The long-term tidal dynamics of differentiated rocky exoplanets

Overview

Tidal interaction plays a unique role in coupling the thermal, orbital, and rotational evolution of close-in moons and exoplanets. The reaction of a planetary body subjected to tidal loading is determined by its interior structure, rheological properties, and orbital parameters. The interior and orbital dynamics are, conversely, affected by the tidal dissipation. In this study, we address the parameter dependence of stable spin states and tidal heating, as well as the long-term coupled thermal-orbital evolution of rocky exoplanets. Special attention is paid to the consistent evolution of spin rate and to the role of an emerging subsurface magma ocean in the secular maintenance of nonzero orbital eccentricity.

 

Model and Methods

Our model consists of several modules and relies on the semi-analytical description of the thermal and orbital dynamics. All processes are interconnected through the tidal dissipation, which drains the energy from the orbit and presents an additional heat source for the planetary interior. The tidal heat rate (e.g., Segatz 1988; Efroimsky and Makarov, 2014) as well as the evolution equations for the orbital parameters and the spin rate are calculated using the Darwin-Kaula expansion into modes corresponding to different tidal frequencies (Kaula, 1964). To quantify the deformation and the additional potential of the deformed planet, we calculate the complex tidal Love numbers k₂^*(ω) (e.g., Castillo-Rogez et al., 2011) using the equations of the normal mode theory (e.g., Sabadini and Vermeersen, 2004). This approach permits us to account for a differentiated interior of the planet.

The thermal evolution of the model planets is described by a parametrized model of stagnant lid convection (e.g., Grott and Breuer, 2008) with basal and volumetric heating, where the major heat source is provided by the tidal dissipation.  The internal dynamics are coupled to the rest of the model through the temperature dependence of mantle viscosity, the evolution of the stagnant lid thickness and the emergence of a stable subsurface magma ocean. All changes in the interior structure and rheological properties are reflected by the tidal Love number, which determines the tidal response and the rate of energy dissipation.

 

Results

The first goal of this study is to explore the parametric dependence of the tidal dissipation and the spin-orbit locking. We study the effect of different rheological parameters, the planet size, the interior structure, and the eccentricity. As a result of the viscoelastic rheology, the model planet can get locked into higher than synchronous spin-orbit resonances, which provide an important source of dissipation especially in the case of relatively low orbital eccentricities. We also find that planets with smaller radii are more likely to get locked into higher resonances than larger or more massive planets. In addition to a parametric study of a generic terrestrial exoplanet, we also apply the model on three currently known low-mass exoplanet candidates: Proxima Centauri b, GJ 411 b, and GJ 625 b.

The second goal is the assessment of the mutual interconnection between the secular thermal and orbital evolution. We investigate the evolution paths of the three mentioned exoplanets and observe that the thermal evolution proceeds as a sequence of thermal equilibria. The equilibrium temperature profiles of the tidally evolving exoplanets are governed primarily by the stable spin-orbit ratio, which is determined by the mantle viscosity and eccentricity. The temperature-driven changes in the interior and the transitions between different spin-orbit resonances also affect the rate of orbital evolution. Specifically, the final despinning of a planet to the synchronous rotation may slow down the circularization of the planetary orbit and help to maintain nonzero orbital eccentricity for a considerable time. Using the coupled model, we also study the effect of different initial eccentricities and mantle viscosities on the current thermal and rotational state of the three exoplanets and compare the resulting eccentricities with their current, empirically given values (Figure 1).

 

Figure 1: Orbital and thermal characteristics of Proxima Centauri b after 5 Gyr of coupled thermal-orbital evolution; adapted from Walterova and Behounkova (2020). Depending on the initial orbital eccentricity (x-axis) and the reference viscosity at temperature T₀=1600 K (y-axis), the individual panels depict the evolved eccentricity (left), the spin-orbit ratio (middle) and the surface tidal heat flux (right). Light blue areas correspond to the model parameters for which the evolved eccentricity complies with observation (Jenkins et al., 2019). The range of the empirically given values is also indicated by a red line in the first colorbar.

 

Conclusion

The orbital evolution of strongly tidally loaded exoplanets, whose thermal state is shaped by the tidal heating, is naturally interconnected with their internal dynamics. The changes in the interior, namely the partial melting and the formation of a subsurface magma ocean, translate into the decrease in the rate of orbital evolution and in the tidal heat generation. The understanding of this complex feedback may help us to better constrain the conditions on extrasolar worlds and to address their hypothetical habitability.

 

Acknowledgements

The research leading to these results received funding from the Czech Science Foundation through project No. 19-10809S and from Charles University through project SVV 115-09/260581.

 

References

[1] M. Segatz, T. Spohn, M. N. Ross, and G. Schubert. Icarus, 75(2):187–206,1988.

[2] M. Efroimsky and V. V. Makarov. The Astrophysical Journal, 795(1):19, 2014.

[3] W. M. Kaula. Reviews of Geophysics, 2(4):661–685, 1964.

[4] J. C. Castillo-Rogez, M. Efroimsky, and V. Lainey. Journal of Geophysical Research, 116(E09008), 2011.

[5] R. Sabadini and B. Vermeersen. Global Dynamics of the Earth: Applications of Normal Mode Relaxation Theory to Solid-Earth Geophysics. Kluwer Academic Publishers, Dordrecht, the Netherlands, 2004.

[6] M. Grott and D. Breuer. Icarus, 193(2):503–515, 2008.

[7] M. Walterova and M. Behounkova. Submitted to The Astrophysical Journal.

[8] J. S. Jenkins, J. Harrington, R. C. Challener, et al. Monthly Notices of the Royal Astronomical Society, 487(1):268–274, 2019.