Three Body Mean Motion Resonances Tabaré Gallardo Departamento de Astronomía Facultad de Ciencias Universidad de la República Uruguay Luchon, September 2016 Tabaré Gallardo Three Body Resonances
preliminaries types of three body resonances (3BRs) semi analytical method numerical studies dynamical maps induced migration Tabaré Gallardo Three Body Resonances
Preliminaries Tabaré Gallardo Three Body Resonances
Preliminaries e = eccentricity a = semimajor axis (in astronomical units) 2 π 1 n = mean motion = mean angular velocity = period ∝ a 3 / 2 Two body resonance: k 0 n 0 + k 1 n 1 ≃ 0 with k 0 , k 1 integers. Tabaré Gallardo Three Body Resonances
Non resonant asteroid: relative positions Mean perturbation is radial: Sun-Jupiter Sun Jupiter Tabaré Gallardo Three Body Resonances
Resonant asteroid Mean perturbation has a transverse component. Sun Jupiter Tabaré Gallardo Three Body Resonances
from Gauss equations T R F perturb = ( R , T , N ) da dt ∝ ( R , T ) asteroid SUN < da dt > ∝ T Resonant Non resonant T � = 0 ⇒ a = oscillating T = 0 ⇒ a = constant Tabaré Gallardo Three Body Resonances
Critical angle σ For resonance k 0 n 0 + k 1 n 1 ≃ 0, is defined: σ = k 0 λ 0 + k 1 λ 1 + γ ( ̟ 0 , ̟ 1 ) the λ ’s are quick varying angles (mean longitudes) γ ( ̟ 0 , ̟ 1 ) is a linear combination of slow varying angles σ ( t ) indicates if the motion is resonant or not: σ ( t ) oscillating means resonance σ ( t ) circulating means NO resonance resonant motion: a ( t ) is correlated with σ ( t ) Tabaré Gallardo Three Body Resonances
Semimajor axis: width Nesvorny et al. in Asteroids III Tabaré Gallardo Three Body Resonances
Two body resonance, restricted case m 0 = 0 k 0 n 0 + k 1 n 1 ≃ 0 P 1 does not feel the resonance, only P 0 Tabaré Gallardo Three Body Resonances
Two body resonance k 0 n 0 + k 1 n 1 ≃ 0 Order: q = | k 0 + k 1 | Strength of resonance is approximately ∝ Cm 1 e q Theories try to obtain expressions for coefficients C Strength is related with amplitude of a ( t ) Tabaré Gallardo Three Body Resonances
Two body resonance, planetary case m 0 � = 0 k 0 n 0 + k 1 n 1 ≃ 0 both P 0 and P 1 feel the resonance Tabaré Gallardo Three Body Resonances
Two body resonance, planetary case Observational evidence in extrasolar systems Tabaré Gallardo Three Body Resonances
Three body resonances Tabaré Gallardo Three Body Resonances
Three body resonance, restricted case m 0 = 0 k 0 n 0 + k 1 n 1 + k 2 n 2 ≃ 0 only P 0 feels the resonance Tabaré Gallardo Three Body Resonances
Three body resonance Order: q = | k 0 + k 1 + k 2 | Strength of resonance is approximately ∝ Cm 1 m 2 e q 3BRs are weaker than 2BRs ( m 1 m 2 << m 1 ) Theories try to obtain expressions for coefficients C Only planar theories have been developed Tabaré Gallardo Three Body Resonances
Strength and eccentricity 1 0.01 q=0 0.0001 q=1 1e-006 q=2 ∆ρ 1e-008 q=3 1e-010 1e-012 q=4 1e-014 0.01 0.1 eccentricity strength ∝ e q for low e Tabaré Gallardo Three Body Resonances
Three body resonance, planetary case m 0 � = 0 k 0 n 0 + k 1 n 1 + k 2 n 2 ≃ 0 all three bodies feel the resonance Tabaré Gallardo Three Body Resonances
3BR 4 n 0 − 1 n 1 − 2 n 2 , planetary case m 0 � = 0 Tabaré Gallardo Three Body Resonances
3BR 4 n 0 − 1 n 1 − 2 n 2 , restricted case m 0 = 0 Tabaré Gallardo Three Body Resonances
Three body resonance k 0 n 0 + k 1 n 1 + k 2 n 2 ≃ 0 It is not necessary to have a chain of 2BRs: P 0 and P 1 not in two body resonance P 0 and P 2 not in two body resonance P 2 and P 1 not in two body resonance but... Tabaré Gallardo Three Body Resonances
1784: Laplacian resonance They are also in commensurability 3 λ Europa − λ Io − 2 λ Ganymede ≃ 180 ◦ by pairs: 2 n Europa − n Io ≃ 0 2 n Ganymede − n Europa ≃ 0 3 n Europa − n Io − 2 n Ganymede ≃ 0 ⇓ It must be the consequence of some physical mechanism. Tabaré Gallardo Three Body Resonances
Two types of 3BRs Three body resonance as... superposition or chain of 2 two-body resonances n I − 2 n E ∼ 0 n E − 2 n G ∼ 0 adding: n I − n E − 2 n G ∼ 0 ⇒ 3BR order 2 substraction: n I − 3 n E + 2 n G ∼ 0 ⇒ 3BR order 0 pure : 3BR that are NOT due to 2BR + 2BR. asteroids + Jupiter + Saturn Tabaré Gallardo Three Body Resonances
Asteroids: histogram of a + 2BRs 2:1 Jup 3:1 Jup 5:2 Jup log (Strength) 1:2 Mars 4:7 Mars 2 2.2 2.4 2.6 2.8 3 3.2 3.4 a (au) Tabaré Gallardo Three Body Resonances
Asteroids: histogram of a + 3BRs log (Strength) 2 2.2 2.4 1-4J+2S 2-1M Tabaré Gallardo 2.6 1-4J+3S 2-7J+4S a (au) 1-3J+1S Three Body Resonances 2.8 2-7J+5S 2-6J+3S 3 1-3J+2S 3-8J+4S 2-5J+2S 3.2 3-7J+2S 3-8J+5S 3.4
Dynamical evidence from AstDyS 0.4 0.35 0.3 0.25 proper e 0.2 0.15 0.1 0.05 0 3 3.05 3.1 3.15 3.2 3.25 3.3 proper a (au) Tabaré Gallardo Three Body Resonances
Thousands of asteroids in 3BRs with Jupiter and Saturn Icarus 222 (2013) 220–228 Contents lists available at SciVerse ScienceDirect Icarus journal homepage: www.elsevier.com/locate/icarus Massive identification of asteroids in three-body resonances Evgeny A. Smirnov, Ivan I. Shevchenko ⇑ Pulkovo Observatory of the Russian Academy of Sciences, Pulkovskoje Ave. 65, St. Petersburg 196140, Russia Smirnov and Shevchenko (2013) See next talk! Tabaré Gallardo Three Body Resonances
Resonance 1 + 1U − 2N, a weird case 263.55 263.54 mean a (au) 263.53 263.52 360 270 σ (1+1U-2N) 180 90 0 0 100 200 300 400 500 time (Myrs) Tabaré Gallardo Three Body Resonances
3BRs are WEAK and numerous Given two planets P 1 and P 2 , an infinite family of 3BRs is defined: n 0 = − k 1 n 1 − k 2 n 2 k 0 Don’t miss the ”TBR Locator” for Android! Each resonance is defined by ( k 0 , k 1 , k 2 ) The question is: how strong are they? They are weak because the perturbation that drives the resonant motion is factorized by m 1 m 2 . There is a huge number of 3BRs: superposition generates chaotic diffusion . Tabaré Gallardo Three Body Resonances
Multiplet resonances and chaos σ = k 0 λ 0 + k 1 λ 1 + k 2 λ 2 + k 4 ̟ 0 + k 5 Ω 0 Figure 8. Separatrices of four multiplet resonances of the 6 1 − 3 three-body resonance. Nesvorny and Morbidelli (1999) Tabaré Gallardo Three Body Resonances
Chaotic diffusion: growing e Morbidelli and Nesvorny (1999) Tabaré Gallardo Three Body Resonances
Chaotic diffusion in the TNR: growing e Nesvorny and Roig (2001) Tabaré Gallardo Three Body Resonances
Three body resonances as... Chains of two body resonances Galilean satellites (Sinclair 1975, Ferraz-Mello, Malhotra, Showman, Peale, Lainey...) Callegari and Yokoyama (2010): satellites of Saturn Extrasolar systems (Libert and Tsiganis 2011; Martí, Batygin, Morbidelli, Papaloizou, Quillen...) Pure three body resonances Lazzaro et al. (1984): satellites of Uranus Aksnes (1988): zero order asteroidal resonances Nesvorny y Morbidelli (1999): theory Jupiter-Saturn-asteroid Cachucho et al. (2010): diffusion in 5J -2S -2. Quillen (2011): zero order extrasolar systems Gallardo (2014), Gallardo et al. (2016): semianalytic Showalter and Hamilton (2015): Pluto satellites 3 n S − 5 n N + 2 n H ∼ 0 Tabaré Gallardo Three Body Resonances
Disturbing function Disturbing function for resonance k 0 + k 1 + k 2 : � R = k 2 m 1 m 2 P j cos ( σ j ) j σ j = k 0 λ 0 + k 1 λ 1 + k 2 λ 2 + γ j γ j = k 3 ̟ 0 + k 4 ̟ 1 + k 5 ̟ 2 + k 6 Ω 0 + k 7 Ω 1 + k 8 Ω 2 P j is a polynomial function depending on the eccentricities and inclinations which its lowest order term is Ce | k 3 | 0 e | k 4 | 1 e | k 5 | sin ( i 0 ) | k 6 | sin ( i 1 ) | k 7 | sin ( i 2 ) | k 8 | 2 Tabaré Gallardo Three Body Resonances
Theories are complicated... it is necessary to consider several P j cos ( σ j ) with several terms in P j calculation of the C s is not trivial only planar theories exist To avoid the difficulties of the analytical methods we proposed to calculate R numerically. Tabaré Gallardo Three Body Resonances
Semi analytical method Tabaré Gallardo Three Body Resonances
Icarus journal homepage: www.elsevier.com/locate/icarus Atlas of three body mean motion resonances in the Solar System Tabaré Gallardo Departamento de Astronomía, Instituto de Física, Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400 Montevideo, Uruguay Tabaré Gallardo Three Body Resonances
Method Disturbing function is a mean over all possible resonant configurations. The point: the disturbing function R must be calculated with the perturbed positions. We cannot assume unperturbed ellipses for the three orbits. Tabaré Gallardo Three Body Resonances
Method For a given resonance: consider a large sample of configurations verifying the resonant condition ( σ = constant) calculate the mutual perturbations ∆ r 0 , ∆ r 1 , ∆ r 2 calculate the effect ∆ R due to (∆ r 0 , ∆ r 1 , ∆ r 2 ) integrate all ∆ R and obtain ρ ( σ ) repeat for several σ ∈ ( 0 , 360 ) obtaining ρ ( σ ) Tabaré Gallardo Three Body Resonances
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