## Prolegomenon

You recognize as a youngster that science, and music, and literature and writing—creative wonders—draw you along comfortable invisible force lines. But not opera. Overbearing, embarrassing falsetto vibrato is just wrong. As your joints grow creaky and more of your pate warms to the Sun, you know that this is a misperception. You stumble upon more of these, as you notice yourself more often assigning past vigorous feats of physical prowess to the unimportant pursuits of the unimportant young. You ponder these, your various misperceptions. And your misperceptions of misperceptions. Recursion tickles you.

$$\dfrac{\mathrm{d}^2\overrightarrow{r}}{\mathrm{d}\theta^2}+2\widehat{z}\times\dfrac{\mathrm{d}\overrightarrow{r}}{\mathrm{d}\theta}+{\left(\widehat{z}\cdot\overrightarrow{r}\right)}\widehat{z}=\frac{1}{{1+e_{p}\mathrm{cos}\mathrm{\theta}}}\overrightarrow{\nabla}\mathrm{\Omega}$$

You realize in the shower one day that your—and others’—universal cognitive foibles smacking into observable reality are an irresistible rabbit hole, wondrously vast and an endless source of material to contemplate. Like a particle in the three-body problem of celestial mechanics, your orbit is a tangled meandering, variously lured into the sphere of influence of first one and then the other of those two massive attractors, science and the creative urge. This resonates, and you realize a re-appreciation of past love.

$$\mathrm{\Omega}=\frac{1}{2}r^{2}+U=\frac{1}{2}r^{2}+\frac{{1-\mathrm{\mu}}}{r_{1}}+\frac{\mathrm{\mu}}{r_{2}}$$

Thus: what shall you write? Unuseful question. The world is big. Where shall you intend your aim? Better. Get thee to the shower!, your ever-reliable Delphic font of nearly every good idea.§ You love nature, and science—especially astronomy and math—and the scientific way of thinking, which come to you with joy and not pain. (This cannot be weird, surely—friends’ and society’s protestations notwithstanding.) The chasm awaits.

$$r_{1}=\sqrt{{{\left(x+\mathrm{\mu}\right)}^{2}+y^{2}+z^{2}}}\hspace{2.222222em}r_{2}=\sqrt{{{\left(x-1+\mathrm{\mu}\right)}^{2}+y^{2}+z^{2}}}$$

On a whim you schlep to a National Association of Science Writers conference, where you are isolated and small, sole introvert amidst a mind-bruising cacophony. Drilling through your crushing discomfort, you meet Roy Peter Clark’s Writing Tools: 50 Essential Strategies for Every Writer (you buy three copies), you hear Jonathan Coulton sing his wistful nerd anthem, “Code Monkey” (you buy three CDs), and a merciful soul tells you to read Lewis Thomas’s classic medley of essays, The Lives of a Cell: Notes of a Biology Watcher (why is there no Kindle version?). This is it. A trigger, an unlatching: your dormant writing compulsion awakens.

Astronomy with math. True stories, precisely told. A worthwhile target.

$$v^{2}-\frac{{2\mathrm{\Omega}}}{{1+e_{p}\mathrm{cos}\mathrm{\theta}}}+z^{2}+C+2\int\frac{{e_{p}\mathrm{sin}\mathrm{\theta}}}{{\left(1+e_{p}\mathrm{cos}\mathrm{\theta}\right)}^{2}}\mathrm{\Omega}\hspace{0.222222em}d\mathrm{\theta}=0$$

Halfway through college, you end the pleasant agony and decide astronomy over music. Seemingly by crazy random utterly naive inevitability, you become a professional astronomer. As your mop grows thinner and your knuckles grow larger, you realize the apparent randomicity is a misperception.

The equations, if you are wondering, tell how a massless particle moves in the combined gravitational fields of two massive objects in orbit about each other.¤ Think, for example, Sun–Jupiter–spacecraft. In astronomy, we call this the restricted three-body problem. It is astonishingly complex.

§ Perhaps only Death is a greater surety—though, surely, only by a little.

¤ For completeness:

$$\mathrm{\mu}=\frac{m_{2}}{{m_{1}+m_{2}}},\hspace{2.2em}r=\sqrt{x^2+y^2+z^2}$$

and

$$\begin{array}{rcl}\overrightarrow{\nabla}\mathrm{\Omega}&=&\left[\begin{array}{l}x-\dfrac{1-\mathrm{\mu}}{r_{1}^{3}}\left(x+\mathrm{\mu}\right)-\dfrac{\mu}{r_{2}^{3}}\left(x-1+\mathrm{\mu}\right)\\\\y\left(1-\dfrac{1-\mathrm{\mu}}{r_{1}^{3}}-\dfrac{\mu}{r_{2}^{3}}\right)\\\\z\left(1-\dfrac{1-\mathrm{\mu}}{r_{1}^{3}}-\dfrac{\mu}{r_{2}^{3}}\right)\end{array}\right]\\\\&=&\left(1-\dfrac{1-\mathrm{\mu}}{r_{1}^{3}}-\dfrac{\mu}{r_{2}^{3}}\right)\overrightarrow{r}-\mathrm{\mu}\left(1-\mathrm{\mu}\right)\left(\dfrac{1}{r_{1}^{3}}-\dfrac{1}{r_{2}^{3}}\right)\widehat{x}\end{array}$$

## 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
• 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 – The onset of dynamical instability and chaos in navigation satellite orbits

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.

Aaron Jay Rosengren (IFAC-CNR)

#### Abstract

Orbital resonances are ubiquitous in the Solar System and are harbingers for the onset of dynamical instability and chaos. It has long been suspected that the Global Navigation Satellite Systems exist in a background of complex resonances and chaotic motion; yet, the precise dynamical character of these phenomena remains elusive. Here we will show that the same underlying physical mechanism, the overlapping of secular resonances, responsible for the eventual destabilization of Mercury and recently proposed to explain the orbital architecture of extrasolar planetary systems (Lithwick Y., Wu Y., 2014, PNAS; Batygin K., Morbidelli A., Holman M.J., 2015, ApJ) is at the heart of the orbital instabilities of seemingly more mundane celestial bodies—the Earth’s navigation satellites. We will demonstrate that the occurrence and nature of the secular resonances driving these dynamics depend chiefly on one aspect of the Moon’s perturbed motion, the regression of the line of nodes. This talk will present analytical models that accurately reflect the true nature of the resonant interactions, and will show how chaotic diffusion is mediated by the web-like structure of secular resonances. We will also present an atlas of FLI stability maps, showing the extent of the chaotic regions of the phase space, computed through a hierarchy of more realistic, and more complicated, models, and compare the chaotic zones in these charts with the analytical estimation of the width of the chaotic layers from the heuristic Chirikov resonance overlap criterion. The obtained results have remarkable practical applications for space debris mitigation and for satellite technology, and are both of essential dynamical and theoretical importance, with broad implications for planetary science.

#### Notes

• Motivation: space debris problem
• Active debris removal is becoming necessary
• New: exploit resonant orbits to obtain relatively stable graveyards or highly unstable disposal orbits
• Resonance overlap & chaos
• asteroid belt resonances: cf. DeMeo & Carry 2014 (Nature Rev)
• What is resonant structure of cislunar space?
• actually less well known than resonant structure of asteroid belt
• Cislunar resonant phenomena:
• tesseral resonances
• MMRs
• lunisolar semi-secular resonances (sun-synchronous, evection resonance)
• secular resonances (crit. inclination, Kxxxx resonance)
• Navsat orbits (European) are unstable!
• Chao 2000, Jenkin & Gick 2002, Chao & Gick 2004
• Also: interference from sats in disposal orbits
• Ref: Mercury’s orbit and secular chaos
• Harmonic analysis of Lunar perturbations
• Tesseral and lunisolar semi-secular resonances cannot be the cause of orbital instabilities observed in numerical surveys
• Role of secular resonances in producing chaos
• simplifications:
• 2nd order in ratio of semimajor axes
• short periodic terms of disturbing function can be averaged out
• resonance: $\dot{\psi} = (2-2p) \dot{\omega} + m \dot{\Omega} \pm s\dot{\Omega}_2 \approx 0$
• chaotic diffusion (~250 yr)
• Daquin et al. CMDA (in prep)
• Chirikov res overlap criterion
• chaotic web
• plot: $e$ vs $i$
• FLI stability maps
• heat map: $e$ vs $i$
• too many dimensions $\rightarrow$ far from understood