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Resonancias orbitales
Tabaré Gallardowww.fisica.edu.uy/∼gallardo
Facultad de CienciasUniversidad de la República
Uruguay
Curso Dinámica Orbital 2018
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Orbital motion
Tabaré Gallardo Resonancias orbitales
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Oscillating planetary orbits
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Orbital Resonances
Commensurability between frequencies associated with orbitalmotion: mean motion, nodes and pericenters
two-body resonances (λ, λp)
three-body resonances(λ, λp1, λp2)
secular resonances (Ω, $)
Kozai-Lidov mechanism (ω)
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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
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Saturn rings
Lissauer and de Pater, Fundamental Planetary Science
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Two-body resonance
k0n0 + k1n1 ' 0
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Three-body resonance
k0n0 + k1n1 + k2n2 ' 0 only the asteroid feels the resonance
SUN asteroid
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Non resonant asteroid: relative positions
Mean perturbation is radial: Sun-Jupiter
Sun Jupiter
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Resonant asteroid
Mean perturbation has a transverse component.
Sun Jupiter
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from Gauss equations
SUN
asteroid
R T Fperturb = (R,T,N)
dadt∝ (R,T)
<dadt>∝ T
Non resonantT = 0⇒ a = constant
ResonantT 6= 0⇒ a = oscillating
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Dynamical effects: a numerical exercise
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
2.3 2.35 2.4 2.45 2.5 2.55 2.6
ecc
entr
icit
y
a (au)
final time: 1 Myrs
orbital statesinitial
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1772: Lagrange equilibrium points
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1906: (588) Achilles by 500 yrs
Sun Jupiter
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1784: Laplacian resonance
3λEuropa−λIo−2λGanymede ' 180
3nEuropa − nIo − 2nGanymede ' 0
They are also in commensurabilityby pairs:
2nEuropa − nIo ' 0
2nGanymede − nEuropa ' 0
⇓
It must be the consequence of some physical mechanism.
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1846: discovery of Neptune
quasi resonance Uranus - Neptune:
nUranus ∼ 2nNeptune
quasi resonance Saturn - Uranus:
nSaturn ∼ 3nUranus
quasi resonance: Jupiter - Saturn
2nJupiter ∼ 5nSaturn
Why the planets are close to resonance?Hint: planetary migration
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1866: Kirkwood gaps
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Kirkwood gaps at present
2 2.2 2.4 2.6 2.8 3 3.2 3.4
log (
Str
ength
)
a (au)
1:2
Mars
3:1
Jup
2:1
Jup
4:7
Mars
5:2
Jup
Main belt of asteroids is sculpted by resonances.
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Resonance locations
k0nast = k1nJup
aast ' (k0
k1)2/3aJup
There is an infinite number of resonances...
which are the relevant ones?
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1875: resonant asteroids (153) Hilda 3:2
Sun Jupiter
2nHilda = 3nJup
aHilda = (23
)2/3aJup = 3.97ua
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Hildas and Trojans
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Temporary satellite capture
the most probable origin of the irregular satellites
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Quasi satellite, resonance 1:1
it is not orbiting around the planet, it is synchronized 1:1 with theplanet
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Quasi satellite
from Wiegert website:
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2004 GU9: Earth quasi satellite, resonance 1:1
Sun Earth
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1999 ND43: Mars horseshoe, resonance 1:1
Sun Mars
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Janus - Epimetheus 1:1
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(134340) Pluto in exterior resonance 2:3
Sun Neptune
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Widths
Murray and Dermott in Solar System Dynamics
Chaos:at resonance borders
superposition ofresonances
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350.000 asteroids (proper elements)
AstDyS database
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Resonant structure (zoom)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
3 3.05 3.1 3.15 3.2 3.25 3.3
prop
er e
proper a (au)
AstDyS database
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Dynamical Maps
take set of initial values (a, e)
integrate for some 10.000 yrs
surface (color) plot of ∆a(a, e)
Model: real SS. Initial i = 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
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
initi
al e
-8.5
-8
-7.5
-7
-6.5
-6
-5.5
-5
-4.5
-4
-3.5
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Asteroid region
Model: real SS
1.5 2 2.5 3 3.5 4
initial a
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
initi
al e
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
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Zoom
Model: real SS.
1.7 1.72 1.74 1.76 1.78 1.8 1.82 1.84 1.86 1.88 1.9
initial a
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
initi
al e
-9
-8
-7
-6
-5
-4
-3
-2
-1
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Zoom
Model: real SS. Initial i = 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
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
initi
al e
-8.5
-8
-7.5
-7
-6.5
-6
-5.5
-5
-4.5
-4
-3.5
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Orbital inclinations of asteroids
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Atlas of resonances in the Solar System, low e
Tabaré Gallardo Resonancias orbitales
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Atlas of resonances in the Solar System, high e
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Atlas for DIRECT orbits
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Atlas for RETROGRADE orbits
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Atlas from 0 to 2 au
Gallardo 2006
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Atlas in the asteroids region
Gallardo 2006
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Atlas in the trans-Neptunian Region
Gallardo 2006
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Stickiness: ability to capture particles
Gallardo et al. 2011
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Bizarre worlds...
Tabaré Gallardo Resonancias orbitales
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Coorbital retrograde
heliocentric motion
SUN
asteroid
planet
relative motion
Morais and Namouni 2013
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2015 BZ509: discovered in January 2015
a = 5.12 au, e = 0.38, i = 163
Sun Jupiter
Tabaré Gallardo Resonancias orbitales
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Bizarre extreme: fictitious particle in resonant polar orbit
0
2
4
6
8
2.04 2.06 2.08 2.1 2.12 2.14 2.16 2.18
q,a
(au
)
0
90
180
270
360
2.04 2.06 2.08 2.1 2.12 2.14 2.16 2.18
i,ω (
degr
ees)
time (Myr)
0
90
180
270
360
2.04 2.06 2.08 2.1 2.12 2.14 2.16 2.18λ J
- λ
Tabaré Gallardo Resonancias orbitales
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Bizarre extreme: fictitious particle in resonant polar orbit
collision with the Sun in the rotating frame
-2.5
-2
-1.5
-1
-0.5
0
0.5
-1.5 -1 -0.5 0 0.5 1 1.5
Z
X
Sun Jupiter
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Resonances of Long Period Comets
Fernandez et al. 2016
Tabaré Gallardo Resonancias orbitales
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Resonances of Long Period Comets
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 10 20 30 40 50
orbi
tal s
tate
s
<a> (au)
1:1
1:6
1:7
1:8
1:2
1:9
1:10
1:11
1:5
1:3
1:4
2:5
2:11
2:7
2:3
2:9
1:12
1:13
1:14
1:15
1:16
1:17
1:18
1:19
1:20
1:21
1:22
1:24
2:13
2:15 2:
17
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Three-body resonance
k0n0 + k1n1 + k2n2 ' 0 only the asteroid feels the resonance
SUN asteroid
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Atlas of TBRs: global view (for e = 0.15)
1e-005
0.0001
0.001
0.01
0.1
1
0.1 1 10 100 1000
Str
en
gth
∆ρ
a (au)
Gallardo 2014
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Effects on the distribution of asteroids
2 2.2 2.4 2.6 2.8 3 3.2 3.4
log (
Str
ength
)
a (au)
1-4
J+
2S
1-4
J+
3S
2-7
J+
4S
1-3
J+
1S
2-7
J+
5S
2-6
J+
3S
1-3
J+
2S
3-8
J+
4S
2-5
J+
2S
3-7
J+
2S
3-8
J+
5S
2-1
M
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Density of resonances versus density of asteroids
0
50
100
150
200
250
1 1.5 2 2.5 3 3.5 4 4.5
dens
ity o
f 3-b
ody
and
2-bo
dy r
eson
ance
s
a (au)
3BRs2BRs
asteroids
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Galilean satellites
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Europa: dynamical map
0.00435 0.0044 0.00445 0.0045 0.00455
initial a (au)
0
0.02
0.04
0.06
0.08
0.1
initi
al e
-8.5
-8
-7.5
-7
-6.5
-6
-5.5
-5
-4.5
-4
E
∆a
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Chronology
1772: Lagrange equilibrium points1799: Lagrange planetary equations, stability of the SS1784: Laplacian resonance 3λE − λI − 2λG ' 180
Laplace quasi resonances: great inequality 2λJup − 5λSat
1846: Neptune discovered1866 Kirkwood gaps1875: first resonant asteroid (153) Hilda 3:21882: secular resonances (Tisserand, ν6)1906: first Trojan asteroid (588) Achilles1930: Pluto and exterior resonance 2:3 Neptune-Pluto1962: Lidov-Kozai mechanism1993: resonant TNOs (2:3 plutino)planetary systems in 2BRs and 3BRsminor bodies in 3BRshigh inclination resonant orbitsretrograde resonances
Tabaré Gallardo Resonancias orbitales
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Bibliografía
Efectos dinámicos de las resonancias orbitales en el SistemaSolar. 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/
Tabaré Gallardo Resonancias orbitales