In 2011, astronomers were saying:
“We’ve crossed a threshold: For the first time, we’ve been able to detect planets smaller than the Earth around another star.”
The planets in question were Kepler-20 e and Kepler-20 f.
In the end six planets were detected: b,e,c,f,d, and g (in order of proximity to their star). Orbit periods range from about 9.38 to 63.55 days, all the planets being closer to the star than Mercury is to the Sun.
A NASA article had the title: Kepler-20, An Unusual Planetary System — referring to the alternate large/small sizes of the planets.
In the search for resonances, it turned out that a very good orbital match existed with the 2nd, 3rd and 4th planets: e, c and f.
16 e = 97.5764 days
9 c = 97.6868
5 f = 97.8879
For the corresponding synodic periods:
7 e-c (16-9) = 97.4347 days
4 c-f (9-5) = 97.4366
11 e-f (16-5) = 97.4354
4,7 and 11 are Lucas numbers.
An analogue in the solar system is found with the synodics of Mars, Earth and Venus:
3 Mars-Earth = 4 Venus-Earth = 7 Mars-Venus (~99.9% true).
3,4, and 7 are Lucas numbers.
Footnote:
The accuracy is increased if bigger numbers are used.
2077 e = 1167 c = 647 f (also = 3427 b) in 12666.55 +/- 0.17 days.
Then:
910 e-c (2077-1167) = 12666.57 days
520 c-f (1167-647) = 12666.75
1430 e-f (2077-647) = 12666.60
Dividing the synodic numbers by 130:
7 e-c = 4 c-f = 11 e-f
Previous finding confirmed i.e. 4:7:11 ratio.
Tallbloke's Talkshop 
Near-resonance: 91 e-c = 135 b-e = 226 b-c
90:135:225 = 2:3:5 ratio (Fibonacci numbers).
These are the first 3 planets in the system.
Fascinating getting both Lucas and Fibonacci series in the system. Nice work OB
Thanks TB. In the solar system:
61 Jupiter-Venus = 100 Venus-Mercury = 161 Jupiter-Mercury
60:100:160 = 2:3:5 ratio
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Take any three planets (or any three moons) which we can call b,c and d in order of distance from the parent body.
The synodics work such that the sum of b-c and c-d (the ‘neighbour’ pairs) conjunctions must equal the b-d conjunctions in any given period. This still applies if there are other bodies orbiting between them, since they would be neighbours if the other bodies were not there.
Since the Fibonacci and Lucas series both rely on incrementing by the sum of the previous two values, the sum of two of them (if consecutive in the series) must always be the next number in the series.
So for example if b-c is 2 and c-d is 3 in the given period, b-d must be 5.
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Note: reading the Lucas series backwards and deducting the number to the left of each pair from the number to its right to get the next in the series, you arrive at ‘1,3’. Since 3-1 = 2 that’s one way of explaining the opening number in the series.
Hmmm. Both TB & JB like OB’s number crunching.
All these B’s.
Reblogged this on Climate- Science.press.
They just updated Kepler-444 system — 5 planets. Found some resonances 🙂
‘Kepler-80 d, e, b, c and g have orbits locked in a resonance. While their periods are in a ~ 1.000: 1.512: 2.296: 3.100: 4.767 ratio, in a frame of reference that rotates with the conjunctions this reduces to a ratio of 4:6:9:12:18. Conjunctions of d and e, e and b, b and c, and c and g occur at relative intervals of 2:3:6:6 in a pattern that repeats about every 191 days. Librations of possible three-body resonances have amplitudes of only about 3 degrees, and modeling indicates the resonant system is stable to perturbations. Triple conjunctions do not occur.’
https://en.m.wikipedia.org/wiki/List_of_multiplanetary_systems
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‘in a frame of reference that rotates with the conjunctions’ – that’s the key.