Dinámica de KozaiTNOs Resonantes
Variaciones orbitales en la región Trans-Neptuniana
Tabaré Gallardowww.fisica.edu.uy/∼gallardo
Departamento de AstronomíaIFFC (UdelaR)
Reunión Anual SUA, 6 octubre 2012
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Dispersión orbital en la Región TN
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Survey of Kozai dynamics beyond Neptune
Tabaré Gallardo ⇑, Gastón Hugo, Pablo PaisDepartamento de Astronomía, Instituto de Física, Facultad de Ciencias, Iguá 4225, 11400 Montevideo, Uruguay
a r t i c l e i n f o
Article history:Received 5 April 2012Revised 14 May 2012Accepted 18 May 2012Available online 30 May 2012
Keywords:Resonances, OrbitalTrans-neptunian objectsKuiper belt
a b s t r a c t
We study the Kozai dynamics affecting the orbital evolution of trans-neptunian objects being captured ornot in MMR with Neptune. We provide energy level maps of the type (x,q) describing the possible orbitalpaths from Neptune up to semimajor axis of hundreds of AU. The dynamics for non-resonant TNOs withperihelion distances, q, outside Neptune’s orbit, aN, is quite different from the dynamics of TNOs withq < aN, already studied in previous works. While for the last case there are stable equilibrium points atx = 0, 90,180 and 270 in a wide range of orbital inclinations, for the former case it appears a familyof stable equilibrium points only at a specific value of the orbital inclination, i 62, that we call criticalinclination. We show this family of equilibrium points is generated by a mechanism analogue to whichdrives the dynamics of an artificial satellite perturbed by an oblate planet. The planetary system also gen-erates an oscillation in the longitude of the perihelion of the TNOs with i 46, being Eris a paradigmaticcase. We discuss how the resonant condition with Neptune modify the energy level curves and the loca-tion of equilibrium points. The asymmetric librations of resonances of the type 1:N generate a distortionin the energy level curves and in the resulting location of the equilibrium points in the phase space (x,q).We study the effect on the Kozai dynamics due to the diffusion process in a that occurs in the ScatteredDisk. We show that a minimum orbital inclination is required to allow substantial variations in periheliondistances once the object is captured in MMR and that minimum inclination is greater for greater semi-major axis.
2012 Elsevier Inc. All rights reserved.
1. Introduction
One of the open problems of the trans-neptunian region (TNR)is to find an explanation for the wide variety of orbits that the dis-covered trans-neptunian objects (TNOs) exhibit. In particular,those objects with perihelion outside Neptune’s orbit and withhigh orbital inclinations and eccentricities that cannot be ex-plained by the diffusive process the Scattered Disk Objects (SDOs)experience, nor by the past dynamical history of the planetarysystem (Tsiganis et al., 2005). It is known (Duncan et al., 1995;Gladman et al., 2002; Fernández et al., 2004) that TNOs with peri-helion q < 36 AU are inside a chaotic region that generates a diffu-sion in semimajor axis enhancing aphelion distances. But thediffusion itself cannot decouple the perihelion from Neptune’s re-gion, then it is necessary to invoke another mechanism to explainthe existence of high perihelion SDOs, also known as DetachedObjects. The origin of these problematic orbits have given rise totheories involving passing stars (Ida et al., 2000), scattered planets(Gladman and Chan, 2006), tides from the star cluster where theSun was formed (Brasser et al., 2006), a stellar companion (Gomeset al., 2006) and others. But, in order to avoid extra hypothesis it is
necessary an abroad panorama of the dynamics generated by theplanetary system itself in the TNR. Secular resonances, mean mo-tion resonances (MMRs) and the Kozai resonance (KR) are dynam-ical mechanisms operating in the TNR and that could explain someeccentric orbits decoupled from encounters with Neptune (Ferná-ndez et al., 2004; Gomes et al., 2005; Gomes, 2011). The presentwork contributes with new results following that line of thinking.
Secular resonances do not affect the TNOs located beyonda 42 (Knezevic et al., 1991) and MMRs with Neptune do not gen-erate themselves a substantial variation in the orbital elements, asfor example the MMRs with Jupiter do (Gallardo, 2006b). The orbi-tal variations observed when a particle is captured in MMR withNeptune are due to a secular dynamics inside the MMR, like Kozaidynamics. The Kozai dynamics, or more properly Kozai–Lidovdynamics, has its roots in a study by Lidov (1962) of the secularevolution of an artificial Earth’s satellite perturbed by the Moon.Applying these ideas to the asteroid belt, Kozai (1962) developedan analytical approximation for the mean secular perturbing func-tion due to Jupiter assumed in circular orbit, valid even for highinclination and high eccentricity asteroidal orbits. The conserva-tion of the semimajor axis, as is usual in secular theories, but alsothe conservation of the parameter H ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 e2p
cos i, and the exis-tence of an energy integral, K(x,e), allowed the calculation of theenergy level curves for this model showing large oscillations of e
0019-1035/$ - see front matter 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.icarus.2012.05.025
⇑ Corresponding author.E-mail address: [email protected] (T. Gallardo).
Icarus 220 (2012) 392–403
Contents lists available at SciVerse ScienceDirect
Icarus
journal homepage: www.elsevier .com/ locate/ icarus
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Mapas de KozaiNuevo punto de equilibrioModelo Sol + anillosEris
Dinámica de Kozai(órbitas planetarias circulares y coplanares)
a = constante
H =√
1− e2 cos i = constante
K(a,H, ω, q) = constante
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Mapas de KozaiNuevo punto de equilibrioModelo Sol + anillosEris
(Greenberg 1982)
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Mapas de KozaiNuevo punto de equilibrioModelo Sol + anillosEris
K = constante
(a) H=0.1, a=50 AU
84.2
69.1
i
0 30 60 90 120 150 180
ω
5
10
15
20
25
30
35
40
45
q (AU)
(e) H=0.1, a=100 AU
83.7
44.9
i
0 30 60 90 120 150 180
ω
10
20
30
40
50
60
q (AU)
(b) H=0.3, a=50 AU
72.5
28.4
i
0 30 60 90 120 150 180
ω
5
10
15
20
25
30
35
40
45
q (AU)
(f) H=0.3, a=100 AU
70.9
16.1
i
0 30 60 90 120 150 180
ω
10
20
30
40
50
60
q (AU)
(c) H=0.5, a=50 AU
60
11.5
i
0 30 60 90 120 150 180
ω
10
15
20
25
30
35
40
45
q (AU)
(g) H=0.5, a=100 AU
56.9
18.3
i
0 30 60 90 120 150 180
ω
15
20
25
30
35
40
45
50
55
60
q (AU)
(d) H=0.7, a=50 AU
45.6
11.4
i
0 30 60 90 120 150 180
ω
15
20
25
30
35
40
45
q (AU)
(h) H=0.7, a=100 AU
40.2
11.4i
0 30 60 90 120 150 180
ω
30
35
40
45
50
55
60
q (AU)
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Mapas de KozaiNuevo punto de equilibrioModelo Sol + anillosEris
K = constante
(a) H=0.1, a=50 AU
84.2
69.1
i
0 30 60 90 120 150 180
ω
5
10
15
20
25
30
35
40
45
q (AU)
(e) H=0.1, a=100 AU
83.7
44.9
i
0 30 60 90 120 150 180
ω
10
20
30
40
50
60
q (AU)
(b) H=0.3, a=50 AU
72.5
28.4
i
0 30 60 90 120 150 180
ω
5
10
15
20
25
30
35
40
45
q (AU)
(f) H=0.3, a=100 AU
70.9
16.1
i
0 30 60 90 120 150 180
ω
10
20
30
40
50
60
q (AU)
(c) H=0.5, a=50 AU
60
11.5
i
0 30 60 90 120 150 180
ω
10
15
20
25
30
35
40
45
q (AU)
(g) H=0.5, a=100 AU
56.9
18.3
i
0 30 60 90 120 150 180
ω
15
20
25
30
35
40
45
50
55
60
q (AU)
(d) H=0.7, a=50 AU
45.6
11.4
i
0 30 60 90 120 150 180
ω
15
20
25
30
35
40
45
q (AU)
(h) H=0.7, a=100 AU
40.2
11.4
i
0 30 60 90 120 150 180
ω
30
35
40
45
50
55
60
q (AU)
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Mapas de KozaiNuevo punto de equilibrioModelo Sol + anillosEris
Nuevo punto de equilibrio en ω = 90
(a) H=0.1, a=200 AU
82
44.9
i
0 30 60 90 120 150 180
ω
10
20
30
40
50
60
q (AU)
(e) H=0.1, a=300 AU
79.6
57
i
0 30 60 90 120 150 180
ω
5
10
15
20
25
30
35
40
45
50
q (AU)
(b) H=0.3, a=200 AU
65.2
16.1
i
0 30 60 90 120 150 180
ω
10
20
30
40
50
60
q (AU)
(f) H=0.3, a=300 AU
65.2
16.1
i
0 30 60 90 120 150 180
ω
20
30
40
50
60
70
80
90
q (AU)
(c) H=0.5, a=200 AU
45.6
4.8
i
0 30 60 90 120 150 180
ω
30
35
40
45
50
55
60
q (AU)
(g) H=0.5, a=300 AU
51.3
7.8
i
0 30 60 90 120 150 180
ω
50
60
70
80
90
100
110
120
q (AU)
(d) H=0.7, a=200 AU
29
6.3
i
0 30 60 90 120 150 180
ω
60
65
70
75
80
q (AU)
(h) H=0.7, a=300 AU
42.1
11.4i
0 30 60 90 120 150 180
ω
100
120
140
160
180
200
q (AU)
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Mapas de KozaiNuevo punto de equilibrioModelo Sol + anillosEris
Metamorfosis de los puntos de equilibrio
H=0.49, a=36 AU
60.7
59.8
i
0 30 60 90 120 150 180
ω
28
29
30
31
32
33
34
35
36
q (AU)
H=0.48, a=41 AU
61.3
60.1
i
0 30 60 90 120 150 180
ω
30
32
34
36
38
40
q (AU)
H=0.46, a=50 AU
62.0
60.2
i
0 30 60 90 120 150 180
ω
31
32
33
34
35
36
37
38
39
40
q (AU)
H=0.46, a=52 AU
61.8
60.9
i
0 30 60 90 120 150 180
ω
35
36
37
38
39
40
q (AU)
H=0.46, a=54 AU
62.4
59.8
i
0 30 60 90 120 150 180
ω
32
34
36
38
40
42
44
46
48
q (AU)
H=0.46, a=80 AU
62.6
61.0
i
0 30 60 90 120 150 180
ω
55
60
65
70
75
80
q (AU)
H=0.36, a=100 AU
65.4
34.3
i
0 30 60 90 120 150 180
ω
10
15
20
25
30
35
40
45
50
q (AU)
H=0.2, a=500 AU
66.9
54.4
i
0 30 60 90 120 150 180
ω
30
35
40
45
50
55
60
65
70
q (AU)
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Mapas de KozaiNuevo punto de equilibrioModelo Sol + anillosEris
Modelo analítico: Sol + anillos
Sustituímos Sol + planetas −→ potencial de esfera + anillos
Tenemos un modelo analítico que depende de C (momento deinercia en la dirección z)
z
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Mapas de KozaiNuevo punto de equilibrioModelo Sol + anillosEris
Mapas de Kozai para nuestro modelo analítico
97.4
81.3
69.2
60.0
0 30 60 90 120 150 180
60
62
64
66
Ω
iHde
gree
sL
qHA
UL
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Mapas de KozaiNuevo punto de equilibrioModelo Sol + anillosEris
Ecuación para ω
dωdt
=3Ck
16a7/2(1− e2)2 (3 + 5 cos 2i) + . . .
i ∼ 63,4 −→ dωdt
= 0
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Mapas de KozaiNuevo punto de equilibrioModelo Sol + anillosEris
Puntos de equilibrio
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Mapas de KozaiNuevo punto de equilibrioModelo Sol + anillosEris
30
35
40
45
50
55
60
65
70
40 60 80 100 120 140
inclination
a (UA)
dω/dt=0
dϖ/dt=0
2005 NU125
2004 XR190
Eris2004 DF77
2006 QR180
2007 TC434
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Mapas de KozaiNuevo punto de equilibrioModelo Sol + anillosEris
90
180
270
360
Ω136199 Eris
90
180
270
ω
43
44
i
70
75
80
85
0 50 100 150 200
ϖ
time (Myr)
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Mapas de KozaiNuevo punto de equilibrioModelo Sol + anillosEris
Eris por 200 MA:
http://youtu.be/CA1XPj_DklY
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Mapas de KozaiNuevo punto de equilibrioModelo Sol + anillosEris
Algunas conclusiones
Para los objetos ubicados más allá de Neptuno el SS esdinámicamente ∼ Sol achatado
Existe una familia de puntos (en i ∼ 63) con ω = 0 que generangrandes oscilaciones en q
No se han descubierto aún objetos reales en esa configuración
Existe una familia de puntos (en i ∼ 46) con $ = 0
Eris es el único objeto conocido en esa configuración
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
TNOs Resonantes
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
H=0.5, a=50.95 AU, 5:11
32.8
45.1
60
i
0 20 40 60 80 100 120 140 160 180
ω
5
10
15
20
25
30
35
40
45
50
q (AU)
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
(a) res 5:11, H=0.1, a=50.95 AU
84.3
68.9
i
0 30 60 90 120 150 180
ω
0
5
10
15
20
25
30
35
40
45
50
q (AU)
(e) res 1:6, H=0.1, a=99.45 AU
84.1
76.8
i
0 30 60 90 120 150 180
ω
0
10
20
30
40
50
60
70
80
q (AU)
(b) res 5:11, H=0.3, a=50.95 AU
72.5
39.4
59.7
65.7
i
0 30 60 90 120 150 180
ω
0
5
10
15
20
25
30
35
40
45
50
q (AU)
(f) res 1:6, H=0.3, a=99.45 AU
72.2
62.6
23.8
i
0 30 60 90 120 150 180
ω
0
10
20
30
40
50
60
70
80
q (AU)
(c) res 5:11, H=0.5, a=50.95 AU
32.8
45.1
60
i 0 30 60 90 120 150 180
ω
5
10
15
20
25
30
35
40
45
50
q (AU)
(g) res 1:6, H=0.5, a=99.45 AU
59.3
33.8
57
i
0 30 60 90 120 150 180
ω
10
20
30
40
50
60
70
80
q (AU)
(d) res 5:11, H=0.7, a=50.95 AU
8.9
28.2
45.6
i
0 30 60 90 120 150 180
ω
15
20
25
30
35
40
45
50
q (AU)
(h) res 1:6, H=0.7, a=99.45 AU
44.5
15.2
29.4
i
0 30 60 90 120 150 180
ω
20
30
40
50
60
70
80q (AU)
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
(a) res 5:11, H=0.1, a=50.95 AU
84.3
68.9
i
0 30 60 90 120 150 180
ω
0
5
10
15
20
25
30
35
40
45
50
q (AU)
(e) res 1:6, H=0.1, a=99.45 AU
84.1
76.8
i
0 30 60 90 120 150 180
ω
0
10
20
30
40
50
60
70
80
q (AU)
(b) res 5:11, H=0.3, a=50.95 AU
72.5
39.4
59.7
65.7
i 0 30 60 90 120 150 180
ω
0
5
10
15
20
25
30
35
40
45
50
q (AU)
(f) res 1:6, H=0.3, a=99.45 AU
72.2
62.6
23.8
i
0 30 60 90 120 150 180
ω
0
10
20
30
40
50
60
70
80
q (AU)
(c) res 5:11, H=0.5, a=50.95 AU
32.8
45.1
60
i
0 30 60 90 120 150 180
ω
5
10
15
20
25
30
35
40
45
50
q (AU)
(g) res 1:6, H=0.5, a=99.45 AU
59.3
33.8
57
i
0 30 60 90 120 150 180
ω
10
20
30
40
50
60
70
80
q (AU)
(d) res 5:11, H=0.7, a=50.95 AU
8.9
28.2
45.6
i
0 30 60 90 120 150 180
ω
15
20
25
30
35
40
45
50
q (AU)
(h) res 1:6, H=0.7, a=99.45 AU
44.5
15.2
29.4
i
0 30 60 90 120 150 180
ω
20
30
40
50
60
70
80
q (AU)
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Más conclusiones
Las resonancias generan grandes cambios en q
Las resonancias tipo 1:N generan curvas de nivel asimétricas
Se requiere una inclinación inicial mínima im para generargrandes cambios en q
im crece con a
Si i > im se generan grandes cambios en q mediante elmecanismo de Kozai
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
Inclinaciones altas
Tabaré Gallardo Variaciones orbitales
Dinámica de KozaiTNOs Resonantes
¡Muchas Gracias!
Tabaré Gallardo Variaciones orbitales
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