Why the Day is 24 Hours Long

Abstract:

Geologic and paleontologic data show that the length of the Lunar month has been increasing for the last 1,300 Myr, consistent with oceanic tidal theory. Similar data show that the number of days per month has been decreasing since that time, conserving the angular momentum of the Earth Moon system, $L_{EM}$. However, tidal rhythmites show that the number of days per month increased between about 2,800 Mya and 1,300 Mya. Combining the two types of data, the length of day was roughly constant, with a length of about 19.5 hours, from ~2,300 to 1,300 Mya. Zahnle and Walker (1987) suggested that the length of day would be fixed at 21 hours as the result of a balance between the Lunar tidal torque and the solar thermal atmospheric torque. We use global circulation models (GCMs) to show that the present day resonant atmospheric period is between 22.8 and 23 hours, substantially longer than the 21 hours predicted by one dimensional models. We find that a resonant period of 19.5 hours corresponds to a mean surface temperature of about 40 C. Our GCM models reach such temperatures despite the lower Solar flux at those epochs if we assume plausible pressures of 1.4 bar of N2, 0.1 bar of CO2, and 2 mbar of methane. The high mean surface temperature is also consistent with the lack of evidence for large scale glaciations from 2,200 Mya to 700 Mya. Dynamical models including ocean and atmospheric tides, when constrained by the data, indicate that the angular momentum budget of the Earth-Moon system, currently 0.350 of the critical (break-up) value $L_s$, was initially significantly smaller, $L_{EM}$ ~ 0.338$L_s$. The 3\% increase was supplied by the Solar thermal tide extracting angular momentum from the Earth's orbit around the Sun, limiting the increase in the length of day. The low initial value of $L_{EM}$ we infer disfavors high angular momentum giant impact models for the formation of the Moon.