Quote:
Originally Posted by tony873004
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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Quote:
Originally Posted by Hornblower
You persist is seeing resonances in cases in which the orbital period ratios are only rough approximations of what are needed for a true resonance. Perhaps you should study some references on the topic of orbital resonance, such as this one: http://en.wikipedia.org/wiki/Orbital_resonance I know Wiki articles should not be taken as gospel, but this one looks good. Perhaps Tony and others would like to comment. |
But why cannot you have a decimal in resonance? The fact is that every 179 years, Jupiter Saturn and Neptune are in the same relative positions, with error around 0.1%, at a point one twelfth of the way further around the sun. Because this is a function of the four biggest objects of our solar system (sun + 3 gas giants), it produces a readily visible pattern in the plot of the solar system barycentre. I don't understand why this is not classed as resonance. Integer resonances apply when you have two planets, but this is looking at three. On the model of this
wiki table of orbital resonances (thank you
Hornblower), putting in the numbers
Tony gave of J:S:N = 15.094: 6.036 : 1.086 produces, as I calculate it, a mismatch after one cycle of only about 0.3° of arc for Jupiter's position relative to both Saturn and Neptune. This is much less than any of the listed binary mismatches, and has a very long randomisation time of around 100,000 years. Adding in Saturn-Neptune, the mismatch is about 1° of arc per cycle with randomisation period for the whole JSN group of about 30,000 years.
Jupiter-Saturn: 9 cycles = 178.65 years
Jupiter-Neptune: 14 cycles = 178.90 years
Saturn-Neptune: 5 cycles = 179.35 years
JS : JN : SN = 9 : 14 : 5. This relationship between the three largest planets is among the closest to an exact integer of all the resonances in the solar system.