tabaré gallardo · uranus satellites asteroids with jupiter, mars, earth, venus... trans neptunian...

Resonancias orbitales

Tabaré Gallardowww.fisica.edu.uy/∼gallardo

Facultad de CienciasUniversidad de la República

Uruguay

Curso Dinámica Orbital 2018

Tabaré Gallardo Resonancias orbitales

http://www.fisica.edu.uy/~gallardo/

http://www.astronomia.edu.uy/depto/mece/

Orbital motion


Oscillating planetary orbits


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)


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

k0n0 + k1n1 ' 0


Three-body resonance

k0n0 + k1n1 + k2n2 ' 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

SUN

asteroid

R T Fperturb = (R,T,N)

dadt∝ (R,T)

<dadt>∝ T

Non resonantT = 0⇒ a = constant

ResonantT 6= 0⇒ a = oscillating


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


1772: Lagrange equilibrium points


1906: (588) Achilles by 500 yrs

Sun Jupiter


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.


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


1866: Kirkwood gaps


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.


Resonance locations

k0nast = k1nJup

aast ' (k0

k1)2/3aJup

There is an infinite number of resonances...

which are the relevant ones?


1875: resonant asteroids (153) Hilda 3:2

Sun Jupiter

2nHilda = 3nJup

aHilda = (23

)2/3aJup = 3.97ua


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 theplanet


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

Murray and Dermott in Solar System Dynamics

Chaos:at resonance borders

superposition ofresonances


350.000 asteroids (proper elements)

AstDyS database


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


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


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


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


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


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

SUN

asteroid

planet

relative motion

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

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


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


Resonances of Long Period Comets

Fernandez et al. 2016


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

Fernandez et al. 2016


Three-body resonance

k0n0 + k1n1 + k2n2 ' 0 only the asteroid feels the resonance

SUN asteroid


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


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

Gallardo 2014


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


Galilean satellites


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

Gallardo 2016


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


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/


http://www.astronomia.edu.uy/depto/mece/ResonanciasTGallardo2016.pdf

http://www.astronomia.edu.uy/depto/mece/ResonanciasTGallardo2016.pdf

https://arxiv.org/abs/1803.06245

http://www.fisica.edu.uy/~gallardo/atlas/

tabaré gallardo · uranus satellites asteroids with jupiter, mars, earth, venus... trans neptunian...

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