Tidal Effects on Planetary Migration in Systems Hosting a Hot Jupiter and an Inner Companion

The formation and evolution of planetary systems are conducted by a series of complex physical processes, with planetary migration playing a crucial role. Traditional models suggested that the presence of gas giants in very close proximity to their host stars—known as hot Jupiters—would be incompatible with the existence of low-mass planets in the inner regions of the system. It was believed that the inward migration of these giants would either expel or destroy smaller planets along the way. However, recent observations have challenged this view, revealing multiple systems where hot Jupiters coexist with super-Earths or mini-Neptunes in inner orbits. This finding suggests the need to rethink current models of system evolution and to further investigate the mechanisms that make such configurations possible.

In this study, we examine the role of tidal forces in the planetary migration of compact multi-planet systems, focusing on TOI-1130. This system consists of two exoplanets: TOI-1130 b, a Neptune-like planet with an orbital period of 4.1 days, and TOI-1130 c, a gas giant with a period of 8.4 days. The close orbital proximity of both bodies and their host star indicates migration processes, and perhaps even resonance interactions must have played an important role in sculpting the current system architecture.

Planetary migration refers to the gradual change in a planet’s semi-major axis over time, altering its distance from the star. This process can explain the presence of gas giants in extremely close-in orbits, regions where they are unlikely to have formed in situ, as well as the compact arrangement of multiple planets and the presence of orbital resonances that lock their periods into specific ratios.

The primary drivers of migration include interactions with the protoplanetary disc during the system’s early stages, when planets are still embedded in the surrounding gas and dust. Gravitational torques between a planet and the disc generate density waves that redistribute angular momentum. After the disc dissipates, tidal effects become increasingly important. Gravitational interactions with the host star deform the planets and star, leading to energy dissipation and a gradual inward drift of the orbits.

Tidal forces, in particular, have a pronounced impact on close-in systems. The difference in gravitational pull between the near and far sides of a planet causes periodic deformation, and the efficiency of energy dissipation depends on the planet’s internal viscosity. This efficiency is characterized by parameters such as Love numbers and relaxation times. Gas giants, with more deformable interiors, tend to dissipate energy more effectively, thus accelerating their inward migration.

In the case of TOI-1130, its current orbital configuration suggests a strong influence from tidal forces, possibly combined with past resonance capture between the two planets. The dynamical history of the system may include phases of convergent migration, resonance capture, and later decoupling, leading to the observed configuration.

To explore this scenario, we employed the TIDYMESS numerical code (Tidal Dynamics of Multi-Body Extra-Solar Systems), developed to simulate the orbital and rotational evolution of exoplanetary systems under tidal influence. TIDYMESS uses a viscoelastic creep-type deformation model, which is well-suited to representing planets with diverse internal structures. The code allows for detailed customization of each body’s physical parameters, such as Love numbers, relaxation times, and moments of inertia, and accounts for mutual gravitational interactions between multiple planets and their host star.

The simulations conducted for the TOI-1130 system aimed to reproduce possible evolutionary trajectories that could explain the current positions of the two planets. We explored scenarios with varying initial configurations to determine a stable configuration similar to the one observed today.