This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Konstantin Batygin (CalTech)

#### Abstract

The early stages of dynamical evolution of planetary systems are often shaped by dissipative processes that drive orbital migration. In multi-planet systems, convergent amassing of orbits inevitably leads to encounters with rational period ratios, which may result in establishment of mean motion resonances. The success or failure of resonant capture yields exceedingly different subsequent evolutions, and thus plays a central role in determining the ensuing orbital architecture of planetary systems. In this talk, we will show how an integrable Hamiltonian formalism for planetary resonances that allows both secondary bodies to have finite masses and eccentricities, can be used to construct a comprehensive theory for resonant capture. Employing the developed analytical model, we shall examine the origins of the dominantly non-resonant orbital distribution of sub-Jovian extrasolar planets, and demonstrate that the commonly observed extrasolar orbital structure can be understood if planet pairs encounter mean motion commensurabilities on slightly eccentric (e ~ 0.02) orbits. Accordingly, we speculate that resonant capture among low-mass planets is typically rendered unsuccessful due to subtle axial asymmetries inherent to the global structure of protoplanetary disks.

#### Notes

- SeeMécaniqueCéleste, Laplace 1805!
- But origins not really understood until Roy & Ovenden 1954, Goldreich 1964 (MNRAS)

- Disk-satellite interactions (Goldreich & Tremaine)
- But what about more than one planet?
- All tend to migrateinward then lock intoMMRs (Pierens 2013 A&A)
- $\Rightarrow$ numerical models predict MMR lock
- BUT only ~15% of observed planet pairs are in resonance

- The real Hamiltonian (planet-planet interactions) is actually probably a mess.
- See Poincare’s book, vol. 2(!)
- Define a canonical rotation that gives an integral of the motion (“generalized reducing transformation” –Poincare)
- Basically, a generalized Tisserand parameter

- Batygin & Morbidelli 2013(?)

- An analytical theory for resonant capture: unrestricted ETB problem.
- Batygin 2015 (MNRAS, submitted)
- Capture prob. only depends on total mass of the planets, NOT the mass ratio
- phase space area occupied by planet is small

- Kepler sample: critical eccentricity is ~0.02 — very small!
- Larger than this, capture fails
- Matches observed value!

- Explains Jupiter-Saturn MMR lock
- Perhaps slight deviations from axial symmetry in protoplanetary disks are responsible for the orbital architecture we observe today.

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