## 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)

#### Abstract

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.

#### Notes

- 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

- without Moon, $\epsilon$ is chaotic (Laskar et al. 1993)
- 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