The "Grand Tour" path Voyager 2 took, flying past Jupiter-Saturn-Uranus-Neptune, repeats every 175 years. As the original paper by Gary Flandro puts it:

Earth – Jupiter – Saturn – Uranus – Neptune mission opportunities occur only every 175 years due to the constraints imposed by the required positions of Uranus and Neptune.

Note how this is not the time from the previous opportunity, or the time to the next. It's a repeating event.

Yet, the synodic period of Uranus and Neptune is 171.4 years. Over time, this will cause significant misalignment from a repeating 175 year event.

Why is the grand tour interval slightly longer? And how can it still be periodic?

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    $\begingroup$ This "175" number is (A) just a one-sentence concluding statement, (B) not implying a regular interval and the statement would be valid if only the next few occurrences appear in this interval, (C) is reasonably close to the actual value and (D) don't forget that alignment with Jupiter and Saturn is needed as well. $\endgroup$ – asdfex Nov 7 '20 at 14:09
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    $\begingroup$ @asdfex Would these points be sufficient to type out as an answer? $\endgroup$ – SE - stop firing the good guys Nov 10 '20 at 10:53

A couple considerations enter here.

  1. We do not expect the alignments to occur every 175 years over the long haul. Rather, they are likely to occur in clusters internally spaced by that interval until the alignment is "permanently" lost, and takes a much longer interval to reset.

  2. The period of 175 years represents a compromise between multiple synodic periods among all the planets involved. The planets do not have identical synodic or multi-synodic period, bit they do have periods close enough for an intermediate value of 175 years to work well, at least temporarily.

Feast or famine

The concept of celestial confluence of three or more objects coming in bunches should be familiar to anyone who sought out the transits of Venus earlier this century (or who, like me in 2004, happened to catch them by dumb luck, see below*). As the planets orbit the Sun, Venus laps Earth at a pentad of points that gradually precess, the entire pentad being covered once in almost exactly eight years. Transits are seen from Earth at those times when one of the conjunction points comes close enough to the line where the orbital planes intersect. Then we see transits every time Venus laps Earth at the appropriate point, thus eight years apart, but typically the precessing conjunction point stays properly aligned only long enough to cover one interval. Hence the alternation between a "feast" where two transits come eight years apart and a century-long "famine" before another lap point gets into proper alignment.

Similarly, getting four or even three outer planets into proper alignment for a "grand tour" requires different lines between the Sun and planets to get into the proper geometric relationship, but they stay in good order long enough to produce multiple opportunities before the (much longer time scale) "famine" inevitably returns.

Like clockwork ... almost

If we were interested only in getting Saturn and Neptune into proper alignment, then the synodic interval between them of 171.4 years would be perfect for the job. Unlike transits of Venus, any synod will do; we don't need the planets to fall on a specific line to get a spacecraft through with gravity assists.

But also unlike the transits of Venus, we are looking for more than one pair of planets to line up as closely as they will, and different pairs have different synodic periods that inevitably do not mesh perfectly. From Wolfram Alpha, the source used by the OP:

Uranus-Neptune = 171.4 years

Saturn-Uranus = 45.34 years; 4×(S-U) = 181.1 years

Jupiter-Saturn = 19.87 years, 9×(J-S) = 178.8 years

We were pretty lucky!

What we have is a set of synodic or multi-synodic periods not identical but in a narrow range, which is what allows bunches of four-planet alignments to occur in the first place. These nearly identical time intervals are clustered around the 175 years quoted for an "average" grand-tour interval, if (as with being around and observing Venus at the right time in the 21st century) we are lucky enough to be residing in the right epoch.

*What happened was this: I had gone out for an early-morning walk along the Chicago lakefront, planning to see just the sunrise. But the weather in Chicago was unusually warm, and a temperature-inversion developed over the cooler waters of Lake Michigan. The eastern horizon was accordingly a beautiful shade of dark gray from the pollution trapped in the inversion. So when the Sun rose the pollution temporarily cut out its glare and rendered Venus, which was still in its transit, visible to the unaided (and carefully glancing only) eyes!

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    $\begingroup$ I hope you don't mind the edits, as always feel free to roll back/edit further $\endgroup$ – uhoh May 14 at 0:32
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    $\begingroup$ Looks good to me. Hope my space knowledge is better than my typing. About that dumb luck: that morning the Sun rose at Chicago through a pollution layer, and the pollution briefly provided a filter to see Venus in front of the Sun. Luck upon luck! $\endgroup$ – Oscar Lanzi May 14 at 0:40
  • $\begingroup$ I've just asked (instead of doing what I should be) Just how much "dumb luck" was reaching the space age during the "Grand Tour" opportunity? Would we have been SOL 1050 or 2100 years before/after? $\endgroup$ – uhoh May 14 at 1:09
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    $\begingroup$ "that morning... the pollution briefly provided a filter..." doesn't ring a bell, can you tell me more (sounds glorious!) :-) $\endgroup$ – uhoh May 14 at 1:11

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