1. Study the dynamics of KBOs --- using restricted three-body model Yeh, Lun-Wen 2007.6.26

2. Outline Introduction and motivation Introduction and motivation The restricted three body model The restricted three body model Some results Some results Future works Future works

3. Outline Introduction and motivation Introduction and motivation The restricted three body model The restricted three body model Some results Some results Future works Future works

4. Up to the present, over 800 Kuiper belt objects (r ≥ several tenth-km) have been discovered. Up to the present, over 800 Kuiper belt objects (r ≥ several tenth-km) have been discovered. Classification: Classification: Resonance KBOs Resonance KBOs Classical KBOs (Non Res., q>a N, e a N, e< 0.2) Scattered KBOs (Non Res., Non Cla., q > a N ) Scattered KBOs (Non Res., Non Cla., q > a N ) (Eugene Chiang, Yoram Lithwick, and Ruth Murray-Clay, 2007, Protostars and Planets V, p895)

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

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

7. Hahn & Malhotra 1999, AJ 117, 3041

8. In their models: planet-planet, planet- planetesimal,. In their models: planet-planet, planet- planetesimal, planetesimal-planetesimal. Reduce computational expense and avoid Reduce computational expense and avoid self-stirring. self-stirring. 10M E -100M E, 1000-10000 planetesimals. 10M E -100M E, 1000-10000 planetesimals.  0.1M E -0.001M E for each planetesimal.  0.1M E -0.001M E for each planetesimal.

9. Is the gravitation between small bodies important? 30-50 AU; 0.1M E ; 200km  l=0.65AU 30-50 AU; 0.1M E ; 200km  l=0.65AU Gm 2 /l 2 ≈ ma ≈ m l/(∆t) 2  ∆t ≈ 1.6*10 4 yr Gm 2 /l 2 ≈ ma ≈ m l/(∆t) 2  ∆t ≈ 1.6*10 4 yr ∆t ≈ l/v ≈ l/((e 2 +i 2 ) 0.5 v K )  ∆t ≈ 6.5(0.1/e)(a/40AU) yr. ∆t ≈ l/v ≈ l/((e 2 +i 2 ) 0.5 v K )  ∆t ≈ 6.5(0.1/e)(a/40AU) yr. Scattering and collision: Scattering and collision: f collision / f scattering ~ (r/r H ) 2 ≈ 10 -8 for Nepunte. f collision / f scattering ~ (r/r H ) 2 ≈ 10 -8 for Nepunte. 10 -6 for Jupiter. 10 -6 for Jupiter.

10. Main purpose of my work: Main purpose of my work: Study the influence of gravity of small bodies in the planet migration scenario. Study the influence of gravity of small bodies in 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.

11. Restricted three-body model Restricted three-body model + N small bodies

12. Step by step …... Planar circular restricted three-body model. Planar circular restricted three-body model. Planar circular restricted three-body model + N small bodies. Planar circular restricted three-body model + N small bodies. 3D circular restricted three-body model. 3D circular restricted three-body model. 3D circular restricted three-body model + N small bodies. 3D circular restricted three-body model + N small bodies.

13. Outline Introduction and motivation Introduction and motivation The restricted three body model The restricted three body model Some results Some results Future works Future works Planar circular

14. η ξ y x nt

15. (ξ,η)  (x,y)

16. Jacobi constant

17. Outline Introduction and motivation Introduction and motivation The restricted three body model The restricted three body model Some results Some results Future works Future works

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

19. a: semi-major axis e: eccentricity θ: true longitude ω : longitude of pericentre f: true anomaly ω f θ (v x, v y, x, y) (a, e, θ, ω)

20. Sun-Jupiter system + one small body Sun-Jupiter system + one small body μ 2 =0.001,μ 1 =1-μ 2 μ 2 =0.001,μ 1 =1-μ 2 x 0 =0.55, y 0 =0.0, v 0 =0.0, C J =3.07 x 0 =0.55, y 0 =0.0, v x0 =0.0, C J =3.07 Example (C. D. Murray, Solar system dynamics)

26. Poincare surface of section: y=0, v y >0.

27. Sun-Jupiter system + one small body Sun-Jupiter system + one small body μ 2 =0.001,μ 1 =1-μ 2 μ 2 =0.001,μ 1 =1-μ 2 x 0 =0.56, y 0 =0.0, v 0 =0.0, C J =3.07 x 0 =0.56, y 0 =0.0, v x0 =0.0, C J =3.07

33. Poincare surface of section: y=0, v y >0.

34. Outline Introduction and motivation Introduction and motivation The restricted three body model The restricted three body model Some results Some results Future works Future works

35. In the page In the page Read more about chaos and resonance. Read more about chaos and resonance. Step by Step……

36. to be continued…. THANKS……