Three body resonances in exoplanet systems

Abstract

3-body resonance, also called Laplace resonance, is the mean motion resonance(MMR) between 3 planets. It comes to people’s interests after the discovery of the TRAPPIST-1 system, which is a special system where the planets have period ratio near 8:5, 5:3, 3:2, 3:2, 4:3, 3:2 from insideout, and all 7 planets are interlocked in a chain of 3-body resonances (Luger et al. 2017; Gillon et al. 2017). Other similar systems are also detected in 3-body resonance/chain of 3-body resonances, such as Kepler-60, Kepler-80 and Kepler-223. It is natural to ask how these systems in chain of commensurabilities form. Improving our understanding of the resonance dynamics in these systems could be the key to this answer.

In this thesis, we study the dynamics of 3-body resonance both numerically and analytically. Our study can be divided into 4 parts. In the first part, we numerically derive the critical migration rate of several first order 2-body MMRs and the linear regime of several first and second order MMRs. These fitted values are necessary for general migration simulations used in 3-body resonance assembling.

In the second part, we study the pure 3-body resonance, which is the libration of Laplace angle in the absence of the libration of 2-body resonant angles. The counter situation is a librating Laplace angle in the presence of librating 2-body resonant angles, which is called double 2-body resonance. Pure 3-body resonance is suspected in some resonant chain systems because the 2-body resonant angles are not well constrained from observations. After our numerical investigations, we reasonably conclude that pure 3-body resonance has the following properties: i.) Pure 3-body resonance only exists for systems o↵set by a certain value from the exact commensurability, unlike double 2-body resonance which can exist for arbitrary o↵set. ii.) Planets in pure 3-body resonance must have free eccentricities greater than forced eccentricities, unlike the double 2-body resonance which have free eccentricities less than forced eccentricities. iii.) All planets in pure 3-body resonance are in a double orbital alignment, while planets in double 2-body resonance are in a double anti-alignment.

In the third part, we study the multiple equilibrium solutions of 3-body resonances, which means the Laplace angle can librate about more than one libration center. We show that the multiple equilibrium solutions originates from the combination of di↵erent geometric phases in 2-body resonance between the inner and the outer planet pair. Later we investigate the probability of capture into di↵erent libration centers. Using 5:4:3 and 4:3:2 3-body resonance as examples, we find that the capture probability profile into different libration centers is roughly an equal probability for general migration parameters and planetary mass ratio less than 2:1, but is highly skewed if the resonance is assembled by non-adiabatic migration.

In the last part, we use an adiabatic disk model adopted from Paardekooper et al. (2010a) to construct an analysis to constrain the protoplanetary disk of resonant chain systems. The results show that it requires rather extreme disk parameters, which is very di↵erent from Minimum Mass Solar Nebula, viscously heated disk, weak or strongly opaque disk, in order to maintain the resonant chains in Kepler-60, Kepler-80, Kepler-223 and TRAPPIST-1. To get around this, we need i) more accurate planetary mass measurement; ii) planetary migration halting mechanism near the inner disk. An empirical power law trend between innermost distance of a resonant chain and stellar mass is observed. Solution (ii) is preferred if future resonant chains observed fall on the empirical trend.