ASTEROIDS, KBOS AND OTHER DEBRIS

IN PLANETARY SYSTEMS

RENU MALHOTRA

UNIVERSITY OF ARIZONA

1. Locating debris in exo-planetary systems

2. Chaotic diffusion of resonant Kuiper Belt objects

@KITP: Planet Formation Conference, March 15-19, 2004

– 2 –

– 3 –

Orbital stability criteria

1 Planet System

• Hill stability: |∆a|/ap > 2.4(mp/m?) 13 ' 3RH

• Resonance overlap: |∆a|/ap > 1.5(mp/m?) 27

• Eccentric planet: r 3 (q − 3RH, Q + 3RH)

2 Planet System

• Direct numerical integration of test particles• Patchwork of 1-planet criteria• Secular stability analysis

– Laplace-Lagrange secular theory for planets: linear modes

∗ e.g., g5, g6 for Sun-Jupiter-Saturn system– Locate secular resonances of test particles

– Nonlinear saturation of test particle eccentricity

∗ resonance Hamiltonian, phase space pictures∗ Initially circular orbits are unstable near g0 ≈ gj∗ maximum eccentricity excitation, emax ∝ |E(j)|,∗ depends strongly on a, weakly on m1/m2.

– 4 –

Secular Resonance Hamiltonian

toy model: planet in eccentric, precessing orbit

Hsec = −mpap

{A0(α) + A(α)e

2 + B(α)e4 − C(α)eep cos($p −$)}. (1)

in canonical variables, −$ and J = √a(1−√1− e2),

Hsec = −mpam

A0(α)− g0J + βJ2 + ε√

2J cos($ −$p), (2)

where

g0 =2A(α)

a12 am

mp, β =A(α)− 4B(α)

aammp, ε =

C(α)

a14 am

mpep. (3)

For initially circular test particle orbits, the maximum perturbation occurs at

g0 = gp + 3(βε2/2)1/3,

and the maximum amplitude is given by

Jmax = 2∣∣∣2εβ

∣∣∣23

or emax ≈ 2∣∣∣ 2CA− 4Bep

∣∣∣13

. (4)

– 5 –

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

-4 -2 0 2 4

-4

-2

0

2

4

–6

–47 Ursae Majoris system

a(AU)

0 5 100

0.2

0.4

0.6

0.8

1

–7

–47 Ursae Majoris system

a(AU)

0 5 100

0.2

0.4

0.6

0.8

1

–8

–upsilon Andromedae system

a(AU)

0 5 100

0.2

0.4

0.6

0.8

1

–9

–upsilon Andromedae system

a(AU)

0 5 100

0.2

0.4

0.6

0.8

1

–10

–Solar system (Sun+Jupiter+Saturn)

inner SS

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

a(AU)

outer SS

0 10 20 300

0.2

0.4

0.6

0.8

1

a(AU)

–11

–Solar system (Sun+Jupiter+Saturn)

inner SS

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

a(AU)

outer SS

0 10 20 300

0.2

0.4

0.6

0.8

1

a(AU)

–12

–Solar system (Jupiter+Saturn only, vs. 4 outer planets)

inner SS

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

a(AU)

outer SS

0 10 20 300

0.2

0.4

0.6

0.8

1

a(AU)

– 13 –

Possible debris locations in:

• 47 Ursae Majoris (two jovian-mass planets)– 0.3-0.6 AU

– 1.0-1.5 AU

– outside 7.5 AU

• υ Andromedae (three jovian-mass planets)– between 0.06 and 0.3 AU

– outside 8 AU

• Sun–Jupiter–Saturn system– inward of 0.5 AU

– between 0.8 and 1.6 AU

– between 2.0 and 4.3 AU

– outside 18 AU

–14

–

Orbital Distribution of Kuiper Belt objects

30 35 40 45 500

0.1

0.2

0.3

0.4

semimajor axis [AU]

2:13:24:35:4 5:37:5

+

Planet Migration and Resonance Sweeping of the Kuiper Belt

3:22:1

5:3

0

50

100

150

– 15 –

– 16 –

Conclusion: Assuming equal initial populations of Plutinos and Twotinos, 4 Gyr

of dynamical diffusion reduces the ratio of Twotinos:Plutinos to < 0.2.

Table 1. Fate of Resonance Escapees

Plutinos Twotinos

Avg. lifetime 150 My 270 Myr

lifetime > 1 Gy 5% 29–36%

Reach inner SS ∼ 25% ∼ 20%