www.fisica.edu.uy/ gallardo Facultad de Ciencias Universidad de la República Uruguay Curso Dinámica Orbital 2018
Orbital motion
Oscillating planetary orbits
Orbital Resonances Commensurability between frequencies associated with orbital motion: mean motion, nodes and pericenters two-body resonances (λ, λ p ) three-body resonances (λ, λ p1, λ p2 ) secular resonances (Ω, ϖ) Kozai-Lidov mechanism (ω)
Some examples Io-Europa-Ganymede Saturn satellites Saturn rings Uranus satellites asteroids with Jupiter, Mars, Earth, Venus... Trans Neptunian Objects with Neptune Pluto - Neptune comets - Jupiter Pluto satellites: Styx, Nix, and Hydra
Saturn rings Lissauer and de Pater, Fundamental Planetary Science
Two-body resonance k 0 n 0 + k 1 n 1 0
Three-body resonance k 0 n 0 + k 1 n 1 + k 2 n 2 0 only the asteroid feels the resonance SUN asteroid
Non resonant asteroid: relative positions Mean perturbation is radial: Sun-Jupiter Sun Jupiter
Resonant asteroid Mean perturbation has a transverse component. Sun Jupiter
from Gauss equations T R F perturb = (R, T, N) SUN Non resonant T = 0 a = constant asteroid da dt (R, T) < da dt > T Resonant T 0 a = oscillating
Dynamical effects: a numerical exercise 0.7 0.6 final time: 1 Myrs orbital states initial 0.5 eccentricity 0.4 0.3 0.2 0.1 0 2.3 2.35 2.4 2.45 2.5 2.55 2.6 a (au)
1772: Lagrange equilibrium points
1906: (588) Achilles by 500 yrs Sun Jupiter
1784: Laplacian resonance 3λ Europa λ Io 2λ Ganymede 180 They are also in commensurability by pairs: 2n Europa n Io 0 3n Europa n Io 2n Ganymede 0 2n Ganymede n Europa 0 It must be the consequence of some physical mechanism.
1846: discovery of Neptune quasi resonance Uranus - Neptune: n Uranus 2n Neptune quasi resonance Saturn - Uranus: n Saturn 3n Uranus quasi resonance: Jupiter - Saturn 2n Jupiter 5n Saturn Why the planets are close to resonance? Hint: planetary migration
1866: Kirkwood gaps
Kirkwood gaps at present log (Strength) 4:7 Mars 1:2 Mars 3:1 Jup 5:2 Jup 2:1 Jup 2 2.2 2.4 2.6 2.8 3 3.2 3.4 a (au) Main belt of asteroids is sculpted by resonances.
Resonance locations k 0 n ast = k 1 n Jup a ast ( k 0 k 1 ) 2/3 a Jup There is an infinite number of resonances... which are the relevant ones?
1875: resonant asteroids (153) Hilda 3:2 2n Hilda = 3n Jup a Hilda = ( 2 3 )2/3 a Jup = 3.97ua Sun Jupiter
Hildas and Trojans
Temporary satellite capture the most probable origin of the irregular satellites
Quasi satellite, resonance 1:1 it is not orbiting around the planet, it is synchronized 1:1 with the planet
Quasi satellite from Wiegert website:
2004 GU9: Earth quasi satellite, resonance 1:1 Sun Earth
1999 ND43: Mars horseshoe, resonance 1:1 Sun Mars
Janus - Epimetheus 1:1
(134340) Pluto in exterior resonance 2:3 Sun Neptune
Widths Chaos: at resonance borders superposition of resonances Murray and Dermott in Solar System Dynamics
350.000 asteroids (proper elements) AstDyS database
Resonant structure (zoom) 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) AstDyS database
Dynamical Maps take set of initial values (a, e) integrate for some 10.000 yrs surface (color) plot of a(a, e) initial e Model: real SS. Initial i = 0 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 1.86 1.861 1.862 1.863 1.864 1.865 1.866 1.867 1.868 1.869 1.87 initial a -3.5-4 -4.5-5 -5.5-6 -6.5-7 -7.5-8 -8.5
Asteroid region initial e Model: real SS 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 1.5 2 2.5 3 3.5 4 initial a 0-1 -2-3 -4-5 -6-7 -8-9
Zoom initial e Model: real SS. 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 1.7 1.72 1.74 1.76 1.78 1.8 1.82 1.84 1.86 1.88 1.9 initial a -1-2 -3-4 -5-6 -7-8 -9
Zoom initial e Model: real SS. Initial i = 0 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 1.86 1.861 1.862 1.863 1.864 1.865 1.866 1.867 1.868 1.869 1.87 initial a -3.5-4 -4.5-5 -5.5-6 -6.5-7 -7.5-8 -8.5
Orbital inclinations of asteroids
Atlas of resonances in the Solar System, low e
Atlas of resonances in the Solar System, high e
Atlas for DIRECT orbits
Atlas for RETROGRADE orbits
Atlas from 0 to 2 au Gallardo 2006
Atlas in the asteroids region Gallardo 2006
Atlas in the trans-neptunian Region Gallardo 2006
Stickiness: ability to capture particles Gallardo et al. 2011
Bizarre worlds...
Coorbital retrograde heliocentric motion relative motion planet asteroid SUN Morais and Namouni 2013
2015 BZ509: discovered in January 2015 a = 5.12 au, e = 0.38, i = 163 Sun Jupiter
Bizarre extreme: fictitious particle in resonant polar orbit λ J - 180 90 0 2.04 2.06 2.08 2.1 2.12 2.14 2.16 2.18 8 q,a (au) 6 4 2 0 2.04 2.06 2.08 2.1 2.12 2.14 2.16 2.18 360 i,ω (degrees) 270 180 90 0 2.04 2.06 2.08 2.1 2.12 2.14 2.16 2.18 time (Myr)
Bizarre extreme: fictitious particle in resonant polar orbit collision with the Sun in the rotating frame 0.5 0 Sun Jupiter -0.5 Z -1-1.5-2 -2.5-1.5-1 -0.5 0 0.5 1 1.5 X
Resonances of Long Period Comets Fernandez et al. 2016
Resonances of Long Period Comets 20000 18000 1:6 1:9 16000 1:3 1:7 14000 1:4 1:10 1:11 12000 1:8 orbital states 10000 8000 6000 4000 1:1 2:3 1:2 2:5 2:7 2:9 1:5 2:11 2:13 2:15 2:17 1:12 1:13 1:14 1:15 1:16 1:17 1:18 1:20 2000 1:19 1:21 1:22 1:24 0 0 10 20 30 40 50 <a> (au) Fernandez et al. 2016
Three-body resonance k 0 n 0 + k 1 n 1 + k 2 n 2 0 only the asteroid feels the resonance SUN asteroid
Atlas of TBRs: global view (for e = 0.15) 1 0.1 Strength ρ 0.01 0.001 0.0001 1e-005 0.1 1 10 100 1000 a (au) Gallardo 2014
Effects on the distribution of asteroids log (Strength) 1-4J+2S 2-1M 1-4J+3S 2-7J+4S 1-3J+1S 2-7J+5S 2-6J+3S 1-3J+2S 3-8J+4S 2-5J+2S 3-7J+2S 3-8J+5S 2 2.2 2.4 2.6 2.8 3 3.2 3.4 a (au) Gallardo 2014
Density of resonances versus density of asteroids 250 3BRs 2BRs asteroids density of 3-body and 2-body resonances 200 150 100 50 0 1 1.5 2 2.5 3 3.5 4 4.5 a (au)
Galilean satellites
Europa: dynamical map 0.1-4 0.08 a -4.5-5 initial e 0.06 0.04 0.02-5.5-6 -6.5-7 -7.5 E 0 0.00435 0.0044 0.00445 0.0045 0.00455 initial a (au) -8-8.5 Gallardo 2016
Chronology 1772: Lagrange equilibrium points 1799: Lagrange planetary equations, stability of the SS 1784: Laplacian resonance 3λ E λ I 2λ G 180 Laplace quasi resonances: great inequality 2λ Jup 5λ Sat 1846: Neptune discovered 1866 Kirkwood gaps 1875: first resonant asteroid (153) Hilda 3:2 1882: secular resonances (Tisserand, ν 6 ) 1906: first Trojan asteroid (588) Achilles 1930: Pluto and exterior resonance 2:3 Neptune-Pluto 1962: Lidov-Kozai mechanism 1993: resonant TNOs (2:3 plutino) planetary systems in 2BRs and 3BRs minor bodies in 3BRs high inclination resonant orbits retrograde resonances
Bibliografía Efectos dinámicos de las resonancias orbitales en el Sistema Solar. Gallardo 2016, BAAA 58, 291. Resonances in the asteroid and TNO belts: a brief review. Gallardo 2018, Planetary and Space Science 157, 96. http://www.fisica.edu.uy/~gallardo/atlas/