Tidal evolution of rocky (exo)planets

The long-term dynamics of close-in terrestrial exoplanets are strongly influenced by tidal interaction with the host star. Periodic tidal loading results in time-varying deformation, which is typically accompanied by energy dissipation. This phenomenon has two important consequences: First, the produced heat might enhance the interior dynamics of the planet and even trigger structural changes in its interior, such as partial melting. Second, the lost (or transferred) energy fuels orbital evolution, i.e., it leads to secular changes in the semi-major axis and eccentricity. Along with the orbital evolution, the spin rate of the planet evolves as well, such that the total angular momentum in the system is conserved. Here, we focus on coupling the effects introduced above. Combining semi-analytical modelling of secular spin-orbital evolution of multilayered planets [1,2,3] with parameterised 1d mantle convection [4,5], we illustrate the feedback between the thermal state of a planet, its spin rate, and the rate of orbit circularisation. As one of the results, we also show how a single planet orbiting a single star can retain nonzero orbital eccentricity by forming a subsurface magma ocean. In addition to the primary topic of this work, we further explore the dependence of stable spin states (spin-orbit resonances) on the planet's interior structure and orbital eccentricity, as well as the parameter dependence of tidal heating. [1] Kaula (1964), Rev. Geophys. and Space Phys., 2:661-685, doi:10.1029/RG002i004p00661. [2] Boué & Efroimsky (2019), CM&DA, 131(7):30, doi:10.1007/s10569-019-9908-2. [3] Sabadini & Vermeersen (2004), Kluwer Academic Publishers, Dodrecht, the Netherlands, ISBN: 9781402012686. [4] Grott and Breuer (2008), Icarus, 193(2):503-515, doi:10.1016/j.icarus.2007.08.015. [5] Tosi et al. (2017), A&A, 605:A71, doi:10.1051/0004-6361/201730728

Coupled thermal and orbital evolution of tidally-loaded exoplanets

The long-term dynamics of close-in terrestrial exoplanets are strongly influenced by tidal interaction with the host star. Periodic tidal loading results in time-varying deformation, which is typically accompanied by energy dissipation. This phenomenon has two important consequences: First, the produced heat might enhance the interior dynamics of the planet and even trigger structural changes in its interior, such as partial melting. Second, the lost (or transferred) energy fuels orbital evolution, i.e., it leads to secular changes in the semi-major axis and eccentricity. Along with the orbital evolution, the spin rate of the planet evolves as well, such that the total angular momentum in the system is conserved. Here, we focus on coupling the effects introduced above. Combining semi-analytical modeling of the long-term spin-orbital evolution of anelastic multilayered planets with parameterised 1d approach to mantle convection, we illustrate the feedback between the thermal state of a planet, its spin rate, and the rate of orbit circularisation. In addition to the primary topic of this work, we explore the parameter dependence of stable spin states (spin-orbit resonances) for layered bodies, the parameter dependence of tidal heat rate, and the possible contribution of inter-planetary tides to the spin rate evolution in tightly-packed multi-planetary systems

Coupled thermal and orbital evolution of tidally-loaded exoplanets

The long-term dynamics of close-in terrestrial exoplanets are strongly influenced by tidal interaction with the host star. Periodic tidal loading results in time-varying deformation, which is typically accompanied by energy dissipation. This phenomenon has two important consequences: First, the produced heat might enhance the interior dynamics of the planet and even trigger structural changes in its interior, such as partial melting. Second, the lost (or transferred) energy fuels orbital evolution, i.e., it leads to secular changes in the semi-major axis and eccentricity. Along with the orbital evolution, the spin rate of the planet evolves as well, such that the total angular momentum in the system is conserved. Here, we focus on coupling the effects introduced above. Combining semi-analytical modeling of the long-term spin-orbital evolution of anelastic multilayered planets with parameterised 1d approach to mantle convection, we illustrate the feedback between the thermal state of a planet, its spin rate, and the rate of orbit circularisation. In addition to the primary topic of this work, we explore the parameter dependence of stable spin states (spin-orbit resonances) for layered bodies, the parameter dependence of tidal heat rate, and the possible contribution of inter-planetary tides to the spin rate evolution in tightly-packed multi-planetary systems

Tidal response of the Moon: with and without a weak basal layer

Interpretation of the data obtained throughout the period of more than 40 years by Lunar Laser Ranging yields an interesting observation: the tidal quality factor of the Moon, which determines the magnitude of ongoing energy dissipation, follows a different frequency dependence than is measured for rocks in laboratory conditions. When the self-gravity of the lunar body is taken into account, the detected frequency dependence can be interpreted as a signal coming from strong dissipation at the lunar base, indicating a deep-seated layer with low viscosity and possibly containing partial melt. Such a layer would be consistent with the non-detection of deep farside moonquakes by nearside seismic stations and is often associated with the ilmenite-bearing cumulates, that are thought to have descended onto the core-mantle boundary during the lunar magma ocean solidification. Alternative models of a melt-free lunar mantle have also been proposed. These models fit the tidal quality factor Q at the monthly frequency but were unable to explain its observed frequency dependence. In this study, we propose a melt-free model, in which the frequency dependence of lunar Q emerges due to elastically accommodated grain-boundary sliding (GBS) in the lunar mantle. We discuss the implications of such a model and compare it with the traditional approach, which assumes a highly dissipative basal layer. For both alternatives, we perform a Bayesian inversion of the measured tidal parameters (tidal quality factor Q at the monthly and the annual frequency, degree-2 Love numbers h2, k2, and degree-3 Love number k3) and predict either the conditions at the base of the lunar mantle or the relaxation time of elastically accommodated GBS. Since the two alternative models prove to be indistinguishable from each other by tidal measurements, we conclude with an outlook on future observations

Is there a semi-molten layer at the base of the lunar mantle?

Interpretation of the data obtained throughout the period of more than 40 years by Lunar Laser Ranging yields an interesting observation: the tidal quality factor of the Moon, which determines the magnitude of ongoing energy dissipation, follows a different frequency dependence than is measured for rocks in laboratory conditions. When the self-gravity of the lunar body is taken into account, the detected frequency dependence can be interpreted as a signal coming from strong dissipation at the lunar base, indicating a deep-seated layer with low viscosity and possibly containing partial melt. Such a layer would be consistent with the non-detection of deep farside moonquakes by nearside seismic stations and is often associated with the ilmenite-bearing cumulates, that are thought to have descended onto the core-mantle boundary during the lunar magma ocean solidification. Alternative models of a melt-free lunar mantle have also been proposed. These models fit the tidal quality factor Q at the monthly frequency but were unable to explain its observed frequency dependence. In this study, we propose a melt-free model, in which the frequency dependence of lunar Q emerges due to elastically accommodated grain-boundary sliding (GBS) in the lunar mantle. We discuss the implications of such a model and compare it with the traditional approach, which assumes a highly dissipative basal layer. For both alternatives, we perform a Bayesian inversion of the measured tidal parameters (tidal quality factor Q at the monthly and the annual frequency, degree-2 Love numbers h2, k2, and degree-3 Love number k3) and predict either the conditions at the base of the lunar mantle or the relaxation time of elastically accommodated GBS. Since the two alternative models prove to be indistinguishable from each other by tidal measurements, we conclude with an outlook on future observations

Is there a semi-molten layer at the base of the lunar mantle?

Interpretation of the data obtained by Lunar Laser Ranging provides an interesting observation: the tidal quality factor of the Moon, which determines the magnitude of ongoing energy dissipation, follows a different frequency dependence than is measured for rocks in laboratory conditions (e.g., [1]). When the self-gravity of the lunar body is taken into account, the detected frequency dependence can be interpreted as a signal coming from strong dissipation at the lunar base [2], indicating a deep-seated layer with low viscosity and possibly containing partial melt (e.g., [3]). Such a layer would be consistent with the non-detection of deep moonquakes originating around the lunar antipode by nearside seismometers [4] and is often associated with ilmenite-bearing cumulates, that are thought to have descended onto the core-mantle boundary during the lunar magma ocean solidification. Alternative models of a melt-free lunar mantle have also been proposed (e.g., [5]). These models fit the tidal quality factor Q at the monthly frequency but do not explain its observed frequency dependence. Here, we propose a melt-free model, in which the frequency dependence of lunar Q emerges due to elastically accommodated grain-boundary sliding (GBS) in the lunar mantle [5,6]. We discuss the implications of such a model and compare it with the traditional approach, which assumes a highly dissipative basal layer. For both alternatives, we perform a Bayesian inversion of the measured tidal parameters (tidal quality factor Q and tidal Love numbers) and predict either the conditions at the base of the lunar mantle or the relaxation time of elastically accommodated GBS. Since the two models prove to be indistinguishable from each other by tidal measurements, we conclude with several suggestions for future missions. [1] Williams & Boggs (2014), JGR: Planets, 120(4):689-724, doi:10.1002/2014JE004755. [2] Harada et al. (2014), Nat. Geosci., 7(8):569-572, doi:10.1038/ngeo2211. [3] Khan et al. (2014), JGR: Planets, 119(10):2197-2221, doi:10.1002/2014JE004661. [4] Nakamura (2005), JGR: Planets, 110(E1):E01001, doi:10.1029/2004JE002332. [5] Nimmo, Faul, & Garnero (2012), JGR: Planets, 117(E9):E09005, doi:10.1029/2012JE004160. [6] Sundberg & Cooper (2010), Philos. Mag., 90(20):2817-2840, doi:10.1080/14786431003746656

Is there a partially molten layer at the base of the lunar mantle?

Interpretation of the data obtained by Lunar Laser Ranging provides an interesting observation: the tidal quality factor of the Moon, which determines the magnitude of ongoing energy dissipation, follows a different frequency dependence than is measured for rocks in laboratory conditions (e.g., [1]). When the self-gravity of the lunar body is taken into account, the detected frequency dependence can be interpreted as a signal coming from strong dissipation at the lunar base [2], indicating a deep-seated layer with low viscosity and possibly containing partial melt (e.g., [3]). Such a layer would be consistent with the non-detection of deep moonquakes originating around the lunar antipode by nearside seismometers [4] and is often associated with ilmenite-bearing cumulates, that are thought to have descended onto the core-mantle boundary during the lunar magma ocean solidification. Alternative models of a melt-free lunar mantle have also been proposed (e.g., [5]). These models fit the tidal quality factor Q at the monthly frequency but do not explain its observed frequency dependence. Here, we propose a melt-free model, in which the frequency dependence of lunar Q emerges due to elastically accommodated grain-boundary sliding (GBS) in the lunar mantle [5,6]. We discuss the implications of such a model and compare it with the traditional approach, which assumes a highly dissipative basal layer. For both alternatives, we perform a Bayesian inversion of the measured tidal parameters (tidal quality factor Q and tidal Love numbers) and predict either the conditions at the base of the lunar mantle or the relaxation time of elastically accommodated GBS. Since the two models prove to be indistinguishable from each other by tidal measurements, we conclude with several suggestions for future missions. [1] Williams & Boggs (2014), JGR: Planets, 120(4):689-724, doi:10.1002/2014JE004755. [2] Harada et al. (2014), Nat. Geosci., 7(8):569-572, doi:10.1038/ngeo2211. [3] Khan et al. (2014), JGR: Planets, 119(10):2197-2221, doi:10.1002/2014JE004661. [4] Nakamura (2005), JGR: Planets, 110(E1):E01001, doi:10.1029/2004JE002332. [5] Nimmo, Faul, & Garnero (2012), JGR: Planets, 117(E9):E09005, doi:10.1029/2012JE004160. [6] Sundberg & Cooper (2010), Philos. Mag., 90(20):2817-2840, doi:10.1080/14786431003746656

Tidal response of the Moon: with and without a weak basal layer

Interpretation of the data obtained throughout the period of more than 40 years by Lunar Laser Ranging yields an interesting observation: the tidal quality factor of the Moon, which determines the magnitude of ongoing energy dissipation, follows a different frequency dependence than is measured for rocks in laboratory conditions (e.g., [1,2]). When the self-gravity of the lunar body is taken into account, the detected frequency dependence can be interpreted as a signal coming from strong dissipation at the lunar base [3], indicating a deep-seated layer with low viscosity and possibly containing partial melt (e.g., [1,4,5]). Such a layer would be consistent with the non-detection of farside moonquakes by nearside seismic stations [6] and is often associated with the ilmenite-bearing cumulates, that are thought to have descended onto the core-mantle boundary during the lunar magma ocean solidification (e.g., [7]). Alternative models of a melt-free lunar mantle have also been proposed [8,9,10]. These models fit the tidal quality factor Q at the monthly frequency but were unable to explain its observed frequency dependence. In this study, we propose a melt-free model, in which the frequency dependence of lunar Q emerges due to elastically accommodated grain-boundary sliding (GBS) in the lunar mantle (e.g., [11]). We discuss the implications of such a model and compare it with the traditional approach, which assumes a highly dissipative basal layer. For both alternatives, we perform a Bayesian inversion of the measured tidal parameters (tidal quality factor Q at the monthly and the annual frequency, degree-2 Love numbers h2, k2, and degree-3 Love number k3) and predict either the conditions at the base of the lunar mantle or the relaxation time of elastically accommodated GBS. Since the two alternative models prove to be indistinguishable from each other by tidal measurements, we conclude with an outlook on future observations. References: [1] Williams, J. G., Boggs, D. H., Yoder, C. F., Ratcliff, J. T., and Dickey, J. O., "Lunar rotational dissipation in solid body and molten core", Journal of Geophysical Research, vol. 106, no. E11, pp. 27933-27968, 2001. doi:10.1029/2000JE001396. [2] Williams, J. G. and Boggs, D. H., "Tides on the Moon: Theory and determination of dissipation", Journal of Geophysical Research (Planets), vol. 120, no. 4, pp. 689-724, 2015. doi:10.1002/2014JE004755. [3] Harada, Y., Goossens, S, Matsumoto, K., Yan, J., Ping, J., Noda, H., Haruyama, J., "Strong tidal heating in an ultralow-viscosity zone at the core-mantle boundary of the Moon", Nature Geoscience, vol. 7, no. 8, pp. 569-572, 2014. doi:10.1038/ngeo2211. [4] Efroimsky, M., "Bodily tides near spin-orbit resonances", Celestial Mechanics and Dynamical Astronomy, vol. 112, no. 3, pp. 283-330, 2012. doi:10.1007/s10569-011-9397-4. [5] Khan, A., Connolly, J. A. D., Pommier, A., and Noir, J., "Geophysical evidence for melt in the deep lunar interior and implications for lunar evolution", Journal of Geophysical Research (Planets), vol. 119, no. 10, pp. 2197-2221, 2014. doi:10.1002/2014JE004661. [6] Nakamura, Y., Lammlein, D., Latham, G., Ewing, M., Dorman, J., Press, F., Toksoz, N., "New Seismic Data on the State of the Deep Lunar Interior", Science, vol. 181, no. 4094, pp. 49-51, 1973. doi:10.1126/science.181.4094.49. [7] Zhao, Y., de Vries, J., van den Berg, A. P., Jacobs, M. H. G., and van Westrenen, W., "The participation of ilmenite-bearing cumulates in lunar mantle overturn", Earth and Planetary Science Letters, vol. 511, pp. 1-11, 2019. doi:10.1016/j.epsl.2019.01.022. [8] Nimmo, F., Faul, U. H., and Garnero, E. J., "Dissipation at tidal and seismic frequencies in a melt-free Moon", Journal of Geophysical Research (Planets), vol. 117, no. E9, 2012. doi:10.1029/2012JE004160. [9] Karato, S.-I., "Geophysical constraints on the water content of the lunar mantle and its implications for the origin of the Moon", Earth and Planetary Science Letters, vol. 384, pp. 144-153, 2013. doi:10.1016/j.epsl.2013.10.001. [10] Matsuyama, I., Nimmo, F., Keane, J. T., Chan, N. H., Taylor, G. J., Wieczorek, M. A., Kiefer, W. S., Williams, J. G., "GRAIL, LLR, and LOLA constraints on the interior structure of the Moon", Geophysical Research Letters, vol. 43, no. 16, pp. 8365-8375, 2016. doi:10.1002/2016GL069952. [11] Sundberg, M. and Cooper, R. F., "A composite viscoelastic model for incorporating grain boundary sliding and transient diffusion creep; correlating creep and attenuation responses for materials with a fine grain size", Philosophical Magazine, vol. 90, no. 20, pp. 2817-2840, 2010. doi:10.1080/14786431003746656