Why Phi? – exoplanetary resonances of Kepler-102

Posted: July 13, 2019 by oldbrew in Analysis, Fibonacci, Lucas, Maths
Tags: , ,

Kepler Space Telescope [credit: NASA]

Star Kepler-102 has five known planets, lettered b,c,d,e,f. These all have short-period orbits between 5 and 28 days. Going directly to the orbit period numbers we find:
345 b = 1824.0012 d
258 c = 1824.4263 d
177 d = 1825.1709 d
113 e = 1824.4629 d
(for comparison: about 1-2 days short of 5 Earth years)

For the purposes of this post planet f (the furthest of the five from its star) is excluded, except to say that in terms of conjunctions 8 e-f = 11 d-e. Now let’s look for some resonances of the inner four planets.

The point about these orbit numbers is the high correlation of the time periods, which allows us to create the chart below showing whole numbers of conjunctions, since those numbers are obtained from the difference in the number of orbits of planet pairs in any given period.

Focussing on b-c, c-e and b-e (i.e. planets b,c and e) we find:
87 b-c = 29*3
145 c-e = 29*5
232 b-e = 29*8

Therefore the ratio of their conjunctions is:
3 b-c: 5 c-e: 8 b-e
(3,5 and 8 are Fibonacci numbers and 29 is a Lucas number)

Turning to d-e, b-d, and b-e (i.e. planets b,d and e):
64 d-e = 8*8
168 b-d = 8*21
232 b-e = 8*29

Therefore the ratio of their conjunctions is:
8 d-e: 21 b-d: 29 b-e
(8 and 21 are Fibonacci numbers and 29 is a Lucas number)

We can see that the conjunction resonances are there, with relatively high rates of repetition, whereas the orbit numbers alone offer nothing obvious.
– – –
Data source: Exoplanet.eu
Note: the Kepler-102 data was last updated on 2019-07-01

  1. tallbloke says:

    Wow. Nice work OB. We could really do with a talented programmer who can suck in exoplanet.eu data and spit out synodic resonance ratios so we can find the percentage of systems which exhibit this sort of behaviour.

  2. oldbrew says:

    Thanks TB. I was a programmer once but mostly on mainframes. Pocket calculators are more my thing these days 😉

    To see the data go here:

    Then click on ‘planets’ in green lettering (lower right of page).
    Or, go to exoplanets.eu and select All catalogs, then put star_name=”Kepler-102″ (no italics needed) in the ‘Filter’ box.

  3. oldbrew says:

    Using star_name=”BD-06 1339″ (see comment above), the newly discovered GJ 221 e has a ratio of 1.6173:1 with planet b in the system (4 known planets) – very close to 55:34 in Fibonacci numbers (2.3946 : 3.8728 in days).

    Why they have different name prefixes is not clear.
    – – –
    One of the closest exoplanet pairs to the 3:2 mean motion resonance: K2-19b and c [2015 paper]

    Another planet ‘d’ has been found. The orbit ratios are now 19d:6b:4c giving 13 b-d = 15 (5*3) d-c = 2 b-c (conjunctions). 2,3,5, and 13 are Fibonacci numbers.

  4. oldbrew says:

    This article in The Astrophysical Journal goes into detail on mean motion resonance theory.

    Masahiro Ogihara and Hiroshi Kobayashi

    Published 2013 August 30 • © 2013. The American Astronomical Society.

    – – –
    However they are concentrating on orbital resonance. It’s usually more productive to look at synodic resonance IMO.

  5. tallbloke says:

    Well, synodic resonances tend to get buried into general discussions on orbital resonance. We’re not the first to understand the importance of synodics (though many researchers seem less aware of their importance than we would like. e.g.

    The three Galilean satellites are involved in the Laplace resonance, in which the orbital periods of Ganymede:Europa:Io are in a near 1:2:4 ratio, but more important, the mutual conjunctions of the Io–Europa pair and of the Europa–Ganymede pair precess around Jupiter at precisely the same rate. Like a child on a swing pushed at the optimal moment, the recurring mutual conjunctions force and maintain eccentricities in their orbits (Fig. 3).

  6. oldbrew says:

    The exoplanetary resonances remind us that equivalents are there to be seen in our own solar system. It’s a lot more than just a numbers game.