Creep tide model for the three-body problem

Abstract

We present a tidal model for treating the rotational evolution in the general three-body problem with arbitrary viscosities, in which all the masses are considered to be extended and all the tidal interactions between pairs are taken into account. Based on the creep tide theory, we present a set of differential equations that describes the rotational evolution of each body, in a formalism that is easily extensible to the N tidally interacting body problem. We apply our model to the case of a circumbinary planet and use a Kepler-38 like binary system as a working example. We find that, in this low planetary eccentricity case, the most likely final stationary rotation state is the 1:1 spin–orbit resonance, considering an arbitrary planetary viscosity inside the estimated range for the Solar System planets. The timescales for reaching the equilibrium state are expected to be approximately millions of years for stiff bodies but can be longer than the age of the system for planets with a large gaseous component. We derive analytical expressions for the mean rotational stationary state, based on high-order power series of the ratio of the semimajor axes a1∕a2 and low-order expansions of the eccentricities. These are found to very accurately reproduce the mean behaviour of the low-eccentric numerical integrations for arbitrary planetary relaxation factors, and up to a1∕a2 ~ 0.4. Our analytical model is used to predict the stationary rotation of the Kepler circumbinary planets and we find that most of them are probably rotating in a subsynchronous state, although the synchrony shift is much less important than our previous estimations. We present a comparison of our results with those obtained with the Constant Time Lag and find that, as opposed to the assumptions in our previous works, the cross torques have a non-negligible net secular contribution, and must be taken into account when computing the tides over each body in an N-extended-body system from an arbitrary reference frame. These torques are naturally taken into account in the creep theory. In addition to this, the latter formalism considers more realistic rheology that proved to reduce to the Constant Time Lag model in the gaseous limit and also allows several additional relevant physical phenomena to be studied.

Orbital evolution of circumbinary planets due to creep tides

“…In the framework of the constant time lag (CTL) model (e.g., Mignard 1979), we have been investigating the tidal evolution of CB planets with the 3BP, and found results remarkably different from those obtained with the 2BP approach (e.g., Correia et al 2016). For example, regarding the spin evolution, we found different pseudo-synchronous stationary states, which may even be subsynchronous for low-eccentricity CB planets (Zoppetti et al 2019(Zoppetti et al , 2020. Moreover, according to our model the orbital evolution is perhaps even more interesting.…”

“…In Zoppetti et al (2019Zoppetti et al ( , 2020, we used the CTL model and found that for Kepler-like CB systems the characteristic tidal timescales of the planetary rotational evolution are typically much lower than those associated with the orbital evolution. Thus, CB planets are expected to have a significant orbital evolution only after reaching rotational stationary states.…”

“…With these two motivations, in what follows we study the orbital evolution of a CB planet whose rotational evolution has reached the pseudo-synchronous state. As initial conditions of our numerical simulations, we chose a Kepler-38-type system (Orosz et al 2012), which has been the working example of several of our previous works (e.g., Zoppetti et al 2018Zoppetti et al , 2019Zoppetti et al , 2021. The nominal values for the system parameters and initial orbital elements are detailed in Table 1.…”

“…Moreover, according to our model the orbital evolution is perhaps even more interesting. Once the CB planet acquires its stationary rotational state, it can migrate outward or inward, depending on the system parameters (Zoppetti et al 2018(Zoppetti et al , 2019(Zoppetti et al , 2020. However, weak-friction models like the CTL are expected to only be valid for stars and gaseous planets, which dissipate small amounts of tidal energy per cycle (e.g., Lainey et al 2009) but not accurate for stiff bodies, whose energy dissipation does not fit with this regime (e.g., Efroimsky 2012Efroimsky , 2015.…”

“…With the aim of having a tidal model able to describe the dynamical evolution of CB planets with arbitrary viscosities, in Zoppetti et al (2021) (hereafter Z2021) we adapted the creep tide model (Ferraz-Mello 2013) to the 3BP. The Newtonian creep theory has proven to be analogous to models that consider the extended masses as Maxwellian bodies (e.g., Correia et al 2014), but without taking into account the elastic component of the tides (Ferraz-Mello 2015).…”