On the Tidal History and Future of the Earth–Moon Orbital System

Abstract

Earth’s rotation rate and the evolution of the Earth–Moon system have been controlled by tidal dissipation in Earth’s ocean. Attempts to model the tidal history have shown incomplete compatibility with observations and unclear isolation of the most important controlling factors. Here it is shown that a relatively simple model with no explicit description of the continents (their effects are instead parameterized) can accurately reproduce the available observations (lunar distance and month, Earth’s rotation and deceleration rates) describing the tidal evolution over the past 2.5 billion yr and also evade the paradox of an early Moon falling within the Roche limit. Notably, the model reproduces an observed dissipation peak 400 million yr ago. The model involves fitting two input parameters (the effective ocean depth h and the nondimensional dissipation timescale $\tilde{T}$ , the latter of which can be related to the more typically used quality factor Q). The best-fit values ( h = 2.3 km and $\tilde{T}=40$ , corresponding to Q ≈ 20–28) derived empirically from the model correspond well with the plausibly expected values ( h ≈ 2.5 km and Q ≈ 20) derived using independent reasoning and observations. The model shows very clearly that the tidal evolution has been primarily controlled by the rate of resonance in the ocean response, not the evolving amplitude of the tidal forces or forcing frequency. More closely, it also shows that the effect of Earth’s rotation on the Lamb parameter (or nondimensional wave speed) has driven the evolution, and that changes in frequency or other parameters have been much less important.