““Chaotic Rotation in the Chaotic Rotation in the Three-Body Coorbital ProblemThree-Body Coorbital Problem””

Alexandre C.M. Alexandre C.M. CorreiaCorreia

IMCCE / Observatoire de IMCCE / Observatoire de ParisParis

gr@av group meetinggr@av group meetingMarch 5March 5thth, 2014 - Aveiro , 2014 - Aveiro

Philippe RobutelPhilippe Robutel

Universidade de AveiroUniversidade de Aveiro

““Chaotic Rotation in the Chaotic Rotation in the Three-Body Coorbital ProblemThree-Body Coorbital Problem””

equilibrium:Lagrange

(1772)

Gascheau (1843)

Wolf (1906)Achilles

stability:

a: semi-major axis

e: eccentricity

ω: longitude of the perihelion

λ: mean anomay

λ

λ = λ0 + n (t – t0)

Two-Body Problem (Kepler problem)Two-Body Problem (Kepler problem)

ω

Two-Body Problem with Two-Body Problem with RotationRotation

Danby (1962)

Circular Orbits with Circular Orbits with RotationRotation

Pendulum phase space:

Eccentric Orbits with Eccentric Orbits with RotationRotation

Eccentric Orbits with Eccentric Orbits with RotationRotation

Phobos Mercury

Moon

Hyperion

Wisdom et al. (1984)

Chirikov (1979)

Three-Body Coorbital Three-Body Coorbital Circular Problem (Circular Problem (3BCP)3BCP)

Tadpole

Horseshoe

Co-rotating frameCo-rotating frame

Tadpole

Horseshoe

Érdi (1977)

3BCP with 3BCP with RotationRotation

Correia & Robutel (2013)

PoincarPoincaré Sections (é Sections ( = 1 = 1))

= 0º = 50º = 10º

Correia & Robutel (2013)

PoincarPoincaré Sections (é Sections ( = 50º = 50º))

log = 1.3

log = 0.4

log = -0.4

Correia & Robutel (2013)

Stability analysisStability analysisSaturnSaturn

Exo-EarthsExo-Earths

Correia & Robutel (2013)

Dissipation & CaptureDissipation & Capture

Tidal evolutionTidal evolution ( ( = 50º = 50º))log =

1.3log =

0.4log = -

0.4

Correia & Robutel (2013)

Tidal evolutionTidal evolution ( ( = 1 = 1)) = 0º = 50º = 10º

Correia & Robutel (2013)

Conclusions:Conclusions:

• Coorbital bodies in quasi-circular orbits may present chaotic rotation for a wide range of mass ratios and body shapes.

• We show the existence of an entirely new family of spin-orbit resonances at the frequencies n k/2.

• The rotational dynamics of a body cannot be dissociated from its orbital environment.