Alice Quillen University of Rochester Department of Physics and Astronomy Oct, 2005 Submillimet er imaging by Greaves and collaborato rs

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 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 Harmonic oscillator Pendulum Stable fixed point Libration Oscillation p Separatrix p q 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 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

Resonance Capture Theoretical Setup shape of resonance depends on b  Non zero resonant angle. Large radius here corresponds to large eccentricity

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  drifting system 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

Rescaling This power sets dependence on initial eccentricity This power sets dependence on drift rate All k-order resonances now look the same

Rescaling 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. 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 Probability of Capture 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  Probability of Capture Second order resonance s High initial particle eccentrici ty allows capture at faster drift rates curves to the right correspond to higher initial eccentricities

When there are two resonant terms drift rate  corotation resonance strength depends on planet eccentricity Capture is prevented by corotation resonance Capture is prevente d by a high drift rate capture escape drif t separation set by secular precession depends on planet eccentricity

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.