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@OscarLanzi's answer to Why is the gas giant grand tour interval 175 years when the synodic period of Uranus and Neptune is 171 years? is an excellent answer describing a subtle situation; each pair of planets have their own synodic period and while there is a cluster of planet-pair synodic periods around 175 years, this is not a magic number.

If you look at the synodic periods between pairs of the outer four planets, we see a lot of numbers near 175 if you take into account that we can look for integer multiples of synodic periods.

                 Saturn                 Uranus                 Neptune
Jupiter    19.86 x  9 = 178.73     13.81 x 13 = 179.56     12.78 x 14 = 178.95
Saturn             --              45.36 x  4 = 181.44     35.87 x  5 = 179.34
Uranus             --                      --             171.41 x  1 = 171.41

Question: Since the cluster of two-planet synodic periods near 175 years have some spread, if we'd hit the space age say 1050 or 2100 years earlier or later1 would we have not had the same opportunity and (among other things) not now have two "interstellar" probes outside the heliopause?

11050 or 2100 years is 6 and 12 times 175 years, rather than "1000 or 2000 years" as I had it originally.

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    $\begingroup$ Hard to tell, really, because this is a spaceflight phenomenon rather than just an astronomical one. A "grand tour" alignment circa 1800 would have been meaningless without spaceflight or even the discovery of Neptune (1846). And is may be meaningless again in the 22nd century if we develop fast interplanetary transport technologies and no longer need gravity assists. $\endgroup$
    – Oscar Lanzi
    May 14, 2021 at 1:47
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    $\begingroup$ @OscarLanzi "reaching the space age" in the title was meant to suggest that things would be pretty much the same as they were in the 1970's but just come about a tiny bit earlier or later. We started making recognizable tools about 2 million years ago so I'm talking about a part-per-thousand difference in the rate of technology development only. $\endgroup$
    – uhoh
    May 14, 2021 at 2:01
  • $\begingroup$ What's so special about 1050 years? Why not 175 or 350? $\endgroup$
    – asdfex
    May 14, 2021 at 9:59
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    $\begingroup$ @asdfex Since all of the big-planet-pair synodic combinations are close but not the same (spread between 171 and 181 years), the slow moving Uranus and Neptune will not be back in just the same place after only 175 years, and each subsequent 175 years the "Grand Tour" gets less and less grand, until at some point it will likely become instead the "Grand Challenge" to do the same basic thing, or one of the tour may end up having to miss a planet entirely. This is the task of an answer-author, to address how the Grandness of the tour degrades for each subsequent 175 year Grand opportunities. $\endgroup$
    – uhoh
    May 14, 2021 at 10:09
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    $\begingroup$ Since you have a four-year offset each 175 years, after eleven such intervals Neptune would have moved 90° out of line with respect to Saturn. Would that be too much? $\endgroup$
    – Oscar Lanzi
    May 14, 2021 at 20:30

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