Why Phi? – resonant exoplanets of star YZ Ceti

Posted: January 19, 2018 by oldbrew in Analysis, exploration, Fibonacci, Phi
Tags: , ,

YZ Ceti is a recently discovered star with three known planets (b,c and d) orbiting very close to it. Although some types of mean motion resonance, or near resonance, are quite common e.g. 2:1 or 3:2 conjunction ratios, this one is a bit different.

The orbit periods in days are:
YZ Ceti b = 1.96876 d
YZ Ceti c = 3.06008 d
YZ Ceti d = 4.65627 d

This gives these conjunction periods:
c-d = 8.9266052 d
b-c = 5.5204368 d
b-d = 3.4109931 d
(Note the first two digits on each line.)

Nearest matching period:
34 c-d = 303.50457 d
55 b-c = 303.62403 d
89 b-d = 303.57838 d

34,55 and 89 are Fibonacci numbers.
Therefore the conjunction ratios are linked to the golden ratio (Phi).

Phi = 1.618034
(c-d) / (b-c) = 1.6170106
(b-c) / (b-d) = 1.618425

Data source: exoplanets.eu

1. tallbloke says:

Wow! Nice work OB.

2. oldbrew says:

These planets are like the moons of Jupiter and Saturn in terms of orbit period and closeness to the main body.

3. tallbloke says:

Ratios of conjunction periods of the principal moons of Jupiter are 1:2:3, which is quite a long way of 33:55:89 though. So what is the rationale for these two different but stable sets of dynamic relationships?

Why Fibonacci?

4. oldbrew says:

YZ Ceti synodic ratios could also be characterised as 1:Phi:Phi²

The Jupiter moons are quite close to a 1:2:4 orbital ratio but YZ Ceti doesn’t have an equivalent.
The nearest whole numbers would be 6:9:14 which gives conjunction ratios of 3:5:8 (Fib. nos.) but it’s less than 98% accurate.

It seems YZ Ceti planets can handle the harmonics they have.

5. tallbloke says:

How accurate is 1:2:3 for the Jovian conjunction ratios?

6. oldbrew says:

TB: accurate, but one extra orbit is included i.e. Eu = 2 Ga + 1, Io = 2 Eu +1

68 Ganymede = 137 Europa = 275 Io (= 486.51~ days) [so 34 * 2 Ga = 55 * 5 Io]

68 * 7.154553 = 486.5096 days
137 * 3.551181 = 486.5118 days
275 * 1.769138 = 486.513 days

https://nssdc.gsfc.nasa.gov/planetary/factsheet/joviansatfact.html

Why Phi? – the resonance of Jupiter’s Galilean moons
https://tallbloke.wordpress.com/2015/11/26/why-phi-the-resonance-of-jupiters-galilean-moons/ 7. tallbloke says:

Better than 99% accurate then?

8. oldbrew says:

A lot better – almost 100%

On the pocket calculator:
1 Ga-Eu = 7.0509267 d
2 Eu-Io = 7.0509288 d
3 Ga-Io = 7.0509279 d

9. oldbrew says:

Puzzling finding raises new questions about atmospheric physics of giant planets
January 22, 2018, McGill University

Wrong-way wind

“We’ve previously studied nine other hot Jupiter, giant planets orbiting super close to their star. In every case, they have had winds blowing to the east, as theory would predict,” says McGill astronomer Nicolas Cowan, a co-author on the study and researcher at MSI and iREx. “But now, nature has thrown us a curveball. On this planet, the wind blows the wrong way. Since it’s often the exceptions that prove the rule, we are hoping that studying this planet will help us understand what makes hot Jupiters tick.”

10. Paul Vaughan says:

Fibonacci only approximates golden ratio stability under discrete constraints, helping sort by presence or absence of discrete constraints.

11. tallbloke says:

Thanks OB.

Paul V: Please expand on the type of discrete constraints you’re referring to. It seems to me that in certain dynamic situations like the Jovian moons, Fibonacci isn’t approximating anything, it is itself stable. Why that should be is itself an important question.

12. Paul Vaughan says:

“[…] Cubism, rather than being an isolated art-form, represented the continuation of a grand tradition: indeed, the golden ratio, or golden section (French: Section d’Or) had fascinated Western intellectuals of diverse interests for at least 2,400 years.”
https://en.wikipedia.org/wiki/Section_d'Or

“The physical universe, then, is a kind of language that invites a privileged spectator to decipher it, although this does not yield a single message so much as a superior network of associations.”
https://en.wikipedia.org/wiki/Symbolism_(arts)

“”the power of the arts is indeed the most immediate and fastest way” to social, political and economic reform.”
https://en.wikipedia.org/wiki/Avant-garde

13. oldbrew says:

Star EPIC 220194974 was discovered in October 2017, with 3 short period planets (b,c,d).

b-c conjunction = 11.954515 days
c-d conjunction = 23.824652 days
Ratio b-c:c-d = 2:1 (1.9929417:1)

Therefore b-d:c-d will be 3:1 and this is another 3:2:1 system.

See: exoplanets.eu — filter = star_name=”EPIC 220194974″

Approx. orbit ratios are 38b:24c:17d

14. tallbloke says:

Paul V: I think we all need to read (re-read in my case) Physics as metaphor by Roger S Jones

https://www.abebooks.co.uk/servlet/SearchResults?sts=t&an=&tn=Physics+as+metaphor+&kn=&isbn=

OB: Right so we have these 3:2:1 conjunction period ratios in several systems now. So lets tabulate the orbit period ratios and see how it looks.

15. oldbrew says:

TB: they can be all over the place really. What counts is the result from the conjunctions IMO.

I found one (LP 358-499, planets b,c,d) that returns the same ratios as Earth, Mars and Jupiter conjunctions.
43 b-d = 22 b-c = 21 c-d
43 J-E = 22 E-Ma = 21 J-Ma

GJ 876 is very similar to the Jupiter moons i.e. near 4:2:1 orbital.
http://en.wikipedia.org/wiki/Gliese_876#Orbital_arrangement

26 e = 53 b = 107 c (orbital)
81 e-c = 54 b-c = 27 e-b (81:54:27 = 3:2:1)

Note also b = 2*e,+1 and c = 2*b, +1 — same as Jupiter moons.

16. tallbloke says:

OB: The +1 just means we need to look at the system as being in a rotating frame of reference I think.

17. oldbrew says:

Kepler 1542 has 4 short period planets of which 3 (c,b,e) are in 21:13:8 (Fibonacci) conjunction ratios.

97c = 71d = 55e therefore:
42 c-e = 26 c-d = 16 d-e
42:26:16 = 21:13:8

Re reference frame – the position of the conjunction could rotate relative to the one before?

18. tallbloke says:

Well yes, they do, but not necessarily at the rate the reference frame would need to rotate in order to get the whole number divisor between the orbits of the two bodies.

19. oldbrew says:

You don’t need whole numbers of orbits, it’s just easier to show what’s going on by using them.

20. tallbloke says:

Sure. It’s the divisor which needs to be a whole number

21. oldbrew says:

Jupiter-Saturn conjunction rotating – Kepler’s trigon (3 sets of 3 conjunctions each) NASA video – J moons, Vangelis soundtrack

22. tallbloke says:

Working out the precession rate of the rotating frame which makes the divisor an exaxt whole number might reveal something about the relationship of a system such as Jupiter’s main moons with the wider solar system or at least other nearby major planets. If that rate also related to conjunction precession rates that would be the cream on the pudding.

We already know that the Jupiter-Saturn conjunction precession rate is in a whole number ratio with the Venus-Earth and Neptune-Uranus conjunction precession rates.

23. oldbrew says:

Not sure what ‘the precession rate of the rotating frame’ means 😦

24. tallbloke says:

How fast it spins round relative to the next frame of reference outside it (fixed stars for example).