The subsection Mercury–Jupiter 1:1 perihelion-precession resonance in the Wikipedia article Stability of the Solar System links to the article Solar system could go haywire before the Sun dies in the NewScientist, which references two articles that were still in press back in 2008 when it was published.

More time (both people-time and CPU-time) has passed. In 2016, is there still thought to be a 2% chance that when Jupiter increases Mercury's eccentricity beyond 0.6, it could get ejected?

I bring this up because the question of weather an orbit is "stable" or "unstable" always dances around the fact that many things are really in the middle ground - stable for months, or eons, but not necessarily actually stable in the sense that two masses orbiting in an otherwise totally empty Euclidian non-relativistic universe would be gravitationally stable. Put a handful (or more) bodies in orbit and chaotic behavior can arise.

So I'm looking for a good example of "stability is an illusion" loosely speaking, and Mercury is the best one I can think of at the minute - only I'm stuck in Wikipedia in 2008.

Has there been any further analysis?

  • $\begingroup$ Is the actual question just about Mercury (as the title states), or are you looking for examples of particularly unstable orbits elsewhere? $\endgroup$
    – Andy
    May 11, 2016 at 9:42
  • $\begingroup$ @Andy within the solar system, artificial or natural. I almost did put this in Astronomy. I waffled a bit, but after considering the excellent answers I get here and the sometimes flaky responses I get there, I thought I would test the water here first. When I think about careful planetary orbit calculations I think of JPL and similar organizations, and that leads me back here. I like to think of the Oort cloud as the soft delimiter between Space SE and Astronomy SE. But that's just space, I guess a billion years might suggest going the other way. $\endgroup$
    – uhoh
    May 11, 2016 at 9:57
  • 3
    $\begingroup$ This question has been reopened in accordance with the new policy established here. $\endgroup$
    – called2voyage
    Apr 10, 2020 at 14:13
  • $\begingroup$ See Laskar and Gastineau theory of mercury orbit: here and here $\endgroup$ Nov 14, 2020 at 9:07
  • $\begingroup$ Some more links: 1, 2, 3, 4, 5 $\endgroup$ Nov 14, 2020 at 9:09

2 Answers 2


Here is the first paragraph of the introduction from Abbot et al. (2023):

Since the landmark study of Laskar (1994), the potential for Mercury’s orbit to destabilize has been widely recognized. The destabilization process has been studied both with simplified test particle secular models (Lithwick & Wu 2011; Boue et al. 2012; Lithwick & Wu 2014; Batygin et al. 2015) and sophisticated, high-order secular models (Laskar 2008; Mogavero & Laskar 2021; Mogavero & Laskar 2022; Hoang et al. 2022; Mogavero et al. 2023) as well as with more computationally intensive and physically realistic N-body codes (Batygin & Laughlin 2008; Laskar & Gastineau 2009; Zeebe 2015a,b; Brown & Rein 2020, 2022, 2023; Abbot et al. 2021, 2023; Hernandez et al. 2022). The secular models have led to the key insight that Mercury’s orbit destabilizes due to resonance between the Solar system’s g1 and g5 secular eigenfrequencies, which are primarily associated with Mercury and Jupiter, respectively.

So, yes, it is still the case that Mercury may be unstable on timescales of billions of years.

  • $\begingroup$ Nice; thanks! The idea that a (secular model of a) solar system can have eigenfrequencies is new to me, but maybe that's not so hard to understand once I figure out exactly what a secular model is, perhaps it's a bit like a group of weakly and nonlinearly-coupled harmonic oscillators? I'll read up first, then post a new question if necessary. $\endgroup$
    – uhoh
    Oct 5, 2023 at 7:43
  • $\begingroup$ I added a supplementary answer. This reminds me to see if I need to ask a question on solar system eigenfrequencies (still have to read first to see if it's a no brainer or if it's an interesting question) $\endgroup$
    – uhoh
    Jan 6 at 3:17

Some additional information, found in a comment in Math Overflow on Breakthroughs in mathematics in 2023

Concerning non-combinatorics: It would be instructive if someone knowledgeable could remark on a theorem concerning 𝑛-body instability.

which links to Quanta Magazine's May 16, 2023 New Proof Finds the ‘Ultimate Instability’ in a Solar System Model:

Now, in three papers that together exceed 150 pages, Guàrdia and two collaborators have proved for the first time that instability inevitably arises in a model of planets orbiting a sun.

Andrew Clarke, Jacques Fejoz, Marcel Guardia (2022, 2022, 2023):


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