Why the day is 24 hours long

Abstract: Geologic data show that the number of days per month decreased over the last 1,300 Myr, while the length of day increased. In contrast, the number of days per month increased between 2,800 Mya and 1,300 Mya, while the length of day was roughly constant, at approximately 19.5 hours. This is consistent with a balance between the gravitational Lunar tidal torque and the resonant solar thermal atmospheric torque. I will describe gravitational and thermal tides, giving order of magnitude or better analytical estimates for the corresponding torques. I will show how to use the data to infer the mean surface temperature T as a function of time; a resonant atmospheric period of 19.5 hours corresponds to T=50C. Similarly high temperatures are reached for plausible atmospheric pressures and compositions, despite the lower Solar flux 1,500 to 2,500 Mya. Absent the thermal tide, the current length of day would be 65 hours.