Resonance Capture When does it Happen?

1. Resonance Capture When does it Happen? Alice Quillen University of Rochester Department of Physics and Astronomy

2. Escape resonant angle fixed -- Capture Resonance CaptureAstrophysical Settings What we would like to know: capture probability as a function of: --initial conditions --migration or drift rate --resonance properties Migrating planets moving inward or outward capturing planets or planetesimals – Dust spiraling inward with drag forces

3. Resonance CaptureTheoretical Setup

4. 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. -- Below a critical eccentricity capture is ensured.

5. Adiabatic Capture Theoryfor Integrable Drifting Resonances Theory introduced by Yoder and Henrard, applied toward mean motion resonances by Borderies and Goldreich

6. Limitations of Adiabatic theory • In complex 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

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

8. Rescaling (continued) Drift rates have units First order resonances have critical 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..

9. Scaling (continued)

10. Probability of Capture drift rate  Numerical Integration of rescaled systemsCapture probability as a function of drift rate and initial eccentricity for first order resonances At low initial eccentricity the transition is very sharp First order Critical drift rate– above this capture is not possible Transition between “low” initial eccentricity and “high” initial eccentric is set by the critical eccentricity below which capture is ensured in the adiabatic limit (depends on rescaling of momentum)

11. Second order resonances Probability of Capture drift rate  Capture probability as a function of initial eccentricity and drift rate High initial particle eccentricity allows capture at faster drift rates

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

13. 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 – but not at high probability. • 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. • Resonance subterm separation (set by the secular precession frequencies) does affect the capture probabilities – mostly at low drift rates – the probability contours are skewed.

14. Lynnette Cook’s graphic for 55Cnc Applications • Neptune’s eccentricity is ~ high enough to prevent 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 can be sorted by size. Capture for dust particles into j:j+1 resonance

15. Applications (continued) • Capture into j:j+1 resonance for drifting planets requires • Formalism is very general – if you look up resonance coefficients critical drift rates can be estimated for any resonance. • Particle distribution could be predicted following migration given an initial eccentricity distribution and a drift rate -- more work needs to be done to relate critical eccentricities to those actually present in simulations