Increased Tidal Dissipation Using Advanced Rheological Models: Implications for Io and Tidally Active Exoplanets

Abstract

The advanced rheological models of Andrade (1910) and Sundberg & Cooper (2010) are compared to the traditional Maxwell model to understand how each affects the tidal dissipation of heat within rocky bodies. We find both the Andrade and Sundberg-Cooper rheologies can produce at least 10× the tidal heating compared to a traditional Maxwell model for a warm (1400-1600 K) Io-like satellite. Sundberg-Cooper can cause even larger dissipation around a critical temperature and frequency. These models allow cooler planets to stay tidally active in the face of orbital perturbations-a condition we term 'tidal resilience.' This has implications for the time evolution of tidally active worlds, and the long-term equilibria they fall into. For instance, if Io's interior is better modeled by the Andrade or Sundberg-Cooper rheologies, the number of possible resonance-forming scenarios that still produce a hot, modern Io is expanded, and these scenarios do not require an early formation of the Laplace resonance. The two primary empirical parameters that define the Andrade anelasticity are examined in several phase spaces to provide guidance on how their uncertainties impact tidal outcomes, as laboratory studies continue to constrain their real values. We provide detailed reference tables on the fully general equations required for others to insert the Andrade and Sundberg-Cooper models into standard tidal formulae. Lastly, we show that advanced rheologies greatly impact the heating of short-period exoplanets and exomoons, while the properties of tidal resilience can mean a greater number of tidally active worlds among all extrasolar systems.

Tides Between the TRAPPIST-1 Planets

“…Viscosities lower than 10 14 Pa s seem unlikely unless the body is largely icy or has an extremely high degree of partial melt (e.g., Bierson & Nimmo 2016;Barr et al 2018;Renaud & Henning 2018). We therefore prefer a high viscosity scenario for these planets, which lowers the effectiveness of planet-forced tidal heating relative to eccentricity-forcing.…”

“…The absolute amount of tidal dissipation due to planet-forcing is very much of interest, but we caution that this strongly depends on the assumed rheological model and interior structure of the body. Maxwellian rheologies can greatly underestimate the amount of possible tidal dissipation, as has been demonstrated for the Jovian satellite Io (Bierson & Nimmo 2016;Renaud & Henning 2018). For that reason we do not focus on the absolute amount of tidal heating, but rather the heating ratio between these two modes of tidal forcing.…”

“…Unfortunately, modeling Andrade materials uses several more free parameters than a Maxwell material does, and some of these are difficult properties to measure in laboratory experiments. The imaginary part of the degree-2 Love number for a homogeneous Andrade body is (e.g., Renaud & Henning 2018, Table 3);…”

The subsurface habitability of small, icy exomoons

“…If a body is sufficiently nonhomogeneous (i.e., has a molten interior or differentiation, such as found in high-mass moons such as Europa), fixed Q becomes less valid and viscoelastic models must be invoked (see Renaud & Henning 2018 for a recent review). These are more complex and describe more underlying physics, accounting for motion in the mantle; however, since we focus on small, icy moons that lack fully molten interiors, a simpler and phenomenological approach is sufficient.…”

Andrade rheology in time-domain. Application to Enceladus' dissipation of energy due to forced libration

“…The Andrade creep model has been introduced in the astronomical community by Efroimsky and Lainey (2007) and Efroimsky (2012b,a). In comparison with more standard rheologies, it can significantly affect the probability of capture in spin-orbit resonance and/or amplify the amount of dissipated energy (Makarov and Efroimsky, 2013;Leconte et al, 2015;Makarov, 2015;Makarov et al, 2016;Walterová and Běhounková, 2017;Renaud and Henning, 2018). The Andrade rheology has been applied in studies of specific celestial bodies such as the Moon (Nimmo et al, 2012;Williams et al, 2014;Williams and Boggs, 2015), Mercury (Padovan et al, 2014;Noyelles et al, 2014;Knibbe and van Westrenen, 2017), Enceladus (Rambaux et al, 2010;Shoji et al, 2013;Běhounková et al, 2013Běhounková et al, , 2015Souček et al, 2019), Iapetus , Io (Bierson and Nimmo, 2016), binary asteroids (Efroimsky, 2015), GJ581 d (Makarov et al, 2012), and Proxima Century b (Ribas et al, 2016).…”