Habitability of the early Earth: liquid water under a faint young Sun facilitated by strong tidal heating due to a closer Moon

Geological evidence suggests liquid water near the Earth’s surface as early as 4.4 gigayears ago when the faint young Sun only radiated about 70% of its modern power output. At this point, the Earth should have been a global snowball if it possessed atmospheric properties similar to those of the modern Earth. An extreme atmospheric greenhouse effect, an initially more massive Sun, release of heat acquired during the accretion process of protoplanetary material, and radioactivity of the early Earth material have been proposed as reservoirs or traps for heat. For now, the faint-young-Sun paradox persists as an important problem in our understanding of the origin of life on Earth. Here, we use the constant-phase-lag tidal theory to explore the possibility that the new-born Moon, which formed about 69 million years (Myr) after the ignition of the Sun, generated extreme tidal friction—and therefore, heat—in the Hadean and possibly the Archean Earth. We show that the Earth–Moon system has lost $${\sim }3~{\times }~10^{31}$$ ∼ 3 × 10 31 J (99% of its initial mechanical energy budget) as tidal heat. Tidal heating of $${\sim }10\,\mathrm{W\,m}^{-2}$$ ∼ 10 W m - 2 through the surface on a time scale of 100 Myr could have accounted for a temperature increase of up to $$5\,^\circ $$ 5 ∘ C on the early Earth. This heating effect alone does not solve the faint-young-Sun paradox but it could have played a key role in combination with other effects. Future studies of the interplay of tidal heating, the evolution of the solar power output, and the atmospheric (greenhouse) effects on the early Earth could help in solving the faint-young-Sun paradox.

The impact of tidal friction evolution on the orbital decay of ultra-short period planets

Unveiling the fate of ultra-short period (USP) planets may help us understand the qualitative agreement between tidal theory and the observed exoplanet distribution. Nevertheless, due to the time-varying interchange of spin-orbit angular momentum in star-planet systems, the expected amount of tidal friction is unknown and depends on the dissipative properties of stellar and planetary interiors. In this work, we couple structural changes in the star and the planet resulting from the energy released per tidal cycle and simulate the orbital evolution of USP planets and the spin-up produced on their host star. For the first time, we allow the strength of magnetic braking to vary within a model that includes photo-evaporation, drag caused by the stellar wind, stellar mass loss, and stellar wind enhancement due to the in-falling USP planet. We apply our model to the two exoplanets with the shortest periods known to date, NGTS-10b and WASP-19b. We predict they will undergo orbital decay in time-scales that depend on the evolution of the tidal dissipation reservoir inside the star, as well as the contribution of the stellar convective envelope to the transfer of angular momentum. Contrary to previous work, which predicted mid-transit time shifts of ∼30 − 190 s over 10 years, we found that such changes would be smaller than 10 s. We note this is sensitive to the assumptions about the dissipative properties of the system. Our results have important implications for the search for observational evidence of orbital decay in USP planets, using present and future observational campaigns.

Convective turbulent viscosity acting on equilibrium tidal flows: new frequency scaling of the effective viscosity

ABSTRACT Turbulent convection is thought to act as an effective viscosity (νE) in damping tidal flows in stars and giant planets. However, the efficiency of this mechanism has long been debated, particularly in the regime of fast tides, when the tidal frequency (ω) exceeds the turnover frequency of the dominant convective eddies (ωc). We present the results of hydrodynamical simulations to study the interaction between tidal flows and convection in a small patch of a convection zone. These simulations build upon our prior work by simulating more turbulent convection in larger horizontal boxes, and here we explore a wider range of parameters. We obtain several new results: (1) νE is frequency dependent, scaling as ω−0.5 when ω/ωc ≲ 1, and appears to attain its maximum constant value only for very small frequencies (ω/ωc ≲ 10−2). This frequency reduction for low-frequency tidal forcing has never been observed previously. (2) The frequency dependence of νE appears to follow the same scaling as the frequency spectrum of the energy (or Reynolds stress) for low and intermediate frequencies. (3) For high frequencies (ω/ωc ≳ 1 − 5), νE ∝ ω−2. 4) The energetically dominant convective modes always appear to contribute the most to νE, rather than the resonant eddies in a Kolmogorov cascade. These results have important implications for tidal dissipation in convection zones of stars and planets, and indicate that the classical tidal theory of the equilibrium tide in stars and giant planets should be revisited. We briefly touch upon the implications for planetary orbital decay around evolving stars.

Tidal evolution of exoplanetary systems hosting potentially habitable exoplanets. The cases of LHS-1140 b-c and K2-18 b-c

ABSTRACT We present a model to study secularly and tidally evolving three-body systems composed by two low-mass planets orbiting a star, in the case where the bodies rotation axes are always perpendicular to the orbital plane. The tidal theory allows us to study the spin and orbit evolution of both stiff Earth-like planets and predominantly gaseous Neptune-like planets. The model is applied to study two recently discovered exoplanetary systems containing potentially habitable exoplanets (PHE): LHS-1140 b-c and K2-18 b-c. For the former system, we show that both LHS-1140 b and c must be in nearly circular orbits. For K2-18 b-c, the combined analysis of orbital evolution time-scales with the current eccentricity estimation of K2-18 b allows us to conclude that the inner planet (K2-18 c) must be a Neptune-like gaseous body. Only this would allow for the eccentricity of K2-18 b to be in the range of values estimated in recent works (e = 0.20 ± 0.08), provided that the uniform viscosity coefficient of K2-18 b is greater than 2.4 × 1019 Pa s (which is a value characteristic of stiff bodies) and supposing that such system has an age of some Gyr.

A method to identify the boundary between rocky and gaseous exoplanets from tidal theory and transit durations

The determination of an exoplanet as rocky is critical for the assessment of planetary habitability. Observationally, the number of small-radius, transiting planets with accompanying mass measurements is insufficient for a robust determination of the transitional mass or radius. Theoretically, models predict that rocky planets can grow large enough to become gas giants when they reach ~10 MEarth, but the transitional mass remains unknown. Here I show how transit data, interpreted in the context of tidal theory, can reveal the critical radius that separates rocky and gaseous exoplanets. Standard tidal models predict that rocky exoplanets’ orbits are tidally circularized much more rapidly than gaseous bodies’, suggesting the former will tend to be found on circular orbits at larger semi-major axes than the latter. Well-sampled transits can provide a minimum eccentricity of the orbit, allowing a measurement of this differential circularization. I show that this effect should be present in the data from the Kepler spacecraft, but is not apparent. Instead, it appears that there is no evidence of tidal circularization at any planetary radius, probably because the publicly-available data, particularly the impact parameters, are not accurate enough. I also review the bias in the transit duration towards values that are smaller than that of planets on circular orbits, stressing that the azimuthal velocity of the planet determines the transit duration. The ensemble of Kepler planet candidates may be able to determine the critical radius between rocky and gaseous exoplanets, tidal dissipation as a function of planetary radius, and discriminate between tidal models.

Paolo Sarpi and the first Copernican tidal theory

Despite his demanding religious responsibilities, Paolo Sarpi maintained an active involvement in science between 1578 and 1598 – as hisPensierireveal. They show that from 1585 onwards he studied the Copernican theory and recorded arguments in its favour. The fact that for 1595 they include an outline of a Copernican tidal theory resembling Galileo'sDialoguetheory is well known. But examined closely, Sarpi's theory is found to be different from that of theDialoguein several important respects. That Sarpi was a Copernican by 1592 is revealed by other of hispensieri, whereas at that time we know that Galileo was not. The examination of Sarpi's tidal theory and of the work of Galileo in this period indicates that the theory Sarpi recorded in 1595 was of his own creation. The appreciation that the theory was Sarpi's and that Galileo subsequently came to change his views on the Copernican theory and adopted the tidal theory has major implications for our understanding of the significance of Sarpi's contribution to the Scientific Revolution. Moreover, it appears that several of the most significant theoretical features of the tidal theory published by Galileo in theDialogue –and which proved of lasting value – were in reality Sarpi's.