## Exoplanet link to Lucas number series

Posted: March 31, 2015 by oldbrew in Astrophysics
Tags: ,

Jupiter-sized exoplanet [Wikipedia]

Tallbloke has spotted a science paper about exoplanets where one system has two planets whose orbital periods are close to 11:4 ratio Lucas numbers.

Paper: ‘We also refine the parameters of two planets announced previously around HD 113538, based on a longer series of measurements. The planets have a period of 663± 8 and 1818 ± 25 days, orbital eccentricities of 0.14 ± 0.08 and 0.20 ± 0.04, and minimum masses of 0.36 ± 0.04 and 0.93 ±0.06 MJup.’
[MJup = mass of Jupiter]

The outer planet is slightly smaller than Jupiter and the inner one is about one third of it, by mass. Noting the uncertainties in the orbital periods, we can see how closely they relate to the Lucas ratio:
663/3 = 221 = 55 x 4, +1
1818/3 = 606 = 55 x 11, +1
difference = 55 x 7

This is very close to 4:11 orbit ratios with 7 conjunctions in that time period, 4-7-11.

The way these two planets have fallen into an almost perfect Lucas-number ratio is interesting because we don’t seem to have any analogue of that in the solar system, although Venus-Earth with a 13:8 orbit ratio and 5 conjunctions is similar to some degree: 5-8-13, all Fibonacci numbers.

The Lucas series follows rounded numeric values of powers of Phi, e.g. 4 is the nearest whole number to Phi³ (4.236~), 7 is near the 4th power of Phi (6.854~) and 11 is near its 5th power (11.09~).

Wikipedia notes that ‘Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.’

This could be a new twist to orbital resonances so we’ll have to keep an eye open for any further examples.

1. oldbrew says:

This could be a new twist to orbital resonances so we’ll have to keep an eye open for any further examples.’

This looks similar: Kepler-11 star has six planets (lettered b-g), and the orbital ratio of c:d is 7:4 (99.53% true).
That would mean 3 conjunctions in the period, and 3-4-7 are all Lucas numbers.

Interestingly the AU values (distance from the star) are in an exact 3:2 ratio i.e. 0.159 AU(d) and 0.106 AU(c).
Data source: http://exoplanet.eu/catalog/?f=%27Kepler-11+%27+in+name

‘Tourist’s Guide to the New Alien Planet System Kepler-11’

http://www.space.com/10748-tour-kepler11-planet-system.html

2. oldbrew says:

Kepler-138 star (aka KOI-314) discovered in late 2014 has planets in both a 3:5 Fibonacci and a 3:4 Lucas ratio.

Planets c and d are 5:3 with 2 conjunctions in the period = 2-3-5 Fibonacci (99.48% true).
Planets b and c are 4:3 with 1 conjunction in the period = 1-3-4 Lucas (99.77% true).

http://en.wikipedia.org/wiki/Kepler-138

So we have direct evidence of 1-3-4, 3-4-7 and 4-7-11 Lucas ratios in exoplanetary systems.

Strange but true: ‘Although b and c have the same size, their masses and densities vary greatly’ (Wikipedia). By a factor of nearly 4:1 for mass.