ALBERT

Description: In a two-planet system, a topological boundary that is created by Sundman (1912, Acta Math., 36, 105) inequality can forbid close encounters between the two planets for an infinite time. A system is said to be Hill stable if it verifies this topological condition. Hill stability is widely used in the study of extrasolar planet dynamics. However, the coplanar and circular orbit approximation is often used. In this paper, we explain how the Hill stability can be understood in the framework of angular momentum deficit (AMD). In the secular approximation, AMD allows us to discriminate between a priori stable systems and systems for which a more in-depth dynamical analysis is required. We show that the general Hill stability criterion can be expressed as a function of only semimajor axes, masses, and total AMD of the system. The proposed criterion is only expanded in the planets-to-star mass ratio ε and not in the semimajor axis ratio, eccentricities, nor the mutual inclination. Moreover, the expansion in ε remains excellent up to values of about 10−3 even for two planets with very different mass values. We performed numerical simulations in order to highlight the sharp change of behavior between Hill stable and Hill unstable systems. We show that Hill stable systems tend to be very regular, whereas Hill unstable systems often lead to rapid planet collisions. We also note that Hill stability does not provide protection from the ejection of the outer planet.

Description: We present a new mixed variable symplectic (MVS) integrator for planetary systems that fully resolves close encounters. The method is based on a time regularisation that allows keeping the stability properties of the symplectic integrators while also reducing the effective step size when two planets encounter. We used a high-order MVS scheme so that it was possible to integrate with large time-steps far away from close encounters. We show that this algorithm is able to resolve almost exact collisions (i.e. with a mutual separation of a fraction of the physical radius) while using the same time-step as in a weakly perturbed problem such as the solar system. We demonstrate the long-term behaviour in systems of six super-Earths that experience strong scattering for 50 kyr. We compare our algorithm to hybrid methods such as MERCURY and show that for an equivalent cost, we obtain better energy conservation.

Description: K2-19 hosts a planetary system composed of two outer planets, b and c, with size of 7.0 ± 0.2 R⊕ and 4.1 ± 0.2 R⊕, and an inner planet, d, with a radius of 1.11 ± 0.05 R⊕. A recent analysis of Transit-Timing Variations (TTVs) suggested b and c are close to but not in 3:2 mean motion resonance (MMR) because the classical resonant angles circulate. Such an architecture challenges our understanding of planet formation. Indeed, planet migration through the protoplanetary disc should lead to a capture into the MMR. Here, we show that the planets are in fact, locked into the 3:2 resonance despite circulation of the conventional resonant angles and aligned periapses. However, we show that such an orbital configuration cannot be maintained for more than a few hundred million years due to the tidal dissipation experienced by planet d. The tidal dissipation remains efficient because of a secular forcing of the innermost planet eccentricity by planets b and c. While the observations strongly rule out an orbital solution where the three planets are on close to circular orbits, it remains possible that a fourth planet is affecting the TTVs such that the four planet system is consistent with the tidal constraints.

Description: The dynamical stability of tightly packed exoplanetary systems remains poorly understood. While a sharp stability boundary exists for a two-planet system, numerical simulations of three-planet systems and higher show that they can experience instability on timescales up to billions of years. Moreover, an exponential trend between the planet orbital separation measured in units of Hill radii and the survival time has been reported. While these findings have been observed in numerous numerical simulations, little is known of the actual mechanism leading to instability. Contrary to a constant diffusion process, planetary systems seem to remain dynamically quiescent for most of their lifetime before a very short unstable phase. In this work, we show how the slow chaotic diffusion due to the overlap of three-body resonances dominates the timescale leading to the instability for initially coplanar and circular orbits. While the last instability phase is related to scattering due to two-planet mean motion resonances (MMRs), for circular orbits the two-planets MMRs are too far separated to destabilise systems initially away from them. The studied mechanism reproduces the qualitative behaviour found in numerical simulations very well. We develop an analytical model to generalise the empirical trend obtained for equal-mass and equally spaced planets to general systems on initially circular orbits. We obtain an analytical estimate of the survival time consistent with numerical simulations over four orders of magnitude for the planet-to-star-mass ratio ε, and 6 to 8 orders of magnitude for the instability time. We also confirm that measuring the orbital spacing in terms of Hill radii is not adapted and that the right spacing unit scales as ε1∕4. We predict that beyond a certain spacing, the three-planet resonances are not overlapped, which results in an increase of the survival time. We confirm these findings with the aid of numerical simulations of three-planet systems with different masses. We finally discuss the extension of our result to more general systems, containing more planets on initially non-circular orbits.

Description: Recent works on three-planet mean motion resonances (MMRs) have highlighted their importance for understanding the details of the dynamics of planet formation and evolution. While the dynamics of two-planet MMRs are well understood and approximately described by a one-degree-of-freedom Hamiltonian, little is known of the exact dynamics of three-body resonances besides the cases of zeroth-order MMRs or when one of the bodies is a test particle. In this work, I propose the first general integrable model for first-order three-planet mean motion resonances. I show that one can generalize the strategy proposed in the two-planet case to obtain a one-degree-of-freedom Hamiltonian. The dynamics of these resonances are governed by the second fundamental model of resonance. The model is valid for any mass ratio between the planets and for every first-order resonance. I show the agreement of the analytical model with numerical simulations. As examples of application, I show how this model could improve our understanding of the capture into MMRs as well as their role in the stability of planetary systems.