Variaciones orbitales en la región Trans-Neptunianagallardo/sem/TGallardoSUA2012.pdf · Tabaré...

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


Dispersión orbital en la Región TN



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



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




(Greenberg 1982)




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)




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)




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)




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)




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




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




Ecuación para ω

dωdt

=3Ck

16a7/2(1− e2)2 (3 + 5 cos 2i) + . . .

i ∼ 63,4 −→ dωdt

= 0




Puntos de equilibrio




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




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)




Eris por 200 MA:

http://youtu.be/CA1XPj_DklY




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



TNOs 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)



(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)



(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)



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



Inclinaciones altas



¡Muchas Gracias!


Variaciones orbitales en la región Trans-Neptunianagallardo/sem/TGallardoSUA2012.pdf · Tabaré...

Documents

Transcript of Variaciones orbitales en la región Trans-Neptunianagallardo/sem/TGallardoSUA2012.pdf · Tabaré...