Exoplanetary system exhibiting Laplace resonance

Posted: February 19, 2014 by tallbloke in Astrophysics, Celestial Mechanics, solar system dynamics, Tides, waves


I came across this recent paper on Arxiv (also published in MNRAS) which nicely confirms some of the work we have been doing in PRP on energy exchange in the solar system via mean motion resonance. Team wassup refer to this well known phenomenon (Observed by Galileo and derived by Laplace) as ‘numerology’. Climate computer games modeler Chris Colose calls it “third rate crank science”.   🙂
An earlier 2010 paper finds a fourth planet in the system as part of the resonant coupling.

Dynamical analysis of the Gliese-876 Laplace resonance

J. G. Marti, C. A. Giuppone, C. Beauge (Submitted on 28 May 2013)

The existence of multiple planetary systems involved in mean motion conmensurabilities has increased significantly since the Kepler mission. Although most correspond to 2-planet resonances, multiple resonances have also been found. The Laplace resonance is a particular case of a three-body resonance where the period ratio between consecutive pairs is n_1/n_2 near to n_2/n_3 near to 2/1. It is not clear how this triple resonance can act in order to stabilize (or not) the systems. The most reliable extrasolar system located in a Laplace resonance is GJ876 because it has two independent confirmations.

However best-fit parameters were obtained without previous knowledge of resonance structure and no exploration of all the possible stable solutions for the system where done. In the present work we explored the different configurations allowed by the Laplace resonance in the GJ876 system by varying the planetary parameters of the third outer planet. We find that in this case the Laplace resonance is a stabilization mechanism in itself, defined by a tiny island of regular motion surrounded by (unstable) highly chaotic orbits. Low eccentric orbits and mutual inclinations from -20 to 20 degrees are compatible with the observations. A definite range of mass ratio must be assumed to maintain orbital stability. Finally we give constraints for argument of pericenters and mean anomalies in order to assure stability for this kind of systems.

Comments:7 pages, 7 figures, accepted in MNRAS
Subjects:Earth and Planetary Astrophysics (astro-ph.EP)DOI:10.1093/mnras/stt765Cite as:arXiv:1305.6768 [astro-ph.EP] (or arXiv:1305.6768v1 [astro-ph.EP] for this version)

  1. tallbloke says:

    US thermophysics
    @USthermophysics follows you
    Thermal/fluid systems scientist. Teach @ The University of Texas. Principal Investigator for NASA, AFRL, NAVAIR, NSF technology projects

    Quite right sir and thank you for your support. Martin Rasmussen, Team Wassup and all trace gas junk scientists take note. 🙂

  2. oldbrew says:

    The same two examples are featured here with interactive animations showing where the conjunctions take place. It’s very easy to see what’s going on and there’s a pause option too.


    Broadly speaking it’s not just a numeric thing, there’s an important positional element too.
    For a very brief guide to the mechanism of resonance with animations, this is easy to follow.


  3. Jupiter is a failure sun, however it’s a mini solar sytem.
    We must study the dynamics vortices [red-(sun)spot :smile] of the jupiter and the three-body resonance

  4. tallbloke says:

    OB: Thinking about this bit: “A definite range of mass ratio must be assumed to maintain orbital stability” – Didn’t we find something in the relative masses or diameters of the Jovian moons?

  5. oldbrew says:

    TB: looks like ‘argument from incredulity’ above 🙂

    “I can’t believe P, therefore not-P.”

    More analysis of Gliese 876 here, with a link to another science paper:


    TB: Re J moons – yes, moons 1+4 have about the same mass as 2+3 (meaning the four major moons).

  6. tallbloke says:

    OB: Thanks, I knew you’d covered Gliese 876 before, but was posting from phone so didn’t have time to hunt it down. The more of these systems we find, the more obvious it is that the Laplace Resonance actively stabilises itself.

  7. oldbrew says:

    This paper (‘Can GJ 876 host four planets in resonance?’) has an interesting graphic (Fig 2).

    Click to access 1202.5865v1.pdf

    The paper says:

    ‘An interesting feature of Fig. 2 is the small island of stability around a = 0.33 AU
    where planet e exists. The value of λ for this region is ∼ 220◦ which places it in a
    Laplace resonance with planets b and c. Given the small size of this area, it is clear
    that this resonance protects planet e against close encounters with the other planets
    of the system.’

    Could the small island of stability near ‘∼ 220◦’ in fact include 222.5◦ which is 360◦ / phi ?

  8. hunter says:

    Gavin Schmidt talking about crank science. Irony meter pegs out.

  9. oldbrew says:

    TB: The first paper you referenced says ‘the Laplace angle is also librating with small amplitude’ so maybe some ‘wiggle room’ for the angle there.

  10. Sparks says:

    I’ve done some research into the resonance of the Jovian moons, I was looking into atmospheric spot formation, I’ve found some interesting leads but nothing substantial as yet, In the Saturn Lunar resonances however, I’ve been looking for a period of 30 years which regularly occurs when the formation of Saturn’s great white spot appears.

  11. tallbloke says:

    Sparks: At the Saturn orbital period?

  12. Sparks says:


    Yes exactly, I know the orbital period of Saturn is 29 years and its great white spot/spots appear or peak with approximately the same periodicity, I’ve been treating the spot activity as having maximum and minimum spot activity which over it’s orbital period which works out as an approximately 14.5 year period between maximum and minimum spot activity, which varies slightly. and been studying if there is a relationship between planetary or lunar resonance.

    I hope this is interesting and makes sense, It’s been a long day here! :))

  13. Sparks says:

    Mostly I’ve been concentrating on Titan and changes in its orbit in relation to Saturn’s orbital period.

  14. tallbloke says:

    Sparks: Seems reasonable that activity should vary over the orbitsl period given the eccentricity of the orbit.

  15. Truthseeker says:

    Hmmm, I don’t think that this science is sexy enough. Maybe if it was “lapdance resonance” …

  16. RoswellJohn says:

    While I see comments on the resonances I never see an explanation of how they come about. Perhaps some astronomers have worked all this out, but the n-body problem is still a problem as far as I know. However, years ago, in the 1960s when I was reading Science and Nature every week there was a paper in one of them about planetary resonances wherein the author converted the periods to frequencies and the added and subtracted all the frequencies from each other. The reason for this approach was that the solar system is a non-linear system (1/(r2)) and modulation theory from the RF world can be used to investigate these periods/frequencies. For our solar system with many bodies the sidebands get to be, well, astronomical. Pun intended. My guess is that a minimal system is one that has resonances and so any many body solar system will tend to move towards a system of resonances in order to minimize the energy in sidebands.

    Is this known to the astronomical world? Maybe in some other terminology?

  17. tallbloke says:

    There’s a reason I’m highlighting Laplace Resonance which will become clear in a followup post.

  18. tallbloke says:

    RJ: Good comment. Up until now, perturbation theory has been a ‘gravity only affair’. Our research in PRP shows that not only are orbital periods linked by resonant ratios, but axial rotation periods are too. Gravity is a straight line force, so it looks like there is a strong possibilty that magnetics also play a role in resonance, providing another field, the IMF, through which energy can propagate between planets (and the sun).

    As well as the inverse square falloff of gravity, we also have another ‘tailed’ distribution to consider. My main paper in PRP shows that the planets are lognormally distributed in the system.

  19. oldbrew says:

    Time to bring on perturbation theory?

    ‘In astronomy, perturbation is the complex motion of a massive body subject to forces other than the gravitational attraction of a single other massive body.’


    It may not have all the answers but goes some of the way at least.

    ‘On-going mutual perturbations of the planets cause long-term quasi-periodic variations in their orbital elements, most apparent when two planets’ orbital periods are nearly in sync.’

  20. Chaeremon says:

    @oldbrew good idea 😎 in my research I (sometimes) separate ‘sync’ appearances from all the other, in order to compare A to B and not A to A (aka. confirmation bias). But this is often a hell lot of work …

  21. hunter says:

    A lognormal distribution of planets concept sounds very interesting.

  22. tallbloke says:

    Hunter: Feel free to check out my papers or use the search facility on the left for posts with ‘why phi’ in the titles

  23. ferdberple says:

    resonance – the crank that drives the solar system.

    Click to access rio97.pdf


  24. tallbloke says:

    Thanks Ferd, that must be the crank Chris Colose is referring to? 😉

  25. oldbrew says:

    If anyone is interested, you can copy the animation here…

    …to Windows Media Player, set it to ‘Slow playback’ and use the pause button to try and figure out the repeating conjunction patterns. They do exist and some are obvious but it’s a geeky job.

    If/when I think I’ve finished I’ll report back unless someone does it all for me first.
    Don’t forget to note where the third moon is for the ‘two-moons-only’ conjunctions 😉

  26. Chaeremon says:

    @oldbrew the author says ratios 1:2:4, what is what you seek “conjunction patterns”? B.t.w. the dashes are the periapses 😎

  27. oldbrew says:

    1-3-5-7-9 etc. are the same two, with Ganymede at right angles but switching sides. It’s the variation in positions I’m looking at. 2-6-10 etc. follow one pattern, 4-8-12 etc. another.

    Those numbers are the values of t/P1. Also things to note at other conjunction points.

  28. Chaeremon says:

    @oldbrew: let’s say I(o) and E(uropa) share their line-of-apse L (and begin in the animation in opposition at opposite sides of L). Then G(anymede)’s periapse is, initially, at ~45° relative to L, and after 1 x G the periapse of G has moved to ~90° relative to L.

    I haven’t found periodicities of periapse for I,E,G (their individual line-of-apse), so I don’t know the following:

    Q[1]: does G’s periapse return to the initial ~45° relation at the time when I,E are again in opposition? if so, has G also returned to the initial (in relation to I,E) orbital position?

    Q[2]: just be sure, is the line-of-apse L shared by I,E for longer time frames?

    Don’t know if this can help me solving your initial question about conjunction pattern, but it looks interesting to me as base for another exercise in underappreciated ‘cyclomania’ 😉

  29. Sparks says:

    oldbrew says:
    February 20, 2014 at 9:41 am

    Time to bring on perturbation theory?

    I had an interesting conversation with Leif awhile back about an idea concerning planetary gates! Leif had an issue with planetary polarity traveling up wind, I understand his doubts. Polarity doesn’t need to travel up wind, it simply has to exist. It’s very basic. and it works. :))

  30. oldbrew says:

    @ Chaeremon

    I can only comment on initial animation results, so the provisional answers are:

    Q1: the periapse of G is about 58 degrees compared to the line of I and E periapse.
    It doesn’t move in relation to them but the whole configuration precesses slowly (a few months).
    Update: it may move slightly, see below*.
    Q2: yes

    It all looks very interesting, so assuming the animation is accurate (which we do at this time) there will be more to say.

    Broadly speaking one orbit of Ganymede = 1 iteration of the 3-body system (+J = 4).

    * Having measured the angle at 11 revolutions of Io, it looks like exactly 60 degrees at that point, with Ganymede itself at right angles to Io and Europa (which are both on their ‘periapse line’). Will post a diagram.

  31. oldbrew says:

    Here’s the diagram (see comment above).


    Every four Io orbits (= 2 Eu, 1 Ga) reaches the same configuration.

  32. Chaeremon says:

    Yay 😎 fascinating surprise 60°, also the iteration. Will watch out for papers which report on periodicities of periapse for I,E,G.

    Where do you go from here [where’s phi]?

  33. oldbrew says:

    ‘Where’s phi?’ – 2:1 Io:Eu and 2:1 Eu:Ga (Fibonacci / resonance).

  34. tallbloke says:

    Also, to a first order approximation, the four Galilean moons orbital distance lie in the ratio phi^3, phi^4, phi^5, phi^6.

  35. Chaeremon says:

    Do you already have a relation between the dance of I,E,G and the dance of the Jovians?

    The J-S (heliocentric) conjunctions appear to dance with intermediate steps of 15 years (the usual jitter applies): S is advancing to the 180° position every step, J is advancing to the next 90° position every step.

  36. oldbrew says:

    No, but there may well be a link to Jupiter’s own orbit.

    Europa completes 1220 orbits of J in the time of one J orbit of the Sun.
    1220 = 610 x 2 (610 is a Fibonacci number).

    In the same period Ganymede does a few less than 610 whereas Io does a few more than 610 x 4.