’re able to find some consistency in its orbit patterns.

Turning to the NASA planetary factsheet for Mars we find:
Sidereal orbit period (days) 686.980
Tropical orbit period (days) 686.973

The timings we’re talking about are all verifiable, or at least testable, using Arnholm’s solar simulator where we find (see graphic below) this:

Elapsed time between the two screenshots = 2470 tropical years [TY] = 2660 minus 194, same calendar date.
(Note: the simulator is limited to years 0-3000).
Conjunctions of Jupiter-Mars in 2470 TY = 1105 (17*65).
Mean conjunction period = 2470/1105 = 2.2352941 TY.
17 J-Mars = 2470/65 = 38 TY exactly.
38 TY = 2*19 TY = 2 Metonic cycles.
(Note: in the Talkshop link we say why we think the Metonic cycle is exactly 19 tropical years, whereas most sources say it’s ~2 hours ‘short’. It depends on what exactly is being measured).

We can calculate a Mars orbit period (J orbits + no. of J-Mars = Mars orbits) from the simulator graphic:
2470/J = 208.31326 J orbits.
J-Mars occurs 1105 times, therefore:
Mars orbits = 208.31326 + 1105 = 1313.3132
2470/1313.3132 = 1.8807395 TY (mean value).
1.8807395 TY in days = 686.9254 (NASA says 686.973).

For the conjunctions with Earth we just use the Metonic cycle and Jupiter.
19/1.8807395 = 10.102409 Mars orbits
In this post we showed Jupiter has 14 orbits per 166 TY in the tropical reference frame (as used by the solar simulator).
10.102409 Mars * 166 = 1677 orbits in 1577 TY (= 83 Metonic).
Since 17 J-Mars is 2 Metonic cycles not 1, we make the full period 2*1577 = 3154 TY = 166*19 or 83*38 TY.
Check: 3154/1677 = 1.8807395 TY = Mars orbit period (as calculated earlier).
The Jupiter-Earth conjunction occurs 3154 – 266(J) = 2888 times in the full period.

The full number chart for the planetary data is on the right. The Metonic cycle (not shown) occurs 166 times in the period.

The tropical timings posts so far have shown these planets align with the 83 TY (166/2) cycle, and therefore with Jupiter and Earth, thus:
Jupiter: 1 cycle (7 J)
Venus: 12 cycles (1619 V)
Mercury: 13 cycles (4480 Me.)
Mars: 19 cycles (1677 Ma.)
Neptune: 144 cycles (73 N)
Uranus: 335 cycles (332 U)

For current ‘tropical timings’ posts see here:
https://tallbloke.wordpress.com/?s=tropical+timings
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Image: Mars [credit: NASA]