Study the dynamics of KBOs Study the dynamics of KBOs --- using restricted three-body --- using restricted three-body modelmodel

Yeh, Lun-WenYeh, Lun-Wen2007.6.262007.6.26

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Introduction and motivationIntroduction and motivation The restricted three body modelThe restricted three body model Some resultsSome results Future worksFuture works

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Introduction and Introduction and motivationmotivation

The restricted three body modelThe restricted three body model Some resultsSome results Future worksFuture works

Up to the present, over 800 Kuiper belt objects Up to the present, over 800 Kuiper belt objects (r (r ≥ several tenth-km≥ several tenth-km) have been discovered. ) have been discovered.

Classification:Classification: Resonance KBOsResonance KBOs Classical KBOs (Non Res., q>aClassical KBOs (Non Res., q>aNN, e< 0.2), e< 0.2) Scattered KBOs (Non Res., Non Cla., q > aScattered KBOs (Non Res., Non Cla., q > aNN))

(Eugene Chiang, Yoram Lithwick, and Ruth Murray-Clay, 2007,Protostars and Planets V, p895)

Eugene Chiang, Yoram Lithwick, and Ruth Murray-Clay, 2007,Protostars and Planets V, p895

One most popular model for explaining tOne most popular model for explaining the spatial distribution of KBOs is planet he spatial distribution of KBOs is planet migration model. migration model. (Malhotra 1993,1995; Hahn & (Malhotra 1993,1995; Hahn & Malhotra1999, 2005; Gomes 2003, 2004; Levison & MorMalhotra1999, 2005; Gomes 2003, 2004; Levison & Morbidelli 2003; Tsiganis et al., 2005)bidelli 2003; Tsiganis et al., 2005)

Hahn & Malhotra 1999, AJ 117, 3041

In their models: planet-planet, planet-In their models: planet-planet, planet-planetesimal, planetesimal, planetesimal-planetesimal..

Reduce computational expense and Reduce computational expense and avoid avoid

self-stirring.self-stirring. 10M10MEE-100M-100MEE, 1000-10000 planetesimals., 1000-10000 planetesimals.

0.1M0.1MEE-0.001M-0.001MEE for each planetesimal. for each planetesimal.

Is the gravitation between small bodies Is the gravitation between small bodies important?important?

30-50 AU; 0.1M30-50 AU; 0.1MEE; 200km ; 200km l=0.65AU l=0.65AU GmGm22/l/l2 2 ≈≈ ma ma ≈≈mm l/( l/(∆t)∆t)2 2 ∆t ∆t ≈≈ 1.6*10 1.6*104 4 yryr ∆ ∆t t ≈≈ l/v l/v ≈≈ l/((e l/((e22+i+i22))0.50.5vvKK) ) ∆t ∆t ≈≈ 6.5(0.1/e)(a/40A 6.5(0.1/e)(a/40A

U) yr.U) yr.

Scattering and collision:Scattering and collision: ffcollisioncollision / f / fscatteringscattering ~ (r/r ~ (r/rHH))22 ≈≈ 10 10-8-8 for Nepunte. for Nepunte. 1010-6-6 for Jupiter. for Jupiter.

Main purpose of my work:Main purpose of my work: Study the influence of gravity of small bodies in Study the influence of gravity of small bodies in

the planet migration scenario.the planet migration scenario.

Beside above I can study:Beside above I can study: Resonance. Resonance.

Chaos.Chaos.

The method:The method: Restricted three-body model + N small bodies.Restricted three-body model + N small bodies.

Restricted three-body model

Restricted three-body model + N small bodies

Step by step…...Step by step…...

Planar circular restricted three-body Planar circular restricted three-body model.model.

Planar circular restricted three-body Planar circular restricted three-body model + N small bodies.model + N small bodies.

3D circular restricted three-body 3D circular restricted three-body model.model.

3D circular restricted three-body 3D circular restricted three-body model + N small bodies. model + N small bodies.

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Introduction and motivationIntroduction and motivation The restricted three body The restricted three body

modelmodel Some resultsSome results Future worksFuture works

Planar circular

η

ξ

y

x

nt

1 21 23 3

1 2

1 21 23 3

1 2

1 1 2 2 1 2

1 2

, , 1

1

( , ) 1

r r

r r

Gm Gm

n

d

(ξ,η) (x,y)

22 2 1 2

1 2

2

2

( )2

Ux ny

xU

y nxy

nU x y

r r

2 2 22 ( ) 2JC U x y U v

22 2 1 2

11 2

2

2

( )2

ii i

i

ii i

i

Nj

i i ij ijj i

Ux ny

x

Uy nx

y

nU x y

r r r

Jacobi constant

2 2 22 ( ) 2Ji i i i i iC U x y U v

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Introduction and motivationIntroduction and motivation The restricted three body modelThe restricted three body model Some resultsSome results Future worksFuture works

2

2

xy

yx

x

y

dv Unv

dt xdv U

nvdt y

dxv

dtdy

vdt

Use 4th-order Runge-Kutta method to solve 4 first order differential equations.

a: semi-major axis

e: eccentricity

θ: true longitude

ω : longitude of pericentre

f: true anomaly

ω

fθ

(vx, vy, x, y) (a, e, θ, ω)

Sun-Jupiter system + one small body Sun-Jupiter system + one small body μμ22=0.001,=0.001,μμ11=1-=1-μμ22

xx00=0.55, y=0.55, y00=0.0, v=0.0, vx00=0.0, C=0.0, CJJ=3.07=3.07

Example (C. D. Murray, Solar system dynamics)

Poincare surface of section: y=0, vy>0.

Sun-Jupiter system + one small body Sun-Jupiter system + one small body μμ22=0.001,=0.001,μμ11=1-=1-μμ22

xx00==0.560.56, y, y00=0.0, v=0.0, vx00=0.0, C=0.0, CJJ=3.07=3.07

Poincare surface of section: y=0, vy>0.

OutlineOutline

Introduction and motivationIntroduction and motivation The restricted three body modelThe restricted three body model Some resultsSome results Future worksFuture works

In the page In the page Read more about chaos and Read more about chaos and

resonance.resonance.

Step by Step……

to be continued….

THANKS……