Alice Quillen University of Rochester

Department of Physics and Astronomy

Oct, 2005

Submillimeter imaging by Greaves and collaborators

Morphology of Circumstellar Disks

•Structure observed in circumstellar disks: arcs, spiral arms, edges, warps, clumps, holes, lopsidedness

•Planets can affect the morphology a planet detection technique,

complimentary to radial velocity and transit searches.

•Jupiter is 10-3 times the Mass of the Sun resonant effects or long timescales (secular) required for a planet to affect the morphology

•Models are sensitive to dust production sites, the gas and planetesimals as well as planetary properties.

•Models are sensitive to the past history –evolution of the planetary systems.

Production of Star Grazing

and Star-Impacting

Planetesimals via Planetary

Orbital Migration

(Quillen & Holman 2000)

FEB

Impact

Simple Hamiltonian systems

2( , ) cos( )

2

pH p K

2 2( , ) ( , )

2 2

is constant

0 is conserved

p qH p q H I I

H dI dtH dI

Idt

Harmonic oscillator

Pendulum

Stable fixed point

Libration

Oscillation

p

Separatrix

pq

I

Resonance Capture vs Stability of Particles in Resonance

• Poynting Robertson drag and that from solar wind cause in-spiral of dust particles. These particles will cross orbital resonances with a planet. If captured, resonances can delay the in-spiral.

• Alternatively dust particles released from planetesimals trapped in resonance are likely to be trapped.

• The particle distribution is dominated by long lived particles. Consequently the morphology can be affected by resonances.

Possible explanations proposed for Epsilon Eridani (me) and Vega disk morphologies (Kuchner and Wyatt and collaborators). For all of these (Deller). Also see Ozernoy et al. 2001. On the dust in the Kuiper Belt (see Liou & Zook and Moro-Martin et al).

Resonance Capture -Astrophysical Settings

Migrating planets or satellites moving inward or outward capturing planets or planetesimals into resonance–

Dust spiraling inward with drag forces

What we would like to know: Capture probability as a function of: --initial conditions--migration or drift rate--resonance properties

resonant angle fixed -- Capture

Escape

2~ e

Resonance CaptureTheoretical Setup

2/ 2

,02

Mean motion resonances can be written

( , ) cos( )

corresponds to order of resonance. : Coefficients depend on time in drifting/ migrating systems sets the distance

t

kk

a bH k

kkk j j k

b

o the resonance

gives the drift rate

Particle eccentricity depends on

dbdt sh

ap

e o

f re

son

an

ce d

ep

en

ds

on

b

Non zero resonant angle. Large radius here corresponds to large eccentricity

2

( ) / 2,

0

cos( ( ) )k

k pk p p

p

K a b c

k p p

p

Resonance Capture in the Adiabatic Limit• Application to tidally drifting satellite

systems by Borderies, Malhotra, Peale, Dermott, based on analytical studies of general Hamiltonian systems by Henrard and Yoder.

• Capture probabilities are predicted as a function of resonance order and coefficients.

• Capture probability depends on initial

particle eccentricity but not on drift rate

d

rift

ing

syste

m

Capture here

Limitations of Adiabatic theory• In complex drifting systems many resonances are available.

Weak resonances are quickly passed by. • At fast drift rates resonances can fail to capture -- the non-adiabatic regime• Subterms in resonances can cause chaotic motion.

temporary capture in a chaotic system

,

* planet mass/ planet eccentricity

pk p e

Me

2 ( ) / 2,

0

cos( ( ) )k

k pk p p

p

K a b c k p p

Rescaling

2/(4 )2,0

(2 ) /(4 )2 /(4 ),0 2

2 / 2

By rescaling momentum and time

The Hamiltonian can be written as

cos

k

k

k kk

k

k

k

a

at

k

K b k

This power sets dependence on initial eccentricity

This power sets dependence on drift rate

All k-order resonances now look the same

Rescaling

2/(4 )2,0

(2 ) /(4 )2 /(4 ),0 2

2 / 2

By rescaling momentum and time

The Hamiltonian can be written as

cos

k

k

k kk

k

k

k

a

at

k

K b k

Drift rates have units

First order resonances have drift rates that scale with planet mass to the power of -4/3. Second order to the power of -2.

2t

Confirming and going beyond previous theory by Friedland and numerical work by Ida, Wyatt, Chiang..

Numerical Integration of rescaled systems

Capture probability as a function of drift rate and initial eccentricity for first order resonances

Pro

bab

ilit

y o

f C

ap

ture

drift rate

First order

Critical drift rate– above this capture is not possible

At low initial eccentricity the transition is very sharp

Capture probability as a function of initial eccentricity and drift rate

drift rate

Pro

bab

ilit

y o

f C

ap

ture

Second order resonances

High initial particle eccentricity allows capture at faster drift rates

curves to the right correspond to higher initial eccentricities

When there are two resonant terms

drift rate

coro

tati

on

reson

an

ce

str

en

gth

dep

en

ds o

n p

lan

et

eccen

tric

ity

Capture is prevented by corotation resonance

Capture is prevented by a high drift rate

capture

escape

drift separation set by secular precession

depends on planet eccentricity

2

1/ 2 cos( ) cos( )p

K b c

Trends• Higher order (and weaker) resonances require slower drift

rates for capture to take place. Dependence on planet mass is stronger for second order resonances.

• If the resonance is second order, then a higher initial particle eccentricity will allow capture at faster drift rates

• If the planet is eccentric the corotation resonance can prevent capture

• For second order resonances e-e’ resonant term behave likes like a first order resonance and so might be favored.

General formulism We have derived a general recipe to predict capture

probabilities for drifting Hamiltonian orbital systems. Coefficients given in appendix of Murray and Dermott can now be used to predict the capture probability for ANY migrating or drifting system into ANY resonance.

First Applications• Neptune’s eccentricity is ~ high enough to

limit capture of TwoTinos.• A migrating extrasolar planet can easily be

captured into the 2:1 resonance but must be moving very slowly to capture into the 3:1 resonance. -- Application to multiple planet extrasolar systems.

• Drifting dust particles are sorted by size. We can predict which size particle will be captured into which resonance.