# 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