Dynamics of Kuiper belt objects

Yeh, Lun-Wen2007.11.8

Outline

MotivationsMotivationsSun-Neptune-Eris-Particle systemSun-Neptune-Eris-Particle systemNumerical methodNumerical methodResultsResultsNext StepNext Step

Motivations Planet migration model is one most popular model for thPlanet migration model is one most popular model for th

e formation of Kuiper belt. (Malhotra 1993,1995; Hahn & e formation of Kuiper belt. (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)

Sun + 4 giant planets + numerous particles.Sun + 4 giant planets + numerous particles.

Because of the computing source the interaction betweeBecause of the computing source the interaction between particles was usually neglected.n particles was usually neglected.

Considering the influence of larger planetesimals in the Considering the influence of larger planetesimals in the model is a way to approach the complete model.model is a way to approach the complete model.

At present there are three dwarf planets by the definition At present there are three dwarf planets by the definition of IAU. (Ceres: D~940km , Pluto: D~2300km, Eris: D~of IAU. (Ceres: D~940km , Pluto: D~2300km, Eris: D~

2400km)2400km) Some papers have considered the effect of Pluto on the Some papers have considered the effect of Pluto on the

Plutinos. (Yu & Tremaine 1999; Nesvorny & Roig 2000)Plutinos. (Yu & Tremaine 1999; Nesvorny & Roig 2000) We would like to know the gravitational influence of Eris We would like to know the gravitational influence of Eris

on the Kuiper belt objects.on the Kuiper belt objects.

a=67 AU, e=0.44, i=44a=67 AU, e=0.44, i=4400.. T=557 years.T=557 years. mass=0.0028 Mmass=0.0028 MEE, diameter=2400 km., diameter=2400 km.

Although we don’t know where Eris was formed and how loAlthough we don’t know where Eris was formed and how long Eris has stayed in the present orbit, but Eris should havng Eris has stayed in the present orbit, but Eris should have some influence on the Kuiper belt.e some influence on the Kuiper belt.

The coagulation model (ex: Kenyon et al. 2007) implies thaThe coagulation model (ex: Kenyon et al. 2007) implies that Eris-size objects were formed in a more massive disk (~3t Eris-size objects were formed in a more massive disk (~30M0MEE) with low e and i (e, i~10) with low e and i (e, i~10-3-3) planetesimals.) planetesimals.

Under the planet migration model (ex: Hahn & Malhotra 20Under the planet migration model (ex: Hahn & Malhotra 2005), the scattered disk objects originated from low e, low i 05), the scattered disk objects originated from low e, low i

and smaller a orbitand smaller a orbit due to Neptune’s migration. The migratidue to Neptune’s migration. The migration timescale of Neptune is about 10 Myr. Therefore Eris-sion timescale of Neptune is about 10 Myr. Therefore Eris-size objects may leave disk between 10 Myr to 4.5 Gyr.ze objects may leave disk between 10 Myr to 4.5 Gyr.

The mass of Eris to the total mass of KBOs is about 0.1 The mass of Eris to the total mass of KBOs is about 0.1 (0.0028/0.03; Soter 2007) hence Eris may still collide an(0.0028/0.03; Soter 2007) hence Eris may still collide and scatter with many other KBOs.d scatter with many other KBOs.

Hahn & Malhotra 2005

Hahn & Malhotra 2005

The main belt isThe main belt is notnot eccentricity enough.eccentricity enough. Absence of higher order resonance. (ex: 5:2)Absence of higher order resonance. (ex: 5:2) High inclination KBOs are deficient.High inclination KBOs are deficient. Simulation: # of (3:2+2:1) / # of MB~2.Simulation: # of (3:2+2:1) / # of MB~2.

Observation: # of (3:2+2:1) / # of MB~0.06. Observation: # of (3:2+2:1) / # of MB~0.06.

Hahn & Malhotra 2005

Hahn & Malhotra 2005

Hahn & Malhotra 2005

Hahn & Malhotra 2005

We consider the Sun-Neptune-Eris-particle system in the We consider the Sun-Neptune-Eris-particle system in the late stage of planet migration model (the last several hunlate stage of planet migration model (the last several hundreds Myrs) and assume Eris at present orbit.dreds Myrs) and assume Eris at present orbit.

Sun-Neptune-Eris-Particle system

S

N E

P

η

ξ

y

x

nt

S

N

EP

3 3 3

3 3 3

, , 1

1, 1

S N ES N E

S N E

S N ES N E

S N E

S S N N S N

E E

N N

r r r

r r r

Gm Gm

Gm

n a

For particles:

3 3

3 3

S E N EE S N

SE NE

S E N EE S N

SE NE

r r

r r

For Eris:

23 3 3

23 3 3

2

2

S N EN N S N E

S N E

S N EN N S N E

S N E

x x x x x xx n y n x

r r r

y y y y y yy n x n y

r r r

For particles:

23 3

23 3

2

2

S E N EE N E N E S N

SE NE

S E N EE N E N E S N

SE NE

x x x xx n y n x

r r

y y y yy n x n y

r r

2 2 2 22( ) ( )S NJE E E E E

SE NE

C x y x yr r

Jacobi constant

For Eris:

We We used Runge-Kutta-Fehlberg method to solve the differential equations. This method combines 4th and

5th order Runge-Kutta method to choose a suitable step

size for each step. We did some modification in the step size determination

in order to have a more significant criterion to choose the step size.

Numerical methodNumerical method

0 0

n+1 n 1 3 4 5

1

1 12 14 4

3 3 93 1 28 32 32

1932 7200 7296124 1 2 313 2197 2197 2197

4395

( , ) ( )

25 1408 2197 1y = y + k + k + k k

216 2565 4101 5

k ( , )

( , )

( , )

( , )

( ,

n n

n n

n n

n n

n n

dyf t y y t y

dt

tf t y

k tf t t y k

k tf t t y k k

k tf t t y k k k

k tf t t y

3680 8451 2 3 4216 513 4104

8 3544 18591 116 1 2 3 4 52 27 2565 4104 40

8 )

( , 2 )n n

k k k k

k tf t t y k k k k k

yi

yi+1

t

yy = f(t,y)

hi

RK4

RK5

yi

yi+1

t

yy = f(t,y)

hi

RK4

RK4

qhi

∆

□

X

R ≡ | ∆ | / (| X |+ | □ |) < TOL

The initial conditions: The initial conditions: Neptune: a=30.07AU (1); e=0.0; : a=30.07AU (1); e=0.0; θθ=0.0=0.0

Eris: a=67.73AU (2.252); e=0.44; : a=67.73AU (2.252); e=0.44; ωω=0.0; =0.0; θθ=0.0=0.0

50 test particles in 3:2 mean motion resonance:: a=39.4 a=39.4 ± 0.3 ± 0.3 AU (1.31 AU (1.31 ± 0.01± 0.01); e=0.0,0.1,0.2,0.3,0.4;); e=0.0,0.1,0.2,0.3,0.4; randomly assign randomly assign ωω,,θθ..

50 test particles in 2:1 mean motion resonance:: a=47.7 a=47.7 ± 0.3 ± 0.3 AU (1.59 AU (1.59 ± 0.01± 0.01); e=0.0,0.1,0.2,0.3,0.4;); e=0.0,0.1,0.2,0.3,0.4; randomly assign randomly assign ωω,,θθ..

50 test particles in main belt:: a=40.3 – 46.8a=40.3 – 46.8 AU (1.34 – 1.56); e=0.0,0.1;AU (1.34 – 1.56); e=0.0,0.1; randomly assign randomly assign ωω,,θθ..

tt00=0, t=0, tff=10=1066TTNN (T (TNN=165 yr)=165 yr)

∆∆ttoutout=10=1022TTNN

∆∆ttmaxmax=10=10-1-1TTNN, ∆, ∆ttminmin=10=10-20-20TTNN

TOL=10TOL=10-12-12

The criterions for stopping the program:The criterions for stopping the program:

when r < 5AU or r > 1000AUwhen r < 5AU or r > 1000AU

when the particle collides with Neptunewhen the particle collides with Neptune

when the particle collides with Eriswhen the particle collides with Eris

when t=twhen t=tff

ResultsResults

a=1.30, e=0.0, curlypi=2.18, theta=6.23

3to2: TS=438 yr

2to1: TS=845 yr

remaining total # / initial total # = 26/50

remaining total # / initial total # = 42/50

remaining total # / initial total # = 48/50

There two reasons cause the number of surviving particlThere two reasons cause the number of surviving particles of 3 :2 smaller than that of 2:1. One is that the synodies of 3 :2 smaller than that of 2:1. One is that the synodic period of 3:2 is smaller than that of 2:1. Another is that c period of 3:2 is smaller than that of 2:1. Another is that the q=30 AU line in the a-e diagram crosses more region the q=30 AU line in the a-e diagram crosses more region in 3:2 than that in 2:1 mean motion resonance.in 3:2 than that in 2:1 mean motion resonance.

remaining total # / initial total # = 116/150

remaining total # / initial total # = 144/150

# in 3to2: 13

# in MB: 57

# in 2to1: 21

(3to2+2to1)/MB = 0.6

Construct three dimensional model.Construct three dimensional model. Run more particles to enhance the statistical Run more particles to enhance the statistical

significance. significance.

From estimation of Hahn 2005 there are ~ 10From estimation of Hahn 2005 there are ~ 1055 KBOs in the Kuiper belt. How KBOs in the Kuiper belt. How many particles I will use depends on the computing sources.many particles I will use depends on the computing sources.

Consider more Eris-size objects. How many? How to Consider more Eris-size objects. How many? How to choose initial conditions? I am still thinking……..choose initial conditions? I am still thinking……..

Next Step

Thank you ^^…..

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