A Criterion for the Onset of Chaos in Systems of Two Eccentric Planets

Sam Hadden, Yoram Lithwick

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We derive a criterion for the onset of chaos in systems consisting of two massive, eccentric, coplanar planets. Given the planets' masses and separation, the criterion predicts the critical eccentricity above which chaos is triggered. Chaos occurs where mean motion resonances overlap, as in Wisdom's pioneering work. But whereas Wisdom considered the overlap of first-order resonances only, limiting the applicability of his criterion to nearly circular planets, we extend his results to arbitrarily eccentric planets (up to crossing orbits) by examining resonances of all orders. We thereby arrive at a simple expression for the critical eccentricity. We do this first for a test particle in the presence of a planet and then generalize to the case of two massive planets, based on a new approximation to the Hamiltonian. We then confirm our results with detailed numerical simulations. Finally, we explore the extent to which chaotic two-planet systems eventually result in planetary collisions.

Original languageEnglish (US)
Article number95
JournalAstronomical Journal
Volume156
Issue number3
DOIs
StatePublished - Sep 2018

Fingerprint

eccentrics
chaotic dynamics
chaos
planets
planet
eccentricity
collision
orbits
collisions
approximation
simulation

Keywords

  • celestial mechanics
  • chaos
  • planets and satellites: dynamical evolution and stability

ASJC Scopus subject areas

  • Astronomy and Astrophysics
  • Space and Planetary Science

Cite this

@article{da613287f1db4ed4822c40dd053d1ef6,
title = "A Criterion for the Onset of Chaos in Systems of Two Eccentric Planets",
abstract = "We derive a criterion for the onset of chaos in systems consisting of two massive, eccentric, coplanar planets. Given the planets' masses and separation, the criterion predicts the critical eccentricity above which chaos is triggered. Chaos occurs where mean motion resonances overlap, as in Wisdom's pioneering work. But whereas Wisdom considered the overlap of first-order resonances only, limiting the applicability of his criterion to nearly circular planets, we extend his results to arbitrarily eccentric planets (up to crossing orbits) by examining resonances of all orders. We thereby arrive at a simple expression for the critical eccentricity. We do this first for a test particle in the presence of a planet and then generalize to the case of two massive planets, based on a new approximation to the Hamiltonian. We then confirm our results with detailed numerical simulations. Finally, we explore the extent to which chaotic two-planet systems eventually result in planetary collisions.",
keywords = "celestial mechanics, chaos, planets and satellites: dynamical evolution and stability",
author = "Sam Hadden and Yoram Lithwick",
year = "2018",
month = "9",
doi = "10.3847/1538-3881/aad32c",
language = "English (US)",
volume = "156",
journal = "Astronomical Journal",
issn = "0004-6256",
publisher = "IOP Publishing Ltd.",
number = "3",

}

A Criterion for the Onset of Chaos in Systems of Two Eccentric Planets. / Hadden, Sam; Lithwick, Yoram.

In: Astronomical Journal, Vol. 156, No. 3, 95, 09.2018.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A Criterion for the Onset of Chaos in Systems of Two Eccentric Planets

AU - Hadden, Sam

AU - Lithwick, Yoram

PY - 2018/9

Y1 - 2018/9

N2 - We derive a criterion for the onset of chaos in systems consisting of two massive, eccentric, coplanar planets. Given the planets' masses and separation, the criterion predicts the critical eccentricity above which chaos is triggered. Chaos occurs where mean motion resonances overlap, as in Wisdom's pioneering work. But whereas Wisdom considered the overlap of first-order resonances only, limiting the applicability of his criterion to nearly circular planets, we extend his results to arbitrarily eccentric planets (up to crossing orbits) by examining resonances of all orders. We thereby arrive at a simple expression for the critical eccentricity. We do this first for a test particle in the presence of a planet and then generalize to the case of two massive planets, based on a new approximation to the Hamiltonian. We then confirm our results with detailed numerical simulations. Finally, we explore the extent to which chaotic two-planet systems eventually result in planetary collisions.

AB - We derive a criterion for the onset of chaos in systems consisting of two massive, eccentric, coplanar planets. Given the planets' masses and separation, the criterion predicts the critical eccentricity above which chaos is triggered. Chaos occurs where mean motion resonances overlap, as in Wisdom's pioneering work. But whereas Wisdom considered the overlap of first-order resonances only, limiting the applicability of his criterion to nearly circular planets, we extend his results to arbitrarily eccentric planets (up to crossing orbits) by examining resonances of all orders. We thereby arrive at a simple expression for the critical eccentricity. We do this first for a test particle in the presence of a planet and then generalize to the case of two massive planets, based on a new approximation to the Hamiltonian. We then confirm our results with detailed numerical simulations. Finally, we explore the extent to which chaotic two-planet systems eventually result in planetary collisions.

KW - celestial mechanics

KW - chaos

KW - planets and satellites: dynamical evolution and stability

UR - http://www.scopus.com/inward/record.url?scp=85053137873&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053137873&partnerID=8YFLogxK

U2 - 10.3847/1538-3881/aad32c

DO - 10.3847/1538-3881/aad32c

M3 - Article

AN - SCOPUS:85053137873

VL - 156

JO - Astronomical Journal

JF - Astronomical Journal

SN - 0004-6256

IS - 3

M1 - 95

ER -