Impending Disturbance

vap’rous backlit wrack


the cold turbulent flow


that baleful winter’s orb

Moon & Overcast 2016-01-19

DDA 2015 – On the Spin-axis Dynamics of the Earth

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Gongjie Li (Harvard) (Duncombe award winner)


The variation of a planet’s obliquity is influenced by the existence of satellites with a high mass ratio. For instance, the Earth’s obliquity is stabilized by the Moon, and would undergo chaotic variations in the Moon’s absence. In turn, such variations can lead to large-scale changes in the atmospheric circulation, rendering spin-axis dynamics a central issue for understanding climate. The relevant quantity for dynamically-forced climate change is the rate of chaotic diffusion. Accordingly, here we reexamine the spin-axis evolution of a Moonless Earth within the context of a simplified perturbative framework. We present analytical estimates of the characteristic Lyapunov coefficient as well as the chaotic diffusion rate and demonstrate that even in absence of the Moon, the stochastic change in the Earth’s obliquity is sufficiently slow to not preclude long-term habitability. Our calculations are consistent with published numerical experiments and illustrate the putative system’s underlying dynamical structure in a simple and intuitive manner. In addition, we examine if at any point in the Earth’s evolutionary history, the obliquity varied significantly. We find that even though the orbital perturbations were different in the past, the system nevertheless avoided resonant encounters throughout its evolution. This indicates that the Earth obtained its current obliquity during the formation of the Moon.


  • Obliquity $\cos \epsilon$ affects climate
    • Mars obliquity variations caused collapse of Martian atmosphere
  • Obliquity variations of a Moonless Earth
    • without Moon, $\epsilon$ is chaotic (Laskar et al. 1993)
      • geostrophic winds
    • but N-body sims: $\epsilon$ constrained to $\epsilon \lesssim 45$ deg — why?
    • Sun and planetary torques: spin precession rate, inclination variation
      • model as superposition of linear modes
      • resonance overlap: two connected chaotic zones — Laskar 1993, Morby 2000, Laskar 1996
    • average over primary resonances $\rightarrow$ secondary resonances
      • overlap of secondary resonances creates the chaotic bridge (Chirikov 1979)
    • Results
      • regular at $\ge 85$ deg
      • less chaotic in bridge
      • analytic and numerical are consistent
      • Li & Batygin 2014a
      • diffusion timescale 10 Myr in primary chaotic zones, 2 Gyr in the bridge
  • Pre-late heavy bombardment evolution of Earth’s obliquity
    • Li & Batygin 2014b
    • solar system starts more compact (Nice model)
    • study evolution of mode freqs and effects on Earth’s inclination
    • also, Moon was closer
    • two freqs match prior to LHB only if $\epsilon \ge 85$ deg
    • $\therefore$ Earth’s obliquity arose during the formation of the Moon

DDA 2015 – Forced libration of tidally synchronized planets and moons

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Session: Moon Formation and Dynamics I

Valeri Makarov (for Bryan Dorland) (USNO)


Tidal dissipation of kinetic energy, when it is strong enough, tends to synchronize the rotation of planets and moons with the mean orbital motion, or drive it into long-term stable spin-orbit resonances. As the actual orbital motion undergoes periodic acceleration due to a finite orbital eccentricity, the spin rate oscillates around the equilibrium mean value too, giving rise to the forced, or eccentricity-driven, librations. We contend that both the shape and amplitude of forced librations of synchronous viscoelastic planets and moons are defined by a combination of two different types of perturbative torque, the tidal torque and the triaxial torque, the latter related to a permanent deformation of the distribution of mass from a perfect rotational symmetry. Consequently, forced librations can be tidally dominated (e.g., Io and possibly Titan) or deformation-dominated (e.g., the Moon) depending on a set of orbital, rheological, and other physical parameters. With small eccentricities, for the former kind, the largest term in the libration angle can be minus cosine of the mean anomaly, whereas for the latter kind, it is minus sine of the mean anomaly. The shape and the amplitude of tidal forced librations determine the rate of orbital evolution of synchronous planets and moons, i.e., the rate of dissipative damping of the semimajor axis and eccentricity. The known super-Earth exoplanets can exhibit both kinds of libration, or a mixture of thereof, depending on, for example, the effective Maxwell time of their rigid mantles. Our approach can be extended to estimate the amplitudes of other libration harmonics, as well as the forced libration in non-synchronous spin-orbit resonances.


  • Numerical sim of Lunarspinlibrations in longitude with polar torques exerted by Earth
    • spectrum of harmonics
    • dominant term: $\dfrac{\dot{\theta}}{n}\, – 1 \propto -\cos M$
  • Geometry offorcedlibrations
    • longest axis of planet tries to align with line of centers (but can’t)
    • triaxial torque (Eckhardt 1981) $\ddot{\theta}\approx 6 n^2 \dfrac{B-A}{C^3}\left[e\, – \dfrac{31}{16}e^2 + O\left(e^5\right)\right] \sin M$
  • Problem: Io
    • striking difference with moon
    • dominant term now $\dfrac{\dot{\theta}}{n}\, – 1 \propto +\sin M$
  • Tidal torques
    • torque = triaxial + tidal
    • tidal = secular + periodic
      • secular can dominate in the “linear” regime
        • plot: $\ddot{\theta}$ vs. $\dfrac{\dot{\theta}}{n}\, – 1$
      • It’s not the amplitude that’s important, but the gradient, which provides a subtle but powerful feedback.
      • tiny interval of pert freqs where secular torque can be important
  • Problem: doesn’t work for Moon
    • because Moon is not sitting on that linear portion of $\dfrac{\dot{\theta}}{n}\, – 1$
    • $\therefore$ need further harmonic terms