Renu Malhotra: Nonlinear Resonances in the Solar System

Posted: January 29, 2016 by tallbloke in solar system dynamics

A very readable paper by NASA scientist Renu Malhotra giving an introduction to the study of orbital resonance as applied to the understanding of the organisation of the solar system. This background validates the study we are undertaking here at the talkshop to understand not only the links between planetary motion, but those links and the variation of the Sun, which correlates strongly with many solar system planetary timings. Many cycles found in paleoproxies also correlate with these periods, implying an effect of solar system timing on variation in Earth’s climate too.

Orbital resonances are ubiquitous in the Solar system. They play a decisive role in the long term dynamics, and in some cases the physical evolution, of the planets and of their natural satellites, as well as the evolution of small bodies (including dust) in the planetary system. The few-body gravitational problem of hierarchical planetary-type systems allows for a complex range of dynamical timescales, from the fast orbital periods to the very slow orbit precession rates. The interaction of fast and slow degrees of freedom produces a rich diversity of resonance phenomena. Weak dissipative eects | such as tides or radiation drag forces | also produce unexpectedly rich dynamical behaviors. This paper provides a mostly qualitative discussion of simple dynamical models for the commonly encountered orbital resonance phenomena in the Solar system.

1. Introduction
Soon after Newton’s formulation of the Universal Law of Gravitation it became evident that a dynamical model of the Solar System based upon a simple superposition of two-body motions was not adequate for the observations. The mutual interactions of the planets (and of their natural satellites) were necessary to t the observations and infer various properties of the system. Theoretical eorts to determine the eects of planetary interactions led to the development of perturbation theory and averaging methods, and to many other developments in mathematics. In recent years, celestial mechanics has evolved from its traditional habitat in the study of orbits of particular Solar System objects to the study of the structure of extended regions of phase space (of planetary systems, of satellite systems, and of other small celestial bodies), as well as of evolution due to dissipative eects. The greatest diculties in these studies | and also the most interesting dynamical behaviors | are associated with orbital resonances.

Even in the simplest system consisting of only two planets, there are six degrees of freedom corresponding to the three spatial degrees of freedom for each planet. Therefore, what at rst glance would appear to be only a two-frequency system (i.e., the frequencies of revolution of the two planets around the Sun) is actually one with six frequencies. Two of these are the obvious ones of revolution around the Sun; the other four (two for each planet) are the much slower frequencies of precession of the orientation of the orbits. The existence of both fast and slow degrees of freedom produces a rich diversity of orbital resonance phenomena.

As a general classication, there are two types of orbital resonances. The intuitively most obvious type, referred to simply as \orbital resonance” or \mean-motion resonance”, occurs when the orbital periods of two planets { or satellites { are nearly commensurate. The second type, called \secular resonance”, involves commensurabilities between the slow frequencies of precession of the orientation of orbits. Classical perturbation theory runs into the notorious problem of \small divisors” in analyzing the mutual gravitational perturbations of two planets near a resonance. In the case of the rst type of resonance, a signicant but subtle diculty is that there are actually several resonances in the vicinity of any orbital period commensurability. This \resonance splitting” arises due to the precession of the orientation of orbits.

The multiplicity of resonances is a consequence of the coupling between perturbations of dierent timescales, and, in general, may produce chaotic behavior in the vicinity of an orbital resonance. However, in many cases, the behavior at a resonance is largely regular, albeit complicated by non-linear eects. Examples include the observed orbital resonances amongst the satellites of Jupiter and Saturn, where \single resonance theory” works [1], and many phenomena in planetary rings where resonant perturbations from satellites are implicated in a bewildering variety of features such as gaps, kinks and sharp edges, and the connement of narrow rings [2, 3]. In other cases, interactions between neighboring resonances become important; these lead to a variety of dierent phenomena associated with secondary resonances and chaotic dynamics. A well-known example is the \Kirkwood gaps” in the asteroid belt, which are decits in the number distribution of asteroids at several locations corresponding to mean motion resonances with Jupiter; chaos due to interacting resonances has been identied as a mechanism for producing at least one of these gaps [4, 5, 6] and for transporting meteorites from the main asteroid belt to the Earth [7, 8]. Chaotic resonances are of great signicance in the formation of planetary systems [9, 10] { in planetesimal dynamics, the transport of planetesimals to a proto-planet, the interactions of proto-planets, and nally the clearing of planetesimal debris from interplanetary space [11, 12, 13]. An interesting and very complicated example is that of Pluto which exhibits several secular resonances in addition to a 3:2 orbital period resonance with Neptune [14]. Secular resonances may play a determining role in the long term dynamical stability of a planetary system [15, 16].

Dissipative effects (such as gas drag and mass loss in the Solar Nebula, tides in the case of planetary satellites, collisions in planetary rings, and radiation forces in the case of dust particles) cause orbits to evolve across resonances, leading to separatrix-crossing phenomena and capture into resonance. Pluto’s very peculiar orbit (it crosses the orbit of Neptune but is dynamically protected from close encounters by the resonance phase locking) may owe its origin to orbit evolution of the outer planets due to mass loss in the late stages of planet formation [17]. The preponderance of orbital resonance locks amongst the Jovian and Saturnian satellites is thought to be due to the dierential orbit evolution induced by tidal dissipation [1, 18, 19, 20, 21, 22]. These satellites probably did not form in resonant orbits; however, once established, the orbital phase locking can be maintained for long times. Similar evolution probably occurred in 2 the Uranian satellite system also. However, owing to a small but crucial dierence in parameters, the orbital resonances in this system were not long-lived [23, 24]. Uranus is less oblate than Jupiter and Saturn; in addition, the relative masses of the satellites and their orbital radii conspire to cause an insucient splitting of resonances; as a result, the interactions between neighboring resonances give rise to an instability of resonant orbits. Tidal evolution and passage through resonance can have a profound eect on the geophysical evolution of satellites [20, 22, 23, 24, 25, 26, 27]. Orbital resonance phenomena are also found in the dynamics of circumplanetary and interplanetary dust particles that are perturbed by radiation and electromagnetic forces. For example, the dusty ring around Jupiter exhibits features that are attributed to \Lorentz resonances” (commensurabilities between the Keplerian orbit period and the spin period of the planetary magnetic eld) that dominate the orbital evolution of charged dust grains [28]. As a nal example, I can mention the recent prediction and detection of a \Solar ring” of asteroidal dust particles that spiral inward due to Poynting-Robertson light drag, and are captured into long-lived orbital resonances with the Earth [29, 30].

The diversity of orbital resonance dynamics precludes a complete treatment in these few short pages. Therefore, this paper is conned to giving a brief review of dynamical models for the simplest of the commonly encountered orbital resonance problems in the Solar System. No attempt has been made to give exhaustive citations to the literature, but the selected references should provide the interested reader with sucient leads to the technical literature.

Full paper here

  1. oldbrew says:

    TB: the abstract and intro seem to have lost some letter ‘f’ combinations, e.g. ‘conned’ should say ‘confined’, ‘rst’ is ‘first’ and ‘eects’ must mean ‘effects’.

  2. Paul Vaughan says:

    Hale Core 101:

  3. Paul Vaughan says:

    When there’s a cross-disciplinary misunderstanding it can be next-to-impossible to resolve it efficiently. (People don’t share a common background.)

    Looking at this conventional mainstream take on resonance I’m left wondering, “Don’t they see the cross-dimensional symmetry??”

    Now with the benefit of 20/20 hindsight, conventional mainstream φ blindness looks spectacularly astounding.

    Downslope from resonance’s sharp rim is resonance-evading φ stability in the naturally antiresonant basin of attraction: (February 6, 2016 at 7:52 am)

  4. tallbloke says:

    From Paul’s linked comment:

    φ indicates not resonance but rather antiresonance and destruction evasion. φ indicates stability. Resonance causes the opposite: instability. In hindsight it’s simple.

    A few sentences of Mae-Wan Ho’s writing instantly corrected my φ vision and I once again thank OB for the link. Without that link the insights shared above may never have arisen

    This is exactly what I’ve been saying since long before Mae Wan’s paper got flagged up. But there is something else too. The actuals are somewhere between φ and resonance. It’s not possible for all relationships to be φ so resonance is required in order to transmit the energy required to push orbits so the system maintains meta stability and synchronisation. The evidence for this is in the near whole number ratios of the conjunction cycle precessions. ~3600yr U-N, 2403yr J-S and 1199yr E-V.

  5. Paul Vaughan says:

    Sometimes it’s the combination of words. Mae-Wan Ho & I have a common background in evolution & population genetics. I suspect that may be why her expression was instantly recognizable. She’s integrating across disciplines and using language I recognize with ease from fields of past immersion.

    Of course that doesn’t mean you weren’t saying something mathematically equivalent that was not recognizable with such ease.

    For example whenever you’ve written of φ resonance I have never interpreted that as antiresonance …and I never would because resonance is the polar opposite of antiresonance.

    So in the most absolute way possible, I was for sure misinterpreting / misunderstanding what you intended by φ resonance IF you meant antiresonance.

    These types of miscommunications are tragic. You’ve been trying to alert me of something important for years. It took longer than we’d prefer and required a link to alternative expression to overcome the miscommunication (thank you OB), but at least the miscommunication has passed.

    TB wrote:
    “But there is something else too.”

    And as I’ve shown that “something else” is balanced in aggregate.
    It’s exactly in line with natural intuition: The system as a whole is stable.

    The really interesting property of the system remains: the solar cycle length differintegral, which represents aberrations of the solar cycle from φ antiresonance. There’s phase coherence with osculating JEV period, but to date we remain in the dark on amplitude. I remain astonished at how extremely ignorant people are of the solar cycle length differintegral. That’s another unfortunate tragedy — an inexplicable & inexcusable one. Maybe Mae-Wan Ho can clear people up on it!


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