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Originally Posted by Robert Tulip
...btw your numbers, ~ 12:30:165 look more like 1:2.5:14 than 1:6:15
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You can't have a decimal in resonance. They have to be integers. Only at integer multiples of their orbits completed will they line up again. For example, when Jupiter and Saturn are at conjunction, 1:2.5 means that after Saturn completes 1 orbit that Jupiter completes 2.5 and is on the opposite side of the Sun from Saturn.
If the pattern repeats every 179 years, then you get the resonance by dividing 179 by the orbital periods:
179/11.85920 = 15.093766864544
179 / 29.657296 = 6.03561430549838
179/164.79 =1.08623096061654
None of these are integers. After 179 years, Neptune has overshot the starting position by 9%, Saturn by 4% and Jupiter by 9%. So there's a 5% difference between Jupiter and Saturn, and between Neptune and Saturn, which is where I got the value of 5% I reported.
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This calculation indicates that as a proportion of Jupiter’s gravity, Saturn’s gravitational effect on the sun is 30% as strong
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Saturn is 1.84 times farther from the Sun than Jupiter (9.58201720 / 5.204267). This makes its gravity (1/1.84^2) = 0.295 as strong as Jupiter's at the distance of the Sun. But Saturn is only 0.3 times as massive as Jupiter, weakening its gravity again.
So at the sun, Saturn's gravity is only 0.295 * 0.3 = 0.088 times as strong as Jupiter's.