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.

### Session: Exoplanet Theory II

Maryame El Moutamid (Cornell)

#### Abstract

Many bodies in the Solar System and some exo-planets are close to or captured in Mean Motion Resonances (MMR). Capture into such resonances has been investigated by many authors. Indeed, the Hamiltonian equations of motion in presence of migration are given by Sicardy and Dubois Cel. Mech. & Dyn. Astron., 86, 321-350 (2003). Fleming and Hamilton, Icarus 148, 479-493 (2000), studied the problem in a less generic context. In these two papers, the authors studied the problem of 1:1 corotation (Lagrange points L4 and L5), rather than m+1:m corotations (El Moutamid et al, Cel. Mech. & Dyn. Astron, 118, 235-252 (2014)). We will present a generic way to analyze details of a successful (or not) capture in the case of an oblate (or not) central body in the context of Restricted Three Body Problem (RTBP) and a more General Three Body Problem in the context of known statistics for captured exoplanets (candidates) observed by Kepler.

#### Notes

- Captures partial near MMR (Fabrycky
*et al.*2012) - No generic study on coupling between associated resonances (ERTB vs. general TB)
- 1) simple model,2DoF — $(m+1) n’ \approx m n$
- splitting the corotation and Lindblad resonances (by $J_2 \neq 0$)
- Lindblad: vary $e$
- corotation: pendular motion (conserves $e$)
- plot: $J_c – J_L$ vs. $\phi_C$

- 2) general case
- can define a constant of motion: $J_{c,relat} = \frac{A^2 \xi}{m} – \frac{A’^2 e’}{m+1} – ?? = const.$

- Add dissipationforMMR capture
- ratio: potential barrier of one vs. other body
- plot: potential energy vs. critical angle of corotation
- probability of capture: very very small

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