# The tidal parameters of TRAPPIST-1 b and c

## TL;DR

This study estimates the tidal parameters of TRAPPIST-1 b and c using dynamical simulations and interior models, providing insights into their tidal evolution and system stability.

## Contribution

It combines dynamical N-body simulations with interior modeling to constrain the tidal parameters of planets in a multi-resonant system, a novel integrated approach.

## Key findings

- Planets typically become unstable around 30 Myr after formation.
- Estimated tidal quality factor ratios are $rac{k_2}{Q} ext{ for b} ext{ and c}$, with specific bounds.
- Interior models provide independent estimates of tidal parameters, aiding in understanding system evolution.

## Abstract

The TRAPPIST-1 planetary system consists of seven planets within 0.05 au of each other, five of which are in a multi-resonant chain. {These resonances suggest the system formed via planet migration; subsequent tidal evolution has damped away most of the initial eccentricities. We used dynamical N-body simulations to estimate how long it takes for the multi-resonant configuration that arises during planet formation to break. From there we use secular theory to pose limits on the tidal parameters of planets b and c. We calibrate our results against multi-layered interior models constructed to fit the masses and radii of the planets, from which the tidal parameters are computed independently.} The dynamical simulations show that the planets typically go unstable 30 Myr after their formation. {Assuming synchronous rotation throughout} we compute $\frac{k_2}{Q} \gtrsim 2\times 10^{-4}$ for planet b and $\frac{k_2}{Q} \gtrsim 10^{-3}$ for planet c. Interior models yield $(0.075-0.37) \times 10^{-4}$ for TRAPPIST-1 b and $(0.4-2)\times 10^{-4}$ for TRAPPIST-1 c. The agreement between the {dynamical and interior} models is not too strong, but is still useful to constrain the dynamical history of the system. We suggest that this two-pronged approach could be of further use in other multi-resonant systems if the planet's orbital and interior parameters are sufficiently well known.