Dynamical Excitation as a probe of planetary origins

TL;DR

The paper investigates how magnetospheric diffusion shapes the inner edge of protoplanetary disks to drive divergent migration and break resonant chains in compact planetary systems. By performing full N-body simulations with disk torques, the authors show that most resonant chains dissolve during disk dispersal, producing final architectures and residual eccentricities consistent with Kepler and transit-timing observations, without requiring long-term dynamical instabilities. Tidal damping is included and found to have limited impact on the qualitative outcomes, particularly for longer-period planets. The study provides a physically motivated mechanism—magnetospheric rebound—for explaining the observed spacing and low-to-moderate eccentricities in Kepler-like systems, and it highlights testable predictions for eccentricity, phase behavior, and three-body resonances in multi-planet configurations.

Abstract

We present a set of numerical simulations of the dynamical evolution of compact planetary systems migrating in a protoplanetary disk whose inner edge is sculpted by the interaction with the stellar magnetic field, as described in Yu et al. (2023). We demonstrate that the resulting final distribution of neighbouring planet period ratios contains only a small surviving fraction of resonant systems, in accordance with observations. The resulting planetary architectures are largely in place by the end of the protoplanetary disk phase (within a few Myr), and do not require significant later dynamical evolution. The divergence of planetary pairs during gas disk dispersal also leads to the excitation of eccentricities when pairs cross mean motion resonances in a divergent fashion. The resulting distribution of remnant free eccentricities is consistent with the values inferred from the observation of transit durations and transit timing variations. We furthermore demonstrate that this conclusion is not significantly altered by tides, assuming standard values for tidal dissipation in Earth or Neptune-class planets. These results demonstrate that the observed spacing and residual dynamical excitation of compact planetary systems can be reproduced by migration through a protoplanetary disk, as long as the inner disk boundary is modelled as a gradual rollover, instead of a sharp transition. Such an effect can be achieved when the model accounts for the diffusion of the stellar magnetic field into the disk. The resulting divergence of planetary pairs during the magnetospheric rebound phase breaks the resonant chains, resulting in a better match to observations than disk models with more traditional inner boundaries.

Figures (20)

• Figure 1: The upper panel shows the evolution of the period ratio of the planetary pair in the system modelled after Kepler-1471. The jump at age 1.345 Myr is the divergence crossing of the 2:1 resonance. In the lower panel, we see a corresponding spike in the eccentricity of both planets, most of which remains after the disk has finally dispersed.
• Figure 2: The solid points show the inner planet eccentricity after 2 Myr, while the open circles show the outer planet eccentricity. The distribution of eccentricities as a function of planetary mass ratio breaks up into three regimes. To the left of the vertical dashed line, i.e. at small $M_2/M_1$, all systems remain in resonance to the end of the simulation. Thus, the eccentricities are determined by the resonant lock of the pair. For mass ratios around unity, the final eccentricities are small, because the pairs break out of resonance and damp eccentricity. On the right, the eccentricities are large again because the pairs diverge far enough to cross another mean motion resonance, resulting in late-time eccentricity excitation that persists to the end of the simulation.
• Figure 3: The upper left plot shows the evolution of the semi-major axes for a three planet system under the influence of the torques derived from the protoplanetary disk. The black curve represents the innermost planet, the red curve the middle planet and the blue curve the outer planet. The dotted line tracks the location of the torque reversal. Important transitions in the system properties are marked by vertical dashed lines. The right-hand plot shows the evolution of the nearest neighbour period ratios (color coded by the outer planet in each pair). The panel in the lower right shows the evolution of the eccentricity of each planet. The remaining panels show the evolution of different resonant angles. The two on the left show the two-body resonant angles for the outer pair (4:3) and inner pair (5:4), color-coded by the planetary precession term in the resonant argument. The magenta curve on the right shows the evolution of the three-body resonant angle.The system masses are in units of $M_{\oplus}$.
• Figure 4: The upper left plot shows the evolution of the semi-major axes for a three planet system under the influence of the torques derived from the protoplanetary disk. The black curve represents the innermost planet, the red curve the middle planet and the blue curve the outer planet. Important transitions in the system properties are marked by vertical dashed lines. The dotted line tracks the location of the torque reversal. The right-hand plot shows the evolution of the nearest neighbour period ratios (color coded by the outer planet in each pair). The panel in the lower right shows the evolution of the eccentricity of each planet. The remaining panels show the evolution of different resonant angles. The two on the left show the two-body resonant angles for the outer pair (2:1) and inner pair (2:1), color-coded by the planetary precession term in the resonant argument. The magenta curve on the right shows the evolution of the three-body resonant angle.The system masses are in units of $M_{\oplus}$.
• Figure 5: The upper panels show the evolution of the semi-major axes (left) and nearest neighbor period ratios (right) for a six planet migrating system. The panel in the lower right shows the corresponding evolution of the eccentricities. The other panels show the cosines of the most important resonant angle for each of the 5 nearest neighbour pairs, with the colours corresponding to the planets in the upper left panel. As before, the dotted line in the upper left shows the location of the torque reversal, and the vertical dashed lines indicate important epochs (when resonances are established or broken).